Balls In Euclidean Space Research Articles

Eigenvalues of the Laplacian on balls with spherically symmetric metrics

In this article we will explore Dirichlet Laplace eigenvalues of balls with spherically symmetric metrics. We will compare any Dirichlet Laplace eigenvalue with the corresponding Dirichlet Laplace eigenvalue on balls in Euclidean space with the same radii. As a special case we shall show that the Dirichlet Laplace eigenvalues of balls with small radii on the sphere are smaller than the corresponding eigenvalues of the Euclidean balls with the same radii. The opposite correspondence is true for the Dirichlet Laplace eigenvalues of hyperbolic spaces.

A step in the Delaunay mosaic of order k

Given a locally finite set X subseteq {{mathbb {R}}}^d and an integer k ge 0, we consider the function {mathbf{w}_{k}^{}} :{mathrm{Del}_{k}{({X})}} rightarrow {{mathbb {R}}} on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space.

Main Concepts in Philosophy of Quantum Information

Quantum mechanics involves a generalized form of information, that of quantum information. It is the transfinite generalization of information and re-presentable by transfinite ordinals. The physical world being in the current of time shares the quality of “choice”. Thus quantum information can be seen as the universal substance of the world serving to describe uniformly future, past, and thus the present as the frontier of time. Future is represented as a coherent whole, present as a choice among infinitely many alternatives, and past as a well-ordering obtained as a result of a series of choices. The concept of quantum information describes the frontier of time, that “now”, which transforms future into past. Quantum information generalizes information from finite to infinite series or collections. The concept of quantum information allows of any physical entity to be interpreted as some nonzero quantity of quantum information. The fundament of quantum information is the concept of ‘quantum bit’, “qubit”. A qubit is a choice among an infinite set of alternatives. It generalizes the unit of classical information, a bit, which refer to a finite set of alternatives. The qubit is also isomorphic to a ball in Euclidean space, in which two points are chosen.

Eigenvalue estimate of the Dirac operator and Rigidity of Poincare–Einstein metrics

We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi-Montiel-Zhang and Raulot and we re-prove them under weaker assumption that a boundary chirality operator exists. Moreover, on these spin manifolds with boundary, we show that any $C^{3,\alpha}$ conformal compactification of some Poincare-Einstein metric must be the standard hemisphere when the first nonzero eigenvalue of the Dirac operator achieves its lowest value, and any $C^{3,\alpha}$ conformal compactification of some Poincare-Einstein metric must be the flat ball in Euclidean space when the first positive eigenvalue of the boundary Dirac operator achieves certain value relating to the second Yamabe invariant.In two cases the Poincare-Einstein metrics are standard hyperbolic metric.

The Besicovitch covering theorem for parabolic balls in Euclidean space

The Besicovitch covering theorem is well known to be the useful tools in many fields of analysis. Federer extended the result of Besicovitch to a directionally limited metric space. In this paper, we prove the Besicovitch covering theorem for parabolic balls in Euclidean space, although the parabolic metric is not directionally limited.

The sphere packing problem in dimension $8$

In this paper we prove that no packing of unit balls in Euclidean space $\mathbb{R}^8$ has density greater than that of the $E_8$-lattice packing.

Extension of the edge tracing algorithm to disconnected Voronoi skeletons

One of the ways how to describe spatial relations among balls in Euclidean space is to use an additively weighted Voronoi diagram, where each ball has a region of closest points assigned to it. The boundary of regions is formed by non-linear faces, edges and vertices. Edges and vertices form a skeleton of the diagram. An edge tracing algorithm exists, which finds an initial vertex and then traces edges until the whole component is discovered. The problem is that the skeleton can have more components but the algorithm does not necessarily discover them all. We introduce an extension of the edge tracing algorithm to discover all components.

Sphere packing bounds via spherical codes

The sphere packing problem asks for the greatest density of a packing of congruent balls in Euclidean space. The current best upper bound in all sufficiently high dimensions is due to Kabatiansky and Levenshtein in 1978. We revisit their argument and improve their bound by a constant factor using a simple geometric argument, and we extend the argument to packings in hyperbolic space, for which it gives an exponential improvement over the previously known bounds. Additionally, we show that the Cohn-Elkies linear programming bound is always at least as strong as the Kabatiansky-Levenshtein bound; this result is analogous to Rodemich's theorem in coding theory. Finally, we develop hyperbolic linear programming bounds and prove the analogue of Rodemich's theorem there as well.

Unboundedness of the Lagrangian Hofer distance in the Euclidean ball

Let L denote the space of Lagrangians Hamiltonian isotopic to the standard Lagrangian in the unit ball in Euclidean space. We prove that the Lagrangian Hofer distance on L is unbounded.

On Lundh’s percolation diffusion

A collection of spherical obstacles in the unit ball in Euclidean space is said to be avoidable for Brownian motion if there is a positive probability that Brownian motion diffusing from some point in the ball will avoid all the obstacles and reach the boundary of the ball. The centres of the spherical obstacles are generated according to a Poisson point process while the radius of an obstacle is a deterministic function. If avoidable configurations are generated with positive probability, Lundh calls this percolation diffusion. An integral condition for percolation diffusion is derived in terms of the intensity of the point process and the function that determines the radii of the obstacles.

Conformal metrics on the unit ball: The Gehring-Hayman property and the volume growth

We continue the study of conformal metrics on the unit ball in Euclidean space. We assume that the density ρ \rho associated with the metric satisfies a Harnack inequality and then consider how much we can relax the volume growth condition from that in [Proc. London Math. Soc. Vol. 77 (3) (1998), 635–664] so that the Gehring-Hayman property still holds along the radii, i.e., if a boundary point can be accessed via a path with ρ \rho -length M > ∞ M>\infty , then the ρ \rho -length of the corresponding radius is bounded by C M CM . It turns out that if the path is inside a Stolz cone, then this result holds irrespective of the volume growth condition. Moreover, even if the path is not inside a Stolz cone, we are able to relax the volume growth condition for large r r , and still conclude that the corresponding radius is ρ \rho -rectifiable. This observation leads to a new estimate on the size of the boundary set corresponding to the ρ \rho -unrectifiable radii.

From the Kneser–Poulsen conjecture to ball-polyhedra

A very fundamental geometric problem on finite systems of spheres was independently phrased by Kneser [M. Kneser, Einige Bemerkungen über das Minkowskische Flächenmass, Arch. Math. 6 (1955) 382–390] and Poulsen [E.T. Poulsen, Problem 10, Math. Scand. 2 (1954) 346]. According to their well-known conjecture if a finite set of balls in Euclidean space is repositioned so that the distance between the centers of every pair of balls is decreased, then the volume of the union (resp., intersection) of the balls is decreased (resp., increased). In the first half of this paper we survey the state of the art of the Kneser–Poulsen conjecture in Euclidean, spherical as well as hyperbolic spaces with the emphases being on the Euclidean case. Based on that it seems very natural and important to study the geometry of intersections of finitely many congruent balls from the viewpoint of discrete geometry in Euclidean space. We call these sets ball-polyhedra. In the second half of this paper we survey a selection of fundamental results known on ball-polyhedra. Besides the obvious survey character of this paper we want to emphasize our definite intention to raise quite a number of open problems to motivate further research.

A Simple Proof of the Restricted Isometry Property for Random Matrices

We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candes and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmically simple proofs of the Johnson–Lindenstrauss lemma; and (ii) covering numbers for finite-dimensional balls in Euclidean space. This leads to an elementary proof of the Restricted Isometry Property and brings out connections between Compressed Sensing and the Johnson–Lindenstrauss lemma. As a result, we obtain simple and direct proofs of Kashin’s theorems on widths of finite balls in Euclidean space (and their improvements due to Gluskin) and proofs of the existence of optimal Compressed Sensing measurement matrices. In the process, we also prove that these measurements have a certain universality with respect to the sparsity-inducing basis.

Refined convexity and special cases of the Blaschke-Santalo inequality

We derive the $l_p$ version of the classical Blaschke-Santalo inequality for polar volumes as a consequence of more subtle convexity estimates for the volume of the $p-$ball in Euclidean space. We also give analogs for the $(p,q)$-substitution norms.

Dirichlet finite harmonic measures on topological balls

Based upon an intuition from electrostatics one might suspect that there is no topological ball in Euclidean space of dimension d≥2 which carries a nonconstant Dirichlet finite harmonic measure. This guess is certainly true for d=2. However, contrary to the above intuition, it is shown in this paper that there does exist a topological ball in Euclidean space of every dimension d≥3 on which there exists a nonconstant Dirichlet finite harmonic measure.

Universal formulas for feedback stabilization with respect to Minkowski balls

This note provides explicit algebraic stabilizing formulas for clf's when controls are restricted to certain Minkowski balls in Euclidean space. Feedbacks of this kind are known to exist by a theorem of Artstein, but the proof of Artstein's theorem is nonconstructive. The formulas are obtained from a general feedback stabilization technique and are used to construct approximation solutions to some stabilization problems.

Conformal Metrics on the Unit Ball in Euclidean Space

Conformal Metrics on the Unit Ball in Euclidean Space

Hitting or avoiding balls in Euclidean space

We investigate the algorithmic complexity of several geometric problems of the following type: given a "feasible" box and a collection of balls in Euclidean space, find a feasible point which is covered by as few or, respectively, by as many balls as possible. We establish that all these problems are NP-hard in their most general version. We derive tight lower and upper bounds on the complexity of their one-dimensional versions. Finally, we show that all these problems can be solved in polynomial time when the dimension of the space is fixed.

Epsilon entropy and the packing of balls in Euclidean space

Let be a sequence of mutually disjoint open balls, with centres xj and corresponding radii aj, each contained in the closed unit ball in d-dimensional euclidean space, ℝd. Further we suppose, for simplicity, that the balls Bj are indexed so that aj≥aj+1. The set obtained by removing, from the balls {Bj} is called the residual set. We say that the balls {Bj} constitute a packing of provided that λ(ℛ)=0, where λ denotes the d-dimensional Lebesgue measure. Thus it follows that henceforth denoted by c(d), whilst the packing restraint ensures that Larman [11] has noted that, under these circumstances, one also has .

The limit sets of some infinitely generated Schottky groups

Let P be a packing of balls in Euclidean space E n having the property that the radius of every ball of P lies in the interval [1 /k, k]. If G is a Schottky group associated to P, then the Hausdorff dimension of the topological limit set of G is less than a uniform constant C(k, n) < n. In particular, this limit set has zero volume