Editorial backlog

  1. 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.


  2. 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.


  3. 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.


  4. 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.


  5. 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.


  6. 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.


  7. 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.


  8. 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.


  9. 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.


  10. 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.


  11. 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.


  12. 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.


  13. 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.


  14. 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.


  15. 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.


  16. 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.


  17. 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.


  18. Малакбозов З.Ш., Shabozov M.Sh. Точные неравенства типа Джексона -- Стечкина в пространстве Харди $H_2$ и поперечники классов функций
    Status: reviewing
    Abstract.
    Exact inequalities of the Jackson -- Stechkin type are obtained in the Hardy space of functions analytic in the unit disc, the modulus of continuity of the function of which is determined using the Steklov function. For the classes of functions given by this characteristic, the exact values of various $n$-widths are found.
    Date of submission: 04 May 2022 г.


  19. 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.


  20. 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.


  21. 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.


  22. 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.


  23. Existence and stability for Ambartsumian equation with $\Xi$-Hilfer generalized proportional fractional derivative
    Status: reviewing
    Abstract.
    The main objective of this paper is to study the Ambartsumian equation in the sense of $\Xi$- Hilfer Generalized proportional fractional derivative(HGPFD). The existence and stability properties of solution are studied. The technique used for study is fixed point theorem and Gronwall inequality. Ulam-Hyers-Rassias stability of the solution is also investigated.
    Date of submission: 08 June 2022 г.


  24. Recent common fixed point results in the setting of bounded metric spaces with an application to nonlinear integral equations
    Status: reviewing
    Abstract.
    In this paper, we prove some common fixed point theorems in the setting of bounded metric spaces without using neither the compactness nor the uniform convexity of the space. Some examples are built to show the superiority of the obtained results compared to the existing ones in the literature. Moreover, we apply the main result to show the existence and uniqueness of a solution for a nonlinear integral system.
    Date of submission: 23 June 2022 г.


  25. Sazonov A.P. About the a priori and asymptotic estimates for Emden-Fowler problem on model Riemannian manifolds
    Status: reviewing
    Abstract.
    This work is devoted to study for Emden-Fowler problem on the model Riemannian manifolds. In particular, it will be obtained a priori and asymptotic estimates for radially symmetric solutions for this problem in the considered manifolds. The results of this work summarize similar statements obtained earlier in the work S.I. Pohozaeva for the space $R^n$.
    Date of submission: 07 July 2022 г.


  26. Rodikova E.G. On Continuous Linear Functionals in Some Spaces of Analytic Functions in a Disk.
    Status: reviewing
    Abstract.
    The question of describing continuous linear functionals on spaces of analytic functions has been studied since the middle of the 20th century. Historically, the structure of linear continuous functionals of the Hardy spaces H p for p ≥ 1 was ?rst found by Taylor in 1951. In the spaces H p (0 < p < 1) this problem was solved by Duren , Romberg, and Shields in 1969. Note that the proof used an estimate of the coe?cient multipliers in these spaces. In the article, developing the method proposed in the work of Duren et al., a description of continuous linear functionals of the area Privalov classes and classes of Nevanlinna-Dzhrbashyan type is obtained.
    Date of submission: 18 July 2022 г.


  27. Danilin A.R. Asymptotics solutions to the problem of optimal distributed control in a convex domain with a small parameter at one of the higher derivatives
    Status: reviewing
    Abstract.
    Рассматривается задача оптимального распределенного управления в плоской строго выпуклой области с гладкой границей и малым параметром при одной из старших производных эллиптического оператора. На границе области в этой задаче задано нулевое условие Дирихле, а управление аддитивно входит в неоднородность. В качестве множества допустимых управлений используется единичный шар в соответствующем пространстве функций, суммируемых с квадратом. Решение получающихся краевых задач рассматриваются в обобщенном смысле как элементы некоторого гильбертова пространства. В качестве критерия оптимальности выступает сумма квадрата нормы отклонения состояния от заданного и квадрата нормы управления с некоторым коэффициентом. Такая структура критерия оптимальности позволяет, при необходимости, усилить роль либо первого, либо второго слагаемого в этом критерии. В первом случае более важным является достижение заданного состояния, а во втором случае --- минимизация ресурсных затрат. Подробно изучена асимптотика задачи, порожденная дифференциальным оператором второго порядка с малым коэффициентом при одной из старших производных, к которому прибавлен дифференциальный оператор нулевого порядка.
    Date of submission: 20 July 2022 г.


  28. Beshtokov M.KH. Numerical solution of initial-boundary value problems for a multidimensional pseudoparabolic equation
    Status: reviewing
    Abstract.
    We consider initial-boundary value problems for a multidi- mensional pseudoparabolic equation with boundary conditions of the first kind and a special form. For an approximate solution of the problems posed, the multidimensional pseudoparabolic equation is reduced to an integro-differential equation with a small parameter. It is shown that as the small parameter tends to zero, the solution of the corresponding modified problem converges to the solution of the original problem. For each of the problems, a locally one-dimensional difference scheme by A.A. Samarsky, the main idea of ??which is to reduce the transition from layer to layer to the sequential solution of a number of one-dimensional problems in each of the coordinate directions. Using the maximum prin- ciple, a priori estimates are obtained, from which the uniqueness, stability, and convergence of the solution of a locally one-dimensional difference scheme in the uniform metric follow. An algorithm for the numerical solution of the modified problem with conditions of a special form is constructed.
    Date of submission: 26 July 2022 г.


  29. Langarshoev M.R Точные неравенства для алгебраических комплексных полиномов и значение поперечников классов функций в весовом пространстве Бергмана
    Status: reviewing
    Abstract.
    В настоящей работе мы решаем некоторые экстремальные задачи связанные c оценка- ми норм производных для алгебраических комплексных полиномов через усредненные значе- ния их модулей непрерывности и гладкости. Обобщаются некоторые результаты Л.В.Тайкова и Н.Айнуллоева полученные для классов дифференцируемых периодических функций на случай аналитических в единичном круге функций f(z) принадлежащих весовому пространству Берг- мана B q,γ , 1 ≤ q ≤ ∞. Вычислены значения поперечников классов аналитических в единичном круге функций в весовом пространстве Бергмана B q,γ , 1 ≤ q ≤ ∞.
    Date of submission: 12 Avgust 2022 г.


  30. Ershov A.A. Bilinear Interpolation of a Program Control in the Approach Problem
    Status: reviewing
    Abstract.
    We consider a control system containing a constant two-dimensional vector parameter, which is reported to the control person only at the moment of the movement start. Only the set of possible values of this indeterminate parameter is known in advance. For this control system, the problem of approaching the target set at a given time is posed. At the same time, the control person does not have the ability to carry out in real time the cumbersome calculations associated with the construction of such permissive structures as reachable sets and integral funnels. Therefore, to solve this problem, it is proposed to calculate in advance several "`nodal"' permissive controls for parameter values, which are grid nodes covering the set of possible parameter values. In the event that at the moment of the movement start it turns out that the parameter value does not coincide with any of the grid nodes, it is supposed to calculate the program control using linear interpolation formulas. However, this procedure can be effective only if a linear com\-bi\-na\-ti\-on of controls is used, corresponding to the same "`guide"' in the terminology of N.N. Krasovsky's method of extreme targeting. For ef\-fec\-ti\-ve application of linear interpolation, for each grid node it is proposed to calculate four "`nodal"' permissive controls and, in addition, use the method of dividing the control into main and compensating. Due to the application of the latter method, the calculated solvability set turns out to be somewhat smaller than the actual one, but the accuracy of transferring the system state to the target set increases. As an example, a nonlinear generalization of the Zermelo navigation problem is considered.
    Date of submission: 23 Avgust 2022 г.


  31. Volchkov V.V., Volchkova N.P. Формула для лапласиана в терминах отклонения функции от ее средних значений
    Status: reviewing
    Abstract.

    Date of submission: 29 Avgust 2022 г.


  32. Vinogradov O.L. Direct and inverse theorems of approximation theory in the Lebesgue spaces with Muckenhoupt weights
    Status: reviewing
    Abstract.
    We establish direct and inverse theorems of approxmation theory in the Lebesgue spaces~$L_{p,w}$ with Muckenhoupt weights~$w$ on the real line and on the period. As moduli of continuity, inter alia of nonitnteger order, we use norms of deviations of Steklov averages. The proofs are based on estimates of norms of convolution operators and do not use the maximal function. This allows to prove the theorems for all $p\in[1,+\infty)$, including $p=1$. All the constants in the estimates depend on~$[w]_p$ (the Muckenhoupt characteristics of the weight~$w$), and any other dependence on $w$ and~$p$ is absent.
    Date of submission: 31 Avgust 2022 г.


  33. Akmanova S.V., Yumagulov M.G. On the Stability of Equilibrium Points of Nonlinear Continuous-Discrete Dynamical Systems
    Status: reviewing
    Abstract.
    The main attention in the work is given Discussion of questions about sufficient stability criteria for Lyapunov equilibrium points of non-linear hybrid (continuous-discrete) system, that is system, processes in which it has several levels of heterogeneous description, and states contain both continuous and discrete components. It is well known that switching modes of a continuous dynamic system can be achieved stability and, conversely, even when all modes of continuous the systems are stable, when switched on, the system can unstable modes will arise. Therefore, important representations research that allows for a detailed analysis of questions stability during the transition from a continuous to a hybrid system. In the present article, new signs of stability in terms of Lyapunov Stationary Regimes of Nonlinear Hybrid System with constant discretization step $h>0$. These signs are based on methods of studying stability at first approximation and perturbation theory formulas that allow analysis stability of equilibrium points and cycles of dynamic systems, depending on a small parameter. The proposed approaches are based on transition from the original hybrid system to an equivalent (in natural sense) dynamic system with a discrete time. The relationship between dynamic characteristic hybrid and discrete system. When studying the main problem of stability with respect to the Lyapunov point of equilibrium The hybrid system is considered in two settings: stability for small $h>0$ and stability for arbitrary fixed $h=h_{0}>0$. In addition, some questions about scenarios for the bifurcation behavior of the hybrid system in loss of stability of the equilibrium point. An example is given, illustrative efficiency of the results obtained in the problem studies of the stability of equilibrium points of hybrid systems.
    Date of submission: 05 September 2022 г.


  34. Voronova Yu.G., Zhiber A.V. On a class of hyperbolic equations with third-order integrals
    Status: reviewing
    Abstract.
    В работе рассмотрен класс нелинейных гиперболических уравнений, обладающих $y$--интегралом первого порядка и $x$--интегралом третьего порядка. Получены формулы для интегралов. Также приведены дифференциальные подстановки, связывающие уравнения Лэне.
    Date of submission: 13 September 2022 г.


  35. Ivanov D.Y. On the uniform convergence of a semi-analytical solution of the Dirichlet problem for the dissipative Helmholtz equation near the boundary of a two-dimensional domain
    Status: reviewing
    Abstract.
    We study an approximate solution of the Dirichlet problem for the two-dimensional dissipative Helmholtz equation, obtained with using a semi-analytical approximation of the double-layer potential. The approxi\-mation of the potential is based on exact integration over the variable $\rho =\left(r^2 -d^2\right)^{1/2}$, where $r$ and $d$ are the distances from the observation point to the integration point and to the boundary of the domain, respectively. It is proved that the semi-analytical approximations of the potential converge uniformly and stably near the boundary of the domain with cubic velocity, and that at the boundary they suffer a discontinuity, the magnitude of which is proportional to the values of the interpolated density function. We also prove the uniform and stable cubic convergence of appropriate approximate solutions of the boundary integral equation and the Dirichlet problem. It is proved that if the quadrature Gauss formulas are used instead of exact integration over the variable $\rho$, then there is no uniform convergence of approximations of the double layer potential near any boundary point. The results of the numerical solution of the Dirichlet problem in the exterior of the circle are presented, confirming the theoretical conclusions.
    Date of submission: 15 September 2022 г.


  36. Fixed point results via a binary relation in the setting of $T$-normed vector spaces
    Status: reviewing
    Abstract.
    In this work, we introduce the notion of $T$-normed vector spaces by extending normed vector spaces. This concept can be considered the first generalization of normed vector spaces satisfying the $T_2$ -separation axiom. Using this axiom, some fixed point theorems are proved via a binary relation in the setting of $T$-normed vector spaces without using neither the compactness nor the uniform convexity. Furthermore, some examples are given to show the superiority of the proven results.
    Date of submission: 18 September 2022 г.


  37. ZUBELEVICH O. Monotone ODEs with Discontinuous Vector Fields in Sequence Spaces
    Status: reviewing
    Abstract.
    We consider a system of ODE in a Fr´ echet space with unconditional Schauder basis. The right side of the ODE is a discontinuous function. Under certain monotonicity conditions we prove an existence theorem for the corresponding initial value prob- lem. We employ an idea of the partial order which seems to be new in this field.
    Date of submission: 25 September 2022 г.


  38. Combinatorial and probabilistic analysis of the scheme placements of particles in indifferent sets
    Status: reviewing
    Abstract.
    The study of this scheme is carried out by the author's enumerative method based on the construction and system analysis of an iterative (step-by-step) random process of direct non-repetitive numbered enumeration of its outcomes in the pre-asymptotic region of parameter variation. Directions of research: finding the number of outcomes of the scheme, establishing the correspondence between the types and numbers of its outcomes, one-to-one determination of the outcomes of the scheme controlled by the probabilistic distribution process and the probabilities of its iterative transitions, and describing the stages of implementing the simulation of the outcomes of the scheme with a given probability distribution
    Date of submission: 29 September 2022 г.


  39. Stabilities of Ulam-Hyers Type for a Class of Nonlinear Fractional Differential Equations with Integral Boundary Conditions in Banach Spaces
    Status: reviewing
    Abstract.
    Motivated by the knowledge of the existence of continuous so- lutions of a certain fractional boundary value problem with integral boundary conditions, we present in here –in a unified manner– new sufficient conditions to conclude the existence and uniqueness of continuously differentiable solutions to this fractional boundary value problem and analyse its stability in the sense of Ulam-Hyers and Ulam-Hyers-Rassias. After presenting the main conclusions, two illustrative examples are provided to verify the effectiveness of the proposed theoretical results.
    Date of submission: 05 October 2022 г.


  40. Mirsaburov M., Ergasheva The problem with the missing Goursat condition for a hyperbolic equation degenerating on the boundary of the domain with a singular coefficient
    Status: reviewing
    Abstract.
    For a hyperbolic equation degenerating on the boundary of the domain with a singular coefficient, theorems of uniqueness and existence of a solution to the problem with the missing Goursat condition on the boundary characteristic and an analogue of the Frankl condition on the segment of degeneration are proved.
    Date of submission: 28 October 2022 г.


  41. Ismoilov A.S. On a problem of integral geometry for a family of parabolas in the plane
    Status: reviewing
    Abstract.
    Integral geometry studies measure invariants with respect to the symmetry group. The definition of integral geometry appears in the works of Luis Antonio Santalo Sors and Wilhelm Blaschke. Hugh Hadwiger, Sigurdur Helgason and Israel Gelfand also made a significant contribution to the development of integral geometry. One of the most important proposals is the Aleksandrov-Fenchel inequality, as well as the Hadwiger theorem. The early results of integral geometry can be attributed to the Buffon problem of throwing needles and the Crofton Formula. In this paper, we consider the problem of recovering a function from a family of parabolas in a strip with a weight function of a new type. A uniqueness theorem is proved and a theorem on the existence of a solution to the problem is introduced. It is shown that the solution to the problem posed is weakly ill-posed, that is, stability estimates in spaces of finite smoothness are obtained. Further, the corresponding problem of integral geometry with perturbation is considered. A uniqueness theorem for its solution in the class of smooth compactly supported functions with support in a strip and an estimate for the stability of the solution in Sobolev spaces are obtained.
    Date of submission: 01 November 2022 г.


  42. Ashurov R.R., M.D. Inverse problem for the subdiffusion equation with fractional Caputo derivative
    Status: reviewing
    Abstract.
    The inverse problem of determining the right-hand side of the subdiffusion equation with the fractional Caputo derivative is considered. The right-hand side of the equation has the form $f(x)g(t)$ and the unknown is function $f(x)$. The condition $ u (x,t_0)= \psi (x) $ is taken as the over-determination condition, where $t_0$ is some interior point of the considering domain and $\psi (x) $ is a given function. It is proved by the Fourier method that under certain conditions on the functions $g(t)$ and $\psi (x) $ the solution of the inverse problem exists and is unique. An example is given showing the violation of the uniqueness of the solution of the inverse problem for some sign-changing functions $g(t)$. For such functions $g(t)$, we find necessary and sufficient conditions on the initial function and on the function from the over-determination condition, which ensure the existence of a solution to the inverse problem.
    Date of submission: 02 November 2022 г.


  43. Градиентные меры Гиббса для модели Блюма-Капеля в случае "петля" на дереве Кэли
    Status: reviewing
    Abstract.
    Работа посвящена градиентным мерам Гиббса (ГМГ) для модели Блюма-Капеля со счетным множеством Z значений спина в случае "петля" на деревьях Кэли. Эта модель определяется потенциалом взаимодействия градиента ближайшего соседа. Используя аргумент Кульске-Шрайвера, основанный на уравнениях граничного закона, мы даем несколько $q$-периодических трансляционно-инвариантных ГМГ для $q = 2,3,4$.
    Date of submission: 24 November 2022 г.


  44. GENETIC ALGORITHM APPLIED TO FRACTIONAL OPTIMAL CONTROL OF A DIABETIC PATIENT
    Status: reviewing
    Abstract.
    Diabetes is a dangerous disease that is increasing in incidence every year. The objective of this paper is to present and analyze the model of diabetes and its complications with the fractional derivative of Caputo. A mathematical model related to the fractional derivative of type 2 diabetes has been proposed. The positivity and boundedness of the solutions were demonstrated by the Laplace transform method. We have studied the existence and uniqueness of the solution of the system. We used the genetic algorithm (GA) to solve the fractional di?erential equation model and to characterize the optimal control, as an e?cient and simple metaheuristic method to implement. Simulations of the total number of diabetics show, with the di?erent values of α chosen, that the combined control strategy leads to a signi?cant decrease. The simulation results also show that the number of uncomplicated diabetics in the fractional model, for the di?erent fractional values of α, decreases more rapidly than the integer derivative model.
    Date of submission: 26 November 2022 г.