Table of contents

This paper investigates the question of removability of singularities of torsion-free sheaves on algebraic surfaces in the universal deformation and the existence in it of a nonempty open set of locally free sheaves, and describes the tangent cone to the set of sheaves having degree of singularities larger than a given one. These results are used to prove that quasitrivial sheaves on an algebraic surface with have a universal deformation whose general sheaf is locally free and stable relative to any ample divisor on , and thereby to find a nonempty component of the moduli space of stable bundles on with and on any algebraic surface.

Bibliography: 11 titles.

The asymptotic expansion

valid outside any angle , , is obtained for the Weyl-Titchmarsh function of the Sturm-Liouville problem on the half-axis with potential .

Bibliography: 8 titles.

A criterion is given, in terms of the Laplace transform of an analytic functional , for the convolution operator , corresponding to this functional, to be an epimorphism, in the case when is an arbitrary convex domain and an arbitrary convex compact set in Cⁿ.

Various types of functional equations for Dirichlet series are examined, and their equivalence is shown.

This paper gives a description of the torsion and cotorsion in the Milnor groups and for an arbitrary field . The main result is that, for any natural number with ,

and the group is uniquely -divisible if . This theorem is a consequence of an analogue of Hilbert's Theorem 90 for relative -groups of extensions of semilocal principal ideal domains. Among consequences of the main result we obtain an affirmative solution of the Milnor conjecture on the bijectivity of the homomorphism , where is the ideal of classes of even-dimensional forms in the Witt ring of the field , as well as a more complete description of the group for all global fields.

A new topological invariant is constructed which classifies integrable Hamiltonian systems with two degrees of freedom (admitting a Bott integral). A criterion for the equivalence of Bott systems is proved: such systems are topologically equivalent if and only if their topological invariants coincide. The topological invariant is effectively calculated for specific integrable Hamiltonian systems in physics and mechanics.

The construction of a general theory of partial differential equations on a superspace is continued in the framework of functional superanalysis. Superanalogues of the spaces and of generalized functions are introduced; a theorem is proved on the existence of a fundamental solution for linear differential equations with constant coefficients on a superspace. In contrast to the scalar case, there exist differential operators not having fundamental solutions. Formulas are obtained for the fundamental solutions of the Laplace operator, the heat conduction operator, the Schrödinger operator, the d'Alembert operator, and the Helmholtz operator on a superspace. There is a discussion of the role of the nilpotence condition for even ghosts in a commutative superalgebra in the construction of a theory of generalized functions.

Stable algebra is developed as needed in Morse theory. New estimates are found for the number of critical points of Morse functions, and conditions are found for the existence of minimal Morse functions on non-simply-connected manifolds.

In this article we consider the problem of uniqueness of so-called normal coordinates for real-analytic generating manifolds of codimension 2 in C⁴. For nonumbilic surfaces we find a class of coordinates which is preserved only by linear transformations.

Weighted estimates

(1)

are considered, where the constant does not depend on , for fractional Riemann- Liouville integrals

and the following problem is examined: find necessary and sufficient conditions on weight functions and under which estimate (1) is valid for all functions for which the right-hand side of (1) is finite. The problem is solved for and . This result is definitive, and it generalizes known results for integral operators when .

Bibliography: 19 titles.