Positive results are derived concerning the long time dynamics of fixed stepsize numerical simulations of stochastic differential equation systems with Markovian switching. Euler–Maruyama and implicit theta-method discretizations are shown to capture exponential mean-square stability for all sufficiently small timesteps under appropriate conditions. Moreover, the decay rate, as measured by the ...

Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lovász’s theta body of the graph. We prove that theta bodies are, up to closure, a version of Lasserre’s rela...

We describe a method for co-estimating r = C/mu (where C is the per-site recombination rate and mu is the per-site neutral mutation rate) and Theta = 4N(e)mu (where N(e) is the effective population size) from a population sample of molecular data. The technique is Metropolis-Hastings sampling: we explore a large number of possible reconstructions of the recombinant genealogy, weighting accordin...

Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lovász’s theta body of the graph. We prove that theta bodies are, up to closure, a version of Lasserre’s rela...

A nonconvex optimal control problem is considered, for systems governed by a parabolic partial differential equation, nonlinear in the state and control variables, with control and state constraints. Since this problem may have no classical solutions, it is reformulated in the relaxed form. The relaxed problem is discretized by using a finite element method in space and an implicit theta-scheme...

In this article we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field Fq of cardinality q with time complexity O(n) and space complexity O(n), where n = log(q). In the latter complexity estimate the genus and the characteristic are assumed as fixed. Our algorithm forms a generalizat...

Let $\theta $ be an irrational number of finite type and let $\psi \ge 0$. We consider numbers in the Beatty sequence integer parts, \[ \mathcal B(x) = \{\lfloor \theta n + \psi \rfloor : 1 \le x\}. \] $C \gt 3$. Writing $P(n)$ for larges

Given a finite alphabet $\mathbb{A}$ and primitive substitution $\theta:\mathbb{A}\to\mathbb{A}^\lambda$ (of constant length $\lambda$), let $(X_\theta,S)$ denote the corresponding dynamical system, where $X_{\theta}$ is closure of orbit via left shift $S$ fixed point natural extension $\theta$ to self-map $\mathbb{A}^{\mathbb{Z}}$. The main result paper that all continuous observables in are o...