Suppose every vertex of a graph G has degree k or k + 1 and at least one vertex has degree k + 1. It is shown that if k > 2q 2 and q is a prime power then G contains a q-regular subgraph (and hence an r-regular subgraph for all r < q. r = q (mod 2)). It is also proved that every simple graph with maximal degree A > 2q 2 and average degree d > ((2q 2)/(2q l))(A + 1), where q is a prime power, co...

We generalize some characterizations of uniformly rectifiable (UR) sets to whose Hausdorff content is lower regular (and in particular, do not need be Ahlfors regular). For example, David and Semmes showed that, given an $d$-regular set $E$, if we consider the $\mathscr{B}$ surface cubes (in sense Christ David) near which $E$ does look approximately like a union planes, then UR only satisfies C...

In 1995, Camillo and Yu showed that an exchange ring has stable range 1 if and only if every regular element is unit regular. An element m in a module RM is called regular if (mλ)m = m for some λ ∈ hom(M,R). In this paper we define stable modules and show that if M has the finite exchange property then M is stable if and only if, for every regular element m ∈ M, (mγ)m = m where γ : M → R is epi...

We study the edit-distance of regular tree languages. The edit-distance is a metric for measuring the similarity or dissimilarity between two objects, and a regular tree language is a set of trees accepted by a finite-state tree automaton or described by a regular tree grammar. Given two regular tree languages L and R, we define the editdistance d(L,R) between L and R to be the minimum edit-dis...