Finite Set Of Points Research Articles

To the Bicommutant Theorem for Algebras Generated by the Symmetries of Finite Point Sets in $${\mathbb{R}}^{3}$$

To the Bicommutant Theorem for Algebras Generated by the Symmetries of Finite Point Sets in $${\mathbb{R}}^{3}$$

Function recovery on manifolds using scattered data

We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold M when given a sample on a finite point set. We prove that the quality of the sample is given by the Lγ(M)-average of the geodesic distance to the point set and determine the value of γ∈(0,∞]. This extends our findings on bounded convex domains [IMA J. Numer. Anal., 2024]. As a byproduct, we prove the optimal rate of convergence of the nth minimal worst case error for Lq(M)-approximation for all 1≤q≤∞.Further, a limit theorem for moments of the average distance to a set consisting of i.i.d. uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with γ<∞. In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput., 29:1203-1214, 2019].

Three-chromatic geometric hypergraphs

We prove that for any planar convex body C there is a positive integer m with the property that any finite point set P in the plane can be three-colored in such a way that no translate of C contains m points of P (or more), all of the same color. As a part of the proof, we show a strengthening of the Erdős–Sands–Sauer–Woodrow conjecture. Surprisingly, the proof also relies on the two-dimensional case of the Illumination Conjecture. The extended abstract of this paper already appeared in the proceedings of SoCG ’22.

The weighted Euclidean one-center problem in $${\mathbb {R}}^n$$

AbstractGiven a finite set of distinct points in $${\mathbb {R}}^n$$ R n and a positive weight for each point, primal and dual algorithms are developed for finding the Euclidean ball of minimum radius so that the weighted Euclidean distance between the center of the ball and each point is less than or equal to the radius of the ball. Each algorithm is based on a directional search method in which the search path at each iteration is either a ray or a two-dimensional circular arc in $${\mathbb {R}}^n$$ R n . At each iteration, a search path is constructed by intersecting bisectors of pairs of points, where the bisectors are either hyperplanes or n-dimensional spheres. Each search path preserves complementary slackness and primal (dual) feasibility for the primal (dual) algorithm. The step size along each search path is determined explicitly. Test problems up to 1000 dimensions and 10,000 points were solved to optimality by both primal and dual algorithms. Computational results also show that these algorithms outperform several open-source SOCP codes.

A finite pointset method for Kuramoto–Sivashinsky equation based on mixed formulation

A finite pointset method for Kuramoto–Sivashinsky equation based on mixed formulation

A uniqueness theorem for meromorphic functions

In this paper, we prove the uniqueness theorem for a special class of meromorphic functions on the complex plane $\mathbb{C}$. In particular, we study the class of meromorphic functions $f$ in the domain $\mathbb{C}\setminus K'$, where $K'$ is the finite set of limit points of simple poles of the function $f$. In this class, we describe non-trivial subclasses in which every function $f$ can be uniquely determined by the residues of the function $f$ at its poles. The result covered in this paper is a part of a problem in a spectral operator theory.

Generic families of finite metric spaces with identical or trivial 1-dimensional persistence

Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of persistent homology provides an upper bound for the change of persistence in the bottleneck distance under perturbations of points, but without giving a lower bound. This paper clarifies the possible limitations persistent homology may have in distinguishing finite metric spaces, which is evident for non-isometric point sets with identical persistence. We describe generic families of point sets in metric spaces that have identical or even trivial one-dimensional persistence. The results motivate stronger invariants to distinguish finite point sets up to isometry.

Representing Infinite Periodic Hyperbolic Delaunay Triangulations Using Finitely Many Dirichlet Domains

The Delaunay triangulation of a set of points P on a hyperbolic surface is the projection of the Delaunay triangulation of the set P~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{P}$$\\end{document} of lifted points in the hyperbolic plane. Since P~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{P}$$\\end{document} 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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$12g-6$$\\end{document} 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.

Using the Ellipsoid Method for Sylvester's Problem and its Generalization

Sylvester's problem or the problem of the smallest bounding circle is the problem of constructing a circle of the smallest radius that contains a finite set of points on the plane. In n-dimensional space, it corresponds to the problem of the smallest bounding hypersphere, which can be formulated as the problem of minimizing a convex piecewise quadratic function. The article is dedicated to study of the ellipsoid method application for solving this problem and the minimax convex programming problem, which is equivalent to the generalized problem of the smallest bounding hypersphere. The generalized problem consists in finding the center of a sphere in an n-dimensional space that has a minimal radius and contains a finite set of n-dimensional spheres given by their centers and radii. The article consists of 3 sections. Section 1 describes the emshor algorithm – the algorithm of the ellipsoid method for problem of minimization of an arbitrary convex function, proves its convergence theorems, gives a geometric interpretation of the algorithm, which is based on the use of a minimum volume ellipsoid. Octave implementation of the emshor algorithm is presented here, which can be successfully applied for non-smooth convex functions minimization if the number of variables is n = 2 - 30. In Section 2, the sylvester1 algorithm is built, which is an application of the emshor algorithm for solving the problem of minimizing a convex piecewise quadratic function, which is equivalent to the problem of finding a sphere of minimum radius for a finite set of points. In Section 3, the sylvester2 algorithm is built, which is an application of the emshor algorithm for solving the problem of minimization of a convex function, which is equivalent to the generalized problem of finding a sphere of minimum radius for a finite set of spheres with given centers and radii. The results of testing the sylvester1 and sylvester2 algorithms demonstrate high working speed for modern computers and high accuracy in terms of optimal value of the objective function when solving problems in n-dimensional spaces for small values n = 2 - 30. Keywords: ellipsoid method, convex function, Sylvester's problem, piecewise smooth function, minimax problem.

A GFDM approach based on the finite pointset method for two-dimensional piezoelectric problems

In this article a novel Generalized Finite Difference Method (GFDM) derived from the so-called Finite Pointset Method (FPM) is presented and discussed for the first time to solve two-dimensional piezoelectric structures. In this approach, the approximation of the field variables depends on both the governing equations and the local problem discretization, and it incorporates the minimization of the errors of the governing equations being solved, which directly leads to its numerical solution. This truly meshfree formulation is applied in order to solve a static piezoelectric problem that consists of the coupled partial differential equations for the deformations and the electric field, in terms of the displacements and the scalar electric potential. The numerical results of some two-dimensional benchmark problems using the proposed approach are reported and compared with analytical solutions, if available, or with reference numerical solutions, in order to demonstrate its accuracy and suitability for this kind of problems. These numerical results suggest that the proposed GFDM formulation is feasible and promising for solving two-dimensional piezoelectric problems.

Extensions of the colorful Helly theorem for d-collapsible and d-Leray complexes

Abstract We present extensions of the colorful Helly theorem for d-collapsible and d-Leray complexes, providing a common generalization to the matroidal versions of the theorem due to Kalai and Meshulam, the ‘very colorful’ Helly theorem introduced by Arocha, Bárány, Bracho, Fabila and Montejano and the ‘semi-intersecting’ colorful Helly theorem proved by Montejano and Karasev. As an application, we obtain the following extension of Tverberg’s theorem: Let A be a finite set of points in ${\mathbb R}^d$ with $|A|&gt;(r-1)(d+1)$ . Then, there exist a partition $A_1,\ldots ,A_r$ of A and a subset $B\subset A$ of size $(r-1)(d+1)$ such that $\cap _{i=1}^r \operatorname {\mathrm {\text {conv}}}( (B\cup \{p\})\cap A_i)\neq \emptyset $ for all $p\in A\setminus B$ . That is, we obtain a partition of A into r parts that remains a Tverberg partition even after removing all but one arbitrary point from $A\setminus B$ .

THE MATHEMATICAL MODEL OF TWO-DIMENSIONAL ELECTROMAGNETHIC FIELD

The purpose of this work is to develop a mathematical model of two-dimensional electromagnetic field, i.e. to construct a system of axioms (assumptions) that we unconditionally accept, although they do not correspond to the real electromagnetic field, but approximate it quite well. A mathematical model of two-dimensional electromagnetic field is constructed based on four assumptions: 1. Boundedness: we assume that on an infinite plane there is a bounded convex domain, outside which the field is zero. 2. Existence of nodes: we assume that in the domain there is a finite set of points, called nodes. All calculations of potential values are carried out only at internal nodes․ 3. Existence of neighbors: we assume that all pairs of nodes are divided into two types: neighboring and non-neighboring, and only neighboring nodes interact. 4. Discretization: we assume that the magnetic permeability of the medium is discrete, i.e. is a piecewise constant function. It is shown that within our mathematical model, from Assumptions 1 – 4, it follows that the electromagnetic potential is necessarily a piecewise linear function. It is shown that within our mathematical model, minimization of the energy functional leads to a system of nonlinear equations, the number of equations is equal to the number of internal nodes. It is proved that within our mathematical model, from Assumptions 1 – 4, it follows that for the numerical solution of the resulting system of equations, the optimal mesh is the Delaunay triangulation. The results are illustrated by an example of numerical solution of a field problem for electromagnetic systems with magnetorheological fluid using the FEMM tool. It is shown that the FEMM application package corresponds to our model.

Scenario of stable transition from diffeomorphism of torus isotopic to identity one to skew product of rough transformations of circle

In this paper, we consider gradient-like diffeomorphisms of a two-dimensional torus isotopic to the identical one. The isotopicity of diffeomorphisms $f_0$, $f_1$ on an $n$-manifold $M^n$ means the existence of some arc $\{f_t:M^n\to M^n,t\in[0,1]\}$ connecting them in the space of diffeomorphisms. If isotopic diffeomorphisms are structurally stable (qualitatively not changing their properties with small perturbations), then it is natural to expect the existence of a stable arc (qualitatively not changing its properties under small perturbations) connecting them. In this case, one says that the isotopic diffeomorphisms $f_0$, $f_1$ are stably isotopic or belong to the same class of stable isotopic connectivity. The simplest structurally stable diffeomorphisms on surfaces are gradient-like transformations having a finite hyperbolic non-wandering set, stable and unstable manifolds of various saddle points of which do not intersect. However, even on a two-dimensional sphere, where all orientation-preserving diffeomorphisms are isotopic, gradient-like diffeomorphisms are generally not stably isotopic. The countable number of pairwise different classes of stable isotopic connectivity is constructed on the base of a rough transformation of the circle $\phi_{\frac{k}{m}}$ with exactly two periodic orbits of the period $m$ and the rotation number $\frac{k}{m}$, which can be continued to a diffeomorphism $F_{\frac k m}:\mathbb S^2\to\mathbb S^2$ with two fixed sources at the North and South poles. On the torus $\mathbb T^2$, the model representative in the considered class is the skew products of rough transformations of a circle. We show that any isotopic gradient-like diffeomorphism of a torus is connected by a stable arc with some model transformation.

Fast approximation algorithms for some maximin clustering problems

In this paper, we consider three cases of an intractable problem of searching for two subsets in a finite set of points of Euclidean space. In all three cases, it is required to maximize the minimum cluster?s cardinality under constraint on each cluster?s scatter. The scatter is the sum of the distances from the cluster elements to the center, which is defined differently in each of the three cases. In the first case, cluster centers are fixed points. In the second case, the centers are unknown points from the input set. In the third case, the centers are defined as the centroids (the arithmetic mean) of clusters. We propose a general scheme that allows us to build a polynomial 1/2-approximation algorithm for a generalized problem and can be used for constructing 1/2-approximation algorithms for the first two cases and for the one-dimensional third case. Also we show how, using precomputed general information, their time complexities can be reduced to the complexity of sorting. Finally, we present the results of computational experiments showing the accuracy of the proposed algorithms on randomly generated input data.

A mechanism for ejecting a horseshoe from a partially hyperbolic chain recurrence class

AbstractWe give a $C^1$ -perturbation technique for ejecting an a priori given finite set of periodic points preserving a given finite set of homo/heteroclinic intersections from a chain recurrence class of a periodic point. The technique is first stated under a simpler setting called a Markov iterated function system, a two-dimensional iterated function system in which the compositions are chosen in a Markovian way. Then we apply the result to the setting of three-dimensional partially hyperbolic diffeomorphisms.

B-spline collocation method for solving Fredholm integral equations with discontinuous right-hand side

In this paper, we propose a method for approximating the solution of the linear Fredholm integral equation of the second kind which is defined on a closed contour Г in the complex plane. The right-hand side of the equation is a piecewise continuous function that is numerically defined on a finite set of points on Г. To approximate the solution, we use a linear combination of B-spline functions and Heaviside step functions defined on Г. We discuss both theoretical and practical aspects of the pointwise convergence of the method, including its performance in the vicinity of the points where discontinuities occur.

A Structural Theorem for Sets with Few Triangles

AbstractWe show that if a finite point set $$P\subseteq {\mathbb {R}}^2$$ P ⊆ R 2 has the fewest congruence classes of triangles possible, up to a constant M, then at least one of the following holds. There is a $$\sigma &gt;0$$ σ &gt; 0 and a line l which contains $$\Omega (|P|^\sigma )$$ Ω ( | P | σ ) points of P. Further, a positive proportion of P is covered by lines parallel to l each containing $$\Omega (|P|^\sigma )$$ Ω ( | P | σ ) points of P. There is a circle $$\gamma $$ γ which contains a positive proportion of P. This provides evidence for two conjectures of Erdős. We use the result of Petridis–Roche–Newton–Rudnev–Warren on the structure of the affine group combined with classical results from additive combinatorics.

Intersecting diametral balls induced by a geometric graph II

For a graph whose vertices are points in Rd, consider the closed balls with diameters induced by its edges. The graph is called a Tverberg graph if these closed balls share a common point.A max-sum tree of a finite point set X⊂Rd is a tree with vertex set X that maximizes the sum of Euclidean distances of its edges among all trees with vertex set X. Similarly, a max-sum matching of an even set X⊂Rd is a perfect matching of X maximizing the sum of Euclidean distances between the matched points among all perfect matchings of X.We prove that a max-sum tree of any finite point set in Rd is a Tverberg graph, which generalizes a recent result of Abu-Affash et al., who established this claim in the plane. Additionally, we provide a new proof of a theorem by Bereg et al., which states that a max-sum matching of any even point set in the plane is a Tverberg graph. Moreover, we proved a slightly stronger version of this theorem.

The average order of a connected induced subgraph of a graph and union‐intersection systems

AbstractBecause connectivity is such a basic concept in graph theory, extremal problems concerning the average order of the connected induced subgraphs of a graph have been of notable interest. A particularly resistant open problem is whether or not, for a connected graph of order , all of whose vertices have degree at least 3, this average is at least . It is shown in this paper that if is a connected, vertex transitive graph, then the average order of the connected induced subgraphs of is at least .The extremal graph theory problems mentioned above lead to a broader theory. The concept of a Union‐Intersection System (UIS) is introduced, being a finite set of points and a set of subsets of called blocks satisfying the following simple property for all : if , then . To generalize results on the average order of a connected induced subgraph of a graph, it is conjectured that if a UIS is, in various senses, “connected and regular,” then the average size of a block is at least half the number of points. We prove that if a union‐intersection set system is regular, completely irreducible, and nonredundant, then the average size of a block is at least half the number of points.

The integer point transform as a complete invariant

The integer point transform $\sigma_\PP$ is an important invariant of a rational polytope $\PP$, and here we show that it is a complete invariant. We prove that it is only necessary to evaluate $\sigma_\PP$ at one algebraic point in order to uniquely determine $\PP$, by employing the Lindemann-Weierstrass theorem. Similarly, we prove that it is only necessary to evaluate the Fourier transform of a rational polytope $\PP$ at a single algebraic point, in order to uniquely determine $\PP$. We prove that identical uniqueness results also hold for integer cones. In addition, by relating the integer point transform to finite Fourier transforms, we show that a finite number of \emph{integer point evaluations} of $\sigma_\PP$ suffice in order to uniquely determine $\PP$. We also give an equivalent condition for central symmetry of a finite point set, in terms of the integer point transform, and prove some facts about its local maxima. Most of the results are proven for arbitrary finite sets of integer points in $\R^d$.