Editorial backlog
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Khaira G.K., Singh G. Coefficient bounds for a class based on class of Starlike function
Status: reviewing
Abstract. In this paper author have defined a new class ${S}^*\left(f(\frac{z}{2})\right)$ and its subclasses with differential subordination, also obtained coefficient bounds for the function $$f(z)= z+a_2z^2+a_3z^3+...$$ (which is analytic and univalent in unit disk $\mathcal{U}=\{z; |z|<1\}$)and its natural to seek relationship between these coefficients which is famously known as Fekete szeg$\ddot{o}$ inequality with their extremal function.
Date of submission: 12 Mart 2019 г.
- 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.
- Akobirshoev M.O., Shabozov M.Sh. Среднеквадратическое
приближение ``углом'' в $L_{2}$ и значение квазипоперечников некоторых классов функции
Status: reviewing
Abstract. In metric $L_{2}$ were obtained the exact
inequalities which relate the best approximation of differentiable
$2\pi$ periodic functions in each of variables of functions $f(x,y)$
by trigonometrical ``angles'' with integrals having in integrant
modules of continuity of higher order mixed derivatives of these
functions. For the some classes of functions defined by modules of
continuity the Kolmogorov's and Linear quasiwidths were calculated.
Date of submission: 01 Aprel 2019 г.
- Kuznetsova M.N. Classification of a Subclass of Quasilinear Two-Dimensional Lattices by means of Characteristic Algebras
Status: reviewing
Abstract. We consider a classification problem of integrable cases of the Toda type two-dimensional lattices $u_{n,xy} = f(u_{n+1},u_n,u_{n-1}, u_{n,x},u_{n,y})$. It is commonly accepted that for a given equation existence of a large class of integrable reductions indicates integrability. Our classification algorithm is based on this observation. We call the lattice integrable if there are cutting off boundary conditions allowing to reduce the lattice to an infinite number of hyperbolic type systems integrable in the sense of Darboux. The study of the obtained finite system is carried out by means of the characteristic Lie-Rinehart algebras. Here we concentrate on a subclass of quasilinear lattices of the form $u_{n,xy}=p(u_{n-1},u_n,u_{n+1}) u_{n,x} + r(u_{n-1},u_n,u_{n+1})u_{n,y} +q(u_{n-1},u_n,u_{n+1})$.
Date of submission: 03 Aprel 2019 г.
- 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.
- Rakhmelevich I.V. О многомерных детерминантных дифференциально-операторных уравнениях
Status: reviewing
Abstract. We consider a class of multi-dimensional determinant differential-operator equations, the left side of which represents a determinant with the elements containing a production of linear one-dimensional differential operators of arbitrary order. The right side of the equation depends on the unknown function and its first derivatives. There are separately investigated both homogeneous and inhomogeneous determinant differential-operator equations. The theorems on decreasing of dimension of equation are prooved. The theorem on interconnection between the solutions of initial equation and the solutions of some auxiliary linear equation is prooved for the homogeneous equation. Also there is obtained the solution of the homogeneous equation for the case when the linear differential operators containing in it, have proportional eigenvalues. There are received the solutions of travelling wave type, the solutions in the form of generalized monomials, and also the solutions expressed through the eigenfunctions of linear operators containing in the equation and the solutions expressed through the functions belong to the kernels of these operators.
Date of submission: 22 Aprel 2019 г.
- Allahverdiev B.P., Tuna H. Existence of the solutions for a nonlinear singular $q$-Sturm-Liouville problems
Status: reviewing
Abstract. In this paper, we investigate a nonlinear $q$-Sturm-Liouville prob-
lem on the semi in…nite interval in which the limit-circle case holds at
in…nity for $q$-Sturm-Liouville expression. We show the existence and
uniqueness of the solutions for this problem.
Date of submission: 24 Aprel 2019 г.