Table of contents

We consider a reaction-diffusion system of two equations, where one equation has a small diffusion coefficient . We construct the trajectory attractor of such a system. We also study the limit system for . In this system one equation is an ordinary differential equation in , but is considered in the domain , where and is the positive time axis, . We construct the trajectory attractor of the limit system. The main result is a convergence theorem: as in the corresponding topology.

Bibliography: 18 titles.

The frequencies of occurrences of elements in linear recurrence sequences of vectors over Galois rings are studied. The study of these frequencies is reduced to the study of the corresponding trigonometric sums over Galois rings. Based on estimates for trigonometric sums, nontrivial estimates for the frequencies of occurrence of elements in linear recurrence sequences are obtained, which generalize some known results for sequences over a finite field. These estimates are asymptotically best possible.

Bibliography: 25 titles.

We investigate problems on a.e. convergence of Riemann sums

with the use of classical maximal functions in . A theorem on the equivalence of Riemann and ordinary maximal functions is proved, which allows us to use techniques and results of the theory of differentiation of integrals in  in these problems. Using this method we prove that for a certain sequence the Riemann sums converge a.e. to , .

Bibliography: 23 titles.

Oskolkov's system of equations with a cubic source is considered; this describes the dynamics of a viscoelastic fluid. Local solvability (with respect to time) of the problem in the weak generalized sense is proved. Some conditions on the initial function which ensure that the solution blows up in finite time are found, and two-sided estimates for the existence time of the solution are obtained. Moreover, sufficient conditions for the global solvability (with respect to time) of the problem are found.

Bibliography: 19 titles.

The structure of the Jacobson radical in the endomorphism ring of a mixed completely decomposable Abelian group is described.

Bibliography: 6 titles.

For multiple orthogonal polynomials with respect to two Pollaczek weight functions weak asymptotics are obtained. It is shown that a solution of a vector equilibrium problem of the theory of logarithmic potential in the presence of an external field and with restriction imposed on the measures is given by the limit measure of the distribution of zeros.

Bibliography: 17 titles.

It is shown that on the average the Frobenius numbers behave like .

Bibliography: 28 titles.