Editorial backlog

  1. Arabov M.K., Mukhamadiev E.M., Nurov I.D., Sobirov H.I. SIGNS OF EXISTENCE OF LIMITING CYCLES IN THE SECOND ORDER DIFFERENTIAL EQUATIONS
    Status: accepted в т.9 №2
    Abstract.
    This article takes consideration into the discovery of the limiting cycles in neighborhood of stationary point of the non - smooth second order differential equation. New conditions are found for coefficient of the equation which ensure the existence of a limiting cycle. On the basis of taken results was made sector division in a plane. Created a package of programs for drawing phase portraits in appropriate sectors.
    Date of submission: 19 Aprel 2016 г.


  2. Klyachin A.A. On continuity and differentiability of the maximum values of functions
    Status: reviewing
    Abstract.
    This article discusses the functions which are the maximum values of continuous functions on compact subsets of families. These functions are used, for example, in the study of the geometric structure of the equilibrium of different surfaces -- minimal surfaces, surfaces of constant mean curvature, etc. In this paper we find conditions under which such functions are continuous and differentiable.
    Date of submission: 17 May 2016 г.


  3. Oreshina M.N. Spectral decomposition of a normal operator in a real Hilbert space
    Status: reviewing
    Abstract.
    Unbounded normal operators acting in a real Hilbert space are considered. In this article we carry over the classic results of the spectral theory to the case of such operators. The questions connected with the complexification and decomplexification of normal operators are discussed. We present two real variants of the spectral decomposition and the real functional calculus theorems for unbounded normal operators acting in a real Hilbert space.
    Date of submission: 22 May 2016 г.


  4. Khasanov Yu.Kh., Shakiriv I.A. BILATERAL EVALUATION OF THE NORM OF THE FOURIER OPERATOR
    Status: reviewing
    Abstract.
    The lower and the upper uniform estimates of the Lebesgue constants of the classical Fourier operator are not final. A new and more simple integral representation received for it, then on the basis of which the problem of its top assessment is completely solved and the known lower assessment is improved.
    Date of submission: 14 July 2016 г.


  5. Salo T.M., Skaskiv O.B. The minimum modulus of gap power series and h-measure of exceptional sets
    Status: reviewing
    Abstract.
    For an entire function of the form $f(z)=\sum_{k=0}^{+\infty}f_kz^{n_k}$, where $(n_k)$ is a strictly increasing sequence of non negative integers, we establish conditions when the relations $$ M_f(r)=(1+o(1)) m_f(r),\quad M_f(r)=(1+o(1))\mu_f(r) $$ is true as $r\to+\infty$ outside some set $E$ such that $\text{\rm h-meas }(E)=\int_{E}\frac{dh(r)}{r}<+\infty$, where $h(r)$ is positive continuous function increasing to $+\infty$ on $[0,+\infty)$ with non-decreasing derivative, and $M_f(r)=\max\{|f(z)|\colon |z|=r\},\ m_f(r)=\min\{|f(z)|\colon |z|=r\},\ \mu_f(r)=\max\{|f_k|r^{n_k}\colon k\geq 0\} $ the maximum modulus, the minimum modulus and the maximum term of $f,$ respectively.
    Date of submission: 22 July 2016 г.


  6. Mitrokhin S.I. Об исследовании дифференциального оператора с суммируемым потенциалом с разрывной весовой функцией
    Status: accepted в т.0 №0
    Abstract.
    В работе предлагается новый подход к исследованию дифференциальных операторов с разрывной весовой функцией. Изучены спектральные свойства дифференциального оператора, заданного на конечном отрезке, с разделенными граничными условиями, с суммируемым потенциалом, с условиями <<сопряжения>> в точке разрыва весовой функции. При больших значениях спектрального параметра получена асимптотика фундаментальной системы решений соответствующего дифференциального уравнения, с помощью которой выведено уравнение на собственные значения изучаемого дифференциального оператора. Изучена индикаторная диаграмма и найдена асимптотика собственных значений исследуемого оператора.
    Date of submission: 25 Avgust 2016 г.


  7. Khushtova F.G. The first boundary value problem in the half-strip for a differential equation with Bessel operator and partial derivative of Riemann-Liouville
    Status: accepted в т.0 №0
    Abstract.
    We study the first boundary value problem in the half-strip for a differential equation with Bessel operator and partial derivative of Riemann-Liouville. The representation of the solution in the case of the zero side condition found in terms of the integral transform with the Wright's function at the kernel. Uniqueness of the solution is proved in the class of functions that satisfy the analogue of Tikhonov's condition.
    Date of submission: 16 September 2016 г.


  8. Khrystiyanyn A.Y., Lukivska D.V. Quasi-elliptic functions
    Status: reviewing
    Abstract.
    We investigate quasi-elliptic functions (i. e. certain generalization of elliptic functions). For this class of functions analogues of $\wp$, $\zeta$ and $\sigma$ Weierstrass functions are constructed and relation between quasi-elliptic and $p$-loxodromic functions is obtained.
    Date of submission: 27 September 2016 г.


  9. Baizaev S., Rakhimova M.A. О некоторых функциональных уравнениях в пространствах Шварца и их приложениях
    Status: reviewing
    Abstract.
    В статье изучается вопросы нетривиальной разрешимости функциональных уравнений вида $$(B+r^{2}E)u(r,\theta)=0$$ где $B -$ постоянная комплексная матрица порядка $n$, $E -$ единичная матрица порядка $n$, $(r,\theta) -$ полярные координаты в пространствах Шварца. Получены многообразия всех решений из указанных пространств и даны приложения результатов к задачам нахождения решений полиномиального роста ряда классов эллиптических систем и переопределенных систем.
    Date of submission: 12 December 2016 г.


  10. Ishkina Sh.Kh. Combinatorial bounds of overfitting for threshold classifiers
    Status: reviewing
    Abstract.
    Tightening generalization bounds is a fundamental objective of statistical learning theory. However, accurate and computationally efficient bounds are still unknown even for very simple cases. In~this paper, we consider one-dimensional threshold decision rules. We~use the framework of combinatorial theory of overfitting, which is based on a single probabilistic assumption that all partitions of a~set of objects into an observed training sample and a~hidden test sample can occur with equal probability. We~propose a~polynomial algorithm for computing both probability of overfitting and complete cross-validation.
    Date of submission: 21 December 2016 г.


  11. Galakhov E.I., Salieva O.A. Условия отсутствия решений некоторых неравенств и систем с функциональными параметрами и сингулярными коэффициентами на границе
    Status: accepted в т.0 №0
    Abstract.
    We obtain sufficient conditions for nonexistence of positive solutions of some nonlinear elliptic inequalities and systems that contain the $p(x)$-Laplace operators with variable power exponents and coefficients possessing singularity on the boundary.
    Date of submission: 28 December 2016 г.


  12. Belaidi B., Saidani M. On The Growth of Solutions of Some Higher Order Linear Differential Equations With Meromorphic Coefficients
    Status: reviewing
    Abstract.
    In this paper, we study the growth of meromorphic solutions of the differential equation ....
    Date of submission: 06 January 2017 г.


  13. Shukla I. Simultaneous Quadruple Series Equations Involving Konhauser Biorthogonal Polynomials
    Status: reviewing
    Abstract.
    Spencer and Fano [11] used the biorthogonal polynomials (for the case of k = 2) in carrying out calculations involving penetration of gamma rays through matter. In the present paper an exact solution of simultaneous quadruple series equations involving Konhauser – biorthogonal polynomials of first kind of different indices is obtained by multiplying factor technique due to Noble [13]. This technique has been modified by Thakare [12] to solve dual series equations involving orthogonal polynomials which led to disprove a possible conjecture of Askey [6] that dual series equations involving Jacobi polynomials of different indices cannot be solved. In this paper the solution of simultaneous quadruple series equations involving generalized Laguerre polynomials also have been discussed in a particular case.
    Date of submission: 28 January 2017 г.


  14. Salimov R.B. The study of the behavior of the singular integral with the Hilbert kernel near a point of the new weak continuity of the density
    Status: accepted в т.0 №0
    Abstract.
    We study the behavior of a singular integral with the Hilbert kernel near a fixed point, where the density vanishes as a negative exponent of the logarithm module of the distance from the fixed point to a variable one at the new conditions.
    Date of submission: 08 February 2017 г.


  15. Murtazina S.A., Fazlytdinov M.F., Shevtsova T.V., Yumagulov M.G. Operator methods for computing Lyapunov quantities in the problems on the local bifurcations of dynamical systems
    Status: accepted в т.0 №0
    Abstract.
    This work proposes new formulas to calculate Lyapunov quantities in problems of major scripts of local bifurcations of dynamical systems. Considered a dynamical system, described by differential equations, and point mappings. The proposed formulas are obtained from the general operational method of the study of local bifurcations and do not require transition to normal forms and use theorems about the central diversity.
    Date of submission: 06 Mart 2017 г.


  16. Kononova A.A. On measures generating orthogonal polynomials with similar asymptotic behavior of the ratio near infinity
    Status: accepted в т.0 №0
    Abstract.
    We consider perturbations of orthogonality measure of the system of polynomials that do not change (in some sense) the asymptotical behavior of the ratio of corresponding orthogonal polynomials. The support of the measure consists of the finite number of Jordan curves and may also contain a finite number of mass-points out of the polynomial convex hull of the support of the absolute continuous part of the measure. The problem is a generalization of the problem of compactness of the perturbation of Jacobi operator generated by the perturbation of its spectral measure. A condition, necessary (or necessary and sufficient under some additional restriction) for the stability of the asymptotical behavior of the corresponding orthogonal polynomials is found.
    Date of submission: 09 Mart 2017 г.


  17. Halder S., Sahoo P. UNIQUENESS OF MEROMORPHIC FUNCTIONS CONCERNING DIFFERENTIAL POLYNOMIALS SHARING A SET
    Status: reviewing
    Abstract.
    In this paper, we investigate the uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a set of values with finite weight and obtain some results that generalize and improve the recent results due to H.Y. Xu [J. Computational Analysis and Applications, 16(2014), 942-954].
    Date of submission: 27 Mart 2017 г.


  18. Rathod A. Nevanlinna’s Five-values Theorems for Algebroid Functions
    Status: accepted в т.0 №0
    Abstract.
    By using the second main theorem of the algebroid function, we inves- tigate the problem on two algebroid functions partially sharing five or more values and that improve and generalize the previous results given by Xuan and Gao.
    Date of submission: 06 Aprel 2017 г.


  19. Khashimov A.R. ЭНЕРГЕТИЧЕСКИЕ ОЦЕНКИ ДЛЯ РЕШЕНИЙ КРАЕВЫХ ЗАДАЧ УРАВНЕНИЯ ТРЕТЬЕГО ПОРЯДКА С КРАТНЫМИ ХАРАКТЕРИСТИКАМИ
    Status: reviewing
    Abstract.
    В статье рассмотрена первая краевая задача для уравнения третьего порядка с кратными характеристиками. Для обобщенного решения уравнения установлена энергетические оценки типа аналога принципа Сен-Венана. С помощью этой оценки выявлены наибольшее класс единственности решений краевых задач в зависимости от геометрических характеристик области.
    Date of submission: 07 Aprel 2017 г.


  20. Stepanova I.V. Symmetries of heat and mass transfer equations in viscous fluids
    Status: reviewing
    Abstract.
    The paper is devoted to description of results of application of classical Lie-Ovsyannikov theory to study of heat and mass transfer equations in viscous liquids. Group properties of equations of convective and molecular heat and mass transfer are under consideration. The author analyzed 124 papers and monographs be concerning to mentioned problem.
    Date of submission: 10 Aprel 2017 г.


  21. Singh D.K., Singh P. Wright Function Associated with Fractional Calculus
    Status: reviewing
    Abstract.
    With the marvelous light of fractional calculus, the study of this paper is based on Wright function and Raizada polynomial.
    Date of submission: 17 Aprel 2017 г.


  22. Zhukova N.I. The influence of stratification on groups of conformal transformations of pseudo-Riemannian orbifolds
    Status: reviewing
    Abstract.
    Groups of conformal transformations of $n$-dimensional pseudo-Riemannian orbifolds $({\mathcal N},g)$ are investigated for $n\geq 3$. It is shown that a conformal pseudo-Riemannian geometry is induced on each stratum of that orbifold. For $k\in\{0,1\}\cup\{3,...,n-1\}$ exact estimates of dimensions of the conformal transformation groups of $n$-dimensional pseudo-Rieman\-ni\-an orbifolds admitting $k$-dimensional strata with essential conformal trans\-for\-ma\-tion groups are obtained.
    Date of submission: 08 May 2017 г.


  23. Kachalov V.I. Pseudoholomorphic functions and their application
    Status: reviewing
    Abstract.
    An analysis of asymptotic methods for solving singularly perturbed problems shows that the solutions obtained by means of these solutions depend in two ways on the small parameter: regularly and singularly. This dependence is particularly clearly demonstrated by the method of regularization of S.A.\,Lomov. Moreover, the regularized solu\-tions of singularly perturbed equations can converge in the usual sense. In this connection, it became necessary to study a special class of functions, pseudoholomorphic functions. This is a very important part of the analy\-sis, it is intended to substantiate the main points of the so-called analytic theory of singular perturbations. On the other hand, the relevance of the theory in question is also dictated by the fact that pseudoholomorphic functions, unlike holomorphic functions, are determined when the condi\-tions of the implicit function theorem are violated.
    Date of submission: 16 May 2017 г.


  24. Trynin A.Yu. Uniform convergence of sync-approximations on the functional class
    Status: accepted в т.0 №0
    Abstract.
    We obtain a uniform convergence inside the interval (0, \ pi) of the values of the Lagrange-Sturm-Liouville operators for functions from the class. The class is defined by means of one-way moduli of continuity and change
    Date of submission: 18 May 2017 г.


  25. Bobodzhanov A.A., Safonov V.F. Regularized asymptotics of solutions of integro-differential partial differential equations with rapidly varying kernels
    Status: reviewing
    Abstract.
    The method of Lomov regularization is generalized to partial differential equations with integral operators, the kernel of which contains a rapidly varying exponential factor. The case when the upper limit of the integral operator coincides with the differentiation variable is investigated. For such problems, an algorithm for constructing regularized asymptotics develops. In contrast to the work of Imanaliev M.I., where for analogous problems with slowly varying nuclei only the limit transition is investigated when the small parameter tends to zero, an asymptotic solution of any order (with respect to the parameter) is constructed here.
    Date of submission: 19 May 2017 г.


  26. Kulaev R.Ch., Shabat A.B. Some properties for Jost functions of a Schr\"odinger equation with potential, which is a distribution
    Status: reviewing
    Abstract.
    Работа посвящена задаче кардинального расширения пространства потенциалов в обратной задаче рассеяния для линейного уравнения Шр\"едингера на числовой прямой. Рассматривается оператор Шр\"едингера с потенциалом из пространства обобщенных функций. Это расширение включает в себя не только потенциалы типа $\delta$-функции, но и экзотику типа функции Кантора. На этом пути устанавливаются условия существования и единственности решений Йоста, изучаются их свойства.
    Date of submission: 23 May 2017 г.


  27. Fedotov A.I. Hermite-Fejer polynomials as an approximate solution of the singular integro-differential equations
    Status: reviewing
    Abstract.
    An approximate method for solving singular integro-differen-tial equations in periodic case is justified. An approximate solution is sought in a form of Hermite-Fejer polynomials. The convergence of the method is proved and the errors are estimated.
    Date of submission: 24 May 2017 г.


  28. Kachalov V.I. On the holomorphic regularization of strongly nonlinear singularly perturbed problems
    Status: reviewing
    Abstract.
    The method of holomorphic regularization, which is a logical extension of the Lomov method, allows one to construct solutions of nonlinear singularly perturbed initial problems in the form of series converging in the usual sense in powers of a small parameter. The method itself is based on a generalization of the Poincare decomposition theorem: in the regular case, solutions depend holomorphically on a small parameter, in the singular case the first integrals inherit this dependence.
    Date of submission: 29 May 2017 г.


  29. Zikkos E. A Taylor-Dirichlet series with no singularities on its abscissa of convergence
    Status: reviewing
    Abstract.
    In this paper it is proved that given any non-negative real number $d$, there exists a Taylor-Dirichlet series of the form \[ \sum_{n=1}^{\infty} \left(\sum_{k=0}^{\mu_n-1}c_{n,k} z^k\right) e^{\lambda_n z},\quad c_{n,k}\in \mathbb{C} \] with no singularities on its abscissa of convergence, such that its associated multiplicity-sequence $\Lambda=\{\lambda_n,\mu_n\}_{n=1}^{\infty}$ has the following properties: \noindent (1) the terms of $\Lambda$ are positive real numbers and uniformly separated, \noindent $(\inf_{n\in\mathbb{N}}(\lambda_{n+1}-\lambda_n)>0)$, \noindent (2) $\Lambda$ has density equal to $d$, $\left(\lim_{t\to\infty}\frac{\sum_{\lambda_n\le t}\mu_n}{t}=d<\infty\right)$, \noindent (3) the multiplicities of the terms of $\Lambda$ are unbounded, $(\mu_n\not=O(1))$. The proof is based on the fact that for this sequence $\Lambda$ its Krivosheev characteristic $S_{\Lambda}$ is negative. We remark that when $\mu_n=1$ for all $n\in\mathbb{N}$ the result is false by a well known theorem of P\'{o}lya.
    Date of submission: 30 May 2017 г.


  30. Vinnitskii B.V., Sharan V.L., Sheparovych I.B. About some interpolation problem in the class of functions of exponential type in a half-plane
    Status: reviewing
    Abstract.
    The conditions of solvability of the interpolation problem $f(\lambda_{k} )=d_{k} $ are found in the class of functions of exponential type. This results are applied to research of some problem of the function's splitting.
    Date of submission: 01 June 2017 г.


  31. Klimentov S.B. About Isomorphism of Some Integro-differential Operators
    Status: accepted в т.0 №0
    Abstract.
    In this work representations of <> for solutions of the linear general elliptic system of the first order in the unit circle are considered. The isomorphism of corresponding operators is established in Banach spaces $C^k_\alpha (\overline D) $ and $W^k_p (\overline D) $, $k\geq $1, $0 <\to alpha <$1, $p> $2. These results develop and supplement B.V. Boyarsky's works where representa\-tions of <> where obtained. Also this work supplements author’s results on representations of <> with more difficult operators.
    Date of submission: 02 June 2017 г.


  32. Rahman S. ON PSEUDO-SLANT SUBMANIFOLDS OF NEARLY LORENTZIAN PARA-SASAKIAN MANIFOLDS
    Status: reviewing
    Abstract.
    The object of the present paper is to study pseudo slant submanifolds of nearly Lorentzian para-Sasakian manifolds. The necessary and sufficient conditions on a totally umbilical proper-slant submanifolds are worked out and obtain some interesting results regarding such submanifolds. The integrability condition of the distribution of pseudo -slant submanifolds of nearly Lorentzian para-Sasakian manifolds are also discussed.
    Date of submission: 10 June 2017 г.


  33. Poluboyarova N.M. On the instability of extremals of the potential energy functional
    Status: reviewing
    Abstract.
    In this paper study has been done on problem of stability and instability of the potential energy functional. By stability we mean the sign-definiteness of the second variation. The expression for the second variation of the functional is calculated. With the capacitive method it is obtain to make the feature of instability extremals. Proved that stability parabolic extremals are planes. We have written the equation of extremals and the second variation of the functional for n-dimensional surfaces of revolution.
    Date of submission: 18 June 2017 г.


  34. Berdellima A. ON A CONJECTURE OF KHABIBULLIN ABOUT A PAIR OF INTEGRAL INEQUALITIES
    Status: accepted в т.9 №2
    Abstract.
    It is known that in general Khabibullin’s conjecture is not true. Sharipov [8] constructed a counterexample when $n = 2$ and $\alpha = 2$. In this paper we develop a method of how to construct a counterexample for the more general case $n > 2$ and $\alpha > 1/2$.
    Date of submission: 24 June 2017 г.


  35. Garayev M., Guediri H., Sadrawi H. New Characterizations of Bloch spaces, Bers-type and Zygmund-type spaces and Related Questions
    Status: reviewing
    Abstract.
    We give in terms of Berezin symbols new characterizations of\ the Bloch spaces $\mathcal{B}$ and $\mathcal{B}_{0},$ Bers-type and the Zygmund-type spaces of analytic functions on the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}.$ Moreover, we discuss some properties of Toeplitz operators on the Bergman space $L_{a}^{2}(\mathbb{D}).$ A new characterization of\ some function space with variable exponents is also given.
    Date of submission: 29 June 2017 г.


  36. Kopachevsky N.D., Tsvetkov D.O. Малые движения идеальной стратифицированной жидкости со свободной поверхностью, полностью покрытой крошеным льдом
    Status: reviewing
    Abstract.
    Let a rigid immovable vessel be partially filled with an ideal incompressible stratified fluid. We assume that in an equilibrium state the density of a fluid is a function of the vertical variable $x_3,$ i.e., $\rho_0=\rho_0(x_3).$ In this case the gravitational field with constant acceleration $\vec g=-g\vec e_3$ acts on the fluid, here $g>0$ and $\vec e_3$ is unit vector of the vertical axis $Ox_3,$ which is directed opposite to $\vec g.$ Let $\Omega$ be the domain filled with a fluid in equilibrium state, $S$ be rigid wall of the vessel adherent to the fluid, $\Gamma$ be a free surface completely covered with a crumbled ice. The initial boundary value problem is reduced to the Cauchy problem \begin{equation*} \begin{split} &\mathcal A \frac{d^2x}{dt^2} + \mathcal C x = f(t), \quad x(0)=x^0, \quad x^{'}(0)=x^1, \\ &0<< \mathcal A= \mathcal A^{*} \in \mathcal L(\mathcal H), \quad 0 \leq \mathcal C = \mathcal C^{*} \in \mathcal L(\mathcal H). \end{split} \end{equation*} in some Hilbert space $\mathcal H$. The theorem on strong solvability of initial boundary value problem is proved.
    Date of submission: 29 June 2017 г.


  37. Baskakov A.G., Uskova N.B. Linear differential operator with an involution as generator of group of operators
    Status: reviewing
    Abstract.
    We consider mixed problem for first order differential equation with involution. By the method of similar operators, he differential operator, which defined by this differential equation, is transformed in orthogonal direct sum of operators. By the main theorem we construct group of operators and we describe the weak solutions of this problem. We use this theorem for Fourier method.
    Date of submission: 29 June 2017 г.


  38. Andriyan S.M., Kroyan A.K., Khachatryan K.A. On Solvability of a Class of Nonlinear Integral Equations in $p$ -adic String Theory
    Status: reviewing
    Abstract.
    In this paper a class of nonlinear integral equations, which has direct application in the $ p $ -adic string theory, is studied. The existence of a nontrivial continuous odd and bounded solution on the whole axis is proved. With some additional conditions, the uniqueness of the constructed solution in the certain class of continuous functions is established as well.
    Date of submission: 15 July 2017 г.


  39. RamReddy T., Shalini D., Vamshee Krishna D. THIRD ORDER HANKEL DETERMINANT FOR STAR LIKE FUNCTIONS OF ORDER $\alpha$
    Status: reviewing
    Abstract.
    The objective of this paper is to obtain best possible upper bound to the third Hankel determinant for the class of starlike functions of order $\alpha$ ($0 \leq \alpha < 1$), using Toeplitz determinants.
    Date of submission: 17 July 2017 г.


  40. Muravnik A.B. On Qualitative Properties of Solutions of Quasilinear Parabolic Equations Admitting Degenerations at Infinity
    Status: reviewing
    Abstract.
    We consider the Cauchy problem для for quasilinear parabolic equations of the kind $\rho(x)u_t=\Delta u + g(u)|\nabla u|^2,$ where the positive coefficient $\rho$ admits a degeneration at infinity, while the coefficient $g$ either is a continuous function or admits power singularities such that the power does not exceed one. The long-time behavior of (classical) solutions of the specified problem is investigated.
    Date of submission: 21 July 2017 г.


  41. Zhapsarbayeva L.K., Kanguzhin B.E., Konyrkulzhayeva M.N. Selfadjoint restriction of the maximal operator on the graph
    Status: reviewing
    Abstract.
    The paper is devoted to the linear differential operators defined on graphs. In this paper Lagrange formula for second-order differential operator on a graph with Kirchhoff conditions at its internal vertices is derived. The conditions for a selfadjointness of the differential operator on the graph are established.
    Date of submission: 24 July 2017 г.


  42. Ehrgashev T.G. Third Double-Layer Potential for a Generalized Bi-Axially Symmetric Helmholtz Equation
    Status: accepted в т.0 №0
    Abstract.
    The double-layer potential plays an important role in solving boundary value problems for elliptic equations, and in the study of which for a certain equation, the properties of the fundamental solutions of the given equation are used. All the fundamental solutions of the generalized bi-axially symmetric Helmholtz equation were known, and only for the first one was constructed the theory of potential. Here, in this paper, we aim at constructing theory of double-layer potentials corresponding to the third fundamental solution. By using some properties of one of Appell's hypergeometric functions in two variables, we prove limiting theorems and derive integral equations concerning a denseness of double-layer potentials.
    Date of submission: 31 July 2017 г.


  43. Bandura A.I., Skaskiv O.B. Exhaustion by balls and entire functions of bounded $\mathbf{L}$-index in joint variables
    Status: reviewing
    Abstract.
    We prove criteria of boundedness of $\mathbf{L}$-index in joint variables which describe local behavior of partial derivatives on sphere in $\mathbb{C}^n.$ Some obtained results are new even for entire functions of bounded index in joint variables, i. e. $\mathbf{L}(z)\equiv 1,$ because we used an exhaustion of $\mathbb{C}^n$ by balls instead an exhaustion of $\mathbb{C}^n$ by polydiscs.
    Date of submission: 08 Avgust 2017 г.


  44. Baskakov A.G., Dikarev E.E. Spectral Theory of Functions in Research of Partial Differential Operators
    Status: reviewing
    Abstract.
    Spectral properties of differential operators with constant coefficients defined on subspaces of space of bounded continuous functions are studied. Necessary and sufficient conditions of invertibility are obtained under condition of regularity at the infinity (ellipticity type conditions) of polynomial which describes such operators. Spectrum, images and kernels are described. Conditions of compactness of resolvent of differential operators are obtained. Main results are obtained by methods of harmonic analysis and spectral theory of Banach modules.
    Date of submission: 10 Avgust 2017 г.


  45. Rubinshtein A.I. On the Bary-Stechkin Theorem
    Status: reviewing
    Abstract.
    We concider the problem on the modulus of continuity for the analogue conjugate functions in the case of functions in the case of functions defined on the diadic group. It is shown that for this case no analogue a Bary--Stechkin theorem.
    Date of submission: 18 Avgust 2017 г.


  46. Il'yasov Y.Sh., Kholodnov E.E. On global instability of solutions to hyperbolic equations with non-Lipschitz non-linearity
    Status: accepted в т.0 №0
    Abstract.
    Stability of ground state type solutions for hyperbolic eguations with non-Lipschitz non-linearity and p-Laplacian is studied.
    Date of submission: 22 Avgust 2017 г.


  47. Das S. ON THE ZEROS OF A POLYNOMIALS
    Status: reviewing
    Abstract.
    In this paper we extend a classical result due to Cauchy [6] for moduli of all zeros of a polynomial of degree $n$. our result is best possible and sharpen some well-known results. In many cases the new bounds are much better than some other known bounds.
    Date of submission: 30 Avgust 2017 г.


  48. Тоtiеvа Zh. THE PROBLEM OF DETERMINING THE PIEZOELECTRIC MODULE OF ELECTROVISCOELASTICITY EQUATION
    Status: reviewing
    Abstract.
    We consider the problem of finding the the piezoelectric module $e(x_3),\ x_3>0$, occurring in the system of integro-differential electroviscoelasticity equations. The medium density and the Lame parameters are assumed to be function of one variable. The integrand $k(t),\ t\in [0,T]$ is known. As additional information is the Fourier transform of the first component of the displacements vector for $x_3 = 0$. The results are the theorems on the existence of a unique solution of the inverse problem and the theorem of stability.
    Date of submission: 31 Avgust 2017 г.


  49. Saks R.S. Operator gradient of divergencie in subspaces of $\mathbf{L}_{2}(G)$ space.
    Status: reviewing
    Abstract.
    Автор изучает структуру пространства $\mathbf{L}_{2}(G)$ вектор-функций, квадратично интегрируемых по ограниченной односвязной области $G$ трехмерного пространства с гладкой границей и роль операторов градиента дивергенции и ротора в построении базисов в подпространствах ${\mathcal{{A}}}$ и ${\mathcal{{B}}}$. Доказана само-сопряженность расширения $\mathcal{N}_d$ оператора $\nabla\mathrm{div}$ в подпро-странство $\mathcal{A} _ {\gamma}\subset {\mathcal{{A}}}$ и базисность системы его собственных функций. Выписаны явные формулы решения спектральной задачи в шаре и условия разложимости вектор-функции в ряд Фурье по собственным функциям градиента дивергенции. Изучена разрешимость краевой задачи:\newline $\nabla\mathrm{div}\,\mathbf{u}+\lambda\,\mathbf{u}=\mathbf{f}$\, в\, $G$, \, $ (\mathbf {n}\cdot\mathbf {u})|_{\Gamma}=g$ в пространствах Соболева $\mathbf{H}^{s}(G)$ порядка $s\geq 0$ и в подпространствах. Попутно изложены аналогичные результаты для оператора ротор и его симметричного расширения $S$ в $\mathcal{B}$.
    Date of submission: 13 September 2017 г.


  50. Gaisin R.A. Pavlov-Korevaar-Dixon Interpolation Problem with Majorant from Convergence Class
    Status: reviewing
    Abstract.
    We study interpolation problem in the class of entire functions of exponential type determined by some majorant from convergence class. Analogous problem in subclass in which majorant carried concavity property was considered by B. Berndtsson but nodes were in the points of some subsequence of natural numbers. He obtained criterion of solvability of this interpolation problem. He was first who used Hermander's $\overline{\partial}$-problem solving method. In the works of A.I. Pavlov, J. Korevaar and M. Dixon interpolation sequences in the sense of B. Berndtsson were succesfully used in series of problems of complex analysis. In addition was found some relation with approximative properties of the system of powers $\{z^{p_n}\}$ and with well known Polya and Macintyre problems. In this paper criterion of interpolationality in more general sense is established for arbitrary sequence of real numbers. In the proof of the main theorem modified method of B. Berndtsson is used.
    Date of submission: 14 September 2017 г.


  51. Startsev S.Ya. On differential substitutions for evolution systems
    Status: reviewing
    Abstract.
    We obtain necessary and sufficient conditions for a differential substitution to be admitted by a family of evolution systems that depends on an arbitrary function. As an illustration, such family is constructed for a multi-component Cole-Hopf substitution and we demonstrate that this family contains all linear systems the right-hands sides of which do not depend on the derivatives of order less than one. As a result, a family of C-integrable systems of arbitrary high order is obtained.
    Date of submission: 14 September 2017 г.


  52. Pavlenko V.A., Suleimanov B.I. <> isomonodromy Hamilton system $H^{\frac{7}{2}+1}$
    Status: reviewing
    Abstract.
    We study compatible linear evolutional equations with times $s_1$, $s_2$, which depend from two space variables. These evolution equations are analogues of the non-stationary Schr\"odinger equations determined by the two Hamiltonian $H^{\frac{7}{2}+1}_{s_k}(s_1,s_2, q_1,q_2, p_1, p_2)$ $(k=1,2)$ of pair compatible Hamilton systems, which can be allowed the method of isomo\-nodromic deformations. We construct the solution of the two evolutional equations in terms of solution corresponding ordinary differential equations of the method of isomonodrome deformations. In this work we also show that solutions of the two Hamilton systems with Hamiltonians $H^{\frac{7}{2}+1}_{s_k}$ explicitly set by common solutions of the Korteweg de Vries equation $u_t+u_{xxx}+uu_x=0$ and of the fifth order ordinary differential equation.
    Date of submission: 15 September 2017 г.


  53. Gadylshin T.R., Mukminov F.Kh. Perturbation of second order nonlinear equations by the delta-like potential
    Status: reviewing
    Abstract.
    Boundary value problems for one-dimensional quasyilinear second order equation are considered, perturbed by the Delta-shaped potential $\varepsilon^{-1}Q\left(\varepsilon^{-1}x\right)$, where $Q(\xi)$ --- finite function, $0<\varepsilon\ll1$. By the method of integral inequalities solutions to these boundary value problems are built with accuracy $O(\varepsilon)$. To prove the existence of a solution the source and limit problems the fixed point theorem is used. For linear boundary value problem all types of boundary conditions are considered.
    Date of submission: 16 September 2017 г.


  54. 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.


  55. Gazizov R.K., Gainetdinova A.A. Operator of invariant differentiation and its application for integrating systems of oedinary differential equations
    Status: reviewing
    Abstract.
    We introduce an algorithm for integrating $n$-th-order systems of ordinary differential equations admitting $n$-dimensional Lie algebras of operators. The algorithm is based on invariant representation of considered system by the invariants of admitting Lie algebra and the application of the operator of invariant differentiation of special type. We proved the existence of the operator of invariant differentiation of suggested type and gave the algorithm for its construction. The suggested algorithm is compared with the known methods of integrating the first and second-order equations. Also we illustrated the application of the algorithm to the integrating systems of ordinary differential equations by the example of a systems of two second-order equations.
    Date of submission: 02 October 2017 г.


  56. Kavitha J., Lellis Thivagar M. Hypergroup via Nano Topology
    Status: reviewing
    Abstract.
    This paper deals with hypergroup theory in nano topology. Also we introduce nano least open sets which forms nano topology. Moreover, we arrive the results hypergroup forms nano topology and vice versa by using particular case. Finally, we obtain invertible subhypergroup nano topology is coarser than nano topology through least open sets.
    Date of submission: 05 October 2017 г.


  57. Nurmagomedov A.A., Rasulov N.K., Umalatov A.A. Approximation Function of the Partial Sums by Fourier on Polynomials Orthogonal on Arbitrary Sets
    Status: reviewing
    Abstract.
    For arbitrary continuous function $f(t)$ on the segment $[-1, 1]$ is constructed discrete sums by Fourier on system polynomials $\{\hat{p}_{k,N}(t)\}_{k=0}^{N-1}$ forming an orthonormals system on any finite non-uniform set $T_N=\{t_j\}_{j=0}^{N-1}$ of $N$ points from segment $[-1, 1]$ with weight $\Delta{t_j}=t_{j+1}-t_j.$ Approximation properties of the constructing partial sums $S_{n,N}(f,t)$ order $n\leq{N-1}$ are investigated. Namely a two-sided pointwise estimate is obtained for the Lebesgue function $L_{n,N}(t)$ discrete Fourier sums for $n=O(\delta_N^{-1/5}), \delta_N=\max_{0\leq{j}\leq{N-1}}\Delta{t_j}$. Coherently also is investigated the question of the convergence of $S_{n,N}(f,t)$ to $f(t).$ In particular, is obtained the estimation deflection partial sums $S_{n,N}(f,t)$ from $f(t)$ for $n=O(\delta_N^{-1/5})$ with is depended from $n$ and position of a point $t$ on the $[-1, 1].$
    Date of submission: 05 October 2017 г.