Integer points on spheres and their orthogonal grids

Half Dirichlet problem for matrix functions on the unit ball in Hermitian Clifford analysis

In the d-dimensional Euclidean space, any set of n + 1 independent random points, uniformly distributed in the interior of a unit ball of center O, determines almost surely a circumsphere of center C and radius Ω (1 ≤ n ≤ d) and an n-flat (1 ≤ n ≤ d − 1). The orthogonal projection of O onto this flat is called O′ while Δ designates the distance O′C. The classical problem of the distance between two random points in a unit ball corresponds to n = 1. The focus is set on the family of circumspheres which are contained in this unit ball. For any d≥2 and 1≤n≤d−1, the joint probability density function of the distance Δ≡O′C and circumradius Ω has a simple closed-form expression. The marginal probability density functions of Δ and Ω are both products of powers and a Gauss hypergeometric function. Stochastic representations of the latter random variables are described in terms of geometric means of two independent beta random variables. For n=d≥1, Δ and Ω have a joint Dirichlet distribution with parameters (d, d2, 1) while Δ and Ω are beta distributed. Results of Monte Carlo simulations are in very good agreement with their calculated counterparts. The tail behavior of the circumradius probability density function has been studied by Monte Carlo simulations for 2≤n=d≤9, where all circumspheres are this time considered, regardless of whether or not they are entirely contained in the unit ball.

In metric Diophantine approximation, one frequently encounters the problem of showing that a limsup set has positive or full measure. Often it is a set of points in m m -dimensional Euclidean space, or a set of n n -by- m m systems of linear forms, satisfying some approximation condition infinitely often. The main results of this paper are bootstraps: if one can establish positive measure for such a limsup set in m m -dimensional Euclidean space, then one can establish positive or full measure for an associated limsup set in the setting of n n -by- m m systems of linear forms. Consequently, a class of m m -dimensional results in Diophantine approximation can be bootstrapped to corresponding n n -by- m m -dimensional results. This leads to short proofs of existing, new, and hypothetical theorems for limsup sets that arise in the theory of systems of linear forms. We present several of these.

Christoffel’s paper [1865] „Ueber die Bestimmung der Gestalt einer krummen Oberflache...“, discussed in W. Klingenberg’s lecture, treats the following questions. Given a function o over the unit sphere Ω of directions u, is there a convex surface E whose sum of principal radii of curvature equals o(u) at that point where the outer normal direction to E is u? To what extent is E unique? Here convex surface means the whole boundary of a convex compactum: the setting is Euclidean space of dimension n ≥ 3. Christoffel answered these questions by solving appropriate Poisson equations by Green’s function methods. For this he required the facts that the integral of uo(u) over Ω to be the zero vector and the integral of o(u) over each hemisphere to be positive. His work was incomplete because it neglected the convexity assumption, which occurs quite indirectly in the paper. Christoffel also showed that E is unique up to a translation.

Let ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> H$ be a closed half-space of $n$-dimensional Euclidean space. Suppose $S$ is a unit sphere in $H$ that touches the supporting hyperplane of $H$. The one-sided kissing number $B(n)$ is the maximal number of unit nonoverlapping spheres in $H$ that can touch $S$. Clearly, $B(2)=4$. It was proved that $B(3)=9$. Recently, K. Bezdek proved that $B(4)=18$ or 19, and conjectured that $B(4)=18$. We present a proof of this conjecture.

We show that an equiangular tight frame of m unit vectors in the n–dimensional Euclidean space is a 4th order semidesign if and only if m = n(n+1)/2. For an important example we present an equiangular tight frame of n + 1 vertices of a regular simplex inscribed into the unit sphere (the Mercedes–Benz frame). We construct a system of the projections of the midpoints of edges of a simplex onto the sphere and show that this system is a tight frame for any $$ n \geqslant 2 $$ and is equiangular only if n = 2 and n = 7. In the last two cases, we find an equiangular 4th order semidesign. Bibliography: 6 titles.

The purpose of this article is to prove the following theorem:Let n be a positive integer larger than or equal to 2, and let S be the unit sphere in the 2n + 1 dimensional Euclidean space. Given a compact Riemann surface, we can always find a conformal and minimal immersion of the surface into S whose image is not lying in any 2n — 1 dimensional hyperplane.This is a partial generalization of the result by R. L. Bryant. In this papers, he demonstrates the existence of a conformal and minimal immersion of a compact Riemann surface into S2n, which is generically 1:1, when n = 2 ([2]) and n = 3 ([1]).

Helices and constant procession curves are special examples of slant curves. However, there is no example of a $k$-slant curve for a positive integer $k\geq 2$ in three dimensional Euclidean spaces. Furthermore, the position vector of a $k$-slant curve for a positive integer $k\geq 2$ has not been known thus far. In this paper, we propose a method for constructing $k$-slant curves in three dimensional Euclidean spaces. We then show that spherical $k$-slant curves and $N_{k}$-constant procession curves can be derived from circles, for $k \in \mathbb{N}$, the set of all nonnegative integers. In addition, we provide a new proof of the spherical curve characterization and define a curve in the sphere called a spherical prime curve. Afterward, we apply $k$-slant curves to magnetic curves. Finally, we discuss the need for further research.

It was recently shown by Aka, Einsiedler and Shapira that if $d>2$ , the set of primitive vectors on large spheres when projected to the $(d-1)$ -dimensional sphere coupled with the shape of the lattice in their orthogonal complement equidistribute in the product space of the sphere with the space of shapes of $(d-1)$ -dimensional lattices. Specifically, for $d=3,4,5$ some congruence conditions are assumed. By using recent advances in the theory of unipotent flows, we effectivize the dynamical proof to remove those conditions for $d=4,5$ . It also follows that equidistribution takes place with a polynomial error term with respect to the length of the primitive points.

We prove two results about the Hadwiger problem of finding the Helly number for line transversals of disjoint unit disks in the plane, and about its higher-dimensional generalization to hyperplane transversals of unit balls in d -dimensional Euclidean space. These consist of (a) a proof of the fact that the Helly number remains 5 even for arbitrarily large sets of disjoint unit disks—thus correcting a 40-year-old error; and (b) a lower bound of d+3 on the Helly number for hyperplane transversals to suitably separated families of unit balls in Rd .

In this paper we study strictly positive definite functions on the unit sphere of the m m -dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are actually strictly positive definite. The study of this problem was initiated recently by Xu and Cheney (Proc. Amer. Math. Soc. 116 (1992), 977–981), where certain sufficient conditions were derived. A new approach, which is based on a critical connection between this problem and that of multivariate polynomial interpolation on spheres, is presented here. The relevant interpolation problem is subsequently analyzed by three different complementary methods. The first is based on the de Boor-Ron general “least solution for the multivariate polynomial interpolation problem”. The second, which is suitable only for m = 2 m=2 , is based on the connection between bivariate harmonic polynomials and univariate analytic polynomials, and reduces the problem to the structure of the integer zeros of bounded univariate exponentials. Finally, the last method invokes the realization of harmonic polynomials as the polynomial kernel of the Laplacian, thereby exploiting some basic relations between homogeneous ideals and their polynomial kernels.

In this paper, we first investigate some properties of an orthogonal complement of a subsemimodule and give some necessary and sufficient conditions for every subsemimodule to be equal to the orthogonal complement of its orthogonal complement. Then we obtain some necessary and sufficient conditions for a set of standard orthogonal vectors to be extended to a standard orthogonal basis of a semimodule. As applications, we present two equivalent descriptions for a semiring to be an exact ring by using terminologies of the orthogonal complements and extending orthogonal subsets of semimodules, respectively.

The solution of many applied problems is to find a solution of nonlinear equations systems in finite- dimensional Euclidean spaces. The problem of finding the solution of a nonlinear system is divided into two problems: 1. The existence of a solution of a nonlinear equations system; in the case of nonunique of the solution, it is necessary to find the number of these solutions and their surroundings. 2. Finding the solution of a system of nonlinear equations with a given accuracy. Many publications are devoted to solving problem 2, namely the construction of iterative methods, their convergence and estimates of the solution accuracy. In contrast to problem 2, for problem 1 there is no general algorithm for solving this task, there are no constructive conditions for the existence of a solution of a nonlinear equations system in Euclidean spaces. In this article, in finite-dimensional Euclidean spaces, the constructive conditions for the existence of a solution of nonlinear systems of polynomial form are found. The connection of these conditions with the linear polynomial interpolant of the minimum norm, generated by a scalar product with Gaussian measure and the conditions of its existence, is given.

It is shown that the conserved magnetic charge discovered by ’t Hooft in non−Abelian gauge theories with spontaneous symmetry breaking is not associated with the invariance of the action under a symmetry group. Rather, it is a topological characteristic of an isotriplet of Higgs fields in a three−dimensional space: the Brouwer degree of the mapping between a large sphere in configuration space and the unit sphere in field space provided by the normalized Higgs field ?a = φa (φbφb)−1/2. The use of topological methods in determining magnetic charge configurations is outlined. A peculiar interplay between Dirac strings and zeros of the Higgs field under gauge transformations is pointed out. The monopole−antimonopole system is studied.

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the map where it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology—that is, the weakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional—if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C ∞ maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C ∞ maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green’s theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction.