We consider in this paper a class of vector valued processes that have the form Yn+1 = An(Yn)+ Bn. Bn is assumed to be stationary ergodic and An is assumed to have a divisibility property. This class includes linear stochastic difference equations as well as multi-type branching processes (with a discrete or with a continuous state space). We derive explicit expressions for the probability dist...

Cellular automaton [Wolfram , 1984] is a modeling and simulation too l in computer science, mathemat­ ics, physics, chemistry, biology, etc. It consists of a grid of cells, wher e each cell has a finite number of states. Every cell has the same rul e for updating, based on the st at es of neighboring cells. Cellular automata can exhibit ma ny emergent phenomena; such as fract als, chaos, random...

The main result is an improvement of previous results on the equation f(x) + f(y)− f(x+ y) = g[φ(x) + φ(y)− φ(x+ y)] for a given function φ. We find its general solution assuming only continuous differentiability and local nonlinearity of φ. We also provide new results about the more general equation f(x) + f(y)− f(x+ y) = g(H(x, y)) for a given function H. Previous uniqueness results required ...

We report on the calculation of multi-loop Feynman integrals for single-scale problems by means of difference equations in Mellin space. The solution to these difference equations in terms of harmonic sums can be constructed algorithmically over difference fields, the so-called ΠΣ∗-fields. We test the implementaion of the Mathematica package Sigma on examples from recent higher order perturbati...

The condition for pressure isotropy, for spherically symmetric gravitational fields with charged and uncharged matter, is reduced to a recurrence equation with variable, rational coefficients. This difference equation is solved in general using mathematical induction leading to an exact solution to the Einstein field equations which extends the isotropic model of John and Maharaj. The metric fu...

MR2864818 39A05 37B99 Sacker, Robert J. (1-SCA; Los Angeles, CA) An invariance theorem for mappings. (English summary) J. Difference Equ. Appl. 18 (2012), no. 1, 163–166. The following theorem is proved. Theorem 2.1. LetD ⊂ R be a bounded subset and f :D→ R be continuous. Suppose f : ◦ D→ R is injective (one-to-one) and f(∂D)⊂D. If D := R rD has no bounded components, then f(D)⊂D. Here D is the...

We develop a new high order accurate time-discretisation technique for initial value problems. We focus on problems that originate from a space discretisation using high order finite difference methods on summation-by-parts form with weak boundary conditions, and extend that technique to the time-domain. The new timediscretisation method is global and together with the approximation in space, i...

Based on multiplicative calculus, the finite difference schemes for the numerical solution of multiplicative differential equations and Volterra differential equations are presented. Sample problems were solved using these new approaches.