Discrete & Computational Geometry最新文献_第4页

Unavoidable Patterns in Complete Simple Topological Graphs 完整简单拓扑图中不可避免的模式

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-05-29 DOI: 10.1007/s00454-024-00658-6

Andrew Suk, Ji Zeng

引用次数: 0

On the Number of Incidences When Avoiding an Induced Biclique in Geometric Settings 论几何设置中避免诱导双斜时的事件数

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-05-23 DOI: 10.1007/s00454-024-00648-8

Timothy M. Chan, Sariel Har-Peled

{"title":"On the Number of Incidences When Avoiding an Induced Biclique in Geometric Settings","authors":"Timothy M. Chan, Sariel Har-Peled","doi":"10.1007/s00454-024-00648-8","DOIUrl":"https://doi.org/10.1007/s00454-024-00648-8","url":null,"abstract":"Given a set of points (P) and a set of regions (mathcal {O}), an incidence is a pair ((p,mathcalligra {o}) in Ptimes mathcal {O}) such that (pin mathcalligra {o}). We obtain a number of new results on a classical question in combinatorial geometry: What is the number of incidences (under certain restrictive conditions)? We prove a bound of (Obigl ( k n(log n/log log n)^{d-1} bigr )) on the number of incidences between n points and n axis-parallel boxes in (mathbb {R}^d), if no k boxes contain k common points, that is, if the incidence graph between the points and the boxes does not contain (K_{k,k}) as a subgraph. This new bound improves over previous work, by Basit et al. (Forum Math Sigma 9:59, 2021), by more than a factor of (log ^d n) for (d >2). Furthermore, it matches a lower bound implied by the work of Chazelle (J ACM 37(2):200–212, 1990), for (k=2), thus settling the question for points and boxes. We also study several other variants of the problem. For halfspaces, using shallow cuttings, we get a linear bound in two and three dimensions. We also present linear (or near linear) bounds for shapes with low union complexity, such as pseudodisks and fat triangles.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146564","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

Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data 揭开拉切夫定理的神秘面纱：从噪声数据中重构流形

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-05-13 DOI: 10.1007/s00454-024-00655-9

Sushovan Majhi

{"title":"Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data","authors":"Sushovan Majhi","doi":"10.1007/s00454-024-00655-9","DOIUrl":"https://doi.org/10.1007/s00454-024-00655-9","url":null,"abstract":"For a closed Riemannian manifold (mathcal {M}) and a metric space S with a small Gromov–Hausdorff distance to it, Latschev’s theorem guarantees the existence of a sufficiently small scale (beta >0) at which the Vietoris–Rips complex of S is homotopy equivalent to (mathcal {M}). Despite being regarded as a stepping stone to the topological reconstruction of Riemannian manifolds from noisy data, the result is only a qualitative guarantee. Until now, it had been elusive how to quantitatively choose such a proximity scale (beta ) in order to provide sampling conditions for S to be homotopy equivalent to (mathcal {M}). In this paper, we prove a stronger and pragmatic version of Latschev’s theorem, facilitating a simple description of (beta ) using the sectional curvatures and convexity radius of (mathcal {M}) as the sampling parameters. Our study also delves into the topological recovery of a closed Euclidean submanifold from the Vietoris–Rips complexes of a Hausdorff close Euclidean subset. As already known for Čech complexes, we show that Vietoris–Rips complexes also provide topologically faithful reconstruction guarantees for submanifolds.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936652","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

Perfectly Packing an Equilateral Triangle by Equilateral Triangles of Sidelengths $$n^{-1/2-epsilon }$$ 用边长为 $$n^{-1/2-epsilon }$ 的等边三角形完美包装等边三角形

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-05-11 DOI: 10.1007/s00454-024-00654-w

Janusz Januszewski, Łukasz Zielonka

引用次数: 0

Representing Infinite Periodic Hyperbolic Delaunay Triangulations Using Finitely Many Dirichlet Domains 用无限多 Dirichlet 域表示无限周期双曲 Delaunay 三角测量

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-05-07 DOI: 10.1007/s00454-024-00653-x

Vincent Despré, Benedikt Kolbe, Monique Teillaud

{"title":"Representing Infinite Periodic Hyperbolic Delaunay Triangulations Using Finitely Many Dirichlet Domains","authors":"Vincent Despré, Benedikt Kolbe, Monique Teillaud","doi":"10.1007/s00454-024-00653-x","DOIUrl":"https://doi.org/10.1007/s00454-024-00653-x","url":null,"abstract":"The Delaunay triangulation of a set of points P on a hyperbolic surface is the projection of the Delaunay triangulation of the set (widetilde{P}) of lifted points in the hyperbolic plane. Since (widetilde{P}) is infinite, the algorithms to compute Delaunay triangulations in the plane do not generalize naturally. Using a Dirichlet domain, we exhibit a finite set of points that captures the full triangulation. We prove that an edge of a Delaunay triangulation has a combinatorial length (a notion we define in the paper) smaller than (12g-6) with respect to a Dirichlet domain. To achieve this, we introduce new tools, of intrinsic interest, that capture the properties of length-minimizing curves in the context of closed curves. We then use these to derive structural results on Delaunay triangulations and exhibit certain distance minimizing properties of both the edges of a Delaunay triangulation and of a Dirichlet domain. The bounds produced in this paper depend only on the topology of the surface. They provide mathematical foundations for hyperbolic analogs of the algorithms to compute periodic Delaunay triangulations in Euclidean space.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140888916","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

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

引用次数: 0

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":null,"pages":null},"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