Table of contents

The present paper is an immediate continuation of the author's paper [22]. Except in the last section, it is implicitly assumed here, as in [22], that the underlying ring contains 1/2 and all the theorems relate to the theory without further comment.

Invariants of nonsimply connected manifolds are investigated of the same type as the signature of a manifold, or of "higher signature" for manifolds with free Abelian fundamental group, which correspond to the possibility of surgery on a manifold. It is proved that these invariants depend only on the homotopy type of the manifold and a bordism class of manifolds.

We consider a nonsingular cubic hypersurface in . We prove that the motif can be expressed by means of the Tate motif and the motif , where is the curve of straight lines on that pass through a fixed line and is an automorphism of that leaves no line coplanar with fixed.

In this work it is proved that, for the regular action of a semisimple irreducible algebraic group G on an affine space, the existence of a closed orbit of maximum dimension is equivalent to the existence of an invariant open set at any point of which the stationary subgroup is reductive. This result is established for the action of G on manifolds of a special type (the so-called factorial manifolds). There are given several other conditions equivalent to the existence of a closed orbit of maximum dimension for the action of G on an arbitrary affine manifold.

The main result of the present paper is that a ring obtained by reduction of Abelian modular functions is a ring of functions on a p-adic symplectic group over a perfect field k of characteristic p, and the ring is induced by a homomorphism of a parabolic subgroup on Aut k.

This work investigates conditions for the possibility of approximating functions f(z) in the pth order mean on a curve C with arbitrary accuracy by polynomials whose coefficients are algebraic integers from a complex quadratic field. The case when f(z) is an analytic function of class E_p in the region bounded by a closed curve C is examined, as is the case when f(z) is integrable of degree p on a curve C which is not closed.

Approximation of functions of n variables is studied by means of infinite-dimensional subspaces of singular functions of a special form (quasipolynomials). In particular, best approximation is examined by means of these subspaces, and both direct and converse theorems are proved.

In this paper a classification of infinitely differentiable functions is developed, based on derivatives introduced earlier by the author. In terms of the classes C_{γ, κ} (0, u] and C_{γ, κ} (0, ∞) considered in this paper, not only is an interrelation between the different classes given, but also the problem of representation of the functions of these classes, in the case of quasi-analyticity (in the sense of Denjoy and Carleman) as well as in the case of non-quasi-analyticity of the class, is completely solved. Some applications of these results are given.

In this work, necessary and sufficient conditions are found in order for a system of measurable functions {f_n(x)} to be a system of convergence for l_p. The assertions set forth here are a strengthening of the work of the author [1] in the case when B = l_p.

In this article, as in the author's article [1] with the same title, we consider questions related to optimal quadrature formulas.

This article considers the problem of spectral synthesis for operators with a complete system of root vectors. Criteria for the existence of spectral synthesis are given. A criterion for spectral synthesis in terms of geometrical properties of the system of eigenvectors is established for operators with simple spectrum. The relationship between the problems of spectral synthesis for operators A and A* is studied.

Existence theorems on optimal controls for systems with distributed parameters and with trajectories given by a general operator of the form u_p(t, w, v) are proved. A set of quasiregular functionals is chosen as a criterion of optimality of the process. Such a choice essentially exhausts the cases which are practically interesting in applications.