scholarly journals Biangular Lines Revisited

2021 ◽  
Author(s):  
Mikhail Ganzhinov ◽  
Ferenc Szöllősi
Keyword(s):  
Euclidean Space ◽  
Association Schemes ◽  
Dimensional Euclidean Space ◽  
Binary Codes ◽  
Maximum Cardinality ◽  
Distance Sets

AbstractLine systems passing through the origin of the d-dimensional Euclidean space admitting exactly two distinct angles are called biangular. It is shown that the maximum cardinality of biangular lines is at least $$2(d-1)(d-2)$$ 2 ( d - 1 ) ( d - 2 ) , and this result is sharp for $$d\in \{4,5,6\}$$ d ∈ { 4 , 5 , 6 } . Connections to binary codes, few-distance sets, and association schemes are explored, along with their multiangular generalization.

scholarly journals Constructions of Maximum Few-Distance Sets in Euclidean Spaces

10.37236/8565 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Ferenc Szöllősi ◽  
Patric R.J. Östergård
Keyword(s):  
Euclidean Space ◽  
Gröbner Basis ◽  
Dimensional Euclidean Space ◽  
Combined Approach ◽  
Euclidean Spaces ◽  
Grobner Basis ◽  
Finite Set ◽  
Distance Set ◽  
Distance Sets ◽  
Basis Computation

A finite set of vectors $\mathcal{X}$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$ is called an $s$-distance set if the set of mutual distances between distinct elements of $\mathcal{X}$ has cardinality exactly $s$. In this paper we present a combined approach of isomorph-free exhaustive generation of graphs and Gröbner basis computation to classify the largest $3$-distance sets in $\mathbb{R}^4$, the largest $4$-distance sets in $\mathbb{R}^3$, and the largest $6$-distance sets in $\mathbb{R}^2$. We also construct new examples of large $s$-distance sets in $\mathbb{R}^d$ for $d\leq 8$ and $s\leq 6$, and independently verify several earlier results from the literature.

scholarly journals On Isosceles Sets in the 4-Dimensional Euclidean Space

2010 ◽  
Vol 2010 ◽  
pp. 1-30
Author(s):  
Hiroaki Kido
Keyword(s):  
Euclidean Space ◽  
Isosceles Triangle ◽  
Dimensional Euclidean Space ◽  
Maximum Cardinality

A subset in the -dimensional Euclidean space that contains points (elements) is called an -point isosceles set if every triplet of points selected from them forms an isosceles triangle. In this paper, we show that there exist exactly two 11-point isosceles sets in up to isomorphisms and that the maximum cardinality of isosceles sets in is 11.

ALGORITHMS FOR POINT SET MATCHING WITH k-DIFFERENCES

2006 ◽  
Vol 17 (04) ◽  
pp. 903-917
Author(s):  
TATSUYA AKUTSU
Keyword(s):  
Euclidean Space ◽  
Related Problem ◽  
Time Algorithm ◽  
Two Dimensions ◽  
Efficient Algorithms ◽  
Common Point ◽  
Dimensional Euclidean Space ◽  
Maximum Cardinality ◽  
Point Set ◽  
Special Case

The largest common point set problem (LCP) is, given two point set P and Q in d-dimensional Euclidean space, to find a subset of P with the maximum cardinality that is congruent to some subset of Q. We consider a special case of LCP in which the size of the largest common point set is at least (|P| + |Q| - k)/2. We develop efficient algorithms for this special case of LCP and a related problem. In particular, we present an O(k3n1.34 + kn2 log n) time algorithm for LCP in two-dimensions, which is much better for small k than an existing O(n3.2 log n) time algorithm, where n = max {|P|,|Q|}.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

Some properties of Wigner coefficients and hyperspherical harmonics

1956 ◽  
Vol 52 (3) ◽  
pp. 424-430 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Angular Momentum ◽  
Euclidean Space ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Hyperspherical Harmonics ◽  
Shift Operators ◽  
Analytical Relations

ABSTRACTGeneral shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

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