Set In R2 Research Articles

Periodicity and decidability of translational tilings by rational polygonal sets

The periodic tiling conjecture asserts that if a region Σ⊂Rd tiles Rd by translations then it admits at least one fully periodic tiling. This conjecture is known to hold in R, and recently it was disproved in sufficiently high dimensions. In this paper, we study the periodic tiling conjecture for polygonal sets: bounded open sets in R2 whose boundary is a finite union of line segments. We prove the periodic tiling conjecture for any polygonal tile whose vertices are rational. As a corollary of our argument, we also obtain the decidability of tilings by rational polygonal sets. Moreover, we prove that any translational tiling by a rational polygonal tile is weakly-periodic, i.e., can be partitioned into finitely many singly-periodic pieces.

Achievement Sets of Series in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document}

We examine the properties of achievement sets of series in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document}. We show several examples of unusual sets of subsums on the plane. We prove that we can obtain any set of P-sums as a cut of an achievement set in R2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2.$$\\end{document} We introduce a notion of the spectre of a set in an Abelian group, which is an algebraic version of the notion of the center of distances. We examine properties of the spectre and we use it, for example, to show that the Sierpiński carpet is not an achievement set of any series.

Rectilinear convex hull of points in 3D and applications

Let P be a set of n points in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document} in general position, and let RCH(P) be the rectilinear convex hull of P. In this paper we obtain an optimal O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log n)$$\\end{document} time and O(n) space algorithm to compute RCH(P). We also obtain an efficient O(nlog2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log ^2 n)$$\\end{document} time and O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log n)$$\\end{document} space algorithm to compute and maintain the set of vertices of the rectilinear convex hull of P as we rotate R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document} around the Z-axis. We study some combinatorial properties of the rectilinear convex hulls of point sets in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}. Finally, as an application of the obtained results, we show an approximation algorithm to an optimization fitting problem in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}.

Stationary solutions to the Boltzmann equation in the plane

The paper proves existence of stationary solutions to the Boltzmann equation in a bounded set of R 2 \mathbb {R}^2 for given indata, hard forces and truncation in the collision kernel for small velocities and close to parallel colliding velocities. It does not use any averaging in velocity lemma. Instead, it is based on stability techniques employing the Kolmogorov-Riesz-Fréchet theorem, from the discrete velocity stationary case, where the averaging in velocity lemmas are not valid.

Spatial decay properties for a model in shear flows posed on the cylinder

We study spatial decay properties for solutions of the Pelinovski–Stepanyants equation posed on the cylinder. We establish the maximum polynomial decay admissible for solutions of such a model. It is verified that the equation on the cylinder propagates polynomial weights with different restrictions than the model set in R2. For example, a local well-posedness theory is deduced which contains the line solitary wave provided by solutions of the Benjamin–Ono equation extended to the cylinder. Key results for our analysis are obtained from a detailed study of the dispersive effects of the linear equation in weighted Sobolev spaces. The results in this paper appear to be one of the first studies of polynomial decay for nonlocal models in the cylinder.

Bounds on piercing and line-piercing numbers in families of convex sets in the plane

A family of sets has the (p,q) property if among any p members of it some q intersect. It is shown that if a finite family of compact convex sets in R2 has the (p+1,2) property then it is pierced by ⌊p2⌋+1 lines. A colorful version of this result is proved as well. As a corollary, the following is proved: Let F be a finite family of compact convex sets in the plane with no isolated sets, and let F′ be the family of its pairwise intersections. If F has the (p+1,2) property and F′ has the (r+1,2) property, then F is pierced by (⌊r2⌋2+⌊r2⌋)p points when r⩾2, and by p points otherwise. The proofs use the topological KKM theorem.

Typed topology and its application to data sets

The concept of typedtopology is introduced. In a typed topological space, some open sets are assigned “types”, and topological concepts such as closure, connectedness can be defined using types. A finite data set in R2 is a typically typed topological space. Clusters calculated by the DBSCAN algorithm for data clustering can be well represented in a finite typed topological space. Other concepts such as tracks, port (starting points), type-p-connectedness, p-closure-connectedness, indexing, branches are also introduced for finite typed topological spaces. Finally, left−r and up−left−r type open sets are introduced for finite data sets in R2, so that tracks, port, branches can be calculated.

On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane

For a locally finite set in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\mathbb {R}}}}^2$$\\end{document}, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\mathbb {R}}}}^2$$\\end{document} is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6, 85–127 (1970)).

A note on sets avoiding rational distances in category bases

Michalski gave a short and elegant proof of a theorem of A. Kumar which states that for each set A⊆R, there exists a set B⊆A which is full in A and such that no distance between points in B is a rational number. He also proved a similar theorem for sets in R2. In this paper, we generalize these results in some special types of category bases.

Nonstandard approximations to Besicovitch sets

A nonstandard approximation to a Besicovitch set in R2 is constructed. It is shown that this approximation easily yields the correct upper Minkowski dimension in the two-dimensional case through a direct computation. By constructing an approximation to a Besicovitch set in three dimensions, we use a theorem of Guth and Katz to obtain an upper bound on the number of intersections of line segments in nonstandard terms.

Continuous maps that preserve Hausdorff measure

We show that the space of homeomorphisms on the unit interval [0,1], equipped with the topology of uniform convergence, contains a dense subspace of functions that preserve the Hausdorff measure of any subset of certain one-dimensional self-similar sets. We extend the results to a class of Cantor dust type self-similar sets in R2.

A Łojasiewicz inequality in hypocomplex structures of ℝ²

For a real analytic complex vector field L L , in an open set of R 2 \mathbb {R}^2 , with local first integrals that are open maps, we attach a number μ ≥ 1 \mu \ge 1 (obtained through Łojasiewicz inequalities) and show that the equation L u = f Lu=f has bounded solution when f ∈ L p f\in L^p with p > 1 + μ p>1+\mu . We also establish a similarity principle between the bounded solutions of the equation L u = A u + B u ¯ Lu=Au+B\overline {u} (with A , B ∈ L p A,B\in L^p ) and holomorphic functions.

Three‐Dimensional Expansion and Graphical Concept of Generalized Triangular Fuzzy Set

We have studied the extended algebraic operations between two fuzzy numbers and calculated Zadeh’s max‐min composition operator for two generalized triangular fuzzy sets in ℝ2. And we generalized the triangular fuzzy numbers from ℝ2 to ℝ3. We prove that the result of the three‐dimensional case is an extension of two‐dimensional case and presented it in a graph. The extension is proved by showing that the result obtained by restricting the three‐dimensional result to two‐dimensional result is consistent with the existing two‐dimensional result.

The Yang-Mills heat equation with finite action in three dimensions

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over R 3 \mathbb {R}^3 and over a bounded open convex set in R 3 \mathbb {R}^3 . The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in the critical gauge group of Sobolev regularity three halves, which is a complete topological group in a natural metric but is not a Hilbert Lie group. The nature of this group must be understood in order to carry out the reconstruction procedure. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation.

Lower Porosity on R2

In this paper, we study the properties of a lower porosity of a set in R2. It turns out that the properties of the lower and upper porosity are symmetrical, except that the main tools for testing the lower porosity are not balls but cones. New families of topologies on R2 generated by the lower porosity are defined. Furthermore, by applying the notion of the lower porosity, we introduce the definition of generalized continuity. Using defined topologies, we study properties of this continuity. We show that the properties of topologies generated by the lower and (upper) porosity are symmetrical.

The box dimensions of exceptional self-affine sets in [formula omitted

We study the box dimensions of self-affine sets in R3 which are generated by a finite collection of generalised permutation matrices. We obtain bounds for the dimensions which hold with very minimal assumptions and give rise to sharp results in many cases. There are many issues in extending the well-established planar theory to R3 including that the principal planar projections are (affine distortions of) self-affine sets with overlaps (rather than self-similar sets) and that the natural modified singular value function fails to be sub-multiplicative in general. We introduce several new techniques to deal with these issues and hopefully provide some insight into the challenges in extending the theory further.

Good elliptic operators on Cantor sets

It is well known that a purely unrectifiable set cannot support a harmonic measure which is absolutely continuous with respect to the Hausdorff measure of this set. We show that nonetheless there exist elliptic operators on (purely unrectifiable) Cantor sets in R2 whose elliptic measure is absolutely continuous, and in fact, essentially proportional to the Hausdorff measure.

Local 𝐶^{1,𝛽}-regularity at the boundary of two dimensional sliding almost minimal sets in ℝ³

In this paper, we will give a C 1 , β C^{1,\beta } -regularity result on the boundary for two dimensional sliding almost minimal sets in R 3 \mathbb {R}^3 . This effect may apply to the regularity of the soap films at the boundary, and may also lead to the existence of a solution to the Plateau problem with sliding boundary conditions proposed by Guy David in the case that the boundary is a 2-dimensional smooth submanifold.

A justification of the Darcy law for a suspension of not self-similar solid particles non-periodically distributed

We extend to a non-periodic framework the classical homogenization result permitting to derive the Darcy law from the Stokes or Navier–Stokes system posed in a perforated domain. Mathematically, we study the asymptotic behavior when ɛ tends to zero of the stationary Stokes system posed in a sequence of varying domains Ωɛ=Ω∖∪k∈NTɛk where Ω is a smooth bounded open subset of R3 and Tɛk are closed sets such that each of them is at a distance of order ɛ of the remaining. Moreover ∪k∈NTɛk is at a distance of order at most ɛ of any point of R3. Each set Tɛk is non-empty, smooth and has a size of order δɛ with ɛ3<<δɛ≤rɛ for some r<1. In the classical periodic case, the sets Tɛk are obtained by repeating periodically with period ɛ the set δɛT with T a non-empty smooth closed set in R3. As in this periodic case we show that the limit problem corresponds to a Darcy system. However, even when the sets Tɛk have all the same shape, we show that for δɛ<<ɛ some strong convergence results for the velocity and some capacity formulae for the limit system do not extend from the periodic framework to the non-periodic one.

A sampling type discernment approach towards reconstruction of a point set in R2

In this paper, we propose a Delaunay triangulation based algorithmic framework for the 2D point set type discernment and its reconstruction. The point set discernment deals with determining whether the given point set S consists of only the points sampled along the boundary of an object (referred to as boundary sample) or S is distributed across the object (referred to as dot-pattern). In general, the existing approaches deal with the reconstruction of an a priori known input type - boundary sample or dot-pattern. Our approach works on both sets of input data type by first identifying the input type that a given point set belongs to. To distinguish the point set type, we introduce the notion of Petal-Ratio(PR) which captures the ratio of the longest edge to the smallest edge emanating from a vertex in Delaunay triangulation. Further, we employ the Petal-Ratio as the essence to design simple reconstruction algorithms pertaining to each input type that unveil the shape of the object hidden in the given point set. The reconstruction algorithms start with computing the Delaunay triangulation of the given point set and for each vertex, the Petal-Ratio is calculated. Following this, the edges are removed based on the value of the PR for every vertex to arrive at the boundaries. Theoretical analysis of determining appropriate values of PRs under ϵ- sampling and minimal reach sampling models have been presented. Unlike many other algorithms, our approach is non-parametric and non-feature specific that can capture features like disconnected components, multiple holes even with the presence of outliers, self-intersection (only for boundary sample) and open curves (only for boundary sample). Moreover, our algorithms take only one pass to reconstruct the hole boundaries as well as outer boundary irrespective of the object features like the number of holes and the number of components. The comparative analysis shows that our algorithms perform equally well or better than their counterparts for the objects with contrasting features. We also demonstrate that the proposed idea can be easily extended to surface reconstruction.