Relative difference sets fixed by inversion (III)—Cocycle theoretical approach

In this paper, we give a characterization of a group G which contains a semiregular relative difference set R relative to a central subgroup N containing the commutator subgroup [G,G] of G such that 1∈R and rRr=R for all r∈R. In particular, these relative difference sets are fixed by inversion and inner automorphisms of the group are multipliers. We also present a construction of such relative difference sets.     

Inertial Manifolds and Forms for Stochastically Perturbed Retarded Semilinear Parabolic Equations

We construct inertial manifolds for a class of random dynamical systems generated by retarded semilinear parabolic equations subjected to additive white noise. These inertial manifolds are finite-dimensional invariant surfaces, which attract exponentially all trajectories. We study the corresponding inertial forms, i.e., the restriction of the stochastic equation to the inertial manifold. These inertial forms are finite-dimensional Ito equations and they completely describe the long-time dynamics of the system under consideration. The existence of inertial manifolds and the properties of inertial forms allow us to show that under mild additional conditions the system has a global (random) attractor in the sense of the theory of random dynamical systems.     

Approximately vanishing of topological cohomology groups

In this paper, we establish the pexiderized stability of coboundaries and cocycles and use them to investigate the Hyers-Ulam stability of some functional equations. We prove that for each Banach algebra A, Banach A-bimodule X and positive integer n,Hn(A,X)=0 if and only if the nth cohomology group approximately vanishes.     

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.     

Co cycle Perturbation on Banach Algebras

Letαbe a flow on a Banach algebra(?),and t(?)u_t a continuous function from Rt into the group of invertible elements of(?)such that u_sα_s(u_t)=u_s+t,s,t∈R.Thenβ_t=Adu_t oα_t,t∈R is also a flow on(?),where Adu_t(B)(?)u_tBu_t~(-1)for any B∈(?).βis said to be a cocycle perturbation ofα.We show that ifα,βare two flows on a nest algebra(or quasi-triangular algebra),thenβis a cocycle perturbation ofα.And the flows on a nest algebra(or quasi-triangular algebra)are all uniformly continuous.     

An ergodic theorem in the qualitative behavior of (non)linear semiflows

Take σ$\sigma$ to be a continuous semiflow on the locally compact metric space Θ$\Theta$, and let _{{A(θ)}θ∈Θ}${\left\{A\left(\theta \right)\right\}}_{\theta \in \Theta }$ be a family of (possibly unbounded) densely defined closed operators on the Banach space X$X$.     

The substitution theorem for semilinear stochastic partial differential equations

In this article we establish a substitution theorem for semilinear stochastic evolution equations (see's) depending on the initial condition as an infinite-dimensional parameter. Due to the infinite-dimensionality of the initial conditions and of the stochastic dynamics, existing finite-dimensional results do not apply. The substitution theorem is proved using Malliavin calculus techniques together with new estimates on the underlying stochastic semiflow. Applications of the theorem include dynamic characterizations of solutions of stochastic partial differential equations (spde's) with anticipating initial conditions and non-ergodic stationary solutions. In particular, our result gives a new existence theorem for solutions of semilinear Stratonovich spde's with anticipating initial conditions.     

Twisted product of cocycles and factorization of semi‐regular relative difference sets

In this article, we introduce what we call twisted Kronecker products of cocycles of finite groups and show that the twisted Kronecker product of two cocycles is a Hadamard cocycle if and only if the two cocycles themselves are Hadamard cocycles. This enables us to generalize some known results concerning products and factorizations of central semi‐regular relative difference sets. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 431–441, 2008     

On Semisimple Hopf Algebras of Dimension pq r

Let p and q be distinct prime numbers. We prove a result on the existence of nontrivial group-like elements in a certain class of semisimple Hopf algebras of dimension pq ^r . We conclude the classification of semisimple Hopf algebras A of dimension pq ² over an algebraically closed field k of characteristic zero, such that both A and A ^* are of Frobenius type. We also complete the classification of semisimple Hopf algebras of dimension pq ²<100.     

The lyapunov spectrum and stable manifolds for stochastic linear delay equations

A class of linear stochastic retarded functional differential equations is considered. These equations have diffusion coefficients that do not look into the past. It is shown that the trajectories of such equations form a continuous linear cocycle on the underlying state space. At times greater than the delay the cocycle is almost surely compact. Consequently, using an infinite-dimensional Oseledec multiplicative ergodic theorem of Ruelle, the existence of a countable non-random Lyapunov spectrum is proved. In the hyperbolic case it is shown that the state space admits an almost sure saddle-point splitting which is cocycle-invariant and corresponds to an exponential dichotomy for the stochastic flow