Volume 11, Issue 2

Four classes of complete semigroups of binary relations are considered. It is proved for the semigroups of these classes that maximal subgroups of these semigroups are groups whose order is not greater than two.

We characterize additive rank-one preserving surjections on the full matrix algebra over any field. As applications, invertibility preservers, determinant preservers and characteristic polynomial preservers are characterized.

The authors apply a certain novel technique based on the combined use of operational methods and of some special multivariable and multiindex polynomials to derive several families of generating functions involving the products of Bessel and related functions. The possibility of extending this technique to the derivation of generating functions of hybrid nature (involving, for example, the product of a Bessel function and Laguerre polynomials) is also investigated.

The existence and uniqueness problems for some quasi-variational inequalities are studied on the basis of the L ∞ -estimates for solutions of the variational inequalities and their differences. An implicit obstacle problem is stated by analogy with one quasi-variational inequality studied by Benoussan and Lions (Méthodes Mathématiques de l'Informatique 11: 1982) and Vescan (1982) and its unique solvability is proved. Some conclusions are given concerning the uniqueness of solutions for an impulse control problem with bilateral restrictions and for a quasi-variational inequality appearing in dynamic programming.

We provide a new variational principle and duality for the problem 𝐿𝑢 = ∇𝐺(𝑢), where 𝐿 is selfadjoint and ∇𝐺 is a gradient mapping such that 𝐺 satisfies superquadratic growth conditions. The results obtained can be applied to Dirichlet problems for both ordinary and partial differential equations.

We investigate the geometric, algebraic and homological properties of Poisson structures on smooth manifolds and introduce noncommutative foundations of these structures for associative Poisson algebras. Noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold, symplectic foliation and symplectic leaf for associative Poisson algebras are given. These structures are considered for the case of the endomorphism algebra of a vector bundle, and a full description of the family of Poisson structures for this algebra is given. An algebraic construction of the reduction procedure for degenerate noncommutative Poisson structures is developed. A noncommutative generalization of Bott connection on foliated manifolds is introduced using the notions of a noncommutative submanifold and a quotient manifold. This definition is applied to degenerate noncommutative Poisson algebras, which allows us to consider Bott connection not only for regular but also for singular Poisson structures.

The aim of the paper is to give, according to Karoubi–Villamayor K -groups, an interpretation of Kasparov KK -groups. It continues the study of KK -theory by the methods of K -theory, focusing attention on the problem posed and discussed in the author's papers published in 2000 and 2001. But the methods used in those papers are based on the excision property and Morita invariance of algebraic and topological K -theories on the category of C* -algebras, which are not applicable to Karoubi–Villamayor K -theory, since excision holds only for some sub class of short exact sequences of C* -algebras. In this paper we introduce and study a weak stability property of the C* -category Rep( A, B ), which is the key to our problem.

It is shown, under Martin's Axiom, that there exist absolutely nonmeasurable additive functions. Starting with this fact, a generalization of the well-known result of Pelc and Prikry (Proc. Amer. Math. Soc. 89: 608–610, 1983) is obtained. Some related questions concerning the measurability of additive functions are also discussed.

We study oscillatory properties of solutions of a functional differential equation of the form 𝑢 (𝑛) (𝑡) + 𝐹(𝑢)(𝑡) = 0, where 𝑛 ≥ 2 and 𝐹 : 𝐶(𝑅 + ; 𝑅) → 𝐿 loc (𝑅 + ; 𝑅) is a continuous mapping. Sufficient conditions for this equation to have the so-called Property A are established. In the case of ordinary differential equation the obtained results lead to an integral generalization of the well-known theorem by Kondrat'ev.

A useful inequality for solutions of linear neutral integrodifferential equations with unbounded delay is established, using a real root of the corresponding characteristic equation. This inequality is used to obtain an estimate for solutions, which leads to a stability criterion.

The purpose of this paper is the construction of invariant regions in which we establish the global existence of solutions for reaction-diffusion systems with a general full matrix of diffusion coefficients without balance law'condition (𝑓 + 𝑔 ≡ 0) and with nonhomogeneous boundary conditions. Our techniques are based on invariant regions and Lyapunov functional methods. The nonlinear reaction term has been supposed to be of polynomial growth.

We establish a sharp estimate for a multilinear Littlewood–Paley operator. As an application, we obtain weighted norm inequalities and a L log L type estimate for a multilinear operator.

An initial-boundary value problem for a fourth order evolution equation with the bilaplacian operator is considered. We prove the existence and uniqueness of a generalized solution. The proofs are mainly based on energy inequalities for the operator generated by the problem under consideration.

The method of independent families of sets elaborated by Tarski [Set theory. With an introduction to descriptiveset theory, PWN–Polish Scientific Publishers, 1976] and Kakutani and Oxtoby [Ann. of Math. 52: 580–590, 1950] is applied to the construction of maximal (in the sense of cardinality) families of pairwise orthogonal nonseparable invariant extensions of Haar measures.

By employing the matrix Riccati technique and the integral averaging technique, new oscillation criteria are established for a second order self-adjoint matrix differential system of the form [𝑌(𝑡)𝑌′]′ + 𝑄(𝑡)𝑌 = 0.