Sets In Euclidean Space Research Articles

Isoperimetric inequalities in Euclidean convex bodies

In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body, i.e., a compact convex set in Euclidean space with interior points. We shall not impose any regularity assumption on the boundary of the convex set. Amongst other results, we shall prove that Hausdorff convergence in the space of convex bodies implies Lipschitz convergence, the continuity of the isoperimetric profile with respect to the Hausdorff distance, and the convergence in Hausdorff distance of sequences of isoperimetric regions and their free boundaries. We shall also describe the behavior of the isoperimetric profile for small volume and the behavior of isoperimetric regions for small volume.

Estimating singularity dimension

AbstractIn this paper we address an interesting question on the computation of the dimension of self-affine sets in Euclidean space. A well-known result of Falconer showed that under mild assumptions the Hausdorff dimension of typical self-affine sets is equal to its Singularity dimension. Heuter and Lalley subsequently presented a smaller open family of non-trivial examples for which there is an equality of these two dimensions. In this article we analyse the size of this family and present an efficient algorithm for estimating the dimension.

Covering sets in

The paper investigates questions related to Borsuk's classical problem of partitioning a set in Euclidean space into subsets of smaller diameter, as well as to the well-known Nelson-Erdős-Hadwiger problem on the chromatic number of a Euclidean space. The results of the work are obtained using combinatorial and geometric methods alike. A new approach to the investigation of such problems is suggested; it leads to a collection of results which significantly improve all results known so far. Bibliography: 58 titles.

Covering sets in

The paper investigates questions related to Borsuk's classical problem of partitioning a set in Euclidean space into subsets of smaller diameter, as well as to the well-known Nelson-Erdős-Hadwiger problem on the chromatic number of a Euclidean space. The results of the work are obtained using combinatorial and geometric methods alike. A new approach to the investigation of such problems is suggested; it leads to a collection of results which significantly improve all results known so far. Bibliography: 58 titles.

A note on gradient-type systems on fractals

The purpose of the paper is to point out that well-established methods to study the multiplicity of solutions of nonlinear elliptic problems defined on open sets in Euclidean spaces can be also used in the case of problems defined on fractals. More exactly, using variational methods and an abstract critical-point theorem by B. Ricceri, we prove the existence of three nonzero weak solutions of certain gradient-type systems defined on a famous fractal, the Sierpinski gasket. The paper emphasizes the way we overcame the difficulties arising from the major structural differences between the highly non-smooth Sierpinski gasket and the open subsets of Euclidean spaces on which gradient-type systems are usually considered.

Fractal zeta functions and complex dimensions of relative fractal drums

The theory of “zeta functions of fractal strings” has been initiated by the first author in the early 1990s and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called “distance zeta functions,” which since then has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta functions is the class of “tube zeta functions,” defined using the tube function of a fractal set. These three classes of zeta functions, under the name of “fractal zeta functions,” exhibit deep connections with Minkowski contents and upper box dimensions, as well as, more generally, with the complex dimensions of fractal sets. Further extensions include zeta functions of relative fractal drums, the box dimension of which can assume negative values, including minus infinity. We also survey some results concerning the existence of the meromorphic extensions of the spectral zeta functions of fractal drums, based in an essential way on earlier results of the first author on the spectral (or eigenvalue) asymptotics of fractal drums. It follows from these results that the associated spectral zeta function has a (nontrivial) meromorphic extension, and we use some of our results about fractal zeta functions to show the new fact according to which the upper bound obtained for the corresponding abscissa of meromorphic convergence is optimal.

On Equidistant Sets and Generalized Conics: The Old and the New

This article is devoted to the study of classical and new results concerning equidistant sets, both from the topological and metric point of view. We include a review of the most interesting known facts about these sets in Euclidean space and we prove two new results. First, we show that equidistant sets vary continuously with their focal sets. We also prove an error estimate result about approximative versions of equidistant sets that should be of interest for computer simulations. Moreover, we offer a viewpoint in which equidistant sets can be thought of as a natural generalization for conics. Along these lines, we show that the main geometric features of classical conics can be retrieved from more general equidistant sets.

Dimension Reduction by Random Hyperplane Tessellations

Given a subset K of the unit Euclidean sphere, we estimate the minimal number m=m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x,y∈K is nearly proportional to the Euclidean distance between x and y. Random hyperplanes prove to be almost ideal for this problem; they achieve the almost optimal bound m=O(w(K)2) where w(K) is the Gaussian mean width of K. Using the map that sends x∈K to the sign vector with respect to the hyperplanes, we conclude that every bounded subset K of $\mathbb{R}^{n}$ embeds into the Hamming cube {−1,1} m with a small distortion in the Gromov–Haussdorff metric. Since for many sets K one has m=m(K)≪n, this yields a new discrete mechanism of dimension reduction for sets in Euclidean spaces.

Invariance of the normalized Minkowski content with respect to the ambient space

It is easy to show that the lower and the upper box dimensions of a bounded set in Euclidean space are invariant with respect to the ambient space. In this article we show that the Minkowski content of a Minkowski measurable set is also invariant with respect to the ambient space when normalized by an appropriate constant. In other words, the value of the normalized Minkowski content of a bounded, Minkowski measurable set is intrinsic to the set.

Efficient processing of optimal meeting point queries in Euclidean space and road networks

Finding an optimal meeting point (OMP) for a group of people (or a set of objects) at different locations is an important problem in spatial query processing. There are many real-life applications related to this problem, such as determining the location of a conference venue, deciding the pick-up location of a tourist bus, and planing tactics of artificial intelligence in real-time strategy games. Formally, given a set $$Q$$ Q of query points in a spatial setting $$P$$ P , an OMP query fetches the point $$o\in P$$ o ? P that minimizes a cost function defined over the distances from $$o$$ o to all points in $$Q$$ Q . Since there are infinitely many locations in a given space setting, it is infeasible to examine all of them to find the OMP, and thus, the problem is challenging. In this paper, we study OMP queries in the following two spatial settings which are common in real-life applications: Euclidean space and road networks. In the setting of Euclidean space, we propose a general framework for answering all OMP query variants and also identify the best algorithms for particular types of OMP queries in the literature. In the setting of road networks, we study how to access only part of the road network and examine part of the candidates. Specifically, we explore two pruning techniques, namely Euclidean distance bound and threshold algorithm, which help improve the efficiency of OMP query processing. Extensive experiments are conducted to demonstrate the efficiency of our proposed approaches on both real and synthetic datasets.

Influence of Fractal Embedding in Three-Dimensional Euclidean Space on Wave Propogation in Electro-Chromodynamics

In this paper two zero-dimensional com- pact sets with equal topological and fractal dimensions but embedded in Euclidean space by different ways are under study. Diffraction of plane electromagnetic wave propagated and reflected by fractal surfaces is considered for each of these compact sets placed in vacuum. It is obtained, that the embedding of com- pact influences on characteristics of wave in final state. Thus, the embedding of Cantor set in Euclidean space is additional property of a fractal which can be important both for applications of fractal electrodynamics and for physics of strong interactions.

Dynamical properties of almost repetitive Delone sets

We consider the collection of uniformly discrete point sets in Euclidean space equipped with the vague topology.For a point set in this collection, we characterise minimality of an associated dynamical system by almost repetitivity of the point set. We also provide linear versions of almost repetitivity which lead to uniquely ergodic systems. Apart from linearly repetitive point sets, examples are given by periodic point sets with almost periodic modulations, and by point sets derived from primitive substitution tilings of finite local complexity with respect to the Euclidean group with dense tile orientations.

Solving nonlinear equations system via an efficient genetic algorithm with symmetric and harmonious individuals

We present an efficient genetic algorithm as a general tool for solving optimum problems. As a specialized application this algorithm can be used to approximate a solution of a system of nonlinear equations on n-dimensional Euclidean space setting. The new idea involves the introduction of some pairs of symmetric and harmonious individuals for the generation of a genetic algorithm. The population diversity is maintained better this way. The elitist model is used to ensure the convergence. Simulation results show that the new method is indeed very effective.

Open Sets Avoiding Integral Distances

We study open point sets in Euclidean spaces $\mathbb{R}^d$ without a pair of points an integral distance apart. By a result of Furstenberg, Katznelson, and Weiss such sets must be of Lebesgue upper density zero. We are interested in how large such sets can be in $d$-dimensional volume. We determine the lower and upper bounds for the volumes of the sets in terms of the number of their connected components and dimension, and also give some exact values. Our problem can be viewed as a kind of inverse to known problems on sets with pairwise rational or integral distances.

Approximate Bayesian estimation of extinction rate in the Finnish Daphnia magna metapopulation

Approximate Bayesian computation (ABC) is useful for parameterizing complex models in population genetics. In this study, ABC was applied to simultaneously estimate parameter values for a model of metapopulation coalescence and test two alternatives to a strict metapopulation model in the well-studied network of Daphnia magna populations in Finland. The models shared four free parameters: the subpopulation genetic diversity (θS), the rate of gene flow among patches (4Nm), the founding population size (N0) and the metapopulation extinction rate (e) but differed in the distribution of extinction rates across habitat patches in the system. The three models had either a constant extinction rate in all populations (strict metapopulation), one population that was protected from local extinction (i.e. a persistent source), or habitat-specific extinction rates drawn from a distribution with specified mean and variance. Our model selection analysis favoured the model including a persistent source population over the two alternative models. Of the closest 750,000 data sets in Euclidean space, 78% were simulated under the persistent source model (estimated posterior probability = 0.769). This fraction increased to more than 85% when only the closest 150,000 data sets were considered (estimated posterior probability = 0.774). Approximate Bayesian computation was then used to estimate parameter values that might produce the observed set of summary statistics. Our analysis provided posterior distributions for e that included the point estimate obtained from previous data from the Finnish D. magna metapopulation. Our results support the use of ABC and population genetic data for testing the strict metapopulation model and parameterizing complex models of demography.

Cusp excursions on parameter spaces

We prove several results for dynamics of SL(d, ℝ)-actions on non-compact parameter spaces by studying associated discrete sets in Euclidean spaces. This allows us to give elementary proofs of logarithm laws for horocycle flows on hyperbolic surfaces and moduli spaces of flat surfaces. We also give applications to quantitative equidistribution and Diophantine approximation.

Certain topological qualities of convex sets in Euclidean space

In this paper, we give four theorems relevant to certain topological qualities of the convex sets in Euclidean space. All results remain to hold in any locally convex space. They show that pair of convex sets under definite conditions satisfies some supplement qualities.

Intersections of descending sequences of affinely equivalent convex bodies

Here it is shown that a set in Euclidean space can be represented as the intersection of a descending sequence of sets affinely equivalent to a given convex body, or arbitrarily closely approximated from above by sets affinely equivalent to the body, if and only if it is affinely equivalent to an affine retract of the body. For the special case in which the body is a simplex, the statement concerning descending sequences is a well-known result of Borovikov.

Decomposition of acyclic normal currents in a metric space

We prove that every acyclic normal one-dimensional real Ambrosio–Kirchheim current in a Polish (i.e. complete separable metric) space can be decomposed in curves, thus generalizing the analogous classical result proven by S. Smirnov in Euclidean space setting. The same assertion is true for every complete metric space under a suitable set-theoretic assumption.

Learning figures with the Hausdorff metric by fractals—towards computable binary classification

We present learning of figures, nonempty compact sets in Euclidean space, based on Gold's learning model aiming at a computable foundation for binary classification of multivariate data. Encoding real vectors with no numerical error requires infinite sequences, resulting in a gap between each real vector and its discretized representation used for the actual machine learning process. Our motivation is to provide an analysis of machine learning problems that explicitly tackles this aspect which has been glossed over in the literature on binary classification as well as in other machine learning tasks such as regression and clustering. In this paper, we amalgamate two processes: discretization and binary classification. Each learning target, the set of real vectors classified as positive, is treated as a figure. A learning machine receives discretized vectors as input data and outputs a sequence of discrete representations of the target figure in the form of self-similar sets, known as fractals. The generalization error of each output is measured by the Hausdorff metric. Using this learning framework, we reveal a hierarchy of learnable classes under various learning criteria in the track of traditional analysis based on Gold's learning model, and show a mathematical connection between machine learning and fractal geometry by measuring the complexity of learning using the Hausdorff dimension and the VC dimension. Moreover, we analyze computability aspects of learning of figures using the framework of Type-2 Theory of Effectivity (TTE).