For any C₂-cofinite vertex operator superalgebra V, the tensor product and the P(z)-tensor product of any two admissible V-modules of finite length are proved to exist, which are shown to be isomorphic, and their constructions are given explicitly in this paper.

We study a class of stochastic differential equation with linear fractal noise. By an auxiliary stochastic differential equation, we prove the existence and uniqueness of the solution under some mild assumptions. We also give some estimates of moments of the solution. The exponential stability of the solution is discussed.

We study the evolution of convex hypersurfaces X(·,?t) with initial X(’,?0)=θX0 at a rate equal to H-f along its outer normal, where H is the inverse of harmonic mean curvature of X(’,?t), X₀ is a smooth, closed, and uniformly convex hypersurface. We find a θ?>0 and a sufficient condition about the anisotropic function f, such that if θ>θ*,? , then X(’,?t) remains uniformly convex and expands to infinity as t→ +∞ and its scaling, X(’,?t)e-nt, converges to a sphere. In addition, the convergence result is generalized to the fully nonlinear case in which the evolution rate is log H-log f instead of H-f.

Constructing some proper functional spaces, we obtain the corresponding norm for the operator (-L)^-1, and then, via spectral theory, we revisit two variational formulas of the spectral gap, given by M. F. Chen [Front. Math. China, 2010, 5(3): 379-515], for transient birth-death processes.

We are concerned with the Cauchy problem of the new integrable three-component system with cubic nonlinearity. We establish the local wellp-osedness in a range of the Besov spaces. Then the precise blow-up scenario for strong solutions to the system is derived.

Hom-Malcev superalgebras can be considered as a deformation of Malcev superalgebras. We give the definition of Hom-Malcev superalgebras. Moreover, we characterize the Hom-Malcev operator and the representation of Hom-Malcev superalgebras. Finally, we study the central extension and the double extension of Hom-Malcev superalgebras.

In this paper, the super O-operators of Jordan superalgebras are introduced and the solutions of super Jordan Yang-Baxter equation are discussed by super O-operators. Then pre-Jordan superalgebras are studied as the algebraic structure behind the super O-operators. Moreover, the relations among Jordan superalgebras, pre-Jordan superalgebras, and dendriform superalgebras are established.

Based on a recent result on linking stochastic differential equations on ?d to (finite-dimensional) Burger-KPZ type nonlinear parabolic partial differential equations, we utilize Galerkin type finite-dimensional approximations to characterize the path-independence of the density process of Girsanov transformation for the infinite-dimensional stochastic evolution equations. Our result provides a link of infinite-dimensional semi-linear stochastic differential equations to infinite-dimensional Burgers-KPZ type nonlinear parabolic partial differential equations. As an application, this characterization result is applied to stochastic heat equation in one space dimension over the unit interval.

We define a wavelet linear estimator for density derivative in Besov space based on a negatively associated stratified size-biased random sample. We provide two upper bounds of wavelet estimations on L^p (1≤p<∞) risk.

Let r= 2^d-1 + 1. We investigate the diophantine inequality|∑i=1rλiΦi(xi,yi)+η||<(max?1≤i≤r{|xi|,|yi|})-σwhere Φ_i(x, y) ∈Z[x, y] (1≤i≤r) are nondegenerate forms of degree d= 3 or 4.

We apply the Monte Carlo, stochastic Galerkin, and stochastic collocation methods to solving the drift-diffusion equations coupled with the Poisson equation arising in semiconductor devices with random rough surfaces. Instead of dividing the rough surface into slices, we use stochastic mapping to transform the original deterministic equations in a random domain into stochastic equations in the corresponding deterministic domain. A finite element discretization with the help of AFEPack is applied to the physical space, and the equations obtained are solved by the approximate Newton iterative method. Comparison of the three stochastic methods through numerical experiment on different PN junctions are given. The numerical results show that, for such a complicated nonlinear problem, the stochastic Galerkin method has no obvious advantages on efficiency except accuracy over the other two methods, and the stochastic collocation method combines the accuracy of the stochastic Galerkin method and the easy implementation of the Monte Carlo method.

We study cotilting comodules and f-cotilting comodules and give a description of localization of f-cotilting comodules and classical tilting comodules. First, we introduce T-cotilting injective comodules and their dimensions which are important for researching cotilting comodules. Then we characterize the localization in f-cotilting comodules, finitely copresented comodules, and classical tilting comodules. In particular, we obtain a localizing property about finitely copresented comodules.