scholarly journals Sums of distances between points of a sphere

1982 ◽  
Vol 5 (4) ◽  
pp. 707-714 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Euclidean Space ◽  
Unit Sphere ◽  
Upper Bound ◽  
Dimensional Euclidean Space ◽  
Spherical Caps

GivenNpoints on a unit sphere ink+1dimensional Euclidean space, we obtain an upper bound for the sum of all the distances they determine which improves upon earlier work by K. B. Stolarsky whenkis even. We use his method, but derive a variant of W. M. Schmidt's results for the discrepancy of spherical caps which is more suited to the present application.

scholarly journals Upper bounds for packings of spheres of several radii

2014 ◽  
Vol 2 ◽  
Author(s):  
DAVID DE LAAT ◽  
FERNANDO MÁRIO DE OLIVEIRA FILHO ◽  
FRANK VALLENTIN
Keyword(s):  
Linear Programming ◽  
Euclidean Space ◽  
Semidefinite Programming ◽  
Unit Sphere ◽  
Upper Bound ◽  
Convex Bodies ◽  
Classical Problem ◽  
Upper Bounds ◽  
Sphere Packings ◽  
Spherical Caps

AbstractWe give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve the bounds for the classical problem of packing identical spheres.

scholarly journals Conformal immersions of compact Riemann surfaces into the 2n-sphere (n ≥ 2)

1996 ◽  
Vol 141 ◽  
pp. 79-105 ◽  
Author(s):  
Jun-Ichi Hano
Keyword(s):  
Riemann Surface ◽  
Positive Integer ◽  
Euclidean Space ◽  
Unit Sphere ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Minimal Immersion ◽  
Dimensional Euclidean Space ◽  
Compact Riemann Surfaces

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]).

scholarly journals The Closest Packing of Spherical Caps in n Dimensions

1955 ◽  
Vol 2 (3) ◽  
pp. 139-144 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Euclidean Space ◽  
Unit Sphere ◽  
Closest Packing ◽  
Dimensional Unit ◽  
Spherical Caps ◽  
Image Position

Let Sn denote the “surface” of an n-dimensional unit sphere in Euclidean space of n dimensions. We may suppose that the sphere is centred at the origin of coordinates O, so that the points P(x1, x2, …, xn) of Sn satisfyWe suppose that n≥2.

scholarly journals An extreme covering of 4-space by spheres

1966 ◽  
Vol 6 (2) ◽  
pp. 179-192 ◽  
Author(s):  
T. J. Dickson
Keyword(s):  
Euclidean Space ◽  
Unit Sphere ◽  
Dimensional Euclidean Space ◽  
Entire Space ◽  
Dimensional Unit

Let Λ be a lattice in n-dimensional Euclidean space En. For any lattice there is a unique minimal positive number μ such that if spheres of radius μ are placed at the points of the lattice then the entire space is covered, i.e. every point in En lies in at least one of the spheres. The density of this covering is defined to be θn(Λ) = Jnμn/d(Λ), where Jn is the volume of an n-dimensional unit sphere and d(Λ) is the determinant of the lattice.

scholarly journals Unique Continuation for Parabolic Equations of Higher Order

1966 ◽  
Vol 26 ◽  
pp. 115-120
Author(s):  
Lu-San Chen ◽  
Tadashi Kuroda
Keyword(s):  
Euclidean Space ◽  
Unit Sphere ◽  
Parabolic Equations ◽  
Unique Continuation ◽  
Higher Order ◽  
Dimensional Euclidean Space

Let x = (xl,…xn) be a point in the n-dimensional Euclidean space and let be the unit sphere In the (n + 1)-dimensional Euclidean space with coordinate (x, t), we putandwhere denotes the boundary of . We also use the following notation:

scholarly journals On the product of distances to a point set on a sphere

1989 ◽  
Vol 47 (3) ◽  
pp. 466-482 ◽  
Author(s):  
Gerold Wagner
Keyword(s):  
Euclidean Space ◽  
Unit Sphere ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Variable Point ◽  
Point Set

AbstractLet S be the surface of the unit sphere in three-dimensional euclidean space, and let WN=(x1x2, xN)be an N-tuple of points on S. We consider the product of mutual distances and, for the variable point x on S, the product of distance from x to the points of ωN. We obtain essentially best possible bounds for maxωN p(ΩN) and for minωN maxx∈sp(x, ωN).

Lattice Coverings ofn-Space By Spheres

1962 ◽  
Vol 14 ◽  
pp. 632-650 ◽  
Author(s):  
M. N. Bleicher
Keyword(s):  
Euclidean Space ◽  
Unit Sphere ◽  
Dimensional Euclidean Space ◽  
Entire Space ◽  
Real Numbers ◽  
Absolute Minimum ◽  
Lattice Covering ◽  
Absolute Value ◽  
Vector Addition ◽  
The Absolute

Alattice∧ in euclideann-space,Enis a group of vectors under vector addition generated bynindependent vectors,X1, X2… ,Xn, called abasisfor the lattice. The absolute value of then×ndeterminant the rows of which are the co-ordinates of a basis is called thedeterminant of the latticeand is denoted byd(∧). For any lattice ∧ there is a unique minimal positive numberrsuch that, if spheres of radiusrare placed with centres at all points of ∧, the entire space is covered. Thedensityof this covering may be defined as(Jnrn)/(d(∧))whereJnis the volume of the unit sphere inn-dimensional euclidean space. This density will be denoted by θn(∧). Thedensity of the most efficient lattice coveringofn-space by spheres,θnis the absolute minimum of θn(∧) considered as a function from the space of all lattices to the real numbers.

scholarly journals Voronoi Diagrams of Moving Points

1998 ◽  
Vol 08 (03) ◽  
pp. 365-379 ◽  
Author(s):  
Gerhard Albers ◽  
Leonidas J. Guibas ◽  
Joseph S. B. Mitchell ◽  
Thomas Roos
Keyword(s):  
Euclidean Space ◽  
Voronoi Diagram ◽  
Upper Bound ◽  
Voronoi Diagrams ◽  
Maximum Length ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Moving Points ◽  
Over Time

Consider a set of n points in d-dimensional Euclidean space, d ≥ 2, each of which is continuously moving along a given individual trajectory. As the points move, their Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Voronoi diagram. In this paper, we present a method of maintaining the Voronoi diagram over time, at a cost of O( log n) per event, while showing that the number of topological events has an upper bound of O(ndλs(n)), where λs(n) is the (nearly linear) maximum length of a (n,s)-Davenport-Schinzel sequence, and s is a constant depending on the motions of the point sites. In addition, we show that if only k points are moving (while leaving the other n - k points fixed), there is an upper bound of O(knd-1λs(n)+(n-k)dλ s(k)) on the number of topological events.

scholarly journals Partial Covering of a Sphere with Random Number of Spherical Caps

2013 ◽  
Vol 54 (1) ◽  
pp. 73-82
Author(s):  
Tomasz Gronek ◽  
Ewa Schmeidel
Keyword(s):  
Euclidean Space ◽  
Difference Equation ◽  
Poisson Distribution ◽  
Random Number ◽  
Dimensional Euclidean Space ◽  
Stochastic Variable ◽  
3 Dimensional ◽  
Spherical Caps ◽  
Random Location ◽  
Partial Covering

Abstract In this paper we study sphere coverage issue. A sphere of radius one in a 3-dimensional Euclidean space is given. We consider random location of N spherical caps on a sphere, assuming that N is a discrete stochastic variable with a Poisson distribution. Using suitable difference equation, the expected area of the covered region is investigated.

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