A numerical approach for the approximation of inertial manifolds of stochastic evolutionary equations with multiplicative noise is presented and illustrated. After splitting the stochastic evolutionary equations into a backward and a forward part, a numerical scheme is devised for solving this backward-forward stochastic system, and an ensemble of graphs representing the inertial manifold is co...

This work is concerned with existence of weak solutions to discontinuous stochastic differential equations driven by multiplicative Gaussian noise and sliding mode control dynamics generated by stochastic differential equations with variable structure, that is with jump nonlinearity. The treatment covers the finite dimensional stochastic systems and the stochastic diffusion equation with multip...

Definition 1.1. Given a ring R, a genus with values in R is a ring homomorphism, Ω ⊗Q→ R, where Ω is the G-bordism ring. For example, the Â-genus and L-genus are ring maps from Ω⊗Q→ Q. The Atiyah-Singer theorem shows that  can be refined to a genus Ω⊗Q→ Z. The L-genus (or signature) is defined on Ω⊗Q, and the Todd genus is defined on the complex cobordism category. We can define genera via mul...

We present a picture of phase transitions of the system with colored multiplicative noise. Considering the noise amplitude as the power– law dependence of the stochastic variable xa we show the way to phase transitions disorder–order and order–disorder. The governed equations for the order parameter and one-time correlator are obtained and investigated in details. The long–time asymptotes in th...

For Fr$acute{mathbf{text{e}}}$chet algebras $(A, (p_n))$ and $(B, (q_n))$, a linear map $T:Arightarrow B$ is textit{almost multiplicative} with respect to $(p_n)$ and $(q_n)$, if there exists $varepsilongeq 0$ such that $q_n(Tab - Ta Tb)leq varepsilon p_n(a) p_n(b),$ for all $n in mathbb{N}$, $a, b in A$, and it is called textit{weakly almost multiplicative} with respect to $(p_n)$ and $(q_n)$...

We prove some properties of analytic multiplicative and sub-multiplicative cocycles. The results allow to construct natural invariant analytic sets associated to complex dynamical systems. AMS classification: 37F, 32H Key-words: multiplicative cocycle, sub-multiplicative cocycle, exceptional set

Abstract We obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.