# Editorial backlog

1. Begmatov A.X., Ochilov Z.X. Uniqueness and existence of solving problems of Integral Geometry of Volterra Type with a Weight Function of a Special Type
Status: reviewing
Abstract.
В работе рассматривается новый класс задач интегральной геометрии вольтеровского типа с весовой функцией специального вида. Доказана теорема единственности, выведены формулы обращения, получены оценки устойчивости в пространствах Соболева, тем самым показана слабая некорректность поставленной задачи. Приводятся формулировка и доказательство теорема существования решения задачи восстановления функции по семейству ломанных с весовой функцией специального вида в полосе.
Date of submission: 27 February 2020 г.

2. Abdulaev O.H., Islomov B. About the non-local problems for a third order equation with operator Caputo and nonlinear loaded term
Status: reviewing
Abstract.
This work devoted to prove of unique solvability of solution of the non-local problems with integral gluing condition for the third order equation with parabolic-hyperbolic operator involving Caputo derivatives and non-linear loaded terms. Solvability of the investigated problem was proved by the method of integral equations.
Date of submission: 01 July 2020 г.

3. Abdullayev J.S., Khudayberganov G. The boundary Morera theorem for the domain ${{\tau }^{+}}\left( n-1 \right)$
Status: reviewing
Abstract.
In this article proved the Morera boundary theorem for the domain ${{\tau }^{+}}\left( n-1 \right)$. An analog of Morera's theorem is given, in which integration is carried out along the boundaries of analytic disks. For this, we use the automorphisms ${{\tau }^{+}}\left( n-1 \right)$ and the invariant Poisson kernel in the domain ${{\tau }^{+}}\left( n-1 \right)$.
Date of submission: 01 July 2020 г.

4. Lebedev P.D., Uspenskii A.A. About the Structure of Singularity of a Minimax Solution of a Dirichlet Problem for the Eikonal Type Equation with the Discontinuous Smoothness of the Curvature of the Target Set
Status: reviewing
Abstract.
The origin of nonsmooth singularities in the minimax (generalized) solution of the Dirichlet problem for an eikonal equation is due to the existence of pseudo-vertices — singular points of the boundary of the boundary set. Finding pseudo-vertices is the first step in the procedure for constructing a singular set for solving a boundary value problem. Finding these points requires the construction of local solutions of an equation such as the golden ratio, which establishes a connection between the eikonal operator and the geometry of the boundary set. Moreover, the problem of identifying local solutions of the equation is associated with the problem of finding fixed points of mappings formed by local reparameterization of the boundary of the boundary set. In this paper, we obtain the necessary conditions for the existence of pseudovertices with a break in the smoothness of curvature of a parametrically specified boundary of a boundary set. The conditions are written in various equivalent forms. In particular, a representation is obtained in the form of a convex combination of one-sided derivatives of curvature. Formulas are presented for the coefficients of a convex combination, which are determined by markers — scalar characteristics of pseudovertices. For markers, a form of the algebraic equation is found whose roots they are. An example of numerical and analytical construction of a minimax solution to the Dirichlet problem is presented, illustrating the effectiveness of the developed methods for solving nonsmooth boundary value problems.
Date of submission: 02 July 2020 г.

5. Artemov M.A., Babkina Yu.N. The first boundary value problem for equations describing flows of a nonlinear viscoelastic fluid in a bounded domain
Status: reviewing
Abstract.
We study the first boundary value problem for a mathematical model describing the steady motion of a viscoelastic fluid with variable viscosity, depending on the shear rate, inside a bounded three-dimensional (or two-dimensional) domain with sufficiently smooth boundary. The concept of a weak solution is introduced. The regularization method is used to formulate this problem in the form of an operator equation with a continuous nonlinear operator satisfying the α-condition. By using of the theorem on the solvability of equations with α-operators and the passage to the limit, the existence of at least one weak solution of the problem is proved and an energy-type estimate for the vector velocity function is derived.
Date of submission: 19 July 2020 г.

6. Zheltukhin K., Zheltukhina N.A. On the discretization of Darboux Integrable Systems admitting second-order integrals
Status: reviewing
Abstract.
The discretization of Darboux integrable systems admitting two first and two second order integrals is considered. The obtained semi-discrete systems possess $n$- integrals that coincide with $x$- or $y$-integrals of the original continuous systems. New examples of semi-discrete Darboux integrable systems are derived.
Date of submission: 03 Avgust 2020 г.

7. Chernov A.V. On differentiation of a functional in the problem of parametric coefficient optimization in the semilinear global electric circuit equation
Status: reviewing
Abstract.
For the problem of parametric optimization with respect to an integral criterion of the coefficient and the right-hand side of the semilinear global electric circuit equation, formulas for the first partial derivatives of the cost functional in control parameters are obtained. In such a form the problem of reconstructing unknown parameters of the equation from data of observation by local sensors can be represented. In the paper it is generalized analogous result obtained by the author formerly for the case of linear global electric circuit equation. But experts believe that right-hand side (treated as the volumetric density of external currents) of the equation actually depends on the gradient (with respect to the collection of space variables) of the unknown electric potential function. In this connection the necessity arises to investigate the case of semilinear equation. Here, we use the conditions for preservation of global solvability of the semilinear global electric circuit equation which we have obtained formerly. The mathematical novelty of presented research is due to the fact that, unlike the earlier-studied linear case, now the right-hand side depends (nonlinearly) on the state (depending, in turn, on the control parameters). This more complicated nonlinear dependence of the state on the control parameters has required, in particular, the development of a special technique to estimate the additional terms arising in the increment of solutions of the controlled equation.
Date of submission: 28 Avgust 2020 г.

8. Komarov M.A. Rate of convergence of a one class of differentiating sums
Status: reviewing
Abstract.
We consider the formula for differentiation of analytic func\-ti\-ons: $azf'(z)=nf(0)-\sum_{k=1}^n f(\lambda_k z)+O(z^{n+1})$, where $a>0$, $\lambda_k=\lambda_{n,k}(a)$, $n=1,2,\dots$. As $n\ge 3\alpha$ ($\alpha:=\max\{a;1\}$), we find an estimate of the rate of convergence of the differentiating sums to $nf(0)-a zf'(z)$ in the disk $|z|<\exp(-3\sqrt{v}-2v)$, $v:=\alpha/(n+1)$.
Date of submission: 02 September 2020 г.

9. Norjigitov A.F., Sharipov O.S. Law of large numbers for weakly dependent random variables with values in $D[0,1]$
Status: reviewing
Abstract.
The paper is devoted to the law of large numbers for the random variables with values in D[0,1] space. The law of large numbers is well well-studied for the sequences of independent D[0,1]-valued random variables. Our main goal is to prove the law of large numbers for the weakly dependent random variables with values in D[0,1] space. In the paper the law of large numbers for $\rho_{m}$-mixing sequences of $D[0,1]$-valued random variables are proved.
Date of submission: 27 September 2020 г.

10. 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.

11. Fedotov A.I. Justification of the Galerkin and collocations methods for one class of singular integro-differential equations on the interval
Status: reviewing
Abstract.
The Galerkin and collocations methods are justified for one class of singular integro-differential equations defined in the pair of the weighted Sobolev spaces. The exact solution of the considered equation is approxi\-mated by the linear combinations of the Chebyshev polynomials of the first kind. According to the Galerkin method the Fourier coeffici\-ents with respect to the second kind Chebyshev polynomials of the right-hand side and the left-hand side of the equation are equated. According to collocations method the values of the right-hand side and the left-hand side of the equation in the roots of the second kind Chebyshev polynomials are equated. The choice of the first kind Chebyshev polynomials as coordinate functions is caused by the possibility to calculate the singular integrals of the products of these polynomials and corresponding weight functions explicitly. It allows to construct simple well-convergent methods for the wide class of singular integro-differential equations on the interval $(-1,1)$. The Galerkin method is justified by the Gabdulkhaev-Kantorovich technique. The convergence of collocations method is proved by the Arnold-Wendland technique as a consequence of convergence of the Ga\-lerkin method. So the covergence of both methods is proved and their errors are estimated.
Date of submission: 01 November 2020 г.

12. Garif'yanov F.N., Strezhneva E.V. The sum-difference equation for analytic functions generated by a triangle and its applications
Status: reviewing
Abstract.
Let $D$ be a triangle and $\Gamma$ the "half" of its boundary $\partial D$. An element-wise linear total-difference equation is considered in the class of functions holomorphic outside $\Gamma$ and disappearing at infinity. The solution is sought in the form of a Cauchy-type integral over $\Gamma$ with unknown density. The boundary values satisfy the H\"{o}lder condition on any compact set from $\Gamma$ that does not contain nodes. At the most, logarithmic singularities are allowed at the nodes. To regularize the equation to $\partial D$, a piecewise linear Carleman shift is introduced. He translates each side into himself with a change in orientation. In this case, the midpoints of the sides are fixed shear points. The equation is regularized and the condition for its solvability is found. Let us consider a particular case when the number of solvability conditions can be precisely counted. Applications to interpolation problems for entire functions of exponential type are indicated. Previously, similar problems were investigated for a quadrilateral, pentagon, and hexagon.
Date of submission: 02 November 2020 г.

13. 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.

14. 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.

15. Niyozov I.E. On the solvability of the Cauchy problem for a system of the moment theory of elasticity
Status: reviewing
Abstract.
We consider the problem on the analytic continuation of the solution of the system of vibration equations in elasticity theory in a spatial domain based on the values of the solution and the stresses on part of the boundary of this domain, i.e., a Cauchy problem. The problem is ill posed. If the part of the domain on which the Cauchy data are given is real analytic, then the problem has a local solution by the Cauchy–Kovalevskaya theorem. The special structure of the vibration equation is used to obtain explicit global solvability conditions and construct approximate solutions.
Date of submission: 25 November 2020 г.

16. 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.

17. Bakirov N.K. Polynomials shrinkage estimators for a multivariate normal mean under the balanced loss function
Status: reviewing
Abstract.
In this paper we propose the shrinkage estimators of a multivariate normal mean and study their minimaxity properties using the balanced loss function. Di?erent classes of shrinkage estimators are presented. In the ?rst step, we consider estima- tors which generalize the James-Stein estimator and show that if the shrinkage function satis?es certain conditions, these estimators dominate the maximum likelihood estimator (MLE), consequently are minimax. In the second step, we deal with estimators of polynomial form and prove that if we increase the de- gree of the polynomial, then we can construct a better estimator from the one constructed previously.
Date of submission: 19 December 2020 г.

18. 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.

19. 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.

20. 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.

21. 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.

22. 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.

23. 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.

24. Rodikova E.G. О коэффициентных мультипликаторах плоских классов Привалова
Status: reviewing
Abstract.
We estimate the growth and Taylor coefficients of analytic functions from the Privalov spaces by area. Also we obtain full descriptions of coe?cient multipliers from $\tilde\Pi_q(q>0)$ to the Hardy spaces $H^p$ (0 < p ≤ +∞).
Date of submission: 31 January 2021 г.

25. Kaptsov O.V., Mirzaokhmedov M.M. General solutions of some linear equations with variable coefficients
Status: reviewing
Abstract.
In this work, we find the general solutions of some classes of linear wave equations with variable coefficients. Such equations describe vibrations of rods, acoustic waves, and also some models of gas dynamics are reduced to them. To construct solutions, Levy-type transformations are used, which are differential substitutions of the first order and their iterations. Specific examples of general solutions that depend on the derivatives of arbitrary functions are given.
Date of submission: 03 February 2021 г.

26. Avkhadiev F.G. Hardy type inequalities containing the gradient of the distance function
Status: reviewing
Abstract.
On domains of the Euclidean space we prove several new Hardy type inequalities containing the gradient of the distance from a point to the boundary of the domain. We study integral inequalities on domains satisfying some geometric properties. In particular, we consider domains with finite inradius. Our proofs have two important ingredients. The first one is connected with an approximation and a special decomposition of the domain and the second ingredient is a theorem on convergence for gradients of the distance functions of approximating domains.
Date of submission: 05 February 2021 г.

27. Bakirov N.K. A note on quasilinear elliptic systems with $L^\infty$-data
Status: reviewing
Abstract.
We prove the existence of a weak energy solution for the boundary-value problem $$−div a(x,u,Du) = f \ \ in\ \Omega,$$ $$u = 0\ on\ \partial\Omega,$$ where Ω is a bounded open domain in $R^ n$ $(n\geq 3)$ and $f\in L^\infty (Ω;R^m ).$ The existence result is proved using the concept of Young measures
Date of submission: 08 February 2021 г.

28. 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.

29. Muravnik A.B. Elliptic differential-difference equations with differently directed translations in half-spaces
Status: reviewing
Abstract.
Исследуется задача Дирихле в полупространстве для эллиптических диф\-фе\-рен\-циально-разностных уравнений с операторами, представляющими собой композиции дифференциальных операторов и операторов сдвига, действующих по пространственноподобным переменным (независимым переменным, изменяющимся на всей вещественной оси). Указанные уравнения, существенно обобщающие классические эллиптические уравнения в частных производных, возникают в разнообразных приложениях математической физики, для которых характерны нелокальные и (или) неоднородные свойства процесса или среды. В теоретическом плане интерес к таким уравнениям обусловлен тем, что они связывают между собой значения неизвестной функции (и ее производных) не в одной точке, а в разных, что делает неприменимыми многие классические методы. Для рассматриваемой задачи устанавливается классическая разрешимость или разрешимость почти всюду (в зависимости от ограничений, наложенных на граничные данные), строится интегральное представление указанного решения формулой пуассоновского типа и доказывается его равномерное стремление к нулю при стремлении времениподобной переменной (единственной независимой переменной, изменяющейся на положительной оси, ортогональной гиперплоскости граничных данных) к бесконечности. Ранее исследовались только случаи, в которых оператор сдвига действует лишь по одной пространственноподобной переменной. В настоящей работе операторы сдвига действуют по каждой пространственноподобной переменной. Для получения ядра Пуассона используется классическая операционная схема Гельфанда---Шилова: к изучаемой задаче применяется преобразование Фурье по всем пространственноподобным переменным (используется тот факт, что операторы сдвига, так же как и дифференциальные операторы, являются мультипликаторами Фурье), и исследуется полученная задача Коши для обыкновенного дифференциального уравнения (зависящего от двойственных переменных, как от параметров).
Date of submission: 10 February 2021 г.

30. Maksimov V.P. Continuous-Discrete Dynamic Models
Status: reviewing
Abstract.
Рассматриваются динамические модели с последействием в форме функционально-дифференциальных уравнений с непрерывным и дискретным временем. Приводится постановка общей задачи управления относительно заданной системы целевых функционалов и дается краткая сводка известных результатов о разрешимости этой задачи при полиэдральных точечных ограничениях на управление. В заключительном разделе представлены результаты об оценке множества достижимости при интегральных ограничениях на управление. Предлагаемый вариант синтеза непрерывных и дискретных систем основан на систематическом использовании теории абстрактного функционально-дифференциального уравнения и обладает определенными преимуществами при исследовании систем и процессов с последействием. Непрерывно-дискретные функционально-дифференциальные модели позволяют учитывать при моделировании эффекты последействия, включая случаи полной памяти, и эффекты, возникающие при учете импульсных возмущений (шоков), приводящих к скачкообразному изменению фазового состояния по компонентам с непрерывным временем.
Date of submission: 15 February 2021 г.

31. Belova A.S., Ibragimova L.S., Yumagulov M.G. Perturbation theory methods in the problem of parametric resonance for linear periodic Hamiltonian systems
Status: reviewing
Abstract.
The new formulas proposed based on the methods of the perturbation theory for linear operators in the problem of approximate construction of multipliers for linear non-autonomous periodic Hamiltonian systems that depend on a small parameter. The main attention is paid to obtaining formulas of the first approximation for perturbations of multiple definite and indefinite multipliers. The proposed formulas lead to new criteria for the Lyapunov stability of linear periodic Hamiltonian systems in critical cases. Applications to the problem of parametric resonance in fundamental resonances are considered. The results obtained are formulated in terms of the original equations and brought to effective formulas and algorithms. The effectiveness of the proposed formulas is illustrated in the problem of constructing the boundaries of the stability regions of triangular libration points of the plane bounded elliptic three-body problem.
Date of submission: 18 February 2021 г.

32. Startsev S.Ya. On Darboux non-integrability of the Hietarinta equation
Status: reviewing
Abstract.
This article employs two-point invertible transformations introduced by R.I.~Yamilov. It is proved that a difference equation on the quad-graph cannot be Darboux integrable if a transformation of the above type maps solutions of this equation into its solutions again. This implies that the generic autonomous Hietarinta equation is not Darboux integrable since the Hietarinta equation in the general case possesses the two-point invertible auto-transformations. Along the way, all Darboux integrable subcases of the Hietarinta equation are found. All of them are reduced by point transformations to already known integrable equations.
Date of submission: 21 February 2021 г.

33. 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.

34. Zaitseva N.V. Hyperbolic differential-difference equations with nonlocal potentials
Status: reviewing
Abstract.
В полуплоскости для двумерного гиперболического дифференциально-разностного уравнения, содержащего сумму дифференциального оператора и операторов сдвига по пространственной переменной, изменяющейся на всей вещественной оси, (или дифференциально-разностного уравнения с нелокальными потенциалами) построено трехпараметрическое семейство гладких решений. Все сдвиги в потенциалах по пространственной переменной --- произвольные вещественные величины, то есть никакие условия соизмеримости на них не накладываются, что является наиболее общим случаем. Для построения решений используется классическая операционная схема Гельфанда---Шилова, согласно которой к уравнению формально применяется сначала прямое преобразование Фурье по пространственной переменной, с учетом того, что в образах Фурье оператор сдвига является мультипликатором, а затем обратное преобразование Фурье. Вообще говоря, данная схема приводит к решениям в смысле обобщенных функций. Однако, в данном случае удается доказать, что полученные решения являются классическими. Доказана теорема, что построенные решения являются классическими, если вещественная часть символа дифференциально-разностного оператора по пространственной переменной, входящего в уравнение, положительна. Приведены классы уравнений, для которых указанное условие выполнено. Получены соотношения, которым должны удовлетворять все коэффициенты и все сдвиги в уравнении, справедливость которых гарантирует требуемую положительность вещественной части символа дифференциально-разностного оператора в уравнении.
Date of submission: 26 February 2021 г.

35. Rautian N.A. Экспоненциальная устойчивость полугрупп, порождаемых вольтерровыми интегро-дифференциальными уравнениями
Status: reviewing
Abstract.
Исследуются абстрактные интегро-дифференциальные уравнения, которые являются операторными моделями задач теории вязкоупругости. Представленные результаты базируются на подходе, связанном с исследованием однопараметрических полугрупп для линейных эволюционных уравнений. Приводится метод сведения исходной начальной задачи для модельного интегро-дифференциального уравнения с операторными коэффициентами в гильбертовом пространстве к задаче Коши для дифференциального уравнения первого порядка. Представлены результаты о существовании сильно непрерывной сжимающей полугруппы, порождаемой вольтерровым интегро-дифференциальным уравнением с операторными коэффициентами в гильбертовом пространстве. Установлено экспонециальное убывание полугруппы при известных предположениях для ядер интегральных операторов. На основе полученных результатов установлена корректная разрешимость исходной начальной задачи для вольтеррова интегро-дифференциального уравнения с соответсвующими оценками решения. Предлагаемый подход может быть также использован для исследования других интегро-дифференциальных уравнений, содержащих интегральные слагаемые вида вольтеррой свертки.
Date of submission: 27 February 2021 г.

36. 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.

37. Borisov A.B. On integrating the $O(3)$--model
Status: reviewing
Abstract.
A differential substitution reducing the equations of the $3D$ $O(3)$--model to a new integrable model is proposed. An explicit solution for this model as an arbitrary function of two variables is found.
Date of submission: 03 Mart 2021 г.

38. 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.

39. Adler V.E. Differential substitutions for non-Abelian equations of KdV type
Status: reviewing
Abstract.
We construct non-Abelian analogs for some KdV type equations, including the (rational form of) exponential Calogero--Degasperis equation and generalizations of the Schwarzian KdV equation. Equations and differential substitutions under study contain arbitrary non-Abelian parameters.
Date of submission: 07 Mart 2021 г.

40. 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.

41. Zamana K.Yu. Усреднение случайных ортогональных преобразований аргумента функции
Status: reviewing
Abstract.
The article discusses the concepts of a random operator, a random operator-valued function given on the Hilbert space, and their averagings. A method for constructing averagings of random orthogonal transformations of the domain of a function which leads to the construction of semigroups describing diffusion on the sphere is presented.
Date of submission: 10 Mart 2021 г.

42. Levi D., Rodr´ıguez M.A. Yamilov’s Theorem for Differential and Difference Equations
Status: reviewing
Abstract.
S-integrable scalar evolutionary differential difference equations in 1+1 dimensions have a very particular form described by Yamilov’s theorem. We look for similar results in the case of S-integrable 2-dimensional partial difference equations and 2-dimensional partial differential equations. To do so on one side we discuss the semi continuous limit of some examples of S-integrable quad equations and on the other we semi discretize partial differential equations. In both cases the difference differential equation we get satisfies Yamilov’s theorem.
Date of submission: 11 Mart 2021 г.

43. 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.

44. On the nonparametric estimation of conditional hazard estimator in the single functional index
Status: reviewing
Abstract.
This paper deals with the conditional hazard estimator of a real response variable given a functional random variable (i.e take values in an infinite dimensional space). Specifically, we focus on the functional index model, this approach represents a good compromise between nonparametric and parametric models. The principal aim of this paper is to give under general conditions and when the variables satisfy the strong mixing dependency to prove the asymptotic normality of the proposed estimator by kernel estima- tor method, based on the single-index structure. Finally, a simulation of our methodology give the good behavior for sample sizes. Their performance is compared with a Nadayara-Watson type kernel estimator via a Monte Carlo .
Date of submission: 19 Mart 2021 г.

45. Matveev V.B., Smirnov A.O. Finite-gap solutions of nonlocal equations from the AKNS hierarchy
Status: reviewing
Abstract.
We consider non-local symmetries that satisfy all the equations from the Ablowitz-Kaup-Newell-Segur hierarchy. Based on the properties of solutions satisfying non-local reduction equations from the AKNS hierarchy, a modification of the theta-functional formula for the Baker-Akhiezer function is proposed. Conditions are found for the parameters of spectral curves associated with multiphase solutions that do not have exponential growth at infinity. It is shown that when these conditions are met, the variables are separated. Examples are given.
Date of submission: 22 Mart 2021 г.

46. Zhiber A.V., Kuznetsova M.N. Integrals and characteristic Lie rings of semi-discrete systems of equations
Status: reviewing
Abstract.
The paper is devoted to the study of systems of semi-discrete equations $\bar{r}_{n+1,x} = \bar{h}(x,n, \bar{r}_n, \bar{r}_{n+1}, \bar{r}_{n,x})$ within the framework of an approach based on the concept of a characteristic Lie ring. Here $\bar{r}_n = (r^1_n, r^2_n, \ldots, r^N_n)$, $\bar{h} = (h^1, h^2, \ldots, h^N)$, $n \in \mathbb{Z}$. Darboux integrable nonlinear hyperbolic equations and systems stand out as a separate broad class among integrable nonlinear partial differential equations and systems. A distinctive property of such equations is the existence of integrals over each characteristic direction (the so-called $x$ - and $y$ -integrals). Darboux integrable equations and systems can be efficiently investigated and classified using characteristic Lie rings. Currently, the algebraic approach is extended to semi-discrete and discrete equations. In this paper, it is proved that the system has functionally independent $N$ $x$ -integrals if and only if the characteristic Lie ring corresponding to a continuous characteristic direction is finite-dimensional.
Date of submission: 23 Mart 2021 г.

47. Novokshenov V.Yu. Discrete Riemann-Hilbert problem and interpolation of entire functions
Status: reviewing
Abstract.
We consider two problems in complex analysis which were developed in Ufa at 1970s years. A progress in recent years led to com\-pre\-hen\-sion that they have much common in subject. It is shown that discrete matrix Riemann-Hilbert (dmRH) problem can be reduced to an interpolation problem of entire functions on a countable set in $\mathbb{C}$ with only condensation point at infinity. In its turn, dmRH provides a way to integrate nonlinear discrete equations of mathematical physics such as discrete Painlev\'e equations. Other applications of dmRH include Fredholm determinants and orthogonal polynomials emerging in combinatorics and representation of groups theory.
Date of submission: 26 Mart 2021 г.

48. Krivosheev A.S., Krivosheeva O.A., Rafikov A.I. Properly distributed subsets
Status: reviewing
Abstract.
В работе рассматриваются комплексные последовательности уточненного порядка $\rho(r)$. Найдены необходимые и достаточные условия, при которых из последовательности $\Lambda^2\supseteq\Lambda^1$ можно выделить правильно распределенное множество $\Lambda$ с заданной угловой плотностью, содержащее $\Lambda^1$. Эти результаты включают в себя большую часть известных результатов, связанных с построением правильно распределенного множества. Рассматриваются различные применения указанных результатов. На их основе получены теоремы о расщеплении целых функций уточненного порядка $\rho(r)$. Кроме того, найдено асимптотическое представление целой функции с измеримой последовательностью нулей. Оно обобщает классическое представление Б.~Я.~Левина функций с правильно распределенным нулевым множеством на случай функций с измеримым нулевым множеством. Указанное представление опирается на полученное представление функций, нулевое множество которых имеет нулевую плотность. Его следствием является усиление известного результата М.~Картрайт о типе функции с нулевым множеством, имеющим нулевую плотность. Другим следствием является способ построения целых функций экспоненциального типа с заданным индикатором и минимально возможной плотностью – нулевой.
Date of submission: 28 Mart 2021 г.

49. Smirnov A.O., Habibullin I.T., Khakimova A.R. Generalized invariant manifolds for integrable equations and their applications
Status: reviewing
Abstract.
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the nonlinear partial differential equations. Its essence is in adding to the nonlinear PDE, a much simpler, as a rule ordinary, differential equation, compatible with the given one. Then any solution of the ODE is a particular solution of the PDE as well. However the main problem is to find this compatible ODE. Our generalization is that we look for an ordinary differential equation that is compatible not with the nonlinear partial differential equation itself, but with its linearization. Such a generalized invariant manifold is effectively sought. Moreover, it allows one to construct such important attributes of integrability theory as Lax pairs and recursion operators for integrable nonlinear equations. In this paper, we show that they provide a way to construct particular solutions to the equation as well.
Date of submission: 31 Mart 2021 г.

50. Suleimanov B.I., Shavlukov A.M. An integrable ordinary differential equation of the first order with a rational right-hand side, arising in the description of the asymptotics of symmetry solutions of the Korteweg-de Vries equation
Status: reviewing
Abstract.
The general solution of an ordinary differential equation of the first order with a rational right-hand side is presented, which arises when constructing asymptotics for large values of the time of simultaneous solutions of the Korteweg - de Vries equation and the stationary part of its higher non-autonomous symmetry. This symmetry is determined by a linear combination of the first higher autonomous symmetry of the Korteweg-de Vries equation and its classical Galileo symmetry. This general solution depends on an arbitrary parameter. By the implicit function theorem, it is locally determined from the first integral explicitly written out in terms of hypergeometric functions. The special case of the general solution defines self-similar solutions of the Whitham equations, found earlier by G.V. Potemin in 1988. (In the well-known works of A.V. Gurevich and L.P. Pitaevsky in the early 1970s, it was established that these solutions of the Whitham equations describe the origination of nondamping oscillating waves in the main order in a wide range of problems with small dispersion.) The result of this article once again confirms the empirical rule according to which, when describing the asymptotics of solutions of the integrable equations, only integrable equations can arise. A general hypothesis is advanced that integrable ordinary differential equations similar to the one considered in this article should also arise in the description of asymptotics for large times of other symmetry solutions of evolution equations that admit the application of the inverse scattering problem method.
Date of submission: 01 Aprel 2021 г.

51. 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.

52. Миллионщиков D.V., Smirnov S.V. Характеристические алгебры и интегрируемые системы экспоненциального типа
Status: reviewing
Abstract.
В данной работе изучаются характеристические алгебры для систем экспоненциального типа, соответствующих вырожденным матрицам Картана и исследуется связь между высшими симметриями этих систем и структурой их характеристических алгебр.
Date of submission: 05 Aprel 2021 г.

53. Gerdjikov V.S. On the mKdV equations related to the Kac-Moody algebras $A_5^{(1)}$ and $A_5^{(2)}$
Status: reviewing
Abstract.
We analyze the spectral properties of the Lax operators generating the mKdV equations related to Kac-Moody algebras $A_5^{(1)}$ and $A_5^{(2)}$. The fundamental analytic solutions of the Lax operators are introduced and shown to satisfy Riemann-Hilbert problems (RHP). We formulate minimal sets of scattering data for both problems and prove that they recover uniquely the potentials and the sewing functions of the RHP.
Date of submission: 12 Aprel 2021 г.