Volume 38, Nº 4 (2017)
- Ano: 2017
- Artigos: 22
- URL: https://journals.rcsi.science/1995-0802/issue/view/12453
Article
On tomographic representation on the plane of the space of Schwartz operators and its dual
Resumo
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.
Effective categoricity for distributive lattices and Heyting algebras
Resumo
We study complexity of isomorphisms between computable copies of lattices and Heyting algebras. For a computable ordinal α, the Δα0dimension of a computable structure S is the number of computable copies of S, up to Δα0 computable isomorphism. The results of Goncharov, Harizanov, Knight, McCoy, Miller, Solomon, and Hirschfeldt, Khoussainov, Shore, Slinko imply that for every computable successor ordinal α and every non-zero natural number n, there exists a computable non-distributive lattice with Δα0 dimension n. In this paper, we prove that for every computable successor ordinal α ≥ 4 and every natural number n > 0, there is a computable distributive lattice with Δα0 dimension n. For a computable successor ordinal α ≥ 2, we build a computable distributive lattice M such that the categoricity spectrum of M is equal to the set of all PA degrees over Ø(α). We also obtain similar results for Heyting algebras.
Generalized Lie problem and differential invariants for the third order ODEs
Resumo
In this work the generalized Lie problem for the third order ODEs ym = F(x, y) is studied. Symmetry algebra of contact transformations for such equations is calculated. Algebra of differential invariants for the action of the symmetry algebra on these equations is described. Classification theorem for two regular ODEs is obtained.
Converting immanants on singular symmetric matrices
Resumo
Let Σn(F) denote the space of all n×n symmetricmatrices over the complex field F, and χ be an irreducible character of Sn and dχ the immanant associated with χ. The main objective of this paper is to prove that the maps Φ: Σn(F) → Σn(F) satisfying dχ(Φ(A) + αΦ(B)) = det(A + αB) for all singular matrices A, B ∈ Σn(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.
Differential invariants of circle foliations on the plane
Resumo
Circle foliations on the plane are considered with respect to the conformal transformations. We derive 3rd order differential equation whose solutions determine such foliations. Lie group of symmetries and differential invariants algebra of this equation are found.
Differential invariants for plane flows of viscid fluids
Resumo
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.
On algebras of the variety B1,1
Resumo
By B1,1 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 B1,1 is a cover for the variety A1,1, where A1,1 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 B1,1 is an automorphism.
On problem of concrete characterization of universal automata
Resumo
In the paper we investigate automata whose state sets and sets of output symbols are endowed with algebraic structures of hypergraphs. Universal hypergraphic automata are universally attracting objects in the category of such automata. The main result is a solving of the problem of concrete characterization of universal hypergraphic automata.
On the algebra generated by projectors with commutator relation
Resumo
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.
Problems on structure of finite quasifields and projective translation planes
Resumo
It is well-known that the constructions and classification of non-Desarguesian projective planes are closely connected with ones for quasifields. We consider the problems on structure of finite quasifields and semifields: automorphisms and autotopisms, maximal subfields and their orders, the spectrum of orders of non-zero elements and hypotheses about generated subsets of the multiplicative loop.
Entropic inequalities for matrix elements of rotation group irreducible representations
Resumo
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.
Algebraic sets of universal algebras and algebraic closure operator
Resumo
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.
On closedness of stationary subgroup of affine transformations group
Resumo
This article deals with Lie algebra g of all infinitesimal affine transformations on a manifold with affine connection and its stationary subalgebra h ⊂ g. Let G be simply connected group generated by algebra g and H ⊂ G be the subgroup generated by subalgebra h ⊂ g and let dimg/h = dimM. Then if algebra g has zero center the subgroup H is closed in G. Thus any infinitesimal affine transformation X ⊂ g on a manifold M = G/H can be extended to affine transformation f: M → M. For Riemannian manifolds the condition dimg/h = dimM can be omitted and the main result can be generalized for algebra g with non-zero center.
Multiple completeness of the root functions for a certain class of ordinary differential pencils with constant coefficients
Resumo
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.
Determinability of Hewitt spaces by the lattices of subalgebras with unit of semifields of continuous positive functions with max-plus
Resumo
The lattice A1(U∨(X)) of subalgebras with unit of the semifield U∨(X) of continuous positive functions defined on a topological space X is considered. A topological space is said to be a Hewitt space if it is homeomorphic to a closed subspace of a Tychonoff power of the real line R. The main achievement of the paper is the proof of the fact that any Hewitt space X is determined by the lattice A1(U∨(X)).
On generalization of Sierpiński gasket in Lobachevskii plane
Resumo
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.
Attractors theory for autonomous systems of hydrodynamics and its application to Bingham model of fluid motion
Resumo
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.