Vol 38, No 4 (2017)

It is shown that the set of optical quantum tomograms can be provided with the topology of Frechet space. In such a case the conjugate space will consist of symbols of quantum observables including all polynomials of the position and momentum operators.

We present an algorithm for construction step wavelets on local fields of positive characteristic.

Let Σ_n(F) denote the space of all n×n symmetricmatrices over the complex field F, and χ be an irreducible character of S_n and d_χ the immanant associated with χ. The main objective of this paper is to prove that the maps Φ: Σ_n(F) → Σ_n(F) satisfying d_χ(Φ(A) + αΦ(B)) = det(A + αB) for all singular matrices A, B ∈ Σ_n(F) and all scalars α ∈ F are linear and bijective. As a corollary of the above result we characterize all such maps Φ acting on the set of all symmetric matrices.

Algebras of symmetries and the corresponding algebras of differential invariants for plane flows of viscid fluids are given. Their dependence on thermodynamical states of media are studied and a classification of thermodynamical states is given.

By B_1,1 we denote the variety of unary algebras of signature f, g which is defined by the identity fg(x) = x. In this note it is proved that B_1,1 is a cover for the variety A_1,1, where A_1,1 is the variety defined by the identities fg(x) = x = gf(x). It is also shown that each endomorphism of a strongly connected algebra from B_1,1 is an automorphism.

In this article we apply methods of representation theory and combinatorial algebra to the different problems related to quantum tomography. For this purpose, we introduce the algebra generated by projectors satisfying some commutator relation. In this paper we study this commutator relation by combinatorialmethods and develop the representation theory of this algebra. Also, we apply our results to the case of mutually unbiased bases in dimension 7.

Using the entropic inequalities for Shannon and Tsallis entropies new inequalities for some classical polynomials are obtained. To this end, an invertible mapping for the irreducible unitary representation of groups SU(2) and SU(1, 1) like Jacoby polynomials and Gauss’ hypergeometric functions, respectively, are used.

The paper is a brief survey of the author’s results connected with the lattices of algebraic sets of universal algebras and with the operator of algebraic closure on the subsets of direct powers of basic sets of algebras.

A class of polynomial pencils of ordinary differential operators with constant coefficients is considered in the article. The pencils from this class are generated by the n-th order ordinary differential expression and twopoint boundary conditions. Coefficients of the differential expression are supposed to be polynomials of the spectral parameter with constant coefficients. The boundary conditions are supposed to depend on the spectral parameter polynomially. It is assumed that the roots of the characteristic equation of the pencils from this class are simple, non-zero and they are located arbitrarily in the complex plane. The author investigates n-fold completeness of the root functions of the pencils from this class in the space of summable with square functions on the main segment. Sufficient conditions of the n-fold completeness of the root functions are obtained. The main idea of the proof of the theorem is a new asymptotics of the characteristic determinant of the pencil. The presented results supplement previous results of the author.

We construct an analogue of Sierpiński gasket in Lobachevskii plane by means of iterated function system with maps from a transformation group of this space. The investigation of a new family of attractors and a Mandelbrot set associated with it reveals higher capacity of Lobachevskii geometry compared to that of Euclid.

In the study of solutions behavior of various mathematical physics equations their limit state when time tends to infinity (so-called attractors) is of particular interest. At the works of russian mathematicians M.I. Vishik and V.V. Chepyzhov and american mathematician G. Sell a new approach for the attractors study based on the consideration of trajectory spaces and trajectory attractors of the corresponding equations was proposed. However, for a large number of hydrodynamics equations this theory could not be applied because of the conditions of the translational invariance and of the closure of a trajectory space. In the introduction of this article a brief summary of the attractors theory for autonomous systems based on the concept of the trajectory space without the assumption of its translational invariance is presented. As an application of this theory, the existence of the global attractor for autonomous model of the Bingham medium motion in the case of periodic spatial variables in proved.