2019/05/06 by Oliver Knill, Knill, Oliver
Computer Science · Mathematics · #05Cxx #05Exx #54F45 #55U10 #68Rxx #Abstract simplicial complex #Cardinality (data modeling) #Combinatorics #Combinatorics (math.CO) #Commutative Algebra and Its Applications #Computer science #Database #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Mathematics #Pure mathematics #Simplex #Simplicial approximation theorem #Simplicial complex #Simplicial set #Topological and Geometric Data Analysis #cs.DM #math.CO #msc:05Cxx #msc:05Exx #msc:54F45 #msc:55U10 #msc:68Rxx
paper · pdf · doi:10.48550/arxiv.1905.02118
19 page, 8 figures
arxiv created 2019/05/06 · openalex publication_date 2019/05/06 · arxiv updated 2019/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the average simplex cardinality Dim+(G) = sumx |x|/(|G|+1) of a finite abstract simplicial complex G. The functional is a homomorphism from the monoid of simplicial complexes to the rationals: the formula Dim+(G + H) = Dim+(G) + Dim+(H) holds for the join + similarly as for the augmented inductive dimension dim+(G) = dim(G)+1 where dim is the inductive dimension dim(G) = 1+ sumx dim(S(x))/|G| with unit sphere S(x) (a recent theorem of Betre and Salinger). In terms of the generating function f(t) = 1+v0 t + v1 t2 + ... +vd t^(d+1) defined by the f-vector (v0,v1, …) of G for which f(-1) is the genus 1-X(G) with Euler characteristic X and f(1)=|G|+1 is the augmented number of simplices, the average cardinality is the logarithmic derivative Dim+(f) = f'(1)/f(1) of f at 1. Beside introducing the average cardinality and establishing its compatibility with arithmetic, we prove two results: 1) the inequality dim+(G)/2 <= Dim+(G) with equality for complete complexes. 2) the limit Cd of Dim+(Gn) for n to infinity is the same for any initial complex G0 of maximal dimension d and the constant cd is explicitly given in terms of the Perron-Frobenius eigenfunction of the universal Barycentric refinement operator and is for positive d always a rational number in the open interval ((d+1)/2,d+1).