# Editorial backlog

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. 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.

15. Allakhyarova N.E., Jabbarov I.Sh. On integral equations of Fredholm type in the space of Bohr almost periodic functions
Status: reviewing
Abstract.
In the present work one considers the question about such an extension of the notion of integral equations of Fredholm type that it is possible to state that the equation has solutions in the class of Bohr almost periodic functions. Since almost periodic functions are defined in all real axes then it is difficult to define them completely in bounded intervals. Close connection of Fredholm equations with differential equations of first order is well known. The special cases of such equations were considered in the space of Bohr almost periodic functions, in which questions on solvability in different classes of almost periodic functions were investigated. Fore some of equations of such kind with almost periodic coefficients there are not solutions in Bohr class of almost periodic functions. There are examples of such almost periodic functions (in Besicovitch sense) that could not be solutions of arbitrary differential equations from sufficiently wide class of equations. In general, it is naturally to wait that ordinary Fredholm integral equations also stood non-solvable in the class of Bohr almost periodic functions. By this reason, in the class of Bohr almost periodic functions the case demands special approach.
Date of submission: 11 January 2022 г.

16. Positivity of $r$-Laplacian fractional boundary value problems for iterative systems
Status: reviewing
Abstract.
The main focus of this research is to seek optimal eigenvalue intervals of $\chi_1,\chi_2,\ldots,\chi_n$ for which the iterative system of Riemann–Liouville type r-Laplacian fractional boundary value problems has positive solutions under appropriate conditions.
Date of submission: 12 January 2022 г.

17. Kulikov V.L., Olekhova E.F., Oseledets V.Iu. Remarks on Garsia entropy and multidimensional Erdös measures
Status: reviewing
Abstract.
he Garsia entropy is calculated for the Garsia numbers. We prove theorem, which generalizes the Garsia theorem on the absolute continuity of the infinite Bernoulli convolution for the Garsia numbers. The proof uses the connection between the multidimensional Erdös problem and the one-dimensional Erdös problem. We discuss the entropy of the invariant Erdös measure and the conditional Ledrappier--Young entropies. We also formulate some hypotheses and get some consequences from them. For 2-numbers, we obtain formulas for the Hausdorff dimension of Erdös measures on an unstable plane.
Date of submission: 14 January 2022 г.

18. Durdiev D.K., Rahmonov A.A. A multi-dimensional diffusion coefficient determination problem for the time-fractional equation
Status: reviewing
Abstract.
In this paper, we consider a multi-dimensional inverse problem for a fractional diffusion equation. The inverse problem is reduced to the equivalent integral equation. For solving this equation the Schauder principle is applied. The local existence and uniqueness results are obtained.
Date of submission: 21 January 2022 г.

19. Absalamov A.T., Ikromov I.A., Safarov A.R. On estimates for trigonometric integrals with quadratic phase.
Status: reviewing
Abstract.
In paper this paper it is considered the summation problem for trigonometric integrals with quadratic phase. This problem considered in the papers [2],[3],[4] in particular cases. Our results generalized the results of that papers to multidimensional trigonometrical integrals.
Date of submission: 26 January 2022 г.

20. Study of the viscoelastic problems with short memory in a thin domain with tresca boundary conditions
Status: reviewing
Abstract.
In this paper, we are interested in the study of the asymptotic behavior of non linear problem in a quasistatic regime in a thin domain with Tresca boundary conditions. In the first step, we derive a variational formu- lation of the mechanical problem and prove the existence and uniqueness of the weak solution. We study the limit when the ε tends to zero, we prove the convergence of the unknowns which are the displacement and the velocity and we obtain the limit problem and the specific Reynolds equation.
Date of submission: 04 February 2022 г.

21. Avkhadiev F.G. Universal inequalities on domains of the Euclidean space and their applications
Status: reviewing
Abstract.
В областях евклидова пространства для пробных функций сконструированы и доказаны несколько новых интегральных неравенств типа Гальярдо-Ниренберга с явными константами. Эти неравенства справедливы в любой области, они являются нелинейными, подынтегральные функции содержат степени от модулей градиента и лапласиана пробной функции $u$, а также множители вида $f(|u(x)|)$, $f'(|u(x)|)$, где $f$--- непрерывно дифференцируемая, неубывающая функция, $f(0)=0$. В качестве весовых функций используются степени расстояния от точки до границы области, а также степени переменного гиперболического (конформного) радиуса. Как применения универсальных неравенств типа Гальярдо-Ниренберга мы получаем новые интегральные неравенства типа Реллиха в плоских областях с равномерно совершенными границами. Для этих $L_p$-неравенств типа Реллиха установлены критерии положительности констант, получены явные двусторонние оценки этих констант в зависимости от евклидова максимального модуля области и от параметра $p\geq 2$. В доказательствах используются несколько числовых характеристик для областей с равномерно совершенными границами.
Date of submission: 06 February 2022 г.

22. Merker J. Inexistence of Non-Product Hessian Rank 1 Affinely Homogeneous Hypersurfaces $H^n \subset \R^{n+1}$
Status: reviewing
Abstract.
Equivalences under the affine group $\Aff(\R^3)$ of constant Hessian rank $1$ surfaces $S^2 \subset \R^3$, sometimes called {\sl parabolic}, were, among other objects, studied by Doubrov, Komrakov, Rabinovich, Eastwood, Ezhov, Olver, Chen, Merker, Arnaldsson, Valiquette. Especially, homogeneous models and algebras of differential invariants in various branches have been fully understood. {\sl Then what about higher dimensions?} We consider hypersurfaces $H^n \subset \R^{n+1}$ graphed as $\big\{ u = F(x_1, \dots, x_n) \big\}$ whose Hessian matrix $\big( F_{x_i x_j} \big)$, a relative affine invariant, is, similarly, of constant rank $1$. {\sl Are there homogeneous models?} Complete explorations were done by the author on a computer in dimensions $n = 2, 3, 4, 5, 6, 7$. The first, expected outcome, was to obtain a complete classification of homogeneous models in dimensions $n = 2, 3, 4$ (forthcoming article, case $n = 2$ already known). The second, unexpected outcome, was that in dimensions $n = 5, 6, 7$, {\em there are {\em no} affinely homogenous models!} (Except those that are affinely equivalent to a product of $\R^m$ with a homogeneous model in dimensions $2, 3, 4$.) The present article establishes such a non-existence result in every dimension $n \geqslant 5$, based on the production of a normal form for $\big\{ u = F(x_1, \dots, x_n) \big\}$, under $\Aff(\R^{n+1})$ up to order $\leqslant n+5$, valid in any dimension $n \geqslant 2$.
Date of submission: 07 February 2022 г.

23. Isakov B.M., Рахматуллаев M.M. Об основных состояниях модели Изинга-Поттса на дереве Кэли.
Status: reviewing
Abstract.
Для модели Изинга-Поттса на дереве Кэли порядка $k\geq2$ описано множество периодических и слабо периодических основных состояний, соответствующих нормальным делителям индекса 2 группового представления дерева Кэли.
Date of submission: 10 February 2022 г.

24. Hardy Type Inequalities Via $(k,\mu)$-Riemann-Liouville Fractional Integral Operators
Status: reviewing
Abstract.
In this study, a new inverse Hardy-type inequality intro- duced via the (k,µ)-Riemann-Liouville fractional integral operators. New results obtained by using two integrability parameters p and q and some particular cases mentioned, according to the choice of the function µ and the reals k,p,q.
Date of submission: 21 February 2022 г.

25. Admasu V.E., Galahov E.I. Условия отсутствия решений для некоторых эллиптических неравенств высокого порядка с сингулярными коэффициентами в $\mathbb{R}^n$
Status: reviewing
Abstract.
In the present paper, we develop Liouville-type theorems for higher order elliptic inequalities with singular coefficients and gradient terms in $\mathbb{R}^n$ by the Pohozaev nonlinear capacity method.
Date of submission: 28 February 2022 г.

26. Belous T.I., Gaisin A.M. Максимальный член ряда Дирихле, сходящегося в полуплоскости: теорема об устойчивости
Status: reviewing
Abstract.
Исследуется задача об эквиалентности логарифмов максимальных членов адамаровской композиции (измененного ряда) $\sum \limits_{n} a_nb_ne^{\lambda_nz}$ рядов Дирихле $\sum \limits_{n} a_ne^{\lambda_nz}$ и $\sum \limits_{n} b_ne^{\lambda_nz}$ с положительными показателями, область сходимости которых есть полуплоскость. Аналогичная задача для целых рядов Дирихле впервые изучалась в А. М. Гайсиным в 2003 году -- им был тогда получен критерий устойчивости максимального члена $\mu(\sigma)=\max \limits_{n\geq 1}\{{\vert a_n\vert} e^{\lambda_n\sigma}\}.$ Этот результат оказался весьма полезным при изучении асимптотических свойств ряда Дирихле на произвольных кривых, уходящих в бесконечность, а именно при доказательстве известной гипотезы Полиа. Как в случае целых рядов Дирихле, и в случае рядов, сходящихся лишь в полуплоскости, в задачах такого типа ключевую роль играют формулы А.Ф. Леонтьева для коэффициентов. Функции соответствующей биортогональной системы содержат множитель -- производную характеристической функции в точках $\lambda_n \; (n\geq 1).$ Это обстоятельство естественным образом и приводит к рассматриваемой здесь постановке задачи об устойчивости максимального члена. Получен критерий того, чтобы логарифм максимального члена ряда Дирихле, область сходимости которого есть полуплоскость, на асимптотическом множестве был эквивалентным логарифму максимального члена изменного ряда.
Date of submission: 02 Mart 2022 г.

27. SOME NEW ESTIMATIONS FOR $m$-CONVEX FUNCTIONS VIA KATUGAMPOLA FRACTIONAL OPERATOR
Status: reviewing
Abstract.
The main motivation of this study is to present new Hermite- Hadamard (HH) type inequalities via a certain fractional operators. We have used an integral identity and give new estimations of HH- type inequalities for differentiable and $m$−convex mapping via Katugampola-fractional operators. Main findings of this study would provide elegant connections and general vari- ants of well known results established recently. These results can be extended to different kinds of convex functions as well as pre-invex functions.
Date of submission: 19 Mart 2022 г.

28. Nasibullin R.G. One-dimensional $L_p$-Hardy-type Inequalities for special weight functions and their applications
Status: reviewing
Abstract.
We establish one-dimensional $L_p$-Hardy inequalities with additional terms. We use these inequalities to obtain their multidimensional analogues in convex domains with finite volume. Variational inequalities with power-law weights are obtained, which are generalizations of the corresponding inequalities presented earlier in articles by M.~Hoffmann-Ostenhof, T.~Hoffmann-Ostenhof, A.~Laptev and J.~Tidblom. We formulate and prove inequalities valid for arbitrary domains, then we simplify them substantially for the class of convex domains. The constants in the additional terms in these spatial inequalities depend on the volume or on the diameter of the domain. As a consequence of the results, we will have estimates for the first eigenvalue of the $p$-Laplacian under the Dirichlet boundary conditions.
Date of submission: 23 Mart 2022 г.

29. Integrable generalized Heisenberg ferromagnet equations with self-consistent potentials and related Yajima-Oikawa type equations
Status: reviewing
Abstract.
We consider some nonlinear models describing interactions of long and short (LS) waves. Such LS models have been derived and proposed with various motivations, which mainly come from fluid and plasma physics. In this paper, we study some of integrable LS models, namely, the Yajima-Oikawa equation, the Newell equation, the Ma equation, the Geng-Li equation and etc. In particular, the gauge equivalent counterparts of these integrable LS models (equations) are found. In fact, these gauge equivalents of the LS equations are integrable generalized Heisenberg ferromagnet equations (HFE) with self-consistent potentials (HFESCP). The associated Lax representations of these HFESCP are given. We also presented several spin-phonon equations which describe nonlinear interactions of spin and lattice subsystems in ferromagnetic materials.
Date of submission: 01 Aprel 2022 г.

30. Braichev G.G., Sherstyukova O.V. On the least type of an entire function with a given subsequence of zeros
Status: reviewing
Abstract.
Настоящая заметка написана по материалам доклада авторов на Международной научной конференции <<Уфимская осенняя математическая школа -- 2021>>. Обсуждается следующая задача. Пусть заданы нецелое число $\rho>0$ и последовательность комплексных чисел $\Lambda$, имеющая конечную верхнюю $\rho$-плотность. Тогда, как известно из классической теоремы Линделефа, существует (отличная от тождественного нуля) целая функция $f$ конечного типа при порядке $\rho$, для которой $\Lambda$ является последовательностью (всех) нулей. Спрашивается, как сильно может измениться тип такой функции, если позволить ей помимо элементов из $\Lambda$ иметь другие нули, причем произвольной кратности. Показаны возможности применения одной общей теоремы, доказанной по означенной задаче Б.\,Н.~Хабибуллиным в 2009 году. С этой целью привлекаются результаты последнего времени, содержащие точные формулы для вычисления экстремального типа в классах целых функций с различными ограничениями на распределение нулей. Случай целого $\rho$ обладает своей спецификой и в данной работе практически не рассматривается.
Date of submission: 04 Aprel 2022 г.

31. Петросян A.S., Khachatryan K.A. On the solvability of a system of nonlinear integral equations with a Hammerstein-Stieltjes operator on the semiaxis
Status: reviewing
Abstract.
In this note, we study a system of nonlinear integral equations of the Hammerstein-Stieltjes type, whose prekernel is a continuous distribution function. The specified system in various special cases has applications in many branches of natural sciences. In particular, such systems are encountered in the theory of Markov processes, in the theory of radiative transfer in inhomogeneous medium, and in the kinetic theory of gases. In this paper, we prove a constructive existence theorem for a nontrivial, nonnegative, and bounded solution. The integral asymptotic behavior of the constructed solution is studied. At the end, specific examples of this system are given for which all the conditions of the theorem are satisfied.
Date of submission: 07 Aprel 2022 г.

32. Gadoev M.G., Iskhokov D.S. On the Abel basis property of the system of root vector-functions of a class of elliptic operators with uncoordinated degeneration
Status: reviewing
Abstract.
В работе изучается класс эллиптических дифференциальных операторов высшего порядка в ограниченной области, коэффициенты которых имеют несогласованное степенное вырождение вдоль всей границы области. Полуторалинейная форма, связанная с исследуемым оператором, может не удовлетворять условию коэрцитивности. Она представляется в виде конечной суммы вспомогательных форм, и среди них выделяются старшие формы. Применением теорем вложения для пространств дифференцируемых функций многих вещественных переменных со степенными весами доказывается, что только вырождения коэффициентов старших форм влияют на область определения рассматриваемого эллиптического оператора. Установлена полнота и суммируемость в смысле Абеля-Лидского системы корневых функций исследуемого оператора.
Date of submission: 11 Aprel 2022 г.

33. Kostin A.B., Sherstyukov V.B. On the Taylor coefficients of an analytic function related to the Euler's number
Status: reviewing
Abstract.
Рассматривается классическая конструкция <<второго замечательного предела>>. Ставится вопрос об асимптотически точном описании характера такой аппроксимации числа~$e$. В связи с этим требуется информация о поведении коэффициентов степенного разложения функции $f(x)=e^{-1}\,(1+x)^{1/x}$, сходящегося в интервале $-1 < x < 1$. Выведено рекуррентное правило, регулирующее формирование означенных коэффициентов. Показано, что коэффициенты образуют знакочередующуюся последовательность рациональных чисел $(-1)^n\,a_n$, где $n\in\NN\cup\{0\}$ и $a_0=1$, модули которых строго убывают. На основе формулы Фаа ди Бруно для производных сложной функции предложен комбинаторный способ вычисления чисел $a_n$ при $n\in\NN$. Исходная функция $f(x)$ есть сужение на вещественный луч $x>-1$ функции $f(z)$, имеющей те же тейлоровские коэффициенты и аналитической в комплексной плоскости $\CC$ с~разрезом $(-\infty,\,-1]$. Методами комплексного анализа получено интегральное представление для $a_n$ при любом значении параметра $n\in\NN$. Доказано, что $a_n\rightarrow 1/e$ при $n\rightarrow\infty$, и~найден порядок стремления к нулю разности $a_n-1/e$. Затронут вопрос о~выборе контура в~интегральной формуле Коши для вычисления тейлоровских коэффициентов $(-1)^n\,a_n$ функции $f(z)$. Посчитаны точные значения возникающих по ходу дела специальных несобственных интегралов. Результаты проведённого исследования позволяют дать серию общих двусторонних оценок уклонения $e-(1+x)^{1/x}$, согласованных с асимптотикой $f(x)$ при $x\rightarrow0$. Обсуждаются возможности применения полученных утверждений.
Date of submission: 12 Aprel 2022 г.

34. Inverse problem of determining two kernels in the integro - differential equation of heat flow
Status: reviewing
Abstract.
The inverse problem of determining the energy-temperature relation $\alpha(t)$ and the heat conduction relation $k(t)$ functions in the one-dimensional integro--differential heat equation are investigated. The direct problem is the initial-boundary problem for this equation. The integral terms have the time convolution form of unknown kernels and direct problem solution. As additional information for solving inverse problem, the solution of the direct problem for $x=x_0$ and $x=x_1$ are given. At the beginning an auxiliary problem, which is equivalent to the original problem is introduced. Then the auxiliary problem is reduced by an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the continuous class of functions, the main result of the article, which is a local existence and uniqueness theorem of inverse problem solutions is proven.
Date of submission: 14 Aprel 2022 г.

35. Ivanova О.А., Melikhov S.N. Hadamard type operators in the space of holomorphic functions on a ball
Status: reviewing
Abstract.
We study Hadamard type operators in spaces of all holomorphic functions on an open ball in $\mathbb C^N$ centered at the point 0. Their representation is obtained in the form of the multiplicative convolution. It is proved that the space of Hadamard type operators from one such space into another with the topology of bounded convergence is linearly topologically isomorphic to the strong dual of the space of all germs of holomorphic functions on a closed polydisk.
Date of submission: 14 Aprel 2022 г.

36. Analysis of a thermo-elasto-viscoplastic contact problem with wear and damage
Status: reviewing
Abstract.
This paper presents a quasistatic problem of a thermo-elaso-visco- plastic body in frictional contact with a moving foundation. The contact is modelled with the normal compliance condition and the associated law of dry friction. The model takes into account wear of the contact surface of the body caused by the friction and which is described by the Archard law. The mechanical damage of the material, caused by excessive stress or strain, is described by the damage function, the evolution of which is determined by a parabolic inclusion. We list the assumptions on the data and derive a variational formulation of the mechanical problem. Existence and uniqueness of the weak solution for the problem is proved using the theory of evolutionary variational inequalities, parabolic variational inequalities, first order evolution equation and Banach fixed point.
Date of submission: 16 Aprel 2022 г.

37. Ishkin Kh.K., Marvanov R.I. Spectral properties of the non-sectoral Sturm--Liouville operator on the semiaxis
Status: reviewing
Abstract.
The present paper is devoted to the study of some spectral properties of the Sturm--Liouville operator on the semiaxis with the complex potential $q$ growing at infinity. Instead of the known conditions, V.~B. Lidsky about $\mathrm{Re}\,q$ being bounded from below or $\mathrm{Im}\,q$ being semibounded, we only require that the range of $q$ does not intersect at some small angle with the bisector along the negative real semiaxis. We construct a special solution of the equation $-y '' + qy = \lambda y$, which decreases at infinity and is an entire function of $\lambda$ for every fixed $x$. Using this solution, a generalization of the well-known results of M.A. Naimark and V.B. Lidskii on conditions under which the spectrum of the corresponding Sturm--Liouville operator is discrete and the system of root vectors is complete and minimal.
Date of submission: 18 Aprel 2022 г.

38. Statistical convergence of double sequences of functions by virtue of difference operator
Status: reviewing
Abstract.
The present paper focus on λ−statistical convergence by means of modulus function and generalized difference operator for double sequences of functions for order γ. Further, we prove that statistical convergence in our newly formed sequence spaces is well defined for γ ∈ (0,1]. In addition to the above result, we establish relation among λ−statistical convergence and strongly λ−summable for our sequence spaces.
Date of submission: 20 Aprel 2022 г.

39. Luu T.H., Shokarev V.A., Budochkina S.A. On an indirect representability of a fourth-order ordinary differential equation in the form of Hamilton-Ostrogradskii equations
Status: reviewing
Abstract.
In the paper, the problem of the representability of a fourth-order ordinary differential equation in the form of Hamilton-Ostrogradskii equations is solved. For this purpose, we obtain necessary and sufficient conditions for a given operator to be potential relative to a local bilinear form, construct the corresponding Hamilton-Ostrogradskii action and define the structure of the considered equation with the potential operator.
Date of submission: 26 Aprel 2022 г.

40. Sukhov A.B. On boundary properties of asymptotically holomorphic functions
Status: reviewing
Abstract.
We prove a Fatou type theorem for bounded functions with $\bar\partial_J$ differential of a controled growth on smoothly bounded domains in an almost complex manifold. In the case of $\mathbb{C}^n$ we obtain results with optimal regularity assumptions.
Date of submission: 28 Aprel 2022 г.

41. Tashpulatov S.M. Spectra of the energy operator of two-electron system in the impurity Hubbard Model
Status: reviewing
Abstract.
We consider two-electron systems for the impurity Hubbard Model and investigate the spectrum of the system in a singlet state for the $\nu-$ dimensional integer valued lattice $Z^{\nu}$. We proved the essential spectrum of the system in the singlet state is consists of union of no more then three intervals, and the discrete spectrum of the system in the singlet state is consists of no mote then five eigenvalues. We show that the discrete spectrum of the system in the triplet and singlet states differ from with each other. In the singlet state the appear additional two eigenvalues. In the triplet state the discrete spectrum of the system can be empty set, or is consists of one-eigenvalue, or is consists of two eigenvalues, or is consists of three eigenvalues.
Date of submission: 05 May 2022 г.

42. Results on two-order fractional boundary value problem under the generalized Riemann-Liouville derivative
Status: reviewing
Abstract.
In this paper we focus our study on the existence, uniqueness and Hyers-Ulam stability for a fractional boundary value problem involving the generalized Riemann-Liouville operators of a function with respect to another non-decreasing function. To prove the uniqueness result we use Banach fixed point Theorem and for the existence result, we apply two classical fixed point Theorems due to Krasnoselskii and Leray-Scauder. Then, we continue our results by studying the Hyers-Ulam stability of solutions.
Date of submission: 12 May 2022 г.

43. A quasistatic electro-elastic contact problem with long memory and slip dependent coefficient of friction
Status: reviewing
Abstract.
In this paper we consider a mathematical model which describes a quasistatic frictional contact problem between a deformable body and an obstacle, say a foundation. We assume that the behavior of the material is described by a linear electro-elastic constitutive law with long memory. The contact is modelled with a version of Coulomb’s law of dry friction in which the normal stress is prescribed on the contact surface. Moreover, we consider a slip dependent coe¢cient of friction. We derive a variational formulation for the model, in the form of a coupled system for the displacements and the electric potential. Under a smallness assumption on the coefficient of friction, we prove an existence result of the weak solution of the model. We can show the uniqueness of the solution by adding another condition. The proofs are based on arguments of time-dependent variational inequalities, differential equations and Banach fixed point theorem.
Date of submission: 18 May 2022 г.

44. Garif'yanov F.N., Strezhneva E.V. О системе производных периодической мероморфной функции
Status: reviewing
Abstract.
We study the approximating properties of a system of successive derivatives of a periodic meromorphic function. A system of functions is constructed that is biorthogonally conjugate to it on the boundary of some rectangle. Here, the Weierstrass theory of elliptic functions is essentially used. The system of derivatives admits a non-trivial expansion of zero in some circular domain. For the constructed biorthogonally conjugate system, an equation of the convolution type is used. They are investigated in a closed form using the discrete Fourier transform. The considered biorthogonal series fundamentally differ from the well-known Appel series.
Date of submission: 26 May 2022 г.

45. Popov A.Yu., Sherstyukov V.B. Lower estimate for the minimum of the modulus of an entire function of genus zero with positive roots through the degree of the maximum modulus in a frequent sequence of points
Status: reviewing
Abstract.
Рассматриваются целые функции нулевого рода, корни которых расположены на одном луче. Выводятся близкие к оптимальным на классе всех таких функций оценки снизу минимума модуля на последовательности окружностей через отрицательную степень максимума модуля на тех же окружностях при ограничении на отношение $a>1$ радиусов соседних окружностей. Введено понятие оптимального показателя $d(a)$ как экстремальной степени максимума модуля в~этой задаче. Для оптимального показателя доказаны двусторонние оценки при <<тестовом>> значении $a=9/4$ и при $a\in(1,\,9/8]$. Найдена асимптотика $d(a)$ при $a\rightarrow1$. Полученные результаты принципиально отличаются от классической $\cos(\pi\rho)$\,-\,теоремы, не содержащей ограничений на частоту радиусов окружностей, на которых минимум модуля целой функции порядка $\rho\in[0,\,1]$ оценивается через степень ее максимума модуля.
Date of submission: 27 May 2022 г.