D-dimensional Euclidean Space Research Articles

Three Cube Packing for All Dimensions

Let Vn(d) denote the least number, such that every collection of n d-cubes with total volume 1 in d-dimensional (Euclidean) space can be packed parallelly into some d-box of volume Vn(d). We show that V3(d)=r1−dd if d≥11 and V3(d)=1r+1rd+1r−rd+1 if 2≤d≤10, where r is the only solution of the equation 2(d−1)kd+dkd−1=1 on 22,1 and (k+1)d(1−k)d−1dk2+d+k−1=kddkd+1+dkd+kd+1 on 22,1, respectively. The maximum volume is achieved by hypercubes with edges x, y, z, such that x=2rd+1−1/d, y=z=rx if d≥11, and x=rd+(1r−r)d+1−1/d, y=rx, z=(1r−r)x if 2≤d≤10. We also proved that only for dimensions less than 11 are there two different maximum packings, and for all dimensions greater than 10, the maximum packing has the same two smallest cubes.

Spatial Voting with Incomplete Voter Information

We consider spatial voting where candidates are located in the Euclidean d-dimensional space, and each voter ranks candidates based on their distance from the voter's ideal point. We explore the case where information about the location of voters' ideal points is incomplete: for each dimension, we are given an interval of possible values. We study the computational complexity of finding the possible and necessary winners for positional scoring rules. Our results show that we retain tractable cases of the classic model where voters have partial-order preferences. Moreover, we show that there are positional scoring rules under which the possible-winner problem is intractable for partial orders, but tractable in the one-dimensional spatial setting. We also consider approval voting in this setting. We show that for up to two dimensions, the necessary-winner problem is tractable, while the possible-winner problem is hard for any number of dimensions.

Approximation error of single hidden layer neural networks with fixed weights

Neural networks with finitely many fixed weights have the universal approximation property under certain conditions on compact subsets of the d-dimensional Euclidean space, where approximation process is considered. Such conditions were delineated in our paper [26]. But for many compact sets it is impossible to approximate multivariate functions with arbitrary precision and the question on estimation or efficient computation of approximation error arises. This paper provides an explicit formula for the approximation error of single hidden layer neural networks with two fixed weights.

On Blaschke–Santaló-Type Inequalities for r-Ball Bodies

Let 𝔼𝑑 denote the 𝑑-dimensional Euclidean space. The 𝑟-ball body generated by a given set in 𝔼𝑑 is the intersection of balls of radius 𝑟 centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke–Santaló-type inequalities for 𝑟-ball bodies: for all 0 < 𝑘 < 𝑑 and for any set of given 𝑑-dimensional volume in 𝔼𝑑 the 𝑘-th intrinsic volume of the 𝑟-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.

Unbounded Regions of High-Order Voronoi Diagrams of Lines and Line Segments in Higher Dimensions

AbstractWe study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions $$\mathbb {S}^{d-1}$$ S d - 1 . We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments and lines is $$O(\min \{k,n-k\}n^{d-1})$$ O ( min { k , n - k } n d - 1 ) , which is tight for $$n-k=O(1)$$ n - k = O ( 1 ) . This exactly reflects the combinatorial complexity of the unbounded features of these diagrams. All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its $$(d-1)$$ ( d - 1 ) -skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of $$n \ge 2$$ n ≥ 2 lines in general position has exactly $$n(n-1)$$ n ( n - 1 ) three-dimensional cells. The Gaussian map of the farthest Voronoi diagram of line segments and lines can be constructed in $$O(n^{d-1} \alpha (n))$$ O ( n d - 1 α ( n ) ) time, for $$d\ge 4$$ d ≥ 4 , while if $$d=3$$ d = 3 , the time drops to worst-case optimal $$\Theta (n^2)$$ Θ ( n 2 ) . We extend the obtained results to bounded polyhedra and clusters of points as sites.

Rudin extension theorems on product spaces, turning bands, and random fields on balls cross time

Characteristic functions that are radially symmetric have a dual interpretation, as they can be used as the isotropic correlation functions of spatial random fields. Extensions of isotropic correlation functions from balls into d-dimensional Euclidean spaces, Rd, have been understood after Rudin. Yet, extension theorems on product spaces are elusive, and a counterexample provided by Rudin on rectangles suggests that the problem is challenging. This paper provides extension theorems for multiradial characteristic functions that are defined in balls embedded in Rd cross, either Rd′ or the unit sphere Sd′ embedded in Rd′+1, for any two positive integers d and d′. We then examine Turning Bands operators that provide bijections between the class of multiradial correlation functions in given product spaces, and multiradial correlations in product spaces having different dimensions. The combination of extension theorems with Turning Bands provides a connection with random fields that are defined in balls cross linear or circular time.

On the Partition of Space by Hyperplanes

We consider the problem of partitioning of space by hyperplanes that arises in many application areas, where the number of regions the space is divided into is required to be determined, such as speech/pattern recognition, various classification problems, data analysis. We obtain some relations for the number of divisions and establish a recurrence relation for the maximum number of regions in d-dimensional Euclidean space cut by n hyperplanes. We also re-derive an explicit formula for the number of regions into which the space can be partitioned by n hyperplanes.

Resonance of ellipsoidal billiard trajectories and extremal rational functions

We study resonant billiard trajectories within quadrics in the d-dimensional Euclidean space. We relate them to the theory of approximation, in particular the extremal rational functions on the systems of d intervals on the real line. This fruitful link enables us to prove fundamental properties of the billiard dynamics and to provide a comprehensive study of a large class of non-periodic trajectories of integrable billiards. A key ingredient is a functional-polynomial relation of a generalized Pell type. Applying further these ideas and techniques to s-weak billiard trajectories, we come to a functional-polynomial relation of the same generalized Pell type.

Linear-time approximation scheme for k-means clustering of axis-parallel affine subspaces

In this paper, we present a linear-time approximation scheme for k-means clustering of incomplete data points in d-dimensional Euclidean space. An incomplete data point with Δ>0 unspecified entries is represented as an axis-parallel affine subspace of dimension Δ. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for k-means clustering of n axis-parallel affine subspaces of dimension Δ that yields an (1+ϵ)-approximate solution in O(nd) time. The constants hidden behind O(⋅) depend only on Δ,ϵ and k. This improves the O(n2d)-time algorithm by Eiben et al. (2021) [7] by a factor of n.

Generalised Wendland functions for the sphere

In this paper, we compute the spherical Fourier expansion coefficients for the restriction of the generalised Wendland functions from d-dimensional Euclidean space to the (d − 1)-dimensional unit sphere. We use results from the theory of special functions to show that they can be expressed in a closed form as a multiple of a certain 3F2 hypergeometric function. We present tight asymptotic bounds on the decay rate of the spherical Fourier coefficients and, in the case where d is odd, we are able to provide the precise asymptotic rate of decay. Numerical evidence suggests that this precise asymptotic rate also holds when d is even and we pose this as an open problem. Finally, we observe a close connection between the asymptotic decay rate of the spherical Fourier coefficients and that of the corresponding Euclidean Fourier transform.

Realizability of iso-g2 processes via effective pair interactions.

An outstanding problem in statistical mechanics is the determination of whether prescribed functional forms of the pair correlation function g2(r) [or equivalently, structure factor S(k)] at some number density ρ can be achieved by many-body systems in d-dimensional Euclidean space. The Zhang-Torquato conjecture states that any realizable set of pair statistics, whether from a nonequilibrium or equilibrium system, can be achieved by equilibrium systems involving up to two-body interactions. To further test this conjecture, we study the realizability problem of the nonequilibrium iso-g2 process, i.e., the determination of density-dependent effective potentials that yield equilibrium states in which g2 remains invariant for a positive range of densities. Using a precise inverse algorithm that determines effective potentials that match hypothesized functional forms of g2(r) for all r and S(k) for all k, we show that the unit-step function g2, which is the zero-density limit of the hard-sphere potential, is remarkably realizable up to the packing fraction ϕ = 0.49 for d = 1. For d = 2 and 3, it is realizable up to the maximum "terminal" packing fraction ϕc = 1/2d, at which the systems are hyperuniform, implying that the explicitly known necessary conditions for realizability are sufficient up through ϕc. For ϕ near but below ϕc, the large-r behaviors of the effective potentials are given exactly by the functional forms exp[ - κ(ϕ)r] for d = 1, r-1/2exp[ - κ(ϕ)r] for d = 2, and r-1exp[ - κ(ϕ)r] (Yukawa form) for d = 3, where κ-1(ϕ) is a screening length, and for ϕ = ϕc, the potentials at large r are given by the pure Coulomb forms in the respective dimensions as predicted by Torquato and Stillinger [Phys. Rev. E 68, 041113 (2003)]. We also find that the effective potential for the pair statistics of the 3D "ghost" random sequential addition at the maximum packing fraction ϕc = 1/8 is much shorter ranged than that for the 3D unit-step function g2 at ϕc; thus, it does not constrain the realizability of the unit-step function g2. Our inverse methodology yields effective potentials for realizable targets, and, as expected, it does not reach convergence for a target that is known to be non-realizable, despite the fact that it satisfies all known explicit necessary conditions. Our findings demonstrate that exploring the iso-g2 process via our inverse methodology is an effective and robust means to tackle the realizability problem and is expected to facilitate the design of novel nanoparticle systems with density-dependent effective potentials, including exotic hyperuniform states of matter.

Compactly-Supported Isotropic Covariances on Spheres Obtained from Matrix-Valued Covariances in Euclidean Spaces

Let p, d be positive integers, with d odd. Let \(\varvec{\phi }:[0,+\infty ) \rightarrow {\mathbb {R}}^{p \times p}\) be the isotropic part of a matrix-valued and isotropic covariance function (a positive semidefinite matrix-valued function) that is defined over the d-dimensional Euclidean space. If \(\varvec{\phi }\) is compactly supported over \([0,\pi ]\), then we show that the restriction of \(\varvec{\phi }\) to \([0,\pi ]\) is the isotropic part of a matrix-valued covariance function defined on a d-dimensional sphere, where isotropy in this case means that the covariance function depends on the geodesic distance. Our result does not need any assumption of continuity for the mapping \(\varvec{\phi }\). Further, when \(\varvec{\phi }\) is continuous, we provide an analytical expression of the d-Schoenberg sequence associated with the compactly-supported covariance on the sphere, which only requires knowledge of the Fourier transform of its isotropic part, and illustrate with the Gauss hypergeometric covariance model, which encompasses the well-known spherical, cubic, Askey and generalized Wendland covariances, and with a hole effect covariance model. Special cases of the results presented in this paper have been provided by other authors in the past decade.

Regularizing effects of the entropy functional in optimal transport and planning problems

We analyze optimal transport problems with additional entropic cost evaluated along curves in the Wasserstein space which join two probability measures m0,m1. The effect of the additional entropy functional results into an elliptic regularization for the (so-called) Kantorovich potentials of the dual problem. Assuming the initial and terminal measures to be positive and smooth, we prove that the optimal curve remains smooth for all time. We focus on the case that the transport problem is set on a convex bounded domain in the d-dimensional Euclidean space (with no-flux condition on the boundary), but we also mention the case of Gaussian-like measures in the whole space. The approach follows ideas introduced by P.-L. Lions in the theory of mean-field games [27]. The result provides with a smooth approximation of minimizers in optimization problems with penalizing congestion terms, which appear in mean-field control or mean-field planning problems. This allows us to exploit new estimates for this kind of problems by using displacement convexity properties in the Eulerian approach.

Asymptotically equivalent prediction in multivariate geostatistics

Cokriging is the common method of spatial interpolation (best linear unbiased prediction) in multivariate geostatistics. While best linear prediction has been well understood in univariate spatial statistics, the literature for the multivariate case has been elusive so far. The new challenges provided by modern spatial datasets, being typically multivariate, call for a deeper study of cokriging. In particular, we deal with the problem of misspecified cokriging prediction within the framework of fixed domain asymptotics. Specifically, we provide conditions for equivalence of measures associated with multivariate Gaussian random fields, with index set in a compact set of a d-dimensional Euclidean space. Such conditions have been elusive for over about 50 years of spatial statistics. We then focus on the multivariate Matérn and Generalized Wendland classes of matrix valued covariance functions, that have been very popular for having parameters that are crucial to spatial interpolation, and that control the mean square differentiability of the associated Gaussian process. We provide sufficient conditions, for equivalence of Gaussian measures, relying on the covariance parameters of these two classes. This enables to identify the parameters that are crucial to asymptotically equivalent interpolation in multivariate geostatistics. Our findings are then illustrated through simulation studies.

Optimal Bound on the Combinatorial Complexity of Approximating Polytopes

This article considers the question of how to succinctly approximate a multidimensional convex body by a polytope. Given a convex body K of unit diameter in Euclidean d -dimensional space (where d is a constant) and an error parameter ε > 0, the objective is to determine a convex polytope of low combinatorial complexity whose Hausdorff distance from K is at most ε. By combinatorial complexity , we mean the total number of faces of all dimensions. Classical constructions by Dudley and Bronshteyn/Ivanov show that O (1/ε ( d -1)/2 ) facets or vertices are possible, respectively, but neither achieves both bounds simultaneously. In this article, we show that it is possible to construct a polytope with O (1/ε ( d -1)/2 ) combinatorial complexity, which is optimal in the worst case. Our result is based on a new relationship between ε-width caps of a convex body and its polar body. Using this relationship, we are able to obtain a volume-sensitive bound on the number of approximating caps that are “essentially different.” We achieve our main result by combining this with a variant of the witness-collector method and a novel variable-thickness layered construction of the economical cap covering.

Precise determination of pair interactions from pair statistics of many-body systems in and out of equilibrium.

The determination of the pair potential v(r) that accurately yields an equilibrium state at positive temperature T with a prescribed pair correlation function g_{2}(r) or corresponding structure factor S(k) in d-dimensional Euclidean space R^{d} is an outstanding inverse statistical mechanics problem with far-reaching implications. Recently, Zhang and Torquato [Phys. Rev. E 101, 032124 (2020)2470-004510.1103/PhysRevE.101.032124] conjectured that any realizable g_{2}(r) or S(k) corresponding to a translationally invariant nonequilibrium system can be attained by a classical equilibrium ensemble involving only (up to) effective pair interactions. Testing this conjecture for nonequilibrium systems as well as for nontrivial equilibrium states requires improved inverse methodologies. We have devised an optimization algorithm to precisely determine effective pair potentials that correspond to pair statistics of general translationally invariant disordered many-body equilibrium or nonequilibrium systems at positive temperatures. This methodology utilizes a parameterized family of pointwise basis functions for the potential function whose initial form is informed by small-, intermediate- and large-distance behaviors dictated by statistical-mechanical theory. Subsequently, a nonlinear optimization technique is utilized to minimize an objective function that incorporates both the target pair correlation function g_{2}(r) and structure factor S(k) so that the small intermediate- and large-distance correlations are very accurately captured. To illustrate the versatility and power of our methodology, we accurately determine the effective pair interactions of the following four diverse target systems: (1) Lennard-Jones system in the vicinity of its critical point, (2) liquid under the Dzugutov potential, (3) nonequilibrium random sequential addition packing, and (4) a nonequilibrium hyperuniform "cloaked" uniformly randomized lattice. We found that the optimized pair potentials generate corresponding pair statistics that accurately match their corresponding targets with total L_{2}-norm errors that are an order of magnitude smaller than that of previous methods. The results of our investigation lend further support to the Zhang-Torquato conjecture. Furthermore, our algorithm will enable one to probe systems with identical pair statistics but different higher-body statistics, which will shed light on the well-known degeneracy problem of statistical mechanics.

On Wendel’s equality for intersections of balls

We study the analogue of Wendel’s equality in random polytope models in which the hull of the random points is formed by intersections of congruent balls, called the spindle (or hyper-) convex hull. According to the classical identity of Wendel the probability that the origin is contained in the (linear) convex hull of n i.i.d. random points distributed according to an origin symmetric probability distribution in the d-dimensional Euclidean space mathbb {R}^{d} that assigns measure zero to hyperplanes is a constant depending only on n and d. While in the classical convex case one gets nonzero probabilities only for nge d+1 points in mathbb {R}^{d}, for the spindle convex hull this happens for all nge 2. We study this question for the uniform and normally distributed random models.

Spectral stability, spectral flow and circular relative equilibria for the Newtonian n-body problem

For the Newtonian (gravitational) n-body problem in the Euclidean d-dimensional space, d≥2, the simplest possible periodic solutions are provided by circular relative equilibria, (RE) for short, namely solutions in which each body rigidly rotates about the center of mass and the configuration of the whole system is constant in time and central (or, more generally, balanced) configuration. For d≤3, the only possible (RE) are planar, but in dimension four it is possible to get truly four dimensional (RE).A classical problem in celestial mechanics aims at relating the (in-)stability properties of a (RE) to the index properties of the central (or, more generally, balanced) configuration generating it. In this paper, we provide sufficient conditions that imply the spectral instability of planar and non-planar (RE) in R4 generated by a central configuration, thus answering some of the questions raised in [14, Page 63]. As a corollary, we retrieve a classical result of Hu and Sun [11] on the linear instability of planar (RE) whose generating central configuration is non-degenerate and has odd Morse index, and fix a gap in the statement of [6, Theorem 1] about the spectral instability of planar (RE) whose (possibly degenerate) generating central configuration has odd Morse index.The key ingredient is a new formula of independent interest that allows to compute the spectral flow of a path of symmetric matrices having degenerate starting point, and a symplectic decomposition of the phase space of the linearized Hamiltonian system along a given (RE) which is inspired by Meyer and Schmidt's planar decomposition [13] and which allows us to rule out the uninteresting part of the dynamics corresponding to the translational and (partially) to the rotational symmetry of the problem.

Functions Tiling with Several Lattices

We study the problem of finding a function f with “small support” that simultaneously tiles with finitely many lattices \(\Lambda _1, \ldots , \Lambda _N\) in d-dimensional Euclidean spaces. We prove several results, both upper bounds (constructions) and lower bounds on how large this support can and must be. We also study the problem in the setting of finite abelian groups, which turns out to be the most concrete setting. Several open questions are posed.

A singularity-free model for a point charge potential in the presence of a momentum cutoff in a D-dimensional Euclidean space (D>=3)

In Maxwell electrostatics the electrostatic potential of a point charge is singular at the position of the point charge. In this paper, a nonsingular model for a point charge potential is presented in the presence of a momentum cutoff pmax based on a one-parameter deformation of the Heisenberg algebra in a 2D-dimensional phase space. For pmax→∞, the results of this paper reduce to the results of ordinary Maxwell electrostatics for a point charge.