Table of contents

In this paper we continue the investigation of the structure of finitely generated modules over rings of general quantum (Laurent) polynomials. We obtain a description of the lattice of submodules of periodic finitely generated modules and describe the irreducible modules. We investigate the problem of Morita equivalence of rings of general quantum polynomials, consider properties of division rings of fractions, and solve Zariski's problem for quantum polynomials.

After the fundamental work of Riesz, Radon and Hausdorff in the period 1909-1914, the following problem of general Radon representation emerged: for any Hausdorff space find the space of linear functionals that are integrally representable by Radon measures. In the early 1950s, a partial solution of this problem (the bijective version) for locally compact spaces was obtained by Halmos, Hewitt, Edwards, Bourbaki and others. For bounded Radon measures on a Tychonoff space, the problem of isomorphic Radon representation was solved in 1956 by Prokhorov.

In this paper we give a possible solution of the problem of general Radon representation. To do this, we use the family of metasemicontinuous functions with compact support and the class of thin functionals. We present bijective and isomorphic versions of the solution (Theorems 1 and 2 of §2.5). To get the isomorphic version, we introduce the family of Radon bimeasures.

In this paper we study the structure of the set of immersed linear networks in that are parallel to a given immersed linear network and whose boundary  coincides with the boundary of . We prove that is a convex polyhedral subset in the configuration space of moving vertices of the graph . We also calculate the dimension of this convex subset and estimate the number of its faces of maximal dimension. The results obtained are used to describe the space of all locally minimal (weighted minimal) networks in  with a fixed topology and a fixed boundary. In the case of planar networks in which the degrees of vertices are at most three (Steiner networks), this dimension is calculated in topological terms.

The notion of constancy of type was introduced by Gray in the study of specific properties of the geometry of six-dimensional nearly Kahlerian manifolds, and has been investigated by many authors. This notion can be generalized in a natural manner to the case of metric -manifolds with the Killing fundamental form (Killing -manifolds). In this paper, the property of constancy of type is studied in the naturally arising class of so-called commutatively Killing -manifolds, and some of their remarkable properties are investigated. An exhaustive description of commutatively Killing -manifolds of constant type is obtained. In particular, it is proved that the constancy of type of commutatively Killing -manifolds is tantamount to their local equivalence to the five-dimensional sphere  endowed with the weakly cosymplectic structure induced by a special embedding of  in the Cayley numbers.

The classical criteria of Kummer, Mirimanov and Vandiver for the validity of the first case of Fermat's theorem for the field  of rationals and prime exponent  are generalized to the field and exponent . As a consequence, some simpler criteria are established. For example, the validity of the first case of Fermat's theorem is proved for the field and exponent  on condition that  does not divide .

We define a new property of finitely presented groups connected with their asymptotic representations. Namely, we say that a group is AGA if each of its almost-representations generates an asymptotic representation. We give examples of groups with and without this property. In particular, free groups, finite groups and free Abelian groups are AGA. In our example of a group Γ that is not AGA, the group K⁰(BΓ) contains elements that are not covered by asymptotic representations of Γ.

We consider non-stationary diffusion problems in a periodic medium with inclusions filled with a material of small conductivity. We propose homogenized equations whose solutions approximate those of the problems under consideration. We prove estimates for the accuracy of this approximation as the period of the medium and the conductivity coefficient tend to zero. The form of the homogenized equations and the accuracy estimates depend essentially on the asymptotic behaviour of the conductivity coefficient in relation to the square of the period.