1 College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 2 Key Laboratory of Network Control and Intelligent Instrument, Chongqing University of Posts and Telecommunications, Ministry of Education, Chongqing 400065, China 3 College of Applied Sciences, Beijing University of Technology, Beijing 100124, China 4 Fundamental Department, Heb...

s Book

We study branching random walks in random i.i.d. environment in Zd, d ≥ 1. For this model, the population size cannot decrease, and a natural definition of recurrence is introduced. We prove a dichotomy for recurrence/transience, depending only on the support of the environmental law. We give sufficient conditions for recurrence and for transience. In the recurrent case, we study the asymptotic...

The solvability of a k-dimensional system of difference equations of interest, which extends several recently studied ones, is investigated. A general sufficient condition for the solvability of the system is given, considerably extending some recent results in the literature.

The Legendre-Stirling numbers are the coeffi cients in the integral Lagrangian symmetric powers of the classical Legendre second-order differential expression. In many ways, these numbers mimic the classical Stirling numbers of the second kind which play a similar role in the integral powers of the classical second-order Laguerre differential expression. In a recent paper, Andrews and Littlejoh...

Differential-difference equations (DDEs) of the form u n (t) = Fn(t, un+a, . . . , un+b) with k ≥ 2 are studied for Lie symmetries and preliminary classification. Explicit forms of equations are given for those admitting at least one intrinsic Lie symmetry. An algorithmic mechanism is also proposed to automate the symmetry calculation for fairly general DDEs via computer algebras.