Journal•ISSN: 0046-5755
Geometriae Dedicata
Springer Science+Business Media
About: Geometriae Dedicata is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Differential geometry & Hyperbolic geometry. It has an ISSN identifier of 0046-5755. Over the lifetime, 4246 publications have been published receiving 59414 citations.
Topics: Differential geometry, Hyperbolic geometry, Algebraic geometry, Projective geometry, Topology (chemistry)
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TL;DR: In this paper, the authors provided an overview of spherical codes and designs, and derived bounds for the cardinality of spherical A-codes in terms of the Gegenbauer coefficients of polynomials compatible with A.
Abstract: Publisher Summary This chapter provides an overview of spherical codes and designs. A finite non-empty set X of unit vectors in Euclidean space R d has several characteristics, such as the dimension d ( X ) of the space spanned by X , its cardinality n = | X |, its degree s( X ), and its strength t ( X ).The chapter presents derivation of bounds for the cardinality of spherical A -codes in terms of the Gegenbauer coefficients of polynomials compatible with A . It also discusses spherical ( d , n , s , i)- configurations X . These are sets X of cardinality n on the unit sphere Ω d , which are spherical t -designs and spherical A -codes with I A I = s ; in other words, the strength t ( X ) is at least t and the degree s ( X ) is at most s . A condition is given for a spherical A -code to be a spherical t -design, in terms of the Gegenbauer coefficients of an annihilator of the set A . The chapter presents many examples of spherical ( d , n , s , t )-configurations; there exist tight spherical t-designs with t = 2, 3, 4, 5, 7, 11, and non-tight spherical ( 2s − 1)-designs. The constructions of these examples use sets of lines with few angles and association schemes, respectively.
1,009 citations
TL;DR: In this article, the authors extend to the not necessarily simply laced case the study of quantum groups whose parameter is a root of 1, and they extend their work to the case where the parameter is fixed.
Abstract: We extend to the not necessarily simply laced case the study [8] of quantum groups whose parameter is a root of 1.
506 citations
418 citations
299 citations
TL;DR: The work of Mess as discussed by the authors gives a classification of flat and anti-de Sitter domains of dependence in 2+1 dimensions, where the dependence is defined as a linear combination of the number of vertices and the dimension of the vertices.
Abstract: This paper is unpublished work of Geoffrey Mess written in 1990, which gives a classification of flat and anti-de Sitter domains of dependence in 2+1 dimensions.
289 citations