Differentiation of set functions using Vitali coverings

Vitali Coverings and Lebesgue’s Differentiation Theorem

The standard techniques used to prove the Lebesgue differentiation theorem (that monotonic functions are a.e. differentiable) are presented in an unusual way that reveals more about their nature and allows greater generality.

Zeta Functions of Graph Coverings

Graphs and digraphs treated here are finite and simple. Let G be a connected graph and D the symmetric digraph corresponding to G. A path P of length n in D(G) is a sequence P=(v0 , v1 , ..., vn&1 , vn) of n+1 vertices and n arcs (edges) such that consecutive vertices share an arc (edge) (we do not require that all vertices are distinct). Also, P is called a (v0 , vn)-path. The subdigraph (subg...

On Set Coverings in Cartesian Product Spaces

Weighted Zeta Functions of Graph Coverings

We present a decomposition formula for the weighted zeta function of an irregular covering of a graph by its weighted L-functions. Moreover, we give a factorization of the weighted zeta function of an (irregular or regular) covering of a graph by equivalence classes of prime, reduced cycles of the base graph. As an application, we discuss the structure of balanced coverings of signed graphs.

L-functions of regular coverings of graphs

Let π1 : K H , π2 : H G and π2π1 : K G be three finite regular coverings of graphs, and let σ be a representation of the covering transformation group of π1. We show that the L-function of G associated to the representation of the covering transformation group of π2π1 induced from σ is equal to that of H associated to σ by means of ordinary voltage assignments. © 2003 Elsevier Science Ltd. All ...