Vol 24, No 128 (2019)
Articles
The Jacobi group and its holomorphic discrete series representations on Siegel-Jacobi domains
Abstract
This is the summary of a part of the talk delivered at the workshop held at the Tambov University in September 2012, reporting several results on Jacobi groups and its holomorphic representations published by the authors.
Russian Universities Reports. Mathematics. 2019;24(128):345-353
345-353
The pseudospectrum of the convention-diffusion operator with a variable reaction term
Abstract
In this paper, we study the spectrum of non-self-adjoint convection-diffusion operator with a variable reaction term defined on an unbounded open set Ω of Rn . Our idea is to build a family of operators that have the same convection-diffusion-reaction formula, but which will be defined on bounded open sets { Ω η } η ∈]0,1[ of Rn . Based on the relationships that link this family to Ω , we obtain relations between the spectrum and the pseudospectrum. We use the notion of the pseudospectrum to build relationships between convection-diffusion operator and its restrictions to bounded domains. Using these relationships we are able to find the spectrum of our operator in R+ . Also, the techniques developed to obtain the spectrum allow us to study the properties of the spectrum of this operator when we go to the limit as the reaction term tends to zero. Indeed, we show a spectral localization result for the same convection-diffusion-reaction operator when a perturbation is carried on the reaction term and no longer on the definition domain.
Russian Universities Reports. Mathematics. 2019;24(128):354-367
354-367
Razlozhenie granichnykh predstavleniy na ploskosti Lobachevskogo v secheniyakh lineynykh rassloeniy
Abstract
Earlier we described canonical (labelled by λ ∈C ) and accompanying boundary representations of the group G = SU (1,1) on the Lobachevsky plane D in sections of linear bundles and decomposed canonical representations into irreducible ones. Now we decompose representations acting on distributions concentrated at the boundary of D . In the generic case 2λ ∉N they are diagonalizable, in the exceptional case Jordan blocks appear.
Russian Universities Reports. Mathematics. 2019;24(128):368-375
368-375
On the existence of a continuously differentiable solution to the Cauchy problem for implicit differential equations
Abstract
We study the question of the existence of a solution to the Cauchy problem for a differential equation unsolved with respect to the derivative of the unknown function. Differential equations generated by twice continuously differentiable mappings are considered. We give an example showing that the assumption of regularity of the mapping at each point of the domain is not enough for the solvability of the Cauchy problem. The concept of uniform regularity for the considered mappings is introduced. It is shown that the assumption of uniform regularity is sufficient for the local solvability of the Cauchy problem for any initial point in the class of continuously differentiable functions. It is shown that if the mapping defining the differential equation is majorized by mappings of a special form, then the solution of the Cauchy problem under consideration can be extended to a given time interval. The case of the Lipschitz dependence of the mapping defining the equation on the phase variable is considered. For this case, estimates of non-extendable solutions of the Cauchy problem are found. The results are compared with known ones. It is shown that under the assumptions of the proved existence theorem, the uniqueness of a solution may fail to hold. We provide examples llustrating the importance of the assumption of uniform regularity.
Russian Universities Reports. Mathematics. 2019;24(128):376-383
376-383
On the implicit and inverse many-valued functions in topological spaces
Abstract
The conditions of continuity of the implicit set-valued map and the inverse setvalued map acting in topological spaces are proposed. For given mappings f : T ×X → Y , y : T → Y , where T,X,Y are topological spaces, the space Y is Hausdorff, the equation f(t,x) = y(t) with the parameter t ∈ T relative to the unknown x ∈ X is considered. It is assumed that for some multi-valued map U : T ⇉ X for all t ∈ T the inclusion f(t,U(t)) ∋ y(t) is satisfied. An implicit mapping R U : T ⇉ X , which associates with each value of the parameter t ∈ T the set of solutions x(t) ∈ U(t) of this equation. It is proved that R U is upper semicontinuous at the point t 0 ∈ T , if the following conditions are satisfied: for any x ∈ X the map f is continuous at ( t 0 ,x) , the map y is continuous at t 0 , a multi-valued map U is upper semicontinuous at the point t 0 and the set U( t 0 ) ⊂ X is compact. If, in addition, with the value of the parameter t 0 , the solution to the equation is unique, then the map R U is continuous at t 0 and any section of this map is also continuous at t 0 . The listed results are applied to the study of a multi-valued inverse mapping. Namely, for a given map g : X → T we consider the equation g(x) = y with respect to the unknown x ∈ X . We obtain conditions for upper semicontinuity and continuity of the map V U : T ⇉ X , V U (t) = {x ∈ U(t) : g(x) = t} , t ∈ T .
Russian Universities Reports. Mathematics. 2019;24(128):384-392
384-392
Universal algorithms for solving discrete stationary Bellman equations
Abstract
This paper investigates algorithms for solving discrete stationary (or) matrix Bellman equations over semirings, in particular over tropical and idempotent semirings, Also there are presented some original algorithms, applications and programmed realization.
Russian Universities Reports. Mathematics. 2019;24(128):393-431
393-431
Radon problems for hyperboloids
Abstract
We offer a variant of Radon transforms for a pair X and Y of hyperboloids in R3 defined by [x,x] = 1 and [y,y] = -1 , y 1 ≥ 1 , respectively, here [x,y] = - x 1 y 1 + x 2 y 2 + x 3 y 3 . For a kernel of these transforms we take δ([x,y]) , δ(t) being the Dirac delta function. We obtain two Radon transforms D(X) → C ∞ (Y) and D(Y) → C ∞ (X) . We describe kernels and images of these transforms. For that we decompose a sesqui-linear form with the kernel δ([x,y]) into inner products of Fourier components.
Russian Universities Reports. Mathematics. 2019;24(128):432-449
432-449
Spectral synthesis on zero-dimensional locally compact Abelian groups
Abstract
Let G be a zero-dimensional locally compact Abelian group whose elements are compact, C(G) the space of continuous complex-valued functions on the group G . A closed linear subspace H ⊆ C(G) is called invariant subspace, if it is invariant with respect to translations τ y : f(x) ↦ f(x + y) , y ∈ G . We prove that any invariant subspace H admits spectral synthesis, which means that H coincides with the closure of the linear span of all characters of the group G contained in H .
Russian Universities Reports. Mathematics. 2019;24(128):450-456
450-456