Algebraic Inequalities Research Articles

Domination results in n$n$‐Fuchsian fibers in the moduli space of Higgs bundles

In this article, we show some domination results on the Hitchin fibration, mainly focusing on the $n$-Fuchsian fibers. More precisely, we show the energy density of associated harmonic map of an $n$-Fuchsian representation dominates the ones of all other representations in the same Hitchin fiber, which implies the domination of topological invariants: translation length spectrum and entropy. As applications of the energy density domination results, we obtain the existence and uniqueness of equivariant minimal (or maximal) surfaces in certain product Riemannian (or pseudo-Riemannian) manifold. Our proof is based on establishing an algebraic inequality generalizing a GIT theorem of Ness on the nilpotent orbits to general orbits.

Hybrid control design for Mittag-Leffler projective synchronization on FOQVNNs with multiple mixed delays and impulsive effects

This article discusses the global Mittag-Leffler projective synchronization (GM-LPS) on fractional-order quaternion-valued neural networks (FOQVNNs) including mixed delays and impulses. The effects of various types of delay factors and impulses on Mittag-Leffler projective synchronization performance are simultaneously explored. Based on Banach contractive mapping principle, the existence result of solution to the considered systems is derived Applying the Lyapunov direct method, inequality analysis technique and generalized fractional Razumikhin theorem, the algebraic criteria for ensuring global Mittag-Leffler projective synchronization by designing the hybrid controllers are proposed. The result of algebraic inequality greatly reduces the computational complexity to some extent. Combining the different orders of Caputo derivative and MATLAB toolbox, the simulations are given to further validate the obtained results.

Finite-time stabilization of discontinuous fuzzy neutral-type neural networks with D operator and multiple time-varying delays

This paper discusses the finite-time stabilization of a class of discontinuous fuzzy neutral-type neural networks with multiple time-varying delays. For the purpose of achieving this stabilization, first of all, by using the differential inclusions and set-valued mapping, the differential inclusion system is established in order to cope with the discontinuities. Then, by means of the introduced sign function, delayed discontinuous control strategies are given. By constructing a modified Lyapunov-Krasovskii functional concerning with the mixed delays, delay-dependent criteria formulated by algebraic inequalities are derived for guaranteeing the finite-time stabilization. Moreover, the settling time is estimated. Finally, numerical examples are carried out to verify the effectiveness of the established results. Since this kind of stabilization analysis of such neural system in this work has not been given much attention, the delay-dependent criteria derived in this paper can be regarded as a leading stabilization result for fuzzy neutral-type neural networks with D operator including multiple time and multiple neutral delays.

Decentralized finite‐time control for a class of large‐scale systems with symmetrically interconnected subsystems

AbstractThe finite‐time stability (FTS) of a symmetrically interconnected system is studied under the consideration of utilizing the system structural properties to reduce the complexity of analysis. The FTS of this kind of large‐scale system can be guaranteed by that of two lower dimensional systems. The concept of mixed stability, taking care of both transient and steady‐state performances of a system, is introduced. Based on the FTS analysis, a decentralized finite‐time controller via state feedback is given to stabilize the large‐scale system. Meanwhile, an observer‐based decentralized output feedback controller is provided to make the closed‐loop system finite‐time stable. The FTS conditions and related decentralized stabilization controller parameters are both derived from solving some differential or algebraic linear matrix inequalities.

Cooperative Time-Varying Formation for Multiple Lypschitz-Type Nonlinear Systems: An Event-Triggered Adaptive Mechanism

This brief investigates the time varying formation problem of Lipschitz-type nonlinear multi-agent systems. An event-triggered adaptive controller is proposed. Comparing to existing related results, in this brief, both the neighbors states information and formation configurations are transmitted via an event-triggered mechanism, which reduce the frequency of communications. Besides, based on state-dependent-gain designs, the proposed protocol can ensure the Lipschitz-type agents achieving the desired final configuration by only solving algebraic Riccati equation (ARE) but linear matrix inequality (LMI), which is not only more practical in engineering scenarios, but also feasible if the system is stabilizable. Also, it follows from continuous adaptive modification, some global information, such as the eigenvalue of the Laplacian matrix and the Lipschitz constant, is not required here.

A general class of algebraic inequalities for generating new knowledge and optimising the design of systems and processes

A special class of general inequalities has been identified that provides the opportunity for generating new knowledge that can be used for optimising systems and processes in diverse areas of science and technology. It is demonstrated that inequalities belonging to this class can always be interpreted meaningfully if the variables and separate terms of the inequalities represent additive quantities. The meaningful interpretation of a new algebraic inequality based on the proposed general class of inequalities led to developing a light-weight design for a supporting structure based on cantilever beams, reducing the maximum force upon impact, generating new knowledge about the deflection of elastic elements connected in parallel and series and optimising the allocation of resources to maximise expected benefit. The interpretation of the new inequality yielded that the deflection of elastic elements connected in parallel is at least \(n^{2}\) times smaller than the deflection of the same elastic elements connected in series, irrespective of the individual stiffness values of the elastic elements. The interpretation of another algebraic inequality from the proposed general class led to a method for decreasing the stiffness of a mechanical assembly by cyclic permutation of the elastic elements building the assembly. The analysis showed that a decrease of stiffness exists only if asymmetry of the stiffness values in the connected elements is present.

On Refinements of Multidimensional Inequalities of Hardy‐Type via Superquadratic and Subquadratic Functions

By utilizing the peculiarities of superquadratic and subquadratic functions, we give the extensions for multidimensional inequalities of Hardy‐type with general kernel. We use some algebraic inequalities such as the Minkowski inequality, the refined Jensen inequality, and the Bernoulli inequality to prove the essential results in this paper. The performance of the superquadratic functions is reliable and effective to obtain new dynamic inequalities on time scales. By utilizing special kernels, we also acquire numerous examples and implementations of the related inequalities.

Optimising processes and generating knowledge by interpreting a new algebraic inequality

Optimising processes and generating knowledge by interpreting a new algebraic inequality

Optimising processes and generating knowledge by interpreting a new algebraic inequality

Optimising processes and generating knowledge by interpreting a new algebraic inequality

Adaptive Neural Model Matching Control for Uncertain Immune Systems via H∞ Approaches

The problem of the robust neural network-based model matching control is considered for a large class of uncertain immune systems. In order to achieve the purpose of therapeutic enhancement, it is essential to deal simultaneously with the effects of plant uncertainties, time-varying perturbations, and continuing environmental pathogens. Neural network control algorithm, robust <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty } $ </tex-math></inline-formula> control theory and VSC technique are combined to construct the hybrid adaptive/robust tracking control scheme such that the controlled immune system achieves a satisfactory model matching control performance. An adaptive neural network system is constructed to learn the behavior of the immune system dynamics. Moreover, an algebraic Riccati-like inequality must be solved to achieve a desired <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty } $ </tex-math></inline-formula> control performance. Consequently, the robust control scheme developed here can be analytically computed and easily implemented. Simulation results are presented to demonstrate the effectiveness of the proposed control scheme.

Homogeneous Lp Stability for Homogeneous Systems

The motivation of this paper comes from the fact that <i>L_p</i>&#x2014;stability and <i>L_p</i>&#x2014;gain, using the classical signal norms, is not well-defined for arbitrary continuous weighted homogeneous systems. However, using homogeneous signal norms it is possible to show that every internally stable homogeneous system has a globally defined finite homogeneous <i>L_p</i>&#x2014;gain, for <i>p</i> sufficiently large. If the system has a homogeneous approximation, the homogeneous <i>L_p</i>&#x2014;gain is inherited locally. Homogeneous <i>L_p</i>&#x2014;stability can be characterized by a homogeneous dissipation inequality, which in the input affine case can be transformed to a homogeneous Hamilton-Jacobi inequality. An estimation of an upper bound for the homogeneous <i>L_p</i>&#x2014;gain can be derived from these inequalities. Homogeneous <i>L</i>_&#x221E;&#x2014;stability is also considered and its strong relationship to Input-to-State stability is studied. These results are extensions to arbitrary homogeneous systems of the well-known situation for linear time-invariant systems, where the Hamilton-Jacobi inequality reduces to an algebraic Riccati inequality. A natural application of finite-gain homogeneous <i>L_p</i>&#x2014;stability is in the study of stability for interconnected systems. An extension of the small-gain theorem for negative feedback systems and results for systems in cascade are derived for different homogeneous norms. Previous results in the literature use classical signal norms, hence, they can only be applied to a restricted class of homogeneous systems. The results are illustrated by several examples.

Convergence conditions for continuous and discrete models of population dynamics

Some classes of continuous and discrete generalized Volterra models of population dynamics are considered. It is supposed that there are relationships of the type "symbiosis", "compensationism" or "neutralism" between any two species in a biological community. The objective of the work is to obtain conditions under which the investigated models possess the convergence property. This means that the studying system admits a bounded solution that is globally asimptotically stable. To determine the required conditions, the V. I. Zubov's approach and its discrete-time counterpart are used. Constructions of Lyapunov functions are proposed, and with the aid of these functions, the convergence problem for the considered models is reduced to the problem of the existence of positive solutions for some systems of linear algebraic inequalities. In the case where parameters of models are almost periodic functions, the fulfilment of the derived conditions implies that limiting bounded solutions are almost periodic, as well. An example is presented illustrating the obtained theoretical conclusions.

Investigation of the existence domain for Dyakonov surface waves in the Sage computer algebra system

Surface electromagnetic waves (Dyakonov waves) propagating along a plane interface between an isotropic substance with a constant dielectric constant and an anisotropic crystal, whose dielectric tensor has a symmetry axis directed along the interface, are considered. It is well known that the question of the existence of such surface waves is reduced to the question of the existence of a solution to a certain system of algebraic equations and inequalities. In the present work, this system is investigated in the Sage computer algebra system. The built-in technique of exceptional ideals in Sage made it possible to describe the solution of a system of algebraic equations parametrically using a single parameter, with all the original quantities expressed in terms of this parameter using radicals. The remaining inequalities were only partially investigated analytically. For a complete study of the solvability of the system of equations and inequalities, a symbolic-numerical algorithm is proposed and implemented in Sage, and the results of computer experiments are presented. Based on these results, conclusions were drawn that require further theoretical substantiation.

Optimised design of systems and processes using algebraic inequalities

A method for optimising the design of systems and processes has been introduced that consists of interpreting the left- and the right-hand side of a correct algebraic inequality as the outputs of two alternative design configurations delivering the same required function. In this way, on the basis of an algebraic inequality, the superiority of one of the configurations is established. The proposed method opens wide opportunities for enhancing the performance of systems and processes and is very useful for design in general. The method has been demonstrated on systems and processes from diverse application domains. The meaningful interpretation of an algebraic inequality based on a single-variable sub-additive function led to developing a light-weight design for a supporting structure based on cantilever beams. The interpretation of a new algebraic inequality based on a multivariable sub-additive function led to a method for increasing the kinetic energy absorbing capacity during inelastic impact. The interpretation of a new inequality has been used for maximising the mass of deposited substance during electrolysis.

Some local properties of subsolution and supersolutions for a doubly nonlinear nonlocal p-Laplace equation

We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional p-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi’s method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse Hölder inequality. Finally, we also prove a logarithmic decay estimate for positive supersolutions.

Optimal Versus Equal Dimensions of Round Bales of Agricultural Materials Wrapped with Plastic Film—Conflict or Compliance?

For the assumed bale volume, its dimensions (diameter, height), minimizing the consumption of the plastic film used for bale wrapping with the combined 3D method, depend on film and wrapping parameters. Incorrect selection of these parameters may result in an optimal bale diameter, which differs significantly from its height, while in agricultural practice bales with diameters equal or almost equal to the height dominate. The aim of the study is to formulate and solve the problem of selecting such dimensions of the bale with a given volume that the film consumption is minimal and, simultaneously, the bale diameter is equal or almost equal to its height. Necessary and sufficient conditions for such equilibria of the optimal bale dimensions are derived in the form of algebraic equations and inequalities. Four problems of the optimal bale dimension design guaranteeing assumed equilibrium of diameter and height are formulated and solved; both free and fixed bale volume are considered. Solutions of these problems are reduced to solving the sets of simple algebraic equations and inequalities with respect to two variables: integer number of film layers and continuous overlap ratio in bottom layers. Algorithms were formulated and examples regarding large bales demonstrate that they can handle the optimal dimensions’ equilibria problems.

An Algebraic Inequality

Summary Construct a cube with edge length 1. By dividing the cube into 8 smaller regions, we give a proof of an algebraic inequality.

Synchronization in Finite/Fixed Time for Markovian Complex-Valued Nonlinear Interconnected Neural Networks With Reaction–Diffusion Terms

This paper considers a novel class of nonlinear interconnected neural networks (INNs) where the Markovian jump parameters and reaction–diffusion terms are both involved. The nonlinear INNs are studied in the complex domain, which is a meaningful attempt in the research on INNs. Then, by designing a suitable nonlinear controller and employing the Lyapunov functional and algebraic inequality technologies, a finite/fixed-time synchronization criterion in probability can be obtained for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> subsystems. Note that the criteria for synchronization in finite time and fixed time are obtained with only one controller and integrated into a unified theorem. A secret communication example is presented, such that the validity and practicability of this paper can be verified.

Finite-time stabilization of discontinuous fuzzy inertial Cohen–Grossberg neural networks with mixed time-varying delays

This article aims to study a class of discontinuous fuzzy inertial Cohen–Grossberg neural networks (DFICGNNs) with discrete and distributed time-delays. First of all, in order to deal with the discontinuities by the differential inclusion theory, based on a generalized variable transformation including two tunable variables, the mixed time-varying delayed DFICGNN is transformed into a first-order differential system. Then, by constructing a modified Lyapunov–Krasovskii functional concerning with the mixed time-varying delays and designing a delayed feedback control law, delay-dependent criteria formulated by algebraic inequalities are derived for guaranteeing the finite-time stabilization (FTS) for the addressed system. Moreover, the settling time is estimated. Some related stability results on inertial neural networks is extended. Finally, two numerical examples are carried out to verify the effectiveness of the established results.

New results on synchronization for second-order fuzzy memristive neural networks with time-varying and infinite distributed delays

Without converting the second order terms to the first order ones, this article deals with the global asymptotic synchronization of second-order fuzzy memristive neural networks (SFMNNs) with infinite distributed and time-varying delays by contriving feedback control and adaptive control schemes. Based on Lyapunov stability theory, Barbalat Lemma and some analysis strategies, several new criteria in lights of algebraic inequalities are directly acquired to assure the global asymptotic synchronization of the concerned SFMNNs. Besides, compared with the existing reduced-order means, the global asymptotic synchronization is directly analyzed via accepting some new Lyapunov–Krasovskii functionals without the reduced-order means. Ultimately, some examples are provided to identify the availability of the theoretical outcomes.