Table of contents

Bijective linear transformations of the matrix algebra over an arbitrary field that preserve simultaneous diagonalizability are characterized. This result is used for the characterization of bijective linear transformations monotone with respect to the - and -orders. Bibliography: 28 titles.

For a smooth one-parameter family of pairs of control systems and profit densities on a circle, the generic transitions between optimal rotations and stationary strategies are studied in the problem of maximization of the time-averaged profit on the infinite horizon. It is shown that there are only two types of such transitions, the corresponding singularities of the average profit as a function of the family parameter are found, and it is proved that these singularities are stable under small perturbations of a generic family. The classification of singularities of the maximum average profit is completed for generic families. Bibliography: 16 titles.

Finite-dimensional problems of finding outer, inner, and uniform estimates for a convex compactum by a ball in an arbitrary norm are considered and compared, as well as the problem of finding estimates of the boundary of a convex compactum by a spherical annulus of the smallest width. It is shown that these problems can be linked by means of the parametric problem of finding the best approximation in the Hausdorff metric of the compactum under consideration by a ball of fixed radius. One can indicate ranges of the fixed radius in which solutions of the latter problem give solutions of the problems mentioned above. However, for some values of the radius this latter problem can be independent. Bibliography: 12 titles.

In a cylindrical domain , where is an unbounded subdomain of , one considers the evolution equation the right-hand side of which is a quasi-elliptic operator with highest derivatives of orders with respect to the variables . For the mixed problem with Dirichlet condition at the lateral part of the boundary of  a uniqueness class of the Täcklind type is described. For domains tapering at infinity another uniqueness class is distinguished, a geometric one, which is broader than the Täcklind-type class. It is shown that for domains with irregular behaviour of the boundary this class is wider than the one described for a second-order parabolic equation by Oleĭnik and Iosif'yan (Uspekhi Mat. Nauk, 1976 [17]). In a wide class of tapering domains non-uniqueness examples for solutions of the first mixed problem for the heat equation are constructed, which supports the exactness of the geometric uniqueness class. Bibliography: 33 titles.

The spectral theory of pseudodifferential operators on ultrametric spaces of general form is investigated with the use of the analysis of ultrametric wavelets. Bases of ultrametric wavelets are constructed on ultrametric spaces of analytic type; it is proved that bases of ultrametric wavelets are bases of eigenvectors for the introduced pseudodifferential operators and the corresponding eigenvalues are calculated. A generalization of the Vladimirov operator of p-adic fractional derivation is introduced for general ultrametric spaces. Bibliography: 32 titles.

The asymptotic model of Oberbeck-Boussinesq convection is considered in the case when the heat conductivity is equal to zero and the viscosity . The global existence and uniqueness results are proved for the basic initial-boundary-value problem; both classical and generalized solutions are considered. It is shown that each solution approaches an equilibrium as . Bibliography: 41 titles.