arXiv - MATH - Mathematical Physics最新文献_第9页

Ocneanu Algebra of Seams: Critical Unitary $E_6$ RSOS Lattice Model 奥克纳努接缝代数：临界单元 $E_6$ RSOS 晶格模型

arXiv - MATH - Mathematical Physics Pub Date : 2024-09-10 DOI: arxiv-2409.06236

Paul A. Pearce, Jorgen Rasmussen

引用次数: 0

Derived algebraic geometry of 2d lattice Yang-Mills theory 二维晶格杨-米尔斯理论的衍生代数几何

arXiv - MATH - Mathematical Physics Pub Date : 2024-09-10 DOI: arxiv-2409.06873

Marco Benini, Tomás Fernández, Alexander Schenkel

引用次数: 0

Limits of spectral measures for linearly bounded and for Poisson distributed random potentials 线性有界和泊松分布随机势的谱量极限

arXiv - MATH - Mathematical Physics Pub Date : 2024-09-10 DOI: arxiv-2409.06508

David Hasler, Jannis Koberstein

引用次数: 0

On the semilinear heat equation with the Grushin operator 关于带有格鲁申算子的半线性热方程

arXiv - MATH - Mathematical Physics Pub Date : 2024-09-10 DOI: arxiv-2409.06578

Geronimo Oliveira, Arlúcio Viana

引用次数: 0

Symmetry Breaking in Neural Network Optimization: Insights from Input Dimension Expansion 神经网络优化中的对称性破坏：输入维度扩展的启示

arXiv - MATH - Mathematical Physics Pub Date : 2024-09-10 DOI: arxiv-2409.06402

Jun-Jie Zhang, Nan Cheng, Fu-Peng Li, Xiu-Cheng Wang, Jian-Nan Chen, Long-Gang Pang, Deyu Meng

{"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}

引用次数: 0

Symmetry generators and quantum numbers for fermionic circularly symmetric motion 费米圆周对称运动的对称发生器和量子数

arXiv - MATH - Mathematical Physics Pub Date : 2024-09-10 DOI: arxiv-2409.06850

V. B. Mendrot, A. S. de Castro, P. Alberto

引用次数: 0

Mathematical foundations of phonons in incommensurate materials 不相容材料中声子的数学基础

arXiv - MATH - Mathematical Physics Pub Date : 2024-09-10 DOI: arxiv-2409.06151

Michael Hott, Alexander B. Watson, Mitchell Luskin

{"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}

引用次数: 0

Quantitative Quantum Zeno and Strong Damping Limits in Strong Topology 强拓扑学中的量子芝诺和强阻尼极限

arXiv - MATH - Mathematical Physics Pub Date : 2024-09-10 DOI: arxiv-2409.06469

Robert Salzmann

引用次数: 0

Basics of Multiple Polyexponential Integrals 多重多项式积分的基础知识

arXiv - MATH - Mathematical Physics Pub Date : 2024-09-10 DOI: arxiv-2409.06760

Gleb Aminov, Paolo Arnaudo

引用次数: 0

A compact QUBO encoding of computational logic formulae demonstrated on cryptography constructions 通过密码学构造展示计算逻辑公式的紧凑 QUBO 编码

arXiv - MATH - Mathematical Physics Pub Date : 2024-09-10 DOI: arxiv-2409.07501

Gregory Morse, Tamás Kozsik, Oskar Mencer, Peter Rakyta

{"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}

引用次数: 0