Given an edge-weighted graph G = (V,E) and a set E0 ⊂ E, the incremental network design problem with minimum spanning trees asks for a sequence of edges e1, . . . , e ′ r ∈ E \ E0 minimizing ∑T t=1 w(Xt) where w(Xt) is the weight of a minimum spanning tree Xt for the subgraph (V,E0 ∪ {e1, . . . , et}) and T = |E \ E0|. We prove that this problem can be solved by a greedy algorithm.

This paper examines the numerical solutions of neutral stochastic functional differential equations (NSFDEs) d[x(t)− u(xt)] = f(xt)dt + g(xt)dw(t), t ≥ 0. The key contribution is to establish the strong mean square convergence theory of the Euler– Maruyama approximate solution under the local Lipschitz condition, the linear growth condition and the contractive mapping. These conditions are gene...

The first attempt to model a process is often a deterministic setup with differential equations. The existing stochastic influence is suppressed and hopefully negligible. However, sometimes the stochastic component is important. We demonstrate and clarify this for a growth process. The deterministic approach is given by Yn + 1 = Yn + g(Yn) or dYt = g(Yt)dt, Y0 = 1, g a positive function. The co...

We prove the existence of regular solutions for a class of quasi-linear partial neutral functional differential equations with unbounded delay that can be described as the abstract retarded functional differential equation d dt(x(t) + F (t, xt)) = Ax(t) + G(t, xt), where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators on a Banach space X and F , G...

This paper deals with some self-interacting diffusions (Xt, t ≥ 0) living on R. These diffusions are solutions to stochastic differential equations: dXt = dBt − g(t)∇V (Xt − μt)dt, where μt is the mean of the empirical measure of the process X , V is an asymptotically strictly convex potential and g is a given function. We study the ergodic behavior of X and prove that it is strongly related to...

A homomorphism from an oriented graph G to an oriented graph H is a mapping φ from the set of vertices of G to the set of vertices of H such that −−−−−→ φ(u)φ(v) is an arc in H whenever −→uv is an arc in G. The oriented chromatic index of an oriented graph G is the minimum number of vertices in an oriented graph H such that there exists a homomorphism from the line digraph LD(G) of G to H (Reca...

The present paper is concerned with some self-interacting diffusions (Xt, t ≥ 0) living on R. These diffusions are solutions to stochastic differential equations: dXt = dBt − g(t)∇V (Xt − μt)dt where μt is the empirical mean of the process X, V is an asymptotically strictly convex potential and g is a given function. The authors have still studied the ergodic behavior of X and proved that it is...

Let (Xt,ot) be a shift of finite type, and G = aut(or) denote the group of homeomorphisms of Xt commuting with ctWe investigate the algebraic properties of the countable group G and the dynamics of its action on Xt and associated spaces. Using "marker" constructions, we show G contains many groups, such as the free group on two generators. However, G is residually finite, so does not contain di...