Discrete & Computational Geometry最新文献_第5页

Curvature Sets Over Persistence Diagrams 持续图上的曲率集

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-04-22 DOI: 10.1007/s00454-024-00634-0

Mario Gómez, Facundo Mémoli

{"title":"Curvature Sets Over Persistence Diagrams","authors":"Mario Gómez, Facundo Mémoli","doi":"10.1007/s00454-024-00634-0","DOIUrl":"https://doi.org/10.1007/s00454-024-00634-0","url":null,"abstract":"We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers (kge 0) and (nge 1) we consider the dimension k Vietoris–Rips persistence diagrams of all subsets of a given metric space with cardinality at most n. We call these invariants persistence sets and denote them as ({textbf{D}}_{n,k}^{textrm{VR}}). We first point out that this family encompasses the usual Vietoris–Rips diagrams. We then establish that (1) for certain range of values of the parameters n and k, computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy’s inequality. We also identify a rich family of metric graphs for which ({textbf{D}}_{4,1}^{textrm{VR}}) fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris–Rips persistence diagrams using Mayer–Vietoris sequences. These yield a geometric algorithm for computing the Vietoris–Rips persistence diagram of a space X with cardinality (2k+2) with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140634134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Inductive Freeness of Ziegler’s Canonical Multiderivations 齐格勒 Canonical Multiderivations 的归纳自由性

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-04-21 DOI: 10.1007/s00454-024-00644-y

Torsten Hoge, Gerhard Röhrle

引用次数: 0

Numerical Semigroups via Projections and via Quotients 通过投影和通过商的数字半群

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-04-15 DOI: 10.1007/s00454-024-00643-z

Tristram Bogart, Christopher O’Neill, Kevin Woods

引用次数: 0

On the Banach–Mazur Distance in Small Dimensions 论小维度中的巴拿赫-马祖尔距离

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-04-13 DOI: 10.1007/s00454-024-00641-1

Tomasz Kobos, Marin Varivoda

引用次数: 0

Affine Stresses: The Partition of Unity and Kalai’s Reconstruction Conjectures 仿应力：统一的分割与卡莱的重构猜想

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-04-12 DOI: 10.1007/s00454-024-00642-0

Isabella Novik, Hailun Zheng

{"title":"Affine Stresses: The Partition of Unity and Kalai’s Reconstruction Conjectures","authors":"Isabella Novik, Hailun Zheng","doi":"10.1007/s00454-024-00642-0","DOIUrl":"https://doi.org/10.1007/s00454-024-00642-0","url":null,"abstract":"Kalai conjectured that if P is a simplicial d-polytope that has no missing faces of dimension (d-1), then the graph of P and the space of affine 2-stresses of P determine P up to affine equivalence. We propose a higher-dimensional generalization of this conjecture: if (2le ile d/2) and P is a simplicial d-polytope that has no missing faces of dimension (ge d-i+1), then the space of affine i-stresses of P determines the space of affine 1-stresses of P. We prove this conjecture for (1) k-stacked d-polytopes with (2le ile kle d/2-1), (2) d-polytopes that have no missing faces of dimension (ge d-2i+2), and (3) flag PL ((d-1))-spheres with generic embeddings (for all (2le ile d/2)). We also discuss several related results and conjectures. For instance, we show that if P is a simplicial d-polytope that has no missing faces of dimension (ge d-2i+2), then the ((i-1))-skeleton of P and the set of sign vectors of affine i-stresses of P determine the combinatorial type of P. Along the way, we establish the partition of unity of affine stresses: for any (1le ile (d-1)/2), the space of affine i-stresses of a simplicial d-polytope as well as the space of affine i-stresses of a simplicial ((d-1))-sphere (with a generic embedding) can be expressed as the sum of affine i-stress spaces of vertex stars. This is analogous to Adiprasito’s partition of unity of linear stresses for Cohen–Macaulay complexes.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Computing Instance-Optimal Kernels in Two Dimensions 计算二维中的实例最优内核

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-04-07 DOI: 10.1007/s00454-024-00637-x

Pankaj K. Agarwal, Sariel Har-Peled

{"title":"Computing Instance-Optimal Kernels in Two Dimensions","authors":"Pankaj K. Agarwal, Sariel Har-Peled","doi":"10.1007/s00454-024-00637-x","DOIUrl":"https://doi.org/10.1007/s00454-024-00637-x","url":null,"abstract":"Let (P) be a set of n points in (mathbb {R}^2). For a parameter (varepsilon in (0,1)), a subset (Csubseteq P) is an (varepsilon )-kernel of (P) if the projection of the convex hull of (C) approximates that of (P) within ((1-varepsilon ))-factor in every direction. The set (C) is a weak (varepsilon )-kernel of (P) if its directional width approximates that of (P) in every direction. Let (textsf{k}_{varepsilon }(P)) (resp. (textsf{k}^{textsf{w}}_{varepsilon }(P))) denote the minimum-size of an (varepsilon )-kernel (resp. weak (varepsilon )-kernel) of (P). We present an (O(ntextsf{k}_{varepsilon }(P)log n))-time algorithm for computing an (varepsilon )-kernel of (P) of size (textsf{k}_{varepsilon }(P)), and an (O(n^2log n))-time algorithm for computing a weak (varepsilon )-kernel of (P) of size (textsf{k}^{textsf{w}}_{varepsilon }(P)). We also present a fast algorithm for the Hausdorff variant of this problem. In addition, we introduce the notion of (varepsilon )-core, a convex polygon lying inside , prove that it is a good approximation of the optimal (varepsilon )-kernel, present an efficient algorithm for computing it, and use it to compute an (varepsilon )-kernel of small size.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Rotation Inside Convex Kakeya Sets 凸刹那集内部的旋转

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-03-30 DOI: 10.1007/s00454-024-00639-9

Barnabás Janzer

引用次数: 0

A Geometric Study of Circle Packings and Ideal Class Groups 圆包和理想类群的几何研究

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-03-27 DOI: 10.1007/s00454-024-00638-w

Daniel E. Martin

引用次数: 0

Bounds on Polarization Problems on Compact Sets via Mixed Integer Programming 通过混合整数编程研究紧凑集上极化问题的界限

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-03-19 DOI: 10.1007/s00454-024-00635-z

Jan Rolfes, Robert Schüler, Marc Christian Zimmermann

引用次数: 0

A Spectral Approach to Polytope Diameter 多面体直径的谱学方法

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-03-16 DOI: 10.1007/s00454-024-00636-y

Hariharan Narayanan, Rikhav Shah, Nikhil Srivastava

引用次数: 0