T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

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.

LU Invariants and Canonical Forms and SLOCC Classification of Pure 3-Qubit States

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.

CONTROLLABILITY CANONICAL FORM FOR NONLINEAR CONTROL SYSTEMS

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.

GH-CONGRUENCE CANONICAL FORM OF QUATERNION MATRIX AND SIMULTANEOUS DIAGONALIZATION OF MATRICES

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.

AN ALGORITHM FOR JORDAN CANONICAL FORM OF A QUATERNION MATRIX

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.

The Construction of Locally D-Optimal Designs by Canonical Forms to an Extension for the Logistic Model

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.

Form Invariance of Raitzin＇s Canonical Equations of Relativistic Mechanical System

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.

ON THE DECOMPOSITION OF COMPLEX VECTOR SPACES AND THE JORDAN CANONICAL FORM OF COMPLEX LINEAR TRANSFORMATIONS

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.

Quasi-Rational Canonical Forms of a Matrix over a Number Field

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.

A Minimality of the Rational Canonical Form

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.

Notes on Canonical Forms of Integrable Vector Nonlinear Schr?dinger Systems

We present canonical forms of integrable vector nonlinear Sehr6dinger systems. Mathematically, it is enough to focus on these canonical forms.

异文形态与择调观念:《玉京秋》清代接受考论

《玉京秋》为宋末周密的自创调,由于周密词集版本较多以及清代词选、词律、词谱中的《玉京秋》多有异文,以至于清人所填《玉京秋》多有不同,并且发生争议。这一历史过程,涉及周密词集在明清时期的传播、相关词选与律谱间的交互影响、清人填词创作时的文本依据等诸多问题,而不同文献所载《玉京秋》之间的异文、清人创作《玉京秋》时的不同体制以及对待《词律》与《钦定词谱》、《草窗词》与《蘋洲渔笛谱》的不同态度、不同选择,也为考察清人的填词过程、词体观念以及《玉京秋》的经典化过程提供了很好的观察视角。

William E.Roth s第二定理的一个新证明

仅利用矩阵的基本理论建立了四个引理,并且在此基础上对W.E.Roth第二定理给出了新的证明.

M-Eigenvalues of the Riemann Curvature Tensor of Conformally Flat Manifolds

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.

ORDERED ANALYTIC REPRESENTATION OF PDES BY HAMILTONIAN CANONICAL SYSTEM

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

Huaitang New Nine Forms of Needles of China Start a New Era of Acupuncture Therapy

"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

Huaitang New Nine Forms of Needles of China Start a New Era of Acupuncture Therapy

"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

Huaitang New Nine Forms of Needles of China Start a New Era of Acupuncture Therapy

'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.

《法显传》与行传体的创立

记载晋唐僧人西域南海巡礼求法的旅行传记专书,以成书于东晋末的《法显传》年代最早,文体最端正。其在文学上的主要功绩,在于行传体的创立和纪行之书六朝范式的确立。自从有了《法显传》,后出诸书就纷纷仿效,直到北宋初,旅行传记仍代有作者,而其最早的文体源头,即是东晋高僧法显的《法显传》。此书乃纪行之书,所用行传体属古行记系列,对此书这两个方面意义的揭示,自然也离不开古行记的文体史研究和辨体研究。

Hurwitz-Radon矩阵方程的华罗庚链

1947年,华罗庚独立研究了Hurwitz-Radon矩阵方程,遗憾的是由于失误他未能得出完整结果。文章补充完善了华罗庚对于Hurwitz-Radon矩阵方程的求解。遵循华罗庚的思路,所采用的方法是矩阵的相似标准型。得到了华罗庚链,并通过酉化技巧得到其酉版本。