Basic and Applied Science OA Books Collection
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Browsing Basic and Applied Science OA Books Collection by Subject "Algebra"
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Item Item A Linear System of Both Equations and Inequalities in Max-Algebra(IntechOpen, 2023) Abdulhadi AminuA Linear System of Both Equations and Inequalities in Max-AlgebraItem Algebraic Theory of Appell Polynomials with Application to General Linear Interpolation Problem(IntechOpen, 2023) Francesco Aldo CostabileAlgebraic Theory of Appell Polynomials with Application to General Linear Interpolation ProblemItem An Interpretation of Rosenbrock's Theorem via Local Rings(IntechOpen, 2023) A. AmparanAn Interpretation of Rosenbrock's Theorem via Local RingsItem Gauge Theory, Combinatorics, and Matrix Models(IntechOpen, 2023) Taro KimuraGauge Theory, Combinatorics, and Matrix ModelsItem Identification of Linear, Discrete-Time Filters via Realization(IntechOpen, 2023) Daniel N. MillerIdentification of Linear, Discrete-Time Filters via RealizationItem Likelihood Ratio Tests in Multivariate Linear Model(IntechOpen, 2023) Yasunori FujikoshiThe aim of this chapter is to review likelihood ratio test procedures in multivariate linear models, focusing on projection matrices. It is noted that the projection matrices to the spaces spanned by mean vectors in hypothesis and alternatives play an important role. Some basic properties are given for projection matrices. The models treated include multivariate regression model, discriminant analysis model, and growth curve model. The hypotheses treated involve a generalized linear hypothesis and no additional information hypothesis, in addition to a usual liner hypothesis. The test statistics are expressed in terms of both projection matrices and sums of squares and products matrices.Item Matrices Which are Discrete Versions of Linear Operations(IntechOpen, 2023) Armando Martinez PerezWe introduce and study a matrix which has the exponential function as one of its eigenvectors. We realize that this matrix represents a set of finite differences derivation of vectors on a partition. This matrix leads to new expressions for finite differences derivatives which are exact for the exponential function. We find some properties of this matrix, the induced derivatives and of its inverse. We provide an expression for the derivative of a product, of a ratio, of the inverse of vectors, and we also find the equivalent of the summation by parts theorem of continuous functions. This matrix could be of interest to discrete quantum mechanics theory.Item Matrices, Moments and Quadrature(IntechOpen, 2023) James V. LambersThe numerical solution of a time-dependent PDE generally involves the solution of a stiff system of ODEs arising from spatial discretization of the PDE. There are many methods in the literature for solving such systems, such as exponential propagation iterative (EPI) methods, that rely on Krylov projection to compute matrix function-vector products. Unfortunately, as spatial resolution increases, these products require an increasing number of Krylov projection steps, thus drastically increasing computational expense.Item Nonnegative Inverse Eigenvalue Problem(IntechOpen, 2023) Ricardo L. SotoNonnegative Inverse Eigenvalue ProblemItem Nonnegative Inverse Elementary Divisors Problem(IntechOpen, 2023) Ricardo L. SotoInverse eigenvalue problems appear in a wide variety of areas in the pure and applied mathematics. They have to do with the construction of a certain matrix from some spectral information. Associated with any inverse eigenvalue problem, there are two important issues: the existence of a solution and the construction of a solution matrix. The purpose of this chapter is to study the nonnegative inverse elementary divisors problem (hereafter, NIEDP) and its state of the art. The elementary divisors of a given matrix A are the characteristic polynomials of the Jordan blocks of the Jordan canonical form of A. The NIEDP looks for necessary and sufficient conditions for the existence of a nonnegative matrix with prescribed elementary divisors. Most of the content of this chapter is based on recent results published by the author and collaborators from the Mathematics Department at Universidad Catolica del Norte, Chile.Item Nullspace of Compound Magic Squares(IntechOpen, 2023) Saleem Al-AshhabIn this chapter, we consider special compound 4 n × 4 n magic squares. We determine a 2 n ? 3 dimensional subspace of the nullspace of the 4 n × 4 n squares. All vectors in the subspaces possess the property that the sum of all entries of each vector equals zero.Item Partition-Matrix Theory Applied to the Computation of Generalized-Inverses for MIMO Systems in Rayleigh Fading Channels(IntechOpen, 2023) P. CervantesPartition-Matrix Theory Applied to the Computation of Generalized-Inverses for MIMO Systems in Rayleigh Fading ChannelsItem Some Recent Advances in Nonlinear inverse Scattering in 2D(IntechOpen, 2023) Valery SerovWe survey our recently published results concerning scattering problems for the nonlinear Schrodinger equationItem Square Matrices Associated to Mixing Problems ODE Systems(IntechOpen, 2023) Victor Martinez-LuacesIn this chapter, mixing problems are considered since they always lead to linear ordinary differential equation (ODE) systems, and the corresponding associated matrices have different structures that deserve to be studied deeply. This structure depends on whether or not there is recirculation of fluids and if the system is open or closed, among other characteristics such as the number of tanks and their internal connections. Several statements about the matrix eigenvalues are analyzed for different structures, and also some questions and conjectures are posed. Finally, qualitative remarks about the differential equation system solutions and their stability or asymptotical stability are included.Item Structured Approaches to General Inverse Eigenvalue Problems(IntechOpen, 2023) Patrick DumondAn inverse eigenvalue problem is one where a set or subset of (generalized) eigenvalues is specified and the matrices that generate it are sought. Many methods for solving inverse eigenvalue problems are only applicable to matrices of a specific type. In this chapter, two recently proposed methods for structured (direct) solutions of inverse eigenvalue problems are presented. The presented methods are not restricted to matrices of a specific type and are thus applicable to matrices of all types. For the first method, the Cayley-Hamilton theorem is developed for the generalized eigenvalue vibration problem. For a given (desired) frequency spectrum, many solutions are possible. Hence, a discussion of the required information and suggestions for including structural constraints are given. An algorithm for solving the inverse eigenvalue problem using the generalized Cayley-Hamilton theorem is then demonstrated. An algorithm for solving partially described systems is also specified. The Cayley-Hamilton theorem algorithm is shown to be a good tool for solving inverse generalized eigenvalue problems. Examples of application of the method are given. A second method, referred to as the inverse eigenvalue determinant method, is also introduced. This method provides another direct approach to the reconstruction of the matrices of the generalized eigenvalue problem, given knowledge of its eigenvalues and various physical parameters. As for the first method, there are no restrictions on the type of matrices allowed for the inverse problem. Examples of application of the method are also given, including application-oriented examples.