@prefix ns1: .
@prefix ns2: .
@prefix rdf: .
@prefix rdfs: .
@prefix xml: .
@prefix xsd: .
a ns1:ScholarlyArticle ;
ns1:about ,
;
ns1:author ( ) ;
ns1:datePublished "2008-02" ;
ns1:datePublishedReg "2008-02-01" ;
ns1:description "Generating functions are commonly used in combinatorics to recover sequences from power series expansions. Convergence of formal power series in Clifford algebras of arbitrary signature is discussed. Given , powers of u are recovered by expanding (1 − tu)−1 as a polynomial in t with Clifford-algebraic coefficients. It is clear that (1 − tu)(1 + tu + t2u2 + ...) = 1, provided the sum (1 + tu + t2u2 + ...) exists, in which case um is the Cliffordalgebraic coefficient of tm in the series expansion of (1 − tu)−1. In this paper, conditions on for the existence of (1 − tu)−1 are given, and an explicit formulation of the generating function is obtained. Allowing A to be an m × m matrix with entries in , a “Clifford-Frobenius” norm of A is defined. Norm inequalities are then considered, and conditions for the existence of (I − tA)−1 are determined. As an application, adjacency matrices for graphs are defined with vectors of as entries. For positive odd integer k > 3, k-cycles based at a fixed vertex of a graph are enumerated by considering the appropriate entry of Ak. Moreover, k-cycles in finite graphs are enumerated and expected numbers of k-cycles in random graphs are obtained from the norm of the degree-2k part of tr(1 − tu)−1. Unlike earlier work using commutative subalgebras of , this approach represents a “true” application of Clifford algebras to graph theory." ;
ns1:genre "research_article" ;
ns1:inLanguage "en" ;
ns1:isAccessibleForFree false ;
ns1:isPartOf [ a ns1:PublicationIssue ;
ns1:issueNumber "1" ],
[ a ns1:PublicationVolume ;
ns1:volumeNumber "18" ],
;
ns1:name "Norms and Generating Functions in Clifford Algebras" ;
ns1:pagination "75-92" ;
ns1:productId [ a ns1:PropertyValue ;
ns1:name "dimensions_id" ;
ns1:value "pub.1006910548" ],
[ a ns1:PropertyValue ;
ns1:name "readcube_id" ;
ns1:value "20eb1e157db4c38fd9c160077c8754c879aa61a9f3390188157ded740a5d619c" ],
[ a ns1:PropertyValue ;
ns1:name "doi" ;
ns1:value "10.1007/s00006-007-0063-6" ] ;
ns1:sameAs ,
;
ns1:sdDatePublished "2019-04-10T19:03" ;
ns1:sdLicense "https://scigraph.springernature.com/explorer/license/" ;
ns1:sdPublisher [ a ns1:Organization ;
ns1:name "Springer Nature - SN SciGraph project" ] ;
ns1:url "http://link.springer.com/10.1007/s00006-007-0063-6" ;
ns2:license ;
ns2:sdDataset "articles" .
a ns1:DefinedTerm ;
ns1:inDefinedTermSet ;
ns1:name "Mathematical Sciences" .
a ns1:DefinedTerm ;
ns1:inDefinedTermSet ;
ns1:name "Pure Mathematics" .
a ns1:Periodical ;
ns1:issn "0188-7009",
"1661-4909" ;
ns1:name "Advances in Applied Clifford Algebras" .
a ns1:Person ;
ns1:affiliation ;
ns1:familyName "Staples" ;
ns1:givenName "G. Stacey" ;
ns1:sameAs .
a ns1:Organization ;
ns1:alternateName "Southern Illinois University Edwardsville" ;
ns1:name "Department of Mathematics and Statistics, Southern Illinois University Edwardsville, 62026-1653, Edwardsville, IL, USA" .