Table of contents

The author investigates the extension of formal arithmetic by the proposition that a concrete "large" natural number is not attainable. It is shown in the article that although the resulting system is inconsistent, the only formulas in the language of arithmetic which can be derived by "short" proofs are those which are theorems of arithmetic.

Bibliography: 10 titles.

The Sapondzhyan-Babuska paradox consists in the fact that, when thin circular plates are approximated by regular polygons with freely supported edges, the limit solution does not satisfy the conditions of free support on the circle. In this article, new effects of the same nature are found. In particular, plates with convex holes are considered. Here, in contrast to the case of convex plates, the boundary conditions on the polygon are not preserved in the limit. Methods of approximating a smooth contour leading to passage to the limit from conditions of free support to conditions of rigid support are discussed.

Bibliography: 20 titles.

The problem of "filling holes" in the spectrum is considered for a given operator when it is perturbed by operators in various symmetrically normed operator ideals. The case of trace-class perturbations of normal operators is allotted a special place.

Bibliography: 22 titles.

The space of Siegel cusp modular forms is embedded as a direct summand in the cohomology space of the direct image of some locally constant sheaf on the toroidal compactification of the corresponding quotient space of the Siegel upper half-plane. This is a generalization of the classical result of Eichler and Shimura.

Bibliography: 11 titles.

It is proved that the Jacobian of a 3-sheeted polynomial mapping cannot be a constant.

Bibliography: 8 titles.

The complex geometry of the future tube is studied, and in particular it is proved that the boundary of the future tube cannot be holomorphically straightened along the complex light rays. Using the general Cauchy-Fantappiè representation we derive the Cauchy-Bochner and Jost-Lehmann-Dyson integral representations and representations with Levi and Cauchy barriers for holomorphic functions and for solutions of the -equation.

Bibliography: 26 titles.

The surfaces of constant energy in integrable Hamiltonian systems which possess Bott integrals are classified. A complete topological classification is given of surgery of Liouville tori in general position in integrable Hamiltonian systems.

Bibliography: 28 titles.

A multilinear identity of degree 3p–2 is given in explicit form, and it is shown that this identity holds in the associated Lie ring of a free group of prime exponent p. It is also shown that if this identity is not a consequence of the known identities of Wall of degree 2p–1 and the (p–1)st Engel identity, there exists a finite p-group in which the index of the (nontrivial) Hughes subgroup is p³.

Bibliography: 12 titles.

The author constructs a pseudoform with negative holomorphic sectional curvature on CP_n less 2n+1 hyperplanes in general position.

Bibliography: 7 titles.