be linear and quasilinear evolution equations of parabolic type in a Banach space X respectively. By "parabolic type" we mean that A(t) and A(t,u) are all the infinitesimal generators of analytic linear semigroups on X we do not necessarily assume that the domains of the operators A(t) and A("t,u) are dense subspaces of X, so the semigroups generated by them may not be of class c0 J. The domain...

A comparison theorem is stated for Markov processes in polish state spaces. We consider a general class of stochastic orderings induced by a cone of real functions. The main result states that stochastic monotonicity of one process and comparability of the infinitesimal generators imply ordering of the processes. Several applications to convex type and to dependence orderings are given. In part...

By using the Lie’s invariance infinitesimal criterion we obtain the continuous equivalence transformations of a class of nonlinear Schrödinger equations with variable coefficients. Starting from the equivalence generators we construct the differential invariants of order one. We apply these latter ones to find the most general subclass of variable coefficient nonlinear Schrödinger equations whi...

In this paper, the extended double (2+1)-dimensional sine-Gorden equation is studied. First of all, using symmetry method, corresponding vector fields, Lie algebra and infinitesimal generators are derived. Then, from generators, reductions presented. addition, these reduced equations converted into partial differential equations, which including classical (1+1)-dimensional equation. Moreover, b...

We investigate the semigroup structure of bosonic Gaussian quantum channels. Particular focus lies on the sets of channels which are divisible, idempotent or Markovian (in the sense of either belonging to one-parameter semigroups or being infinitesimal divisible). We show that the non-compactness of the set of Gaussian channels allows for remarkable differences when comparing the semigroup stru...

We study the averaging of a diffusion process living in simplex K Rn, n≥1. assume that its infinitesimal generator can be decomposed as sum two generators corresponding to distinct timescales and one fastest timescale is pure noise with coefficient vanishing exactly on vertices K. show this averages jump Markov for Meyer–Zheng topology. The role geometric assumptions done also discussed.

Let X be a normal variety endowed with an algebraic torus action. An additive group action $\alpha$ on is called vertical if general orbit of contained in the closure and image normalizes Aut(X). Our first result this paper classification actions under assumption that proper over affine variety. Then we establish criterion as to when infinitesimal generators finite collection generate finite-di...

In this paper, we consider the eigenvalue PDE problem of infinitesimal generators metastable diffusion processes. We propose a numerical algorithm based on training artificial neural networks for solving leading eigenvalues and eigenfunctions such high-dimensional problem. The is able to find multiple eigenpairs by single task. It useful in understanding dynamical behaviors processes large time...

We define the dual of a set of generators of the fundamental group of an oriented two-surface S g,n of genus g with n punctures and the associated surface S g,n \D with a disc D removed. This dual is another set of generators related to the original generators via an involution and has the properties of a dual graph. In particular, it provides an algebraic prescription for determining the inter...