Subsequent Work on Christoffel’s Problem about Determining a Surface from Local Measurements

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.

The set of primitive vectors on large spheres in the euclidean space of dimension d>2 equidistribute when projected on the unit sphere. We consider here a refinement of this problem concerning the direction of the vector together with the shape of the lattice in its orthogonal complement. Using unipotent dynamics we obtained the desired equidistribution result in dimension d>5 and in dimension d=4,5 under a mild congruence condition on the square of the radius. The case of d=3 is considered in a separate paper.

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.

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.

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.

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.

The best constant of Sobolev inequality in an $n$ dimensional Euclidean space is found by means of the theory of reproducing kernel and Green function. The concrete form of the best constant is also found in the case of Sobolev space $W^2(\mathbf{R}^n)$ ($n=2,3$).

This work develops the notion of a kernel function for the heat equation in certain regions of n +1 -dimensional Euclidean space and applies that notion to the study of the boundary behavior of nonnegative temperatures.The regions in question are bounded between spacelike hyperplanes and satisfy a parabolic Lipschitz condition at points on the lateral boundary.Kernel functions (normalized, nonnegative temperatures which vanish on the parabolic boundary except at a single point) are shown to exist uniquely.A representation theorem for nonnegative temperatures is obtained and used to establish the existence of finite parabolic limits at the boundary (except for a set of heat-related measure zero).0. Introduction.The notion of a kernel function has been developed in the case of the Laplace operator by Hunt and Wheeden [4], who prove existence and uniqueness for such functions in Lipschitz domains of -dimensional Euclidean space Rn and use these functions to study the nontangential boundary behavior of harmonic functions which have a one-sided bound in the domain.In [7] we reported analogous results for the heat equation in certain regions of the plane.These results had been obtained in [6].There, difficulties in the application of the techniques of [4] to the heat equation were overcome and some simplification of those techniques was achieved, notably in the proof of uniqueness of kernel functions.However, technical problems prevented treatment of the heat equation in more than one space dimension.The present work extends those results to the case of several space variables.In 1 existence and uniqueness of kernel functions is established and a representation theorem is obtained for temperatures with a one-sided bound in a region satisfying a certain mixed-Lipschitz condition (conditions LI and L2 below).The main result of 2 is the existence almost everywhere (with respect to caloric measure) on the parabolic boundary of finite parabolic limits for temperatures with a one-sided bound in the region.This generalizes the work of Jones and Tu [5] and Hattemer [2], who considered regions less general than those dealt with here.

In this paper we define Weyl semiparallel ( ·h= 0) and Weyl 2-semiparallel ( ·∇̅h= 0) submanifolds. We considern-dimensional normally flat submanifolds satisfying these curvature conditions in (n+d)-dimensional Euclidean space n+d. We also consider normally flat submanifolds in (n+d)-dimensional Euclidean space n+dsatisfying the condition  ·h=LQ(g,h) and  ·∇̅h=LQ(g,∇̅h).