# 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. Bakirov N.K. Duhamel Banach algebra structure of some space and related topics
Status: reviewing
Abstract.
Let ? be a …xed complex number, and let ? be a simply connected region in complex plane C that is starlike with respect to ? 2 ?: We de…ne some Banach space of analytic functions on ? and prove that, with the ?-Duhamel product de…ned by ? f ~ ? g ? (z) := d dz z Z ? f(z + ? ? t)g(t)dt; this space is a Banach algebra. We prove that its maximal ideal space consists of the homomorphism h ? de…ned by h ? (f) = f (?): Moreover, we describe in terms of ?-Duhamel operators the extended eigenvectors of the non-de…nite integration operator J ? ; J ? f (z) = z R ? f(t)dt: Some other related questions are also discussed.
Date of submission: 21 December 2020 г.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. 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.

15. The proof of the RH with the integral representative of the zeta function
Status: reviewing
Abstract.
We will put forth a proof for the RH by using one of the integral representatives of the Riemann zeta function and by using L’Hopital rule
Date of submission: 07 Mart 2021 г.

16. Nazarov S.A. Волны Рэлея для эллиптических систем в областях с периодическими границами
Status: reviewing
Abstract.
Рассмотрены формально самосопряженные эллиптические системы дифференциальных уравнений в частных производных, порождающие формально положительные операторы и обладающие полиномиальным свойством. Найдены достаточные условия, обеспечивающие существование поверхностных волн Рэлея в задаче Неймана на полупространстве с периодической границей. Приведены примеры конкретных задач математической физики, в которых полученные достаточные условия упрощаются или превращаются в критерий, а также изучены не обслуживаемые общими результатами задачи теории пластин и пьезоэлектрики, причем последняя требует серьезной модификации подхода.
Date of submission: 16 Mart 2021 г.

17. 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.

18. 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.

19. 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.

20. 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.

21. 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.

22. 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.

23. 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.

24. 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.

25. 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.

26. 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.

27. 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.

28. 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.

29. Cluster analysis on various cluster validity indexes with average linkage method and Euclidean distance
Status: reviewing
Abstract.
The aim of this study is to investigate and explain the differences in the use of different indicators of cluster validity in the application of KPR client subgroups at Bank X Malan City using the associative Euclidean approach, mean and distance. Cluster analysis is used to classify samples into relatively homogeneous subsamples called clusters. The sub-instances of each group tend to be similar and very different (dissimilar) from objects in other clusters. In cluster analysis, different links can be used to form a cluster. The association diagram is includes simple, full, and medium links. This study considers of the application of the mean association method to seven types of cluster validity indices. Measure the distance of each link in this study using the Euclidean distance. Determining the number of clusters with valid indexes from different clusters will inevitably yield different results. In this study, we want to examine the impact of using a cluster validity index on the problem of aggregating Bank X's customers.
Date of submission: 13 Avgust 2021 г.

30. Existence and uniqueness solution for fractional Volterra equation with fractional anti-periodic boundary conditions
Status: reviewing
Abstract.
In this paper, some existence results for a first kind Volterra differential equation of fractional order with fractional anti-periodic boundary conditions are presented. Our results are based on some fixed point theorems, Leray–Schauder degree theory.Some illustrative examples are discussed.
Date of submission: 20 Avgust 2021 г.

31. Abduragimov G.E. On the existence and uniqueness of a positive solution to a boundary value problem for a nonlinear functional fractional order differential equation
Status: reviewing
Abstract.
В настоящей статье рассматривается двухточечная краевая задача для одного нелинейного функционально - дифференциального уравнения дробного порядка на отрезке $[0,1]$ с нулевыми условиями Дирихле на границе. С помощью специальных топологических средств доказано существование единственного положительного решения рассматриваемой задачи. Приведен пример, иллюстрирующий выполнение достаточных условий, обеспечивающую однозначную разрешимость поставленной задачи. Полученные результаты являются продолжением исследований автора, посвященным вопросам существования и единственности положительных решений краевых задач для нелинейных функционально - дифференциальных уравнений.
Date of submission: 23 Avgust 2021 г.

32. Khasanov A.B., Yakhshimuratov A.B. Интегрирование модифицированного уравнения Кортевега-де Фриза с интегральным источником в классе периодических функций
Status: reviewing
Abstract.
In this work, the method of the inverse spectral problem is applied to the integration of the modified Korteweg-de Vries equation with a self-consistent integral source in the class of periodic functions. Important corollaries are obtained about the analyticity and the period of the solution with respect to the spatial variable.
Date of submission: 23 Avgust 2021 г.