Square-mean weighted pseudo almost automorphic solutions for non-autonomous stochastic evolution equations

In this paper, we first introduce the concepts and properties of the square-mean weighted pseudo almost automorphy and the square-mean bi-almost automorphy for a stochastic process. With these preliminary settings and by virtue of the theory of the semigroups of the operators, the Banach fixed point theorem and the stochastic analysis techniques, we investigate the well-posedness of the square-mean weighted pseudo almost automorphic solutions for a general class of non-autonomous stochastic evolution equations that satisfy either global or only local Lipschitz condition. Moreover, we estimate the boundedness of attractive domain for the case where the only local Lipschitz condition is taken into account. Finally, we provide two illustrative examples to show the practical usefulness of the analytical results that we establish in the paper.     

A Prüfer angle approach to semidefinite Sturm–Liouville problems with coupling boundary conditions

It is shown how to reduce the periodic/antiperiodic Sturm–Liouville problems to analysis of the Prüfer angle. This provides an alternative to the more usual approaches via operator theory or the Hill discriminant in the definite case, and leads to new results in the semidefinite case. An extension to more general coupling boundary conditions is also given.     

ON THE ADJOINT SITUATIONS BETWEEN BEHAVIOUR AND REALIZATION

ABSTRACT

Following the Arbib-Manes approach to machines in categories, the categories Mach(C) of machines and TBeh(C) of total behaviours are constructed, where c is a class of state-behaviour processes. It is shown that the total external behaviour functor E: Mach(C) → TBeh(C) has a left adjoint and that the free realization of any total behaviour is reachable. Furthermore, the restriction of E to the full subcategory of Mach(C) with all reachable machines as objects has as right adjoint the minimal realization functor.     

RESULTS ON SUBGROUPS OF THE GROUP OF HOMEOMORPHISMS OF THE RATIONAL NUMBERS

Abstract

If G is the group of homeomorphisms of the rationals with usual topology and H ? G is such that |G:H| < 2^x0 then there exists a finite, non-empty subset Y of the rational numbers such that G(Y) ? H ? G^{Y} where G(Y) is the group o all homeomorphisms in G fixing a neighbourhood of Y and G^{Y} is the set-wise stabilizer of Y in G.     

ESSENTIALLY FINITELY INDECOMPOSABLE ABELIAN p-GROUPS

Abstract

An abelian p-group C is said to be essentially finitely indecomposable (efi) if given any decomposition of G as the direct sum of a family of subgroups, there exists a positive integer n such that all but at moat a finite number of subgroups of this family are bounded by n. We look at examples and related questions. We prove that a reduced abelian p-group G is efi if and only if G modulo its elements of infinite height is efi. In the proof of this we obtain the following result which is of independent interest: Let A be a reduced p-group with a summand K such that K is a direct sum of cyclic groups. Let B be a basic subgroup of A. Then B contains a subgroup C such that C is a summand of A and the final rank of C is equal to the final rank of K.     

Existance and Uniqueness for Boussinesq type equations on a circle

We establish local and global existence results for Boussinesq type equations on a circle, employing Fourier series and a fixed point argument.     

On the category of quotient Banach spaces after Wegner

We study Waelbroeck's category of Banach quotients after Wegner, focusing on its basic homological and functional analytic properties.     

Every order is the endomorphism ring of a projective module over a quasi–hereditary order

We give in this note some new bounds on p _c(G) in terms of σ _c for several classes of finite groups, in particular, we prove that ρ _c ≤4σ _c (G) for any finite solvable group G, which improves some known results. We also pose some related open problems.     

Penalization methods for reflecting stochastic differential equations with jumps

The problem of approximation of a solution to a reflecting stochastic differential equation (SDE) with jumps by a sequence of solutions to SDEs with penalization terms is considered. The approximating sequence is not relatively compact in the Skorokhod topology J ¹ and so the methods of approximation based on the J ¹-topology break down. In the paper, we prove our convergence results in the S-topology on the Skorokhod space D(R⁺,?R ^d ) introduced recently by Jakubowski. The S-topology is weaker than J ¹ but stronger than the Meyer-Zheng topology and shares many useful properties with J ¹.     

On the simplicity of some kac-moody groups

Recently, Marcuson extended the classical construction of Tits systems in Steinberg groups to include the Kac-Moody Steinberg groups associated with the infinite dimensional versions of the great Lie algebras. If these Lie algebras and their Kac-Moody groups are viewed as limits of their finite dimensional counterparts, more direct methods may be employed. In fact, the Kac-Moody Chevalley groups of these Lie algebras are seen to be simple.