Algebraic Inequalities Research Articles

Synchronization of Clifford-valued neural networks with leakage, time-varying, and infinite distributed delays on time scales

<abstract><p>Neural networks (NNs) with values in multidimensional domains have lately attracted the attention of researchers. Thus, complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and their generalization, Clifford-valued neural networks (ClVNNs) have been proposed in the last few years, and different dynamic properties were studied for them. On the other hand, time scale calculus has been proposed in order to jointly study the properties of continuous time and discrete time systems, or any hybrid combination between the two, and was also successfully applied to the domain of NNs. Finally, in real implementations of NNs, time delays occur inevitably. Taking all these facts into account, this paper discusses ClVNNs defined on time scales with leakage, time-varying delays, and infinite distributed delays, a type of delays which have been relatively rarely present in the existing literature. A state feedback control scheme and a generalization of the Halanay inequality for time scales are used in order to obtain sufficient conditions expressed as algebraic inequalities and as linear matrix inequalities (LMIs), using two general Lyapunov-like functions, for the exponential synchronization of the proposed model. Two numerical examples are given in order to illustrate the theoretical results.</p></abstract>

Intermediate quantifiers and valid syllogisms on EQ-algebras

Intermediate quantifiers are expressions of natural language, for example “most, almost all, many, a few” using which we quantify a number of some objects in a given universe. We have shown in [23] that all valid syllogisms with intermediate quantifiers are a consequence of only two algebraic inequalities and one equality. The result was obtained in the formalism of Lukasiewicz fuzzy type theory whose truth values form a linearly ordered complete MV-algebra. In this paper we will prove that the same holds if we replace MV-algebra by a much more general IEQ-algebra (involutive EQ-algebra).

Understanding the Capability of PD Control for Uncertain Stochastic Systems

In this article, we focus on the global stabilizability problem for a class of uncertain stochastic control systems, where both the drift term and the diffusion term are nonlinear functions of the state variables and the control variables. We will show that the widely applied proportional-derivative (PD) control in engineering practice has the ability to globally stabilize such systems in the mean square sense, provided that the upper bounds of partial derivatives of the nonlinear functions satisfy a certain algebraic inequality. It will also be proved that the stabilizing PD parameters can only be selected from a two dimensional bounded convex set, which differs significantly from the existing results for PD controlled stochastic systems where the diffusion term does not depend on the control variables. Moreover, a particular polynomial on these bounds is introduced, which can be used to determine under what conditions the system is not stabilizable by the PD control, and thus demonstrating the fundamental limitations of PD control.

Adaptive Drive-Response Synchronization of Timescale-Type Neural Networks With Unbounded Time-Varying Delays.

In recent years, adaptive drive-response synchronization (DRS) of two continuous-time delayed neural networks (NNs) has been investigated extensively. For two timescale-type NNs (TNNs), how to develop adaptive synchronization control schemes and demonstrate rigorously is still an open problem. This article concentrates on adaptive control design for synchronization of TNNs with unbounded time-varying delays. First, timescale-type Barbalat lemma and novel timescale-type inequality techniques are first proposed, which provides us practical methods to investigate timescale-type nonlinear systems. Second, using timescale-type calculus, novel timescale-type inequality, and timescale-type Barbalat lemma, we demonstrate that global asymptotic synchronization can be achieved via adaptive control under algebraic and matrix inequality criteria even if the time-varying delays are unbounded and nondifferentiable. Adaptive DRS is discussed for TNNs, which implies our control schemes are suitable for continuous-time NNs, their discrete-time counterparts, and any combination of them. Finally, numerical examples on TNNs and timescale-type chaotic Ikeda-like oscillator with unbounded time-varying delays are carried out to verify the adaptive control schemes.

Mittag-Leffler projective synchronization of uncertain fractional-order fuzzy complex valued neural networks with distributed and time-varying delays

<p>To study the Mittag-Leffler projective synchronization (MLPS) problem of fractional-order fuzzy neural networks (FOFNNs), in this work we introduced the FOFNNs model. On this basis, we discussed the MLPS of uncertain fractional-order fuzzy complex valued neural networks (FOFCVNNs) with distributed and time-varying delays. Utilizing Banach contraction mapping principle, we proved the existence and uniqueness of the model solution. Moreover, employing the construction of a new hybrid controller, an adaptive hybrid controller, and the fractional-order Razumikhin theorem, algebraic criteria was obtained for implementing MLPS. The algebraic inequality criterion obtained in this article improves and extends the previously published results on MLPS, making it easy to prove and greatly reducing the computational complexity. Finally, different Caputo derivatives of different orders were given, and four numerical examples were provided to fully verify the accuracy of the modified criterion.</p>

Exponential Stability of Impulsive Timescale-Type Nonautonomous Neural Networks With Discrete Time-Varying and Infinite Distributed Delays.

Global exponential stability (GES) for impulsive timescale-type nonautonomous neural networks (ITNNNs) with mixed delays is investigated in this article. Discrete time-varying and infinite distributed delays (DTVIDDs) are taken into consideration. First, an improved timescale-type Halanay inequality is proven by timescale theory. Second, several algebraic inequality criteria are demonstrated by constructing impulse-dependent functions and utilizing timescale analytical techniques. Different from the published works, the theoretical results can be applied to GES for ITNNNs and impulsive stabilization design of timescale-type nonautonomous neural networks (TNNNs) with mixed delays. The improved timescale-type Halanay inequality considers time-varying coefficients and DTVIDDs, which improves and extends some existing ones. GES criteria for ITNNNs cover the stability conditions of discrete-time nonautonomous neural networks (NNs) and continuous-time ones, and these theoretical results hold for NNs with discrete-continuous dynamics. The effectiveness of our new theoretical results is verified by two numerical examples in the end.

Improving reconstructions in nanotomography for homogeneous materials via mathematical optimization.

Compressed sensing is an image reconstruction technique to achieve high-quality results from limited amount of data. In order to achieve this, it utilizes prior knowledge about the samples that shall be reconstructed. Focusing on image reconstruction in nanotomography, this work proposes enhancements by including additional problem-specific knowledge. In more detail, we propose further classes of algebraic inequalities that are added to the compressed sensing model. The first consists in a valid upper bound on the pixel brightness. It only exploits general information about the projections and is thus applicable to a broad range of reconstruction problems. The second class is applicable whenever the sample material is of roughly homogeneous composition. The model favors a constant density and penalizes deviations from it. The resulting mathematical optimization models are algorithmically tractable and can be solved to global optimality by state-of-the-art available implementations of interior point methods. In order to evaluate the novel models, obtained results are compared to existing image reconstruction methods, tested on simulated and experimental data sets. The experimental data comprise one 360° electron tomography tilt series of a macroporous zeolite particle and one absorption contrast nano X-ray computed tomography (nano-CT) data set of a copper microlattice structure. The enriched models are optimized quickly and show improved reconstruction quality, outperforming the existing models. Promisingly, our approach yields superior reconstruction results, particularly when only a small number of tilt angles is available.

Roots of Descent Polynomials and an Algebraic Inequality on Hook Lengths


 
 
 By reinterpreting the descent polynomial as a function enumerating standard Young tableaux of a ribbon shape, we use Naruse's hook-length formula to express the descent polynomial as a product of two polynomials: one is a trivial part which is a product of linear factors, and the other comes from the excitation factor of Naruse's formula. We expand the excitation factor positively in a Newton basis which arises naturally from Naruse's formula. Under this expansion, each coefficient is the weight of a certain combinatorial object, which we introduce in this paper. We introduce and prove the "Slice and Push Inequality", which compares the weights of such combinatorial objects. As a consequence, we establish a proof of a conjecture by Diaz-Lopez et al. that bounds the roots of descent polynomials.
 
 

On the Construction of the Graph of Discrete States of a Switched Affine System

The problem of constructing the graph of states of a switched affine system closed by a static state feedback is considered. To solve this problem, a constructive algorithm based on the study of the consistency of systems of linear algebraic inequalities is proposed.

Guided e-Assessment of Math Proofs with the Halomda Platform

The mode of assessment is one of the most important factors influencing learning. E-assessment usually includes only checking the final answer, thus limiting teacher's ability to check the complete solution, and it does not allow inclusion of math proofs problems that constitute an important part of math content. The e-assessment module of Halomda elearning platform allows assessment of proof problems from different areas of school and high Math, like algebraic identities and inequalities, geometry proofs, trigonometrical identities, analytic geometry, vector algebra, etc. Although eassessment of a proof of a mathematical statement is generally a difficult task, in this paper we show through some working examples how guided e-assessment can be implemented using the Halomda platform.

Reliability-Related Interpretations of Algebraic Inequalities

New results related to maximizing the reliability of common systems with interchangeable redundancies at a component level have been obtained by using the method of algebraic inequalities. It is shown that for systems with independently working components with interchangeable redundancies, the system reliability corresponding to a symmetric arrangement of the redundant components is always inferior to the system reliability corresponding to an asymmetric arrangement of the redundant components, irrespective of the probabilities of failure of the different types of components. It is also shown that for series–parallel systems, the system reliability is maximized by arranging the main components in ascending order of their probabilities of failure, whereas the redundant components are arranged in descending order of their probabilities of failure. Finally, this article derives rigorously the highly counterintuitive result that if two components must be selected from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> batches containing reliable and faulty components with unknown proportions, the likelihood that both components will be reliable is maximized by selecting both components from a randomly selected batch.

Fault-Tolerant Consensus Control of Positive Networked Systems

In this paper, we explore the consensus of positive networked systems with actuator faults. Firstly, the undirected and strongly connected topology is established with graph theory. The positive system theory is used to analyze the positive consensus of the closed-loop networked systems. State feedback gains are derived utilizing Algebraic Riccati Inequalities. Bounded multiplicative faults are regarded as uncertainties in the system matrix, while treating additive faults as external disturbances. Further, this transformation refocuses the analysis on the consensus problem with an L2-gain. Subsequently, the Genetic Algorithm is employed to optimize the L2 performance criteria. Finally, the effectiveness of the proposed theory is validated through simulations involving both single-input electric circuit systems and multi-input networked systems.

Asymptotic and Mittag–Leffler Synchronization of Fractional-Order Octonion-Valued Neural Networks with Neutral-Type and Mixed Delays

Very recently, a different generalization of real-valued neural networks (RVNNs) to multidimensional domains beside the complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and Clifford-valued neural networks (ClVNNs) has appeared, namely octonion-valued neural networks (OVNNs), which are not a subset of ClVNNs. They are defined on the octonion algebra, which is an 8D algebra over the reals, and is also the only other normed division algebra that can be defined over the reals beside the complex and quaternion algebras. On the other hand, fractional-order neural networks (FONNs) have also been very intensively researched in the recent past. Thus, the present work combines FONNs and OVNNs and puts forward a fractional-order octonion-valued neural network (FOOVNN) with neutral-type, time-varying, and distributed delays, a very general model not yet discussed in the literature, to our awareness. Sufficient criteria expressed as linear matrix inequalities (LMIs) and algebraic inequalities are deduced, which ensure the asymptotic and Mittag–Leffler synchronization properties of the proposed model by decomposing the OVNN system of equations into a real-valued one, in order to avoid the non-associativity problem of the octonion algebra. To accomplish synchronization, we use two different state feedback controllers, two different types of Lyapunov-like functionals in conjunction with two Halanay-type lemmas for FONNs, the free-weighting matrix method, a classical lemma, and Young’s inequality. The four theorems presented in the paper are each illustrated by a numerical example.

An efficient and provable sequential quadratic programming method for American and swing option pricing

A sequential quadratic programming numerical method is proposed for American option pricing based on the variational inequality formulation. The variational inequality is discretized using the θ-method in time and the finite element method in space. The resulting system of algebraic inequalities at each time step is solved through a sequence of box-constrained quadratic programming problems, with the latter being solved by a globally and quadratically convergent, large-scale suitable reflective Newton method. It is proved that the sequence of quadratic programming problems converges with a constant rate under a mild condition on the time step size. The method is general in solving the variational inequalities for the option pricing with many styles of optimal stopping and complex underlying asset models. In particular, swing options and stochastic volatility and jump diffusion models are studied. Numerical examples are presented to confirm the effectiveness of the method.

Predefined-time synchronization of incommensurate fractional-order competitive neural networks with time-varying delays

This paper focuses on the predefined-time synchronization (PTS) of incommensurate fractional-order competitive neural networks (IFCNNs) with bounded or unbounded time-varying delays in the sense of Caputo fractional derivative (CFD). To this end, based on the piecewise Lyapunov function, a novel predefined-time stability theorem (PTST) is presented, where the setting of an adjustable time parameter in the deduced results makes it to be more versatile and more flexible, which refines former studies and can infer a series of new results. Besides, a double-layers predefined-time controller with two different fractional-order integrals is proposed, where the predefined time (PT) is predefined during the control design and can ensure that before a known time. Using this new PTST and some inequality techniques, an effective criteria for ensuring the PTS for two IFCNNs is derived in terms of algebraic inequalities. The PT is set to any positive parameter in the controllers and wholly independent of the initial values. Finally, examples are given to verify the proposed results.

A general class of linear unconditionally energy stable schemes for the gradient flows, II

This paper continues to study linear and unconditionally modified-energy stable (abbreviated as SAV-GL) schemes for the gradient flows. The schemes are built on the SAV technique and the general linear time discretizations (GLTD) as well as the extrapolation for the nonlinear term. Different from [44], the GLTDs with three parameters discussed here are not necessarily algebraically stable. Some algebraic identities are derived by using the method of undetermined coefficients and further used to establish the modified-energy inequalities for the unconditional modified-energy stability of the semi-discrete-in-time SAV-GL schemes. It is worth emphasizing that those algebraic identities or energy inequalities are not necessarily unique for some choices of three parameters in the GLTDs. Numerical experiments on the Allen-Cahn, the Cahn-Hilliard and the phase field crystal models with the periodic boundary conditions are conducted to validate the unconditional modified-energy stability of the SAV-GL schemes, where the Fourier pseudo-spectral method is employed in space with the zero-padding to eliminate the aliasing error and the time stepsizes for ensuring the original-energy monotonous decay are estimated by using the stability regions of our SAV-GL schemes for the test equation. The resulting time stepsize constraints for the SAV-GL schemes are almost consistent with the numerical results on the above gradient flow models.

Stability of nonlinear impulsive systems with non-uniformly distributed packet loss

This study focuses mainly on the stability of a class of nonlinear systems with impulsive disturbance. In particular, we consider the possible non-uniformly distributed packet loss phenomenon in the controller-to-actuator channel. This packet loss is described by a characterization scenario in the average sense, known as the average packet loss interval (APLI). In this scenario, an average allocation strategy between the number of impulsive sampling intervals and the length of packet loss intervals is established. This implies that the limitations on the proportion or the upper bound of packet loss intervals can be relaxed. A sampled-data control is put forward based on impulsive disturbance, which not only brings better control performance but also reduces conservatism. In view of the Halanay inequality, the number of subsystem switchings caused by packet loss is allowed to be arbitrary. Based on the elastic constraint relationships established between system parameters and APLI, a series of conditions guaranteeing the globally uniformly exponential stability (GUES) are provided. Further, the admissible packet loss intervals can be estimated on an average basis by solving an algebraic inequality. Finally, a numerical example is discussed to demonstrate the validity of our study.

Computation of the Hausdorff Distance between Two Compact Convex Sets.

The Hausdorff distance between two closed sets has important theoretical and practical applications. Yet apart from finite point clouds, there appear to be no generic algorithms for computing this quantity. Because many infinite sets are defined by algebraic equalities and inequalities, this a huge gap. The current paper constructs Frank-Wolfe and projected gradient ascent algorithms for computing the Hausdorff distance between two compact convex sets. Although these algorithms are guaranteed to go uphill, they can become trapped by local maxima. To avoid this defect, we investigate a homotopy method that gradually deforms two balls into the two target sets. The Frank-Wolfe and projected gradient algorithms are tested on two pairs of compact convex sets, where: (1) is the box translated by and is the intersection of the unit ball and the non-negative orthant; and (2) is the probability simplex and is the unit ball translated by . For problem (2), we find the Hausdorff distance analytically. Projected gradient ascent is more reliable than the Frank-Wolfe algorithm and finds the exact solution of problem (2). Homotopy improves the performance of both algorithms when the exact solution is unknown or unattained.

Can system reliability be predicted from average component reliabilities?

The paper reveals that a prediction of system reliability on demand based on average reliabilities on demand of components is a fundamentally flawed approach. A physical interpretation of algebraic inequalities demonstrated that assuming average component reliabilities on demand entails an overestimation of the system reliability on demand for systems with components logically arranged in series and series-parallel and underestimation of the reliability on demand for systems with components logically arranged in parallel. The key reason for these discrepancies is the variability of components from the same type. Techniques for countering variability by promoting asymmetric response through inversion have also been introduced. The paper demonstrates that variability during assembly operations can affect negatively the reliability of mechanical systems. Accordingly, techniques for reducing the variability of stresses during assembly operations have been discussed. Finally, the paper provides a discussion related to the reasons for the relatively slow adoption of domain-independent methods for improving reliability despite their numerous advantages.

Event-triggered impulsive control for synchronization in finite time of fractional-order reaction–diffusion complex networks

This paper is concerned with the finite-time synchronization (FTS) issue for fractional-order reaction–diffusion complex networks (RDCNs). A finite-time stability principle, which plays a key role in the synchronization analysis later, is developed for fractional-order nonlinear impulsive system. A novel hybrid controller, which consists of a event-triggered controller and a impulsive controller, is designed to realize the global FTS objective for the considered network. By applying the Lyapunov stability theory and the fractional calculus, the global FTS conditions are addressed in the form of algebraic inequalities. In addition, the exclusion of Zeno behavior are proved for the designed event-triggered strategy. Finally, a numerical example is provided to illustrate the feasibility of the proposed control approach and the correctness of the theoretical results.