# Editorial backlog

1. Alsarori N.A., Ghadle K.P. New results for infinite functional differential inclusions with impulses effect and sectorial operators in Banach spaces
Status: reviewing
Abstract.
New results for infinite functional differential inclusions with impulses effect and sectorial operators in Banach spaces
Date of submission: 02 October 2020 г.

2. Osipchuk T.M. On linear convexity generalized to commutative algebras
Status: reviewing
Abstract.
A commutative associative algebra $\mathcal{A}$ with identity over the field of real numbers which has a basis $\{\boldsymbol{e}_k\}_{k=1}^{m}$, where all elements $\boldsymbol{e}_k$ are invertible, is considered in the work. Moreover, among matrixes $\Gamma^p=(\gamma_{lk}^p)$, $p=\overline{1,m}$, consisting of the structure constants $\gamma_{lk}^p$ of $\mathcal{A}$, defined as $\boldsymbol{e}_l\boldsymbol{e}_k=\sum_{p=1}^{m}\gamma_{lk}^p\boldsymbol{e}_p$, $l,k=\overline{1,m}$, there is at least one that is non-degenerate. The notion of linearly convex domains in the finite-dimensional complex space $\mathbb{C}^n$ and some of their properties are generalized to the finite-dimensional space $\mathcal{A}^n$, $n\ge 2$, that is the Cartesian product of $n$ algebras $\mathcal{A}$. Namely, a domain in $\mathcal{A}^n$ is said to be \emph{\textbf{(locally) $\mathcal{A}$-linearly convex}} if for every boundary point $\boldsymbol{w}$ of the domain there exists a hyperplane in $\mathcal{A}^n$ passing through $\boldsymbol{w}$ but not intersecting the domain (in some neighborhood of $\boldsymbol{w}$). The main result of the work is the separate necessary and sufficient conditions of the local $\mathcal{A}$-linear convexity of domains with smooth boundary. The conditions are obtained in terms of nonnegativity and positivity of the differential of the second order of a real function defining the domain, respectively. Moreover, the sign of the differential is determined on the boundary of the domain and on the vectors of the hyperplane tangent to the domain. These conditions are a generalization of well-known conditions of the local linear convexity of a domain with a smooth boundary, obtained by B.~Zinoviev.
Date of submission: 07 November 2020 г.

3. Durdiev D.Q., Nuriddinov J.Z. Multidimensional kernel determination problems from heat equations with memory
Status: reviewing
Abstract.
We study two problems of determining the kernel of the integral terms in a parabolic integro-differential equation. In the first problem the kernel depends on time $t$ and $x=(x_1, ..., x_n)$ spatial variables in the multidimensional integro-differential equation of heat conduction. In the second problem the kernel it is determined from one dimensional integro-differential heat equation with a time-variable coefficient of thermal conductivity. In both cases it is supposed that the initial condition for this equation depends on a parameter $y=(y_1, ..., y_n)$ and the additional condition is given with respect to a solution of direct problem on the hyperplanes $x=y.$ It is shown that if the unknown kernel has the form $k(x, t)=\sum_{i=o}^N a_i(x)b_i(t),$ then it can be uniquely determined.
Date of submission: 23 November 2020 г.

4. Juraev F.M., Islomov B. Local boundary value problems for a loaded equation of parabolic-hyperbolic type, degenerating inside the domain
Status: reviewing
Abstract.
As we know, boundary value problems for non-degenerate equations of hyperbolic, parabolic, hyperbolic-parabolic and elliptic-hyperbolic type was studied at the beginning of the 21 st century. In recent years, this direction has been intensively developed and it has been clarified, that very important problems of mathematical physics and biology lead to boundary value problems for non-degenerate loaded partial differential equations. Fact, that the boundary value problems for a degenerate loaded mixed-type equation of the second order have not been studied previously. This is due to, that, first of all, to the lack of representation of the general solution for such equations; on the other hand, such problems are reduced to little-studied integral equations with a shift. Based on the above, this work is devoted to formulation and investigation of local boundary value problems for a loaded parabolic-hyperbolic equation, degenerating inside the domain. In this work, a new approach is found for an obtaining representation of the general solution for a degenerate loaded mixed type equation. The uniqueness of solution of the formulated problems is proved by the method of integral energy. The existence of solutions of the investigated problems is equivalently reduced to the Fredholm and Volterra types integral equations of the second kind with a shift. The unique solvability of the obtained integral equations is proved.
Date of submission: 12 December 2020 г.

5. Khamdamov I.M. Central limit theorem for the perimeter of a convex hull generated by an inhomogeneous Poisson point process
Status: reviewing
Abstract.
This article is devoted to the study of the properties of the vertex process of convex hulls generated by independent observations of a two-dimensional random vector with a Poisson distribution inside a parabola. In this study, under the conditions that the measure of the intensity of the Poisson law behaves like a regularly varying function near the boundary of the support, a central limit theorem is obtained for the difference between the perimeter of the convex hull and the boundary of the support of the distribution. Here we apply a method developed by P. Groeneboom [4] to prove the central limit theorem for the number of vertices of a convex hull, based on martingality with the property of strong mixing of stationary vertex processes of the convex hull in the case when the support of the original uniform distribution is either a convex polygon or an ellipse.
Date of submission: 19 January 2021 г.

6. Baltaeva I.I., Urazboev G.U. Интегрирование уравнения Камассы-Холма с самосогласованным источником интегрального типа
Status: reviewing
Abstract.
In this paper, the evolution of scattering data for the spectral problem is determined, the potential of which is a solution of the Camassa-Holm equation with a self-consistent source of integral type.
Date of submission: 22 January 2021 г.

7. Testici A. MAXIMAL CONVERGENCE OF FABER SERIES IN WEIGHTED REARRANGEMENT INVARIANT SMIRNOV CLASSES
Status: reviewing
Abstract.
Let $G$\ be a simply connected domain on the complex plane $\mathbb{C}$ and let $G_{R}$, $R>1$\ be its canonical domain constructed via conformal mapping of $G^{-}:=\mathbb{C}\setminus \overline{G}$ onto $\left\{ w\in \mathbb{C}:\left\vert w\right\vert >1\right\}$. In this work, the maximal convergence of the partial sums of the Faber series in weighted rearrangement invariant Smirnov class $E_{X}\left( {\small G}_{R}% {\small ,\omega }\right)$\ are investigated where $\omega$ belongs to Muckenhoupt class of weights.
Date of submission: 24 January 2021 г.

8. To investigate on existence of a solution of the integral equation on tripled quasi-dislocated spaces and new tripled Hausdorff quasi-dislocated metric space
Status: reviewing
Abstract.
The purpose of this study is to introduce the concept of a tripled Hausdorff quasi-dislocated metric and we investigate to the existence of a solution of the integral equation by using some fixed point theorems for multi-valued mappings on on tripled quasi-dislocated spaces and new tripled Hausdorff quasi-dislocated metric space. We give some example and application of our main results.
Date of submission: 26 January 2021 г.

9. Commutativity Conditions in Pseudo-Michael algebras
Status: reviewing
Abstract.
In this paper, we first derive some specific results regarding the differentiable and entire functions in pseudo-Michael algebras. Then we show how can be applied such results in order to obtain commutativity conditions for these algebras.
Date of submission: 28 January 2021 г.

10. Abdrasheva G.K., Biyarov B.N., Zakariyeva Z.A. Non self-adjoint correct restrictions and extensions with real spectrum
Status: reviewing
Abstract.
The work is devoted to the study of the similarity of a correct restriction to some self-adjoint operator in the case when the minimal operator is symmetric. The resulting theorem was applied to the Sturm-Liouville operator and the Laplace operator. It is shown that the spectrum of a non self-adjoint singularly perturbed operator is real and the corresponding system of eigenvectors forms a Riesz basis.
Date of submission: 09 February 2021 г.

11. Sedov A.I. Prediction of multidimensional time series by method of inverse spectral problem
Status: reviewing
Abstract.
The work develops a new method for predicting time series by the inverse spectral problem. It is shown that it is possible to construct such a differential operator that its eigennumbers coincide with a given numerical sequence. The paper gives a theoretical justification of the proposed method. The algorithm for finding a solution and an example of constructing a differential operator with partial derivatives are given. In the presented work, a generalization is made into multidimensional time series.
Date of submission: 23 February 2021 г.

12. Kaverina V.K., Loboda A.V. On the degeneracy of orbits of nilpotent Lie algebras
Status: reviewing
Abstract.
In connection with the problem of describing holomorphically homogeneous real hyper\-surfaces the orbits in $\Bbb C^4$ of two families of nilpo\-tent 7-dimensional Lie algebras are discussed in this article. Like nilpotent 5-dimensional algebras of holomorphic vector fields in $\Bbb C^3$, most of the algebras considered in this article have no orbits nondegenerate in the Levi sense. In particular, the absence of such orbits was proved for the family of decomposable 7-dimensional nilpotent Lie algebras (31 algebras). At the same time, in the family of 12 indecomposable 7-dimensional nilpotent Lie algebras, each of which contains at least three abelian 4-dimensional ideals, four algebras have non-degenerate orbits. For the two algebras these hypersurfaces are holomorphically equivalent to the quadrics, and for two others to the nonspherical generalizations (to the case of 4-dimensional space) of the well-known Winkelmann surface. All the orbits of algebras from the second family admit tubular realizations.
Date of submission: 02 Mart 2021 г.

13. Mukminov T.F., Khabirov S.V. Simple waves of conic motions
Status: reviewing
Abstract.
Модели сплошной среды газодинамического типа допускают 11-мерную алгебру Ли группы Галилея, расширенную равномерным растяжением всех независимых переменных. Объектом исследования является построение подмоделей цепочки вложенных подалгебр размерностей от 1 до 4, описывающие конические движения газа. Для выбранной цепочки найдены согласованные инварианты в цилиндрической системе координат. На их основе получены представления инвариантного решения для каждой подмодели из цепочки. Подстановкой их в систему уравнений газовой динамики получены вложенные инвариантные подмодели рангов от 0 до 3. Доказано, что решения подмодели, построенной по подалгебре большей размерности, будут являться решениями подмоделей, построенных по подалгебрам меньших размерностей. Из выбранной цепочки рассмотрена 4-х мерная подалгебра, производящая нерегулярные частично инвариантные решения ранга 1 дефекта 1 в цилиндрических координатах. В газовой динамике такие решения называются простыми волнами. Изучена совместность соответствующей подмодели с помощью системы альтернативных предположений, получаемых из уравнений подмодели. Получены решения, зависящие от произвольных функций, а также частные решения, которые могут быть инвариантными относительно подалгебр, вложенных в рассматриваемую подалгебру, но не обязательно из рассматриваемой цепочки.
Date of submission: 05 Mart 2021 г.

14. Atanov A.V. Orbits of decomposable 7-dimensional Lie algebras with $\mathfrak{sl}(2)$ subalgebra
Status: reviewing
Abstract.
Motivated by the problem of holomorphic classification of (locally) homogeneous real hypersurfaces in $\mathbb{C}^4$, we consider orbits of action of one family of 7-dimensional Lie algebras. Each of these Lie algebras is the direct sum of $\mathfrak{sl}(2)$ and a 4-dimensional Lie algebra. Moreover, all the considered 7-dimensional Lie algebras have at most 3-dimensional abelian subalgebras. Using the technique of simultaneous straightening of vector fields, we describe all Levi-nondegenerate ho\-lo\-morphi\-cal\-ly homogeneous real hypersurfaces that are the orbits of the considered 7-dimensional Lie algebras in $\mathbb{C}^4$. Many of these orbits are tubular, potential local equivalence of other orbits to tubes requires further investigation. The sphericity property was studied for one family of orbits.
Date of submission: 02 Aprel 2021 г.

15. Zhuikov K.N., Savin A.Yu. Eta-invariant for parameter-dependent families with periodic coefficients
Status: reviewing
Abstract.
On a closed smooth manifold, we consider operator families equal to linear combinations of parameter-dependent pseudodifferential operators with periodic coefficients. For this class of families, we introduce the $\eta$-invariant (of Atiyah-Patodi-Singer type) as a regularized winding number. To this end, certain regularizations for the trace of the operator and the integral are introduced. Further, we establish main properties of the $\eta$-invariant and present a formula for the variation of the $\eta$-invariant as the family changes.
Date of submission: 21 Aprel 2021 г.

16. Generalized Hausdorff operator on Hardy spaces of the unit disk
Status: reviewing
Abstract.
In this paper, we give brief idea about generalized Hausdorff matrix act as a operator on Hardy spaces of the unit disk. Under certain conditions on $\mu$ a positive Borel measure on $(0,1],$ we prove the operator is bounded linear on $H^p(\mathbb{\mathbb{D}}),$ for different cases of $p.$
Date of submission: 04 May 2021 г.

17. Univalence and Boundedness Stipulations for Fractional Integrodifferential Operator via Pre-Schwarzian Derivatives
Status: reviewing
Abstract.
Complex-valued regular functions that are normalized in the open unit disk are vastly studied. The current study introduces a new fractional integrodifferential (non-linear) operator. Based on the pre-Schwarzian derivative, certain appropriate stipulations on the pa- rameters included in this constructed operator to be univalent and bounded are investigated and determined.
Date of submission: 05 May 2021 г.

18. Култураев Д.Ж., Эшкабилов Ю.Х. О дискретном спектре одного двухчастичного решетчатого гамильтониана
Status: reviewing
Abstract.
The discrete spectrum of one two-particle Hamiltonian $Q(\varepsilon), \ \varepsilon>0$ on the lattice $\mathbb{Z^{\nu}}\times\mathbb{Z^{\nu}}$ is studied. In the case $\nu=1,2$ for all $\varepsilon>0$ , the existence of an infinite number of negative eigenvalues of the Hamiltonian $Q(\varepsilon)$ is proved. In the case of $\nu\geq3$ , it is proved that for sufficiently small $\varepsilon$ , the Hamiltonian $Q(\varepsilon)$ absent a negative eigenvalue.
Date of submission: 21 May 2021 г.

19. Глазатов V.A., Sakbaev V.Zh. Меры на гильбертовом пространстве, инвариантные относительно гамильтоновых потоков
Status: reviewing
Abstract.
В настоящей статье исследуются гамильтоновы потоки в наделенном симплектической структурой вещественном сепарабельном гильбертовом пространстве. Исследованы меры на гильбертовом пространстве, инвариантные относительно потоков вполне интегрируемых гамильтоновых систем, и позволяющие описывать гамильтоновы потоки в фазовом пространстве посредством унитарных групп в пространстве квадратично интегрируемых по инвариантной мере функций. Введенные инвариантные относительно вполне интегрируемых потоков меры применяются к изучению модельных линейных гамильтоновых систем (гиперболических осцилляторов), допускающих особенности типа неограниченного возрастания за конечное время кинетической энергии. Благодаря такому подходу решения уравнений Гамильтона, допускающие особенности, могут быть описаны посредством соответствующей фазовому потоку унитарной группы в пространстве квадратично интегрируемых функций на расширении фазового пространства.
Date of submission: 27 May 2021 г.

20. Common fixed points for $\alpha - \psi - \phi$-contractions in generalized tripled metric space with application in Lebesgue integral
Status: reviewing
Abstract.
In this paper, by using fixed point techniques, we establish some common fixed point theorems for mappings satisfying an $\alpha - \psi - \phi$-contractive condition in generalized tripled metric space. Finally, we give an example to illustrate our main outcome.
Date of submission: 17 June 2021 г.

21. An Approximate Model for Total Amount of Non-life Insurance Claims using Generalized Gamma Distribution and $H$-function
Status: reviewing
Abstract.
This article proposes an analytical method to approximate the probability density function (PDF) and the cumulative distribution function (CDF) of the total amount of non-life claims to be paid by the insurer over a financial period considered. The individual claims amounts are independent positive random variables following the generalized gamma distribution (GGD) and distributed in a non-identical manner. The proposed analytical method is based on the Fox H-function which has several implementations available in the literature. The method, thus developed, has shown its effectiveness both in terms of the result obtained (compared to the Monte- Carlo method), and in terms of simplicity (easily accessible for the most common distributions of the amount of claims). The resulting PDF expression can be directly used to estimate the technical benefit, total cost, and ruin probability of the non-life insurance company.
Date of submission: 20 June 2021 г.

22. Leontiev V.L. The Fourier method associated with orthogonal splines in parabolic initial-boundary value problem for region with a curvilinear boundary
Status: reviewing
Abstract.
In a parabolic initial boundary-value problem for a region with a curvilinear boundary, we study the algorithm of the Fourier method associated with the use of orthogonal splines [1]. The sequence of finite generalized Fourier series formed by the algorithm of the method converges at each moment of time to the exact solution of the problem -- the infinite Fourier series. With an increase in the number of grid nodes in the considered region with a curvilinear boundary, the structure of the finite Fourier series approaches the structure of the infinite Fourier series, which is an exact solution to the initial boundary value problem. The method provides arbitrarily accurate approximate analytical solutions to the problem in the form of orthogonal series - generalized Fourier series, opens up new possibilities of the classical Fourier method.
Date of submission: 23 June 2021 г.

23. Boundary value problems for parabolic equations with involution
Status: reviewing
Abstract.
The paper consists of two parts. The first one studies the solvability of a nonlocal boundary value problems (including problems with integral conditions) for linear parabolic equations with time variable involution in lower terms. Existence and uniqueness of regular solutions (having all required generalized Sobolev derivatives) are proved. In the second part we study some spectral problems for parabolic equations with involution. In particular we discuss the influence of low order terms on the solution uniqueness.
Date of submission: 24 July 2021 г.

24. Nikonorova R.F. Invariant solutions on 4-dimensional subalgebras with a projective operator for the gas dynamics equations
Status: reviewing
Abstract.
We consider the gas dynamics equations with the state equation of the monatomic gas. The equations admits a group of trans- formations with a 14-dimensional Lie algebra. We consider 4-dimensional subalgebras containing the projective operator from the optimal system of subalgebras. The invariants of the basic operators are computed. Eight simple invariant solutions of rank 0 are obtained. Of these, 4 physical solutions specify gas motion with a linear velocity field and one physical solutions with a nonlinear velocity field. All solutions have variable entropy except one. The motion of gas particles as a whole is constructed for it. The solutions obtained have a density singularity on a constant or moving plane: boundary with a vacuum or wall.
Date of submission: 27 July 2021 г.

25. Kakushkin S.N. Finding eigenfunctions of perturbed discrete semi-bounded operators given on compact graphs
Status: reviewing
Abstract.
The article presents a new numerical method for finding eigenfunctions of perturbed discrete semi-bounded operators given on compact graphs. Theorems are given according to which the eigenfunctions of an unperturbed problem form a basis in the energy space under consideration, and the convergence of the method to the exact solution is also proved. A convenient, computationally efficient method for finding the coefficients of an approximate solution is proposed. An example of a computational experiment of applying the described method to find the eigenfunctions of a perturbed problem given on a three-edge compact graph is given.
Date of submission: 08 Avgust 2021 г.

26. Mkrtchyan A.J. Multiple power series continuability into a sectorial domain by means of interpolation of coefficients
Status: reviewing
Abstract.
We consider the problem of continuability into a sectorial domain for multiple power series centered at the origin of $\mathbb{C}^n$. The condition of the mentioned continuability is given in terms of entire function interpolating the coefficients of power series.
Date of submission: 15 September 2021 г.

27. On the speed of approximation in the classes of $\overline{\psi}$-integrals
Status: reviewing
Abstract.
The concept of the $\overline{\psi}$-integrals introduced by A. I. Stepanets who is an Ukrainian mathematician has brought a new perspective in the theory of Fourier series, in especially approximation theory. The main objective of this study is to get the speed of approach to the functions of the class $\overline{\psi}$-integrals by generalized Zygmund sums, Woronoi$-$N\"{o}rlund and Riesz means, responding to the solution of the Kolmogorov$-$Nikol'skii problem under the uniform norm.
Date of submission: 18 September 2021 г.

28. Khuddush K., Prasad K.R. Positive Radial Solutions for an Iterative System of Nonlinear Elliptic Equations on an Exterior Domain
Status: reviewing
Abstract.
This paper deals with the existence of single and coupled positive radial solutions to the iterative system of nonlinear elliptic equations of the form \begin{aligned} &\triangle{\mathtt{z}_{{\mathtt{j}} }}-\frac{(\mathtt{N}-2)^2r_0^{2\mathtt{N}-2}}{\vert x\vert^{2\mathtt{N}-2}}\mathtt{z}_{\mathtt{j}} +\upchi(\vert x\vert)\mathtt{g}_{{\mathtt{j}} }(\mathtt{z}_{{\mathtt{j}} +1})=0,~x\in\Omega,\\ &\hskip1.85cm \mathtt{z}_\mathtt{j}\vert_{\partial\Omega}=0,~ \lim_{\vert x\vert\to\infty}\mathtt{z}_\mathtt{j}(x)=0, \end{aligned} where $\mathtt{j}\in\{1,2,3,\cdot\cdot\cdot,\mathtt{n}\},$ $\mathtt{z}_1=\mathtt{z}_{\mathtt{n}+1},$ $\Delta\mathtt{z}=\mathtt{div}(\nabla\mathtt{z}),$ $\mathtt{N}>2,$ $\Omega=\{ \mathtt{z}\in\mathbb{R}^\mathtt{N}|~\vert\mathtt{z}\vert>r_0\},$ $\upchi=\prod_{i=1}^{m}\upchi_i,$ each $\upchi_i:(r_0,+\infty)\to(0,+\infty)$ is continuous, $r^{\mathtt{N}-1}\upchi$ is integrable, and $\mathtt{g}_\mathtt{j}:[0,+\infty)\to\mathbb{R}$ is continuous, by an application of Krasnoselskii and Avery-Henderson fixed point theorems in a Banach space. Further, we also establish uniqueness of solution to the addressed system by using Rus's theorem in a complete metric space.
Date of submission: 04 October 2021 г.

29. Klyachin A.A. On the $C^1$-convergence of piecewise polynomial solutions of a fourth-order variational equation
Status: reviewing
Abstract.
In this paper a fourth-order variational equation is considered. For this equation, the concept of a piecewise polynomial solution on a triangular grid is introduced. A theorem on the existence and uniqueness of such a solution is proved, as well as conditions for the convergence of piecewise polynomial solutions of a fourth-order equation are obtained when the fineness of the grid partition tends to zero.
Date of submission: 05 October 2021 г.

30. Allahverdiev B.P., Tuna H. Singular Hahn-Hamiltonian systems
Status: reviewing
Abstract.
We study a Hahn-Hamiltonian system in the singular case. For this system, the Titchmarsh-Weyl theory is established.
Date of submission: 12 October 2021 г.

31. Balkizov Zh.A. Внутреннекраевые задачи со смещением для одного смешанно-гиперболического уравнения второго порядка
Status: reviewing
Abstract.
В работе исследованы внутреннекраевые задачи со смещением для одного смешанно-гиперболического уравнения второго порядка, состоящего из волнового оператора в одной части области и вырождающегося гиперболического оператора первого рода в другой части.
Date of submission: 27 October 2021 г.

32. Grishin S.V. Applying generating functions to problems of random walk
Status: reviewing
Abstract.
We consider a problem of defining the first positive half-line achievement time for a homogenous discrete integer random walk on a line. More precisely, the object of our research is a graph of the generating function of the mentioned random value. For the random walk with the maximal positive increase value 1 the equation is derived on the implicit generating function, which implies rationality of the inverse generating function. In this case, mathematical expectation and dispersion are computed for the first positive half-line achievement time. A general method of deriving systems of equations for the first positive half-line achievement time for a homogenous discrete integer random walk on a line is described. The algebraic equation is derived on the implicit generating function for the random walk with increase values -1, 0, 1, 2. The corresponding flat algebraic curve is shown to be rational. Several general propositions are formulated and proved about the genera$\-$ting function of the first positive half-line achievement time for a homoge$\-$nous discrete integer random walk on a line.
Date of submission: 29 October 2021 г.

33. Dexkonov J.D. On $(k_0)$ - translation invariant and $(k_0)$ - periodic Gibbs measures for the Potts model on the Cayley tree
Status: reviewing
Abstract.
The Potts model on the Cayley tree is considered. For the ferromagnetic Potts model, in the case $k_0 = 3$, existence of $(k_0)$ - translation-invariant (i.e., $(3)$ - translation-invariant) Gibbs measures is proved. For the antiferromagnetic Potts model, also in the case $k_0 = 3$, existence of $(k_0)$ - periodic ($(3)$ - periodic) Gibbs measures on the Cayley tree is proved.
Date of submission: 10 November 2021 г.

34. Akishev G. On estimates of the order of the best $M$ - term approximations of functions of several variables in the anisotropic Lorentz - Karamata space
Status: reviewing
Abstract.
The article consider the anisotropic Lorentz-Karamata space of pe\-rio\-dic functions of several variables and the Nikol'skii-Besov class in this space. The order-sharp estimates are established for the best $M$ - term trigonometric approximations of functions from the Nikol'skii-Besov class in the norm of another Lorentz-Karamata space.
Date of submission: 11 November 2021 г.

35. Adukov V.M. Normalization of the Wiener--Hopf Factorization for $2\times 2$ Matrix Functions and Its Application
Status: reviewing
Abstract.
We fill the gap that exists in the general theory of Wiener - Hopf factorization of matrix functions. It is known that the factorization factors are not uniquely, and in the general case, there are no ways to normalize the factorization that guaran\-tee its uniqueness. In the paper, the notion of $P$-normalized factorization is introduced. This normaliza\-tion ensures the uniqueness of the Wiener--Hopf factorization and makes it possible to find the Birkhoff factorization. For matrix functions of the second order, it is shown that the factorization of any matrix function can be reduced to $P$-normalized factorization. All possible types of such factorizations are described, conditions are obtained under which this normalization exists, and the form of factoriza\-tion factors for this type of normalization is found. The stability of the $P$-normalization under a small perturbation of the original matrix function is studied. The results are applied to refine Shubin's theorem on the continuity of factorization factors and to obtain explicit estimates of the absolute errors of factors for an approximate factorization.
Date of submission: 12 November 2021 г.

36. Kapkaev N.V., Tikhov M.S. Negative binomial regression in dose-effect relationships.
Status: reviewing
Abstract.
This paper is concern to the problem of estimating the dis\-tribution function and its quantiles in the dose-effect relationships with nonparametric negative bino\-mial regression in the dose-effect relationships. A kernel-based estimators of the distribution function are proposed, of which kernel is weighted by the negative binomial random variable at each covariate. Nonparametric quantiles estimators obtained by inverting a kernel estimator of the distribution function are offered. It is shown that the asymptotic normality of this bias-adjusted estimator holds under some regularity conditions. The proposed estimators are compared through their asymptotic MSEs. From these considerations advantages of our estimating scheme are repor\-ted.
Date of submission: 18 November 2021 г.

37. Levenshtam V.B. Reconstruction of a rapidly oscillating junior coefficient and the source of a hyperbolic equation from the partial asymptotics of the solution
Status: reviewing
Abstract.
The Cauchy problem is considered for a one-dimensional hyperbolic equation, the junior coefficient and the right part of which oscillate in time with a high frequency, and the amplitude of the junior coefficient is small. The question of recovering these high oscillating functions from the partial asymptotics of the solution given at some point in space is investigated.
Date of submission: 27 November 2021 г.

38. Zhukova N.I., Sheina K.I. The structure of foliations with integrable Ehresmann connection
Status: reviewing
Abstract.
We investigate foliations of arbitrary codimension $q$ with integrable Ehresmann connection on $n$-dimensional smooth manifolds. The category of Сartan foliations is considered, where isomor\-phisms preserve not only foliations, but also their Ehresmann connection. It is shown that this category it can be considered as a category of bifoliations covered by products. The concept of a canonical bifoliation is defined and it is proved that any foliation $(M, F)$ with integrable Ehresmann connection is isomorphic to some canonical foliation. The notion of a struc\-t\-ural group of $(M, F)$ is introduced. The category of triples is const\-ructed and its equivalence to the category of foliations with integrable Ehresmann connections is proved. Thus, the classification of foliations with integrable Ehresmann connections reduces to the classification of associated actions of discrete groups diffeomorphisms on the product of respective manifolds. Classes of foliations with an integrable Ehresmann connections are indicated. The pplication to $G$-foliations is considered.
Date of submission: 30 November 2021 г.

39. Haliullin S.G. Ultraproducts of quantum mechanical systems
Status: reviewing
Abstract.
The article discusses stochastic properties of so-called quan\-tum mechanical systems in a rather abstract form. Such systems (structu\-res) are found in probability theory, in the theory of operator algebras, in the theory of topological vector spaces. Ultraproducts of sequences of these structures are also considered and some properties of such ultrapro\-ducts are investigated.
Date of submission: 30 November 2021 г.

40. Lipacheva E.V. Trivial extensions of semigroups and semigroup $C^*$-algebras
Status: reviewing
Abstract.
The paper deals with the normal extensions of semigroups and the reduced semigroup $C^*$-algebras corresponding to the semigroups which constitute those extensions. We study the question on functoriality for morphisms of semigroup $C^*$-algebras. This is the question on the existence of the canonical embedding of semigroup $C^*$-algebras which is induced by the embedding of semigroups. It is also shown that if a semigroup $L$ is a trivial extension of a semigroup $S$ by means of a finite group then there exists a structure of a free Banach module over the reduced semigroup $C^*$-algebra for $S$ on the underlying space of the semigroup $C^*$-algebra for $L$.
Date of submission: 02 December 2021 г.

41. Gekkieva S.Kh., Kerefov M.A., Nakhusheva F.M. Local and nonlocal boundary value problems for the generalized Aller - Lykov equation
Status: reviewing
Abstract.
The paper investigates boundary value problems for the Aller~-- Lykov inhomogeneous moisture transfer equation with variable coefficients Riemann – Liouville time fractional derivative. The considered equation is a generalization for the Aller~-- Lykov equation obtained by introducing the concept on the rate of change in fractal fluid, which explains the presence of flows against the moisture potential. Using the method of energy inequalities for local and nonlocal problems, a priori estimates are obtained in terms of the Riemann – Liouville fractional derivative implying the uniqueness of the solution to the considered boundary value problems and the solution stability with respect to the right-hand side and initial data.
Date of submission: 02 December 2021 г.

42. Gumerov R.N., Khazhin R.L. On divisible quantum dynamical mappings
Status: reviewing
Abstract.
The paper is concerned with quantum dynamical mappings, which are also called quantum processes, with values in a set of comp\-letely positive trace-preserving linear operators. We consider bijective completely positive divisible quantum processes. It is shown that a com\-pound quantum process constructed by means of two such processes, which satisfy a commutativity condition, is comp\-letely positive divisible as well. Endowing a set of quantum channels with the norm topology we consider continuous pro\-cesses and evolutions. We prove that a quantum process generates a continuous completely positive evolution. Examples of quantum processes are given to illustrate the concepts under the consideration and the results on them.
Date of submission: 03 December 2021 г.

43. Mukhsinov Y.M. About one differential game of neutral type with integral restrictions in Hilbert space
Status: reviewing
Abstract.
In the field of the theory of differential games, when a game is specified in a finite-dimensional space, fundamental work was carried out by academicians L.S. Pontryagin and N.N. Krasovsky. The works of N.N. Krasovsky and his students are devoted mainly to positional games. And in the works of L.S. Pontryagin and his students, the differential game is considered separately from the point of view of the pursuer and from the point of view of the evader, which inevitably links the differential game with two different problems. In the future, it is important to investigate games in infinite-dimensional spaces, because many important problems of optimal control, under conditions of conflict or uncertainty, controlled by distributed systems whose motion is described by integral-differential equations and partial differential equations, can be formulated and studied as differential games in suitable Banach spaces. In this paper, in a Hilbert space, we consider the pursuit problem in the sense of L.S. Pontryagin for a quasilinear differential game, when the dynamics of the game is described by a functional differential equation of neutral type. An auxiliary lemma and four theorems on sufficient conditions for the solvability of the pursuit problem are proved.
Date of submission: 07 December 2021 г.

44. Proskurnin I.A. Minimal morsifications of invariant functions.
Status: reviewing
Abstract.
We consider a problem of producing a deformation of a function in two variables with the smallest possible number of real critical points. The function in quaestion is considered to be invariant with respect to a finite group action. We construct a morsification with the smallest possible number of critical point permitted by equivariant topology for every semihomogenous invariant function.
Date of submission: 07 December 2021 г.

45. Kudaybergenov К.К., Nurjanov B.O. Partial orders on $\ast$-regular rings
Status: reviewing
Abstract.
We introduce some new partial orders on $\ast$-regular rings. Let $\mathcal{A}$ be a $\ast$-regular ring and let $a,b \in \mathcal{A}.$ We define on $\mathcal{A}$ the following three partial orders: $a\prec_s b \Longleftrightarrow b=a+c,\, a \perp c;$ $a\prec_l b \Longleftrightarrow l(a)b=a;$ $a\prec_r b \Longleftrightarrow br(a)=a.$ If $\mathcal{A}$ is a $\ast$-regular algebra with a rank-metric $\rho,$ then the order topologies associated with these partial orders are stronger than the topology generated by the metric $\rho.$ We also consider the restrictions of these partial orders on the subsets of projections, unitaries and partial isometries of the $\ast$-regular algebra $\mathcal{A}.$
Date of submission: 25 December 2021 г.

46. Inverse source problem for the heat equation associated with the singular Laplacian and Dunkl operator
Status: reviewing
Abstract.
The purpose of this paper is to establish the solvability results to direct and inverse problems for the heat equation associated with the singular Laplacian and Dunkl operator. We prove existence and uniqueness results for the solution of the direct and inverse problems. Also, some explicit formulas are derived for the considered direct and inverse problems.
Date of submission: 30 December 2021 г.

47. Ishankulov T., Fozilov D., Kholmurzaev Kh. Continuation of bianalytic functions of several complex variables
Status: reviewing
Abstract.
It is considered the problem of continuation of bianalytic function of several complex variables in domain by it values and values of first derivatives on a part of boundary. The uniqueness theorem is proved, the conditionally stability estimate is obtained, Carleman's formula is constructed.
Date of submission: 31 December 2021 г.