In this paper the entanglement of pure 3-qubit states is discussed. The local unitary （LU） polynomial invariants that are closely related to the canonical forms are constructed and the relations of the coefficients ...In this paper the entanglement of pure 3-qubit states is discussed. The local unitary （LU） polynomial invariants that are closely related to the canonical forms are constructed and the relations of the coefficients of the canonical forms are given. Then the stochastic local operations and classlcal communication （SLOCC） classification of the states are discussed on the basis of the canonical forms, and the symmetric canonical form of the states without 3-tangle is discussed. Finally, we give the relation between the LU polynomial invariants and SLOCC classification.展开更多
General nonlinear control systems are studied in this paper with the goal to transform them into the so-called controllability canonteal form via state transformation only. The conditions of transformability are given...General nonlinear control systems are studied in this paper with the goal to transform them into the so-called controllability canonteal form via state transformation only. The conditions of transformability are given for both single input and multiple input cases. Besides, by an algebraic approach the procedure for constructing the state transformation is established. This paper is formulated in the framework of calculus rather than differential geometry approach.展开更多
In this paper,the GH-congruence canonical forms of positive semidefinite and definte inite and definite(need not be self-conjugate)quaternion matrices are given,and a neccessary and sufficientcondition of GH-congruenc...In this paper,the GH-congruence canonical forms of positive semidefinite and definte inite and definite(need not be self-conjugate)quaternion matrices are given,and a neccessary and sufficientcondition of GH-congruence for two positive semidifinite(definite)quaternion matrices isgiven also.Then simultaneous GH-congruence reduced forms for two self-conjugate matri-ces and some result about the simultaneous GH-congruence diagonalization of quaternionmatrices are obtained.展开更多
In this paper, we first introduce a concept of companion vector, and studythe Jordan canonical forms of quaternion matrices by using the methods of complex representation and companion vector, not only give out a prac...In this paper, we first introduce a concept of companion vector, and studythe Jordan canonical forms of quaternion matrices by using the methods of complex representation and companion vector, not only give out a practical algorithm for Jordancanonical form J of a quaternion matrix A, but also provide a practical algorithm forcorresponding nonsingular matrix P with P- 1 AP = J.展开更多
Logistic regression models for binary response problems are present in a wide variety of industrial, biological, social and medical experiments;therefore, optimum designs are a valuable tool for experimenters, leading...Logistic regression models for binary response problems are present in a wide variety of industrial, biological, social and medical experiments;therefore, optimum designs are a valuable tool for experimenters, leading to estimators of parameters with minimum variance. Our interest in this contribution is to provide explicit formulae for the D-optimal designs as a function of the unknown parameters for the logistic model where q is an indicator variable. We have considered an experiment based on the dose-response to a fly insecticide in which males and females respond in different ways, proposed in Atkinson et al. (1995) [1]. To find the D-optimal designs, this problem has been reduced to a canonical form.展开更多
A form invariance of Raitzin's canonical equations of relativistie mechanical system is studied. First, the Raitzin's canonical equations of the system are established. Next, the definition and criterion of th...A form invariance of Raitzin's canonical equations of relativistie mechanical system is studied. First, the Raitzin's canonical equations of the system are established. Next, the definition and criterion of the form invariance in the system under infinitesimal transformations of groups are given. Finally,the relation between the form invariance and the conserved quantity of the system is obtained and an example is given to illustrate the application of the result.展开更多
New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations a...New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations and provide all Jordan basesby which the Jordan canonical form is constructed. Accordingly, they can result in thecelebrated Jordan theorem and the third decomposition theorem of space directly. and,moreover, they can give a new deep insight into the exquisite and subtle structure ofthe Jordan form. The latter indicates that the Jordan canonical form of a complexlinear transformation is an invariant structure associated with double arbitrary. choices.展开更多
A matrix is similar to Jordan canonical form over the complex field and the rational canonical form over a number field, respectively. In this paper, we further study the rational canonical form of a matrix over any n...A matrix is similar to Jordan canonical form over the complex field and the rational canonical form over a number field, respectively. In this paper, we further study the rational canonical form of a matrix over any number field. We firstly discuss the elementary divisors of a matrix over a number field. Then, we give the quasi-rational canonical forms of a matrix by combining Jordan and the rational canonical forms. Finally, we show that a matrix is similar to its quasi-rational canonical forms over a number field.展开更多
The rational canonical form theorem is very essential basic result of matrix theory, which has been proved by different methods in the literature. In this note, we provide an effcient direct proof, from which the mini...The rational canonical form theorem is very essential basic result of matrix theory, which has been proved by different methods in the literature. In this note, we provide an effcient direct proof, from which the minimality for the decomposition of the rational canonical form can be found.展开更多
Based on the method of symplectic geometry and variational calculation,the method for some PDEs to be ordered and analytically represented by Hamiltonian canonical system is discussed.Meanwhile some related necessar...Based on the method of symplectic geometry and variational calculation,the method for some PDEs to be ordered and analytically represented by Hamiltonian canonical system is discussed.Meanwhile some related necessary and sufficient conditions are obtained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
In this paper,the authors aim to study Kronecker canonical form theory for T-type digraphs,which can be used to construct trees by tensor product with some directed paths.Firstly,the authors show that some bicyclic di...In this paper,the authors aim to study Kronecker canonical form theory for T-type digraphs,which can be used to construct trees by tensor product with some directed paths.Firstly,the authors show that some bicyclic digraphs and multicyclic digraphs are T-type digraphs.Secondly,the authors provide a characterization for T-type digraphs by their Kronecker canonical form.Moreover,the authors present an algorithm for computing the Kronecker canonical form,which can be used to determine whether or not a digraph is a T-type digraph.Lastly,for a class of T-type digraphs,the authors show that their incidence matrix pair can be transformed into Kronecker canonical form using unimodular matrices.The authors also present an algorithm related to finding such unimodular matrices.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
"Nine forms of needles"were the nine kinds of ancientacupuncture tools,which were made of different stones and met-als in China,firstly recorded in the anctient monumental work,The Yellow Emperor’s Canon of..."Nine forms of needles"were the nine kinds of ancientacupuncture tools,which were made of different stones and met-als in China,firstly recorded in the anctient monumental work,The Yellow Emperor’s Canon of Internal Medicine(722-221 B.C.).After that,except the filiform needle,the others had beenlost gradually.Fortunately,the eminent clinical acupuncturist展开更多
"Nine forms of needles"were the nine kinds of ancientacupuncture tools,which were made of different stones and met-als in China,firstly recorded in the anctient monumental work,The Yellow Emperor’s Canon of..."Nine forms of needles"were the nine kinds of ancientacupuncture tools,which were made of different stones and met-als in China,firstly recorded in the anctient monumental work,The Yellow Emperor’s Canon of Internal Medicine(722-221 B.C.).After that,except the filiform needle,the others had beenlost gradually.Fortunately,the eminent clinical acupuncturist inChina,world well-known scholar of acupuncture,director of theInstitute of Shanxi Huaitang Nine Forms of Needles,chief Dr Shihuaitang has developed new nine forms of needles using mordenmedicine,morden science and technologey on the basiss ofstuding on the ancient nine forms of needles,and estabished u-nique new nine forms of needles therapies.These therapies展开更多
'Nine forms of needles' were the nine kinds of ancientacupuncture tools, Which were made of different stones and met-als in China, firstly recorded in the anctient monumental work,The Yellow Emperor’s Canon o...'Nine forms of needles' were the nine kinds of ancientacupuncture tools, Which were made of different stones and met-als in China, firstly recorded in the anctient monumental work,The Yellow Emperor’s Canon of Internal Medicine(722-221 B.C.). After that, except the filiform needle, the others had beenlost gradually. Fortunately, the eminent clinical acupuncturist inChina, world well-known scholar of acupuncture, director of theInstitute of Shanxi Huaitang Nine Forms of Needles, chief Dr Shihuaitang has developed new nine forms of needles using mordenmedicine, morden science and technologey on the basiss ofstuding on the ancient nine forms of needles, and estabished u-nique new nine forms of needles therapies. These therapies haveextended the indications of acupuncture and improved the thera-peutic effect.展开更多
基金The project supported by National Natural Science Foundation of China under Grant No. 6J3433050 and the Natural Science Foundation of Xuzhou Normal University （Key Project） under Grant No. 03XLA04
文摘In this paper the entanglement of pure 3-qubit states is discussed. The local unitary （LU） polynomial invariants that are closely related to the canonical forms are constructed and the relations of the coefficients of the canonical forms are given. Then the stochastic local operations and classlcal communication （SLOCC） classification of the states are discussed on the basis of the canonical forms, and the symmetric canonical form of the states without 3-tangle is discussed. Finally, we give the relation between the LU polynomial invariants and SLOCC classification.
文摘General nonlinear control systems are studied in this paper with the goal to transform them into the so-called controllability canonteal form via state transformation only. The conditions of transformability are given for both single input and multiple input cases. Besides, by an algebraic approach the procedure for constructing the state transformation is established. This paper is formulated in the framework of calculus rather than differential geometry approach.
文摘In this paper,the GH-congruence canonical forms of positive semidefinite and definte inite and definite(need not be self-conjugate)quaternion matrices are given,and a neccessary and sufficientcondition of GH-congruence for two positive semidifinite(definite)quaternion matrices isgiven also.Then simultaneous GH-congruence reduced forms for two self-conjugate matri-ces and some result about the simultaneous GH-congruence diagonalization of quaternionmatrices are obtained.
基金Supported by the National Natural Science Foudnation of China and Shanghai Priority Academic Discipline Foundation,Shanghai,China.
文摘In this paper, we first introduce a concept of companion vector, and studythe Jordan canonical forms of quaternion matrices by using the methods of complex representation and companion vector, not only give out a practical algorithm for Jordancanonical form J of a quaternion matrix A, but also provide a practical algorithm forcorresponding nonsingular matrix P with P- 1 AP = J.
文摘Logistic regression models for binary response problems are present in a wide variety of industrial, biological, social and medical experiments;therefore, optimum designs are a valuable tool for experimenters, leading to estimators of parameters with minimum variance. Our interest in this contribution is to provide explicit formulae for the D-optimal designs as a function of the unknown parameters for the logistic model where q is an indicator variable. We have considered an experiment based on the dose-response to a fly insecticide in which males and females respond in different ways, proposed in Atkinson et al. (1995) [1]. To find the D-optimal designs, this problem has been reduced to a canonical form.
文摘A form invariance of Raitzin's canonical equations of relativistie mechanical system is studied. First, the Raitzin's canonical equations of the system are established. Next, the definition and criterion of the form invariance in the system under infinitesimal transformations of groups are given. Finally,the relation between the form invariance and the conserved quantity of the system is obtained and an example is given to illustrate the application of the result.
文摘New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations and provide all Jordan basesby which the Jordan canonical form is constructed. Accordingly, they can result in thecelebrated Jordan theorem and the third decomposition theorem of space directly. and,moreover, they can give a new deep insight into the exquisite and subtle structure ofthe Jordan form. The latter indicates that the Jordan canonical form of a complexlinear transformation is an invariant structure associated with double arbitrary. choices.
文摘A matrix is similar to Jordan canonical form over the complex field and the rational canonical form over a number field, respectively. In this paper, we further study the rational canonical form of a matrix over any number field. We firstly discuss the elementary divisors of a matrix over a number field. Then, we give the quasi-rational canonical forms of a matrix by combining Jordan and the rational canonical forms. Finally, we show that a matrix is similar to its quasi-rational canonical forms over a number field.
文摘The rational canonical form theorem is very essential basic result of matrix theory, which has been proved by different methods in the literature. In this note, we provide an effcient direct proof, from which the minimality for the decomposition of the rational canonical form can be found.
基金Supported in part by the National Natural Science Foundation of China (1 0 0 71 0 2 1 ) the Foundationfor University Key Teacher by MEC and Shanghai Priority Academic Discipline Foundation
文摘Based on the method of symplectic geometry and variational calculation,the method for some PDEs to be ordered and analytically represented by Hamiltonian canonical system is discussed.Meanwhile some related necessary and sufficient conditions are obtained
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supported by the National Natural Science Foundation of China under Grant Nos.11001064,11601105,and 11871175。
文摘In this paper,the authors aim to study Kronecker canonical form theory for T-type digraphs,which can be used to construct trees by tensor product with some directed paths.Firstly,the authors show that some bicyclic digraphs and multicyclic digraphs are T-type digraphs.Secondly,the authors provide a characterization for T-type digraphs by their Kronecker canonical form.Moreover,the authors present an algorithm for computing the Kronecker canonical form,which can be used to determine whether or not a digraph is a T-type digraph.Lastly,for a class of T-type digraphs,the authors show that their incidence matrix pair can be transformed into Kronecker canonical form using unimodular matrices.The authors also present an algorithm related to finding such unimodular matrices.
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.
文摘"Nine forms of needles"were the nine kinds of ancientacupuncture tools,which were made of different stones and met-als in China,firstly recorded in the anctient monumental work,The Yellow Emperor’s Canon of Internal Medicine(722-221 B.C.).After that,except the filiform needle,the others had beenlost gradually.Fortunately,the eminent clinical acupuncturist
文摘"Nine forms of needles"were the nine kinds of ancientacupuncture tools,which were made of different stones and met-als in China,firstly recorded in the anctient monumental work,The Yellow Emperor’s Canon of Internal Medicine(722-221 B.C.).After that,except the filiform needle,the others had beenlost gradually.Fortunately,the eminent clinical acupuncturist inChina,world well-known scholar of acupuncture,director of theInstitute of Shanxi Huaitang Nine Forms of Needles,chief Dr Shihuaitang has developed new nine forms of needles using mordenmedicine,morden science and technologey on the basiss ofstuding on the ancient nine forms of needles,and estabished u-nique new nine forms of needles therapies.These therapies
文摘'Nine forms of needles' were the nine kinds of ancientacupuncture tools, Which were made of different stones and met-als in China, firstly recorded in the anctient monumental work,The Yellow Emperor’s Canon of Internal Medicine(722-221 B.C.). After that, except the filiform needle, the others had beenlost gradually. Fortunately, the eminent clinical acupuncturist inChina, world well-known scholar of acupuncture, director of theInstitute of Shanxi Huaitang Nine Forms of Needles, chief Dr Shihuaitang has developed new nine forms of needles using mordenmedicine, morden science and technologey on the basiss ofstuding on the ancient nine forms of needles, and estabished u-nique new nine forms of needles therapies. These therapies haveextended the indications of acupuncture and improved the thera-peutic effect.
