Table of contents

Let  be a normed linear space, let be a finite-dimensional subspace, and let . We define a multiplicative -selection to be a map such that

We prove that there is an -selection  whose smoothness coincides with that of the norm in . We show that, generally speaking, it is impossible to find an -selection of greater smoothness in .

If  is a variety of groups and  is a subvariety, then the symbol denotes the complete lattice of varieties  such that . Let , where  is the lattice of subspaces of the -dimensional vector space over the field of two elements, and let be the Cartesian product operation. A non-empty subset  of a complete lattice  is called a complete sublattice of  if and for any non-empty .

We prove that  is isomorphic to a complete sublattice of . On the other hand, it is obvious that is isomorphic to a complete sublattice of  for any locally finite variety . We deduce criteria for the existence of an isomorphism onto a (complete) sublattice of for some locally finite variety . We also prove that there is a sublattice generated by four elements and containing an infinite chain.

Natural Seifert bundles for plesiocompact (in particular, compact) homogeneous spaces  are considered. Conditions are given for the realizability of compact forms of Thurston geometries of dimension less than or equal to four on the bases  of these bundles. The structure of these  is discussed in detail.

We suggest natural conditions for embedding the weight classes of meromorphic functions N{ω} of M. M. Dzhrbashyan. Some of these conditions are sufficient for the solubility of his moment problem and the Volterra integral equation of the first kind.

We prove a formula that expresses the square of the statistical sum for the two-dimensional Ising model for an arbitrary plane lattice in terms of the determinant of a matrix similar to the Kac-Ward matrix.

The problem of spectral synthesis for subspaces of analytic functions invariant under the operators of partial differentiation is reduced to the problem of the local description of closed submodules in the module of entire functions of exponential type over the ring of polynomials.

This paper deals with real algebraic varieties without real points. In particular, the Picard, Brauer, and Witt groups are calculated for these varieties.

A study was made in [1] of the class of one-dimensional singular integral-functional equations closely connected with elliptic boundary-value problems on the plane in domains with piecewise-smooth boundaries. A criterion for the Fredholm property of these operators was formulated there in terms of the so-called end symbol. This symbol is a semi-almost periodic matrix-valued function assigned to the equation under consideration. In this paper we study this situation from a more general point of view.

We prove that the functors and  of Radon and -additive probability measures, respectively, preserve neither the real-completeness nor the Dieudonne completeness of Tychonoff spaces. We suggest conditions under which Martin's axiom implies that  preserves real-complete spaces, absolute extensors, and Tychonoff bundles. These last results cannot be obtained without additional set-theoretic assumptions.