Numerical Methods for a Class of Quadratic Matrix Equations

Quadratic matrix equations arise in many elds of scienti c computing and engineering applications.In this paper,we consider a class of quadratic matrix equations.Under a certain condition,we rst prove the existence of minimal nonnegative solution for this quadratic matrix equation,and then propose some numerical methods for solving it.Convergence analysis and numerical examples are given to verify the theories and the numerical methods of this paper.

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.

A FIXED POINT APPROACH TO THE STABILITY OF A GENERALIZED APOLLONIUS TYPE QUADRATIC FUNCTIONAL EQUATION

Using the fixed point method, this article proves the Hyers-Ulam-Rassias stability of a generalized Apollonius type quadratic functional equation in Banach spaces. The conditions of these results are demonstrated by the quadratic functional equation of Apollonius type.

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.

Forward-backward Stochastic Differential Equations and Backward Linear Quadratic Stochastic Optimal Control Problem

In this paper, we use the solutions of forward-backward stochastic differential equations to get the optimal control for backward stochastic linear quadratic optimal control problem. And we also give the linear feedback regulator for the optimal control problem by using the solutions of a group of Riccati equations.

Independence of the Residual Quadratic Sums in the Dispersion Equation with Noncentral χ²-Distribution

A model adequacy test should be carried out on the basis of accurate aprioristic ideas about a class of adequate models, as in solving of practical problems this class is final. In article, the quadratic sums entering into the equation of the dispersive analysis are considered and their independence is proved. Necessary and sufficient conditions of existence of adequate models are resulted. It is shown that the class of adequate models is infinite.

Higher-Order WHEP Solutions of Quadratic Nonlinear Stochastic Oscillatory Equation

This paper introduces higher-order solutions of the quadratic nonlinear stochastic oscillatory equation. Solutions with different orders and different number of corrections are obtained with the WHEP technique which uses the WienerHermite expansion and perturbation technique. The equivalent deterministic equations are derived for each order and correction. The solution ensemble average and variance are estimated and compared for different orders, different number of corrections and different strengths of the nonlinearity. The solutions are simulated using symbolic computation software such as Mathematica. The comparisons between different orders and different number of corrections show the importance of higher-order and higher corrected WHEP solutions for the nonlinear stochastic differential equations.

Anisotropic Standard Decelerating Cosmological Model with Quadratic Equation of State

Spatially homogeneous and anisotropic Bianchi type-I cosmological model containing perfect fluid with quadratic equation of state has been diagnosed in general theory of relativity. To obtain a deterministic solution, we have used a relation between metric potentials. The exact solution of Einstein’s field equations thus obtained represents an expanding and decelerating universe. The physical and kinematical parameters of the model have also been analyzed with certain constrained between the parameters of the quadratic equation of state.

Extended Generalized Riccati Equation Mapping for Thermal Traveling-Wave Distribution in Biological Tissues through a Bio-Heat Transfer Model with Linear/Quadratic Temperature-Dependent Blood Perfusion

Analytical thermal traveling-wave distribution in biological tissues through a bio-heat transfer (BHT) model with linear/quadratic temperature-dependent blood perfusion is discussed in this paper. Using the extended generalized Riccati equation mapping method, we find analytical traveling wave solutions of the considered BHT equation. All the travelling wave solutions obtained have been used to explicitly investigate the effect of linear and quadratic coefficients of temperature dependence on temperature distribution in tissues. We found that the parameter of the nonlinear superposition formula for Riccati can be used to control the temperature of living tissues. Our results prove that the extended generalized Riccati equation mapping method is a powerful tool for investigating thermal traveling-wave distribution in biological tissues.

Novel Method to Deal with Interval Quadratic Equations via Sign-Variation Analysis

In this article, analytical results are obtained apparently for the first time in the literature, for the lower and upper bounds of the roots of quadratic equations when two or all three coefficients a, b, c constitute an interval, with a method called the sign-variation analysis. The results are compared with the parametrization technique offered by Elishakoff and Miglis, and with the solution yielded by minimization and maximization commands of the Maple software. Solutions for some interval word problems are also provided to edulcorate the methodology. This article only focuses on the real roots of those quadratic equations, complex solutions being beyond this investigation.

The Appearance of Noise Terms in Modified Adomian Decomposition Method for Quadratic Integral Equations

In this paper, we apply the modified Adomian Decomposition Method to get the numerical solutions of Quadratic integral equations. The appearance of noise terms in Decomposition Method was investigated. The method was described along with several examples.

Numerical Solutions for Quadratic Integro-Differential Equations of Fractional Orders

In this article, variational iteration method (VIM) and homotopy perturbation method (HPM) solve the nonlinear initial value problems of first-order fractional quadratic integro-differential equations (FQIDEs). We use the Caputo sense in this article to describe the fractional derivatives. The solutions of the problems are derived by infinite convergent series, and the results show that both methods are most convenient and effective.

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:

Quadratic nonconforming finite element method for 3D Stokes equations on cuboid meshes

In this paper, the quadratic nonconforming brick element （MSLK element） intro- duced in [10] is used for the 3D Stokes equations. The instability for the mixed element pair MSLK-P1 is analyzed, where the vector-valued MSLK element approximates the velocity and the piecewise P1 element approximates the pressure. As a cure, we adopt the piecewise P1 macroelement to discretize the pressure instead of the standard piecewise P1 element on cuboid meshes. This new pair is stable and the optimal error estimate is achieved. Numerical examples verify our theoretical analysis.

Unification of the Quadratic Model Equations of the Inhibition Characteristics of Acidified Ocimum Basilicum on the Corrosion Behaviour of Mild Steel

An attempt has been made at unifying the resulting quadratic models from the study of the correlation behavior of the inhibition characteristics of acidified ocimum basilicum on conventional mild steel. Weight-loss corrosion technique was employed in obtaining the corrosion penetration rate using the equation: cpr = . Subsequently, the quadratic models were developed by using a computer-aided statistical modeling technique (International Business Machine (IBM)’s SPSS version 17.0). The results obtained showed a nearly perfect positive correlation with a correlation coefficient in the range of 0.986 ≤ R ≤ 0.996 which depicts that R ≥ 1. Also, the coefficient of determination fell within the range of 0.972 ≤ R2 ≤ 0.992 showing that approximately 97% to 99% of the total variation in passivation rate was accounted for by corresponding variation in exposure time, leaving out only between 3% and 1% to extraneous factors that are not incorporated into the model equations. The equations were further unified into a generalized form using MathCAD 7.0 and the resulting equation was y = 1.032 - 0.002t + 1.899?× 10-6t2 with a R2 value of 0.935 indicating a well-correlated relationship. With this, a new frontier on corrosion studies has emerged typifying a classical departure from previously long-held assumption that corrosion behaviours at room temperature were only logarithmic.

On Polynomials Solutions of Quadratic Diophantine Equations

Let P:=P(t) be a polynomial in Z[X]\{0,1} In this paper, we consider the number of polynomial solutions of Diophantine equation E:X2–(P2–P)Y2–(4P2–2)X+(4P2–4P)Y=0. We also obtain some formulas and recurrence relations on the polynomial solution (Xn,Yn) of

Solvability of Chandrasekhar’s Quadratic Integral Equations in Banach Algebra

In this paper, we prove some results concerning the existence of solutions for some nonlinear functional-integral equations which contain various integral and functional equations that are considered in nonlinear analysis. Our considerations will be discussed in Banach algebra using a fixed point theorem instead of using the technique of measure of noncompactness. An important special case of that functional equation is Chandrasekhar’s integral equation which appears in radiative transfer, neutron transport and the kinetic theory of gases [1].

Obtaining a New Representation for the Golden Ratio by Solving a Biquadratic Equation

In the present work we show how different ways to solve biquadratic equations can lead us to different representations of its solutions. A particular equation which has the golden ratio and its reciprocal as solutions is shown as an example.

Inverse Problems for Difference Equations with Quadratic Eigenparameter Dependent Boundary Conditions-II

The following inverse problem is solved—given the eigenvalues and the potential b(n) for a difference boundary value problem with quadratic dependence on the eigenparameter, λ, the weights c(n) can be uniquely reconstructed. The investi-gation is inductive on m where represents the number of unit intervals and the results obtained depend on the specific form of the given boundary conditions. This paper is a sequel to [1] which provided an algorithm for the solution of an analogous inverse problem, where the eigenvalues and weights were given and the potential was uniquely reconstructed. Since the inverse problem considered in this paper contains more unknowns than the inverse problem considered in [1], an additional spectrum is required more often than was the case in [1].

Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equation with Lévy Process

This paper investigates a linear-quadratic mean-field stochastic optimal control problem under both positive definite case and indefinite case where the controlled systems are mean-field stochastic differential equations driven by a Brownian motion and Teugels mar-tingales associated with Lévy processes.In either case,we obtain the optimality system for the optimal controls in open-loop form,and by means of a decoupling technique,we obtain the optimal controls in closed-loop form which can be represented by two Riccati differen-tial equations.Moreover,the solvability of the optimality system and the Riccati equations are also obtained under both positive definite case and indefinite case.