comparisons of nonnegative splittings of matrices abstract the comparison of the asymptotic rates of convergence of two iteration matrices induced by two splitting of the same matrix has arisen in the work of many authors in this article some new comparison theorems for two nonnegative splittings a splitting a m n is nonnegative if m 1 exist and m 1 n is nonnegative are derived they extend the known results in the literature in addition we also point out three incorrect conditions in a paper by beauwens furthermore we give some reasonable conditions ensuring the strict inequality between the asymptotic convergence rates this also answers the open question which additional and appropriate conditions should be imposed on miller neumann splittings to obtain strict inequality finally some applications to a class of generalized aor sor and jor iterative methods whose special cases imply block also point aor sor and jor iterative methods for solving linear systems are discussed

comments on explicit criterion for the positive definiteness of a general quartic form the purpose of this note is to point out that the result in the above paper w h ku ieee trans autom control vol ac 10 no 3 pp 372 373 jul 1965 is incomplete and to give a complete and improved result

z eigenvalue methods for a global polynomial optimization problem as a global polynomial optimization problem the best rank one approximation to higher order tensors has extensive engineering and statistical applications different from traditional optimization solution methods in this paper we propose some z eigenvalue methods for solving this problem we first propose a direct z eigenvalue method for this problem when the dimension is two in multidimensional case by a conventional descent optimization method we may find a local minimizer of this problem then by using orthogonal transformations we convert the underlying supersymmetric tensor to a pseudo canonical form which has the same e eigenvalues and some zero entries based upon these we propose a direct orthogonal transformation z eigenvalue method for this problem in the case of order three and dimension three in the case of order three and higher dimension we propose a heuristic orthogonal transformation z eigenvalue method by improving the local minimum with the lower dimensional z eigenvalue methods and a heuristic cross hill z eigenvalue method by using the two dimensional z eigenvalue method to find more local minimizers numerical experiments show that our methods are efficient and promising

letter to the editor comment on a comparison theorem of the sor iterative method in this work we point out that there are incorrect assertions in the article by li ying sun li ying sun a comparison theorem for the sor iterative method j comput appl math 181 2005 336 341 moreover we answer an open problem

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

a procedure for the positive definiteness of forms of even order a positivity method for forms of even order is reported it is shown that testing for positivity of such forms can be reduced to a test for positive definiteness of parametric matrices special emphasis is placed on quartic and sixth order forms examples are given to illustrate the proposed criterion

positive cones and convergence conditions for iterative methods based on splittings the study of convergence conditions to solve large and sparse linear systems ax b by iterative methods has been discussed by many authors in this paper by using the partial order induced by positivity cone of matrices and conditions on the matrices and splittings we obtain the convergence of the iterative method the usual partial orders of nonnegative matrices and nonnegative definite matrices satisfy the conditions presented for this new order and hence we unify the theory moreover we also show that the new conditions apply to other partial orders as well thus extending the usual convergence theory to a new situation

iterative criteria for identifying strong h tensors abstract strong h tensors play an important role in the theories and applications of numerical linear algebra it is necessary to identify whether a given tensor is a strong h tensor or not in this paper we establish some iterative criteria for identifying strong h tensors these criteria depend on the elements of the tensors therefore they are easy to be verified the results obtained in this paper extend the corresponding conclusions for strictly generalized diagonally dominant matrices as an application some sufficient conditions for the positive definiteness of an even order real symmetric tensor are presented some numerical experiments show the feasibility and efficiency of the results which are obtained in this paper

comments on a procedure for the positive definiteness of forms of even order hasan and hasan ieee trans automat control vol 41 p 615 17 1996 give a necessary and sufficient condition for the positive definiteness of a quadratic programming problem the purpose of this paper is to point out that the main result in the above mentioned paper by is erroneous

on the z eigenvalues of the adjacency tensors for uniform hypergraphs abstract the adjacency matrices for graphs are generalized to the adjacency tensors for uniform hypergraphs and some fundamental properties for the adjacency tensor and its z eigenvalues of a uniform hypergraph are obtained in particular some bounds on the smallest and the largest z eigenvalues of the adjacency tensors for uniform hypergraphs are presented

comparisons of regular splittings of matrices in this article new comparison theorems for regular splittings of matrices are derived in so doing the initial results of varga in 1960 on regular splittings of matrices and the subsequent unpublished results of wo nicki in 1973 on regular splittings of matrices will be seen to be special cases of these new comparison theorems

a note on comparison theorems for splittings and multisplittings of hermitian positive definite matrices abstract we discuss iterative methods for the solution of the linear system ax b which are based on a single splitting or a multisplitting of a in order to compare different methods it is common to compare the spectral radius of the iterative matrix for m matrices a and weak regular splittings there exist well known comparison theorems here we give a comparison theorem for splittings of hermitian positive definite matrices furthermore we establish a comparison theorem for multisplittings of a hermitian positive definite matrix

an eigenvalue method for testing positive definiteness of a multivariate form in this paper we present an eigenvalue method for testing positive definiteness of a multivariate form this problem plays an important role in the stability study of nonlinear autonomous systems via lyapunov s direct method in automatic control at first we apply the d andrea dickenstein version of the classical macaulay formulas of the resultant to compute the symmetric hyperdeterminant of an even order supersymmetric tensor by using the supersymmetry property we give detailed computation procedures for the bezoutians and specified ordering of monomials in this approach we then use these formulas to calculate the characteristic polynomial of a fourth order three dimensional supersymmetric tensor and give an eigenvalue method for testing positive definiteness of a quartic form of three variables some numerical results of this method are reported

d eigenvalues of diffusion kurtosis tensors diffusion kurtosis imaging dki is a new model in medical engineering where a diffusion kurtosis dk tensor is involved a dk tensor is a fourth order three dimensional fully symmetric tensor in this paper we introduce d eigenvalues for a dk tensor the largest the smallest and the average d eigenvalues of a dk tensor correspond with the largest the smallest and the average apparent kurtosis coefficients akc of a water molecule in the space respectively we present their computational methods and discuss related anisotropy value of akc

on the connected components of the spectrums of the character rings of a finite group abstract the numbers of connected components of the spectrums of three kinds of rings of characters of a finite group are determined in terms of the numbers of special kinds of conjugacy classes of the group

on the connected components of the spectrum of the extended character ring of a finite group abstract let r g be the character ring of a finite group g for any subring s of the complex field let be the set of such rational primes whose inverse do not lie in s and consider the prime spectrum of the coefficient extended character ring s r g by means of a generalized version of brauer s induction theorem we show that the number of the connected components of the prime spectrum of s r g equals the number of the regular conjugacy classes of g

some comparison theorems for weak nonnegative splittings of bounded operators abstract the comparison of the asymptotic rates of convergence of two iteration matrices induced by two splittings of the same matrix has arisen in the works of many authors in this paper we derive new comparison theorems for weak nonnegative splittings and weak splittings of bounded operators in a general banach space and rather general cones and in a hilbert space which extend some of the results obtained by wo nicki japan j indust appl math 11 1994 289 342 and marek and szyld numer math 44 1984 23 35 furthermore we present new theorems also for bounded operator which extend some results by csordas and varga numer math 44 1984 23 35 for weak nonnegative splittings of matrices

eigenvalues and invariants of tensors abstract a tensor is represented by a supermatrix under a co ordinate system in this paper we define e eigenvalues and e eigenvectors for tensors and supermatrices by the resultant theory we define the e characteristic polynomial of a tensor an e eigenvalue of a tensor is a root of the e characteristic polynomial in the regular case a complex number is an e eigenvalue if and only if it is a root of the e characteristic polynomial we convert the e characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor i e they are invariant under co ordinate system changes we call them principal invariants of that tensor the maximum number of principal invariants of mth order n dimensional tensors is a function of m and n we denote it by d m n and show that d 1 n 1 d 2 n n d m 2 m for m 3 and d m n m n 1 m for m n 3 we also define the rank of a tensor all real eigenvectors associated with nonzero e eigenvalues are in a subspace with dimension equal to its rank

comparisons of weak regular splittings and multisplitting methods comparison results for weak regular splittings of monotone matrices are derived as an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings this yields a very simple proof of results of neumann plemmons on upper bounds and establishes lower bounds which has in special cases been conjectured by these authors

a note on comparison theorems for nonnegative matrices in a recent paper 4 csordas and varga have unified and extended earlier theorems of varga in 10 and wo nicki in 11 on the comparison of the asymptotic rates of convergence of two iteration matrices induced by two regular splittings the main purpose of this note is to show a connection between the csordas varga paper and a paper by beauwens 1 in which a comparison theorem is developed for the asymptotic rate of convergence of two nonnegative iteration matrices induced by two splittings which are not necessarily regular monotonic norms already used in 1 play an important role in our work here

pseudospectra localization sets of tensors with applications abstract a localization set and an exclusion set for the pseudospectrum of tensors are given as applications we first use the pseudospectra localization sets to locate asymptotically stable sets for homogeneous dynamical systems and then two upper bounds for the largest modulus of the asymptotically stable set are given to estimate the asymptotic decay rate for homogeneous dynamical systems we also provide two lower bounds for the preservation measure to the positive definiteness of the homogeneous polynomial form in addition some sufficient conditions for the global uniqueness and solvability of tensor complementarity problems are presented the last application is to test copositivity of some potential fields in particle physics

high order contrasts for independent component analysis this article considers high order measures of independence for the independent component analysis problem and discusses the class of jacobi algorithms for their optimization several implementations are discussed we compare the proposed approaches with gradient based techniques from the algorithmic point of view and also on a set of biomedical data

criterions for the positive definiteness of real supersymmetric tensors the positive definiteness of an even degree homogeneous polynomial form f x plays an important role in the stability study of nonlinear autonomous systems via lyapunov s direct method in automatic control and the positive definiteness of f x is equivalent to that of an even order supersymmetric tensor which defines f x in this paper we provide some criterions for identifying the positive definiteness of an even order real supersymmetric tensor moreover an iterative algorithm for identifying the positive definiteness of an even order real supersymmetric tensor is obtained numerical examples are given to verify the corresponding results

comparison theorems for weak splittings of bounded operators comparison theorems for weak splittings of bounded operators are presented these theorems extend the classical comparison theorem for regular splittings of matrices by varga the less known result by wo nicki and the recent results for regular and weak regular splittings of matrices by neumann and plemmons elsner and lanzkron rose and szyld the hypotheses of the theorems presented here are weaker and the theorems hold for general banach spaces and rather general cones hypotheses are given which provide strict inequalities for the comparisons it is also shown that the comparison theorem by alefeld and volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators

the cartan ring of a finite group abstract we determine the structure of the cartan ring of a finite group g defined as the grothendieck ring of g modulo its ideal generated by projective modules and we identify the ideal of the cartan ring arising from all relatively q projective modules for a p subgroup q of g

on stein rosenberg type theorems for nonnegative and perron frobenius splittings abstract the stein rosenberg theorem is extended and generalized to the class of nonnegative splittings a m 1 n 1 m 2 n 2 as well as to the most generalized class of perron frobenius splittings two types of the stein rosenberg theorem are stated and proved for both classes these theorems allow us to obtain comparison results for the rate of convergence of the associated iterative methods specific assumptions are given under which the inequalities of the spectral radii become equalities or strict inequalities the theoretical results are confirmed by numerical examples

induction theorems for finite groups including a common generalization of two classical theorems abstract roquette proved results involving induction about rings related to the character rings of finite groups and consisting of functions that vanish outside a p section kletzing did something similar for q characters this paper gives new generalizations of these results involving k characters for any field k of characteristic 0 as well as corresponding theorems for a set of primes that these facts are useful is illustrated by showing that they imply a common generalization of the witt berman and artin induction theorems as well as results of chen fan hu robinson and sin

the degree of the e characteristic polynomial of an even order tensor the e characteristic polynomial of an even order supersymmetric tensor is a useful tool in determining the positive definiteness of an even degree multivariate form in this paper for an even order tensor we first establish the formula of its e characteristic polynomial by using the classical macaulay formula of resultants then give an upper bound for the degree of that e characteristic polynomial examples illustrate that this bound is attainable in some low order and dimensional cases