D-dimensional Euclidean Space Research Articles

Simulation of intrinsic random fields of order k with a continuous spectral algorithm

Intrinsic random fields of order k, defined as random fields whose high-order increments (generalized increments of order k) are second-order stationary, are used in spatial statistics to model regionalized variables exhibiting spatial trends, a feature that is common in earth and environmental sciences applications. A continuous spectral algorithm is proposed to simulate such random fields in a d-dimensional Euclidean space, with given generalized covariance structure and with Gaussian generalized increments of order k. The only condition needed to run the algorithm is to know the spectral measure associated with the generalized covariance function (case of a scalar random field) or with the matrix of generalized direct and cross-covariances (case of a vector random field). The algorithm is applied to synthetic examples to simulate intrinsic random fields with power generalized direct and cross-covariances, as well as an intrinsic random field with power and spline generalized direct covariances and Matern generalized cross-covariance.

From r-dual sets to uniform contractions

Let $$\mathbb {M}^d$$ denote the d-dimensional Euclidean, hyperbolic, or spherical space. The r-dual set of a given set in $$\mathbb {M}^d$$ is the intersection of balls of radii r centered at the points of the a given set. In this paper we prove that for any set of given volume in $$\mathbb {M}^d$$ the volume of the r-dual set becomes maximal if the set is a ball. As an application we prove the following. The Kneser–Poulsen Conjecture states that if the centers of a family of N congruent balls in Euclidean d-space is contracted, then the volume of the intersection does not decrease. A uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers, that is, when the pairwise distances of the two sets are separated by some positive real number. We prove a special case of the Kneser–Poulsen conjecture namely, we prove the conjecture for uniform contractions (with sufficiently large N) in $$\mathbb {M}^d$$ .

An exact algorithm for finding a vector subset with the longest sum

We consider the problem: Given a set of n vectors in the d-dimensional Euclidean space, find a subsetmaximizing the length of the sum vector.We propose an algorithm that finds an optimal solution to this problem in time O(nd−1(d + logn)). In particular, if the input vectors lie in a plane then the problem is solvable in almost linear time.

Approximating set multi-covers

Johnson and Lovász and Stein proved independently that any hypergraph satisfies τ≤(1+lnΔ)τ∗, where τ is the transversal number, τ∗ is its fractional version, and Δ denotes the maximum degree. We prove τf≤3.153τ∗max{lnΔ,f} for the f-fold transversal number τf. Similarly to Johnson, Lovász and Stein, we also show that this bound can be achieved non-probabilistically, using a greedy algorithm.As a combinatorial application, we prove an estimate on how fast τf∕f converges to τ∗. As a geometric application, we obtain an upper bound on the minimal density of an f-fold covering of the d-dimensional Euclidean space by translates of any convex body.

From Tarski's Plank Problem to Simultaneous Approximation

A slab (or plank) is the part of the d-dimensional Euclidean space that lies between two parallel hyperplanes. The distance between the these hyperplanes is called the width of the slab. It is conjectured that the members of any infinite family of slabs with divergent total width can be translated so that the translates together cover the whole d-dimensional space. We prove a slightly weaker version of this conjecture, which can be regarded as a converse of Bang's theorem, also known as Tarski's plank problem.This result enables us to settle an old conjecture of Makai and Pach on simultaneous approximation of polynomials. We say that an infinite sequence S of positive numbers controls all polynomials of degree at most d if there exists a sequence of points in the plane whose x-coordinates form the sequence S, such that the graph of every polynomial of degree at most d passes within vertical distance 1 from at least one of the points. We prove that a sequence S has this property if and only if the sum of the reciprocals of the dth powers of its elements is divergent.

A note on the equioscillation theorem for best ridge function approximation

We consider the approximation of a continuous function, defined on a compact set of the d-dimensional Euclidean space, by sums of two ridge functions. We obtain a necessary and sufficient condition for such a sum to be a best approximation. The result resembles the classical Chebyshev equioscillation theorem for polynomial approximation.

Canonical ensemble of particles in a self-avoiding random walk

We consider an ensemble of particles not interacting with each other and randomly walking in the d-dimensional Euclidean space ℝ d . The individual moves of each particle are governed by the same distribution, but after the completion of each such move of a particle, its position in the medium is “marked” as a region in the form of a ball of diameter r 0, which is not available for subsequent visits by this particle. As a result, we obtain the corresponding ensemble in ℝ d of marked trajectories in each of which the distance between the centers of any pair of these balls is greater than r 0. We describe a method for computing the asymptotic form of the probability density W n (r) of the distance r between the centers of the initial and final balls of a trajectory consisting of n individual moves of a particle of the ensemble. The number n, the trajectory modulus, is a random variable in this model in addition to the distance r. This makes it necessary to determine the distribution of n, for which we use the canonical distribution obtained from the most probable distribution of particles in the ensemble over the moduli of their trajectories. Averaging the density W n (r) over the canonical distribution of the modulus n allows finding the asymptotic behavior of the probability density of the distance r between the ends of the paths of the canonical ensemble of particles in a self-avoiding random walk in ℝ d for 2 ≤ d < 4.

A central limit theorem for Lipschitz–Killing curvatures of Gaussian excursions

This paper studies the excursion set of a real stationary isotropic Gaussian random field above a fixed level. We show that the standardized Lipschitz–Killing curvatures of the intersection of the excursion set with a window converges in distribution to a normal distribution as the window grows to the d-dimensional Euclidean space. Moreover a lower bound for the asymptotic variance is derived.

Recognising Multidimensional Euclidean Preferences

Euclidean preferences are a widely studied preference model, in which decision makers and alternatives are embedded in d-dimensional Euclidean space. Decision makers prefer those alternatives closer to them. This model, also known as multidimensional unfolding, has applications in economics, psychometrics, marketing, and many other fields. We study the problem of deciding whether a given preference profile is d-Euclidean. For the one-dimensional case, polynomial-time algorithms are known. We show that, in contrast, for every other fixed dimension d &gt; 1, the recognition problem is equivalent to the existential theory of the reals (ETR), and so in particular NP-hard. We further show that some Euclidean preference profiles require exponentially many bits in order to specify any Euclidean embedding, and prove that the domain of d-Euclidean preferences does not admit a finite forbidden minor characterisation for any d &gt; 1. We also study dichotomous preferences and the behaviour of other metrics, and survey a variety of related work.

Regularizing transformations of polygons

We start with a generic n-gon Q_0 with vertices q_{j,0} (j = 0, dots , n-1) in the d-dimensional Euclidean space {mathbb {E}}^d. Additionally, m+1 real numbers u_0, ldots , u_m in {mathbb {R}} , (m < n) with sum _{mu = 0}^m u_mu = 1 are given. From these initial data we iteratively define generations of n-gons Q_k in {mathbb {E}}^d for k in {mathbb {N}} with vertices q_{j,k}:= sum _{mu = 0}^m u_mu , q_{j+mu , k-1}. We can show that this affine iteration generally regularizes in an affine sense.

On the Combinatorial Complexity of Approximating Polytopes

Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body K of diameter \(\mathrm {diam}(K)\) is given in Euclidean d-dimensional space, where d is a constant. Given an error parameter \(\varepsilon > 0\), the objective is to determine a polytope of minimum combinatorial complexity whose Hausdorff distance from K is at most \(\varepsilon \cdot \mathrm {diam}(K)\). By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. A well-known result by Dudley implies that \(O(1/\varepsilon ^{(d-1)/2})\) facets suffice, and a dual result by Bronshteyn and Ivanov similarly bounds the number of vertices, but neither result bounds the total combinatorial complexity. We show that there exists an approximating polytope whose total combinatorial complexity is \(\widetilde{O}(1/\varepsilon ^{(d-1)/2})\), where \(\widetilde{O}\) conceals a polylogarithmic factor in \(1/\varepsilon \). This is a significant improvement upon the best known bound, which is roughly \(O(1/\varepsilon ^{d-2})\). Our result is based on a novel combination of both old and new ideas. First, we employ Macbeath regions, a classical structure from the theory of convexity. The construction of our approximating polytope employs a new stratified placement of these regions. Second, in order to analyze the combinatorial complexity of the approximating polytope, we present a tight analysis of a width-based variant of Bárány and Larman’s economical cap covering. Finally, we use a deterministic adaptation of the witness-collector technique (developed recently by Devillers et al.) in the context of our stratified construction.

Support function machine for set-based classification with application to water quality evaluation

In some applications, measurement errors and multiple repeated measurements often lead to a set-based classification task where objects are represented with a set of samples, and the traditional support vector machines (SVMs) do not work in these settings. To deal with this problem, we construct a new classifier called support function machine (SFM) in this work. First, sets in d-dimensional Euclidean space Rd are mapped into an infinite-dimensional Banach space C(S) (whose elements are functions) via support functions, and then set-based classification in Rd is converted into function-based classification in C(S). Second, we define the hyperplane via the Riesz representation theorem in Banach space, and discuss the Hausdorff distance of hyperpalnes and maximum margin principle (MMP) in C(S). Based on MMP, we construct an optimal problem and discuss some of its properties. Thereafter, we establish an SFM to solve set-based classification. Experiments about water quality evaluation and set-valued data classifications show the superiority of SFM.

Can exotic disordered "stealthy" particle configurations tolerate arbitrarily large holes?

The probability of finding a spherical cavity or "hole" of arbitrarily large size in typical disordered many-particle systems in the infinite-system-size limit (e.g., equilibrium liquid states) is non-zero. Such "hole" statistics are intimately linked to the thermodynamic and nonequilibrium physical properties of the system. Disordered "stealthy" many-particle configurations in d-dimensional Euclidean space [Doublestruck R]d are exotic amorphous states of matter that lie between a liquid and crystal that prohibit single-scattering events for a range of wave vectors and possess no Bragg peaks [Torquato et al., Phys. Rev. X, 2015, 5, 021020]. In this paper, we provide strong numerical evidence that disordered stealthy configurations across the first three space dimensions cannot tolerate arbitrarily large holes in the infinite-system-size limit, i.e., the hole probability has compact support. This structural "rigidity" property apparently endows disordered stealthy systems with novel thermodynamic and physical properties, including desirable band-gap, optical and transport characteristics. We also determine the maximum hole size that any stealthy system can possess across the first three space dimensions.

Extreme lattices: symmetries and decorrelation

We study statistical and structural properties of extreme lattices, which are the local minima in the density landscape of lattice sphere packings in d-dimensional Euclidean space . Specifically, we ascertain statistics of the densities and kissing numbers as well as the numbers of distinct symmetries of the packings for dimensions 8 through 13 using the stochastic Voronoi algorithm. The extreme lattices in a fixed dimension of space d () are dominated by typical lattices that have similar packing properties, such as packing densities and kissing numbers, while the best and the worst packers are in the long tails of the distribution of the extreme lattices. We also study the validity of the recently proposed decorrelation principle, which has important implications for sphere packings in general. The degree to which extreme-lattice packings decorrelate as well as how decorrelation is related to the packing density and symmetry of the lattices as the space dimension increases is also investigated. We find that the extreme lattices decorrelate with increasing dimension, while the least symmetric lattices decorrelate faster.

Disordered hyperuniform heterogeneous materials

Disordered hyperuniform many-body systems are distinguishable states of matter that lie between a crystal and liquid: they are like perfect crystals in the way they suppress large-scale density fluctuations and yet are like liquids or glasses in that they are statistically isotropic with no Bragg peaks. These systems play a vital role in a number of fundamental and applied problems: glass formation, jamming, rigidity, photonic and electronic band structure, localization of waves and excitations, self-organization, fluid dynamics, quantum systems, and pure mathematics. Much of what we know theoretically about disordered hyperuniform states of matter involves many-particle systems. In this paper, we derive new rigorous criteria that disordered hyperuniform two-phase heterogeneous materials must obey and explore their consequences. Two-phase heterogeneous media are ubiquitous; examples include composites and porous media, biological media, foams, polymer blends, granular media, cellular solids, and colloids. We begin by obtaining some results that apply to hyperuniform two-phase media in which one phase is a sphere packing in d-dimensional Euclidean space . Among other results, we rigorously establish the requirements for packings of spheres of different sizes to be ‘multihyperuniform’. We then consider hyperuniformity for general two-phase media in . Here we apply realizability conditions for an autocovariance function and its associated spectral density of a two-phase medium, and then incorporate hyperuniformity as a constraint in order to derive new conditions. We show that some functional forms can immediately be eliminated from consideration and identify other forms that are allowable. Specific examples and counterexamples are described. Contact is made with well-known microstructural models (e.g. overlapping spheres and checkerboards) as well as irregular phase-separation and Turing-type patterns. We also ascertain a family of autocovariance functions (or spectral densities) that are realizable by disordered hyperuniform two-phase media in any space dimension, and present select explicit constructions of realizations. These studies provide insight into the nature of disordered hyperuniformity in the context of heterogeneous materials and have implications for the design of such novel amorphous materials.

Discrete directional Gabor frames

Abstract We develop a theory of discrete directional Gabor frames for functions defined on the d-dimensional Euclidean space. Our construction incorporates the concept of ridge functions into the theory of isotropic Gabor systems, in order to develop an anisotropic Gabor system with strong directional sensitivity. We present sufficient conditions on a window function g and a sampling set Λ ω for the corresponding directional Gabor system { g m , t , u } ( m , t , u ) ∈ Λ ω to form a discrete frame. Explicit estimates on the frame bounds are developed. A numerical implementation of our scheme is also presented, and is shown to perform competitively in compression and denoising schemes against state-of-the-art multiscale and anisotropic methods, particularly for images with significant texture components.

Inequalities for hyperconvex sets

Abstract An r-hyperconvex body is a set in the d-dimensional Euclidean space 𝔼 d that is the intersection of a family of closed balls of radius r. We prove the analogue of the classical Blaschke–Santaló inequality for r-hyperconvex bodies, and we also establish a stability version of it. The other main result of the paper is an r-hyperconvex version of the reverse isoperimetric inequality in the plane.

Maximum volume polytopes inscribed in the unit sphere

In this paper we investigate the problem of finding the maximum volume polytopes, inscribed in the unit sphere of the d-dimensional Euclidean space, with a given number of vertices. We solve this problem for polytopes with \(d+2\) vertices in every dimension, and for polytopes with \(d+3\) vertices in odd dimensions. For polytopes with \(d+3\) vertices in even dimensions we give a partial solution.

Spectral simulation of vector random fields with stationary Gaussian increments in d-dimensional Euclidean spaces

This paper addresses the problem of simulating multivariate random fields with stationary Gaussian increments in a d-dimensional Euclidean space. To this end, one considers a spectral turning-bands algorithm, in which the simulated field is a mixture of basic random fields made of weighted cosine waves associated with random frequencies and random phases. The weights depend on the spectral density of the direct and cross variogram matrices of the desired random field for the specified frequencies. The algorithm is applied to synthetic examples corresponding to different spatial correlation models. The properties of these models and of the algorithm are discussed, highlighting its computational efficiency, accuracy and versatility.

Two applications of Brouwer’s fixed point theorem: in insurance and in biology models

In the first part of the article, a new interesting system of difference equations is introduced. It is developed for re-rating purposes in general insurance. A nonlinear transformation φ of a d-dimensional (d ≥ 2) Euclidean space is introduced that enables us to express the system in the form ft+1:=φ( ft), t = 0, 1, 2,. …. Under typical actuarial assumptions, existence of solutions of that system is proven by means of Brouwer’s fixed point theorem in normed spaces. In addition, conditions that guarantee uniqueness of a solution are given. The second, smaller part of the article is about Leslie–Gower’s system of d ≥ 2 difference equations. We focus on the system that satisfies conditions consistent with weak inter-specific competition. We prove existence and uniqueness of the equilibrium of the model under surprisingly simple and very general conditions. Even though the two parts of this article have applications in two different sciences, they are connected with similar mathematics, in particular by our use of Brouwer’s fixed point theorem.