Discrete & Computational Geometry最新文献_第5页

On the Maximal Distance Between the Centers of Mass of a Planar Convex Body and Its Boundary 论平面凸体质量中心与其边界之间的最大距离

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-05-06 DOI: 10.1007/s00454-024-00650-0

Fedor Nazarov, Dmitry Ryabogin, Vladyslav Yaskin

引用次数: 0

Convex Bodies of Constant Width with Exponential Illumination Number 具有指数照明数的恒宽凸面体

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-05-04 DOI: 10.1007/s00454-024-00647-9

Andrii Arman, Andrii Bondarenko, Andriy Prymak

引用次数: 0

Finitary Affine Oriented Matroids 有限仿射定向矩阵

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-05-01 DOI: 10.1007/s00454-024-00651-z

Emanuele Delucchi, Kolja Knauer

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The Schwarzian Octahedron Recurrence (dSKP Equation) II: Geometric Systems 施瓦兹八面体递归（dSKP 等式）II：几何系统

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-04-26 DOI: 10.1007/s00454-024-00640-2

Niklas Christoph Affolter, Béatrice de Tilière, Paul Melotti

{"title":"The Schwarzian Octahedron Recurrence (dSKP Equation) II: Geometric Systems","authors":"Niklas Christoph Affolter, Béatrice de Tilière, Paul Melotti","doi":"10.1007/s00454-024-00640-2","DOIUrl":"https://doi.org/10.1007/s00454-024-00640-2","url":null,"abstract":"We consider nine geometric systems: Miquel dynamics, P-nets, integrable cross-ratio maps, discrete holomorphic functions, orthogonal circle patterns, polygon recutting, circle intersection dynamics, (corrugated) pentagram maps and the short diagonal hyperplane map. Using a unified framework, for each system we prove an explicit expression for the solution as a function of the initial data; more precisely, we show that the solution is equal to the ratio of two partition functions of an oriented dimer model on an Aztec diamond whose face weights are constructed from the initial data. Then, we study the Devron property (Glick in J Geom Phys 87:161–189, 2015), which states the following: if the system starts from initial data that is singular for the backwards dynamics, this singularity is expected to reoccur after a finite number of steps of the forwards dynamics. Again, using a unified framework, we prove this Devron property for all of the above geometric systems, for different kinds of singular initial data. In doing so, we obtain new singularity results and also known ones (Glick in J Geom Phys 87:161–189, 2015; Yao in Glick’s conjecture on the point of collapse of axis-aligned polygons under the pentagram maps (2014). Preprint arXiv:1410.7806). Our general method consists in proving that these nine geometric systems are all related to the Schwarzian octahedron recurrence (dSKP equation), and then to rely on the companion paper (Affolter et al. in Comb Theory 3(2), 2023), where we study this recurrence in general, prove explicit expressions and singularity results.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806042","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

Triangle Percolation on the Grid 网格上的三角渗透

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-04-25 DOI: 10.1007/s00454-024-00645-x

Igor Araujo, Bryce Frederickson, Robert A. Krueger, Bernard Lidický, Tyrrell B. McAllister, Florian Pfender, Sam Spiro, Eric Nathan Stucky

引用次数: 0

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":"17 1","pages":""},"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":"107 1","pages":""},"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