Table of contents

In this paper the author presents an approach to the problem of classifying quasihomogeneous singularities, based on the use of simple properties of deformation theories of such singularities. By means of Grothendieck local duality the Poincaré series of the space of the first cotangent functor T' of a one-dimensional singularity is computed. Lists of normal forms and monomial bases of the spaces of T' are given for one-dimensional quasihomogeneous complete intersections with inner modality 0 and 1, and also with Milnor number less than seventeen. An adjacency diagram is constructed for all singularities that have been found. Bibliography: 33 titles.

In this paper the author establishes a sharp order estimate, in the mixed norm of for and in ( is the -dimensional torus), of the derivatives of order of the multidimensional Dirichlet -kernel and the function (, ), which are sums of exponentials lying respectively inside and outside a "graduated hyperbolic cross", i.e., the set , where , . Bibliography: 11 titles.

In this paper a multidimensional Tauberian theorem is proved that establishes a connection between the behavior of a generalized function in a cone and the behavior of its Laplace transform in the neighborhood of zero in the tube domain over the cone. Here it is assumed that the Laplace transform has nonnegative imaginary part or, more generally, bounded argument. The theorem is used to illuminate sufficient conditions for the existence of an angular limit of holomorphic functions of bounded argument. An example is constructed of a holomorphic function with bounded nonnegative imaginary part in , having a limit over a countable set of rays coming into the origin, but without an angular limit. In addition, a number of theorems on the existence of quasi-asymptotic limits of the solutions of multidimensional convolution equations are proved, and examples are considered of finding quasi-asymptotic limits of fundamental solutions of hyperbolic operators with constant coefficients, as well as of passive systems. The quasi-asymptotic limit of a fundamental solution of the system of equations governing a rotating compressible fluid is found, and similarly for other systems. Bibliography: 10 titles.

In this paper the group SK₁(A) is computed for a division ring A whose center has a nontrivial Henselian valuation. Bibliography: 17 titles.

The equation

generalizing the well-known Hutchinson equation, is used to describe the dynamics of change in the population of a species, with the age structure taken into account. Conditions are found for bifurcation of a periodic solution of this equation. The results obtained here are applied to explain some natural phenomena. Bibliography: 12 titles.

In this paper the author studies the connection between smoothness, expressed in terms of the integral modulus of continuity, and the existence of a derivative, understood in some sense, for functions in , ; an analogous question is considered for boundary values of analytic functions in the Hardy classes , . A connection is established between the derivatives of an analytic function in and the derivatives of its boundary value; both global and pointwise derivatives are considered. Bibliography: 25 titles.

Let be the best uniform approximation of by rational fractions of degree at most , and let be the set of monotone convex functions such that and . Theorem. Suppose the function is absolutely continuous on the interval , and let and . If is summable on , then . Various applications and generalizations of this result are given, and the periodic case is also considered. Bibliography: 23 titles.

The main result of this paper is the following theorem: If is a Noetherian ring, then the canonical homomorphism is surjective when and injective when . Bibliography: 9 titles.