Coefficient Matrix Research Articles

The fundamental theorem of calculus in differential rings

In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such integro-differential rings and discuss many examples. We also construct corresponding integro-differential operators and provide normal forms via rewrite rules. They are then used to derive several identities and properties in a purely algebraic way, generalizing well-known results from analysis. In identities like shuffle relations for nested integrals and the Taylor formula, additional terms are obtained that take singularities into account. Another focus lies on treating basics of linear ODEs in this framework of integro-differential operators. These operators can have matrix coefficients, which allow to treat systems of arbitrary size in a unified way. In the appendix, using tensor reduction systems, we give the technical details of normal forms and prove them for operators including other functionals besides evaluation.

Legendre Galerkin spectral collocation least squares method for the Darcy flow in homogeneous medium and non-homogeneous medium

The Darcy's equation consists of the mass conservation equation and the Darcy's law that involves the hydraulic potential (or called pressure) and the fluid velocity, which governs the flow of an incompressible fluid through a porous medium. In this paper, we investigate the Legendre Galerkin spectral collocation least squares method for approximating the problem of Darcy flow in homogeneous medium and non-homogeneous medium, respectively. The proposed scheme can be solved the approximate solutions of the hydraulic potential and the average velocity of the fluid simultaneously. A symmetric positive definite coefficient matrix of the corresponding linear algebra equation is obtained by applying our scheme. Numerical examples are presented to validate the efficiency and accuracy of the proposed scheme.

WITHDRAWN: Accurate and efficient analytical simulation of free vibration for embedded nonlocal CNTRC beams with general boundary conditions

This study aims to evaluate the free vibrational response of restrained nanobeam enriched by nanocomposites based on an exact Fourier series approach. In order to capture the small size effects on the dynamical response, Eringen’s differential form of nonlocal elasticity is used which employs the one size (nonlocal) parameter. Within the framework of Rayleigh and Bernoulli-Euler beam theories to include the effect of nonlocality and by employing Fourier sine series together with Stokes’ transformation, a system of linear equations is obtained then solved by using the coefficient matrix. The combined effects of elastic boundary conditions, elastic medium, dispersion patterns and nonlocal effects are examined by solving this eigen-value problem constructed with Fourier infinite series. Free vibration frequencies are calculated for carbon nanotube reinforced nanobeam under different rigid or restrained boundary conditions, including an elastic medium Winkler-Pasternak type. A comprehensive parametric study is performed, with the particular focus on the influences of distribution pattern, elastic boundary and medium parameter on the free vibrational response of the composite nanobeam with different graded distributions. It is concluded that the addition of a small amount of carbon nanotube material can reinforce the stiffness of the nanobeam, and its free vibration performance is significantly affected by the distribution patterns, elastic medium and boundary conditions.

Multivariate Meixner polynomials related to holomorphic discrete series representations of [formula omitted]

We show that Griffiths’ multivariate Meixner polynomials occur as matrix coefficients of holomorphic discrete series representations of the group SU(1,d). Using this interpretation we derive several fundamental properties of the multivariate Meixner polynomials, such as orthogonality relations and difference equations. Furthermore, we also show that matrix coefficients for specific group elements lead to degenerate versions of the multivariate Meixner polynomials and their properties.

A phase-splitting approach to describe macroscopically non-equilibrium transport in porous media

Classical macro-scale dispersion equation cannot correctly represent non-local, non-equilibrium effects, and alternatives had to be proposed to overcome these discrepancies. Two-equation models have been widely used and proved useful to incorporate, to some extent, local non-equilibrium effects. More accurate descriptions may be obtained through the introduction of additional complex terms in the one-equation dispersion equation, e.g. spatial and time convolutions. An alternative more practical route rely on the introduction of a N-equations description. For instance, instead of a two-temperature model, transfers in the solid and/or fluid phase maybe represented by several equations, each one defined by some splitting framework. Splitting of the solid phase has been proposed in the past (multi-rate models). Splitting of the flowing phase has a rustic illustration in the case of the Coats and Smith model, i.e., ”flowing” and ”stagnant” fluid equations. In this paper, we explore further the possibilities offered by N-equations splitting of the flowing phase. The N-equations macro-scale model is developed applying an averaging technique to N-pseudo phases obtained by splitting the flowing phase based on different criteria, e.g. the velocity histogram. Most important model properties are the matrices of dispersion tensors and exchange coefficients, which are provided by a system of specific ”closure” problems. To illustrate this approach, the methodology is applied to Taylor’s dispersion problem, in which non-local effects are produced by pore-scale spatially distributed inlet conditions. The proposed N-equations description was compared to estimates of the N-temperatures obtained from direct numerical simulations. The N-equations model was able to reproduce accurately the dynamic of pore-scale computations, for such reputedly non-homogenizable problems, with a relatively small number of equations.

Electromagnetic Analysis for Axial-Flux Induction Planar Motor Based on Reduced Order Coefficient Matrix

Electromagnetic Analysis for Axial-Flux Induction Planar Motor Based on Reduced Order Coefficient Matrix

Approximation of reliabilities for random-regression single-step genomic best linear unbiased predictor models

Random-regression models (RRM) are used in national genetic evaluations for longitudinal traits. The outputs of RRM are an index based on random regression coefficients and its reliability. The reliabilities are obtained from the inverse of the coefficient matrix of mixed model equations (MME). The reliabilities must be approximated for large data sets because it is impossible to invert the MME. There is no extensive literature on methods to approximate the reliabilities of RRM when genomic information is included by single-step GBLUP. We developed an algorithm to approximate such reliabilities. Our method combines the reliability of the index without genomic information with the reliability of a GBLUP model in terms of effective record contributions. We tested our algorithm in the 3-lactation model for milk yield from the Czech Republic. The data had 30 million test-day records, 2.5 million animals in the pedigree, and 54,000 genotyped animals. The correlation between our approximation and the reliabilities obtained from the inversion of the MME was 0.98, and the slope and intercept of the regression were 0.91 and 0.02, respectively. The elapsed time to approximate the reliabilities for the Czech data was 21 min.

Evaluation of Goalkeepers’ Performance via Grey Relational Analysis Method: Example of Africa Cup of Nations

Aim: The aim of the study is to evaluate the technical performance of goalkeepers via the gray relational analysis method. Materials and Method: The sample of the study consisted of 21 goalkeepers competing in the African Cup of Nations 2023. Data was obtained from the official website of the Confederation of African Football and Transfermarkt. Technical performance indicators were determined as “save percentage”, “claimed crosses” and “passing accuracy”. The data were evaluated by the grey relational analysis method among the quantitative research methods. In this context, six steps of the grey relational analysis method were applied respectively. These are "preparation of the data set and generation of the decision matrix", "normalization of the data", "generation of the reference series", "generation of the absolute value table", "generation of the grey relational coefficient matrix", "calculation and ranking of the grey relational degrees". Results: According to the analysis results, the highest technical performance value was calculated as 0.7654, while the lowest technical performance value was calculated as 0.3949. Thus, the top 3 goalkeepers were determined as Ronwen Williams (South Africa), Edouard Mendy (Senegal), and Babacar Niass (Mauritania). The last 3 goalkeepers were determined as Richard Ofori (Ghana), Ibrahim Koné (Guinea), and Baboucarr Gaye (Gambia). Conclusion: According to the results of the research, Ronwen Williams (South Africa), who won the Golden Glove Award and was selected for the team of the tournament at the Africa Cup of Nations, was determined as the goalkeeper with the highest performance. His national team South Africa completed the tournament in 3rd place, despite being ranked 19th among 24 teams in terms of market value. Similar conclusions were obtained for the remained goalkeepers. It is expected that this conclusion will provide significant contributions for stakeholders such as researchers, goalkeepers, coaches, scouts, and trainers.

Prediction Model of Coal Gas Permeability Based on Improved DBO Optimized BP Neural Network.

Accurate measurement of coal gas permeability helps prevent coal gas safety accidents effectively. To predict permeability more accurately, we propose the IDBO-BPNN coal body gas permeability prediction model. This model combines the Improved Dung Beetle algorithm (IDBO) with the BP neural network (BPNN). First, the Sine chaotic mapping, Osprey optimization algorithm, and adaptive T-distribution dynamic selection strategy are integrated to enhance the DBO algorithm and improve its global search capability. Then, IDBO is utilized to optimize the weights and thresholds in BPNN to enhance its prediction accuracy and mitigate the risk of overfitting to some extent. Secondly, based on the influencing factors of gas permeability, effective stress, gas pressure, temperature, and compressive strength, they are chosen as the coupling indicators. The SPSS 27 software is used to analyze the correlation among the indicators using the Pearson correlation coefficient matrix. Additionally, the Kernel Principal Component Analysis (KPCA) is employed to extract the original data. Then, the original data is divided into principal component data for the model input. The prediction results of the IDBO-BPNN model are compared with those of the PSO-BPNN, PSO-LSSVM, PSO-SVM, MPA-BPNN, WOA-SVM, BES-SVM, and DPO-BPNN models. This comparison assesses the capability of KPCA to enhance the accuracy of model predictions and the performance of the IDBO-BPNN model. Finally, the IDBO-BPNN model is tested using data from a coal mine in Shanxi. The results indicate that the predicted outcome closely aligns with the actual value, confirming the reliability and stability of the model. Therefore, the IDBO-BPNN model is better suited for predicting coal gas permeability in academic research writing.

ИНДЕКС ФОНДООТДАЧИ: СОВРЕМЕННОЕ СОСТОЯНИЕ И ФАКТОРЫ РОСТА

The scientific article contains an analysis of scientific literature on ensuring the efficient use of fixed assets in enterprises, research on capital productivity and capital intensity, and the development of main directions for the efficient use of fixed capital. The dynamics of the coefficients of renewal and disposal of fixed assets in the economy are presented in graphical form. The information base for the study was data from the official website of the Federal State Statistics Service. The maximum values of the renewal coefficient were observed in 2018, 2019 and 2022, and the minimum in 2015, 2020 and 2021. The maximum values of the retirement rate are presented in 2014, 2015 and 2016, and the minimum in 2019, 2020 and 2021. An econometric study of the capital productivity index showed an inverse and very weak relationship with the value of gross domestic product and the cost of fixed assets (the value of the correlation coefficient (index) (-0.1)): with an increase in the value of gross domestic product and the value of fixed assets in the period under study, no growth was observed capital productivity index. The value of the gross domestic product and the cost of fixed assets have a direct and close relationship (the value of the correlation coefficient (index) is 0.9): the annual growth of the gross domestic product is accompanied by an increase in the cost of fixed assets. A matrix of correlation coefficients (indices) was constructed to calculate the total correlation coefficient (index). The result of calculating the total correlation coefficient (index) (its value is 0.4) showed that there is a direct and weak connection between the capital productivity index, the value of the gross domestic product and the cost of fixed assets: the growth of the capital productivity index is weakly related to the growth of the gross domestic product and the cost of fixed assets funds.

High-Dimensional Multivariate Linear Regression with Weighted Nuclear Norm Regularization

We consider a low-rank matrix estimation problem when the data is assumed to be generated from the multivariate linear regression model. To induce the low-rank coefficient matrix, we employ the weighted nuclear norm (WNN) penalty defined as the weighted sum of the singular values of the matrix. The weights are set in a nondecreasing order, which yields the non-convexity of the WNN objective function in the parameter space. Although the objective function has been widely applied, studies on the estimation properties of its resulting estimator are limited. We propose an efficient algorithm under the framework of the alternative directional method of multipliers (ADMM) to estimate the coefficient matrix. The estimator from the suggested algorithm converges to a stationary point of an augmented Lagrangian function. Under the orthogonal design setting, the effects of the weights for estimating the singular values of the ground-truth coefficient matrix are derived. Under the Gaussian design setting, a minimax convergence rate on the estimation error is derived. We also propose a generalized cross-validation (GCV) criterion for selecting the tuning parameter and an iterative algorithm for updating the weights. Simulations and a real data analysis demonstrate the competitive performance of our new method. Supplementary materials for this article are available online.

Properties of spectral characteristics of quasi-stationary electron states in an open multi-cascade nanostructure

The exact expressions for scattering matrix and transmission coefficient in open multi-cascade nanostructure are obtained. The theory of spectral characteristics of electron quasi-stationary states is developed in two approaches. For the nanostructure with GaAs wells and AlGaAs barriers, it is established that resonance bands appear in the N-cascade structure. Each one is formed by the energies of N states. Only the scattering matrix allows to unambiguously determine the resonance energies and resonance widths of all electron states in the multi-cascade structure. The approach of transmission coefficient, due to the collapse of resonances, gives the incomplete information about the spectral characteristics.

The Potential of Multi-Task Learning in CFDST Design: Load-Bearing Capacity Design with Three MTL Models.

Concrete-filled double steel tubes (CFDSTs) are a load-bearing structure of composite materials. By combining concrete and steel pipes in a nested structure, the performance of the column will be greatly improved. The performance of CFDSTs is closely related to their design. However, existing codes for CFDST design often focus on how to verify the reliability of a design, but specific design parameters cannot be directly provided. As a machine learning technique that can simultaneously learn multiple related tasks, multi-task learning (MTL) has great potential in the structural design of CFDSTs. Based on 227 uniaxial compression cases of CFDSTs collected from the literature, this paper utilized three multi-task models (multi-task Lasso, VSTG, and MLS-SVR) separately to provide multiple parameters for CFDST design. To evaluate the accuracy of models, four statistical indicators were adopted (R2, RMSE, RRMSE, and ρ). The experimental results indicated that there was a non-linear relationship among the parameters of CFDSTs. Nevertheless, MLS-SVR was still able to provide an accurate set of design parameters. The coefficient matrices of two linear models, multi-task Lasso and VSTG, revealed the potential connection among CFDST parameters. The latent-task matrix V in VSTG divided the prediction tasks of inner tube diameter, thickness, strength, and concrete strength into three groups. In addition, the limitations of this study and future work are also summarized. This paper provides new ideas for the design of CFDSTs and the study of related codes.

On the asymptotics of matrix coefficients of representations of [formula omitted]

We study matrix coefficients of the unitary (and also the completely bounded) representations of SL(2,R) and its universal covering group. We describe the asymptotic distribution of column vectors in terms of Whittaker functions, exhibiting also a relationship to the Fourier transform.

On the boundary conditions for a thin circular plate conjugated to a massive body

The problem of deformation under the action of uniform pressure of a circular plate coupled with a massive base is considered, while the condition for the coupling of the plate with the base is modeled using boundary conditions of the generalized elastic embedding type, i.e. the relationship between the bending moment and forces at the edge of the plate with displacements and rotation angles through the compliance matrix. The main goal of the work is to study the influence of the elasticity of the embedding on the elastic response of the plate. The solution to the problem was obtained in the formulation of the linear theory of plates, the theory of membranes in the approximation of homogeneity of longitudinal forces, and the Foppl — von Karman theory, also in the approximation of the assumption of homogeneity of longitudinal forces. The values of the coefficients of the compliance matrix were obtained using the finite element method for the auxiliary problem and compared with the values of the coefficients obtained for related problems by analytical methods. Numerical results were obtained for an aluminum wafer on a silicon base. The obtained solution was compared with the solution obtained for the rigid embedment condition for all three models used. It is shown that in the case of large deflections (several plate thicknesses), taking into account the compliance of the embedment becomes essential.

ПАКЕТ GAP ДЛЯ РОЗЩЕПЛЕННЯ ЛІНІЙНИХ СИСТЕМ

There are a number of problems, the solution of which requires the breakdown of the initial system of equations using algebraic decoupling methods. This means reducing the matrix of coefficients to block-diagonal (or block-triangular) form by means of substitution of variables. The main computational tasks when using such methods are finding the centralizer of several matrices or compiling the algebra generated by these matrices. For calculations, it is convenient to use the GAP computer algebra system because the system itself is designed for discrete algebra calculations. The problem is that the GAP program does not support calculations with real numbers. For practical problems you can try to replace them (with some accuracy) by rational numbers. At the same time, the decision may turn out to be excessively cumbersome. On the other hand, the advantage of GAP is the complete absence of rounding errors.

Robust sparse concept factorization with graph regularization for subspace learning

Concept factorization (CF) is a powerful tool in subspace learning. Recently graph-based CF and local coordinate CF have been proposed to exploit the intrinsic geometrical structure of data, and have been shown quite successful in improving performance. However, these methods have limited robustness and might be sensitive to noises and disturbances in practical applications. In this paper, we propose a novel robust sparse CF framework (RSCF) for subspace learning. Specifically, we present a robust loss function to effectively eliminate the impact of the large outliers. The local coordinate constraint and the graph regularization term are incorporated into RSCF to simultaneously guarantee the sparsity of the coefficient matrix and maintain the local structure of the data. We prove that the local coordinate constraint implies the orthogonality of the coefficient matrix. By using the half-quadratic optimization technique, we transform the objective function of RSCF into a quadratic form and provide the iterative updating rules and the convergence analysis. Extensive experiments demonstrate the robustness and superiority of the proposed RSCF in comparison to state-of-the-art CF methods.

Comprehensive multi-view self-representations for clustering

Subspace learning-based methods have shown excellent performance for multi-view clustering, yet have the following problems: (1) most existing methods obtain the subspace representation from the original space, which might contain noises and cannot guarantee a clean enough subspace representation; (2) existing methods mainly focus on the consistency of the subspace representation, while the unique information of each view is not sufficiently exploited. To solve these two problems, we propose a novel multi-view subspace clustering method called comprehensive multi-view self-representations (CMSR). Specifically, we learn the original coefficient matrix of each view through the self-representation, which can reduce the noise of the original space to some extent. Then, we learn the subspace representation of the original coefficient matrix and decompose it into a consistent coefficient matrix and multiple diverse coefficient matrices, which can exploit the consistent and complementary information of multi-view data. Further, we impose the Schatten p-norm constraint on the consistent coefficient matrix to capture robust consistent information. Finally, the comprehensive results on eight real datasets demonstrate the versatility and effectiveness of the proposed method.

Statistical Inference For Noisy Matrix Completion Incorporating Auxiliary Information

This article investigates statistical inference for noisy matrix completion in a semi-supervised model when auxiliary covariates are available. The model consists of two parts. One part is a low-rank matrix induced by unobserved latent factors; the other part models the effects of the observed covariates through a coefficient matrix which is composed of high-dimensional column vectors. We model the observational pattern of the responses through a logistic regression of the covariates, and allow its probability to go to zero as the sample size increases. We apply an iterative least squares (LS) estimation approach in our considered context. The iterative LS methods in general enjoy a low computational cost, but deriving the statistical properties of the resulting estimators is a challenging task. We show that our method only needs a few iterations, and the resulting entry-wise estimators of the low-rank matrix and the coefficient matrix are guaranteed to have asymptotic normal distributions. As a result, individual inference can be conducted for each entry of the unknown matrices. We also propose a simultaneous testing procedure with multiplier bootstrap for the high-dimensional coefficient matrix. This simultaneous inferential tool can help us further investigate the effects of covariates for the prediction of missing entries. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Average-case analysis of the Gaussian elimination with partial pivoting

The Gaussian elimination with partial pivoting (GEPP) is a classical algorithm for solving systems of linear equations. Although in specific cases the loss of precision in GEPP due to roundoff errors can be very significant, empirical evidence strongly suggests that for a typical square coefficient matrix, GEPP is numerically stable. We obtain a (partial) theoretical justification of this phenomenon by showing that, given the random n×n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\ imes n$$\\end{document} standard Gaussian coefficient matrix A, the growth factor of the Gaussian elimination with partial pivoting is at most polynomially large in n with probability close to one. This implies that with probability close to one the number of bits of precision sufficient to solve Ax=b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Ax = b$$\\end{document} to m bits of accuracy using GEPP is m+O(logn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m+O(\\log n)$$\\end{document}, which improves an earlier estimate m+O(log2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m+O(\\log ^2 n)$$\\end{document} of Sankar, and which we conjecture to be optimal by the order of magnitude. We further provide tail estimates of the growth factor which can be used to support the empirical observation that GEPP is more stable than the Gaussian Elimination with no pivoting.