Stochastic partial differential equations have wide applications in various fields of science and engineering. This paper addresses the optical stochastic solitons other exact solutions through birefringent fibers for Biswas–Arshed equation with multiplicative white noise using modified extended mapping method. model contains many kinds soliton solutions, which are always symmetric or anti-symm...

In this paper, we take into account the coupled stochastic Korteweg–De Vries (CSKdV) equations in Itô sense. Using mapping method, new trigonometric, rational, hyperbolic, and elliptic solutions are obtained. These obtained can be applied to analysis of a wide variety crucial physical phenomena because KdV have important applications various fields physics engineering. Also, it is used design o...

A differential analysis framework of longitudinal FLAIR MRI volumes is proposed, based on non-linear gray value mapping, to quantify low-grade glioma growth. First, MRI volumes were mapped to a common range of gray levels via a midway-based histogram mapping. This mapping enabled direct comparison of MRI data and computation of difference maps. A statistical analysis framework of intensity dist...

The Johnson–Lindenstrauss lemma asserts that an n-point set in any Euclidean space can be mapped to a Euclidean space of dimension k = O(ε−2 log n) so that all distances are preserved up to a multiplicative factor between 1 − ε and 1 + ε. Known proofs obtain such a mapping as a linear map Rn → Rk with a suitable random matrix. We give a simple and self-contained proof of a version of the Johnso...

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...