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 $\unicode[STIX]{x1D6FC}$.

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.

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 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 $d^2$ .

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 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

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

A Displacement Metric for Finite Sets of Rigid Body Displacements

2008 ◽  
Author(s):  
Venkatesh Venkataramanujam ◽  
Pierre Larochelle
Keyword(s):  
Euclidean Space ◽  
Rigid Body ◽  
Orthogonal Group ◽  
Polar Decomposition ◽  
Coordinate Frame ◽  
Invariant Metric ◽  
Motion Generation ◽  
Euclidean Group ◽  
Finite Set ◽  
Potential Applications

There are various useful metrics for finding the distance between two points in Euclidean space. Metrics for finding the distance between two rigid body locations in Euclidean space depend on both the coordinate frame and units used. A metric independent of these choices is desirable. This paper presents a metric for a finite set of rigid body displacements. The methodology uses the principal frame (PF) associated with the finite set of displacements and the polar decomposition to map the homogenous transform representation of elements of the special Euclidean group SE(N-1) onto the special orthogonal group SO(N). Once the elements are mapped to SO(N) a bi-invariant metric can then be used. The metric obtained is thus independent of the choice of fixed coordinate frame i.e. it is left invariant. This metric has potential applications in motion synthesis, motion generation and interpolation. Three examples are presented to illustrate the usefulness of this methodology.

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 On Euclidean Designs and Potential Energy

10.37236/8 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Tsuyoshi Miezaki ◽  
Makoto Tagami
Keyword(s):  
Linear Programming ◽  
Lower Bound ◽  
Euclidean Space ◽  
Potential Energy ◽  
Harmonic Analysis ◽  
Upper Bound ◽  
Type Inequality ◽  
Linear Programming Bound ◽  
Concentric Spheres ◽  
Finite Set

We study Euclidean designs from the viewpoint of the potential energy. For a finite set in Euclidean space, we formulate a linear programming bound for the potential energy by applying harmonic analysis on a sphere. We also introduce the concept of strong Euclidean designs from the viewpoint of the linear programming bound, and we give a Fisher type inequality for strong Euclidean designs. A finite set on Euclidean space is called a Euclidean $a$-code if any distinct two points in the set are separated at least by $a$. As a corollary of the linear programming bound, we give a method to determine an upper bound on the cardinalities of Euclidean $a$-codes on concentric spheres of given radii. Similarly we also give a method to determine a lower bound on the cardinalities of Euclidean $t$-designs as an analogue of the linear programming bound.

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