Vol 24, No 127 (2019)
Articles
Asymptotics for the Radon transform on hyperbolic spaces
Abstract
Let G/ H be a hyperbolic space over R ; C or H ; and let K be a maximal compact subgroup of G . Let D denote a certain explicit invariant differential operator, such that the non-cuspidal discrete series belong to the kernel of D . For any L 2 -Schwartz function f on G/ H we prove that the Abel transform A ( Df) of Df is a Schwartz function. This is an extension of a result established in [2] for K -finite and K∩ H -invariant functions.
Russian Universities Reports. Mathematics. 2019;24(127):241-251
241-251
Core of a matrix in max algebra and in nonnegative algebra: A survey
Abstract
This paper presents a light introduction to Perron-Frobenius theory in max algebra and in nonnegative linear algebra, and a survey of results on two cores of a nonnegative matrix. The (usual) core of a nonnegative matrix is defined as ∩ k≥1 span+ (A k ) , that is, intersection of the nonnegative column spans of matrix powers. This object is of importance in the (usual) Perron-Frobenius theory, and it has some applications in ergodic theory. We develop the direct max-algebraic analogue and follow the similarities and differences of both theories.
Russian Universities Reports. Mathematics. 2019;24(127):252-271
252-271
On the extension of Chaplygin’s theorem to the differential equations of neutral type
Abstract
We consider functional-differential equation x(gt = f t; xht , t ∈ 0; 1 , where function f satisfies the Caratheodory conditions, but not necessarily guarantee the boundedness of the respective superposition operator from the space of the essentially bounded functions into the space of integrable functions. As a result, we cannot apply the standard analysis methods (in particular the fixed point theorems) to the integral equivalent of the respective Cauchy problem. Instead, to study the solvability of such integral equation we use the approach based not on the fixed point theorems but on the results received in [A.V. Arutyunov, E.S. Zhukovskiy, S.E. Zhukovskiy. Coincidence points principle for mappings in partially ordered spaces // Topology and its Applications, 2015, v. 179, № 1, 13-33] on the coincidence points of mappings in partially ordered spaces. As a result, we receive the conditions on the existence and estimates of the solutions of the Cauchy problem for the corresponding functional-differential equation similar to the well-known Chaplygin theorem. The main assumptions in the proof of this result are the non-decreasing function f(t; ·) and the existence of two absolutely continuous functions v, w, that for almost each t ∈ [0; 1] satisfy the inequalities vgt ≥ f t; vht , wgt ≤ f t;wht . The main result is illustrated by an example.
Russian Universities Reports. Mathematics. 2019;24(127):272-280
272-280
Star product and star function
Abstract
We give a brief review on star products and star functions [8, 9]. We introduce a star product on polynomials. Extending the product to functions on complex space, we introduce exponential element in the star product algebra. By means of the star exponential functions we can define several functions called star functions in the algebra.We show certain examples.
Russian Universities Reports. Mathematics. 2019;24(127):281-292
281-292
On the solvability of causal functional inclusions with infinite delay
Abstract
In the present article we develop the results of works devoted to the study of problems for functional differential equations and inclusions with causal operators, in case of infinite delay. In the introduction of the article we substantiates the relevance of the research topic and provides links to relevant works A. N. Tikhonov, C. Corduneanu, A. I. Bulgakov, E. S. Zhukovskii, V. V. Obukhovskii and P. Zecca. In section two we present the necessary information from the theory of condensing multivalued maps and measures of noncompactness, also introduced the concept of a multivalued causal operator with infinite delay and illustrated it by examples. In the next section we formulate the Cauchy problem for functional inclusion, containing the composition of multivalued and single-valued causal operators; we study the properties of the multiopera-tor whose fixed points are solutions of the problem. In particular, sufficient conditions under which this multioperator is condensing on the respective measures of noncompacness. On this basis, in section four we prove local and global results and continuous dependence of the solution set on initial data. Next the case of inclusions with lower semicontinuous causal multioperators is considered. In the last section we generalize some results for semilinear differential inclusions and Volterra integrodifferential inclusions with infinite delay.
Russian Universities Reports. Mathematics. 2019;24(127):293-315
293-315
Bergman-Hartogs domains and their automorphisms
Abstract
For Cartan-Hartogs domains and also for Bergman-Hartogs domains, the determination of their automorphism groups is given for the cases when the base is any bounded symmetric domain and a general bounded homogeneous domain respectively.
Russian Universities Reports. Mathematics. 2019;24(127):316-323
316-323
About a complex operator exponential function of a complex operator argument main property
Abstract
Operator functions eA , sin B , cos B of the operator argument from the Banach algebra of bounded linear operators acting from E to E are considered in the Banach space E . For trigonometric operator functions sin B , cos B , formulas for the sine and cosine of the sum of the arguments are derived that are similar to the scalar case. In the proof of these formulas, the composition of ranges with operator terms in the form of Cauchy is used. The basic operator trigonometric identity is given. For a complex operator exponential function eZ of an operator argument Z from the Banach algebra of complex operators, using the formulas for the cosine and sine of the sum, the main property of the exponential function is proved. Operator functions eAt , sin Bt , cos Bt , eZt of a real argument t∈(-∞;∞) are considered. The facts stated for the operator functions of the operator argument are transferred to these functions. In particular, the group property of the operator exponent eZt is given. The rule of differentiation of the function eZt is indicated. It is noted that the operator functions of the real argument t listed above are used in constructing a general solution of a linear n th order differential equation with constant bounded operator coefficients in a Banach space.
Russian Universities Reports. Mathematics. 2019;24(127):324-332
324-332
On a dilation of a some class of completely positive maps
Abstract
In this article we investigate sesquilinear forms defined on the Cartesian product of Hilbert C* -module M over C* -algebra B and taking values in B . The set of all such defined sesquilinear forms is denoted by SBM . We consider completely positive maps from locally C* -algebra A to SB M. Moreover we assume that these completely positive maps are covariant with respect to actions of a group symmetry. This allow us to view these maps as generalizations covariant quantum instruments which are very important for the modern quantum mechanic and the quantum field theory. We analyze the dilation problem for these class of maps. In order to solve this problem we construct the minimal Stinespring representation and prove that every two minimal representations are unitarily equivalent.
Russian Universities Reports. Mathematics. 2019;24(127):333-339
333-339