Relaxed Stability Criteria for Time-Delay Systems:A Novel Quadratic Function Convex Approximation Approach

This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays.By introducing two adjustable parameters and two free variables,a novel convex function greater than or equal to the quadratic function is constructed,regardless of the sign of the coefficient in the quadratic term.The developed lemma can also be degenerated into the existing quadratic function negative-determination(QFND)lemma and relaxed QFND lemma respectively,by setting two adjustable parameters and two free variables as some particular values.Moreover,for a linear system with time-varying delays,a relaxed stability criterion is established via our developed lemma,together with the quivalent reciprocal combination technique and the Bessel-Legendre inequality.As a result,the conservatism can be reduced via the proposed approach in the context of constructing Lyapunov-Krasovskii functionals for the stability analysis of linear time-varying delay systems.Finally,the superiority of our results is illustrated through three numerical examples.

Enhancing Proficiency in Quadratic Equations and Functions Through the MILAPlus Strategy

This study investigates the efficacy of the Mathematics Independent Learning Activity Practice and Play Unite Scheme(MILAPlus)as an instructional strategy to improve the proficiency levels of Grade 9 students in quadratic equations and functions through a study carried out at Quezon National High School.The research involved 116 Grade 9 students and utilized a quantitative approach,incorporating both pre-assessment and post-assessment measures.The research utilizes a quasi-experimental design,examining the academic performance of students before and after the introduction of MILAPlus.The pre-assessment establishes a baseline,and the subsequent post-assessment measures the impact of the instructional strategy.Statistical analyses,including t-tests,assess the significance of differences in mean scores and mean percentage scores,providing quantitative insights into the effectiveness of MILAPlus.Findings from the study revealed a statistically significant improvement in both mean scores and mean percentage scores after the utilization of MILAPlus,indicating enhanced proficiency in quadratic equations and functions.The Mean Proficiency Scores(MPS)also showed a substantial increase,demonstrating a marked improvement in overall proficiency levels among Grade 9 students.In light of the results,recommendations were given including the continued utilization of MILAPlus as an instructional strategy and aligning its development with prescribed learning competencies.Emphasizing the consistent adherence to policies and guidelines for MILAPlus implementation is suggested for sustaining positive effects on students’long-term performance in mathematics.This research contributes valuable insights into the practical application and effectiveness of MILAPlus within the context of Grade 9 mathematics education at Quezon National High School.

Nonparametric Statistical Feature Scaling Based Quadratic Regressive Convolution Deep Neural Network for Software Fault Prediction

The development of defect prediction plays a significant role in improving software quality. Such predictions are used to identify defective modules before the testing and to minimize the time and cost. The software with defects negatively impacts operational costs and finally affects customer satisfaction. Numerous approaches exist to predict software defects. However, the timely and accurate software bugs are the major challenging issues. To improve the timely and accurate software defect prediction, a novel technique called Nonparametric Statistical feature scaled QuAdratic regressive convolution Deep nEural Network (SQADEN) is introduced. The proposed SQADEN technique mainly includes two major processes namely metric or feature selection and classification. First, the SQADEN uses the nonparametric statistical Torgerson–Gower scaling technique for identifying the relevant software metrics by measuring the similarity using the dice coefficient. The feature selection process is used to minimize the time complexity of software fault prediction. With the selected metrics, software fault perdition with the help of the Quadratic Censored regressive convolution deep neural network-based classification. The deep learning classifier analyzes the training and testing samples using the contingency correlation coefficient. The softstep activation function is used to provide the final fault prediction results. To minimize the error, the Nelder–Mead method is applied to solve non-linear least-squares problems. Finally, accurate classification results with a minimum error are obtained at the output layer. Experimental evaluation is carried out with different quantitative metrics such as accuracy, precision, recall, F-measure, and time complexity. The analyzed results demonstrate the superior performance of our proposed SQADEN technique with maximum accuracy, sensitivity and specificity by 3%, 3%, 2% and 3% and minimum time and space by 13% and 15% when compared with the two state-of-the-art methods.

A Modified Lagrange Method for Solving Convex Quadratic Optimization Problems

In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.

Orthogonal Stability of Mixed Additive-Quadratic Jensen Type Functional Equation in Multi-Banach Spaces

In this paper, we prove the Hyers-Ulam stability of the following mixed additive-quadratic Jensen type functional equation:

Constructing Common Quadratic Lyapunov Functions for a Class of Stable Matrices

Narendra 和 Balakrishnan 建议了一个方法构造普通二次的 Lyapunov 功能(CQLF )[1 ] ，稳定的矩阵的 whena 集合是可交换的。这份报纸的目的是概括方法到非可交换、非可解决的大小写。一个修改构造算法被建议，某些条件被提供向是的矩阵保证结果 CQLF。下次，讨论的问题是一个稳定的矩阵什么时候能与 CQLF 被加到一套矩阵为扩大集合构造新 CQLF。

Global Practical Stabilization of Discrete-Time Switched Affine Systems via a General Quadratic Lyapunov Function and a Decentralized Ellipsoid

This paper addresses the problem of global practical stabilization of discrete-time switched affine systems via statedependent switching rules.Several attempts have been made to solve this problem via different types of a common quadratic Lyapunov function and an ellipsoid.These classical results require either the quadratic Lyapunov function or the employed ellipsoid to be of the centralized type.In some cases,the ellipsoids are defined dependently as the level sets of a decentralized Lyapunov function.In this paper,we extend the existing results by the simultaneous use of a general decentralized Lyapunov function and a decentralized ellipsoid parameterized independently.The proposed conditions provide less conservative results than existing works in the sense of the ultimate invariant set of attraction size.Two different approaches are proposed to extract the ultimate invariant set of attraction with a minimum size,i.e.,a purely numerical method and a numerical-analytical one.In the former,both invariant and attractiveness conditions are imposed to extract the final set of matrix inequalities.The latter is established on a principle that the attractiveness of a set implies its invariance.Thus,the stability conditions are derived based on only the attractiveness property as a set of matrix inequalities with a smaller dimension.Illustrative examples are presented to prove the satisfactory operation of the proposed stabilization methods.

ON THE HYERS-ULAM STABILITY OF AN EQUATION CHARACTERIZING MULTI-QUADRATIC MAPPINGS

In this article, we prove, both in complete non-Archimedean normed spaces and in 2-Banach spaces, the generalized Hyers-Ulam stability of an equation characterizing multi- quadratic mappings. Our results generalize some known outcomes.

SPEECH EMOTION RECOGNITION USING MODIFIED QUADRATIC DISCRIMINATION FUNCTION

Quadratic Discrimination Function (QDF) is commonly used in speech emotion recognition, which proceeds on the premise that the input data is normal distribution. In this paper, we propose a transformation to normalize the emotional features, then derivate a Modified QDF (MQDF) to speech emotion recognition. Features based on prosody and voice quality are extracted and Principal Component Analysis Neural Network (PCANN) is used to reduce dimension of the feature vectors. The results show that voice quality features are effective supplement for recognition, and the method in this paper could improve the recognition ratio effectively.

QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN BANACH SPACES

In this paper, we solve the quadratic p-functional inequalities ……where p is a fixed complex number with ｜P｜ 〈 1, and^where p is a fixed complex number with ｜P｜ 〈 2^-1.Using the direct method, we prove the Hyers-Ulam stability of the quadratic p-functional inequalities （0.1） and （0.2） in complex Banach spaces and prove the Hyers-Ulam stability of quadratic p-functional equations associated with the quadratic p-functional inequalities （0.1） and （0.2） in complex Banach spaces.

Three-dimensional free vibration analysis of multi-directional functionally graded piezoelectric annular plates on elastic foundations via state space based differential quadrature method

The three-dimensional free vibration analysis of a multi-directional func- tionally graded piezoelectric （FGP） annular plate resting on two parameter （Pasternak） elastic foundations is investigated under different boundary conditions. The material properties are assumed to vary continuously along the radial and thickness directions and have exponent-law distribution. A semi-analytical approach named the state space based differential quadrature method （SSDQM） is used to provide an analytical solution along the thickness using the state space method （SSM） and an approximate solution along the radial direction using the one-dimensional differential quadrature method （DQM）. The influence of the Winkler and shear stiffness of the foundation~ the material property graded variations, and the circumferential wave number on the nomdimensional natural frequency of multi-directional FGP annular plates is studied.

Synchronising chaotic Chua's circuit using switching feedback control based on piecewise quadratic Lyapunov functions

This paper investigates the chaos synchronisation between two coupled chaotic Chua＇s circuits. The sufficient condition presented by linear matrix inequalities （LMIs） of global asymptotic synchronisation is attained based on piecewise quadratic Lyapunov functions. First, we obtain the piecewise linear differential inclusions （pwLDIs） model of synchronisation error dynamics, then we design a switching （piecewise-linear） feedback control law to stabilise it based on the piecewise quadratic Laypunov functions. Then we give some numerical simulations to demonstrate the effectiveness of our theoretical results.

Lipschitz continuity of the optimal value function and KKT solution set in indefinite quadratic programs

When all the involved data in indefinite quadratic programs change simultaneously, we show the locally Lipschtiz continuity of the KKT set of the quadratic programming problem firstly, then we establish the locally Lipschtiz continuity of the KKT solution set. Finally, the similar conclusion for the corresponding optimal value function is obtained.

Continuity of the optimal value function and optimal solutions of parametric mixed-integer quadratic programs

To properly describe and solve complex decision problems, research on theoretical properties and solution of mixed-integer quadratic programs is becoming very important. We establish in this paper different Lipschitz-type continuity results about the optimal value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of the linear constraints. The obtained results extend some existing results for continuous quadratic programs, and, more importantly, lay the foundation for further theoretical study and corresponding algorithm analysis on mixed-integer quadratic programs.

A branch-and-bound algorithm for multi-dimensional quadratic 0-1 knapsack problems

In this paper, a branch-and-bound method for solving multi-dimensional quadratic 0-1 knapsack problems was studied. The method was based on the Lagrangian relaxation and the surrogate constraint technique for finding feasible solutions. The Lagrangian relaxations were solved with the maximum-flow algorithm and the Lagrangian bounds was determined with the outer approximation method. Computational results show the efficiency of the proposed method for multi-dimensional quadratic 0-1 knapsack problems.

ESTIMATES ON THE FINITE ELEMENT APPROXIMATIONS OF A QUADRATIC FUNCTIONAL

This paper gives some estimates of a quadratic functional related with Rie- mann Hypothesis.

Quadratic Lyapunov Function and Exponential Dichotomy on Time Scales

In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation x△=A（t）x on time scales. Moreover, for the nonlinear perturbed equation x△= A（t）x ＋ f（t, x） we give the instability of the zero solution when f is sufficiently small.

Optimal control of quadratic functionals for affine nonlinear systems

In this paper we analyze the optimal control problem for a class of afflne nonlinear systems under the assumption that the associated Lie algebra is nilpotent. The Lie brackets generated by the vector fields which define the nonlinear system represent a remarkable mathematical instrument for the control of affine systems. We determine the optimal control which corresponds to the nilpotent operator of the first order. In particular, we obtain the control that minimizes the energy of the given nonlinear system. Applications of this control to bilinear systems with first order nilpotent operator are considered.

SAR Imaging of Moving Target Based on Quadratic Phase Function

In this paper, a novel signal processing technique hasbeen developed to refocus moving targets image from their smeared responses in the Synthetic Aperture Radar (SAR) image according to the characteristics of the received signals for moving targets. Quadratic Phase Function is introduced to the parameters estimation for moving target echo and SAR imaging. Our method is available even under a low SNR environment and acquiring an exact SAR image of moving targets. The simulated results demonstrated the validity of the algorithm proposed.

Asymptotically Necessary and Sufficient Quadratic Stability Conditions of T-S Fuzzy Systems Using Staircase Membership Function and Basic Inequality

Asymptotically necessary and sufficient quadratic stability conditions of Takagi-Sugeno (T-S) fuzzy systems are obtained by utilizing staircase membership functions and a basic inequality. The information of the membership functions is incorporated in the stability analysis by approximating the original continuous membership functions with staircase membership functions. The stability of the T-S fuzzy systems was investigated based on a quadratic Lyapunov function. The asymptotically necessary and sufficient stability conditions in terms of linear matrix inequalities were derived using a basic inequality. A fuzzy controller was also designed based on the stability results. The derivation process of the stability results is straightforward and easy to understand. Case studies confirmed the validity of the obtained stability results.