arXiv - MATH - Metric Geometry最新文献_第6页

Orthogonal Fourier Analysis on Domains 域上的正交傅里叶分析

arXiv - MATH - Metric Geometry Pub Date : 2024-08-27 DOI: arxiv-2408.15361

Mihail N. Kolountzakis

引用次数: 0

Sobolev spaces via chains in metric measure spaces 通过度量空间链的索波列夫空间

arXiv - MATH - Metric Geometry Pub Date : 2024-08-27 DOI: arxiv-2408.15071

Emanuele Caputo, Nicola Cavallucci

{"title":"Sobolev spaces via chains in metric measure spaces","authors":"Emanuele Caputo, Nicola Cavallucci","doi":"arxiv-2408.15071","DOIUrl":"https://doi.org/arxiv-2408.15071","url":null,"abstract":"We define the chain Sobolev space on a possibly non-complete metric measure\u0000space in terms of chain upper gradients. In this context, $varepsilon$-chains\u0000are a finite collection of points with distance at most $varepsilon$ between\u0000consecutive points. They play the role of discrete versions of curves. Chain\u0000upper gradients are defined accordingly and the chain Sobolev space is defined\u0000by letting the size parameter $varepsilon$ going to zero. In the complete\u0000setting, we prove that the chain Sobolev space is equal to the classical\u0000notions of Sobolev spaces in terms of relaxation of upper gradients or of the\u0000local Lipschitz constant of Lipschitz functions. The proof of this fact is\u0000inspired by a recent technique developed by Eriksson-Bique. In the possible\u0000non-complete setting, we prove that the chain Sobolev space is equal to the one\u0000defined via relaxation of the local Lipschitz constant of Lipschitz functions,\u0000while in general they are different from the one defined via upper gradients\u0000along curves. We apply the theory developed in the paper to prove equivalent\u0000formulations of the Poincar'{e} inequality in terms of pointwise estimates\u0000involving $varepsilon$-upper gradients, lower bounds on modulus of chains\u0000connecting points and size of separating sets measured with the Minkowski\u0000content in the non-complete setting. Along the way, we discuss the notion of\u0000weak $varepsilon$-upper gradients and asymmetric notions of integral along\u0000chains.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187639","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

A characterisation of graphs quasi-isometric to $K_4$-minor-free graphs 与 $K_4$ 无小图准等距的图的特征描述

arXiv - MATH - Metric Geometry Pub Date : 2024-08-27 DOI: arxiv-2408.15335

Sandra Albrechtsen, Raphael W. Jacobs, Paul Knappe, Paul Wollan

引用次数: 0

Holographic Foliations: Self-Similar Quasicrystals from Hyperbolic Honeycombs 全息叶形：来自双曲蜂巢的自相似准晶体

arXiv - MATH - Metric Geometry Pub Date : 2024-08-27 DOI: arxiv-2408.15316

Latham Boyle, Justin Kulp

{"title":"Holographic Foliations: Self-Similar Quasicrystals from Hyperbolic Honeycombs","authors":"Latham Boyle, Justin Kulp","doi":"arxiv-2408.15316","DOIUrl":"https://doi.org/arxiv-2408.15316","url":null,"abstract":"Discrete geometries in hyperbolic space are of longstanding interest in pure\u0000mathematics and have come to recent attention in holography, quantum\u0000information, and condensed matter physics. Working at a purely geometric level,\u0000we describe how any regular tessellation of ($d+1$)-dimensional hyperbolic\u0000space naturally admits a $d$-dimensional boundary geometry with self-similar\u0000''quasicrystalline'' properties. In particular, the boundary geometry is\u0000described by a local, invertible, self-similar substitution tiling, that\u0000discretizes conformal geometry. We greatly refine an earlier description of\u0000these local substitution rules that appear in the 1D/2D example and use the\u0000refinement to give the first extension to higher dimensional bulks; including a\u0000detailed account for all regular 3D hyperbolic tessellations. We comment on\u0000global issues, including the reconstruction of bulk geometries from boundary\u0000data, and introduce the notion of a ''holographic foliation'': a foliation by a\u0000stack of self-similar quasicrystals, where the full geometry of the bulk (and\u0000of the foliation itself) is encoded in any single leaf in a local invertible\u0000way. In the ${3,5,3}$ tessellation of 3D hyperbolic space by regular\u0000icosahedra, we find a 2D boundary quasicrystal admitting points of 5-fold\u0000symmetry which is not the Penrose tiling, and record and comment on a related\u0000conjecture of William Thurston. We end with a large list of open questions for\u0000future analytic and numerical studies.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187634","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

Products of hyperbolic spaces 双曲空间的乘积

arXiv - MATH - Metric Geometry Pub Date : 2024-08-26 DOI: arxiv-2408.14093

Andrei Sipos

引用次数: 0

Glued spaces and lower curvature bounds 胶合空间和曲率下限

arXiv - MATH - Metric Geometry Pub Date : 2024-08-23 DOI: arxiv-2408.13137

Christian Ketterer

引用次数: 0

A 4-Dimensional Peabody of Constant Width 恒定宽度的四维皮博迪

arXiv - MATH - Metric Geometry Pub Date : 2024-08-23 DOI: arxiv-2408.13241

Isaac Arelio, Luis Montejano, Deborah Oliveros