A parallel nonlinear energy sink(NES) is proposed and analyzed. The parallel NES is composed of a vibro-impact(VI) NES and a cubic NES. The dynamical equation is given, and the essential analytical investigation is ca...A parallel nonlinear energy sink(NES) is proposed and analyzed. The parallel NES is composed of a vibro-impact(VI) NES and a cubic NES. The dynamical equation is given, and the essential analytical investigation is carried out to deal with the cubic nonlinearity and impact nonlinearity. Multiple time-scale expansion is introduced, and the zeroth order is derived to give a rough outline of the system. The underlying Hamilton dynamic equation is given, and then the optimal stiffness is expressed. The clearance is regarded as a critical factor for the VI. Based on the periodical impact treatment by analytical investigation, the relationships of the cubic stiffness, the clearance, and the zeroth-order attenuation amplitude of the linear primary oscillator(LPO) are obtained.A cubic NES under the optimal condition is compared with the parallel NES. Harmonic signals, harmonic signals with noises, and the excitation generated by a second-order?lter are considered as the potential excitation forces on the system. The targeted energy transfer(TET) in the designed parallel NES is shown to be more e?cient.展开更多
In this paper,we investigate the tensor similarity and propose the T-Jordan canonical form and its properties.The concepts of the T-minimal polynomial and the T-characteristic polynomial are proposed.As a special case...In this paper,we investigate the tensor similarity and propose the T-Jordan canonical form and its properties.The concepts of the T-minimal polynomial and the T-characteristic polynomial are proposed.As a special case,we present properties when two tensors commute based on the tensor T-product.We prove that the Cayley-Hamilton theorem also holds for tensor cases.Then,we focus on the tensor decompositions:T-polar,T-LU,T-QR and T-Schur decompositions of tensors are obtained.When an F-square tensor is not invertible with the T-product,we study the T-group inverse and the T-Drazin inverse which can be viewed as the extension of matrix cases.The expressions of the T-group and T-Drazin inverses are given by the T-Jordan canonical form.The polynomial form of the T-Drazin inverse is also proposed.In the last part,we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverses can be given by a limit process.展开更多
We investigate the M-eigenvalues of the Riemann curvature tensor in the higher dimensional conformally flat manifold.The expressions of Meigenvalues and M-eigenvectors are presented in this paper.As a special case,M-e...We investigate the M-eigenvalues of the Riemann curvature tensor in the higher dimensional conformally flat manifold.The expressions of Meigenvalues and M-eigenvectors are presented in this paper.As a special case,M-eigenvalues of conformal flat Einstein manifold have also been discussed,and the conformal the invariance of M-eigentriple has been found.We also reveal the relationship between M-eigenvalue and sectional curvature of a Riemannian manifold.We prove that the M-eigenvalue can determine the Riemann curvature tensor uniquely.We also give an example to compute the Meigentriple of de Sitter spacetime which is well-known in general relativity.展开更多
The vibration propagates through the shaft in the form of elastic waves. The propagation characteristics of the elastic waves are a ected by the axial loads. The influence of the axial loads to the propagation charact...The vibration propagates through the shaft in the form of elastic waves. The propagation characteristics of the elastic waves are a ected by the axial loads. The influence of the axial loads to the propagation characteristics of the elastic waves is studied in this paper. Firstly, the transfer matrix of the elastic waves for the non-uniform shaft with axial loads is deduced by combining the transfer matrix without axial load and the additional equation caused by the axial load. And then, a numerical method is used to study the influence of the axial load, non-uniformity and the rotating speed to the propagation characteristics of the elastic waves. It’s found that a new Stop Band will appear due to the axial force, and the central frequency of which will decrease as the increase of the force, while the band width of which remains the same. The central frequency of the new Stop Band will also increase as the increase of the cross-section area ratio;however, the rotating speed of the shaft doesn’t a ect the propagation characteristics of the elastic waves obviously. Finally, an experimental rig is built up for further study, even though there are some small local errors, the results of experiments match well with the numerical ones, which indicates the validation of the theoretical results. The result can help to study the influence of the axial load to the dynamics of a non-uniform shaft and help to reveal the vibration propagating mechanism in such a shaft.展开更多
We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the mill spaces and the ranges of tensors, and study their relationship. We extend the fundamental theore...We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the mill spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares （M） solutions to a multilineax system and establish the relationship between the minimum-norm （N） leastsquares （M）solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.展开更多
We study the constrained systemof linear equations Ax=b,x∈R(A^(k))for A∈C^(n×n)and b∈Cn,k=Ind(A).When the system is consistent,it is well known that it has a unique A^(D)b.If the system is inconsistent,then we...We study the constrained systemof linear equations Ax=b,x∈R(A^(k))for A∈C^(n×n)and b∈Cn,k=Ind(A).When the system is consistent,it is well known that it has a unique A^(D)b.If the system is inconsistent,then we seek for the least squares solution of the problem and consider min_(x∈R(A^(k)))||b−Ax||2,where||·||2 is the 2-norm.For the inconsistent system with a matrix A of index one,it was proved recently that the solution is A^(■)b using the core inverse A^(■)of A.For matrices of an arbitrary index and an arbitrary b,we show that the solution of the constrained system can be expressed as A^(■)b where A^(■)is the core-EP inverse of A.We establish two Cramer’s rules for the inconsistent constrained least squares solution and develop several explicit expressions for the core-EP inverse of matrices of an arbitrary index.Using these expressions,two Cramer’s rules and one Gaussian elimination method for computing the core-EP inverse of matrices of an arbitrary index are proposed in this paper.We also consider the W-weighted core-EP inverse of a rectangular matrix and apply the weighted core-EP inverse to a more general constrained system of linear equations.展开更多
In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem： minx ｜｜Ax- b｜｜2, where A is an m-by-n （m ≥ n） rank deficient matrix...In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem： minx ｜｜Ax- b｜｜2, where A is an m-by-n （m ≥ n） rank deficient matrix. We first derive an explicit expression for the condition number in the weighted Frobenius norm ｜｜ [AT,βb] ｜｜F of the data A and b, where T is a positive diagonal matrix and β is a positive scalar. We then discuss the sensitivity of the standard 2-norm condition numbers for the generalized matrix inversion and rank deficient least squares and establish relations between the condition numbers and their condition numbers called level-2 condition numbers.展开更多
In this work,we try to build a theory for random double tensor integrals(DTI).We begin with the definition of DTI and discuss how randomness structure is built upon DTI.Then,the tail bound of the unitarily invariant n...In this work,we try to build a theory for random double tensor integrals(DTI).We begin with the definition of DTI and discuss how randomness structure is built upon DTI.Then,the tail bound of the unitarily invariant norm for the random DTI is established and this bound can help us to derive tail bounds of the unitarily invariant norm for various types of two tensors means,e.g.,arithmetic mean,geometric mean,harmonic mean,and general mean.By associating DTI with perturbation formula,i.e.,a formula to relate the tensor-valued function difference with respect the difference of the function input tensors,the tail bounds of the unitarily invariant norm for the Lipschitz estimate of tensor-valued function with random tensors as arguments are derived for vanilla case and quasi-commutator case,respectively.We also establish the continuity property for random DTI in the sense of convergence in the random tensor mean,and we apply this continuity property to obtain the tail bound of the unitarily invariant norm for the derivative of the tensor-valued function.展开更多
The H-matrices are an important class in the matrix theory, and have many applications. Recently, this concept has been extended to higher order H-tensors. In this paper, we establish important properties of diagonall...The H-matrices are an important class in the matrix theory, and have many applications. Recently, this concept has been extended to higher order H-tensors. In this paper, we establish important properties of diagonally dominant tensors and H-tensors. Distributions of eigenvalues of nonsingular symmetricH-tensors are given. An J（t%-tensor is semi-positive, which enlarges the area of semi-positive tensor from H-tensor to H＋-tensor. The spectral radius of Jacobi tensor of a nonsingular （resp. singular） H-tensor is less than （resp. equal to） one. In particular, we show that a quasi-diagonally dominant tensor is a nonsingular H-tensor if and only if all of its principal sub-tensors are nonsingular H-tensors. An irreducible tensor H is an H-tensor if and only if it is quasi-diagonally dominant.展开更多
We define the {i}-inverse （i = 1,2, 5） and group inverse of tensors based on a general product of tensors. We explore properties of the generalized inverses of tensors on solving tensor equations and computing formu...We define the {i}-inverse （i = 1,2, 5） and group inverse of tensors based on a general product of tensors. We explore properties of the generalized inverses of tensors on solving tensor equations and computing formulas of block tensors. We use the {1}-inverse of tensors to give the solutions of a multilinear system represented by tensors. The representations for the {1}-inverse and group inverse of some block tensors are established.展开更多
A class of structured multi-linear system defined by strong M_(z)-tensors is considered.We prove that the multi-linear system with strong M_(z)-tensors always has a nonnegative solution under certain condition by the ...A class of structured multi-linear system defined by strong M_(z)-tensors is considered.We prove that the multi-linear system with strong M_(z)-tensors always has a nonnegative solution under certain condition by the fixed point theory.We also prove that the zero solution is the only solution of the homogeneous multi-linear system for some structured tensors,such as strong M-tensors,H^(+)-tensors,strictly diagonally dominant tensors with positive diagonal elements.Numerical examples are presented to illustrate our theoretical results.展开更多
The main propose of this paper is devoted to study the solvability of the generalized order tensor complementarity problem.We define two problems:the generalized order tensor complementarity problem and the vertical t...The main propose of this paper is devoted to study the solvability of the generalized order tensor complementarity problem.We define two problems:the generalized order tensor complementarity problem and the vertical tensor comple-mentarity problem and show that the former is equivalent to the latter.Using the degree theory,we present a comprehensive analysis of existence,uniqueness and stability of the solution set of a given generalized order tensor complementarity problem.展开更多
The aim of this paper is to systematize solutions of some systems of linear equations in terms of generalized inverses.As a significant application of the Moore-Penrose inverse,the best approximation solution to linea...The aim of this paper is to systematize solutions of some systems of linear equations in terms of generalized inverses.As a significant application of the Moore-Penrose inverse,the best approximation solution to linear matrix equations(i.e.both least squares and the minimal norm)is considered.Also,characterizations of least squares solution and solution of minimum norm are given.Basic properties of the Drazin-inverse solution and the outer-inverse solution are present.Motivated by recent research,important least square properties of composite outer inverses are collected.展开更多
The 2014 International Conference on Tensors, Matrices and Their Applications （TMA 2014） was held at Suzhou University of Science and Technology （USTS）, Suzhou, China, December 17-19, 2014. The academic committee ...The 2014 International Conference on Tensors, Matrices and Their Applications （TMA 2014） was held at Suzhou University of Science and Technology （USTS）, Suzhou, China, December 17-19, 2014. The academic committee of TMA 2014 is co-chaired by Professors Richard A. Brauldi,展开更多
Bom on 8 August 1931,Sir Roger Penrose is a British mathematician,mathematical physicist,philosopher of science and Nobel Laureate in Physics.He is an Emeritus Rouse Ball Professor of Mathematics at the University of ...Bom on 8 August 1931,Sir Roger Penrose is a British mathematician,mathematical physicist,philosopher of science and Nobel Laureate in Physics.He is an Emeritus Rouse Ball Professor of Mathematics at the University of Oxford.展开更多
Since Kilmer et al.introduced the new multiplication method between two third-order tensors around 2008 and third-order tensors with such multiplication structure are also called as T-product tensors,T-product tensors...Since Kilmer et al.introduced the new multiplication method between two third-order tensors around 2008 and third-order tensors with such multiplication structure are also called as T-product tensors,T-product tensors have been applied to many fields in science and engineering,such as low-rank tensor approximation,signal processing,image feature extraction,machine learning,computer vision,and the multi-view clustering problem,etc.However,there are very few works dedicated to exploring the behavior of random T-product tensors.This work considers the problem about the tail behavior of the unitarily invariant norm for the summation of random symmetric T-product tensors.Majorization and antisymmetric Kronecker product tools are main techniques utilized to establish inequalities for unitarily norms of multivariate T-product tensors.The Laplace transform method is integrated with these inequalities for unitarily norms of multivariate T-product tensors to provide us with Bernstein bound estimation of Ky Fan k-norm for functions of the symmetric random T-product tensors summation.Finally,we also apply T-product Bernstein inequality to bound Ky Fan norm of covariance T-product tensor induced by hypergraph signal processing.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.11632011,11702170,11472170,51421092,and 11572189)
文摘A parallel nonlinear energy sink(NES) is proposed and analyzed. The parallel NES is composed of a vibro-impact(VI) NES and a cubic NES. The dynamical equation is given, and the essential analytical investigation is carried out to deal with the cubic nonlinearity and impact nonlinearity. Multiple time-scale expansion is introduced, and the zeroth order is derived to give a rough outline of the system. The underlying Hamilton dynamic equation is given, and then the optimal stiffness is expressed. The clearance is regarded as a critical factor for the VI. Based on the periodical impact treatment by analytical investigation, the relationships of the cubic stiffness, the clearance, and the zeroth-order attenuation amplitude of the linear primary oscillator(LPO) are obtained.A cubic NES under the optimal condition is compared with the parallel NES. Harmonic signals, harmonic signals with noises, and the excitation generated by a second-order?lter are considered as the potential excitation forces on the system. The targeted energy transfer(TET) in the designed parallel NES is shown to be more e?cient.
基金the National Natural Science Foundation of China(Grant No.11771099)the Hong Kong Research Grant Council(Grant Nos.PolyU 15302114,15300715,15301716 and 15300717)the Innovation Program of Shanghai Municipal Education Commission.
文摘In this paper,we investigate the tensor similarity and propose the T-Jordan canonical form and its properties.The concepts of the T-minimal polynomial and the T-characteristic polynomial are proposed.As a special case,we present properties when two tensors commute based on the tensor T-product.We prove that the Cayley-Hamilton theorem also holds for tensor cases.Then,we focus on the tensor decompositions:T-polar,T-LU,T-QR and T-Schur decompositions of tensors are obtained.When an F-square tensor is not invertible with the T-product,we study the T-group inverse and the T-Drazin inverse which can be viewed as the extension of matrix cases.The expressions of the T-group and T-Drazin inverses are given by the T-Jordan canonical form.The polynomial form of the T-Drazin inverse is also proposed.In the last part,we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverses can be given by a limit process.
基金the National Natural Science Foundation of China(Grant No.11771099)supported by the Hong Kong Research Grant Council(Grant Nos.PolyU 15302114,15300715,15301716,15300717)supported by the Innovation Program of Shanghai Municipal Education Commission。
文摘We investigate the M-eigenvalues of the Riemann curvature tensor in the higher dimensional conformally flat manifold.The expressions of Meigenvalues and M-eigenvectors are presented in this paper.As a special case,M-eigenvalues of conformal flat Einstein manifold have also been discussed,and the conformal the invariance of M-eigentriple has been found.We also reveal the relationship between M-eigenvalue and sectional curvature of a Riemannian manifold.We prove that the M-eigenvalue can determine the Riemann curvature tensor uniquely.We also give an example to compute the Meigentriple of de Sitter spacetime which is well-known in general relativity.
基金Supported by National Natural Science Foundation of China(Grant Nos.U1709210,51505430)Key Research and Development Project of Zhejiang Province(Grant No.2019C03108)+1 种基金Public Project of Zhejiang Province(Grant No.LGG18E050021)Research Foundation from Zhejiang Sci-Tech University(Grant No.15022013-Y)
文摘The vibration propagates through the shaft in the form of elastic waves. The propagation characteristics of the elastic waves are a ected by the axial loads. The influence of the axial loads to the propagation characteristics of the elastic waves is studied in this paper. Firstly, the transfer matrix of the elastic waves for the non-uniform shaft with axial loads is deduced by combining the transfer matrix without axial load and the additional equation caused by the axial load. And then, a numerical method is used to study the influence of the axial load, non-uniformity and the rotating speed to the propagation characteristics of the elastic waves. It’s found that a new Stop Band will appear due to the axial force, and the central frequency of which will decrease as the increase of the force, while the band width of which remains the same. The central frequency of the new Stop Band will also increase as the increase of the cross-section area ratio;however, the rotating speed of the shaft doesn’t a ect the propagation characteristics of the elastic waves obviously. Finally, an experimental rig is built up for further study, even though there are some small local errors, the results of experiments match well with the numerical ones, which indicates the validation of the theoretical results. The result can help to study the influence of the axial load to the dynamics of a non-uniform shaft and help to reveal the vibration propagating mechanism in such a shaft.
文摘We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the mill spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares （M） solutions to a multilineax system and establish the relationship between the minimum-norm （N） leastsquares （M）solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.
文摘We study the constrained systemof linear equations Ax=b,x∈R(A^(k))for A∈C^(n×n)and b∈Cn,k=Ind(A).When the system is consistent,it is well known that it has a unique A^(D)b.If the system is inconsistent,then we seek for the least squares solution of the problem and consider min_(x∈R(A^(k)))||b−Ax||2,where||·||2 is the 2-norm.For the inconsistent system with a matrix A of index one,it was proved recently that the solution is A^(■)b using the core inverse A^(■)of A.For matrices of an arbitrary index and an arbitrary b,we show that the solution of the constrained system can be expressed as A^(■)b where A^(■)is the core-EP inverse of A.We establish two Cramer’s rules for the inconsistent constrained least squares solution and develop several explicit expressions for the core-EP inverse of matrices of an arbitrary index.Using these expressions,two Cramer’s rules and one Gaussian elimination method for computing the core-EP inverse of matrices of an arbitrary index are proposed in this paper.We also consider the W-weighted core-EP inverse of a rectangular matrix and apply the weighted core-EP inverse to a more general constrained system of linear equations.
基金The first author is supported by the National Natural Science Foundation of China Under grant 10471027 Shanghai Education＇Committee. The third author is partially supported by Natural ScienceEngineering Research Council of Canada and supported by Shanghai Key Laboratory of Contemporary Applied Mathematics of Fudan University during Sanzheng Qiao＇s visit.
文摘In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem： minx ｜｜Ax- b｜｜2, where A is an m-by-n （m ≥ n） rank deficient matrix. We first derive an explicit expression for the condition number in the weighted Frobenius norm ｜｜ [AT,βb] ｜｜F of the data A and b, where T is a positive diagonal matrix and β is a positive scalar. We then discuss the sensitivity of the standard 2-norm condition numbers for the generalized matrix inversion and rank deficient least squares and establish relations between the condition numbers and their condition numbers called level-2 condition numbers.
基金supported by the National Natural Science Foundation of China under grant No.12271108Shanghai Municipal Science and Technology Commission under grant No.22WZ2501900Innovation Program of Shanghai Municipal Education Commission
文摘In this work,we try to build a theory for random double tensor integrals(DTI).We begin with the definition of DTI and discuss how randomness structure is built upon DTI.Then,the tail bound of the unitarily invariant norm for the random DTI is established and this bound can help us to derive tail bounds of the unitarily invariant norm for various types of two tensors means,e.g.,arithmetic mean,geometric mean,harmonic mean,and general mean.By associating DTI with perturbation formula,i.e.,a formula to relate the tensor-valued function difference with respect the difference of the function input tensors,the tail bounds of the unitarily invariant norm for the Lipschitz estimate of tensor-valued function with random tensors as arguments are derived for vanilla case and quasi-commutator case,respectively.We also establish the continuity property for random DTI in the sense of convergence in the random tensor mean,and we apply this continuity property to obtain the tail bound of the unitarily invariant norm for the derivative of the tensor-valued function.
基金Acknowledgements The authors would like to thank Professors Liqun Qi and Yiju Wang for their comments and the preprint [14]. They would like to thank two referees for their detailed suggestions which greatly improve the presentation. They also thank Prof. Liqun Qi for kindly reminding them of the very recent paper [12] after their first revision in February, 2015. This work was supported by the National Natural Science Foundation of China （Grant Nos. 11171371, 11271084.）
文摘The H-matrices are an important class in the matrix theory, and have many applications. Recently, this concept has been extended to higher order H-tensors. In this paper, we establish important properties of diagonally dominant tensors and H-tensors. Distributions of eigenvalues of nonsingular symmetricH-tensors are given. An J（t%-tensor is semi-positive, which enlarges the area of semi-positive tensor from H-tensor to H＋-tensor. The spectral radius of Jacobi tensor of a nonsingular （resp. singular） H-tensor is less than （resp. equal to） one. In particular, we show that a quasi-diagonally dominant tensor is a nonsingular H-tensor if and only if all of its principal sub-tensors are nonsingular H-tensors. An irreducible tensor H is an H-tensor if and only if it is quasi-diagonally dominant.
基金The authors are very grateful to the referees for their valuable suggestions, which have considerably improved the paper. Yimin Wei was supported by the International Cooperation Project of Shanghai Municipal Science and Technology Commission （Grant No. 16510711200） and the National Natural Science Foundation of China （Grant No. 11771099） Changjiang Bu was supported by the National Natural Science Foundation of China （Grant No. 11371109）.
文摘We define the {i}-inverse （i = 1,2, 5） and group inverse of tensors based on a general product of tensors. We explore properties of the generalized inverses of tensors on solving tensor equations and computing formulas of block tensors. We use the {1}-inverse of tensors to give the solutions of a multilinear system represented by tensors. The representations for the {1}-inverse and group inverse of some block tensors are established.
基金C.Mo is supported in part by Promotion Program of Excellent Doctoral Research,Fudan University(SSH6281011/001)Y.Wei is supported by National Natural Science Foundations of China under grant 11771099Innovation Program of Shanghai Mu-nicipal Education Commission.
文摘A class of structured multi-linear system defined by strong M_(z)-tensors is considered.We prove that the multi-linear system with strong M_(z)-tensors always has a nonnegative solution under certain condition by the fixed point theory.We also prove that the zero solution is the only solution of the homogeneous multi-linear system for some structured tensors,such as strong M-tensors,H^(+)-tensors,strictly diagonally dominant tensors with positive diagonal elements.Numerical examples are presented to illustrate our theoretical results.
基金The first author is supported by the Fundamental Research Funds for the Central Universities under grant No.JBK1801058Partial work is fin-ished during the author’s visiting at Shanghai Key Laboratory of Contemporary Ap-plied Mathematics+2 种基金The second author is supported by the Hong Kong Research Grant Council(Grant Nos.PolyU 501913,15302114,15300715 and 15301716)The third author is supported by the National Natural Science Foundation of China under grant No.11771099Innovation Program of Shanghai Municipal Education Commission.We would like to thank the editor and two anonymous reviewers for very helpful com-ments.
文摘The main propose of this paper is devoted to study the solvability of the generalized order tensor complementarity problem.We define two problems:the generalized order tensor complementarity problem and the vertical tensor comple-mentarity problem and show that the former is equivalent to the latter.Using the degree theory,we present a comprehensive analysis of existence,uniqueness and stability of the solution set of a given generalized order tensor complementarity problem.
基金P.S.Stanimirovic was supported by the Ministry of Education and Science,Republic of Serbia(Grant 174013/451-03-9/2021-14/200124)D.Mosic was supported by the Ministry of Education,Science and Technological Development,Republic of Serbia(Grant 174007/451-03-9/2021-14/200124)Y.Wei was supported by the bilateral project between China and Serbia The theory of tensors,operator matrices and applications(no.4-5)'.
文摘The aim of this paper is to systematize solutions of some systems of linear equations in terms of generalized inverses.As a significant application of the Moore-Penrose inverse,the best approximation solution to linear matrix equations(i.e.both least squares and the minimal norm)is considered.Also,characterizations of least squares solution and solution of minimum norm are given.Basic properties of the Drazin-inverse solution and the outer-inverse solution are present.Motivated by recent research,important least square properties of composite outer inverses are collected.
文摘The 2014 International Conference on Tensors, Matrices and Their Applications （TMA 2014） was held at Suzhou University of Science and Technology （USTS）, Suzhou, China, December 17-19, 2014. The academic committee of TMA 2014 is co-chaired by Professors Richard A. Brauldi,
文摘Bom on 8 August 1931,Sir Roger Penrose is a British mathematician,mathematical physicist,philosopher of science and Nobel Laureate in Physics.He is an Emeritus Rouse Ball Professor of Mathematics at the University of Oxford.
基金supported by Innovation Program of Shanghai Municipal Education Commissionthe National Natural Science Foundation of China under grant No.11771099
文摘Since Kilmer et al.introduced the new multiplication method between two third-order tensors around 2008 and third-order tensors with such multiplication structure are also called as T-product tensors,T-product tensors have been applied to many fields in science and engineering,such as low-rank tensor approximation,signal processing,image feature extraction,machine learning,computer vision,and the multi-view clustering problem,etc.However,there are very few works dedicated to exploring the behavior of random T-product tensors.This work considers the problem about the tail behavior of the unitarily invariant norm for the summation of random symmetric T-product tensors.Majorization and antisymmetric Kronecker product tools are main techniques utilized to establish inequalities for unitarily norms of multivariate T-product tensors.The Laplace transform method is integrated with these inequalities for unitarily norms of multivariate T-product tensors to provide us with Bernstein bound estimation of Ky Fan k-norm for functions of the symmetric random T-product tensors summation.Finally,we also apply T-product Bernstein inequality to bound Ky Fan norm of covariance T-product tensor induced by hypergraph signal processing.
