The Necessary and Sufficient Conditions for Two Kinds of Matrix Equtions Having Semipositive Subdefinite Solutions

Finding solutions of matrix equations in given set SR n×n is an active research field. Lots of investigation have done for these cases, where S are the sets of general or symmetric matrices and symmetric positive definite or sysmmetric semiposite definite matrices respectively . Recently, however, attentions are been paying to the situation for S to be the set of general(semi) positive definite matrices(called as semipositive subdefinite matrices below) . In this paper the necessary and sufficient conditions for the following two kinds of matrix equations having semipositive, subdefinite solutions are obtained. General solutions and symmetric solutions of the equations (Ⅰ) and (Ⅱ) have been considered in in detail.

POSITIVE DEFINITE KERNEL IN SUPPORT VECTOR MACHINE(SVM)

The relationship among Mercer kernel, reproducing kernel and positive definite kernel in support vector machine （SVM） is proved and their roles in SVM are discussed. The quadratic form of the kernel matrix is used to confirm the positive definiteness and their construction. Based on the Bochner theorem, some translation invariant kernels are checked in their Fourier domain. Some rotation invariant radial kernels are inspected according to the Schoenberg theorem. Finally, the construction of discrete scaling and wavelet kernels, the kernel selection and the kernel parameter learning are discussed.

A Remark on the Positive Definite Problem of a Binary Quartic Form

In this paper,we discussed the positive definite problem of binary quartic forms and obtained a necessary and sufficient conditions.

SYMMETRIC POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATIONS (AX,XB)=(C,D) AND AXB=C

The symmetric positive definite solutions of matrix equations (AX,XB)=(C,D) and AXB=C are considered in this paper. Necessary and sufficient conditions for the matrix equations to have symmetric positive definite solutions are derived using the singular value and the generalized singular value decompositions. The expressions for the general symmetric positive definite solutions are given when certain conditions hold.

The Classification of Positive Definite Unimodular Lattices Over Z[(1+21^(1/2))/2]

In this paper, the author applies adjacent lattice method and Siegel mass formula to determine the classes of positive definite unimodular lattices of rank 4 over Z , and obtains that the class number of unit genus gen( I 4 ) is nine and the class number of even unimodular lattices is three, and also gives the representative lattices of each class.

Completely Positive Definite Maps Over Topological-algebras

In this paper, we discuss completely positive definite maps over topological algebras. A Schwarz type inequality for n-positive definite maps, and the Stinespring representation theorem for completely positive definite maps over topological algebras are given.

The Classification of Positive Definite Unimodular Lattices Over Z[(1+211/2)/2]

In this paper, the author applies adjacent lattice method and Siegel mass formula to determine the classes of positive definite unimodular lattices of rank 4 over Z , and obtains that the class number of unit genus gen( I 4 ) is nine and the class number of even unimodular lattices is three, and also gives the representative lattices of each class.

Study on the Relation of Definite Integral and Indefinite Integral

Relation of definite integral and indefinite integral was discussed and an important result was gotten. If f(x) is bounded and has primary function, the formal definite integral x s f(t)dt is the indefinite integral of f(x), where x is a self-variable, s is a parameter,~f(x) is a function defined in(-∞, +∞), which comes from f(x) by restriction and extension. In other words, the indefinite integral is a special form of definite integral, its lower integral limit and upper integral limit are all indefinite.

The Metapositive Definite Self－Conjugate Solution of the Matrix Equation AXB＝C over a Skew Field

Let F be the strong p-division ring [4]. This paper is sequel to [1]. Metapositive definite self-conjugate matrix over F is defined and the necessary and sufficient conditions for determining whether a partitioned matrix over F is metapositive definite self-conjugate are given.Moreover,a decomposition of pairwise matrices over F with the same numbers of columns is also presented. Whence some necessary and sufficient conditions for the existence of and the explicit expression for the metapositive definite self-conjugate solution of the matrix equation AXB=C over F are derived.

Some new bound estimates of the Hermitian positive definite solutions of the nonlinear matrix equation X^s+ A~*X^(-t) A = Q

The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian positive definite matrix and parameters s,t>0. Based on the matrix geometry theory, relevant matrix inequality and linear algebra technology, according to the different value ranges of the parameters s,t, the existence intervals of the Hermitian positive definite solution and the necessary conditions for equation solvability are presented, respectively. Comparing the existing correlation results, the proposed upper and lower bounds of the Hermitian positive definite solution are more accurate and applicable.

Two Structure-Preserving-Doubling Like Algorithms to Solve the Positive Definite Solution of the Equation X-A^(H)X^(-1)A=Q

In this paper,we study the nonlinear matrix equation X-A^(H)X^(-1)A=Q,where A,Q∈C^(n×n),Q is a Hermitian positive definite matrix and X∈C^(n×n)is an unknown matrix.We prove that the equation always has a unique Hermitian positive definite solution.We present two structure-preserving-doubling like algorithms to find the Hermitian positive definite solution of the equation,and the convergence theories are established.Finally,we show the effectiveness of the algorithms by numerical experiments.

DIAGONALLY COMPENSATED REDUCTION AND MULTISPLITTING OF A SYMMETRIC POSITIVE DEFINITE MATRIX

To solve the symmetric positive definite linear system Ax = b on parallel and vector machines, multisplitting methods are considered. Here the s.p.d. (symmetric positive definite) matrix A need not be assumed in a special form (e.g. the dissection form [11]). The main tool for deriving our methods is the diagonally compensated reduction (cf. [1]). The convergence of such methods is also discussed by using this tool. [WT5,5”HZ]

SEMI-DEFINITE RELAXATION ALGORITHM OF MULTIPLE KNAPSACK PROBLEM

The multiple knapsack problem denoted by MKP (B,S,m,n) can be defined as fol- lows.A set B of n items and a set Sof m knapsacks are given such thateach item j has a profit pjand weightwj,and each knapsack i has a capacity Ci.The goal is to find a subset of items of maximum profit such that they have a feasible packing in the knapsacks.MKP(B,S,m,n) is strongly NP- Complete and no polynomial- time approximation algorithm can have an approxima- tion ratio better than0 .5 .In the last ten years,semi- definite programming has been empolyed to solve some combinatorial problems successfully.This paper firstly presents a semi- definite re- laxation algorithm (MKPS) for MKP (B,S,m,n) .It is proved that MKPS have a approxima- tion ratio better than 0 .5 for a subclass of MKP (B,S,m,n) with n≤ 1 0 0 ,m≤ 5 and maxnj=1{ wj} minmi=1{ Ci} ≤ 2 3 .

Thermal Sense Prediction Model for Definite Human Group in Indoor Case

In order to build a prediction model of the indoor thermal comfort for a given human group, the original predicted mean vote （PMV） equation is reconstructed and simplified, the modified PMV equation is named PMVR （PMV for region） , where five variables are needed to be fitted with the dataset of actual thermal sense of a definite human group. As the fitting algorithm, the particle swarm optimization algorithm is used to optimize the solution, and a fixed PMVR can be finally determined. Experiment results indicate that for a definite human group, PMVR is more accurate on the prediction of thermal sense compared with some other models.

ON RUBIN'S HARMONIC ANALYSIS AND ITS RELATED POSITIVE DEFINITE FUNCTIONS

In this paper,a new formulation of the Rubin’s q-translation is given,which leads to a reliable q-harmonic analysis.Next,related q-positive definite functions are introduced and studied,and a Bochner’s theorem is proved.

THE PARALLEL MULTISPLITTING METHOD FOR CONSISTENT SYMMETRIC POSITIVE(SEMI-)DEFINITE SYSTEMS

Main resultsTheorem 1 Let A be an n×n symmetric positive semidefinite matrix and let

Some Notes on the Positive Definite Problem of a Binary Quartic Form

In this paper, we discuss the positive definite problem of a binary quartic form and obtain a necessary and sufficient condition. In addition we give two examples to show that there are some errors in the paper [1].

DEFINITE PURSUITS IN AN INDEFINITE ERA

Today’s world has witnessed economic changes,political upheavals and restructuring of international relations.Europe since 2008 has experienced many problems including those from Greek debt crisis to euro zone crisis,from ownership of Crimea to Ukraine crisis,from long-lasting illegal migra-

Hermite Positive Definite Solution of the Quaternion Matrix Equation Xm + B*XB = C

This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation Xm+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .

An Analysis of the Misuse of Definite Article THE of English Majors

The acquisition and usage of definite article THE has always gained attention from scholars from domestic and overseas due to its difficulty of learning and mastering. As a result, the relative study of definite article THE acquisition has been one of the the most significant tasks of second language acquisition all the time. For English majors, the acquisition of definite articles THE runs through the whole process of language learning and application. Learner＇ s version of foreign language, known as interlanguage, is a kind of language that exists between the mother tongue （Chinese） and the target language （English）. Interlanguage is a systematic unit which changes and develops constantly. Since definite article THE has rather complex and frequently altered usage, it is inevitable that there are a throng of misuse and problems for English majors when they are using definite article THE. This paper intends to do a survey focusing on English majors＇ application situation of definite article THE. Through analyzing the common misuse and mistakes appeared when English majors are using definite article THE, this paper aims to summerize the FAQ （frequently asked questions） and causes of those misuses. At the same time, the 100 experimental subjects of this survey are equally and randomly chosen from freshmen and third year students of foreign languages college, OUC （Ocean University of China）. By comparing the experimental results of two different grade English major students, this paper aims to demonstrate the influence of different grades on the mastering of definite articles THE of English majors.