Editorial backlog
- Balkizov Zh.A. Внутреннекраевые задачи со смещением для одного смешанно-гиперболического уравнения второго порядка
Status: reviewing
Abstract. В работе исследованы внутреннекраевые задачи со смещением для одного смешанно-гиперболического уравнения второго порядка, состоящего из волнового оператора в одной части области и вырождающегося гиперболического оператора первого рода в другой части.
Date of submission: 27 October 2021 г.
- Hardy Type Inequalities Via $(k,\mu)$-Riemann-Liouville
Fractional Integral Operators
Status: reviewing
Abstract. In this study, a new inverse Hardy-type inequality intro-
duced via the (k,µ)-Riemann-Liouville fractional integral operators. New
results obtained by using two integrability parameters p and q and some
particular cases mentioned, according to the choice of the function µ and
the reals k,p,q.
Date of submission: 21 February 2022 г.
- Admasu V.E., Galahov E.I. Условия отсутствия решений для некоторых эллиптических неравенств высокого порядка с сингулярными коэффициентами в $\mathbb{R}^n$
Status: reviewing
Abstract. In the present paper, we develop Liouville-type theorems for higher order elliptic inequalities with singular coefficients and gradient terms in $\mathbb{R}^n$ by the Pohozaev nonlinear capacity method.
Date of submission: 28 February 2022 г.
- Inverse problem of determining two kernels in the integro - differential equation of heat flow
Status: reviewing
Abstract. The inverse problem of determining the energy-temperature relation $\alpha(t)$ and
the heat conduction relation $k(t)$ functions in the one-dimensional
integro--differential heat equation are investigated. The direct
problem is the initial-boundary problem for this equation. The
integral terms have the time convolution form of unknown kernels
and direct problem solution. As additional information for solving
inverse problem, the solution of the direct problem for $x=x_0$ and $x=x_1$
are given. At the beginning an auxiliary problem, which is
equivalent to the original problem is introduced. Then the
auxiliary problem is reduced by an equivalent closed system of
Volterra-type integral equations with respect to unknown
functions. Applying the method of contraction mappings to this
system in the continuous class of functions,
the main result of the article, which is a local
existence and uniqueness theorem of inverse problem solutions is proven.
Date of submission: 14 Aprel 2022 г.
- Analysis of a thermo-elasto-viscoplastic contact
problem with wear and damage
Status: reviewing
Abstract. This paper presents a quasistatic problem of a thermo-elaso-visco-
plastic body in frictional contact with a moving foundation. The contact is
modelled with the normal compliance condition and the associated law of
dry friction. The model takes into account wear of the contact surface of
the body caused by the friction and which is described by the Archard law.
The mechanical damage of the material, caused by excessive stress or strain,
is described by the damage function, the evolution of which is determined
by a parabolic inclusion. We list the assumptions on the data and derive a
variational formulation of the mechanical problem. Existence and uniqueness
of the weak solution for the problem is proved using the theory of evolutionary
variational inequalities, parabolic variational inequalities, first order evolution
equation and Banach fixed point.
Date of submission: 16 Aprel 2022 г.
- Statistical convergence of double sequences of functions by virtue of difference operator
Status: reviewing
Abstract. The present paper focus on λ−statistical convergence by means of modulus
function and generalized difference operator for double sequences of functions for order
γ. Further, we prove that statistical convergence in our newly formed sequence spaces
is well defined for γ ∈ (0,1]. In addition to the above result, we establish relation
among λ−statistical convergence and strongly λ−summable for our sequence spaces.
Date of submission: 20 Aprel 2022 г.
- Luu T.H., Shokarev V.A., Budochkina S.A. On an indirect representability of a fourth-order ordinary differential equation in the form of Hamilton-Ostrogradskii equations
Status: reviewing
Abstract. In the paper, the problem of the representability of a fourth-order ordinary differential equation in the form of Hamilton-Ostrogradskii equations is solved. For this purpose, we obtain necessary and sufficient conditions for a given operator to be potential relative to a local bilinear form, construct
the corresponding Hamilton-Ostrogradskii action and define the structure of the considered equation with the potential operator.
Date of submission: 26 Aprel 2022 г.
- Малакбозов З.Ш., Shabozov M.Sh. Точные неравенства типа Джексона -- Стечкина в пространстве Харди
$H_2$ и поперечники классов функций
Status: reviewing
Abstract. Exact inequalities of the Jackson --
Stechkin type are obtained in the Hardy space of functions analytic
in the unit disc, the modulus of continuity of the function of which
is determined using the Steklov function. For the classes of
functions given by this characteristic, the exact values of various
$n$-widths are found.
Date of submission: 04 May 2022 г.
- Results on two-order fractional boundary value problem under the generalized Riemann-Liouville derivative
Status: reviewing
Abstract. In this paper we focus our study on the existence, uniqueness and
Hyers-Ulam stability for a fractional boundary value problem involving the
generalized Riemann-Liouville operators of a function with respect to another
non-decreasing function. To prove the uniqueness result we use Banach fixed
point Theorem and for the existence result, we apply two classical fixed point
Theorems due to Krasnoselskii and Leray-Scauder. Then, we continue our
results by studying the Hyers-Ulam stability of solutions.
Date of submission: 12 May 2022 г.
- A quasistatic electro-elastic contact problem with long memory and slip dependent coefficient of friction
Status: reviewing
Abstract. In this paper we consider a mathematical model which describes
a quasistatic frictional contact problem between a deformable body and an
obstacle, say a foundation. We assume that the behavior of the material is
described by a linear electro-elastic constitutive law with long memory. The
contact is modelled with a version of Coulomb’s law of dry friction in which
the normal stress is prescribed on the contact surface. Moreover, we consider
a slip dependent coe¢cient of friction. We derive a variational formulation
for the model, in the form of a coupled system for the displacements and the
electric potential. Under a smallness assumption on the coefficient of friction,
we prove an existence result of the weak solution of the model. We can show the
uniqueness of the solution by adding another condition. The proofs are based
on arguments of time-dependent variational inequalities, differential equations
and Banach fixed point theorem.
Date of submission: 18 May 2022 г.
- Garif'yanov F.N., Strezhneva E.V. О системе производных периодической мероморфной функции
Status: reviewing
Abstract. We study the approximating properties of a system of successive derivatives of a periodic meromorphic function. A system of functions is constructed that is biorthogonally conjugate to it on the boundary of some rectangle. Here, the Weierstrass theory of elliptic functions is essentially used. The system of derivatives admits a non-trivial expansion of zero in some circular domain. For the constructed biorthogonally conjugate system, an equation of the convolution type is used. They are investigated in a closed form using the discrete Fourier transform. The considered biorthogonal series fundamentally differ from the well-known Appel series.
Date of submission: 26 May 2022 г.
- Recent common fixed point results in the setting of bounded metric spaces with an application to nonlinear integral equations
Status: reviewing
Abstract. In this paper, we prove some common fixed point theorems in the setting of bounded metric spaces without using neither the compactness nor the uniform convexity of the space. Some examples are built to show the superiority of the obtained results compared to the existing ones in the literature.
Moreover, we apply the main result to show the existence and uniqueness of a solution for a nonlinear integral system.
Date of submission: 23 June 2022 г.
- Sazonov A.P. About the a priori and asymptotic estimates for Emden-Fowler problem on model Riemannian manifolds
Status: reviewing
Abstract. This work is devoted to study for Emden-Fowler problem on the model Riemannian manifolds. In particular, it will be obtained a priori and asymptotic estimates for radially symmetric solutions for this problem in the considered manifolds. The results of this work summarize similar statements obtained earlier in the work
S.I. Pohozaeva for the space $R^n$.
Date of submission: 07 July 2022 г.
- Rodikova E.G. On Continuous Linear Functionals in Some Spaces of Analytic
Functions in a Disk.
Status: reviewing
Abstract. The question of describing continuous linear functionals on
spaces of analytic functions has been studied since the middle of the 20th
century. Historically, the structure of linear continuous functionals of the
Hardy spaces H p for p ≥ 1 was ?rst found by Taylor in 1951. In the spaces
H p (0 < p < 1) this problem was solved by Duren , Romberg, and Shields
in 1969. Note that the proof used an estimate of the coe?cient multipliers
in these spaces. In the article, developing the method proposed in the
work of Duren et al., a description of continuous linear functionals of
the area Privalov classes and classes of Nevanlinna-Dzhrbashyan type is
obtained.
Date of submission: 18 July 2022 г.
- Danilin A.R. Asymptotics solutions to the problem of optimal distributed
control in a convex domain with a small
parameter at one of the higher derivatives
Status: reviewing
Abstract. Рассматривается задача оптимального распределенного
управления в плоской строго выпуклой области с гладкой границей и
малым параметром при одной из старших производных эллиптического оператора.
На границе области в этой задаче задано нулевое условие Дирихле,
а управление аддитивно входит в неоднородность. В качестве множества
допустимых управлений используется единичный шар в соответствующем
пространстве функций, суммируемых с квадратом. Решение получающихся
краевых задач рассматриваются в обобщенном смысле как элементы
некоторого гильбертова пространства. В качестве критерия оптимальности
выступает сумма квадрата нормы отклонения состояния от заданного и
квадрата нормы управления с некоторым коэффициентом. Такая структура
критерия оптимальности позволяет, при необходимости, усилить роль
либо первого, либо второго слагаемого в этом критерии. В первом случае
более важным является достижение заданного состояния, а во втором случае
--- минимизация ресурсных затрат. Подробно изучена асимптотика задачи,
порожденная дифференциальным оператором второго порядка с малым
коэффициентом при одной из старших производных, к которому прибавлен
дифференциальный оператор нулевого порядка.
Date of submission: 20 July 2022 г.
- Beshtokov M.KH. Numerical solution of initial-boundary value problems for a multidimensional pseudoparabolic equation
Status: reviewing
Abstract. We consider initial-boundary value problems for a multidi- mensional pseudoparabolic equation with boundary conditions of the first kind and a special form. For an approximate solution of the problems posed, the multidimensional pseudoparabolic equation is reduced to an integro-differential equation with a small parameter. It is shown that as the small parameter tends to zero, the solution of the corresponding modified problem converges to the solution of the original problem. For each of the problems, a locally one-dimensional difference scheme by A.A. Samarsky, the main idea of ??which is to reduce the transition from layer to layer to the sequential solution of a number of one-dimensional problems in each of the coordinate directions. Using the maximum prin- ciple, a priori estimates are obtained, from which the uniqueness, stability, and convergence of the solution of a locally one-dimensional difference scheme in the uniform metric follow. An algorithm for the numerical solution of the modified problem with conditions of a special form is constructed.
Date of submission: 26 July 2022 г.
- Langarshoev M.R Точные неравенства для алгебраических комплексных полиномов
и значение поперечников классов функций в весовом
пространстве Бергмана
Status: reviewing
Abstract. В настоящей работе мы решаем некоторые экстремальные задачи связанные c оценка-
ми норм производных для алгебраических комплексных полиномов через усредненные значе-
ния их модулей непрерывности и гладкости. Обобщаются некоторые результаты Л.В.Тайкова и
Н.Айнуллоева полученные для классов дифференцируемых периодических функций на случай
аналитических в единичном круге функций f(z) принадлежащих весовому пространству Берг-
мана B q,γ , 1 ≤ q ≤ ∞. Вычислены значения поперечников классов аналитических в единичном
круге функций в весовом пространстве Бергмана B q,γ , 1 ≤ q ≤ ∞.
Date of submission: 12 Avgust 2022 г.
- Ershov A.A. Bilinear Interpolation of a Program Control in the Approach Problem
Status: reviewing
Abstract. We consider a control system containing a constant two-dimensional vector parameter, which is reported to the control person only at the moment of the movement start. Only the set of possible values of this indeterminate parameter is known in advance.
For this control system, the problem of approaching the target set at a given time is posed.
At the same time, the control person does not have the ability to carry out in real time the cumbersome calculations associated with the construction of such permissive structures as reachable sets and integral funnels. Therefore, to solve this problem, it is proposed to calculate in advance several "`nodal"' permissive controls for parameter values, which are grid nodes covering the set of possible parameter values.
In the event that at the moment of the movement start it turns out that the parameter value does not coincide with any of the grid nodes, it is supposed to calculate the program control using linear interpolation formulas.
However, this procedure can be effective only if a linear com\-bi\-na\-ti\-on of controls is used, corresponding to the same "`guide"' in the terminology of N.N. Krasovsky's method of extreme targeting.
For ef\-fec\-ti\-ve application of linear interpolation, for each grid node it is proposed to calculate four "`nodal"' permissive controls and, in addition, use the method of dividing the control into main and compensating.
Due to the application of the latter method, the calculated solvability set turns out to be somewhat smaller than the actual one, but the accuracy of transferring the system state to the target set increases.
As an example, a nonlinear generalization of the Zermelo navigation problem is considered.
Date of submission: 23 Avgust 2022 г.
- Belov Yu.S., Kulikov A.I., Lyubarskii Yu.I. Gabor frame operator for the Cauchy kernel
Status: reviewing
Abstract. We obtain frame bounds estimates and
the Gabor frame operator $S=S^{\alpha,\beta}$ for Gabor frames generated by the Cauchy kernel.
In addition we find the explicit expression for the canonical dual window for all values of the
lattice parameters $\alpha,\beta$, $\alpha\beta\leq 1$.
Date of submission: 25 Avgust 2022 г.
- Volchkov V.V., Volchkova N.P. Формула для лапласиана в терминах отклонения функции от ее средних значений
Status: reviewing
Abstract.
Date of submission: 29 Avgust 2022 г.
- Vinogradov O.L. Direct and inverse theorems of approximation theory in the Lebesgue spaces with Muckenhoupt weights
Status: reviewing
Abstract. We establish direct and inverse theorems of approxmation theory in the
Lebesgue spaces~$L_{p,w}$ with Muckenhoupt weights~$w$ on
the real line and on the period. As moduli of continuity, inter alia of nonitnteger order, we use
norms of deviations of Steklov averages.
The proofs are based on estimates of norms of convolution operators
and do not use the maximal function. This allows to prove the theorems
for all $p\in[1,+\infty)$, including $p=1$. All the constants in the
estimates depend on~$[w]_p$
(the Muckenhoupt characteristics of the weight~$w$), and any other
dependence on $w$ and~$p$ is absent.
Date of submission: 31 Avgust 2022 г.
- Akmanova S.V., Yumagulov M.G. On the Stability of Equilibrium Points of Nonlinear Continuous-Discrete Dynamical Systems
Status: reviewing
Abstract. The main attention in the work is
given Discussion of questions about sufficient stability
criteria for Lyapunov equilibrium points of non-linear hybrid
(continuous-discrete) system, that is system, processes in
which it has several levels of heterogeneous description, and
states contain both continuous and discrete components. It is
well known that switching modes of a continuous dynamic system
can be achieved stability and, conversely, even when all modes
of continuous the systems are stable, when switched on, the
system can unstable modes will arise. Therefore, important
representations research that allows for a detailed analysis of
questions stability during the transition from a continuous to
a hybrid system.
In the present article, new signs of stability in terms of
Lyapunov Stationary Regimes of Nonlinear Hybrid System with
constant discretization step $h>0$. These signs are based on
methods of studying stability at first approximation and
perturbation theory formulas that allow analysis stability of
equilibrium points and cycles of dynamic systems, depending on
a small parameter. The proposed approaches are based on
transition from the original hybrid system to an equivalent (in
natural sense) dynamic system with a discrete time. The
relationship between dynamic characteristic hybrid and discrete
system. When studying the main problem of stability with
respect to the Lyapunov point of equilibrium The hybrid system
is considered in two settings: stability for small $h>0$ and
stability for arbitrary fixed $h=h_{0}>0$. In addition, some
questions about scenarios for the bifurcation behavior of the
hybrid system in loss of stability of the equilibrium point. An
example is given, illustrative efficiency of the results
obtained in the problem studies of the stability of equilibrium
points of hybrid systems.
Date of submission: 05 September 2022 г.
- Voronova Yu.G., Zhiber A.V. On a class of hyperbolic equations with third-order integrals
Status: reviewing
Abstract. В работе рассмотрен класс нелинейных гиперболических уравнений, обладающих $y$--интегралом первого порядка и $x$--интегралом третьего порядка. Получены формулы для интегралов. Также приведены дифференциальные подстановки, связывающие уравнения Лэне.
Date of submission: 13 September 2022 г.
- Ivanov D.Y. On the uniform convergence of a semi-analytical solution of the Dirichlet problem for the dissipative Helmholtz equation near the boundary of a two-dimensional domain
Status: reviewing
Abstract. We study an approximate solution of the Dirichlet problem for the two-dimensional dissipative Helmholtz equation, obtained with using a semi-analytical approximation of the double-layer potential. The approxi\-mation of the potential is based on exact integration over the variable $\rho =\left(r^2 -d^2\right)^{1/2}$, where $r$ and $d$ are the distances from the observation point to the integration point and to the boundary of the domain, respectively. It is proved that the semi-analytical approximations of the potential converge uniformly and stably near the boundary of the domain with cubic velocity, and that at the boundary they suffer a discontinuity, the magnitude of which is proportional to the values of the interpolated density function. We also prove the uniform and stable cubic convergence of appropriate approximate solutions of the boundary integral equation and the Dirichlet problem. It is proved that if the quadrature Gauss formulas are used instead of exact integration over the variable $\rho$, then there is no uniform convergence of approximations of the double layer potential near any boundary point. The results of the numerical solution of the Dirichlet problem in the exterior of the circle are presented, confirming the theoretical conclusions.
Date of submission: 15 September 2022 г.
- Fixed point results via a binary relation in
the setting of $T$-normed vector spaces
Status: reviewing
Abstract. In this work, we introduce the notion of $T$-normed vector
spaces by extending normed vector spaces. This concept can be considered the first generalization of normed vector spaces satisfying the
$T_2$ -separation axiom. Using this axiom, some fixed point theorems are
proved via a binary relation in the setting of $T$-normed vector spaces
without using neither the compactness nor the uniform convexity. Furthermore, some examples are given to show the superiority of the proven
results.
Date of submission: 18 September 2022 г.
- Stabilities of Ulam-Hyers Type for a Class of
Nonlinear Fractional Differential Equations with
Integral Boundary Conditions in Banach Spaces
Status: reviewing
Abstract. Motivated by the knowledge of the existence of continuous so-
lutions of a certain fractional boundary value problem with integral boundary
conditions, we present in here –in a unified manner– new sufficient conditions
to conclude the existence and uniqueness of continuously differentiable solutions
to this fractional boundary value problem and analyse its stability in the sense
of Ulam-Hyers and Ulam-Hyers-Rassias. After presenting the main conclusions,
two illustrative examples are provided to verify the effectiveness of the proposed
theoretical results.
Date of submission: 05 October 2022 г.
- Mirsaburov M., Ergasheva The problem with the missing Goursat condition for a hyperbolic equation degenerating on the boundary of the domain with a singular
coefficient
Status: reviewing
Abstract. For a hyperbolic equation degenerating on the boundary of the domain with a singular coefficient, theorems of uniqueness and existence of a solution to the problem with the missing Goursat
condition on the boundary characteristic and an analogue of the Frankl condition on the segment of
degeneration are proved.
Date of submission: 28 October 2022 г.
- Ashurov R.R., M.D. Inverse problem for the subdiffusion equation with fractional Caputo derivative
Status: reviewing
Abstract. The inverse problem of determining the right-hand side of the subdiffusion equation with the fractional Caputo derivative is considered. The right-hand side of the equation has the form $f(x)g(t)$ and the unknown is function $f(x)$. The condition $ u (x,t_0)= \psi (x) $ is taken as the over-determination condition, where $t_0$ is some interior point of the considering domain and $\psi (x) $ is a given function. It is proved by the Fourier method that under certain conditions on the functions $g(t)$ and $\psi (x) $ the solution of the inverse problem exists and is unique. An example is given showing the violation of the uniqueness of the solution of the inverse problem for some sign-changing functions $g(t)$. For such functions $g(t)$, we find necessary and sufficient conditions on the initial function and on the function from the over-determination condition, which ensure the existence of a solution to the inverse problem.
Date of submission: 02 November 2022 г.
- Baranov A.D., Lishanskii A.A. Point spectrum and hypercyclicity problem for a class of truncated Toeplitz operators
Status: reviewing
Abstract. In this note we discuss an open problem whether a truncated Toeplitz operator on a model space can be hypercyclic. We compute point spectrum and eigenfunctions for a class of truncated Toeplitz operators with polynomial analytic and antianalytic parts. We show that, for a class of model spaces, truncated Toeplitz operators with symbols of the form $\Phi(z) =a \bar{z} +b + cz$, $|a| \ne |c|$, have complete sets of eigenvectors, and, in particular, are not hypercyclic.
Date of submission: 03 November 2022 г.
- Градиентные меры Гиббса для модели Блюма-Капеля в случае "петля"
на дереве Кэли
Status: reviewing
Abstract. Работа посвящена градиентным мерам Гиббса (ГМГ) для модели Блюма-Капеля со
счетным множеством Z значений спина в случае "петля" на деревьях Кэли. Эта модель
определяется потенциалом взаимодействия градиента ближайшего соседа. Используя
аргумент Кульске-Шрайвера, основанный на уравнениях граничного закона, мы даем
несколько $q$-периодических трансляционно-инвариантных ГМГ для $q = 2,3,4$.
Date of submission: 24 November 2022 г.
- GENETIC ALGORITHM APPLIED TO FRACTIONAL OPTIMAL
CONTROL OF A DIABETIC PATIENT
Status: reviewing
Abstract. Diabetes is a dangerous disease that is increasing in incidence
every year. The objective of this paper is to present and analyze the
model of diabetes and its complications with the fractional derivative
of Caputo. A mathematical model related to the fractional derivative of
type 2 diabetes has been proposed. The positivity and boundedness of
the solutions were demonstrated by the Laplace transform method. We
have studied the existence and uniqueness of the solution of the system.
We used the genetic algorithm (GA) to solve the fractional di?erential
equation model and to characterize the optimal control, as an e?cient
and simple metaheuristic method to implement. Simulations of the total
number of diabetics show, with the di?erent values of α chosen, that the
combined control strategy leads to a signi?cant decrease. The simulation
results also show that the number of uncomplicated diabetics in the
fractional model, for the di?erent fractional values of α, decreases more
rapidly than the integer derivative model.
Date of submission: 26 November 2022 г.
- Caplieva A.A., Smirnov A.O. The vector form of Kundu-Eckhaus equation and its simplest solutions
Status: reviewing
Abstract. In our work a hierarchy of integrable vector nonlinear differential equations depending on the functional parameter $r$ is constructed using a monodromy matrix. The first equation of this hierarchy for $r=\alpha(\bp^t\bq)$ is vector analogue of the Kundu-Eckhaus equation. When $\alpha=0$, the equations of this hierarchy turn into equations of the Manakov system hierarchy. New elliptic solutions to vector analogue of the Kundu-Eckhaus and Manakov system are presented. In conclusion, it is shown that there exist linear transformations of solutions to vector integrable nonlinear equations into other solutions to the same equations.
Date of submission: 09 December 2022 г.
- Об обратимости и спектре интегрального оператора Винера-Хопфа в счетно-нормированном пространстве функций со степенным характером убывания на бесконечности.
Status: reviewing
Abstract. In a countable normalized space of measurable functions on the real axis, decreasing faster then any power, the Wiener-Hopf integral operator considered. It is shown that the class of bounded Wiener-Hopf operators contains with discontinuous symbols of a special form. The questions of boundedness, Noetherian property and invertibility of such operators in the given countably normed space are considered. In particular, criteria for Noetherianity and invertibility in terms of a symbol are obtained. As a corollary, the spectrum of such an operator in the topological space under consideration is described. Some relations are given that connect the spectra of the Wiener-Hopf integral operator with the same symbol in the countably normed spaces of measurable functions decreasing at infinity faster than any power.
Date of submission: 12 December 2022 г.
- Gaisin R.A. Rate of decreasing of the extremal function in Carleman class
Status: reviewing
Abstract. Исследуются вопросы, связанные с теоремами типа Левинсона-Шёберга-Волфа в комплексном анализе, в частности, обсуждается известный вопрос, поставленный в 70-е годы Е.\,М.~Дынькиным об эффективной оценке мажоранты роста аналитической функции вблизи множества особых точек и другая близкая проблема о скорости стремления к нулю экстремальной функции в неквазианалитическом классе Карлемана в окрестности точки, где все производные функций из этого класса обращаются в нуль.
Точные асимптотические оценки наилучшей мажоранты роста вблизи особенностей были найдены В. Мацаевым и М. Содиным в 2002 году.
Некоторые оценки (как сверху, так и снизу) для экстремальной функции в классе Карлемана в 2018 году были получены А.М. Гайсиным, но они оказались не очень близкими к истинной величине этой функции. В настоящей статье получены точные двусторонние оценки для экстремальной функции.
Date of submission: 12 December 2022 г.
- Kalmetev R.Sh., Orlov Yu.N., Sakbaev V.Zh. Averaging of random affine transformations of functions domain
Status: reviewing
Abstract. We study averagings of Feynman-Chernov iterations of random operator-valued strongly continuous functions whose values are bounded operators on a separable Hilbert space. The operators under consideration are given by random affine transformations of the domain of functions, and their compositions are a non-commutative analog of random walks.
Sufficient conditions are obtained for the convergence of the mathematical expectation of the sequence of Feynman-Chernov iterations to a semigroup solving the Cauchy problem for the corresponding Fokker-Planck equation.
Date of submission: 21 December 2022 г.
- Nazarov S.A. The influence of the Winkler--Steklov conditions
on natural oscillations of an elastic weighty body.
Status: reviewing
Abstract. We consider the spectral problem for the spacial system of equations of the elasticity theory.
%Small parts of the body surface are supplied with the Winkler--Steklov conditions which
%model spring mount while the remaining part of the boundary is traction-free.
%In several cases (the relative stiffness of springs and their disposition are varied)
%we consider asymptotics of eigenfrequencies of the body. Certain cases are analysed in detail,
%open questions are formulated and pathological situations leading to the loss of the customary
%properties by the spectrum, are discussed.
Date of submission: 24 December 2022 г.
- Makhmudov O.I., Niyozov I.E. The Cauchy problem for the system of elasticity theory
Status: reviewing
Abstract. In this paper, we considered the problem of analytical continuation of the solution of the system equations of the moment theory of elasticity in spacious bounded domain from its values and values of its strains on part of the boundary of this domain, i.e., the Cauchy's problem. The condition of solvability of this problem is considered.
Date of submission: 28 December 2022 г.
- Krivosheev A.S., Krivosheeva O.A. A necessary condition for the fundamental principle to hold for invariant subspaces on unbounded convex domain
Status: reviewing
Abstract. In this paper we study the spaces of analytic functions in convex domains of the complex plane.
Subspaces of such spaces invariant with respect to the differentiation operator are considered. The
problem of the fundamental principle for an invariant subspace is investigated, i.e. we investigate the problem of representing all its elements
using a series of eigenfunctions and associated functions of the differentiation operator in this subspace --- exponents and exponential monomials.
Simple geometric conditions are obtained, which are necessary for the existence of a fundamental principle.
These conditions are formulated in terms of the arc length of the convex domain and the maximum density of the exponent sequence.
Date of submission: 06 January 2023 г.
- Balakhnev M.Yu. On the symmetry classification
of integrable evolution equations of the 3rd order
Status: reviewing
Abstract. New results within classification of nonlinear evolution vector equations of the 3rd order are obtained
Date of submission: 10 January 2023 г.
- Inequalities for Meromorphic functions with Prescribed Poles
Status: reviewing
Abstract. For a rational function $P\in \mathcal{P}_n,$ Dewan et al.[J. Math Anal. Appl.\textbf{363(1),}(2010), 38-41] proved:
$$\bigg|zP'(z)+\dfrac{n\beta}{2}P(z)\bigg|\geq n\bigg|1+\dfrac{\beta}{2}\bigg|\min \limits_{z\in T_k}{|P(z)|}.$$
In this paper we prove some refinements of Bernstein-type inequalities for meromorphic functions with prescribed poles and restricted zeros. These results not only generalize some inequalities for rational functions but also improve as well as generalize some polynomial inequalities too.
Date of submission: 10 January 2023 г.
- EXISTENCE AND FINITE-TIME STABILITY OF SOLUTIONS
FOR A CLASS OF NONLINEAR HILFER FUZZY
FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATIONS
WITH TIME-DELAYS
Status: reviewing
Abstract. In the current paper, we investigate a novel class of nonlinear Hil-
fer fuzzy fractional stochastic differential equations with time-delays. Firstly,
we convert the system under consideration into an analogous integral system.
Secondly, using Schauder and Banach fixed point theorem, the existence and
uniqueness results of solutions for nonlinear Hilfer fuzzy fractional stochastic
differential equations are then established. Additionally, we explore the finite-
time stability result of solution for the system under consideration. Lastly, an
example is provided to visualize the theoretical results.
Date of submission: 11 January 2023 г.
- Topological degree method for a new class of $\psi$-Hilfer fractional
differential Langevin equation
Status: reviewing
Abstract. This paper investigates the existence and uniqueness of solution for a new class of $\psi$-Hilfer fractional differential Langevin equation. The suggested study is based on some basic definitions of fractional calculus and topological degree theory. We established the existence result by utilizing the topological degree method for condensing maps, and by making use of Banach’s fixed point theorem we deal with the uniqueness result. As application, we give an illustrative example to demonstrate our theoretical result.
Date of submission: 23 January 2023 г.
- Aitkuzhina N.N., Gaisin A.M., Gaisin R.A. Behavior of entire Dirichlet series of the class $\underline{D}(\Phi)$ on curves of bounded $K$-slope
Status: reviewing
Abstract. Изучается асимптотическое поведение суммы целого ряда Дирихле $F(s)=\sum\limits_{n}a_{n}e^{\lambda_{n}s}$, $0<\lambda_{n}\uparrow\infty$, на кривых ограниченного $K$-наклона, естественным образом уходящих в бесконечность. Для целых трансцендентных функций конечного порядка, имеющих вид $f(z)=\sum\limits_{n}a_{n}z^{p_{n}}$, $p_{n}\in\mathbb{N}$, Полиа показал, что если плотность последовательности $\left\{p_{n}\right\}$ равна нулю, то для любой кривой $\gamma$, уходящей в бесконечность, существует неограниченная последовательность $\{\xi_{n}\}\subset\gamma$, такая, что при $\xi_{n}\rightarrow\infty$ имеет место соотношение:
$$\ln M_{f}(|\xi_{n}|)\sim \ln\left|f(\xi_{n})\right|$$
($M_{f}(r)$ --- максимум модуля функции $f$). Позже эти результаты были полностью перенесены И.Д. Латыповым на целые ряды Дирихле конечного порядка и конечного нижнего порядка по Ритту. Дальнейшее обобщение было получено в работах Н.Н. Юсуповой--Аиткужиной на более общие классы $D(\Phi)$ и $\underline{D}(\Phi)$, определяемые выпуклой мажорантой $\Phi$. В настоящей статье получены необходимые и достаточные условия на показатели $\lambda_{n}$ для того, чтобы логарифм модуля суммы любого ряда Дирихле из класса $\underline{D}(\Phi)$ на кривой $\gamma$ ограниченного $K$--наклона был эквивалентен логарифму максимального члена, когда $\sigma=Re s\rightarrow +\infty$ по некторому асимптотическому множеству, верхняя плотность которого не меньше $\frac{1}{\sqrt{K^{2}+1}$.
Date of submission: 31 January 2023 г.
- Allahverdiev B.P., Tuna H. On a $\phi$-fractional Dirac equation
Status: reviewing
Abstract. In this paper, we construct a $\phi$-fractional Dirac system by using $\phi$-
Riemann-Liouville and $\phi$-Caputo derivatives: Some properties of this system investigated. Finally, a sufficient condition on eigenvalues for the
existence and uniqueness of the associated eigenfunctions is given.
Date of submission: 05 February 2023 г.
- Ikromov I.A., Muranov Sh.A. On estimates for oscillatory integrals with phase mixed homogeneous polynomial of degree one
Status: reviewing
Abstract. В этой статье рассматриваются оценки преобразования Фурье мер,
сосредоточенных на аналитических гиперповерхностях содержащих множитель гашения.
В качестве гасителя выбирается степень гауссовой кривизны
гиперповерхности. Известно, что если степень гауссовой кривизны достаточно большое
положительное число, то преобразование Фурье соответствующей меры убывает
оптимально. С.Д. Согги и И.М. Стейном поставлена задача о минимальной степени
гауссовой кривизны, гарантирующей оптимальное убывание преобразования Фурье с
множителем гашения. В работе приведено решение задачи С.Д.Согги и И.М.Стейна
об оптимальном убывании преобразования Фурье мер с множителем гашения для
поверхностей в $\mathbb{R}^3$, заданных графиком квазиоднородного полинома степени единица.
Date of submission: 10 February 2023 г.
- Shamaev A.S., Shumilova V.V. Homogenization of the equations of motion for a medium consisting
of an elastic material and an incompessible Kelvin-Voigt fluid
Status: reviewing
Abstract. We consider an initial-boundary problem
describing the motion of a two-phase medium with a periodic structure.
The first phase of the medium is an isotropic elastic material
and the second phase is an incompressible viscoelastic Kelvin-Voigt fluid.
For the stated problem, we deduce the corresponding homogenized
problem describing the motion of a homogeneous viscoelastic medium
with memory. We find the explicit analytical expressions
for the coefficients and convolution kernels of the homogenized
equations corresponding to the case of a layered medium.
Date of submission: 14 February 2023 г.
- Timergaliev S.N. Solvability of Nonlinear Boundary Value Problems for Non-Sloping Timoshenko-Type Isotropic Shells of Zero Principal Curvature
Status: reviewing
Abstract. We study the solvability of a boundary value problem for a system of nonlinear
partial differential equations of the second order
under given boundary conditions, which describes the equilibrium state of elastic non-sloping isotropic inhomogeneous shells with loose edges in the framework of the Timoshenko shear model.
The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in the Sobolev space, the solvability of which is established from
using the contraction mapping principle.
Date of submission: 22 February 2023 г.
- Об алгебраической модификации
метода неопределенных коэффициентов для решения неоднородных линейных разностных уравнений
Status: reviewing
Abstract. In the present work we propose an algebraic modification to method of undetermined coefficients for solving nonhomogeneous linear stationary difference equations with quasipolynomial right-hand sides. Although the classical method of undetermined coefficients is well-knon in both differential and difference case, it is in the difference case that its application is essentially limited. For instance, it is hard to apply to rather complex expressions which may appear in case of complex quasipolynomial and resonance. Novelty of the research lies in proposing algebraic modification of the given method which eliminates its main drawbacks, and also allows to modifty the superposition principle for applying the method of undetermined coefficients to the entire difference equation at once without dividing the problem into more simple ones. The principle of superposition is stated in matrix form.
Date of submission: 28 February 2023 г.
- Braichev G.G., Khabibullin B.N., Sherstyukov V.B. The Sylvester problem, coverings by shifts, and uniqueness theorems for entire functions
Status: reviewing
Abstract. Идея написать заметку возникла в ходе обсуждения, последовавшего за докладом первого автора
на Международной научной конференции <<Уфимская осенняя математическая школа -- 2022>>.
Предложены три общих способа построения множеств единственности в классах целых функций с ограничениями на рост.
Во всех трех случаях в качестве такого множества выбирается последовательность нулей целой функции со специальными свойствами.
Первый способ связан с известной проблемой Сильвестра о наименьшем круге, содержащем заданный набор точек на плоскости, и теоремами выпуклой геометрии.
Второй исходно опирается на теорему Хелли о пересечении выпуклых множеств и ее применения к возможности покрытия одного множества сдвигом другого.
Третий способ основан на классической формуле Иенсена, позволяющей оценить тип целой функции через усредненную верхнюю плотность последовательности ее нулей. Мы даем сейчас только базовые результаты. Развитие наших подходов предполагается изложить в последующих работах.
Date of submission: 06 Mart 2023 г.