Related papers: The average simplex cardinality of a finite abstra…

Let $R$ be a finite ring with unity, $\psi: R \to \mathbb{C}^\times$ be an additive character of $R$, and \( \chi_0 \) be the principal multiplicative character ($i.e.$, $\chi_0(x) = 1 \quad \text{for all } x \in R^\times$), then the Gauss…

Let $G\leqslant {\rm Sym}(\Omega)$ be a finite transitive permutation group with point stabiliser $H$. A base for $G$ is a subset of $\Omega$ whose pointwise stabiliser is trivial, and the minimal cardinality of a base is called the base…

Given the Cayley graph of a finitely generated group $G$, with respect to a presentation $G^{\alpha}$ with $n$ generators, the quotient of the rank of the fundamental group of subgraphs of the Cayley graph by the cardinality of the set of…

Let G be a finite group and let F be a field of characteristic zero. In this paper we construct a generic G-crossed product over F using generic graded matrices. The center of this generic G-crossed product, denoted by F(G), is then the…

Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This…

A subset $S$ of the vertices $V$ of a connected graph $G$ resolves $G$ if no two vertices of $V$ share the same list of distances (shortest-path metric) with respect to the vertices of $S$ listed in a given order. The choice of such an $S$…

We provide two independent systematic methods of performing $D$-dimensional physical-state sums in gauge theory and gravity in such a way so that spurious light-cone singularities are not introduced. A natural application is to generalized…

Given a periodic function $f$, we study the almost everywhere and norm convergence of series $\sum_{k=1}^\infty c_k f(kx)$. As the classical theory shows, the behavior of such series is determined by a combination of analytic and number…

The famous Erdos-Heilbronn conjecture plays an important role in the development of additive combinatorics. In 2007 Z. W. Sun made the following further conjecture (which is the linear extension of the Erdos-Heilbronn conjecture): For any…

We study a generalization of the classical Dedekind sum that incorporates two Dirichlet characters and develop properties that generalize those of the classical Dedekind sum. By calculating the Fourier transform of this generalized Dedekind…

The Flag Complex Conjecture of Charney and Davis states that for a simplicial complex $S$ which triangulates a $(2n - 1)$-generalized homology sphere as a flag complex one has $(-1)^n \sum_{\sigma \in S}…

We extend the notion of generalised Cesaro summation/convergence developed previously to the more natural setting of what we call "remainder" Cesaro summation/convergence and, after illustrating the utility of this approach in deriving…

Let $\kappa$,$\lambda$ be regular uncountable cardinals such that $\lambda > \kappa^+$ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with $s(\kappa) = \lambda$ starting from a ground model in…

We introduce a Dirichlet-series framework for studying the asymptotic behavior of generalized factorial functions defined by Legendre-type valuation formulas. Let $K$ be a number field and let $S$ be a finite set of prime ideals. For a…

The class of Lambert series generating functions (LGFs) denoted by $L_{\alpha}(q)$ formally enumerate the generalized sum-of-divisors functions, $\sigma_{\alpha}(n) = \sum_{d|n} d^{\alpha}$, for all integers $n \geq 1$ and fixed real-valued…

Let $q\ge3$ be an integer, $\chi$ denote a Dirichlet character modulo $q$, for any real number $a\ge 0$, we define the generalized Dirichlet $L$-functions $$ L(s,\chi,a)=\sum_{n=1}^{\infty}\frac{\chi(n)}{(n+a)^s}, $$ where $s=\sigma+it$…

It is known that a lattice simplex of dimension $d$ corresponds a finite abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$. Conversely, given a finite abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$ such that the sum of all entries of…

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[T] of degree d for which s consecutive coefficients a_{d-1},...,a_{d-s} are fixed. Our estimate asserts that…

Let A_1,...,A_n be finite subsets of a field F, and let f(x_1,...,x_n)=x_1^k+...+x_n^k+g(x_1,...,x_n)\in F[x_1,...,x_n] with deg g<k. We obtain a lower bound for the cardinality of {f(x_1,...,x_n): x_1\in A_1,...,x_n\in A_n, and x_i\not=x_j…

Let $(G, +)$ be an abelian group. In 2004, Eliahou and Kervaire found an explicit formula for the smallest possible cardinality of the sumset $A+A$, where $A \subseteq G$ has fixed cardinality $r$. We consider instead the smallest possible…