We consider the $A$ series and exceptional $E_6$ Restricted Solid-On-Solid lattice models as prototypical examples of the critical Yang-Baxter integrable two-dimensional $A$-$D$-$E$ lattice models. We focus on type I theories which are characterized by the existence of an extended chiral symmetry in the continuum scaling limit. Starting with the commuting family of column transfer matrices on the torus, we build matrix representations of the Ocneanu graph fusion algebra as integrable seams for arbitrary finite-size lattices with the structure constants specified by Petkova and Zuber. This commutative seam algebra contains the Verlinde, fused adjacency and graph fusion algebras as subalgebras. Our matrix representation of the Ocneanu algebra encapsulates the quantum symmetry of the commuting family of transfer matrices. In the continuum scaling limit, the integrable seams realize the topological defects of the associated conformal field theory and the known toric matrices encode the twisted conformal partition functions.
{"title":"Ocneanu Algebra of Seams: Critical Unitary $E_6$ RSOS Lattice Model","authors":"Paul A. Pearce, Jorgen Rasmussen","doi":"arxiv-2409.06236","DOIUrl":"https://doi.org/arxiv-2409.06236","url":null,"abstract":"We consider the $A$ series and exceptional $E_6$ Restricted Solid-On-Solid\u0000lattice models as prototypical examples of the critical Yang-Baxter integrable\u0000two-dimensional $A$-$D$-$E$ lattice models. We focus on type I theories which\u0000are characterized by the existence of an extended chiral symmetry in the\u0000continuum scaling limit. Starting with the commuting family of column transfer\u0000matrices on the torus, we build matrix representations of the Ocneanu graph\u0000fusion algebra as integrable seams for arbitrary finite-size lattices with the\u0000structure constants specified by Petkova and Zuber. This commutative seam\u0000algebra contains the Verlinde, fused adjacency and graph fusion algebras as\u0000subalgebras. Our matrix representation of the Ocneanu algebra encapsulates the\u0000quantum symmetry of the commuting family of transfer matrices. In the continuum\u0000scaling limit, the integrable seams realize the topological defects of the\u0000associated conformal field theory and the known toric matrices encode the\u0000twisted conformal partition functions.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A derived algebraic geometric study of classical $mathrm{GL}_n$-Yang-Mills theory on the $2$-dimensional square lattice $mathbb{Z}^2$ is presented. The derived critical locus of the Wilson action is described and its local data supported in rectangular subsets $V =[a,b]times [c,d]subseteq mathbb{Z}^2$ with both sides of length $geq 2$ is extracted. A locally constant dg-category-valued prefactorization algebra on $mathbb{Z}^2$ is constructed from the dg-categories of perfect complexes on the derived stacks of local data.
{"title":"Derived algebraic geometry of 2d lattice Yang-Mills theory","authors":"Marco Benini, Tomás Fernández, Alexander Schenkel","doi":"arxiv-2409.06873","DOIUrl":"https://doi.org/arxiv-2409.06873","url":null,"abstract":"A derived algebraic geometric study of classical $mathrm{GL}_n$-Yang-Mills\u0000theory on the $2$-dimensional square lattice $mathbb{Z}^2$ is presented. The\u0000derived critical locus of the Wilson action is described and its local data\u0000supported in rectangular subsets $V =[a,b]times [c,d]subseteq mathbb{Z}^2$\u0000with both sides of length $geq 2$ is extracted. A locally constant\u0000dg-category-valued prefactorization algebra on $mathbb{Z}^2$ is constructed\u0000from the dg-categories of perfect complexes on the derived stacks of local\u0000data.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show the existence of infinite volume limits of resolvents and spectral measures for a class of Schroedinger operators with linearly bounded potentials. We then apply this result to Schroedinger operators with a Poisson distributed random potential.
{"title":"Limits of spectral measures for linearly bounded and for Poisson distributed random potentials","authors":"David Hasler, Jannis Koberstein","doi":"arxiv-2409.06508","DOIUrl":"https://doi.org/arxiv-2409.06508","url":null,"abstract":"We show the existence of infinite volume limits of resolvents and spectral\u0000measures for a class of Schroedinger operators with linearly bounded\u0000potentials. We then apply this result to Schroedinger operators with a Poisson\u0000distributed random potential.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we study the heat equation with Grushin's operator. We present an expression for its heat kernel and get regularity properties and decay on $L^p$ spaces for both heat Kernel and semigroup associated to Grushin's operator. Next, we use the results to prove the existence, uniqueness, continuous dependence and blowup alternative of mild solutions of a nonlinear Cauchy's problem associated to Grushin's operator.
{"title":"On the semilinear heat equation with the Grushin operator","authors":"Geronimo Oliveira, Arlúcio Viana","doi":"arxiv-2409.06578","DOIUrl":"https://doi.org/arxiv-2409.06578","url":null,"abstract":"In this work, we study the heat equation with Grushin's operator. We present\u0000an expression for its heat kernel and get regularity properties and decay on\u0000$L^p$ spaces for both heat Kernel and semigroup associated to Grushin's\u0000operator. Next, we use the results to prove the existence, uniqueness,\u0000continuous dependence and blowup alternative of mild solutions of a nonlinear\u0000Cauchy's problem associated to Grushin's operator.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216511","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Understanding the mechanisms behind neural network optimization is crucial for improving network design and performance. While various optimization techniques have been developed, a comprehensive understanding of the underlying principles that govern these techniques remains elusive. Specifically, the role of symmetry breaking, a fundamental concept in physics, has not been fully explored in neural network optimization. This gap in knowledge limits our ability to design networks that are both efficient and effective. Here, we propose the symmetry breaking hypothesis to elucidate the significance of symmetry breaking in enhancing neural network optimization. We demonstrate that a simple input expansion can significantly improve network performance across various tasks, and we show that this improvement can be attributed to the underlying symmetry breaking mechanism. We further develop a metric to quantify the degree of symmetry breaking in neural networks, providing a practical approach to evaluate and guide network design. Our findings confirm that symmetry breaking is a fundamental principle that underpins various optimization techniques, including dropout, batch normalization, and equivariance. By quantifying the degree of symmetry breaking, our work offers a practical technique for performance enhancement and a metric to guide network design without the need for complete datasets and extensive training processes.
{"title":"Symmetry Breaking in Neural Network Optimization: Insights from Input Dimension Expansion","authors":"Jun-Jie Zhang, Nan Cheng, Fu-Peng Li, Xiu-Cheng Wang, Jian-Nan Chen, Long-Gang Pang, Deyu Meng","doi":"arxiv-2409.06402","DOIUrl":"https://doi.org/arxiv-2409.06402","url":null,"abstract":"Understanding the mechanisms behind neural network optimization is crucial\u0000for improving network design and performance. While various optimization\u0000techniques have been developed, a comprehensive understanding of the underlying\u0000principles that govern these techniques remains elusive. Specifically, the role\u0000of symmetry breaking, a fundamental concept in physics, has not been fully\u0000explored in neural network optimization. This gap in knowledge limits our\u0000ability to design networks that are both efficient and effective. Here, we\u0000propose the symmetry breaking hypothesis to elucidate the significance of\u0000symmetry breaking in enhancing neural network optimization. We demonstrate that\u0000a simple input expansion can significantly improve network performance across\u0000various tasks, and we show that this improvement can be attributed to the\u0000underlying symmetry breaking mechanism. We further develop a metric to quantify\u0000the degree of symmetry breaking in neural networks, providing a practical\u0000approach to evaluate and guide network design. Our findings confirm that\u0000symmetry breaking is a fundamental principle that underpins various\u0000optimization techniques, including dropout, batch normalization, and\u0000equivariance. By quantifying the degree of symmetry breaking, our work offers a\u0000practical technique for performance enhancement and a metric to guide network\u0000design without the need for complete datasets and extensive training processes.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The planar dynamics of spin-1/2 quantum relativistic particles is important for several physical systems. In this paper we derive, by a simple method, the generators for the continuous symmetries of the 3+1 Dirac equation for planar motion, when there is circular symmetry, i.e., the interactions depend only on the radial coordinate. We consider a general set of potentials with different Lorentz structures. These generators allow for several minimal complete sets of commuting observables and their corresponding quantum numbers. We show how they can be used to label the general eigenspinors for this problem. We also derive the generators of the spin and pseudospin symmetries for this planar Dirac problem, which arise when the vector and scalar potentials have the same magnitude and tensor potential and the space components of the four-vector potential are absent. We investigate the associated energy degeneracies and compare them to the known degeneracies in the spherically symmetric 3+1 Dirac equation.
{"title":"Symmetry generators and quantum numbers for fermionic circularly symmetric motion","authors":"V. B. Mendrot, A. S. de Castro, P. Alberto","doi":"arxiv-2409.06850","DOIUrl":"https://doi.org/arxiv-2409.06850","url":null,"abstract":"The planar dynamics of spin-1/2 quantum relativistic particles is important\u0000for several physical systems. In this paper we derive, by a simple method, the\u0000generators for the continuous symmetries of the 3+1 Dirac equation for planar\u0000motion, when there is circular symmetry, i.e., the interactions depend only on\u0000the radial coordinate. We consider a general set of potentials with different\u0000Lorentz structures. These generators allow for several minimal complete sets of\u0000commuting observables and their corresponding quantum numbers. We show how they\u0000can be used to label the general eigenspinors for this problem. We also derive\u0000the generators of the spin and pseudospin symmetries for this planar Dirac\u0000problem, which arise when the vector and scalar potentials have the same\u0000magnitude and tensor potential and the space components of the four-vector\u0000potential are absent. We investigate the associated energy degeneracies and\u0000compare them to the known degeneracies in the spherically symmetric 3+1 Dirac\u0000equation.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Michael Hott, Alexander B. Watson, Mitchell Luskin
In some models, periodic configurations can be shown to be stable under, both, global $ell^2$ or local perturbations. This is not the case for aperiodic media. The specific class of aperiodic media we are interested, in arise from taking two 2D periodic crystals and stacking them parallel at a relative twist. In periodic media, phonons are generalized eigenvectors for a stability operator acting on $ell^2$, coming from a mechanical energy. The goal of our analysis is to provide phonons in the given class of aperiodic media with meaning. As rigorously established for the 1D Frenkel-Kontorova model and previously applied by one of the authors, we assume that we can parametrize minimizing lattice deformations w.r.t. local perturbations via continuous stacking-periodic functions, for which we previously derived a continuous energy density functional. Such (continuous) energy densities are analytically and computationally much better accessible compared to discrete energy functionals. In order to pass to an $ell^2$-based energy functional, we also study the offset energy w.r.t. given lattice deformations, under $ell^1$-perturbations. Our findings show that, in the case of an undeformed bilayer heterostructure, while the energy density can be shown to be stable under the assumption of stability of individual layers, the offset energy fails to be stable in the case of twisted bilayer graphene. We then establish conditions for stability and instability of the offset energy w.r.t. the relaxed lattice. Finally, we show that, in the case of incommensurate bilayer homostructures, i.e., two equal layers, if we choose minimizing deformations according to the global energy density above, the offset energy is stable in the limit of zero twist angle. Consequently, in this case, one can then define phonons as generalized eigenvectors w.r.t. the stability operator associated with the offset energy.
{"title":"Mathematical foundations of phonons in incommensurate materials","authors":"Michael Hott, Alexander B. Watson, Mitchell Luskin","doi":"arxiv-2409.06151","DOIUrl":"https://doi.org/arxiv-2409.06151","url":null,"abstract":"In some models, periodic configurations can be shown to be stable under,\u0000both, global $ell^2$ or local perturbations. This is not the case for\u0000aperiodic media. The specific class of aperiodic media we are interested, in\u0000arise from taking two 2D periodic crystals and stacking them parallel at a\u0000relative twist. In periodic media, phonons are generalized eigenvectors for a\u0000stability operator acting on $ell^2$, coming from a mechanical energy. The\u0000goal of our analysis is to provide phonons in the given class of aperiodic\u0000media with meaning. As rigorously established for the 1D Frenkel-Kontorova\u0000model and previously applied by one of the authors, we assume that we can\u0000parametrize minimizing lattice deformations w.r.t. local perturbations via\u0000continuous stacking-periodic functions, for which we previously derived a\u0000continuous energy density functional. Such (continuous) energy densities are\u0000analytically and computationally much better accessible compared to discrete\u0000energy functionals. In order to pass to an $ell^2$-based energy functional, we\u0000also study the offset energy w.r.t. given lattice deformations, under\u0000$ell^1$-perturbations. Our findings show that, in the case of an undeformed\u0000bilayer heterostructure, while the energy density can be shown to be stable\u0000under the assumption of stability of individual layers, the offset energy fails\u0000to be stable in the case of twisted bilayer graphene. We then establish\u0000conditions for stability and instability of the offset energy w.r.t. the\u0000relaxed lattice. Finally, we show that, in the case of incommensurate bilayer\u0000homostructures, i.e., two equal layers, if we choose minimizing deformations\u0000according to the global energy density above, the offset energy is stable in\u0000the limit of zero twist angle. Consequently, in this case, one can then define\u0000phonons as generalized eigenvectors w.r.t. the stability operator associated\u0000with the offset energy.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Frequent applications of a mixing quantum operation to a quantum system slow down its time evolution and eventually drive it into the invariant subspace of the named operation. We prove this phenomenon, the quantum Zeno effect, and its continuous variant, strong damping, in a unified way for infinite-dimensional open quantum systems, while merely demanding that the respective mixing convergence holds pointwise for all states. Both results are quantitative in the following sense: Given the speed of convergence for the mixing limits, we can derive bounds on the convergence speed for the corresponding quantum Zeno and strong damping limits. We apply our results to prove quantum Zeno and strong damping limits for the photon loss channel with an explicit bound on the convergence speed.
{"title":"Quantitative Quantum Zeno and Strong Damping Limits in Strong Topology","authors":"Robert Salzmann","doi":"arxiv-2409.06469","DOIUrl":"https://doi.org/arxiv-2409.06469","url":null,"abstract":"Frequent applications of a mixing quantum operation to a quantum system slow\u0000down its time evolution and eventually drive it into the invariant subspace of\u0000the named operation. We prove this phenomenon, the quantum Zeno effect, and its\u0000continuous variant, strong damping, in a unified way for infinite-dimensional\u0000open quantum systems, while merely demanding that the respective mixing\u0000convergence holds pointwise for all states. Both results are quantitative in\u0000the following sense: Given the speed of convergence for the mixing limits, we\u0000can derive bounds on the convergence speed for the corresponding quantum Zeno\u0000and strong damping limits. We apply our results to prove quantum Zeno and\u0000strong damping limits for the photon loss channel with an explicit bound on the\u0000convergence speed.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a set of special functions called multiple polyexponential integrals, defined as iterated integrals of the exponential integral $text{Ei}(z)$. These functions arise in certain perturbative expansions of the local solutions of second-order ODEs around an irregular singularity. In particular, their recursive definition describes the asymptotic behavior of these local solutions. To complement the study of the multiple polyexponential integrals on the entire complex plane, we relate them with two other sets of special functions - the undressed and dressed multiple polyexponential functions - which are characterized by their Taylor series expansions around the origin.
{"title":"Basics of Multiple Polyexponential Integrals","authors":"Gleb Aminov, Paolo Arnaudo","doi":"arxiv-2409.06760","DOIUrl":"https://doi.org/arxiv-2409.06760","url":null,"abstract":"We introduce a set of special functions called multiple polyexponential\u0000integrals, defined as iterated integrals of the exponential integral\u0000$text{Ei}(z)$. These functions arise in certain perturbative expansions of the\u0000local solutions of second-order ODEs around an irregular singularity. In\u0000particular, their recursive definition describes the asymptotic behavior of\u0000these local solutions. To complement the study of the multiple polyexponential\u0000integrals on the entire complex plane, we relate them with two other sets of\u0000special functions - the undressed and dressed multiple polyexponential\u0000functions - which are characterized by their Taylor series expansions around\u0000the origin.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"283 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gregory Morse, Tamás Kozsik, Oskar Mencer, Peter Rakyta
We aim to advance the state-of-the-art in Quadratic Unconstrained Binary Optimization formulation with a focus on cryptography algorithms. As the minimal QUBO encoding of the linear constraints of optimization problems emerges as the solution of integer linear programming (ILP) problems, by solving special boolean logic formulas (like ANF and DNF) for their integer coefficients it is straightforward to handle any normal form, or any substitution for multi-input AND, OR or XOR operations in a QUBO form. To showcase the efficiency of the proposed approach we considered the most widespread cryptography algorithms including AES-128/192/256, MD5, SHA1 and SHA256. For each of these, we achieved QUBO instances reduced by thousands of logical variables compared to previously published results, while keeping the QUBO matrix sparse and the magnitude of the coefficients low. In the particular case of AES-256 cryptography function we obtained more than 8x reduction in variable count compared to previous results. The demonstrated reduction in QUBO sizes notably increases the vulnerability of cryptography algorithms against future quantum annealers, capable of embedding around $30$ thousands of logical variables.
{"title":"A compact QUBO encoding of computational logic formulae demonstrated on cryptography constructions","authors":"Gregory Morse, Tamás Kozsik, Oskar Mencer, Peter Rakyta","doi":"arxiv-2409.07501","DOIUrl":"https://doi.org/arxiv-2409.07501","url":null,"abstract":"We aim to advance the state-of-the-art in Quadratic Unconstrained Binary\u0000Optimization formulation with a focus on cryptography algorithms. As the\u0000minimal QUBO encoding of the linear constraints of optimization problems\u0000emerges as the solution of integer linear programming (ILP) problems, by\u0000solving special boolean logic formulas (like ANF and DNF) for their integer\u0000coefficients it is straightforward to handle any normal form, or any\u0000substitution for multi-input AND, OR or XOR operations in a QUBO form. To\u0000showcase the efficiency of the proposed approach we considered the most\u0000widespread cryptography algorithms including AES-128/192/256, MD5, SHA1 and\u0000SHA256. For each of these, we achieved QUBO instances reduced by thousands of\u0000logical variables compared to previously published results, while keeping the\u0000QUBO matrix sparse and the magnitude of the coefficients low. In the particular\u0000case of AES-256 cryptography function we obtained more than 8x reduction in\u0000variable count compared to previous results. The demonstrated reduction in QUBO\u0000sizes notably increases the vulnerability of cryptography algorithms against\u0000future quantum annealers, capable of embedding around $30$ thousands of logical\u0000variables.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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