Table of contents

Let be a nondegenerate real analytic hypersurface in , let , and let consist of the automorphisms of fixing the point . Then, as follows from a theorem of Moser, the real dimension of does not exceed 5. Here it is shown that 1) dimensions 2, 3, and 4 cannot be realized, but for 0, 1, and 5 examples are given; 2) if the point is not umbilical, then consists of not more than two mappings. Bibliography: 4 titles.

This paper is devoted to the realization of certain types of Chevalley groups as the Galois group of extensions of certain cyclotomic fields. In addition, a criterion for an algebraic curve to be defined over an algebraic number field is given. Bibliography: 11 titles.

Let there be given numbers , , , and , , , , , , and let be the class of rational functions of degree , analytic for , with

Bibliography: 6 titles.

In this article the Lie algebra of a Galois group which operates on the Tate module of a two- or three-dimensional Abelian variety is calculated. It is assumed that the Abelian variety does not have nontrivial endomorphisms and is defined over a global field with characteristic greater than two. Bibliography: 14 titles.

Let be a topological module (over the ring of polynomials) of vector-valued functions holomorphic in a domain . In the first part of this article it was established that to solve the local description problem for closed submodules of it suffices to verify two properties - stability and saturation. In the present second part various methods of carrying out this verification are elaborated. Conditions are presented under which closed submodules are stable and saturated and, consequently, local. In particular a wide class of locally convex algebras of analytic functions which admit local description of their closed ideals is isolated. Bibliography: 15 titles.

The following equation is considered:

where the functions are of bounded variation on , the function having jumps at the end points. A series of elementary solutions is associated with the solution by a certain rule (MR 32 #7986). The convergence of this series is investigated. The results of Sedleckiĭ (MR 43 #3734) for the case are used. Bibliography: 5 titles.

The behavior of analytic functions of several complex variables with respect to analytic continuation along certain sets in Cⁿ is studied. Bibliography: 4 titles.

Unilateral and bilateral multiple weighted shift operators whose weight sequences consist of invertible operators are studied. Under a certain condition on the decrease of the weight sequence the invariant subspaces of unilateral shift operators with finite multiplicity are found, and a reducibility condition for bilateral weighted shift operators (of any multiplicity) is obtained. In some cases the reducing subspaces of these operators are described. Bibliography: 11 titles.

Suppose that the profinite group G is an extension of A by H. In this paper the profinite subgroups of the topological group of continuous maps from H to A are investigated. The results obtained are used to prove topological analogues for profinite groups of the Frobenius and Magnus embedding theorems. Moreover, a sufficient condition is formulated for a pro-p-group that is an extension of an abelian group by a finitely presented group to be finitely presented, in the language of complete tensor products of abelian pro-p-groups; and this condition is used to prove that a finitely generated metabelian pro-p-group is a subgroup of a finitely presented metabelian pro-p-group. Bibliography: 15 titles.

Let be a simple 4-dimensional abelian variety of the first or second type in Albert's classification (i.e. all simple factors of the -algebra are isomorphic to or ). In this case the algebra over is generated by divisor classes. If , and the Hodge group has type or , then and the -space is not generated by classes of intersections of divisors. Bibliography: 6 titles.

Smoothness of the general anticanonical divisor of a Fano 3-fold is proved. In addition, an analogous result is established for the linear system , where for some natural number . The results obtained in the paper can be used to investigate projective imbeddings of Fano 3-folds. Bibliography: 6 titles.

In this paper the author studies spaces in which one can define a "distance" from points to canonically closed sets (the -metric). It is proved that products of metric spaces and locally compact groups are examples of such spaces, and in these cases the -metric can be constructed so that an analogue of the triangle axiom is satisfied. The topological structure of -metrizable compact Hausdorff spaces is studied. Bibliography: 26 titles