Discrete & Computational Geometry最新文献_第2页

Stability and Inference of the Euler Characteristic Transform. 欧拉特征变换的稳定性与推理。

IF 0.7 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2026-02-09 DOI: 10.1007/s00454-025-00763-0

Lewis Marsh, David Beers

引用次数: 0

Error Resilient Space Partitioning. 错误弹性空间分区。

IF 0.7 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2025-11-28 DOI: 10.1007/s00454-025-00804-8

Orr Dunkelman, Zeev Geyzel, Chaya Keller, Nathan Keller, Eyal Ronen, Adi Shamir, Ran J Tessler

{"title":"Error Resilient Space Partitioning.","authors":"Orr Dunkelman, Zeev Geyzel, Chaya Keller, Nathan Keller, Eyal Ronen, Adi Shamir, Ran J Tessler","doi":"10.1007/s00454-025-00804-8","DOIUrl":"10.1007/s00454-025-00804-8","url":null,"abstract":"A major research area in discrete geometry is to consider the best way to partition the d-dimensional Euclidean space <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> under various quality criteria. In this paper we introduce a new type of space partitioning that is motivated by the problem of rounding noisy measurements from the continuous space <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> to a discrete subset of representative values. Specifically, we study partitions of <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> into bounded-size tiles colored by one of k colors, such that tiles of the same color have a distance of at least t from each other. Such tilings allow for error-resilient rounding, as two points of the same color and distance less than t from each other are guaranteed to belong to the same tile, and thus, to be rounded to the same point. The main problem we study in this paper is characterizing the achievable tradeoffs between the number of colors k and the distance t, for various dimensions d. On the qualitative side, we show that in <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> , using <math><mrow><mi>k</mi> <mo>=</mo> <mi>d</mi> <mo>+</mo> <mn>1</mn></mrow> </math> colors is both sufficient and necessary to achieve <math><mrow><mi>t</mi> <mo>></mo> <mn>0</mn></mrow> </math> . On the quantitative side, we achieve numerous upper and lower bounds on t as a function of k. In particular, for <math><mrow><mi>d</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>24</mn></mrow> </math> , we obtain sharp asymptotic bounds on t, as <math><mrow><mi>k</mi> <mo>→</mo> <mi>∞</mi></mrow> </math> . We obtain our results with a variety of techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, Bapat's connector-free lemma, and Čech cohomology.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"75 3","pages":"839-870"},"PeriodicalIF":0.7,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13050337/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147628884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On Subsets of Lattice Cubes Avoiding Affine and Spherical Degeneracies. 避免仿射和球简并的格立方子集。

IF 0.7 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2026-07-16 DOI: 10.1007/s00454-026-00853-7

Anubhab Ghosal, Ritesh Goenka, Peter Keevash

引用次数: 0

Subword Complexes and Kalai's Conjecture on Reconstruction of Spheres. 子词复合体与卡莱关于球的重构猜想。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2025-01-01 Epub Date: 2025-05-13 DOI: 10.1007/s00454-025-00733-6

Cesar Ceballos, Joseph Doolittle

引用次数: 0

The χ -Binding Function of d-Directional Segment Graphs. d向线段图的χ -绑定函数。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2025-01-01 Epub Date: 2025-05-17 DOI: 10.1007/s00454-025-00737-2

Lech Duraj, Ross J Kang, Hoang La, Jonathan Narboni, Filip Pokrývka, Clément Rambaud, Amadeus Reinald

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">The <ns0:math><ns0:mi>χ</ns0:mi></ns0:math> -Binding Function of d-Directional Segment Graphs.","authors":"Lech Duraj, Ross J Kang, Hoang La, Jonathan Narboni, Filip Pokrývka, Clément Rambaud, Amadeus Reinald","doi":"10.1007/s00454-025-00737-2","DOIUrl":"https://doi.org/10.1007/s00454-025-00737-2","url":null,"abstract":"Given a positive integer d, the class d-DIR is defined as all those intersection graphs formed from a finite collection of line segments in <math> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </math> having at most d slopes. Since each slope induces an interval graph, it easily follows for every G in d-DIR with clique number at most <math><mi>ω</mi></math> that the chromatic number <math><mrow><mi>χ</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo></mrow> </math> of G is at most <math><mrow><mi>d</mi> <mi>ω</mi></mrow> </math> . We show for every even value of <math><mi>ω</mi></math> how to construct a graph in d-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvořák and Noorizadeh. Furthermore, we show that the <math><mi>χ</mi></math> -binding function of d-DIR is <math><mrow><mi>ω</mi> <mo>↦</mo> <mi>d</mi> <mi>ω</mi></mrow> </math> for <math><mi>ω</mi></math> even and <math><mrow><mi>ω</mi> <mo>↦</mo> <mi>d</mi> <mo>(</mo> <mi>ω</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> <mo>+</mo> <mn>1</mn></mrow> </math> for <math><mi>ω</mi></math> odd. This extends an earlier result by Kostochka and Nešetřil, which treated the special case <math><mrow><mi>d</mi> <mo>=</mo> <mn>2</mn></mrow> </math> .","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"74 3","pages":"758-770"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12484362/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145214309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Compact Metric Spaces with Infinite Cop Number. 具有无限Cop数的紧度量空间。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2025-01-01 Epub Date: 2024-10-14 DOI: 10.1007/s00454-024-00696-0

Agelos Georgakopoulos

引用次数: 0

Monochromatic Infinite Sets in Minkowski Planes. 闵可夫斯基平面上的单色无限集。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2025-01-01 Epub Date: 2024-11-12 DOI: 10.1007/s00454-024-00702-5

Nóra Frankl, Panna Gehér, Arsenii Sagdeev, Géza Tóth

引用次数: 0

Rigidity of Symmetric Frameworks on the Cylinder. 圆柱对称框架的刚度。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2025-01-01 Epub Date: 2025-03-06 DOI: 10.1007/s00454-025-00723-8

Anthony Nixon, Bernd Schulze, Joseph Wall

{"title":"Rigidity of Symmetric Frameworks on the Cylinder.","authors":"Anthony Nixon, Bernd Schulze, Joseph Wall","doi":"10.1007/s00454-025-00723-8","DOIUrl":"https://doi.org/10.1007/s00454-025-00723-8","url":null,"abstract":"A bar-joint framework (G, p) is the combination of a finite simple graph <math><mrow><mi>G</mi> <mo>=</mo> <mo>(</mo> <mi>V</mi> <mo>,</mo> <mi>E</mi> <mo>)</mo></mrow> </math> and a placement <math><mrow><mi>p</mi> <mo>:</mo> <mi>V</mi> <mo>→</mo> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </mrow> </math> . The framework is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of the space. This article combines two recent extensions of the generic theory of rigid and flexible graphs by considering symmetric frameworks in <math> <msup><mrow><mi>R</mi></mrow> <mn>3</mn></msup> </math> restricted to move on a surface. In particular necessary combinatorial conditions are given for a symmetric framework on the cylinder to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. In every case when the symmetry group is cyclic, which we prove restricts the group to being inversion, half-turn or reflection symmetry, these conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"73 3","pages":"629-673"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11914369/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143665211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets. 基于根子集的零维持久性模块分解。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2025-01-01 Epub Date: 2024-11-16 DOI: 10.1007/s00454-024-00700-7

Ángel Javier Alonso, Michael Kerber

{"title":"Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets.","authors":"Ángel Javier Alonso, Michael Kerber","doi":"10.1007/s00454-024-00700-7","DOIUrl":"https://doi.org/10.1007/s00454-024-00700-7","url":null,"abstract":"We study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and first study the decomposition problem at the level of sets. This approach allows us to define the combinatorial notion of rooted subsets. In the case of a filtered metric space M, rooted subsets relate the clustering behavior of the points of M with the decomposition of the associated persistence module. In particular, we can identify intervals in such a decomposition quickly. In addition, rooted subsets can be understood as a generalization of the elder rule, and are also related to the notion of constant conqueror of Cai, Kim, Mémoli and Wang. As an application, we give a lower bound on the number of intervals that we can expect in the decomposition of zero-dimensional persistence modules of a density-Rips filtration in Euclidean space: in the limit, and under very general circumstances, we can expect that at least 25% of the indecomposable summands are interval modules.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"74 4","pages":"818-838"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12662898/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145649970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The Complexity of Order Type Isomorphism 阶式同构的复杂性

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-09-18 DOI: 10.1007/s00454-024-00687-1

Greg Aloupis, John Iacono, Stefan Langerman, Özgür Özkan, Stefanie Wuhrer

引用次数: 0