A connection between Brownian motion and birth–death chains is explored. Several results concerning birth–death chains are shown to be consequences of well-known results on Brownian motion.

In general, multiplication of operators is not essentially commutative in an algebra generated by integral-type operators and composition operators. In this paper, we characterize the essential commutativity of the integral operators and composition operators from a mixed-norm space to a Bloch-type space, and give a complete description of the universal set of integral operators. Corresponding results for boundedness and compactness are also obtained.

Using a recent result on the sum–product problem, we estimate the number of elements γ in a prime finite field such that both γ and γ+γ−1 are of small order.

Let k,r≥2 be two integers. We prove an asymptotic formula for the number of k-free values of the r variables polynomial t1⋯tr−1 over [1,x]r∩ℤr.

Let G be a connected simple graph. The degree distance of G is defined as D′(G)=∑ u∈V (G)dG(u)DG(u), where DG(u) is the sum of distances between the vertex u and all other vertices in G and dG(u) denotes the degree of vertex u in G. In contrast to many established results on extremal properties of degree distance, few results in the literature deal with the degree distance of composite graphs. Towards closing this gap, we study the degree distance of some composite graphs here. We present explicit formulas for D′ (G) of three composite graphs, namely, double graphs, extended double covers and edge copied graphs.

In this paper, the amenability and approximate amenability of weighted ℓp-direct sums of Banach algebras with unit, where 1≤p<∞, are completely characterized. Applications to compact groups and hypergroups are given.