General Linear Equation Research Articles

Hyperstability Results for the General Linear Functional Equation in Non-Archimedean 2-Banach Spaces

Hyperstability Results for the General Linear Functional Equation in Non-Archimedean 2-Banach Spaces

Stability estimates for some parabolic inverse problems with the final overdetermination via a new Carleman estimate

This paper is about Hölder and Lipschitz stability estimates and uniqueness theorems for some coefficient inverse problems and associated inverse source problems for a general linear parabolic equation of the second order with variable coefficients. The data for the inverse problem are given at the final moment of time . In addition, both Dirichlet and Neumann boundary conditions are given either on a part or on the entire lateral boundary. Thus, if these boundary conditions are given only at a part of the boundary, then even if the target coefficient is known, still the forward problem is not a classical initial boundary value problem.

HHL algorithm with mapping function and enhanced sampling for model predictive control in microgrids

This paper presents a refined quantum Harrow Hassidim Lloyd (HHL) algorithm for microgrid control. The first novelty of the developed method is that a mapping shift function enables the original HHL algorithm to handle general linear equations with non-singular and indefinite matrix. Second, a method of Matrix Extension for Amplifying Sampling Probabilities of Intended Solution (ME-ASPI) is proposed to design the reformulated linear algebraic equations, allowing for improved sampling efficiency of the quantum tomography in the refined HHL algorithm. Then, we applied the method to solve the model predictive control (MPC) problem in nonlinear dynamical microgrids. Specifically, with the ME-ASPI method, the refined HHL algorithm can effectively obtain the intended partial optimal control inputs for MPC. The optimization of quadratic programming problem in each time step of MPC is transformed into a linear system problem, which is addressed by the proposed quantum solver through using only partial information, with the time complexity improved from O(N2.37286) classically to O(N2logN×plogp) in quantum. Numerical examples have validated the effectiveness of the refined HHL algorithm with the proposed mapping function and the ME-ASPI method. By leveraging quantum properties, the proposed method provides a hybrid quantum–classical framework for microgrid control. This generic method can also potentially tackle many other challenges in analyzing and controlling general complex engineered systems.

Hyperstability of the General Linear Functional Equation in Non-Archimedean Banach Spaces

Hyperstability of the General Linear Functional Equation in Non-Archimedean Banach Spaces

Correlation between Static and Dynamic Young’s Modulus of Elasticity of Different Types of Rocks

Correlation between static and dynamic elastic modulus in rocks has often been addressed in various studies. In this study, the correlation between the static and dynamic elastic modulus of six different sedimentary, metamorphic and igneous rock types namely coarse-grained sandstone, fine-grained sandstone, schist, quartzite, basalt and granite are studied. Physical properties such as bulk density and porosity are evaluated of all samples. Static elastic modulus is determined by uniaxial compression test and dynamic elastic modulus is found out by ultrasonic pulse velocity testing. Scanning Electron Microscope experiment, X-Ray Diffraction experiment are also conducted on all the samples for mineralogical and petrological studies. The individual correlation between the static and dynamic elastic modulus of all the rock types have been proposed using linear regression with zero intercept and high values of coefficient of determination. These equations are useful in calculating the static elastic modulus using non-destructive techniques, in a wide range of rock materials. Finally, a general linear equation has been developed to relate ratio of static and dynamic elastic modulus with density of the rock.

Association between residential environment and emotional wellbeing among older adults in China: the mediating effect of health lifestyle.

The association between the residential environment and emotional wellbeing (EWB) in older adults has received extensive attention from gerontologists, especially during the COVID-19 pandemic; however, the mediating mechanism of how residential environment affects emotional wellbeing has not been fully explored. This study examined the effects of the residential environment on EWB and the mediating role of health lifestyle. This study analyzed the survey data of 493 rural and 515 urban older adults from 2021 Chinese General Social Survey. General linear regression and structural equation models were used to examine the effects of residential environment and health lifestyle. Urban participants exhibited clear advantages in EWB, residential environment, and physical activity. Residential environment significantly affected the EWB of older adults, and health lifestyle played a mediating role in this relationship. The residential environment and health lifestyle did not significantly affect EWB in rural participants. This study revealed differences in the effects of health lifestyles and residential environments on EWB among older adults in rural and urban settings in China. This study provided empirical evidence of mental health disparities between older rural and urban Chinese residents.

The adjoint map of Euclidean plane curves and curvature problems

The adjoint map of a pair of naturally parametrized curves in the Euclidean plane is studied from the point of view of the curvature. A main interest is when the given curve and its adjoint curve share the same natural parameter and the same curvature. For the general linear second order differential equation we introduce a function expressing the deformation of curvatures induced by the adjoint map.

Linear Programming Using ABS Method

Nowadays, we face many equations in everyday life, where many attempts have been made to find their solutions, and various methods have been introduced. Many complex problems often lead to the solution of systems of equations. In mathematics, linear programming problems is a technique for optimization of a linear objective function that must impose several constraints on linear inequality. Linear programming emerged as a mathematical model. In this study, we introduce the category of ABS methods to solve general linear equations. These methods have been developed by Abafi, Goin, and Speedicato, and the repetitive methods are of direct type, which implicitly includes LU decomposition, Cholesky decomposition, LX decomposition, etc. Methods are distinguished from each other by selecting parameters. First, the equations system and the methods of solving the equations system, along with their application, are examined. Introduction and history of linear programming and linear programming problems and their application were also discussed.

Deformation of an elastic second gradient spherical body under equatorial line density of dead forces

We consider deformations of an elastic body having initially a spherical shape. Assumed deformation energy depends on the first and second gradient of displacements. We apply an equatorial line density of dead loads, that are forces per unit line length directed in radial direction and applied along the equator of the sphere. We restrict ourselves our analysis to the case of linearized second strain gradient isotropic elasticity (for which the more general energy was determined by Mindlin) with only one characteristic length. Differently to what happens in first gradient continua, i.e. in classic linear elasticity, we show that for the particular class second gradient continua considered here these forces do not determine infinite displacements in the direction of applied dead line forces. Instead, using a series method for the solution of the considered boundary-value problem, we demonstrate that the displacements are finite. So in the deformed configuration there is not the formation of an edge at the material points where the forces are applied. Further investigations are therefore needed for establishing if this elastic-regime edge formation is made possible: (I) either in the case of more general linear elastic constitutive equations or (II) only when large deformations are considered or (III) if non-elastic phenomena are involved.

On the dynamics of first and second order GeCo and gBBKS schemes

In this paper we investigate the stability properties of the so-called gBBKS and GeCo methods, which belong to the class of nonstandard schemes and preserve the positivity as well as all linear invariants of the underlying system of ordinary differential equations for any step size. A stability investigation for these methods, which are outside the class of general linear methods, is challenging since the iterates are always generated by a nonlinear map even for linear problems. Recently, a stability theorem was derived presenting criteria for understanding such schemes.For the analysis, the schemes are applied to general linear equations and proven to be generated by C1-maps with locally Lipschitz continuous first derivatives. As a result, the above mentioned stability theorem can be applied to investigate the Lyapunov stability of non-hyperbolic fixed points of the numerical method by analyzing the spectrum of the corresponding Jacobian of the generating map. In addition, if a fixed point is proven to be stable, the theorem guarantees the local convergence of the iterates towards it.In the case of first and second order gBBKS schemes the stability domain coincides with that of the underlying Runge–Kutta method. Furthermore, while the first order GeCo scheme converts steady states to stable fixed points for all step sizes and all linear test problems of finite size, the second order GeCo scheme has a bounded stability region for the considered test problems. Finally, all theoretical predictions from the stability analysis are validated numerically.

A mild-slope formulation based on Weyl rule of association with application to coastal wave modelling

Weyl rule of association, proposed by Hermann Weyl for quantum mechanics applications (Weyl, 1931), can be used to associate between the dispersion relation of water waves and a non-local pseudo-differential operator. The central result of this study is that this operator correctly approximates the Dirichlet-to-Neumann operator derived for linear waves over a slowly varying bathymetry. This opens the door to a formal use of Weyl’s operational calculus, and consequently, allowing straightforward derivations and generalizations of water waves’ models over mild slopes. Specifically, within the framework of linear wave theory, the formulation based on Weyl rule of association provides a generalized mild-slope model which does not impose a limit on the spectral width. Most significantly, the mild-slope formulation based on Weyl rule of association allows to derive a general linear kinetic equation for which the widely used energy balance equation (the central equation of forecasting models such as SWAN and WAVEWATCH) serves as a special case. This result not only provides a formal link between the deterministic description (i.e., Euler equations) and the stochastic description (i.e., the energy balance equation), but also establishes the theoretical foundations for the statistical description of bathymetry-induced wave interferences. Such a statistical description is especially important over coastal waters, where through the interaction with the bathymetry, waves are rapidly scattered and tend to form focal zones and associated interference patterns.

A New Complex Double Integral Transform and It’s Applications in Partial Differential Equations

In this paper, we present a new complex double integral transform namely “Complex Double Sadik Transform”, to solve general linear partial differential equations. Several functions are used (applied) to show the usefulness of this new double transformation.

Two for the GOES: Exploring Gambling Outcome Expectancies Scores Across Mixed and Offline-Only Gamblers in Relation to Problem Gambling Risk Status

As online gambling becomes more prevalent, understanding the motives of online gamblers has become a key focus for research and practice. The aim of this study was to understand differences in gambling-related outcome expectancies between mixed (both online and offline) gamblers and offline-only gamblers, by incorporating gambling harm risk categories from the problem gambling severity index (PGSI). This study comprised a secondary data analysis of the 2015 Northern Territory Gambling Prevalence and Wellbeing Survey. A sample of 1207 individuals in the Northern Territory who had reported gambling at least once in the previous 12 months were used in the analyses. General linear and structural equation modelling were used to ascertain differences in gambling outcome expectancies, in relation to gambling modality (i.e., mixed, offline-only) and PGSI scores. Mixed gamblers tended to score higher on all outcome expectancies than their offline-only counterparts. Outcome expectancy scores were higher in individuals in higher-risk PGSI categories. The escape outcome expectancy was dependent on both modality and risk category. Invariance testing of a low and problem gambling risk subsample revealed differential relationships for both the escape and excitement outcome expectancies for mixed and offline-only gamblers. The results provide an important contribution to the existing literature regarding motivation and outcome expectancies in relation to gambling modality and problem gambling severity. The findings highlight the importance of considering both gambling outcome expectancies and modality when considering problem gambling.

PENGARUH KECERDASAN EMOSIONAL TERHADAP PRESTASI BELAJAR PAI SISWA SMP DI JOMBANG

Emotional intelligence is a person's ability to recognize himself, control himself, motivate himself, empathize, and the ability to build relationships (cooperation) with others. Most teachers only consider students to excel if their intelligence scores are high, even though emotional intelligence influences success more. With good emotional intelligence, it is hoped that it can influence student achievement in PAI learning. The method used in this research is a quantitative survey method. With Linear Regression analysis technique. The sample of this research is the students of the one-roof SMPN Pengampon Kabuh Jombang with the results of the general simple linear regression equation: Y = a + bX. Y = 55.461 + 0.429X this equation can be translated: a constanta of 55.461 and a regression coefficient of X of 0.429 states that the regression coefficient is positive, so it can be said that the direction of the influence of variable X is positive. And seen from the r2 value of 0.472 or 47.2%, the amount of contribution (contribution) of intelligence to learning achievement in PAI and 52.8% is influenced by other factors.

Late-time asymptotics for geometric wave equations with inverse-square potentials

We introduce a new, physical-space-based method for deriving the precise leading-order late-time behaviour of solutions to geometric wave equations on asymptotically flat spacetime backgrounds and apply it to the setting of wave equations with asymptotically inverse-square potentials on Schwarzschild black holes. This provides a useful toy model setting for introducing methods that are applicable to more general linear and nonlinear geometric wave equations, such as wave equations for electromagnetically charged scalar fields, wave equations on extremal Kerr black holes and geometric wave equations in even space dimensions, where existing proofs for deriving precise late-time asymptotics might not apply. The method we introduce relies on exploiting the spatial decay properties of time integrals of solutions to derive the existence and precise genericity properties of asymptotic late-time tails and obtain sharp, uniform decay estimates in time.

Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs

Abstract We consider a general nonsymmetric second-order linear elliptic partial differential equation in the framework of the Lax–Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive meshrefinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, for example, an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e., the total computational time. Numerical experiments underline the theory.

Weak dual generalized inverse of a dual matrix and its applications

Recently, the dual Moore-Penrose generalized inverse has been applied to study the linear dual equation when the dual Moore-Penrose generalized inverse of the coefficient matrix of the linear dual equation exists. Nevertheless, the dual Moore-Penrose generalized inverse exists only in partially dual matrices. In this paper, to study more general linear dual equation, we introduce the weak dual generalized inverse described by four dual equations, and is a dual Moore-Penrose generalized inverses for it when the latter exists. Any dual matrix has the weak dual generalized inverse and is unique. We obtain some basic properties and characterizations of the weak dual generalized inverse. Also, we investigate relationships among the weak dual generalized inverse, the Moore-Penrose dual generalized inverses and the dual Moore-Penrose generalized inverses, give the equivalent characterization and use some numerical examples to show that the three are different dual generalized inverse. Afterwards, by applying the weak dual generalized inverse we solve two special linear dual equations, one of which is consistent and the other is inconsistent. Neither of the coefficient matrices of the above two linear dual equations has dual Moore-Penrose generalized inverses.

Hyers-Ulam stability of a general linear partial differential equation

In this paper we will study Hyers-Ulam stability for a general linear partial differential equation of first order in a Banach space.

Nonlinear function activated GNN versus ZNN for online solution of general linear matrix equations

In this work, the gradient-based design method is generalized to establish a finite-/fixed-time GNN (gradient-based neural network) to obtain an online solution of the general linear matrix equation (LME): P1X(t)Q1+P2X(t)Q2=G. Two nonlinear activation functions (AFs), which are broadly investigated in ZNN (zeroing neural network), are introduced and put to use for the construction of the GNN model. Theoretical analyses and comparisons are given to illustrate their impacts on the GNN and the corresponding ZNN models in terms of convergence. It is shown that both the GNN model and the ZNN model exhibit the same convergence property (i.e., finite-/fixed-time convergence) under the action of the same AF. A tighter estimate of the upper bound of fixed convergence time is also obtained for the ZNN model. Comparative verification is conducted to confirm the validation of theoretical analysis.

Operator estimates for non-periodically perforated domains: disappearance of cavities

ABSTRACT We consider a boundary value problem for a general second-order linear equation in a perforated domain. The perforation is made by small cavities, a minimal distance between the cavities is also small. We impose minimal natural geometric conditions on the shapes of the cavities and no conditions on their distribution in the domain. On the boundaries of the cavities, a nonlinear Robin condition is imposed. The sizes of the cavities and the minimal distance between them are supposed to satisfy a certain simple condition ensuring that under the homogenization the cavities disappear and we obtain a similar problem in a non-perforated domain. Our main results state the convergence of the solution of the perturbed problem to that of the homogenized one in W 2 1 - and L 2 -norms uniformly in L 2 -norm of the right-hand side in the equation and provide the estimates for the convergence rates.