Editorial backlog

1. 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.

2. 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.

3. Andriyan S.M., Kroyan A.K., Khachatryan K.A. On Solvability of a Class of Nonlinear Integral Equations in $p$ -adic String Theory
Status: accepted в т.0 №0
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.

4. 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.

5. 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.

6. 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.

7. Baskakov A.G., Dikarev E.E. Spectral Theory of Functions in Research of Partial Differential Operators
Status: accepted в т.0 №0
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.

8. 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.

9. 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.

10. 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.

11. Rathod A. CHARACTERISTIC FUNCTION AND DEFICIENCY OF ALGEBROID FUNCTIONS ON ANNULI
Status: reviewing
Abstract.
In this paper, the value distribution theory for meromorphic functions with maximal deficiency sum will be considered for algebroid functions on annuli and also the relationship between the deficiency of algebroid function on annuli and that of their derivatives is studied.
Date of submission: 26 October 2017 г.

12. Gorbatkov S.A., Polupanov D.V. ИССЛЕДОВАНИЕ УСТОЙЧИВОСТИ РЕШЕНИЯ НЕЛИНЕЙНОЙ КРАЕВОЙ ЗАДАЧИ ДЛЯ ПАРАБОЛИЧЕСКОГО УРАВНЕНИЯ
Status: reviewing
Abstract.
Получено аналитическое решение задачи анализа устойчивости решений нелинейной начально-краевой задачи теплопроводности в твердых телах, описываемой параболическим уравнением. Использован разработанный ранее авторами итеро-аппроксимативный метод (ИАМ) и метод функций Ляпунова. ИАМ позволяет выразить решение на каждом шаге итерации в виде рядов по собственным функциям линейной части параболического оператора задачи и создает все предпосылки для применения математического аппарата функций Ляпунова. Приведены результаты расчетов устойчивости теплофизического процесса в трехмерном металлическом теле с переменными по объему теплофизическими свойствами при возмущении начального состояния.
Date of submission: 28 December 2017 г.

13. Singh G., Singh G., A New Subclass of Univalent Functions
Status: reviewing
Abstract.
In this paper, a new subclass $\chi_t(A,B)$ of close-to-convex functions, defined by means of subordination is investigated. Some results such as coefficient estimates, inclusion relations, distortion theorems, radius of convexity and Fekete-Szego problem for this class are derived. The results obtained here is extension of earlier known work.
Date of submission: 02 January 2018 г.

14. 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.

15. 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.

16. 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.

17. Biswas T. RELATIVE ORDER AND RELATIVE TYPE ORIENTED GROWTH PROPERTIES OF GENERALIZED ITERATED ENTIRE FUNCTIONS
Status: reviewing
Abstract.
The main aim of this paper is to study some growth properties of generalized iterated entire functions in the light of their relative orders, relative types and relative weak types.
Date of submission: 21 February 2018 г.

18. Абдуллаева З.~Ш., Фаязов К.С. Условная корректность внутренней краевой задачи для псевдо-дифференциального уравнения с меняющимся направлением времени
Status: reviewing
Abstract.
We consider a problem with data inside of the regularity domain for a pseudo-differential equation, spectral problems related to like thise equations. The uniqueness of the solution of the problem is proved, and the conditional stability of the solution of the problem on the set of correctness is obtained. Using the results of the generalized spectral problem, the form of the solution of the unknown problem is constructed and the incorrectness is proved, namely, the lack of stability of the solution from the data. The conditional stability of the solution on the set of correctness is proved by the methods of functional analysis. The obtained estimates characterizing the conditional stability of the solution of the required problem.
Date of submission: 28 February 2018 г.

19. Khusnullin I.Kh. The perturbation of the quantum and the acoustic waveguide narrow potential
Status: reviewing
Abstract.
We consider boundary value problems in an n-dimensional cylinder, Modeling quantum and acoustic waveguides with a potential- which depend on two parameters, small and large. Small the parameter corresponds to the diameter of the carrier of the potential, and the large - its maximum value. The ratios of the parameters are as follows: the product of a small parameter by a square root of a large parameter tends to zero. In this formulation, the problem is different from the previously investigated topics, that the ratio of the parameters of the tax- wives are weaker, and different types of boundary conditions. The main content of the work is constructing a special transformation that translates the operator to an operator with a small localized perturbation. Moreover, this transformation does not change the spectrum of the original operation. Torah. The condition on the potential is obtained, under which from the edge of the non- an eigenvalue arises, as well as condition absence of such an intrinsic value. In case of occurrence, the principal terms of its asymptotics are constructed. Results are formulated as a theorem.
Date of submission: 01 Mart 2018 г.

20. 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.

21. Chourdhary A., Raj K. ORLICZ DIFFERENCE TRIPLE LACUNARY IDEAL SEQUENCE SPACES OVER N-NORMED SPACES
Status: reviewing
Abstract.
In the present article, we introduce and study some Lacunary I−convergent and Lacunary I−bounded triple difference sequence spaces defined by Orlicz function over n−normed spaces. We shall investigate some algebraic and topological properties of newly formed sequence spaces. We also make an effort to obtain some inclusion results between these spaces.
Date of submission: 26 Mart 2018 г.

22. Godase A.D. ON GENERALIZED $k$- LUCAS SEQUENCES
Status: reviewing
Abstract.
The k- Lucas sequence is companion sequence of k- Fibonacci sequence defined with the k- Lucas numbers which are defined with the recurrence relation L k,n = kL k,n−1 + L k,n−2 with the initial conditions L k,0 = 2 and L k,1 = k. In this paper, we introduce a new generalisation M k,n of k-Lucas sequence. We present generating functions and Binet formulas for generalized k-Lucas sequence, and establish binomial and congruence sums of generalized k-Lucas sequence.
Date of submission: 27 Mart 2018 г.

23. Kalyakin L.A. Capture and holding of resonance far from equilibrium
Status: reviewing
Abstract.
A nonlinear oscillating system with small perturbation is considered. It is supposed that the perturbation corresponds to external pumping which has a slow varying frequency. An asymptotics with respect to small parameter is constructed for the solutions capturing in resonance. The time interval of resonance is defined.
Date of submission: 05 Aprel 2018 г.

24. Sabitov K.B., Sidorov S.N. Inverse problems for a mixed parabolic-degenerate hyperbolic equation in finding the factors of right-hand sides that depend on time
Status: reviewing
Abstract.
For a mixed parabolic-hyperbolic equation with a degenerate hyperbolic part, we consider a direct initial-boundary value problem and inverse problems for determining the factors of right-hand sides that depend on time. On the basis of the formula for solving a direct problem, the solution of inverse problems is equivalent to reducing the solvability of the loaded integral equations. Using the theory of integral equations, we prove the corresponding uniqueness and existence theorems for solutions of the inverse problems and give explicit formulas for the solution.
Date of submission: 07 Aprel 2018 г.

25. Vil'danova V.F. On the uniqueness of a weak solution for the integro-differential aggregation equation
Status: reviewing
Abstract.
In a well-known paper A.~Bertozzi, D.~Slepcev (2010) the existence and uniqueness of solution to a mixed problem for the aggregation equation $$u_t - \triangle A(x, u) + {\rm div} (u\nabla K \ast u)=0$$ are established. The equation describe the evolution of a colony of bacteria in a bounded convex domain $\Omega$. In this paper we prove the existence and uniqueness of the solution of a mixed problem for the more general equation $$\beta(x,u)_t={\rm div}(\nabla A(x,u)-\beta(x,u)G(u))+f (x,u).$$ Summand $f (x,u)$ in the equation models the processes of "birth-destruc\-tion" of bacteria. The class of integral operators $G (v)$ is wide enough and contains, in particular, convolution operators $\nabla K \ast u$. The vector kernel $g (x,y)$ of the $G(u)$ operator can have singularities: $$|\nabla g (x,y)| \le C(1+|x-y| ^{- \lambda}),\ \lambda\in (0,n),\ x,y\in\Omega.$$ Proof of the uniqueness of the solution from the work of A.~Bertozzi, D.~Slepcev is based on the fact of conservation of "masses" $\int_\Omega u(x,t)dx=const$ of bacteria and uses the convexity of $\Omega$ and properties of the convolution operator. Presence in equation the "inhomogeneity" $f (x,u)$ violates the conservation of "mass". The proof of uniqueness proposed in the paper is suitable for a nonuniform equation that does not use the convexity of $\Omega$ and properties of the convolution operator.
Date of submission: 19 Aprel 2018 г.

26. OZTURK O. SOLUTIONS IN THE DIFFERINTEGRAL FORMS OF THE RADIAL SCHR ¨ ODINGER EQUATION FOR TWO DIFFERENT POTENTIALS
Status: reviewing
Abstract.

Date of submission: 03 May 2018 г.

27. Kulaev R.Ch., Shabat A.B. The Darboux system and separation of variables in the Goursat problem for the third-order equation
Status: reviewing
Abstract.
В работе строится редукция трехмерной системы Дарбу для символов Кристоффеля, описывающей ортогональные криволинейные системы координат. Показывается, что соответствующий класс решений системы Дарбу параметризуется шестью функциями одной переменной (по две на каждую из трех независимых переменных). Даются явные формулы для символов Кристоффеля. Изучается ассоциированная с системой Дарбу линейная система, которая сводится к трехмерной задаче Гурса для уравнения третьего порядка с данными на координатных плоскостях. Показывается, что решение задачи Гурса допускает разделение переменных и определяется своими значениями на координатных прямых.
Date of submission: 04 May 2018 г.

28. Dontsova М.V. The solvability of the Cauchy problem for a system of first order quasilinear equations with right-hand sides $f_1={a_2}u(t,x) + {b_2}(t)v(t,x),$ \ $f_2={g_2}v(t,x)$
Status: reviewing
Abstract.
We consider a Cauchy problem for a system of two first order quasilinear differential equations with right-hand sides $f_1={a_2}u(t,x) + {b_2}(t)v(t,x),$ \ $f_2={g_2}v(t,x).$ We obtain the sufficient conditions for the local and the nonlocal solvability of the Cauchy problem in the original coordinates. The study of the solvability of the Cauchy problem is based on the method of an additional argument. The proof of the nonlocal solvability of the Cauchy problem for a system of two quasilinear first order partial differential equations with right-hand sides $f_1={a_2}u(t,x) + {b_2}(t)v(t,x),$ \ $f_2={g_2}v(t,x)$ relies on global estimates.
Date of submission: 10 May 2018 г.

29. Khabirov S.V. Simple partial invariant solutions
Status: reviewing
Abstract.
Вычислены инварианты 4-х мерных подалгебр 11-и мерной алгебры Ли, допускаемой уравнениями гидродинамического типа. Для некоторых подалгебр получены обобщения простых решений : регулярные и нерегулярные частично инвариантные подмодели ранга 1 дефекта 1.
Date of submission: 14 May 2018 г.

30. 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.

31. 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.

32. 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.

33. Valiullina L.G., Ishkin Kh.K., Marvanov R.I. Spectral asymptotics of fourth order differential operator with two turning points}
Status: reviewing
Abstract.
In this paper we consider the operator $Ly=y^{[4]}:=y^{(4)}-2(p(x)y')'+q(x)y,\ D(L)=\{y\in L^2(0,+\infty):\ y^{[k]}(k=\overline{0,3})\in AC[0,+infty), y^{[4]}\in L^2(0,+\infty), \ y(0)=y''(0)=0 \}$ in case when the equation $y^{[4]}=\lambda y,\ \lambda\gg1$ has two turning points: $a_\lambda>0$ and $+\infty$. We derived the asymptotic equation for the spectrum under the assumption of power growth at infinity of functions $p$ and $q$, and under some additional conditions such as smoothness and regularity. This equation allows us to write the first few terms of the asymptotic expansion of eigenvalues $\lambda_n$ as $n\to+\infty$. We note that in the case under consideration the roots of the corresponding characteristic equation grow "not in one force," which leads to additional difficulties in investigating the asymptotics of $N(\lambda)$ by the traditional Carleman-Kostyuchenko method. A series of works by Ya.,T.~Sultanaev was devoted to this occasion in due time.
Date of submission: 07 June 2018 г.

34. 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.

35. 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.

36. 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.

37. 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.

38. Kondratiev G.V. Characteristic numbers of data in a measure metric space - an approach to a weak data equivalence
Status: reviewing
Abstract.
The paper introduces invariant numbers attached to the data, characterizing the density and its derivatives along the direction of maximal density growth at each point. Natural definitions of data maps seem to be non verifiable. The proposed characteristic numbers provide an acceptable approach to a weak data equivalence.
Date of submission: 02 July 2018 г.

39. 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.

40. 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.

41. Sukhov A.B. Discs and boundary uniqueness for psh functions on an almost complex manifold
Status: reviewing
Abstract.
We prove that a totally real submanifold (of maximal dimension) of an almost complex manifold is a boundary uniqueness set for plurisubharmonic functions
Date of submission: 09 July 2018 г.

42. Arif M., Riaz U., Zada A. Stability Analysis of Discrete Linear Time–Varying Systems via Summation Function Approach
Status: reviewing
Abstract.
This article presents a new approach for the exponential stability analysis of discrete linear time–varying systems, which is widely used to study control systems in aerospace engineering. With the introduction of summation function for the discrete linear time–varying systems and satisfying some of its characteristics, a necessary and sufficient condition is obtained for the exponential stability of discrete linear time–varying systems.
Date of submission: 10 July 2018 г.

43. 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.

44. 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.

45. Khan N.U., Khan S.W. A study of unified integrals involving generalized Mittag-Leffler function(GMLF)
Status: reviewing
Abstract.
Many authors have developed integrals, involving a variety of special functions. Recently Khan et.al. have developed many integral formulas involving Whittaker function, MLF, Bessel function and generalized Bessel function. This paper deals with the integrals involving GMLF which are explicitly written in terms of GWHF. Several special cases are obtained as the application of our main results. In view of diverse applications of MLF in mathematical physics, the results here may be potentially applicable in some related areas.
Date of submission: 17 July 2018 г.

46. Startsev S.Ya. Structure of a set of symmetries for hyperbolic systems of the Liouville type and generalized Laplace invariants
Status: reviewing
Abstract.
The present paper is devoted to hyperbolic systems which consist of $n$ partial differential equations and possess symmetry drivers (i.e. differential operators that map any function of one independent variable into a symmetry of the corresponding system). The existence of the symmetry drivers is a hallmark of the Liouville equation and systems similar to it. The composition of a differential operator with a symmetry driver is a symmetry driver again if the coefficients of the differential operator belong to the kernel of a total derivative. We prove that the entire set of the symmetry drivers is generated via the above compositions from a basis set consisting of no more than $n$ symmetry drivers whose sum of orders is smallest of possible ones. It is also proved that if a system admits a symmetry driver of order $k-1$ and generalized Laplace invariants are well-defined for this system, then the leading coefficient of the symmetry driver belongs to the kernel of the $k$-th Laplace invariant. Basing on this statement, we can, after calculating the Laplace invariants of a system, obtain the lower-bound estimates for the smallest orders of the symmetry drivers of this system. This allows us to check whether we can guarantee that a particular set of the drivers is a basis.
Date of submission: 17 July 2018 г.

47. 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.

48. 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.

49. Gaisin A.M., Gaisina G.A. The order of a Dirichlet series with a regular distribution of the exponents in the half-strips
Status: reviewing
Abstract.
We study Dirichlet series that converge only in the half-plane, whose sequence of exponents has the density $b$ and in a sense the regular distribution. The equality of the orders of the Dirichlet series in any closed semi-strips is proved, the width of each of which is not less $2 \pi b$. It is shown that if the width of one of the two semi-strips is less than $2 \pi b$, then the orders in these semi-strips are not equal.
Date of submission: 27 July 2018 г.

50. Pavlenko V.A., Suleimanov B.I. The solutions of analogues of non-stationary Schr\"odinger equations defeined by isomonodromic Hamilton system $H^{2+1+1+1}$
Status: reviewing
Abstract.
We construct solutions for two analogues of the non-stationary Schr\"odinger equations determined by the two Hamiltonian $H^{2+1+1+1}_{s_k}(s_1,s_2, q_1,q_2, p_1, p_2)$ $(k=1,2)$ of the Наmilton system $H^{2+1+1+1}$. This system is the first representative of the famous hierarchy of degenerations of the Garnier system, which was described in 1986 by H. Kimura. (By an explicit symplectic transformation this system reduces to a symmetric Hamilton system. In the constructions of this paper we rely heavily on linear systems of equations of the method of isomonodomi deformations for the system $H^{2+1+1+1}$, written out in 2012 in the article by A. Kavakami, A. Nakamura and H. Sakai.) These analogues of the non-stationary Schr\"odinger equations are evolutionary equations with times $s_1$ and $s_2$, which depend from two space variables. From the canonical non-stationary Schr\"odinger equations %defined by the Hamiltonians $H^{2+1+1+1}_{s_k}$ these analogues arise as a result of the formal replacement of the Planck constant by $-2\pi i$. We construct the exact solutions of the two evolution equations in terms of the solutions corresponding linear ordinary differential equations of the method of isomonodromic deformations. Тhe prospects for constructing similar solutions of analogs of the non-stationary Schr\"odinger equations corresponding to the Hamiltonians of the entire degeneracy hierarchy of the Garnier system are discussed.
Date of submission: 01 Avgust 2018 г.

51. 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.

52. Sharipov R. A. On simultaneous approximation of several eigenvalues of a semi-definite serlf-adjoint linear operator in a Hilbert space
Status: reviewing
Abstract.
Рассматривается самосопряжённый ограниченный снизу линейный оператор в гильбертовом пространстве, дискретный спектр которого непуст и содержит по крайней мере несколько собственных значений $\lambda_{\text{min}}=\lambda_1\leqslant\ldots\leqslant\lambda_m$. Рассматривается и решается задача аппроксимации этих собственных чисел собственными числами некоторого линейного оператора в конечномерном пространстве размерности $s$. Точность аппроксимации может быть сделана сколь угодно большой при $s\to\infty$.
Date of submission: 21 Avgust 2018 г.

53. Abdelwanis A.Y., Boua A. ON GENERALIZED DERIVATIONS OF PARTIALLY ORDERED SETS
Status: reviewing
Abstract.
Let P be a poset and d be a derivation on P . In this research, the notion of generalized d -derivation on partially ordered sets is presented and studied. Several characterization theorems on generalized d - derivations are introduced. The properties of the fixed points based on the generalized d -derivations are examined. The properties of ideals and operations related with generalized d-derivations are studied
Date of submission: 21 Avgust 2018 г.

54. 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.

55. Musin I.Kh. On some linear operators on Fock type space
Status: reviewing
Abstract.
A weighted Hilbert space of entire functions of $n$ variables $F^2_{\varphi}$ is considered in the paper. The weight function $\varphi$ is a continuous function on ${\mathbb C}^n$ depending on modules of variables. Functions of $F^2_{\varphi}$ are described in terms of coefficients of power series expansions. An integral formula for orthogonal projection operator from $L^2_{\varphi}$ to $F^2_{\varphi}$ is obtained. There are found conditions under which a weighted composition operator on $F^2_{\varphi}$ is a Hilbert-Schmidt operator.
Date of submission: 24 Avgust 2018 г.

56. Saks R.S. Оператор градиент дивергенции и пространства Соболева
Status: reviewing
Abstract.
Автор изучает структуру пространства $\mathbf{L}_{2}(G)$ вектор-функций, квадратично интегрируемых по области $G$ трехмерного прост-ранства, его подпространства ${\mathcal{{A}}}$-потенциальных и ${\mathcal{{B}}}$-соленоидальных полей, пространства Соболева в них, и взаимодействие с ними операторов: градиент дивергенции $\nabla\mathrm{div}$, ротор (вихрь) и их обратных.% в построении базисов в. В ограниченной области $G$ с гладкой границей изучается (в пространствах Соболева) краевая задача для оператора %ротор (вихрь) и градиент дивергенции с младшим членом $\lambda \mathbf{u}$. Особенность этого матричного оператора состоит в том, что при $\lambda\neq 0$ он приводим к эллиптическому оператору методом Б.Вайнберга и В.Грушина, а краевая задача удовлетворяет условиям эллиптичености В.Солонникова. Откуда вытекают свойства решений спектральной задачи градиента дивер-генции: а)каждое ненулевое собстенное значение имеет конечную кратность, б)любая обобщенная собственная функция бесконечно дифференцируема вплоть до границы области. Оператор градиент дивергенции имеет самосопряженное расширение $\mathcal{N}_d$ %$\mathbf{V}^0$ в подпространство $\mathcal{A}_{\gamma}$ в $\mathcal{A}$, где он обратим. Его обратный оператор - вполне непрерывен, а собственные векторы образуют полный ортогональный базис в $\mathcal{A}_{\gamma}$. Изучены свойства рядов Фурье градиента дивергенции и его расширения $\mathcal{N}_d$, действующего в $\mathcal{A}_{\gamma}$ и в его подпространствах $\mathbf{A}^{2k}_{\gamma}$, - пространствах Соболева в $\mathcal{A}_{\gamma}$. Выделены шкалы пространств Соболева и доказано, что оператор $\nabla\text{div}+\lambda I$ при почти всех $\lambda$ отображает их взаимно однозначно и непрерывно. Приведены формулы базисных полей градиента дивергенции в шаре. Попутно изложены аналогичные результаты для оператора ротор и его симметричного расширения $S$ в $\mathcal{B}$.
Date of submission: 24 Avgust 2018 г.