Deviations Of Argument Research Articles

Dirichlet Solutions of Functional Differential Equations without Delay

Necessary and sufficient conditions for the existence of a valid Dirichlet solution were obtained. A method was developed to find Dirichlet solutions of the functional differential equation with non-delayed linear argument deviation.

Synchronization robustness analysis of memristive-based neural networks with deviating arguments and stochastic perturbations

<abstract><p>In this article, we investigate the robustness of memristive-based neural networks (MNNs) with deviating arguments (DAs) and stochastic perturbations (SPs). Based on the set-valued mapping method, differential inclusion theory and Gronwall inequalities, we derive the upper bounds for the width of DAs and the intensity of SPs. When the DAs and SPs are smaller than these upper bounds, the MNNs maintains exponential synchronization. Finally, several specific simulation examples demonstrate the effectiveness of the results.</p></abstract>

The Cauchy Problem for a Stochastic Parabolic Equation with an Argument Deviation

The Cauchy Problem for a Stochastic Parabolic Equation with an Argument Deviation

Existence and global exponential stability of periodic solution for Cohen-Grossberg neural networks model with piecewise constant argument

In this paper, we introduce a Cohen-Grossberg neural networks model with piecewise alternately advanced and retarded argument. Some sufficient conditions are established for the existence and global exponential stability of periodic solutions. The approaches are based on employing Brouwer's fixed-point theorem and an integral inequality of Gronwall type with deviating argument. The criteria given are easily verifiable, possess many adjustable parameters, and depend on piecewise constant argument deviations, which provide flexibility for the design and analysis of Cohen-Grossberg neural networks model. Several numerical examples and simulations are also given to show the feasibility and effectiveness of our results.

Optimal conditions of solvability of periodic problem for systems of differential equations with argument deviation

Abstract In this paper, we study, on the interval I = [ a , b ] {I=[a,b]} , the problem of the solvability and unique solvability of systems of higher-order differential equations with argument deviation u i ( m i ) ( t ) = p i ( t ) u i + 1 ( τ i ( t ) ) + q i ( t ) ( i = 1 , … , n ) u_{i}^{(m_{i})}(t)=p_{i}(t)u_{i+1}(\tau_{i}(t))+q_{i}(t)\quad(i=1,\ldots,n) and u i ( m i ) ( t ) = f i ( t , u i + 1 ( τ i ( t ) ) ) + q 0 ⁢ i ( t ) ( i = 1 , … , n ) , u_{i}^{(m_{i})}(t)=f_{i}\bigl{(}t,u_{i+1}(\tau_{i}(t))\bigr{)}+q_{0i}(t)\quad(% i=1,\ldots,n), under the periodic boundary conditions u i ( j ) ( b ) - u i ( j ) ( a ) = c i ⁢ j ( i = 1 , … , n , j = 0 , … , m i - 1 ) , u_{i}^{(j)}(b)-u_{i}^{(j)}(a)=c_{ij}\quad(i=1,\ldots,n,\,j=0,\ldots,m_{i}-1), where u n + 1 = u 1 {u_{n+1}=u_{1}} , n ≥ 2 {n\geq 2} , m i ≥ 1 {m_{i}\geq 1} , p i ∈ L ∞ ⁢ ( I ; R ) {p_{i}\in L_{\infty}(I;R)} , q 0 ⁢ i ∈ L ⁢ ( I ; R ) {q_{0i}\in L(I;R)} , f i : I × R → R {f_{i}:I\times R\to R} are Carathéodory class functions, and τ i : I → I {\tau_{i}:I\to I} are measurable functions. The optimal conditions are obtained, which guarantee the unique solvability of the linear problem and take into account the effect of argument deviation. Based on these results, the optimal conditions of the solvability and unique solvability are proved for the nonlinear problem.

A precise high-dimensional asymptotic theory for boosting and minimum-ℓ1-norm interpolated classifiers

This paper establishes a precise high-dimensional asymptotic theory for boosting on separable data, taking statistical and computational perspectives. We consider a high-dimensional setting where the number of features (weak learners) $p$ scales with the sample size $n$, in an overparametrized regime. Under a class of statistical models, we provide an exact analysis of the generalization error of boosting when the algorithm interpolates the training data and maximizes the empirical $\ell_1$-margin. Further, we explicitly pin down the relation between the boosting test error and the optimal Bayes error, as well as the proportion of active features at interpolation (with zero initialization). In turn, these precise characterizations answer certain questions raised in \cite{breiman1999prediction, schapire1998boosting} surrounding boosting, under assumed data generating processes. At the heart of our theory lies an in-depth study of the maximum-$\ell_1$-margin, which can be accurately described by a new system of non-linear equations; to analyze this margin, we rely on Gaussian comparison techniques and develop a novel uniform deviation argument. Our statistical and computational arguments can handle (1) any finite-rank spiked covariance model for the feature distribution and (2) variants of boosting corresponding to general $\ell_q$-geometry, $q \in [1, 2]$. As a final component, via the Lindeberg principle, we establish a universality result showcasing that the scaled $\ell_1$-margin (asymptotically) remains the same, whether the covariates used for boosting arise from a non-linear random feature model or an appropriately linearized model with matching moments.

Robustness Analysis of Control Laws in Complex Dynamical Networks Evoked by Deviating Argument

In recent years, robust performance of the system has been broadly studied as a trending topic among a vast array of scholars. This paper discusses the robustness of control laws for complex dynamic networks (CDNs) with a deviation argument. We design two categories of control laws (linear control law and nonlinear control law) for the undisturbed CDNs to achieve exponential synchronization. It is intractable to ascertain the range of the deviation function exactly. Hence, some corresponding sufficient criteria are put forward to ensure exponential synchronization of CDNs with deviation argument when control laws are not changed. By adopting the Gronwall–Bellman lemma and solving the transcendental equation, we can obtain the admissible upper limits of the deviating function, to keep the corresponding control laws. In comparison with previous research findings, robustness, deviating argument, and control laws are all considered in this study, which enhances the previous findings. Finally, two emulation examples verify the validity of the analysis.

GROUP MOVEMENT OF ROBOTS WHEN TAKING INTO ACCOUNT THE INFLUENCE OF CONNECTIONS WITH A DEVIATING ARGUMENT

GROUP MOVEMENT OF ROBOTS WHEN TAKING INTO ACCOUNT THE INFLUENCE OF CONNECTIONS WITH A DEVIATING ARGUMENT

THE NONLOCAL PROBLEM FOR FRACTAL DIFFUSION EQUATION

Over the past few decades, the theory of pseudodifferential operators (PDO) and equations with such operators (PDE) has been intensively developed. The authors of a new direction in the theory of PDE, which they called parabolic PDE with non-smooth homogeneous symbols (PPDE), are Yaroslav Drin and Samuil Eidelman. In the early 1970s, they constructed an example of the Cauchy problem for a modified heat equation containing, instead of the Laplace operator, PDO, which is its square root. Such a PDO has a homogeneous symbol |σ|, which is not smooth at the origin. The fundamental solution of the Cauchy problem (FSCP) for such an equation is an exact power function. For the heat equation, FSCP is an exact exponential function. The Laplace operator can be interpreted as a PDO with a smooth homogeneous symbol |σ|^2, σ ∈ Rn. A generalization of the heat equation is PPDE containing PDO with homogeneous non-smooth symbols. They have an important application in the theory of random processes, in particular, in the construction of discontinuous Markov processes with generators of integro-differential operators, which are related to PDO; in the modern theory of fractals, which has recently been rapidly developing. If the PDO symbol does not depend on spatial coordinates, then the Cauchy problem for PPDE is correctly solvable in the space of distribution-type generalized functions. In this case, the solution is written as a convolution of the FSCP with an initial generalized function. These results belong to a number of domestic and foreign mathematicians, in particular S. Eidelman and Y. Drin (who were the first to define PPDO with non-smooth symbols and began the study of the Cauchy problem for the corresponding PPDE), M. Fedoruk, A. Kochubey, V. Gorodetsky, V . Litovchenko and others. For certain new classes of PPDE, the correct solvability of the Cauchy problem in the space of Hölder functions has been proved, classical FSCP have been constructed, and exact estimates of their power-law derivatives have been obtained [1–4]. Of fundamental importance is the interpretation of PDO proposed by A. Kochubey in terms of hypersingular integrals (HSI). At the same time, the HSI symbol is constructed from the known PDO symbol and vice versa [6]. The theory of HSI, which significantly extend the class of PDO, was developed by S. Samko [7]. We extends this concept to matrix HSI [5]. Generalizations of the Cauchy problem are non-local multipoint problems with respect to the time variable and the problem with argument deviation. Here we prove the solvability of a nonlocal problem using the method of steps. We consider an evolutionary nonlinear equation with a regularized fractal fractional derivative α ∈ (0, 1] with respect to the time variable and a general elliptic operator with variable coefficients with respect to the second-order spatial variable. Such equations describe fractal properties in real processes characterized by turbulence, in hydrology, ecology, geophysics, environment pollution, economics and finance.

ON ROOT FUNCTIONS OF NONLOCAL DIFFERENTIAL SECOND-ORDER OPERATOR WITH BOUNDARY CONDITIONS OF PERIODIC TYPE

In this paper we consider one class of spectral problems for a nonlocal ordinary differential operator (with involution in the main part) with nonlocal boundary conditions of periodic type. Such problems arise when solving by the method of separation of variables for a nonlocal heat equation. We investigate spectral properties of the problem for the nonlocal ordinary differential equation Ly (x) ≡ −y 00 (x) + εy00 (−x) = λy (x), −1 < x < 1. Here λ is a spectral parameter, |ε| < 1. Such equations are called nonlocal because they have a term y 00 (−x) with involutional argument deviation. Boundary conditions are nonlocal y 0 (−1) + ay0 (1) = 0, y (−1) − y (1) = 0. Earlier this problem has been investigated for the special case a = −1. We consider the case a 6= −1. A criterion for simplicity of eigenvalues of the problem is proved: the eigenvalues will be simple if and only if the number r = sqrt{p (1 − ε) / (1 + ε)} is irrational. We show that if the number r is irrational, then all the eigenvalues of the problem are simple, and the system of eigenfunctions of the problem is complete and minimal but does not form an unconditional basis in L2(−1, 1). For the case of rational numbers r, it is proved that a (chosen in a special way) system of eigen- and associated functions forms an unconditional basis in L2(−1, 1).

Robustness of Hopfield Neural Networks Described by Differential Algebraic Systems of Index-1 under the Conditions of Deviation Argument and Stochastic Disturbance

Robustness refers to the ability of a system to maintain its original state under a continuous disturbance conditions. The deviation argument (DA) and stochastic disturbances (SDs) are enough to disrupt a system and keep it off course. Therefore, it is of great significance to explore the interval length of the deviation function and the intensity of noise to make a system remain exponentially stable. In this paper, the robust stability of Hopfield neural network (VPHNN) models based on differential algebraic systems (DAS) is studied for the first time. By using integral inequalities, expectation inequalities, and the basic control theory method, the upper bound of the interval of the deviation function and the noise intensity are found, and the system is guaranteed to remain exponentially stable under these disturbances. It is shown that as long as the deviation and disturbance of a system are within a certain range, there will be no unstable consequences. Finally, several simulation examples are used to verify the effectiveness of the approach and are described below.

Исследование влияния запаздывания в одной математической модели динамики мирового развития

Динамике мирового развития посвящено достаточное количество работ. Но очень немногие из них имеют четкие абстрактные математические модели соответствующих процессов. Данная работа посвящена дальнейшему углублению и математической абстрактизации исследования процессов мирового развития. Проведен качественный анализ линейной и модифицированной нелинейной модели посредством систем неоднородных дифференциальных уравнений. Вычислены их стационарные состояния, записаны явные аналитические решения. Впервые предложена модель с учетом фактора временного запаздывания, записанная в виде функционально-дифференциальных уравнений с отклонением аргумента. Показано, что при таком введении в модель запаздывающего аргумента систему можно свести к системе линейных неоднородных дифференциальных уравнений с постоянными коэффициентами без запаздывания, и на устойчивость стационарного состояния изучаемого равновесия системы будут влиять только линейные члены уравнений. , не содержащие отклонения аргумента. Этот факт хорошо соотносится с социально-экономической интерпретацией данной задачи. В дальнейшем работа будет направлена на изучение влияния не одного, а нескольких факторов временного запаздывания, когда модель будет представлена в виде системы функционально-дифференциальных уравнений с несколькими разными отклоняющимися аргументами в уравнениях, отвечающих за динамику конкретного процесса, являющегося отдельной составляющей общей динамики мирового развития.

Some two-point boundary value problems for systems of higher order functional differential equations

In the paper we study the question of the solvability and unique solvability of systems of the higher order differential equations with the argument deviations \begin{equation*} u_i^{(m_i)}(t)=p_i(t)u_{i+1}(\tau _{i}(t))+ q_i(t), (i=\overline {1, n}), \text {for $t\in I:=[a, b]$}, \end{equation*} and \begin{equation*}u_i^{(m_i)} (t)=F_{i}(u)(t)+q_{0i}(t), (i = \overline {1, n}), \text {for $ t\in I$}, \end{equation*} under the conjugate $u_i^{(j_1-1)}(a)=a_{i j_1}$, $u_i^{(j_2-1)}(b)=b_{i j_2}$, $j_1=\overline {1, k_i}$, $j_2=\overline {1, m_i-k_i}$, $i=\overline {1, n}$, and the right-focal $u_i^{(j_1-1)}(a)=a_{i j_1}$, $u_i^{(j_2-1)}(b)=b_{i j_2}$, $j_1=\overline {1, k_i}$, $j_2=\overline {k_i+1,m_i}$, $i=\overline {1, n}$, boundary conditions, where $u_{n+1}=u_1, $ $n\geq 2, $ $m_i\geq 2, $ $p_i \in L_{\infty }(I; R), $ $q_i, q_{0i}\in L(I; R), $ $\tau _i\colon I\to I$ are the measurable functions, $F_i$ are the local Caratheodory's class operators, and $k_i$ is the integer part of the number $m_i/2$.In the paper are obtained the efficient sufficient conditions that guarantee the unique solvability of the linear problems and take into the account explicitly the effect of argument deviations, and on the basis of these results are proved new conditions of the solvability and unique solvability for the nonlinear problems.

Dirichlet-Neumann problem for the partial differential equations with deviation over the space argument

Dirichlet-Neumann problem for the typeless high order partial differential equation with deviating over the space argument is studied in the domain, which is the Cartesian product of the segment $(0,T)$ and the unit circle $\Omega=\mathbb R/(2\pi \mathbb Z)$. Dirichlet-Neumann problem for hyperbolic equations and their systems in case with absent argument deviation $h$ has been studied by the authors before. Correct solvability conditions have been established for these problems for almost all (with respect to Lebesgue measure) numbers $T>0$ and for almost all (with respect to Lebesgue measure) vectors, constructed by coefficients of the equation.
 In this paper, the solvability conditions of the problem for $h\ne0$ are described and the influence of the deviation $h$ on the solvability of the problem is studied. The solution of the problem is constructed in the form of the series with respect to the systems of orthogonal functions. Metric estimations (of exponential type) are proved for small denominators appearing during construction of the problem solution. These estimations guarantee the correctness of the problem in Sobolev spaces for almost all (with respect to Lebesgue measure) values $ T> 0 $ and for almost all (with respect to Lebesgue measure) values $ h \in [0,2\pi) $. The obtained results are based on the fact that the corresponding characteristic determinant permits factorization in the form of the product of hyperbolic functions with integer parameters.

НЕЛІНІЙНА МОДЕЛЬ ПОВЕДІНКИ ДВОХ КОНКУРЕНТНИХ ФІРМ

The practical task of economics lies in applying the methods of substantiating its decisions. For economics, the main method is the modeling of economic phenomena and processes and, above all, mathematical modeling, which has been stipulated by the presence of stable MATHEMATICAL METHODS, MODELS AND INFORMATION TECHNOLOGIES IN ECONOMY Issue I (81), 2021 117 quantitative patterns and the possibility of a formalized description of many economic processes. The economic-mathematical model contains a system of equations of linear and nonlinear units that promote a mathematical description of economic processes and phenomena, consists of a set of variables and parameters and serves to study these processes and control them. Dynamic models of the economy describe it in development, as well as provide a detailed description of technological methods of production. Mathematical description of dynamic models is carried out with the use of a system of differential equations (in models with continuous time), difference equations (in models with discrete time), as well as systems of algebraic equations. It is important that the investigation of various economic issues has led to the development of the mathematical apparatus. In linear algebra, productive matrices are caused by the studies of intersectoral balance, whereas mathematical programming arose in the course of researching the optimal plan for the distribution of limited resources. In a similar way, there emerged the theory of economic indices and econometrics, the theory of production functions and the theory of consumption, the theory of general economic balance and social welfare, the theory of optimal economic growth. The paper under studies deals with the dynamic economic behavior of two competing objects, whose mathematical model is a nonlinear nonlocal problem for a system of ordinary differential equations with variable coefficients and argument deviation. The dynamic mathematical model is based on the assumption that the volume of output of both firms is determined by such factors on which output depends linearly. The model under discussion includes nonlinear factors, which describe the level of distrust of the competitors and depend on the time of observations and production volumes in previous moments, because the latter significantly affect the production activities of the firm. Such mathematical models are called time-delayed models.

EXISTENCE RESULTS FOR FIXED AND FREE DISCONTINUITIES PROBLEMS WITH DEVIATING ARGUMENTS

EXISTENCE RESULTS FOR FIXED AND FREE DISCONTINUITIES PROBLEMS WITH DEVIATING ARGUMENTS

Моделювання крайових задач для лінійних диференціально-різницевих рівнянь нейтрального типу

In space navigation problems, optimal control of systems with aftereffect, ecology and immunology problems, boundary value problems for differential-difference and integro-differential equations with delays arise, which are an important part of modern theory of differential-functional equations. Finding precise solutions of boundary value problems for differential-difference equations is possible only for the simplest classes of such problems. At present, projection-iterative methods, numerical-analytical method and others are suggested for boundary value problems with delay and of neutral type. The spline-collocation method for solving boundary value problems for differential-difference equations is one of the most efficient algorithms that allows building simple computational schemes. In this paper, we investigate the scheme of modeling boundary value problems for linear differential-difference equations of neutral type with many variable deviations of the argument. A functional space is defined to which the solutions of the considered boundary value problems belong, the properties of the solution smoothness are investigated depending on the structure of the argument deviations. Simple and verifiable sufficient conditions for the boundary value problem solution existence are given. For finding the solution of the boundary value problem, an iterative computational scheme based on the spline approximation method is described. In order to take into account possible discontinuities of the boundary value problem solution derivatives, cubic splines of defect two are used for neutral-type equations. Coefficient conditions for the initial equation which ensure the convergence of the iterative process are obtained. An estimate of the approximate solution error is conducted. A model example of a boundary value problem for a neutral type differential-difference equation is presented on which the iterative scheme is demonstrated. Numerical experiments confirm the obtained theoretical results.

On Construction of Solutions of Linear Differential Systems with Argument Deviations of Mixed Type

We show the use of parametrization techniques and successive approximations for the effective construction of solutions of linear boundary value problems for differential systems with multiple argument deviations. The approach is illustrated with a numerical example.

A Precise High-Dimensional Asymptotic Theory for Boosting and Minimum-L1-Norm Interpolated Classifiers

This paper establishes a precise high-dimensional asymptotic theory for boosting on separable data, taking statistical and computational perspectives. We consider the setting where the number of features (weak learners) p scales with the sample size n, in an over-parametrized regime. Under a broad class of statistical models, we provide an exact analysis of the generalization error of boosting, when the algorithm interpolates the training data and maximizes the empirical L1-margin. The relation between the boosting test error and the optimal Bayes error is pinned down explicitly. In turn, these precise characterizations resolve several open questions raised in [15, 81] surrounding boosting. On the computational front, we provide a sharp analysis of the stopping time when boosting approximately maximizes the empirical L1 margin. Furthermore, we discover that the larger the overparametrization ratio p/n, the smaller the proportion of active features (with zero initialization), and the faster the optimization reaches interpolation. At the heart of our theory lies an in-depth study of the maximum L1-margin, which can be accurately described by a new system of non-linear equations; we analyze this margin and the properties of this system, using Gaussian comparison techniques and a novel uniform deviation argument. Variants of AdaBoost corresponding to general Lq geometry, for q > 1, are also presented, together with an exact analysis of the high-dimensional generalization and optimization behavior of a class of these algorithms.

ON COMPLETENESS OF ROOT VECTORS OF THE CAUCHY PROBLEM OF THE FIRST ORDER EQUATION WITH DEVIATING ARGUMENT

ON COMPLETENESS OF ROOT VECTORS OF THE CAUCHY PROBLEM OF THE FIRST ORDER EQUATION WITH DEVIATING ARGUMENT