Cauchy Inequality Research Articles

The application of convex function and GA-convex function

A convex function is a function that maps from a convex subset of a vector space to the set of real numbers. Convex functions have some important properties, such as non-negativity, monotonicity, and convexity, which can help us derive and prove inequalities. This paper explores the concepts of convex functions and GA-convex functions, demonstrating their utility in proving a variety of common and complex inequalities. Beginning with an overview of convex functions and their extension to GA-convex functions, the study shows how these mathematical tools can be effectively utilized in the context of inequality proofs. By leveraging the properties of these functions, the paper successfully establishes rigorous proofs for a range of inequalities, highlighting the versatility and applicability of convex and GA-convex functions in mathematical analysis. The properties convex and GA-convex functions allow us to use it to determine the direction of inequalities, prove inequalities, determine the optimal solution of inequalities, and even prove Cauchy inequalities.

Robustness analysis of exponential stability of fuzzy inertial neural networks through the estimation of upper limits of perturbations

This paper characterizes the robustness of exponential stability of fuzzy inertial neural network which contains time delays or stochastic disturbance through the estimation of upper limits of perturbations. By utilizing Gronwall-Bellman lemma, stochastic analysis, Cauchy inequality, the mean value theorem of integrals, as well as the properties of integrations, the limits of both time delays and stochastic disturbances are derived in this paper which can make the disturbed system keep exponential stability. The constraints between the two types of disturbances are provided in this paper. Examples are offered to validate our results.

Nonsingular Fast Terminal Sliding Mode Neural Network Decentralized Control of a Quadrotor Unmanned Aerial Vehicle

A nonsingular terminal sliding mode decentralized controller that can ensure the tracking errors of the trajectories and attitude rapid convergence in finite time is proposed for an insufficient driven and strongly coupled nonlinear four-rotor unmanned aerial vehicle (UAV). The total lift of the UAV system is decomposed into three virtual drive separation forces corresponding to the three positions. The insufficient drive UAV system is transformed into a virtual full-drive model for research. The three position states and the three attitude states of UAV are placed correspondingly to the six subsystems by variable substitution. The model uncertainty and unknown disturbance term for each subsystem serves as total coupling terms among the subsystems. The upper bounds of the total coupling terms are considered as unknown ordinary higher order polynomials varying with the six states of the system under the action of time change. With the help of Cauchy inequality, the estimates of the upper bounds are obtained from the approximation performance of the RBF neural network. Finally, the decentralized controller is designed for each attitude subsystem and the virtual decentralized controller for each position subsystem. It is also mapped to the tracking total lift controller by using the virtual decentralized position controller. The controller design process uses the nonsingular terminal sliding mode control technology to ensure that the quadrotor attitude and position variables can quickly converge to the desired value in a short time. Simulation experiments verify that the proposed control method is effective and feasible.

Systematic and Durable Analysis of the Rule based Cauchy’s Inequality in solving the n-order Differential Equations

This paper instigates the multivariate generalization of the widely used Cauchy inequality 1 + x ≤ ex where x can be any non-negative real number. The results of this study can be the solution for the Cauchy’s problem for particular Ordinary Differential Equation (ODE). This is also related to study of the complete monotone function and the divided differences theory. The proof is based on the empty product convention notion and the Beppo Levi theorem of Monotone convergence. This study is also extended to multivariate generalization of the simultaneous inequalities. Key Words: ODEs, Inequalities, Population Dynamics, Simultaneous Inequalities, Divided differences.

Adaptive containment control of nonlinear multi‐agent systems about privacy preservation with multiple attacks

SummaryThis paper focuses on the problem of adaptive containment control about privacy preservation in nonlinear multi‐agent systems, where followers suffer from multiple cyber‐attacks and leaders have non‐zero control inputs. A multiple attacks mold consisting of denial‐of‐service (DoS) attacks and false data‐injection attacks is constructed for nonlinear systems. The containment control protocol is updated by applying an event‐triggered scheme. This scheme is proposed to dispatch the updating information in the presence of multiple attacks. Meanwhile, to deal with the information insecurity that may exist in the control process, the protocol for privacy preservation about initial information is developed by introducing time‐varying functions as masks. Tracking errors of agents are described by taking the inputs of leaders as the external disturbances. This method does not depend on the widths of boundary layers and reduce the computational complexity. The stability of multi‐agent systems is analyzed by applying the Cauchy inequality, the Young's inequality and the Lyapunov function theory. Finally, two simulation examples are shown to illustrate the effectiveness of the proposed control strategy.

STABILITY OF FUNCTIONAL INEQUALITIES WITH 3K-VARIABLE BASED ON JORDAN-VON NEUMANN TYPE ADDITIVE FUNCTIONAL EQUATIONS IN BANACH SPACE

In this paper, we study to solve the Cauchy, Jensen and Cauchy-Jensen additive function inequalities with 3k-variables related to Jordan-von Neumann type in Banach space. These are the main results of this paper.

Derivation and Properties of the Angular Momentum Relative Amended Potential

AbstractThe paper concerns the determination of the Angular Momentum Relative Amended Potential (AMR potential) in the framework of the Full n-Body problem and some of its basic properties are discussed. The AMR potential is derived using two different approaches : first using a Routh reduction of the system relative to rotation about the total angular momentum, second as a variation of the Sundman Inequality, using the Cauchy inequality.

Synchronization of reaction–diffusion Hopfield neural networks with s-delays through sliding mode control

Synchronization of reaction–diffusion Hopfield neural networks with s-delays via sliding mode control (SMC) is investigated in this paper. To begin with, the system is studied in an abstract Hilbert space C([–r; 0];U) rather than usual Euclid space Rn. Then we prove that the state vector of the drive system synchronizes to that of the response system on the switching surface, which relies on equivalent control. Furthermore, we prove that switching surface is the sliding mode area under SMC. Moreover, SMC controller can also force with any initial state to reach the switching surface within finite time, and the approximating time estimate is given explicitly. These criteria are easy to check and have less restrictions, so they can provide solid theoretical guidance for practical design in the future. Three different novel Lyapunov–Krasovskii functionals are used in corresponding proofs. Meanwhile, some inequalities such as Young inequality, Cauchy inequality,Poincaré inequality, Hanalay inequality are applied in these proofs. Finally, an example is given to illustrate the availability of our theoretical result, and the simulation is also carried out based on Runge–Kutta–Chebyshev method through Matlab.

Gradient estimates of a nonlinear elliptic equation for the $V$-Laplacian on noncompact Riemannian manifolds

In this paper, we consider gradient estimates for positive solutions to the following equation $$\triangle_V u+au^p\log u=0$$ on complete noncompact Riemannian manifold with $k$-dimensional Bakry-Emery Ricci curvature bounded from below. Using the Bochner formula and the Cauchy inequality, we obtain upper bounds of $ \nabla u $ with respect to the lower bound of the Bakry-Emery Ricci curvature.

Joint Polarimetric Subspace Detector Based on Modified Linear Discriminant Analysis

Polarimetric synthetic aperture radar (PolSAR) is widely used in remote sensing and has important applications in the detection of ships. Although many polarimetric detectors have been proposed, they are not well combined. Recently, a polarimetric detection optimization filter (PDOF) was proposed, which performs well in most environments. In this study, a novel subspace form of the PDOF [strict PDOF (SPDOF)] was further developed based on the Cauchy inequality and matrix decomposition theories, enhancing detection performance. Furthermore, a simple method to determine the optimal dimension of the subspace detector based on the trace ratio form was proposed by calculating the area under the receiver operating characteristic (ROC) curve, reaching the best detection performance among the subspaces of the detector. Moreover, to combine different subspace detectors, a modified linear discriminant analysis was proposed and developed for the diagonal loading detector (DLD) based on polarimetric subspaces. The experimental results demonstrate the superiority of these joint polarimetric subspace detectors. Most importantly, DLD solves for previous limitations due to the complex clutter background and achieves a performance comparable to that of the Wishart (Gaussian) distribution, particularly in the low target-to-clutter ratio (TCR) case.

НАУКОВО-МЕТОДИЧНИЙ ПІДХІД ДО ОЦІНЮВАННЯ БЮДЖЕТНИХ ДИСБАЛАНСІВ

Анотація. Зосереджено увагу на неоднозначності ставлення вчених-економістів до незбалансованості державного бюджету. Обґрунтовано розуміння збалансованого за доходами і видатками державного бюджету відповідно до концепції «здорових фінансів». Виявлено недоліки концепції «здорових фінансів», зокрема те, що щорічне балансування доходів і видатків бюджету виключає, чи значною мірою зменшує можливості антициклічного, стабілізувального впливу фіскальної політики держави. Актуалізовано увагу на особливостях концепції хронічного бюджетного дефіциту і специфіці дефіцитного фінансування державних витрат у рамках концепції контрциклічного регулювання. Водночас доведено, що збалансованість бюджету є другорядною проблемою за концепцією функціональних фінансів. З’ясовано, що дефіцит бюджету може спричинювати як позитивні, так і негативні соціально-економічні ефекти, залежно від обставин, у яких дефіцит сформований. Виділено та охарактеризовано різновиди дефіциту державного бюджету, а саме: номінальний дефіцит, реальний дефіцит, операційний дефіцит первинний дефіцит, фактичний дефіцит, прихований дефіцит, квазіфіскальний дефіцит, сукупний дефіцит, активний дефіцит, пасивний дефіцит, вимушений дефіцит, свідомий дефіцит, структурний дефіцит і циклічний дефіцит. Доведено, що основним завданням держави є досягнення його збалансованості, тобто підтримка балансу доходів і витрат держави. Розглянуто підхід до оцінювання дисбалансу бюджету за допомогою застосування нерівності Коші та сформовано алгоритм оцінювання дисбалансу бюджету. Доведено, що забезпечення бюджетної збалансованості можливо в таких напрямах: поліпшення якості планування основних показників бюджету; розширення дохідної бази; оптимізація / секвестр видатків бюджету; цільове фінансування; підвищення ефективності управління державною власністю, бюджетного контролю; запровадження бюджетування, орієнтованого на результат, та аудиту ефективності використання бюджетних коштів; удосконалення міжбюджетних відносин; управління державним боргом. Розвинуто науково-методичний підхід до оцінювання дисбалансу, зокрема Зведеного бюджету в частині обґрунтування граничного значення показника міри дисбалансу, що ґрунтується на застосуванні нерівності Коші, а також визначення впливу факторів динамічних змін дисбалансу бюджету на основі застосування методу ланцюгових підстановок. Перевагами такого науково-методичного підходу є можливість порівняння оцінки дисбалансу бюджетів, що суттєво різняться за обсягом фінансових ресурсів, як перерозподіляються через бюджет, оскільки такі оцінки є відносними, що дозволяє нівелювати ефект масштабу бюджету. Ключові слова: бюджетна система, бюджет, бюджетний дефіцит, дефіцитне фінан­сування, бюджетна збалансованість, оцінювання дисбалансу, контрциклічне регулювання економіки. Формул: 3; рис.: 1; табл.: 1; бібл.: 10.

Difference schemes for dynamics problems

To solve dynamics problems in the LIRA 10.12 software package, the difference scheme is used, known as modified central difference method. The goal of dynamic problems solving is to get a good approximation of actual dynamic response of a given structure. It's a matter of convergence conditions of the difference scheme used in numerical integration of motion equations.
 The solution of the linear dynamic problem for all possible displacements satisfies the equations
 ,
 , , – symmetric positive-definite bilinear functionals of possible work of the internal and inertial forces and motion resistance forces, they correspond to stiffness matrices, mass and damping matrices, – linear functional of possible work of the external forces.
 After approximation in spatial variables (usually the finite element method is applied) we obtain the system of ordinary differential equations
 ,
 , , – stiffness matrices, mass and damping matrices, – external forces.
 Difference scheme of modified central difference method we obtain by replacing the values of functions and derivatives with corresponding difference relations. Difference relations, which are applied in modified central difference method, approximate accelerations , velocities and displacements with an error proportional to the square of the time step.
 Approximation and stability are the convergence conditions of the difference scheme. To study the stability of difference schemes, we assume that and apply the obvious equations
 ,
 .
 Let us denote , apply the Cauchy inequality on the right-hand side and integrate the resulting inequality. Then
 ,
 where are positive constants. The boundedness follows from the Gronwall's lemma. In addition to the continual version of study of the difference schemes stability, the discrete version is also considered.
 Unconditional practical efficiency of the modified central difference method and the simplicity of its underlying principles allow it to be successfully applied to a wide range of dynamic problems.

A Dynamic Discriminative Canonical Correlation Analysis via Adaptive Weight Scheme

In multi-view learning, massive literature is devoted to exploring the intrinsic structure between cross-views. It is well known that canonical correlation analysis (CCA) is a conventional multi-view learning method, which considers the correlation between two views. However, it fails to utilize class information and is difficult to suit different issues to extract discriminative features. In this paper, we propose a novel cross-view discriminative feature learning method called dynamic discriminative canonical correlation analysis, which captures class information to yield discriminative features. More specifically, we develop an adaptive weight scheme of cross-view within-class and between-class scatters to make full use of distribution class information. In addition, an iterative algorithm with Cauchy inequalities and the Lagrange multiplier is proposed to handle the non-smooth objective function. Our method is applied to face recognition and multi-linguistic text classification tasks. Extensive experimental results reveal that the adaptive weight scheme plays a beneficial role and our method is an effective feature learning.

Non-standard methods for solving inequalities

This article discusses some interesting methods for solving inequalities. And there are some inequalities that will help with this. For example: Cauchy inequality, Cauchy-Schwarz inequality, Titu's Lemma, Holder's inequality, Chebyshev's inequality. Examples of inequalities are also consideredfor better understanding.

Uncertainty Relations for the Number of Photons and the Phase Operator of an Electromagnetic Field for Phase Quantum Superpositions of Coherent States

We examine various uncertainty relations: (electromagnetic-field phase (the so-called “Pegg–Barnett operator”))–(number of photons), (trigonometric functions of the phase)–(number of photons), and (trigonometric functions of the phase)–(field phase). An electromagnetic field in quantum states of phase superpositions of coherent states, which are general superpositions of two coherent states with identical moduli but arbitrary phases, is considered. The rigorous uncertainty relation (Cauchy inequality) and the soft uncertainty relation (Heisenberg inequality) are examined and compared.

Some Opial type inequalities in (<i>p, q</i>)-calculus

In this paper, we establish 5 kinds of integral Opial-type inequalities in (p, q)-calculus by means of H?lder's inequality, Cauchy inequality, an elementary inequality and some analysis technique. First, we investigated the Opial inequalities in (p, q)-calculus involving one function and its (p, q) derivative. Furthermore, Opial inequalities in (p, q)-calculus involving two functions and two functions with their (p, q) derivatives are given. Our results are (p, q)-generalizations of some known inequalities, such as Opial-type integral inequalities and (p, q)-Wirtinger inequality.

The probability of the correct majority made decision

Aim: the probability of correctness of the collegial decision, which is made by a majority vote of some collective (board), consisting of an odd number of members is investigated, if the probability of correctness of the individual decision of each member of Board is known.
 Мaterials and methods: Bernoulli scheme, asymptotic representation, estimation via geometric series, power series expansion, the formula of Wallis, a power scale of averages, average of Kolmogorov.
 Result: it is established, that if for each member of the board the probability of correctness of the individual decision is more than ½, then with an unlimited increase in the number of members of the Board the probability of correctness of the collegial decision tends to 1. The asymptotic representation and a number of bilateral estimates characterizing the speed of this aspiration are obtained. For heterogeneous Board (that is a Board, whose members make the right individual decision with different probability) introduced the concept of collegial average as an average characteristics, which can replaced the individual probability of each member of the board with the preservation of the probability of a collegial decision. The existence and uniqueness of the collegial average are proved.
 We derive a collegial inequality showing that the collegial average of some a set of numbers is not less than the geometric average of the same numbers with the equality takes place in the case and only if all the numbers are equal to each other. The collegial inequality serves as an analogue and complement to known set of inequalities establishing a connection between different averages (for example, Cauchy inequality for arithmetic average and geometric average).
 Conclusion: thus, the results of the study fully meet the aim of determining the probability of correctness of collegial decision taken by a majority of votes under the assumptions. As a result we obtain an asymptotic representation and bilateral estimates characterizing the rate of striving for the correct solution. For a heterogeneous board, the existence uniqueness of the concept of collegial average as an average characteristics is introduced and strictly proved, which can be replaced by an individual probability of each with preserving the probability of correctness of the collegial decision. It is established that the collegial average is not less than the geometric average. Possible applications of the results obtained can be the quantitative evaluation of election procedures and the solution of problems associated with improving the reliability of recognition of weak signals of control sensors of various transport systems, including high-speed transport systems on magnetic suspension.

When the Cauchy Inequality Becomes a Formula

In this note, we revisit the classical geometric–arithmetic mean inequality and find a formula for the difference of the arithmetic and the geometric means of n given nonnegative numbers . The formula yields new stronger versions of the geometric-arithmetic mean inequality. We also find a second version of a strong geometric-arithmetic mean inequality and show that all inequalities are optimal in some sense. Another striking novelty is that the equality in all new inequalities holds not only in the case when all n numbers are equal, but also in other cases.

A Cauchy type inequality for M\xf6bius operations

In this article, we show two fundamental features of the restriction of Möbius operations to the real numbers, that is, a Cauchy type inequality and a criterion for convergence of series.

Incidence Results and Bounds Of Trilinear and Quadrilinear Exponential Sums

We give a new bound on the number of collinear triples for two arbitrary subsets of a finite field. This improves on existing results which rely on the Cauchy inequality. We then use this to provide a new bound on trilinear and quadrilinear exponential sums.