An upper bound for the cardinality of ans-distance subset in real euclidean space

COMBINATORICA ◽  
1981 ◽  
Vol 1 (2) ◽  
pp. 99-102 ◽  
Author(s):  
Eiichi Bannai ◽  
Etsuko Bannai
Keyword(s):  
Euclidean Space ◽  
Upper Bound ◽  
Real Euclidean Space

An upper bound for the cardinality of ans-distance subset in real euclidean space, II

COMBINATORICA ◽  
1983 ◽  
Vol 3 (2) ◽  
pp. 147-152 ◽  
Author(s):  
Eiichi Bannai ◽  
Etsuko Bannai ◽  
Dennis Stanton
Keyword(s):  
Euclidean Space ◽  
Upper Bound ◽  
Real Euclidean Space

On the total mean curvature of a compact space-like submanifold in Lorentz–Minkowski spacetime

2017 ◽  
Vol 148 (1) ◽  
pp. 199-210 ◽  
Author(s):  
Oscar Palmas ◽  
Francisco J. Palomo ◽  
Alfonso Romero
Keyword(s):  
Euclidean Space ◽  
Compact Space ◽  
Upper Bound ◽  
Mean Curvature ◽  
Light Cone ◽  
Minkowski Spacetime ◽  
Totally Umbilical ◽  
Total Mean Curvature ◽  
Lower Estimation ◽  
Compact Submanifold

By means of several counterexamples, the impossibility to obtain an analogue of the Chen lower estimation for the total mean curvature of any compact submanifold in Euclidean space for the case of compact space-like submanifolds in Lorentz–Minkowski spacetime is shown. However, a lower estimation for the total mean curvature of a four-dimensional compact space-like submanifold that factors through the light cone of six-dimensional Lorentz–Minkowski spacetime is proved by using a technique completely different from Chen's original one. Moreover, the equality characterizes the totally umbilical four-dimensional round spheres in Lorentz–Minkowski spacetime. Finally, three applications are given. Among them, an extrinsic upper bound for the first non-trivial eigenvalue of the Laplacian of the induced metric on a four-dimensional compact space-like submanifold that factors through the light cone is proved.

On Certain Functional Identities in EN

1971 ◽  
Vol 23 (2) ◽  
pp. 315-324 ◽  
Author(s):  
A. McD. Mercer
Keyword(s):  
Euclidean Space ◽  
Taylor Series ◽  
Dimensional Space ◽  
The Real ◽  
Higher Dimensional ◽  
Isolated Points ◽  
Functional Identities ◽  
Image Position ◽  
Special Case ◽  
Real Euclidean Space

1. If f is a real-valued function possessing a Taylor series convergent in (a — R, a + R), then it satisfies the following operational identity1.1in which D2 = d2/du2. Furthermore, when g is a solution of y″ + λ2y = 0 in (a – R, a + R), then g is such a function and (1.1) specializes to1.2In this note we generalize these results to the real Euclidean space EN, our conclusions being Theorems 1 and 2 below. Clearly, (1.2) is a special case of (1.1) but in higher-dimensional space it is of interest to allow g, now a solution of1.3to possess singularities at isolated points away from the origin. It is then necessary to consider not only a neighbourhood of the origin but annular regions also.

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.

PRODUCTS OF ROTATIONS BY A GIVEN ANGLE IN THE ORTHOGONAL GROUP

2017 ◽  
Vol 97 (2) ◽  
pp. 308-312
Author(s):  
M. G. MAHMOUDI
Keyword(s):  
Euclidean Space ◽  
Upper Bound ◽  
Orthogonal Group

For every rotation $\unicode[STIX]{x1D70C}$ of the Euclidean space $\mathbb{R}^{n}$ ($n\geq 3$), we find an upper bound for the number $r$ such that $\unicode[STIX]{x1D70C}$ is a product of $r$ rotations by an angle $\unicode[STIX]{x1D6FC}$ ($0<\unicode[STIX]{x1D6FC}\leq \unicode[STIX]{x1D70B}$). We also find an upper bound for the number $r$ such that $\unicode[STIX]{x1D70C}$ can be written as a product of $r$ full rotations by an angle $\unicode[STIX]{x1D6FC}$.

Inequalities and separation for Schrödinger type operators in L2(Rn)

1978 ◽  
Vol 79 (3-4) ◽  
pp. 257-265 ◽  
Author(s):  
W. N. Everitt ◽  
M. Giertz
Keyword(s):  
Euclidean Space ◽  
Differential Expression ◽  
Integral Inequalities ◽  
Laplacian Operator ◽  
Generalized Derivatives ◽  
Maximal Domain ◽  
Definition Of ◽  
Schrödinger Type Operators ◽  
Real Euclidean Space

SynopsisThe symmetric differential expression M determined by Mf = − Δf;+qf on G, where Δ is the Laplacian operator and G a region of n-dimensional real euclidean space Rn, is said to be separated if qfϵL2(G) for all f ϵ Dt,; here D1 ⊂ L2(G) is the maximal domain of definition of M determined in the sense of generalized derivatives. Conditions are given on the coefficient q to obtain separation and certain associated integral inequalities.

scholarly journals Large Equiangular Sets of Lines in Euclidean Space

10.37236/1533 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
D. De Caen
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Euclidean Space ◽  
Upper Bound ◽  
Equiangular Lines

A construction is given of ${{2}\over {9}} (d+1)^2$ equiangular lines in Euclidean $d$-space, when $d = 3 \cdot 2^{2t-1}-1$ with $t$ any positive integer. This compares with the well known "absolute" upper bound of ${{1}\over {2}} d(d+1)$ lines in any equiangular set; it is the first known constructive lower bound of order $d^2$ .

The convergence of a sum of independent random variables

1963 ◽  
Vol 59 (2) ◽  
pp. 411-416
Author(s):  
G. De Barra ◽  
N. B. Slater
Keyword(s):  
Distribution Function ◽  
Euclidean Space ◽  
Rate Of Convergence ◽  
Radial Distribution Function ◽  
Random Variables ◽  
Covariance Matrices ◽  
Independent Random Variables ◽  
Second Moments ◽  
Image Position ◽  
Real Euclidean Space

Let Xν, ν= l, 2, …, n be n independent random variables in k-dimensional (real) Euclidean space Rk, which have, for each ν, finite fourth moments β4ii = l,…, k. In the case when the Xν are identically distributed, have zero means, and unit covariance matrices, Esseen(1) has discussed the rate of convergence of the distribution of the sumsIf denotes the projection of on the ith coordinate axis, Esseen proves that ifand ψ(a) denotes the corresponding normal (radial) distribution function of the same first and second moments as μn(a), thenwhere and C is a constant depending only on k. (C, without a subscript, will denote everywhere a constant depending only on k.)

An Upper Bound for the Cardinality of an s -Distance Set in Euclidean Space

COMBINATORICA ◽  
2003 ◽  
Vol 23 (4) ◽  
pp. 535-557 ◽  
Author(s):  
Etsuko Bannai ◽  
Kazuki Kawasaki ◽  
Yusuke Nitamizu ◽  
Teppei Sato
Keyword(s):  
Euclidean Space ◽  
Upper Bound ◽  
Distance Set
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