Equiangular spherical semidesigns

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.

A complex equiangular tight frame (ETF) is a tight frame consisting of N unit vectors in C d whose absolute inner products are identical. One may view complex ETFs as a natural geometric generalization of an orthonormal basis. Numerical evidence suggests that these objects do not arise for most pairs ( d , N). The goal of this paper is to develop conditions on ( d , N) under which complex ETFs can exist. In particular, this work concentrates on the class of harmonic ETFs, in which the components of the frame vectors are roots of unity. In this case, it is possible to leverage field theory to obtain stringent restrictions on the possible values for ( d , N).

An equiangular tight frame (ETF) is an M N matrix which has orthogonal equal norm rows, equal normcolumns, and the inner products of all pairs of columns have the same modulus. In this paper we studyETFs in which all of the entries are unimodular, and in particular pth roots of unity. A new constructionof unimodular ETFs based on resolvable Steiner systems is presented. This construction gives many newexamples of unimodular ETFs. In particular, an new in nite class of ETFs with entries in f1; 1g ispresented.Keywords: tight frames, equiangular, Hadamard matrices, Steiner systems, resolvable 1. INTRODUCTION Let F be an M N matrix with real or complex entries. If there exists A > 0 such that FF = AI , where Iis the identity matrix, then F is called a tight frame . If F F has constant diagonal and all of the o -diagonalentries have the same modulus, then F is called equiangular. If all of the entries of F have modulus 1 thenwe call F unimodular . If all of the entries of F are pth roots of unity, then we call F unimodular of degree p.This article is concerned with equiangular tight frames (ETFs) which are unimodular of degree p. Inparticular, we are concerned with nding unimodular ETFs of degree 2, which we shall also refer to asHadamard ETFs. Note that in the usual de nition of ETFs the columns are taken to have norm 1. However,since we are concerned with unimodular ETFs we will avoid normalizing, thus the columns of an M Nunimodular ETF have norm M .The paper is organized as follows. In Section 2 we present some necessary conditions on the existence ofunimodular ETFs. In Section 3 we review the construction of harmonic ETFs, which is an important classof ETFs that also happen to be unimodular. In Section 4 we review the construction of the Steiner ETFs ofFickus, Mixon, and Tremain,

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.

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.

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.

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.

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.

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.

Transmit beamforming can improve the performance of multiple-input multiple-output (MIMO) antenna systems. However, the feedback overhead of beamforming matrixes may significantly reduce the FDD system throughput. Efficient feedback techniques were reported recently. However, their complexities can be prohibitively high for systems with large numbers of transmit antennas (M/sub t/) and spatial streams (M/sub s/) since numerous large matrix codebooks of Grassman manifold packing G(M/sub f/,M/sub s/) are required. In Roh et al. (2004), it is shown that the beamforming matrix can be parameterized by independent unit vectors. This implies that quantizing the matrix jointly is equivalent to quantizing the vectors separately. Firsts we show that the distributions of the unit vectors are isotropic on unit spheres. Secondly, a low-complexity, scalable quantization scheme is derived. The key innovation lies in the quantization of the large dimension unit vectors. Each large dimension unit vector is partitioned into smaller dimension vectors and quantized by small size codebooks designed for low dimension unit vectors, and the partition itself is quantized by a 3-bit codebook. Finally, an interpolation scheme is proposed for beamforming matrixes downsampled in frequency domain, which significantly reduces overhead for MIMO-OFDM systems. The missing beamforming vectors are interpolated along the geodesic connecting two fed back vectors on a unit sphere. Simulations demonstrate that the overhead in IEEE 802.16d/e draft can be reduced by a factor of three using the proposed scheme.

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.

31 Nonparametric methods in directional data analysis