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Based on some problems in the punitive damages system that are not conducive to economic development, we take the perspective of economic law to play a positive role in promoting the further improvement of punitive damages. In this paper, the rank of punitive damages is first defined by the singular value analysis algorithm, and the function is set up as a diagonal matrix composed of different vectors, and the square root of the decomposition eigenvalue is the singular value of the matrix. Then the matrix nodes are calculated, and each node is divided into two column blocks and predicted, and the number of nodes is input for the eigenvalue decomposition. The matrix regression coefficients of the independent variables are examined after the decomposition is performed, and the fit of the singular value analysis algorithm in dealing with the compensation problem is derived from this coefficient. The experimental results from a region of the United States show that among 100 large punitive damages cases exceeding $100 million between January 2009 and December 2018, the number of cases of punitive damages in 2010, 2011, and 2013 was only one, and the highest number reached 12 in 2018 illustrating the accuracy of singular value analysis in locating punitive damages, which helps to promote the construction and development of the punitive compensation system and contribute to the stable development of the social economy.

In this letter, we propose a novel multiple minor components (MCs) extraction algorithm by adding a diagonal matrix into the Douglas minor subspace trascking algorithm. Through analyzing all the characteristics of the fixed points, it is proven that the proposed algorithm is stable if and only if the state matrix is composed by the desired MCs. Compared with traditional algorithms, the proposed algorithm has two advantages: (1) The initial conditions can be easily satisfied. (2) The permutation of the estimated MCs is explicitly given in prior. Besides, the norm of the state matrix has stability on the manifold, which illustrates that the proposed algorithm has the virtues of anti-jamming capability. Simulation results are presented to further demonstrate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

For a subspace S of C n and a fixed basis, we study the compact and convex set m S = c o n v e x h u l l { | s | 2 ∈ R ≥ 0 n : s ∈ S a n d ‖ s ‖ = 1 } ≃ { Diag (Y) ∈ M n h (C) : Y ≥ 0 , t r (Y) = 1 , P S Y P S = Y } that we call the moment of S, where | s | 2 = (| s 1 | 2 , | s 2 | 2 , ... , | s n | 2). This set is relevant in the determination of minimal hermitian matrices ( M ∈ M n h such that ‖ M + D ‖ ≤ D for every diagonal D and the spectral norm ‖ ⋅ ‖). We describe extremal points and certain curves of m S in terms of principal vectors that minimize the angle between S and the coordinate axes of the fixed basis. We also relate m S to the joint numerical range W of n rank one n × n hermitian matrices constructed with orthogonal projection P S and the fixed basis { e i } i = 1 n used. This connection provides a new approach to the description of m S and to minimal matrices. As a consequence, the intersection of two of these joint numerical ranges corresponding to orthogonal subspaces allows the construction or detection of a minimal matrix. [ABSTRACT FROM AUTHOR]

Abstract For massive multiple-input multiple-output (m-MIMO) uplink, the performances of the linear minimum mean-square error (MMSE) detector are considered near optimal, and they occupy benchmark place for most linear iterative detectors. However, the MMSE algorithm is known by its load computational complexity due to the implication of large-scale matrix inversions, and in other hand, iterative methods are often preferred in signal detection because of its low complexity. In this paper, we propose a New Damped Jacobi (NDJ) detector in order to improve the performance of the classical Jacobi linear algorithm. Starting from the classical Jacobi technique to our new proposal, we go through the development of two variants; one uses a damping factor and the other uses a stair-matrix. However, the NDJ incorporates a damping factor in its construction and basing also on stair matrix instead of diagonal matrix. The performances in terms of convergence and low complexity of each Jacobi variant studied in this paper are analyzed. Finally, some simulation examples are given to illustrate the advantages of the new proposed algorithm.

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Abstract n-widths in approximation theory characterize how well one can approximate a subset by some “good” subsets of a normed linear space. Especially, n-widths of sets of RN $\mathbb{R}^{N}$ have been studied deeply. Now the following problem is posed: we know that RN $\mathbb{R} ^{N}$ can be embedded in the fuzzy number space EN $E^{N}$. Is it then possible to define n-widths of set A in EN $E^{N}$ and obtain asymptotic estimates for these n-widths? In this paper, we shall introduce four n-widths of A in EN $E^{N}$ and determine these n-widths of Zadeh’s extension of diagonal matrices.

For massive multiple-input multiple-output (m-MIMO) uplink, the performancesof thelinearminimummean-square error (MMSE) detector are considered near optimal, and they occupy benchmark place for most linear iterative detectors. However, the MMSE algorithm is known by its load computational complexity due to the implication of large-scale matrix inversions, and in other hand, iterative methods are often preferred in signal detection because of its low complexity. In this paper, we propose a New Damped Jacobi (NDJ) detector in order to improve the performance of the classical Jacobi linear algorithm. Starting from the classical Jacobi technique to our new proposal, we go through thedevelopment of two variants; one uses adamping factor and the other uses a stair-matrix. However, the NDJ incorporates a damping factor in its construction and basing also on stair matrix instead of diagonal matrix. The performances in terms of convergence and low complexity of each Jacobi variant studied in this paper are analyzed. Finally, some simulation examples are giventoillustratetheadvantagesofthenewproposedalgorithm. [ABSTRACT FROM AUTHOR]

In order to reveal the sectoral composition between the years (2003-2020) with the inflation-free (real) values of the Payments made by Debit Card, Credit Card and Internet Card, and to increase the efficiency of payment systems by identifying similar and differentiated sectors, Binary Loads (Orthogonal (Vertical) and Diagonal (Diagonal) matrices are created. The eigenvalues, eigenvectors (charges) and ordinary correlation results obtained from the matrices are combined in the plane of the binary load graph and the shapes of the columns are equalized to the eigenvalues using Mahalanobis distances, and without observation scaling so that the cosines of the angles between the vectors are equal to the correlations between the variables, the results are interpreted only in a constant proportionality. In the light of results such as eigenvalues, values, forward difference in eigenvalues, and the total variance ratio explained, it was determined that some of the sectors paid by card differ significantly from other sectors and clustered by carrying positive or negative loads (variances). [ABSTRACT FROM AUTHOR]

Given a Hermitian matrix M ∈ M3(ℂ) we describe explicitly the real diagonal matrices DM such that ║M + DM║ ≤ ║M + D║ for all real diagonal matrices D ∈ M3(ℂ), where ║ · ║ denotes the operator norm. Moreover, we generalize our techniques to some n × n cases.

In mathematics, matrices have many uses, they are finding solutions of a linear equation system, looking for specific solutions of differential equations, determining state classification on Markov chains, and so on. There is a special matrix in matrix theory, that is a diagonal matrix. The diagonal matrix is a matrix whose all non-diagonal entries are primarily zero so that the product of the diagonal matrix can be computed by considering only the components along the main diagonal. A square matrix can sometimes be formed into a diagonal matrix. If a non-diagonal square matrix A can be conjugated with a diagonal matrix, then there is an invertible matrix P so PAP-1=D, where D is a diagonal matrix and P is said to diagonalize A. To find a square matrix diagonalizable or not, many researchers usually use eigenvalues and eigenvectors evaluation. In this study, we discuss that the other way to form a diagonal matrix by using Minimum Polynomial Algorithm.

In massive multiple-input multiple-output (M-MIMO) systems, a detector based on maximum likelihood (ML) algorithm attains optimum performance, but it exhaustively searches all possible solutions, hence, it has a very high complexity and realization is denied. Linear detectors are an alternative solution because of low complexity and simplicity in implementation. Unfortunately, they culminate in a matrix inversion that increases the computational complexity in high loaded systems. Therefore, several iterative methods have been proposed to approximate or avoid the matrix inversion, such as the Neuamnn series (NS), Newton iterations (NI), successive overrelaxation (SOR), Gauss−Siedel (GS), Jacobi (JA), and Richardson (RI) methods. However, a detector based on iterative methods requires a pre-processing and initialization where good initialization impresses the convergence, the performance, and the complexity. Most of the existing iterative linear detectors are using a diagonal matrix ( D ) in initialization because the equalization matrix is almost diagonal. This paper studies the impact of utilizing a stair matrix ( S ) instead of D in initializing the linear M-MIMO uplink (UL) detector. A comparison between iterative linear M-MIMO UL detectors with D and S is presented in performance and computational complexity. Numerical Results show that utilization of S achieves the target performance within few iterations, and, hence, the computational complexity is reduced. A detector based on the GS and S achieved a satisfactory bit-error-rate (BER) with the lowest complexity.

Several approximate matrix inversion methods have been used in linear massive MIMO uplink detectors where their convergence rate, performance, and complexity are greatly affected by the initial solution. In this Letter, the authors exploit a stair matrix, instead of a diagonal matrix, in initialising iterative linear minimum mean square error massive MIMO detector based on several approximate matrix inversion methods, namely, the Gauss–Seidel, successive over relaxation, Richardson iteration, and Newton iteration methods. Numerical results show a significant performance enhancement without a burden of extra complexity using a stair matrix over a diagonal matrix in all methods. [ABSTRACT FROM AUTHOR]

This study addresses the l2−l∞ filter design issue for fuzzy Markov stochastic systems with sensor failures and packet dropouts. A more general failure model is used for Markov systems to better describe the sensor failures, and the packet dropouts are assumed to obey a series of Bernoulli processes which are formed into a diagonal matrix. Sufficient condition is established to guarantee that the filtering error system is stochastically stable and achieves the given l2−l∞ performance. The parameters of the filter are then derived by solving linear matrix inequalities. Simulation results are given to demonstrate the usefulness and availability of the proposed method. [ABSTRACT FROM AUTHOR]

Different values describing both the efficiency of rectification column and the efficiency of mass-transfer are represented in the brief literature review. It is also showed that the approach used for the calculation of mass-transfer efficiency characteristics in binary mixtures is not very acceptable for multi-component mixtures.

A generic matrix $$A\in \,\mathbb {C}^{n \times n}$$ is shown to be the product of circulant and diagonal matrices with the number of factors being $$2n-1$$ at most. The demonstration is constructive, relying on first factoring matrix subspaces equivalent to polynomials in a permutation matrix over diagonal matrices into linear factors. For the linear factors, the sum of two scaled permutations is factored into the product of a circulant matrix and two diagonal matrices. Extending the monomial group, both low degree and sparse polynomials in a permutation matrix over diagonal matrices, together with their permutation equivalences, constitute a fundamental sparse matrix structure. Matrix analysis gets largely done polynomially, in terms of permutations only. [ABSTRACT FROM AUTHOR]

A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It studies the finite - step realization of the joint generalized spectral radius of a pair of real square matrices. If S = | S1, S2 % where S1, S2 are n xn matrices and S1 = ( aij ) with aij 0 and S2 is similar to a diagonal matrix whose nonzero elements have the same sign, S has the finitenes property. [ABSTRACT FROM AUTHOR]

Abstract: We describe properties of a Hermitian matrix having minimal quotient norm in the following sense:for all real diagonal matrices . Here denotes the operator norm. We show a constructive method to obtain all the minimal matrices of any size. [Copyright &y& Elsevier]

Abstract: In this paper, the approximation characteristic of a diagonal matrix in probabilistic and average case settings is investigated. And the asymptotic degree of the probabilistic linear -width and -average linear -width of diagonal matrix are determined. [Copyright &y& Elsevier]

The paper addresses the forced flexural-and-torsional vibrations of a cantilever beam of constant cross section. The relevant boundary-value problem is solved. The system of two partial differential equations of the fourth order that describes these vibrations is analyzed in a vector-function space and is subjected to an equivalent transformation to obtain one vector equation of the fourth order with two matrices as coefficients. One is an idempotent matrix; the other is a diagonal matrix. This makes it much easier to construct a Cauchy vector function as an analytic function of these matrices [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider decidability questions that are related to the membership problem in matrix semigroups. In particular, we consider the membership of a given invertible diagonal matrix in a matrix semigroup and then a scalar matrix, which has a separate geometric interpretation. Both problems have been open for any dimensions and are shown to be undecidable in dimension 4 with integral matrices by a reduction of the Post Correspondence Problem (PCP). Although the idea of PCP reduction is standard for such problems, we suggest a new coding technique to cover the case of diagonal matrices. [Copyright &y& Elsevier]

Abstract: To enable large-scale 3D simulation of metal forming processes, the 3D rigid-plastic finite element method using a diagonal matrix is extended to parallel processing. In the present method, uncoupled equilibrium equations expressed by a matrix with three rows and three columns at each nodal point are repeatedly calculated in turn without solving the large global simultaneous equations. The method using a diagonal matrix is suitable for parallel processing because of the iterative calculation at each nodal point. The convergence of the solution is examined in the simulation of upsetting of a rectangular block with an inclined tool. The speedup of the computation in the parallel processing considerably approaches the ideal one using the odd–even method in the nodal point assignment approach and the intermittent correction of non-linearity in equilibrium equations. It is found that the present parallel method is suitable for the large-scale three-dimensional simulation of metal forming processes. [Copyright &y& Elsevier]

We consider the system of linear equations (C + iD)x=b, where C is a circulant matrix and D is a real diagonal matrix. We study the technique for constructing the normal/skew-Hermitian splitting for such coefficient matrices. Theoretical results show that if the eigenvalues of C have positive real part, the splitting method converges to the exact solution of the system of linear equations. When the eigenvalues of C have non-negative real part, the convergence conditions are also given. We present a successive overrelaxation acceleration scheme for the proposed splitting iteration. Numerical examples are given to illustrate the effectiveness of the method. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: Variable air volume (VAV) air conditioning control system has the feature of multi-control loops. While all the control loops are working together, they interfere and influence each other. This paper designs the decoupling unit in VAV air conditioning system in the method of diagonal matrix decoupling. Lonworks technology is adopted in VAV air conditioning decoupling control system so that data could be exchanged among multi-loops. It is suitable that decoupling compensation coefficients among multi-loops are handled as network variables of Lonworks technology. Experimental results demonstrate that the combination of the diagonal matrix decoupling and Lonworks technology can improve the performance of the VAV air conditioning control system. [Copyright &y& Elsevier]

Maximum-likelihood (ML) data-aided frequency estimation in multipath Rayleigh-fading channels with sparse impulse responses is investigated. We solve this problem under the assumption that the autocorrelation matrix of the pilot signal can be approximated by a diagonal matrix, the fading of different path amplitudes are independent from each other, and the additive noise is white and Gaussian. The ML frequency estimator is shown to be based on combining nonlinearly transformed path periodograms. We have derived the nonlinear function for the two cases: known and unknown fading variances. The new frequency estimators lead, in particular cases, to known ML frequency estimators for nonsparse multipath fading channels. The use of a priori information about the mean number of paths in the channel allows a significant improvement of the accuracy performance. Exploiting the sparseness of the channel impulse response is shown to significantly reduce the threshold signal-to-noise ratio at which the frequency error departs from the Cramer-Rao lower bound. However, precise knowledge of the channel sparseness is not required in order to realize this improvement. [ABSTRACT FROM PUBLISHER]

The problem inverse to the synthesis problem of a controller optimal with respect to quadratic functional is considered for linear stationary system. This problem is formulated as follows. For the given matrices which describe the system dynamics and the controller matrix it is necessary to determine the weight matrices of quadratic functional. The offered algorithm of this problem solution includes the procedure of finding the set of solutions to select the solution satisfying certain criteria. [ABSTRACT FROM AUTHOR]

In massive multiple-input multiple-output (M-MIMO) systems, a detector based on maximum likelihood (ML) algorithm attains optimum performance, but it exhaustively searches all possible solutions, hence, it has a very high complexity and realization is denied. Linear detectors are an alternative solution because of low complexity and simplicity in implementation. Unfortunately, they culminate in a matrix inversion that increases the computational complexity in high loaded systems. Therefore, several iterative methods have been proposed to approximate or avoid the matrix inversion, such as the Neuamnn series (NS), Newton iterations (NI), successive overrelaxation (SOR), Gauss–Siedel (GS), Jacobi (JA), and Richardson (RI) methods. However, a detector based on iterative methods requires a pre-processing and initialization where good initialization impresses the convergence, the performance, and the complexity. Most of the existing iterative linear detectors are using a diagonal matrix (D) in initialization because the equalization matrix is almost diagonal. This paper studies the impact of utilizing a stair matrix (S) instead of D in initializing the linear M-MIMO uplink (UL) detector. A comparison between iterative linear M-MIMO UL detectors with D and S is presented in performance and computational complexity. Numerical Results show that utilization of S achieves the target performance within few iterations, and, hence, the computational complexity is reduced. A detector based on the GS and S achieved a satisfactory bit-error-rate (BER) with the lowest complexity. [ABSTRACT FROM AUTHOR]

Finite-element time-domain (FETD) simulations of Maxwell equations in irregular simplicial grids require the solution of a sparse linear system involving the Hodge (mass) matrix at each time step. This can be avoided by mass lumping techniques that approximate the mass matrix by a diagonal matrix, but not without shortcomings. In this communication, we propose an alternative approach to yield a conditionally stable, fully explicit, and sparse FETD by using a sparse approximate inverse of the mass matrix. [ABSTRACT FROM PUBLISHER]