Editorial backlog
- Rathod A. Uniqueness and Value Sharing of Meromomorphic
Functions on Annuli
Status: reviewing
Abstract. In this paper, we study meromorphic functions that share only one
value on annuli and prove the following results. Let f(z) and g(z) two non
constant meromorphic functions on annli and For n ≥ 11, if f n f 0 and g n g 0
share the same nonzero and finite value a with the same multiplicities on an-
nuli, then f ≡ dg or g = c 1 e cz and f = c 2 e −cz , where d is an (n + 1) th root of
unity, c, c 1 and c 2 being constants.
Date of submission: 04 June 2019 г.
- Serikbaev D. INVERSE PROBLEM FOR FRACTIONAL ORDER
PSEUDO-PARABOLIC EQUATION WITH INVOLUTION
Status: reviewing
Abstract. In this paper we consider an inverse problem of recovering the right-
hand side of a fractional pseudo-parabolic equation with involution. The results on
existence and uniqueness of solutions of this problem are presented by using Fourier
analysis. The classical and generalized solutions of the inverse problem are studied.
Moreover, the direct problem is also investigated.
Date of submission: 04 February 2020 г.
- Ahmadova A.N., Aliev R.A. Boundedness of the discrete Hilbert transform on discrete Morrey spaces
Status: reviewing
Abstract. The Hilbert transform has been well studied on classical Lebesgue and Morrey spaces. But its discrete version, which also has numerous applications, has not been fully studied. In this paper, we prove that the discrete Hilbert transform is a bounded operator in discrete Morrey spaces.
Date of submission: 14 February 2020 г.
- Begmatov A.X., Ochilov Z.X. Uniqueness and existence of solving problems of Integral Geometry of Volterra Type with a Weight Function of a Special Type
Status: reviewing
Abstract. В работе рассматривается новый класс задач интегральной геометрии вольтеровского типа с весовой функцией специального вида. Доказана теорема единственности, выведены формулы обращения, получены оценки устойчивости в пространствах Соболева, тем самым показана слабая некорректность поставленной задачи. Приводятся формулировка и доказательство теорема существования решения задачи восстановления функции по семейству ломанных с весовой функцией специального вида в полосе.
Date of submission: 27 February 2020 г.
- ABDALLAH A.M. , RASLAN K.R., SOLIMAN A.A. ANALYSIS FOR FRACTIONAL INTEGRO-DIFFERENTIAL
EQUATION WITH TIME DELAY
Status: reviewing
Abstract. The main purpose of this paper is to solve some linear and non-
linear Caputo fractional Volterra-Fredholm integro-differential equations with
delay. In addition, we investigate the convergence analysis for these equations.
A semi-analytical method is described and illustrated with examples.
Date of submission: 05 Mart 2020 г.
- Zhukovskiy E.S., Merchela W. On covering mappings in generalized metric spaces in the study of implicit differential equations
Status: reviewing
Abstract. Пусть на множестве $X\neq \emptyset$
задана метрика $\rho_X :X\times X \to [0,\infty],$ а на $Y\neq\emptyset$~---
расстояние $d_Y :Y\times Y \to [0,\infty],$ удовлетворяющее только
аксиоме тождества. Для отображений $X\to Y$ определены понятие накрывания
и липшицевости. Сформулированы условия
существования решения $x\in X$ уравнений вида $F(x,x)=y,$ $y \in Y,$
с~отображением $F:X\times X \to Y,$ являющимся накрывающим по одному из
аргументов и липшицевым по другому. Полученное утверждение применено для
исследования разрешимости функционального уравнения с отклоняющимся аргументом
и задачи Коши для неявного дифференциального уравнения.
Для этого исследования на пространстве $S$ измеримых по (Лебегу) функций
$z:[0,1]\to \mathbb{R}$ определено расстояние
\begin{equation*}
d (z_1,z_2)=\mathrm{vrai}\sup_{t\in[0,1]}\theta(z_1(t),z_2(t)),\,\,\, z_1,z_2\in S,
\end{equation*}
где для непрерывной функции $\theta:\mathbb{R}\times \mathbb{R} \to [0,\infty] $
выполнено $\theta(z_1,z_2)=0$ тогда и только тогда, когда $z_1=z_2.$
Date of submission: 23 Mart 2020 г.
- , Emin Özdemir M. Generalization Of Hadamard {Type Trapezoid Inequalities For Fractional Integral Operators
Status: reviewing
Abstract. The article formulates and proves the identity with the participation of the fractional integration operator.
Based on this identity, new generalized Hadamard-type integral inequalities are obtained for functions for which the second derivatives are convex and take values at intermediate points the integration interval. It is shown that the upper limit of the absolute error of inequality decreases by approximately $n^{2}$ times ($n$ is the number of intermediate points). In a particular case, the estimates obtained are consistent with those available in the literature.
Date of submission: 01 Aprel 2020 г.
- Budochkina S.A. On connection between variational symmetries and algebraic structures
Status: reviewing
Abstract. The present paper deals with a general approach to establish connection between symmetries of $B_u$-potentials (variational symmetries) and algebraic structures (Lie-admissible algebras and Lie algebras). For this purpose, such bilinear operations as $({\EuScript S}, {\EuScript T})$-product, ${\EuScript G}$-commutator, commutator are defined in the space of symmetry generators of functionals. In addition, the connection between symmetries of functionals in the case of bipotentiality of their gradients and Lie algebras is also established.
Date of submission: 15 Aprel 2020 г.
- ABDALLAH A.M. , RASLAN K.R., SOLIMAN A.A. ON SOME MODIFIED METHODS ON FRACTIONAL DELAY
AND NONLINEAR INTEGRO-DIFFERENTIAL EQUATION
Status: reviewing
Abstract. The fundamental objective of this work is to construct a compar-
ative study of some modified methods with Sumudu transform on fractional
Delay integro-differential equation. The existed solution in the equation is very
accurately computed. The aforesaid methods is presented by an illustrative example.
Date of submission: 27 Aprel 2020 г.
- Эрмаматова З.Э. Регуляризация решения задачи Коши для уравнений Пуассона в
ограниченной области типа конуса
Status: reviewing
Abstract. In this paper, we investigate the continuation of
the solution of the Poisson equations in a bounded space domain from
the values of the solution on part of the boundary of this domain,
i.e., a Cauchy problem is studied. We construct an approximate
solution of this problem based on the Carleman-Yarmuhamedov function
method.
Date of submission: 29 Aprel 2020 г.
- Langarshoev M.R , Khorazmshoev S.S. Exact inequalities of Jackson-Stechkin
type and the widths of classes of functions in $L_2$.
Status: reviewing
Abstract. The paper considers new exact inequalities of the Jackson-
Stechkin type, connecting best approximations of differentiable periodic
functions by trigonometric polynomials with integrals containing generalized
modulus of continuity. For classes of functions determined using these
characteristics, the exact values of some known n-diameters are calculated.
Date of submission: 04 May 2020 г.
- БАХШАЛЫЕВА М.Н., ХАЛИЛОВ Э.Г. Исследование приближенного решения интегрального уравнения смешанной краевой задачи для уравнения Лапласа
Status: reviewing
Abstract. In this work, the approximate solution of the curvilinear integral equation of the mixed boundary value problem for the Laplace equation is considered. We construct a quadrature formula for one class of curvilinear integrals at some chosen control points, and then we replace our equation with the system of algebraic equations. We also establish the existence and uniqueness of the solution of this system, prove its convergence to the exact solution of integral equation and determine the convergence rate of the method. Besides, we construct a sequence which converges to the solution of the mixed boundary value problem for the Laplace equation.
Date of submission: 06 May 2020 г.
- Gunawan H., Hakim D.I., Putri A.S. On Geometric Properties Of Morrey Spaces
Status: reviewing
Abstract. In this article, we show constructively that Morrey spaces are not uniformly non-ℓ 1
n for
any n ≥ 2. This result is sharper than those previously obtained in [4, 11], which show that Morrey
spaces are not uniformly non-square and also not uniformly non-octahedral. We also discuss the
n-th James constant C (n)
J
(X) and the n-th Von Neumann-Jordan constant C (n)
NJ (X) for a Banach
space X, and obtain that both constants for any Morrey space M p q (R d ) with 1 ≤ p < q < ∞ are
equal to n.
Date of submission: 07 May 2020 г.
- Akgün R. Weighted norm inequalities in Lebesgue spaces with Muckenhoupt
weights and some applications
Status: reviewing
Abstract. In the present work we give a simple method to obtain
weighted norm inequalities in Lebesgue spaces $L_{p,\gamma }$ with
Muckenhoupt weights $\gamma $. This method is different from celebrated
Extrapolation or Interpolation Theory. In this method starting point is
uniform norm estimates of special form. Then a procedure give desired
weighted norm inequalities in $L_{p,\gamma }.$ We apply this method to
obtain several convolution type inequalities. As an application we consider
a difference operator of type $\Delta _{v}^{r}:=\left( \mathbb{I}-\mathfrak{T%
}_{v}\right) ^{r}$ where $\mathbb{I}$ is the identity operator, $r\in
\mathbb{N},$ $v\geq 0$ and%
\begin{equation*}
\mathfrak{T}_{v}f\left( x\right) :=\frac{1}{v}\int\nolimits_{x}^{x+v}f\left(
t\right) dt,\quad x\in \lbrack -\pi ,\pi ),\text{\quad }v>0,\quad \mathfrak{T%
}_{0}:=\mathbb{I}.
\end{equation*}%
We obtain main properties of $\Delta _{v}^{r}f$ for functions $f$ given in $%
L_{p,\gamma }$, $1\leq p<\infty $, with weights $\gamma $ satisfying the
Muckenhoupt's $A_{p}$ condition. Also we consider some applications of
difference operator $\Delta _{v}^{r}$ in these spaces. In particular, we
obtain that difference $\left\Vert \Delta _{v}^{r}f\right\Vert _{p,\gamma }$
\ is a useful tool for computing the smoothness properties of functions
these spaces. It is obtained that $\left\Vert \Delta _{v}^{r}f\right\Vert
_{p,\gamma }$ is equivalent to Peetre's \textit{K}-functional. Jackson
inequalities of trigonometrical approximation are hold. Obtained Jackson
inequalities are refine some similar inequalities proven earlier. We obtain
some trigonometric polynomial of best approximation to a given function $f$.
Simultaneous approximation inequalities are proved.
Date of submission: 15 June 2020 г.
- Meyliev A.Kh., Imomov A.A. ON ASYMPTOTIC STRUCTURE OF CONTINUOUS-TIME
MARKOV BRANCHING PROCESSES ALLOWING
IMMIGRATION AND WITHOUT HIGH-ORDER MOMENTS
Status: reviewing
Abstract. We observe the continuous-time Markov Branching Process
without high-order moments and allowing Immigration. Limit properties
of transition functions and their convergence to invariant measures are
investigated. Main mathematical tool is regularly varying generating
functions with remainder.
Date of submission: 25 June 2020 г.
- Abdulaev O.H., Islomov B. About the non-local problems for a third order equation with operator Caputo and nonlinear loaded term
Status: reviewing
Abstract. This work devoted to prove of unique solvability of solution of the non-local problems with integral gluing condition for the third order equation with parabolic-hyperbolic operator involving Caputo derivatives and non-linear loaded terms. Solvability of the investigated problem was proved by the method of integral equations.
Date of submission: 01 July 2020 г.
- Abdullayev J.S., Khudayberganov G. The boundary Morera theorem for the domain ${{\tau }^{+}}\left( n-1 \right)$
Status: reviewing
Abstract. In this article proved the Morera boundary theorem for the domain ${{\tau }^{+}}\left( n-1 \right)$. An analog of Morera's theorem is given, in which integration is carried out along the boundaries of analytic disks. For this, we use the automorphisms ${{\tau }^{+}}\left( n-1 \right)$ and the invariant Poisson kernel in the domain ${{\tau }^{+}}\left( n-1 \right)$.
Date of submission: 01 July 2020 г.
- Lebedev P.D., Uspenskii A.A. About the Structure of Singularity of a Minimax Solution of a Dirichlet Problem for the Eikonal Type Equation with the Discontinuous Smoothness of the Curvature of the Target Set
Status: reviewing
Abstract. The origin of nonsmooth singularities in the minimax (generalized) solution of the Dirichlet problem for an eikonal equation is due to the existence of pseudo-vertices — singular points of the boundary of the boundary set. Finding pseudo-vertices is the first step in the procedure for constructing a singular set for solving a boundary value problem. Finding these points requires the construction of local solutions of an equation such as the golden ratio, which establishes a connection between the eikonal operator and the geometry of the boundary set. Moreover, the problem of identifying local solutions of the equation is associated with the problem of finding fixed points of mappings formed by local reparameterization of the boundary of the boundary set. In this paper, we obtain the necessary conditions for the existence of pseudovertices with a break in the smoothness of curvature of a parametrically specified boundary of a boundary set. The conditions are written in various equivalent forms. In particular, a representation is obtained in the form of a convex combination of one-sided derivatives of curvature. Formulas are presented for the coefficients of a convex combination, which are determined by markers — scalar characteristics of pseudovertices. For markers, a form of the algebraic equation is found whose roots they are. An example of numerical and analytical construction of a minimax solution to the Dirichlet problem is presented, illustrating the effectiveness of the developed methods for solving nonsmooth boundary value problems.
Date of submission: 02 July 2020 г.
- Ergashev T.G. Potentials for elliptic equation with several singular
coefficients and their applications to the solving a Dirichlet
problem
Status: reviewing
Abstract. Potentials play an important role in
solving boundary value problems for elliptic equations. In the
middle of the last century, a potential theory was constructed for
a two-dimensional elliptic equation with one singular coefficient.
In the study of potentials, the properties of the fundamental
solutions of the given equation are essentially and fruitfully
used. At the present time, fundamental solutions of a
multidimensional elliptic equation with several singular
coefficients are already known. In this paper, we investigate the
double- and simple-layer potentials for this kind of elliptic
equations. Results from potential theory allow us to represent the
solution of the boundary value problems in integral equation form.
By using a decomposition formula and other identities for the
Lauricella's hypergeometric function in many variables, we prove
limiting theorems and derive integral equations concerning a
densities of the double- and simple-layer potentials. The obtained
results are applied to find an explicit solution of the Dirichlet
problem for the generalized singular elliptic equation in the some
part of the multidimensional ball.
Date of submission: 03 July 2020 г.
- Gaisina G.A. The growth order of the sum of the Dirichlet series: relation on coefficients and exponents
Status: reviewing
Abstract. We study the optimality of conditions under which the order of the sum of the Dirichlet series, converging only in some half-plane, can be calculated using a certain formula (it depends only on the coefficients and exponents). A formula of this type in different years was independently obtained by many specialists, including N.V. Govorov (1959), MacLane (1966) and M.N. Sheremeta (1968) for unbounded $ $ analytic functions in the unit circle. Later, an analog of this notion was introduced for Dirichlet series that absolutely converge in some half-plane. But the corresponding formula for the order of the Dirichlet series was established by most authors under essential restrictions. In all previous works, conditions were indicated that were only sufficient for the validity of this formula. In this paper, we find conditions that are not only sufficient, but also necessary so that the order of any Dirichlet series from the considered class can be calculated using the same formula.
Date of submission: 11 July 2020 г.
- Jafari H. Short-time behavior in arithmetic Asian option under a stochastic volatility model with jumps
Status: reviewing
Abstract. This article studies the short-time behavior of the arithmetic Asian option pricing in a general class of the stochastic volatility model with jumps. The arithmetic Asian options that rarely has the explicit expression, can reduce the volatility in the price option because of the average of the underlying asset price over the time interval. The future average processes in the model are the non-adapted process. By using the Malliavin calculus operators, we get an anticipating It\^{o} formula and a Hull-White decomposition formula of the price in the model. We apply the decomposition of the price formula to find the short-time limit of the arithmetic Asian option price and the implied volatility.
Date of submission: 11 July 2020 г.
- Artemov M.A., Babkina Yu.N. The first boundary value problem for equations describing flows of a nonlinear viscoelastic fluid in a bounded domain
Status: reviewing
Abstract. We study the first boundary value problem for a mathematical model describing the steady motion of a viscoelastic fluid with variable viscosity, depending on the shear rate, inside a bounded three-dimensional (or two-dimensional) domain with sufficiently smooth boundary. The concept of a weak solution is introduced. The regularization method is used to formulate this problem in the form of an operator equation with a continuous nonlinear operator satisfying the α-condition. By using of the theorem on the solvability of equations with α-operators and the passage to the limit, the existence of at least one weak solution of the problem is proved and an energy-type estimate for the vector velocity function is derived.
Date of submission: 19 July 2020 г.
- Avdeev F.S., Yaremko O.E., Yaremko N.N. Operator Transmutation Method for Mixing Laplace and Mellin Transfoms
Status: reviewing
Abstract. The article contributes to the theory of integral transformations. The purpose of the article is to construct an operational calculus for usage in research of transient events. ~The article develops the Mixing Integral Laplace and Mellin transforms. An analog of the Laplace-Mellin and Mellin-Laplace operational calculus is constructed. Analogs of classical concepts such as original (pre-image) function, Laplace transform image, and convolution are introduced. Analogs of classical theorems on differentiating the original function (Time - domain differentiation) and on differentiating the Laplace transform (Frequency-domain derivative), on changing the scale, and others are proved. We set the formula for convolution. Based on the concept of convolution, the definition of a fractal integral is given. The direct and inverse transforms are defined in this paper. With their help, the connection of the Laplace-Mellin integral transform with the Laplace integral transform is established. Solutions of three mixed boundary value problems for modeling transient events are found.
Date of submission: 27 July 2020 г.
- Zheltukhin K., Zheltukhina N.A. On the discretization of Darboux Integrable Systems admitting second-order integrals
Status: reviewing
Abstract. The discretization of Darboux integrable systems admitting two first and two second order integrals is considered. The obtained semi-discrete systems possess $n$- integrals that coincide with $x$- or $y$-integrals of the original continuous systems. New examples of semi-discrete Darboux integrable systems are derived.
Date of submission: 03 Avgust 2020 г.
- Abuzyarova N.F., Fazullin Z.Yu. On the necessary and aufficient condition in the theory of regularized traces
Status: reviewing
Abstract. Настоящая работа посвящена изучению формул регуляризованных следов симметрических $L_0$-компактных возмущений дискретного самосопряженного полуограниченного снизу оператора $L_0$ в сепарабельном гильбертовом пространстве.
Исследования формул регуляризованных следов возмущений абстрактных самосопряженных дискретных операторов до сих пор, в основном, были направлены на нахождение
достаточного условия, при котором равна нулю регуляризованная сумма со скобками с вычетом первой или нескольких поправок теории возмущений.
Это условие формулируется в терминах спектральных характеристик невозмущенного оператора $L_0$ в зависимости от принадлежности определенному классу
оператора возмущения $V$.
В частности, в последнее время интенсивно изучаются формулы следов двумерных модельных операторов математической физики, возмущенных оператором умножения на
функцию.
Здесь мы исследуем необходимое и достаточное условие для двух случаев: равенства нулю и равенства конечному числу --- суммы регуляризованного следа со скобками с вычетом первой поправки теории возмущений. При этом рассматривается конкретная скобка суммирования, которая, как правило, возникает при исследовании формул регуляризованных следов возмущений дифференциальных операторов в частных производных.
Date of submission: 21 Avgust 2020 г.
- Chernov A.V. On differentiation of a functional in the problem of
parametric coefficient optimization in the
semilinear global electric circuit equation
Status: reviewing
Abstract. For the problem of parametric optimization with respect to an
integral criterion of the coefficient and the right-hand side
of the semilinear global electric circuit equation,
formulas for the first partial derivatives of the cost functional
in control parameters are obtained.
In such a form the problem of reconstructing unknown parameters of
the equation from data of observation by local sensors can be represented.
In the paper it is generalized analogous result obtained by the
author formerly for the case of linear global electric circuit equation.
But experts believe that right-hand side
(treated as the volumetric density of external currents) of
the equation actually depends on the gradient
(with respect to the collection of space variables)
of the unknown electric potential function.
In this connection the necessity arises to investigate
the case of semilinear equation.
Here, we use the conditions for preservation of
global solvability of the semilinear global electric circuit equation
which we have obtained formerly.
The mathematical novelty of presented research is due
to the fact that, unlike the earlier-studied linear case, now
the right-hand side depends (nonlinearly) on the state
(depending, in turn, on the control parameters).
This more complicated nonlinear dependence of the state on
the control parameters has required, in particular,
the development of a special technique to estimate the additional
terms arising in the increment of solutions
of the controlled equation.
Date of submission: 28 Avgust 2020 г.
- Poluboyarova N.M. Relationship between length and instability of tubular extreme surfaces
Status: reviewing
Abstract. The article investigates surfaces that are extremals of the potential energy functional. For example, for the area functional, the extremals are minimal surfaces. Instability conditions are obtained based on the estimate of the second variation of the functional. It was found that the length of a tubular extreme surface can be estimated using the minimum and maximum (n-1) -dimensional measure of the surface section by planes. The statement is proved that too long tubes with nonzero mean curvature are unstable. The physical aspects of this phe\-no\-menon are considered in the work of V.A. Saranina.
Date of submission: 30 Avgust 2020 г.
- Khabibullin B.N. Liouville-type theorems for functions of finite order
Status: reviewing
Abstract. We prove that convex functions of finite order on the real line and subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some set of zero relative Lebesgue density, are bounded from above everywhere. It follows that subharmonic functions of finite order on the complex plane, entire and plurisubharmonic functions of finite order, and convex or harmonic functions of finite order bounded from above outside some set of zero relative Lebesgue density are constant.
Date of submission: 01 September 2020 г.
- Komarov M.A. Rate of convergence of a one class of differentiating sums
Status: reviewing
Abstract. We consider the formula for
differentiation of analytic func\-ti\-ons:
$azf'(z)=nf(0)-\sum_{k=1}^n f(\lambda_k z)+O(z^{n+1})$,
where $a>0$, $\lambda_k=\lambda_{n,k}(a)$, $n=1,2,\dots$.
As $n\ge 3\alpha$ ($\alpha:=\max\{a;1\}$), we find an estimate
of the rate of convergence of the differentiating sums to
$nf(0)-a zf'(z)$ in the disk $|z|<\exp(-3\sqrt{v}-2v)$,
$v:=\alpha/(n+1)$.
Date of submission: 02 September 2020 г.
- Borisov D.I., Konyrkulzhayeva M.N. On infinite system of resonances and eigenvalues with exponential asymptotics generated by distant perturbations
Status: reviewing
Abstract. Рассматривается одномерный оператор Шрёдингера с четырьмя потенциалами, разнесёнными на большие расстояния друг от друга. Все расстояния пропорциональны одному большому параметру. Исходные потенциалы имеют форму кинков, которые склеиваются друг с другом таким образом, что финальный потенциал обращается в нуль на бесконечности и между вторым и третьим потенциалами, и равен единице между первым и вторым, а также между третьим и четвертым потенциалами. Потенциалы не предполагаются вещественными и могут быть комплекснозначными. Показано, что при
определенных, достаточно естественных и легко реализуемых условиях на исходные четыре потенциала, оператор с разбегающимися потенциалами имеет неограниченное число резонансов и/или собственных значений вида $\l=k_n^2$, $n\in\mathds{Z}$, которые накапливаются вдоль заданного отрезка существенного спектра. Расстояние между соседними числами $k_n$ есть величина порядка обратной степени расстояния между потенциала, а мнимые части этих величин экспоненциально малы. Для чисел $k_n$ получены представления в виде пределов явно вычисляемых последовательностей и сумм бесконечных рядов и доказаны явные эффективные оценки на скорость сходимости последовательностей и для остатков рядов.
Date of submission: 02 September 2020 г.
- Musin I.Kh. On Fourier-Laplace transformation of a class of generalized functions
Status: reviewing
Abstract. A subspace of Schwartz space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set of multidimensional real space with a topology defined by a countable family of norms constructed with a help of some logarithmically convex sequences of positive numbers is considered in the article. Thanks to conditions on the indicated sequences this space is a Fr\'echet-Schwartz space. We study the problem of description of the strong dual for this space in terms of the Fourier-Laplace transformatation of functionals. Special cases of this problem were considered by J.W. De Roever while studying problems of mathematical physics, complex analysis in frames of developed by him theory of ultradistributions with carriers on an unbounded closed convex set and also by P.V. Fedotova and the author. The main result of the article is Theorem I. It states that the Fourier-Laplace transformation establishes an isomorphism between the strong dual for functional space under consideration and some space of holomorphic functions in tube domain of a form ${\mathbb R}^n + iC$ (where $C$ is an open convex proper cone in ${\mathbb R}^n$ with vertex at the origin) with prescribed growth majorants at infinity and near the boundary of the tube domain. The work is close to researches of V.S. Vladimirov devoted to the theory of the Fourier-Laplace transformatation of of tempered distributions and spaces of holomorphic functions in tube domains.
In the proof of Theorem I we apply the scheme taken from M. Neymark and B.A. Taylor. Also some previous results of P.V. Yakovleva (Fedotova) and the author devoted to Paley-Wiener type theorems for ultradistributions are essentially used.
Date of submission: 03 September 2020 г.
- Ershov A.A., Ushakov V.N. On Determination of an Unknown Constant Parameter by Several Test Controls
Status: reviewing
Abstract. We consider a control system contains a constant vector parameter unknown to the control person. Only the many possible values of the unknown parameter are known.
For this control system, the problem is posed of approaching the target set at a given time.
To solve the control problem at the very beginning of the movement, the
unknown parameter is determined by sequential short-term application to the control system
several test control vectors and observation of the corresponding reaction of the control system.
The choice of a set of the test control vectors is proposed to be carried out in order to minimize the error in the determination of the unknown constant parameter.
Unlike previous works,
a more general case is considered when
one test control constant vector is not enough for an unambiguous determination of the unknown parameter, and, in addition, to approximate the speed of motion, the central difference derivative of the system is used instead of the right-hand difference derivative.
As an example, the problem of controlling a pendulum with unknown coefficient of friction and coefficient of elasticity of the spring is considered.
Date of submission: 13 September 2020 г.
- Isaev K.P., Yulmukhametov R.S. The geometry of radial Hilbert spaces with unconditional bases of reproducing kernels
Status: reviewing
Abstract. In this paper we consider the geometry of abstract radial functional Hilbert spaces having the division property and possessing unconditional basis of reproducing kernels. We obtained a simple necessary condition for the existence of such bases in terms of the sequence $\| z^n\| , \ n\in \mathbb N\cup \{ 0\}$. Also we obtained a sufficient condition for the norm and the Bergman function of the space to be restored through the sequence of monomial norms. Two main statements are proved. Let $ H $ be a radial functional Hilbert space of entire functions having the division property, and the system of monomials $\{z^n\} ,\ n\in \mathbb N\cup \{ 0\},$ is complete in it.
1. If the space $H$ possesses an unconditional basis of reproducing kernels, then
\begin{equation*}
\|z^n\| \asymp e^{u(n)},\ n\in \mathbb N\cup \{0\},
\end{equation*}
where the sequence $u(n)$ is convex, i.e.
\begin{equation*}
u(n+1)+u(n-1)-2u(n)\ge 0,\ n\in \mathbb N.
\end{equation*}
2. Let $u_{n,k}=u(n)-u(k)-(u(n)-u(n-1))(n-k)$. If $\mathcal U$ is the matrix with elements $e^{2u_{n,k}}$, $n,k\in \mathbb N\cup \{ 0\}$, and
\begin{equation*}
\left \| \mathcal U\right \| :=\sup _n\left (\sum _ke^ {2u_{n,k}}\right )^{\frac 12}<\infty ,
\end{equation*}
then
2.1. the space $H$ as a Banach space is isomorphic to the space of entire functions with the norm
\begin{equation*}
\|F\|^2=\frac 1 {2\pi }\int _0^\infty \int _0^{2\pi }|F(re^{i\varphi }) |^2e^{-2\widetilde u(\ln r)}d\varphi dr,
\end{equation*}
where $\widetilde u$ is the Young conjugate of the piecewise-linear function $u(t)$;
2.2. the Bergman function of the space $H$ satisfies the condition
\begin{equation*}
K(z)\asymp e^{2\widetilde u(\ln |z|)},\ z\in \mathbb C.
\end{equation*}
Date of submission: 17 September 2020 г.
- Aouaouda M., Ayadi A., Fujita Yashima H. Уравнение переноса-диффузии в полуплоскости и модель испарения и диффузии водяного пара
Status: reviewing
Abstract. In this paper we consider a family
of approximate solutions to the transport-diffusion equation.
The approximate solutions are constructed by the application of
fundamental solution to the heat equation on each step of time
discretization. We present also a numerical model of evaporation
and diffusion of water vapor, model which corresponds to an
approximate solution considered in the first part of the paper.
Date of submission: 17 September 2020 г.
- Amosov G.G., Baitenov E.L. On rank one perturbations of the semigroup of shifts on the half axis
Status: reviewing
Abstract. We study a special case of perturbations of the semigroup of shifts on the half-axis that change the domain of its generator. Rank one perturbations of generator defined by the exponent are considered. It is shown that such a perturbation of the generator always leads to the generator of some $C_0$-semigroup, the action of which is obtained explicitly. The criterion of isometricity and contractivity of the perturbed semigroup is obtained. For the contractive case, it is shown that the considered generator perturbation leads to a rank one perturbation of the cogenerator. The studied special case is used to build the model of perturbation for the semigroup of shifts defined by an integral equation with respect to some operator-valued measure. In a situation where the domain of the generator is not changed, such an integral equation is reduced to the well-known equation of perturbation theory, where integration is carried out using the usual Lebesgue measure (as described, for example, in the book by T. Kato). In this paper, it is proved that if the domain of the generator is changed, the perturbation will never satisfy the equation where integration is carried out with respect to the Lebesgue measure. If the domain is changed, the perturbation will already satisfy an integral equation with a nontrivial measure that has no density with respect to the Lebesgue measure. Here such questions are studied in a model situation where a rank one perturbation is defined by the exponent. The problem of selecting an operator-valued measure that defines the integral equation connecting the perturbed semigroup with the original one is fully investigated. The measure, when it exists, is obtained explicitly. It is shown that it is defined ambiguously. We study the possibility of choosing an operator-valued measure with values in the set of self-adjoint and positive operators.
Date of submission: 21 September 2020 г.
- Norjigitov A.F., Sharipov O.S. Law of large numbers for weakly dependent random variables with values in $D[0,1]$
Status: reviewing
Abstract. The paper is devoted to the law of large numbers for the random variables with values in D[0,1] space. The law of large numbers is well well-studied for the sequences of independent D[0,1]-valued random variables.
Our main goal is to prove the law of large numbers for the weakly dependent random variables with values in D[0,1] space. In the paper the law of large numbers for $\rho_{m}$-mixing
sequences of $D[0,1]$-valued random variables are proved.
Date of submission: 27 September 2020 г.
- Alsarori N.A., Ghadle K.P. New results for infinite functional differential
inclusions with impulses effect and sectorial
operators in Banach spaces
Status: reviewing
Abstract. New results for infinite functional differential
inclusions with impulses effect and sectorial
operators in Banach spaces
Date of submission: 02 October 2020 г.
- Certain Class of Non-Bazilevic Functions Associated with Exponential Function
Status: reviewing
Abstract. Its well-known that a Hankel matrix is one whose entries of the reverse
diagonals are constant, i.e.
Mathematician, physicists and engineers are attracted to this matrix because of
their computational properties and appearances in different areas: dynamical
systems, dynamical systems, quantum mechanics, and partial differential
equations.
The main object in this paper is give an upper bound for the determinant of the
third Hankel matrix for which the entries are belong to a new certain class of
Non-Bazilević functions in the open disk associated with exponential
function.
Date of submission: 07 October 2020 г.
- MAJUMDER S., Saha S. Power of entire function sharing non-zero polynomials with it's Linear Differential Polynomial
Status: reviewing
Abstract. In this paper we mainly investigate the power $f^{n}(n\in \mathbb{N})$ of a transcendental entire function $f$ and its
linear differential polynomial $L(f^{n})$ sharing non-zero polynomials, where $L(f^{n})$ is defined by
$L(f^{n})=(f^{n})^{(k)}+a_{k-1} (f^{n})^{(k-1)}+\ldots +a_{1} (f^{n})'$, $a_{j}(j=1,\ldots, k-1)$ are rationals. Also we exhibit some examples to show that some conditions of our results are the best possible.
Date of submission: 08 October 2020 г.
- Akhmetshina A.D., Ishkin Kh.K. On localization conditions for the spectrum of the model operator for the Orr--Sommerfeld equation
Status: reviewing
Abstract. For the model operator $ L_\varepsilon $, connected with the Orr--Sommerfeld equation, the question of the necessity of the well-known A.\,A.~Shkalikov conditions sufficient for localizing the spectrum near the graph of the form << Y >> is studied. We have considered two types of potentials for which the limiting spectral graph (LSG) of the operator $ L_\varepsilon $ can be constructed explicitly. The first of them is a piecewise constant potential with a countable number of jumps, at which the LSG consists of a countable number of rays. The second type is a potential glued together from two holomorphic functions. In the case when some derivative of the potential undergoes a jump at the gluing point, the LSG is constructed. Further, it is proved that even the infinite differentiability of the potential is insufficient for localizing the spectrum around one curve. This fact is a simple consequence of Theorem~3, which is the main result of the article: if the potential does not decrease on $[0,1]$, is differentiable, its derivative is absolutely continuous and the unbounded component of the LSG consists of one curve when the small parameter tends to 0 along any piecewise smooth curve with bounded slope, then the potential admits an analytic continuation to some neighborhood of the interval $(0,1)$.
Date of submission: 20 October 2020 г.
- Shakirov I.A. Approximation of the fundamental characteristic of the Fourier operator by a logarithmic function
Status: reviewing
Abstract. The Lebesgue constant $L_n$ of the classical Fourier operator is uniformly approximated by a family of logarithmic functions that depend on two parameters. The case where the residual term has non-monotonic behavior is considered. The obtained result of approximation $L_n$ by indicated family of functions strengthens the known results corresponding to cases of strict decrease and increase of the residual term.
Date of submission: 21 October 2020 г.
- Rastogi A., Rathore G.P. Some growth analysis of entire function in the form of vector valued Dirichlet series in terms of $(p,q)$th relative Ritt $L$-order and $(p,q)$th relative Ritt $L$-lower order
Status: reviewing
Abstract. After the recent works of Biswas [11] on the idea of (p,q)-th relative ritt order
and (p,q)-th relative ritt type, we introduce in this paper to established some results
of the growth analysis of entire function represented by vector valued Dirichlat series
f(s) =
P ∞
n=1 a n e
sλ n
on the basis of (p,q)-th relative ritt L-order and (p,q)-th relative
ritt L-lower order of an entire function represented by vector valued Dirichlat series.
Date of submission: 22 October 2020 г.