Editorial backlog

  1. Salakhudinov R.G. Some properties of domain functionals on level sets
    Status: reviewing
    Abstract.
    For a plane domain $G$ we consider special functionals that are constructed with the help of domain functions, such as the distance function from a point to the boundary $\partial G$, and the warping function of $G$. Functionals that depend on the distance function are considered in the case of simply-connected domains. Functionals that depend on the distance function are considered in the case of simply-connected domains. Functionals depending on the warping function of a finitely connected domain are also studied. We prove that isoperimetric monotonicity properties with respect to a free parameter of the functionals generate another monotonicity of the functionals. Namely, we consider the functionals as functions defined on subdomains of $G$. Some special cases of inequalities were obtained earlier by Payne. We note that the inequalities have been successfully applied to justify new estimates of the torsional rigidity of simply connected and multiply connected domains. In particular, new functionals of the domain with monotonic property in both their arguments are constructed.In addition, exact estimates of the rate of change of the functionals are found, that is, exact estimates of the derivatives are obtained.
    Date of submission: 26 September 2017 г.


  2. Alhouzani M., Chuprunov A.N. ПУАССОНОВСКИЕ ПРЕДЕЛЬНЫЕ ТЕОРЕМЫ В СХЕМАХ РАЗМЕЩЕНИЯ РАЗЛИЧИМЫХ ЧАСТИЦ
    Status: reviewing
    Abstract.
    Рассматривается случайная величина - число ячеек, содержащих $r$ частиц, среди первых $K$ ячеек в равновероятной схеме размещения не более $n$ различимых частиц по $N$ различным ячейкам. Найдены условия, обеспечивающие сходимость этих случайных величин к пуассоновской случайной величине. Получено описание предельного распределения. Показано, что эти результаты переносятся на схему размещения различимых частиц по различным ячейкам.
    Date of submission: 18 January 2018 г.


  3. Galkina V.S., Polyntseva S.V. Two problems of identification of two lower coefficients in the many-dimensional parabolic equation of a special type
    Status: reviewing
    Abstract.
    We consider two problems of identification of two lower coefficients of the many-dimensional parabolic equa\-tions of a special type. In the first problem the overdetermination conditions are given on the same hyperplane, and in the second problem this conditions are given on two various hyperplanes. The inverse problems are reduced to Cauchy's direct auxiliary problems by means of the overdetermination conditions. The resolvability of direct auxiliary problems are proved. The theorems of existence and uniqueness of classical solutions of the inverse problems are proved in the classes of smooth bounded functions. The solutions of the inverse problems are represented explicitly in terms of the solutions of the direct problems.
    Date of submission: 29 January 2018 г.


  4. Gekkieva S.Kh., Kerefov M.A. First boundary-value problem for Aller – Lykov moisture transfer equation with time fractional derivative
    Status: reviewing
    Abstract.
    In this paper we consider the first boundary value problem for the Aller – Lykov moisture transfer equation with the Riemann – Liouville fractional derivative with respect to time. The equation under study presents generalization for the Aller – Lykov equation employing the idea of the fractal speed change in humidity that explains the existence of moisture flows opposing the humidity potential. The existence of a solution to the first boundary-value problem is proved by the Fourier method. With the method of energy inequalities, an a priori estimate is obtained for the solution to the problem in terms of the Riemann – Liouville fractional derivative that implies the uniqueness of the solution.
    Date of submission: 20 February 2018 г.


  5. Aldweby H., Darus M., Elhaddad S. A Subclass of Harmonic Univalent Functions Defined by a Generalized Differential Operator Involving $q$-Mittag-Leffler function
    Status: reviewing
    Abstract.
    The starlike class of complex-valued harmonic univalent functions is defined in this paper by using a rather generalized operator that involve q-Mittag-Leffler function. In a more precise approach, a necessary and sufficient coefficient for functions f is given to be included in this class. Growth bounds and neighborhoods are also consider.
    Date of submission: 18 Mart 2018 г.


  6. Beshtokov M.KH. Boundary value problems for degenerate and degenerate fractional order differential equations with non-local linear source and difference methods for their numerical implementation
    Status: reviewing
    Abstract.
    In this paper, we obtain a priori estimates in the differential and difference interpretations for solutions of non-local boundary value problems for degenerate and degenerate fractional differential equations order with variable coefficients with non-local linear the source, from which the uniqueness and stability of the solution for initial data and the right-hand side, and the convergence of the solution the difference problem to the solution of the differential problem.
    Date of submission: 29 May 2018 г.


  7. Bazzaev A.K., Tsopanov I.D. Difference schemes for partial differential equations of fractional order
    Status: reviewing
    Abstract.
    In this paper we consider difference schemes of higher order of approximation for differential equations with fractional-order derivatives with respect to both space and time variables. Using the maximum principle, a priori estimates are obtained, stability and uniform convergence of difference schemes are proved.
    Date of submission: 31 May 2018 г.


  8. Meshkov A.G. Векторные эволюционные интегрируемые уравнения 3-го порядка, допускающие частичное разделение переменных
    Status: reviewing
    Abstract.
    We present a complete list of the nonlinear integrable evolution vectorial equations in N dimensions of the third order with two independent variables, that admit a partial separation of varables in the spherical coordinates.
    Date of submission: 05 June 2018 г.


  9. Iskhokov S.A., Rakhmonov B.A. On solvability and smoothness of a solution of the variational Dirichlet problem in the whole space associated with a noncoercive form
    Status: reviewing
    Abstract.
    We study the Variational Dirichlet problem for a class of higher order degenerate elliptic operators in the whole $n$-dimensional Euclidean space. A theorem on unique solvability of the problem is proved and under some additional condition on smoothness of coefficients and the right-hand side of the equation, differential properties of the solution are studied. A case when a solution of the variational Dirichlet problem stabilizes to a given polynomial at infinity. Formulation of the problem under consideration is connected with integro-differential sesquilinear form that may not satisfy the coercitivity condition.
    Date of submission: 15 June 2018 г.


  10. Malyutin K.G., Malyutina T.I., Shevtsova T.V. Limiting sets of Azarin of functions and asymptotic representation of integrals
    Status: reviewing
    Abstract.
    Мы доказываем аналог леммы Римана-Лебега для тригонометрических интегралов. Применение этой леммы позволяет получить асимптотические формулы для интегралов с абсолютно непрерывной функцией. Рассматриваются случаи, когда в качестве абсолютно непрерывной функции берется произведение степенной функции на ядро Пуассона или сопряженное ядро Пуассона для полуплоскости, а в качестве промежутка интегрирования берется мнимая полуось. Вещественные и мнимые части этих интегралов представляют собой гармонические функции в комплексной плоскости разрезанной по положительному лучу. Находим предельное множество Азарина для таких функций.
    Date of submission: 18 June 2018 г.


  11. DEBNATH S., Esi A., SUBRAMANIAN N. On extremal rough I- convergence limit point of triple sequence spaces defined by a metric function
    Status: reviewing
    Abstract.
    We introduce and study some basic properties of rough I− convergent of triple sequence spaces and also study the set of all rough $I$− limits of a triple sequence spaces.
    Date of submission: 21 June 2018 г.


  12. Mitrokhin S.I. ОБ ИССЛЕДОВАНИИ АСИМПТОТИКИ СПЕКТРА СЕМЕЙСТВА ФУНКЦИОНАЛЬНО-ДИФФЕРЕНЦИАЛЬНЫХ ОПЕРАТОРОВ С СУММИРУЕМЫМ ПОТЕНЦИАЛОМ
    Status: reviewing
    Abstract.
    The paper investigates a high-order functional-differential operator with a summable potential. The boundary conditions are separated. Operators of this type are called loaded. The method of studying operators with summable potential is an extension of the method of studying operators with piecewise smooth coefficients. To solve the functional-differential equation that defines a differential operator, the method of variation of constants is used. The solution of the original functional-differential equation is reduced to the study of the Volterra integral equation. The solution of the obtained Volterra integral equation is found by the method of successive Picard approximations. As a result of the study of the integral equation for large values of the spectral parameter, asymptotic formulas and estimates for solutions of the functional-differential equation that defines the differential operator are found. Boundary conditions are studied by the help of the obtained asymptotic formulas. To find the eigenvalues of the operator under study, we arrive at the study of the roots of the function represented in the form of a determinant of high order. To find the roots of this function, it is necessary to study the indicator diagram. The roots of the eigenvalue equation are in twelve sectors of an infinitesimal angles, determined by the indicator diagram. The behavior of the roots of this equation is studied in each of the sectors of the indicator diagram. The asymptotics of the eigenvalues of the studied differential operator is obtained. The formulas found for the asymptotic behavior of the eigenvalues are sufficient for studying the spectral properties of the eigenfunctions of the differential operator. In the case of a piecewise smooth potential of the obtained formulas for the asymptotic behavior of the eigenvalues it is sufficient to derive the formula for the first regularized trace of the studied functional-differential operator. Functional-differential operators of this kind arise in the study of vibrations of bridges and beams composed of materials of different density.
    Date of submission: 22 June 2018 г.


  13. Mironova L.B. On a class of integral equations with partial integrals and its applications
    Status: reviewing
    Abstract.
    We prove the existence and uniqueness of the solution for a class of systems of integral equations with partial integrals containing integrals with variable and constant limits of integration. Based on this result, we get sufficient conditions for the unique solvability of the problem for a hyperbolic system with multiple characteristics.
    Date of submission: 03 July 2018 г.


  14. DAM A., MAJUMDER S. SOME FURTHER DIFFERENCE RESULTS ON HAYMAN CONJECTURE AND VALUE-SHARING
    Status: reviewing
    Abstract.
    In this paper we investigate the zeros distributions of difference polynomials of entire functions of finite order, which can be viewed as the Hayman Conjecture for difference. We also study the uniqueness of difference polynomials of entire functions of finite order sharing a common value and obtain uniqueness theorems for difference.
    Date of submission: 05 July 2018 г.


  15. Ghayasuddin M., Khan W.A., Srivastava D. On Hadamard product of extended Gauss and Confluent hypergeometric functions
    Status: reviewing
    Abstract.
    In the present research note, we establish a new class of generating functions associated with the extended Gauss and Confluent hypergeometric functions by using the concept of Hadamard product. Some deductions of our main results are also indicated.
    Date of submission: 06 July 2018 г.


  16. Shakirov I.A. OPTIMAL APPROXIMATE REPLACEMENT OF THE LEBESGUE CONSTANTS OF THE FOURIER OPERATOR THE LOGARITHMIC FUNCTION
    Status: reviewing
    Abstract.
    The classical Fourier operator defined in the space of continuous  2 -periodic functions is considered. Its Lebesgue constant n L approximates by logarithmic functions depending on two parameters. Originally, the influence on this process of the parameter defining the shift of an argument of a logarithm is studied. Then for each chosen value of parameter from some area unimprovable bilateral assessment of a constant n L is defined, among which the best and worst estimates are distinguished. Quite defined values of parameters are specified, at which the best logarithmic approximation of a constant n L is reached. The value of the best approximation is established. The class of the extremum problems is considered that allows to reduce sequentially tentative value of the best approximation.
    Date of submission: 13 July 2018 г.


  17. Turmetov B.Kh. On the Green's function of an analogue of the third boundary-value problem for the polyharmonic equation
    Status: reviewing
    Abstract.
    In this paper the analogue of the third boundary value problem for a polyharmonic equation is studied. For this problem the explicit representation of the Green's function is given. In finding the Green's function of this problem the Green's function of the Dirichlet problem for the polyharmonic equation is essentially used.
    Date of submission: 14 July 2018 г.


  18. Zhukova N.I. Graphs of totally geodesic foliations on pseudo-Riemannian manifolds
    Status: reviewing
    Abstract.
    We investigate totally geodesic foliations $(M, F)$ of arbitrary codimen\-sion $q$ on $n$-dimensional pseudo-Riemannian manifolds for which the induced metrics on leaves don't degenerate. We assume that the $q$-dimensional orthogonal distribution $\mathfrak{M}$ to $(M, F)$ is an Ehresmann connection for this foliation. Since the usual graph $G(F)$ is not Hausdorff manifold in general, we investigate the graph $G_{\mathfrak{M}}(F)$ of a foliation with an Ehresmann connec\-ti\-on $\mathfrak M$ introduced early by the author. This graph is always Hausdorff ma\-ni\-fold. We prove that on the graph $G_{\mathfrak{M}}(F)$ a pseudo-Riemannian metric is defined, with respect to which the induced foliation and simple foliations formed by the fibers of the canonical projections are totally geodesic. It is proved that the leaves of the induced foliation on the graph are reducible pseudo-Riemannian manifolds and their structure is described. The application to parallel foliations on non-degenerate reducible pseudo-Riemannian manifolds is considered. It is shown that every foliation defined by the suspension of a homomorphism of the fundamental group of a pseudo-Riemannian manifold belongs to the investigated class of foliations.
    Date of submission: 19 July 2018 г.


  19. MANDAL R. ENTIRE SOLUTIONS OF ZERO ORDER OF $q$-SHIFT DIFFERENCE EQUATIONS
    Status: reviewing
    Abstract.
    We investigate the possible uniqueness solutions when the q-shift difference poly- nomials P(f)(z) P λ∈J b λ (z) Q τ λ j=1 f(q λ,j z+δ λ,j ) µ λ,j and P(g)(z) P λ∈J b λ (z) Q τ λ j=1 g(q λ,j z+ δ λ,j ) µ λ,j of entire functions of zero order share a small function under relaxed sharing hy- potheses, which improve a number of existing results.
    Date of submission: 23 July 2018 г.


  20. Kaliev I.A., Sabitova G.S. The second boundary-value problem for the system of equations non-equilibrium sorption
    Status: reviewing
    Abstract.
    The second boundary-value problem for the system of equations non-equilibrium sorption
    Date of submission: 07 Avgust 2018 г.


  21. Bikchentaev A.M. Renormalizations of measurable operator ideal spaces, affiliated to a semifinite von Neumann algebra
    Status: reviewing
    Abstract.
    Let ${\mathcal M}$ be a von Neumann algebra of operators on a Hilbert space $\mathcal H$ and $\tau$ be a faithful normal semifinite trace on $\mathcal{M}$. Let $\mathcal{E}$, $\mathcal{F}$ be an ideal spaces on $(\mathcal{M}, \tau )$. We present the construction method of the mapping $ \tilde{\rho} \colon \mathcal{E}\to [0, +\infty]$ with ``nice'' properties based on the function $\rho$ defined on the positive cone $ \mathcal{E}^+$. Moreover, if $\mathcal{E}= \mathcal{M}$ and $\rho = \tau$ then $ \tilde{\rho}(X)=\tau (|X|)=\|X\|_1$ for all $X\in \mathcal{E}$. With the help of mappings on $\mathcal{E}$ and $\mathcal{F}$ we construct the new mapping with ``nice'' properties on the sum $\mathcal{E}+\mathcal{F}$. We give the examples of such mappings. The results are new even for *-algebra $\mathcal{M}=\mathcal{B}(\mathcal{H})$ of all bounded linear operators on $\mathcal{H}$, equipped with the canonical trace $\tau =\text{\rm tr}$.
    Date of submission: 22 Avgust 2018 г.


  22. Aitzhanov S.E., Zhanuzakova D.T. Blow up of solutions to an inverse problem for a parabolic equation with a double nonlinearity
    Status: reviewing
    Abstract.
    In this article we consider the inverse problem with an integral condition by redefinition for a parabolic type equation. In a bounded domain with a homogeneous Dirichlet condition, sufficient conditions for the destruction of its solution in a finite time are obtained, and also the stability of the solution for the inverse problem with the opposite sign on the nonlinearity of the power type.
    Date of submission: 30 Avgust 2018 г.


  23. Kuznetsov D.F. Expansion of iterated Stratonovich stochastic integrals, based on generalized multiple Fourier series
    Status: reviewing
    Abstract.
    The article is devoted to expansions of iterated Stratonovich stochastic integrals of multiplicities 1-4 on the base of the method of generalized multiple Fourier series. Mean-square convergence of expansions for the case of Legendre polynomials as well as for the case of trigonometric functions is proven. Considered expansions contain only one operation of the limit transition in contrast to its existing analogues. This property is comfortable for the mean-square approximation of iterated stochastic integrals. Results of the article can be applied to numerical integration of Ito stochastic differential equations.
    Date of submission: 01 September 2018 г.


  24. Danilin A.R., Shaburov A.A. Asymptotic expansion of a solution to a singularly perturbed optimal control problem with a convex integral performance index whose terminal part depends on slow and fast variables.
    Status: reviewing
    Abstract.
    We consider an optimal control problem with a convex integral performance index for a linear system with fast and slow variables in the class of piecewise continuous controls with smooth constraints on the control $$ \left\{ \begin{array}{lll} \dot{x}_{\varepsilon} = A_{11}x_{\varepsilon} + A_{12}y_{\varepsilon}+B_{1}u,\quad t\in[0,T],\quad \|u\|\leqslant 1,\\[2ex] \varepsilon\dot{y}_{\varepsilon} = A_{22}y_{\varepsilon} + B_{2}u,\quad x_{\varepsilon}(0)=x^{0},\quad y_{\varepsilon}(0)=y^{0},\quad \nabla\varphi_2(0)=0,\\[2ex] J(u)\mathop{:=}\nolimits \varphi_1\left(x_\varepsilon(T)\right) + \varphi_2\left(y_\varepsilon(T)\right) + \int\limits_{0}^{T}\|u(t)\|^2\,dt\rightarrow \min, \end{array} \right. $$ where $x\in\mathbb{R}^{n}$, $y\in\mathbb{R}^{m}$, $ u\in\mathbb{R}^{r}$; $A_{ij}$ and $B_{i}$ for $i,j=1,2$ are constant matrices of corresponding dimension, and the functions $\varphi_{1}(\cdot), \varphi_{2}(\cdot)$ are continuously differentiable in $\mathbb{R}^{n}, \mathbb{R}^{m},$ strictly convex, and cofinite in the sense of convex analysis. In the general case, Pontryagin's maximum principle is applied as a necessary and sufficient optimality condition in this problem, and there exist unique vectors $l_\varepsilon$ and $\rho_\varepsilon$ that define an optimal control by the formula $$ u_{\varepsilon}(T-t):= \frac{C_{1,\varepsilon}^{*}(t)l_\varepsilon + C_{2,\varepsilon}^{*}(t)\rho_\varepsilon} {S\left(\|C_{1,\varepsilon}^{*}(t)l_\varepsilon + C_{2,\varepsilon}^{*}(t)\rho_\varepsilon\|\right)}, $$ where $$ C_{1,\varepsilon}^{*}(t):= B^*_1 e^{A^*_{11}t} + \varepsilon^{-1}B^*_2\mathcal{W^*}_\varepsilon(t),\quad C_{2,\varepsilon}^{*}(t):= \varepsilon^{-1} B^*_2 e^{A^*_{22} t/\varepsilon}, $$ $$ \mathcal{W}_\varepsilon(t):= e^{A_{11}t}\int\limits_{0}^{t} e^{-A_{11}\tau}A_{12}e^{A_{22} \tau/\varepsilon}\,d\tau, \quad S(\xi)\mathop{:=}\nolimits \left\{ \begin{array}{ll} 2, & 0\leqslant \xi\leqslant2,\\[1ex] \xi, & \xi>2. \end{array} \right. $$ The main difference of this problem from the previous papers is that the terminal part of the performance index depends on the slow and fast variables. It is proved that, in the case of a finite number of points where the type of the control is changed, a power asymptotic expansion can be constructed for the initial vector $\lambda_\varepsilon=\left(l_\varepsilon^*\: \rho_\varepsilon^*\right)^*$ of the conjugate system that defines the type of the optimal control.
    Date of submission: 09 September 2018 г.


  25. SOME PROPERTIES OF THREE DIMENSIONAL $f$ -KENMOTSU MANIFOLDS WITH A SEMI-SYMMETRIC METRIC CONNECTION
    Status: reviewing
    Abstract.
    In this paper, we study 3-dimensional $f$ -Kenmotsu manifold with semi-symmetric metric connection. We also obtain condition for the man- ifold to be ...
    Date of submission: 27 September 2018 г.


  26. BOUA A. HOMODERIVATIONS AND JORDAN RIGHT IDEALS IN 3-PRIME NEAR-RINGS
    Status: reviewing
    Abstract.
    In this paper, we study the commutativity of 3-prime near-rings admitting homoderivations which satisfy certain differential identities on near- ring.
    Date of submission: 02 October 2018 г.


  27. On estimates for oscillatory integrals with phase depending on parameters
    Status: reviewing
    Abstract.
    In this work there are considered estimates for Fourier transform of measure, concentrated to analytic hypersurfaces, containing mitigating factor. In this paper it is given a solution of C.D.Sogge and E.M.Stein problem on optimal decaying of Fourier transform of measures with mitigating factor for partial class of family of analytic surfaces of three-dimensional Euclidian spaces.
    Date of submission: 08 October 2018 г.


  28. Benallia M., Realization of homogeneous Triebel-Lizorkin spaces with $p=\infty $ and characterizations via differences
    Status: reviewing
    Abstract.
    We study the commuting translations and dilations of realizations in the homogeneous Triebel-Lizorkin spaces $\dot{F}_{\infty,q}^{s}(\R)$, then we will give a characterization of the realized spaces of $\dot{F}_{\infty,q}^{s}(\R)$ via differences.
    Date of submission: 11 October 2018 г.


  29. Darus M., Dustov S.T., Lakaev S.N. Threshold phenomenon for a family of the Generalized Friedrichs models with the perturbation of rank one
    Status: reviewing
    Abstract.
    A family $H_\mu(p),$ $\mu>0,$ $p\in\mathbb{T}^3$ of the Generalized Friedrichs models with the perturbation of rank one, associated to a system of two particles, moving on the three dimensional lattice $\mathbb{Z}^3,$ is considered. The existence or absence of the unique eigenvalue of the operator $H_\mu(p)$ lying outside the essential spectrum, depending on the values of $\mu>0$ and $p\in U_{\delta}(p_{\,0})\subset\mathbb{T}^3$ is proven. Moreover, the analyticity of the eigenvalue and associated eigenfunction are shown.
    Date of submission: 06 November 2018 г.


  30. Abdo M.S., Panchal S.K., Wahash H.A. Fractional integro-differential equations with nonlocal conditions and $\psi-$Hilfer fractional derivative
    Status: reviewing
    Abstract.
    Considering a fractional integro-differential equation with nonlocal conditions involving a general form of Hilfer fractional derivative with respect to another function. We show that weighted Cauchy-type problem is equivalent to a Volterra integral equation, we also prove the existence, uniqueness of solutions and Ulam- Hyers stability of this problem by employing a variety of tools of fractional calculus including Banach fixed point theorem and Krasnoselskii’s fixed point theorem. An example is provided to illustrate our main results.
    Date of submission: 11 November 2018 г.


  31. Abdelwanis A.Y. ON TRIPLE DERIVATIONS OF PARTIALLY ORDERED SETS
    Status: accepted в т.0 №0
    Abstract.
    In this paper, as a generalization of derivation on a partially ordered set, the notion of triple derivation is presented and some fundamental properties are investigated for the triple derivation on partially ordered sets. Furthermore, it is shown that the image of an ideal and the set of fixed points under triple derivation are ideals under certain conditions. Finally, the properties of ideals and operations related with triple derivations are examined.
    Date of submission: 15 November 2018 г.


  32. Budak H., SARIKAYA M.Z. ON INEQUALITIES FOR PRODUCTS OF TRIGONOMETRICALLY $\rho$-CONVEX FUNCTIONS
    Status: reviewing
    Abstract.
    In this study, we obtain some new inequalities for products of two trigonometrically $\rho$-convex mappings.
    Date of submission: 20 November 2018 г.


  33. Tursunov F.R. Задачи Коши для линейных эллиптических систем первого порядка с постоянными коэффициентами в трехмерной ограниченной области
    Status: reviewing
    Abstract.
    В статье изучается задача продолжения решения линейных систем эллипти-ческого типа первого порядка с постоянными коэффициентами в области $G$ по ее известным значениям на гладкой части $S$ границы $\partial G$. Рассматриваемая задача относится к некорректным задачам математической физики, т.к. отсутствует непрерывная зависимость решений от начальных данных. Предпо-лагается, что решение задачи существует и непрерывно дифференцируемо в замкнутой области и данные Коши заданы точны. Для этого случая устанавли-вается явная формула продолжения решения. Предлагается также явная формула регуляризации для случая, когда при указанных условиях вместо данных Коши заданы их непрерывные приближения с заданной погрешностью (уклонением) в равномерной метрике.
    Date of submission: 25 November 2018 г.


  34. Volchkov V.V., Volchkova N.P. A one-radius theorem on a sphere with pricked point
    Status: reviewing
    Abstract.
    We study functions on a sphere with pricked point having zero integrals over all admissible spherical caps and circles of a single fixed radius. For such functions a new one-radius theorem is established giving an injectivity condition of corresponding integral transform. An intermediate result of the article is strengthening of the well-known Ungar theorem on spherical means.
    Date of submission: 03 December 2018 г.


  35. Klimentov D.S. Stochastic analogue of the main theorem of the theory of surfaces for surfaces of bounded distortion and positive curvature
    Status: reviewing
    Abstract.
    Stochastic analogue of the main theorem of the theory of surfaces for surfaces of bounded distortion and positive curvature are under consideration. In this note a stochastic analogue of the Gauss--Peterson--Codazzi equations is derived and a stochastic analog of the main theorem of the theory of surfaces for surfaces of positive curvature of bounded distortion is given. In 1956, I.Ya. Bakelman derived the Gauss--Peterson--Codazzi equations for surfaces of bounded distortion. These surfaces are defined by functions with continuous first derivatives and summable with a square generalized second derivatives in the sense of Sobolev. In 1988, Yu.E. Borovsky proved that the Gauss--Peterson--Codazzi equations (derived by I.Ya. Bakelman) uniquely determine the surface of a limited curvature. The purpose of this paper is to present the results of I. Ya. Bakelman. and Borovsky Y.E. in the terms of the theory of random processes in the case of a surface of positive bounded distortion. With the help of two main forms of the surface, two random processes are constructed and the system of equations relating the characteristics (transition functions) of these processes is derived. The resulting system is a stochastic analogue of the system of Gauss--Peterson--Codazzi equations and is a necessary and sufficient condition for the uniquely determination of the surface (up to motion). Note that the generators of random processes are second-order operators generated by the main forms of the surface. For example, if the surface metric is given by the expression $ I = ds^2 = g_{ij} dx^i dx^j$, then the generator of the corresponding process is $ A = g^{ij} \partial_i \partial_j $. Next, a relationship between the transition functions of the random process and the generator coefficients is established. The obtained expressions are substituted into the generalized Gauss – Peterson -- Codazzi equations, which leads to the desired result.
    Date of submission: 25 December 2018 г.


  36. EL-AZHAR H., IDRISSI K., E.H. ZEROUALI E.H. A NOTE ON WEAK POSITIVE MATRICES, FINITE MASS MEASURES AND HYPONORMAL WEIGHTED SHIFTS
    Status: reviewing
    Abstract.
    We study the class of Hankel matrices for which the k × k-block- matrices are positive semi-definite. We prove that a k × k-block-matrix has non zero determinant if and only if all k × k-block matrices have non zero determinant. We use this result to extend the notion of propagation phenomena to k-hyponormal weighted shifts. Finally we give a study on invariance of k- hyponormal weighted shifts under one rank perturbation.
    Date of submission: 29 December 2018 г.


  37. Borisov D.I., Konyrkulzhayeva M.N. Simplest models of quantum graphs with small edges: asymptotics of resolvents and holomorphic dependence for spectrum
    Status: reviewing
    Abstract.
    В работе рассматривается простейший граф, состоящий из двух ребер конечной длины и малого ребра с общей внутренней вершиной. Длина малого ребра считается малым параметром в задаче, описывающем возмущение. На таком графе ребре рассматривается оператор Шрёдингера с условием Кирхгофа во внутренней вершине, условиями Дирихле на внешних вершинах конечных ребер и условием Дирихле либо условием Неймана на внешней вершине малого ребра. Показано, что такие операторы в смысле равномерной резольвентной сходимости сходится к оператору Шредингеру на графе без малого ребра, для которого во внутренней вершине следует поставить условие Дирихле, если на внешней вершине малого ребра исходно ставилось условие Дирихле. Если же на внешней вершине малого ребра исходно ставилось условие Неймана, то в пределе во внутренней вершине сохраняется условие Кирхгофа, в котором тем не менее может измениться коэффициент. Основной полученный результат для резольвент -- выяснение вида первой поправки в их асимптотике и получение оценки остатка. Вторая часть работы посвящена изучению зависимости собственных значений от малого параметра. Несмотря на по сути сингулярное возмущение графа, собственные значения зависят от малого параметра голоморфно и представляется сходящимися степенными рядами. Обнаружено, что при возмущении могут возникать неподвижные собственные значения, остающиеся на месте и не зависящие от малого параметра. Приведён критерий, определяющий возникновение таких собственных значений. Для подвижных собственных значений выписаны формулы для коэффициента в первом члене в их ряде Тейлора.
    Date of submission: 03 January 2019 г.


  38. Huk K., Dilnyi V.N. Hilbert transform on $W_{\sigma}^1$
    Status: reviewing
    Abstract.
    In this article we obtain the criterion of boundless of the Hilbert transform on the Paley - Wiener space in the terms of decomposition. Since we have a simple method of evaluation of the Hilbert transform.
    Date of submission: 06 January 2019 г.


  39. Garif'yanov F.N., Strezhneva E.V. On applications of summary equation induced by a quadrilateral
    Status: reviewing
    Abstract.
    A linear functional equation is investigated in the class of solutions that are holomorphic outside the quadrangle and disappear at infinity. A system of entire functions of a completely regular growth biorthogonal with a piecewise exponential weight system of degrees on three rays is constructed.
    Date of submission: 08 January 2019 г.


  40. Кокунин P.A., Чикрин Д.Е., Chuprunov A.N. ПРЕДЕЛЬНЫЕ ТЕОРЕМЫ ДЛЯ ЧИСЛА ЧАСТИЦ ИЗ ФИКСИРПОВАННОГО МНОЖЕСТВА ЯЧЕЕК
    Status: reviewing
    Abstract.
    Мы рассматриваем случайные величины - количества частиц в первых $K$ ячейках в неоднородной схеме размещения $n$ различимых частиц по $N$ различным ячейкам, где $K$ - фиксированное число. Мы показывает, что при некоторых условиях эти случайные величины ведут себя как независимые пуассоновские случайные величины. В честности, найдены условия, при которых суммы квадратов этих случайных величин, центрированных математическими ожиданиями и нормированных средними квадратическими отклонениями, сходятся по распределению к случайной величине, имеющей хи-квадрат распределение с $K$ степенями свободы, суммы этих случайных величин, центрированных математическими ожиданиями и нормированных средними квадратическими отклонениями, сходятся к гауссовской случайной величине с нулевым средним и единичной дисперсией. Даны приложения этих результатов к математической статистике.
    Date of submission: 24 January 2019 г.


  41. NAVEENKUMAR S. H., WAGHAMORE H.P. UNIQUENESS OF DIFFERENTIAL-DIFFERENCE POLYNOMIALS OF MEROMORPHIC FUNCTIONS SHARING ONE VALUE
    Status: reviewing
    Abstract.
    In this paper, We investigate the uniqueness problems of differential-difference polynomials of hyper order(≥ 1) meromorphic functions sharing a one value and prove some results, which generalise and improve the results due to Y.Liu, J.P.Wang and F.H.Liu [5].
    Date of submission: 25 January 2019 г.


  42. CIHAT DAGLI M. Some Relations On Universal Bernoulli polynomials
    Status: reviewing
    Abstract.
    In this paper, we derive a formula on the integral of products of higher-order Universal Bernoulli polynomials. As an application of this formula, the Laplace transform of periodic Universal Bernoulli polynomials is presented. Moreover, we obtain the Fourier series expansion of higher-order Universal Bernoulli function.
    Date of submission: 28 January 2019 г.


  43. Tursunov F.R., Of the Cauchy problem for the Laplace equation
    Status: reviewing
    Abstract.
    The article studies the problem of continuation of the solution and the stability estimate of the Cauchy problem for the Laplace equation in a domain $G$ by its known values on the smooth part $S$ of the boundary$\partial G$. The considered problem belongs to the problems of mathematical physics, in which there is no continuous dependence of solutions on the initial data. It is assumed that the solution to the problem exists and is continuously differentiable in a closed domain with exactly given Cauchy data. For this case, an explicit formula for the continuation of the solution is established, as well as a regularization formula for the case when, under these conditions, instead of the Cauchy data, their approximations are given with a given error in the uniform metric. We obtain estimates for the stability of the solution of the Cauchy problem in the classical sense .
    Date of submission: 04 February 2019 г.


  44. Langarshoev M.R Exact inequalities of Jackson-Stechkin type and the widths of classes of functions in $L_{2}$
    Status: reviewing
    Abstract.
    In the space $L_{2}^{(r)},$ \, $r\in\mathbb{Z}_{+}$ we obtain exact inequalities of Jackson-Stechkin type, which connect the best approximations of differentiable periodic functions by trigonometric polynomials and integrals containing averaged with weight generalized moduli of continuity. On the basis of the obtained inequalities, classes of functions are introduced that are defined by generalized higher order moduli of modules and satisfy certain constraints. The exact values of various widths of given classes of functions are calculated.
    Date of submission: 08 February 2019 г.


  45. Braeutigam I.N., Polyakov D.M. Asymptotics of the eigenvalues of infinite block matrices
    Status: reviewing
    Abstract.
    We consider operators with a compact resolvent generated by infinite tridiagonal block matrices . Using the method of similar operators we find an asymptotic behavior of arithmetic means of the eigenvalues of such operators. This result is applied to find the asymptotics of eigenvalues of operators generates by classical Jacobi matrices and generalized Jacobi matrices.
    Date of submission: 18 February 2019 г.


  46. Avkhadiev F.G., Nasibullin R.G., Shafigullin I.K. Conformal invariants of hyperbolic plain domains
    Status: reviewing
    Abstract.
    We consider plane domains of hyperbolic type and conformally invariant functionals, defined as best constants for Hardy type inequalities. We study relationships between these functionals and optimal constants in hyperbolic isoperimetric inequalities. We deal with conformally invariant Hardy type inequalities with weight functions depending on the hyperbolic radius of the domain. It is proved that the positivity of Hardy constants is connected with existence of some hyperbolic isoperimetric inequalities of special kind. Also, we prove a comparison theorem for Hardy constants with different numerical parameters, and investigate relationships between the linear hyperbolic isoperimetric inequality in a domain and Euclidean maximum modulus of the domain. In proofs of theorems we use several numerical characteristics of domains with uniformly perfect boundary.
    Date of submission: 20 February 2019 г.


  47. Babajanov B.A., Yakhshimuratov A.B. Integration of equation of Kaup system kind with a self-consistent source in the class of periodic functions
    Status: reviewing
    Abstract.
    In this paper, the inverse spectral problem is applied to the equation of Kaup system kind with a self-consistent source in the class of periodic functions.
    Date of submission: 25 February 2019 г.


  48. Khasanov Yu.Kh. About the climate investment classes almost periodic functions of Besicovitch
    Status: reviewing
    Abstract.
    In the article some conditions of attachment are established the so-called class of almost periodic in the sense of Besicovitch functions with arbitrary Fourier exponents. The results identified are analogous to the known results on the embedding of classes $L_p\,\,\,(1\leq p<\infty)$ are periodic functions.
    Date of submission: 03 Mart 2019 г.


  49. HAMOUDA S. ESTIMATES OF THE LOGARITHMIC DERIVATIVE NEAR A SINGULAR POINT AND APPLICATIONS
    Status: reviewing
    Abstract.
    In this paper, we will give estimates near 0 for the logarithmic derivative $\left| \frac{f^{(k)}(z)}{f(z)}\right|$ where $f$ is a meromorphic function in a region of the form $D(0,R) = {z \in \mathbf{C} : 0 < |z| < R}$. Some applications on the growth of solutions of linear differential equations near a singular point are given.
    Date of submission: 08 Mart 2019 г.


  50. Dyavanal R.S., Muttagi J.B. Uniqueness of a meromorphic function and its linear difference polynomial sharing values partially
    Status: reviewing
    Abstract.
    Uniqueness of a meromorphic function and its linear difference polynomial sharing values partially
    Date of submission: 16 Mart 2019 г.


  51. Morozov A.N. On Continuity and Differentiability in Spaces $ L_p, 0 < p\leq \infty$
    Status: reviewing
    Abstract.
    A function $f\in L_p[I],\;0 < p \le\infty,$ is called $(k,p)$-differentiable at the point $x~\!\in~\!I$ if there is an algebraic polynom $\pi(t)= a_0+a_1 t+ \cdots + a_k t^k,$ for which is true % $ \Vert f-\pi \Vert_{L_p[J_{x,h}]} = o(h^{k+\frac{1}{p}}), \; $ where $ \;J_{x,h}=[x-h; x+h]\cap I. $ The number $k!\cdot a_{k}$ we consider as the value of the corresponding derivative. In this article we are constructed an integral-difference expressions for calculation of derivatives in spaces $L_p,\; p\ge 1,$ and on their basis -- sequences $\{\Lambda_n^k[f]\}$ of piecewise constant functions subordinate to a uniform partition of the segment into $n$ parts. It is shown that for $f\in W_p^k[I] $ the sequence $\{\Lambda_n^k[f] \}$ converges to $ f^{(k)} $ in the norm of the space $ L_p[I], $ also is shown the close connection between the discussed sequences and the spreading of the $k$-times differentiation operator, originally considered on the space $ C^k[I],$ to other spaces. For $ k=0 $ the convergence character of the sequence $ \{\Lambda_n^k[f] \} $ is a criterion for the membership of the function $f$ to the corresponding space.
    Date of submission: 10 Aprel 2019 г.