condensed cramer rule for some restricted quaternion linear equations abstract in this paper we derive an alternative more condensed cramer rule for the unique solution of some restricted left and right systems of quaternion linear equations the findings of this paper extend some known results in the literature

rank equalities related to the generalized inverses at s 2 bt 1 s1 2 of two matrices a and b for given matrices a and b by applying the group inverse expressions of generalized inverses a t s 2 and b t 1 s 1 2 we present some rank equalities related to a t s 2 and b t 1 s 1 2 as applications we obtain a variety of rank equalities related to the moore penrose inverses the drazin inverses the group inverses and the weighted moore penrose inverses of a and b

determinantal representation of the generalized inverses over the quaternion skew field with applications we first present some determinantal representations of one 1 5 inverse of a quaternion matrix within the framework of a theory of the row and column determinants as applications we show some new explicit expressions of generalized inverses a r t 1 s 1 2 a l t 2 s 2 2 and a t 1 t 2 s 1 s 2 2 over the quaternion skew field finally we give the representations of the unique solution to some restricted left and right systems of quaternionic linear equations the findings of this paper extend some known results in the literature

the block independence in the generalized inverse at s 2 for some ordered matrices and applications abstract in this paper the definition of block independence in the generalized inverse a t s 2 is firstly given and then a necessary and sufficient condition for some ordered matrices to be block independent in the generalized inverse a t s 2 is derived as an application a necessary and sufficient condition for a 1 a 2 a k t s 2 a 1 t 1 s 1 2 a 2 t 2 s 2 2 a k t k s k 2 is proved moreover some results are shown with respect to the moore penrose inverse the weighted moore penrose inverse and the drazin inverse respectively

convergence of newton like methods for singular operator equations using outer inverses we present a semilocal kantorovich type analysis for newton like methods for singular operator equations using outer inverses we establish sharp generalizations of the kantorovich theory and the mysovskii theory for operator equations when the derivative is not necessarily invertible the results reduce in the case of an invertible derivative to well known theorems of kantorovich and mysovskii with no additional assumptions unlike earlier theorems which impose strong conditions the strategy of the analysis is based on banach type lemmas and perturbation bounds for outer inverses which show that the set of outer inverses to a given bounded linear operator admits selections that behave like bounded linear inverses in contrast to inner inverses or generalized inverses which do not depend continuously on perturbations of the operator we give two examples to illustrate our results and compare them with earlier results and another numerical example to relate our results to computational issues

the perturbation theory for the drazin inverse and its applications abstract let a and e be n n matrices and b a e denote the drazin inverse of a by ad we give an upper bound for the relative error b d a d a d under certain circumstances an error bound for the solution for the singular equations ax b b r a is also considered

a family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non square matrix based on a quadratical convergence method a family of iterative methods to compute the approximate inverse of square matrix are presented the theoretical proofs and numerical experiments show that these iterative methods are very effective and more importantly these methods can be used to compute the inner inverse and their convergence proofs are given by fundamental matrix tools

characterizations and representations of the p q outer generalized inverse we present a new explicit expression for the p q outer generalized inverse based on the idempotent operators p and q we derive properties of the p q outer generalized inverse and analyse relations among the various p q outer generalized inverses

on integral representation of the generalized inverse at s 2 we present a general integral representation for the generalized inverse a t s 2 which extends earlier result on the moore penrose inverse weighted moore penrose inverse and drazin inverse

a characterization of the moore penrose inverse abstract if a is a nonsingular matrix of order n the inverse of a is the unique matrix x for which rank a i i x rank a we present a generalization of this fact to singular or rectangular matrices a to obtain an analogous result for the moore penrose inverse a of a we then give a geometric interpretation for the case that a is a positive semidefinite matrix

the iterative methods for computing the generalized inverse at s 2 of the bounded linear operator between banach spaces in this paper we present the iterative methods for computing the generalized inverse a t s 2 over banach spaces and also for computing the generalized drazin inverses of banach algebra elements moreover we estimate the error bounds of the iterative methods for approximating a t s 2

the 2 inverse with applications in statistics an inverse g of a given matrix a which satisfies the property gag g is known as a 2 inverse this paper presents a three phase inversion procedure for which the 2 inverse is a special case we present the geometry of 2 inverses and show that starting from 2 inverses various types of generalized inverses can be derived two examples of the occurrence of 2 inverses in statistics are given one concerning the constrained least squares estimator the other concerning a necessary and sufficient condition for a quadratic form of singular multivariate normal variates to follow a chi square distribution

asymptotic behavior of varying discrete jacobi sobolev orthogonal polynomials in this contribution we deal with a varying discrete sobolev inner product involving the jacobi weight our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros we are interested in mehler heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these sobolev orthogonal polynomials and the jacobi ones moreover this asymptotic behavior provides an approximation of the zeros of the sobolev polynomials in terms of the zeros of other well known special functions we generalize some results appeared in the literature very recently

asymptotic properties of the positive equilibrium of a discrete survival model this paper is concerned with the discrete wazewska and lasota model x n 1 x email protected n pe n x n k where email protected 0 1 n email protected 0 and email protected 0 1 2 we show that every positive solution of is persistent and then we provide sufficient conditions for oscillation of all solutions about x finally we give sufficient conditions for x to be stable or attractive

global stability for a class of nonlinear nonautonomous delay equations it is easy to imagine that more realistic models for single species growths should take into account both the changing of the environment and the effects of time delay thus it is neces sary and interesting to study the following general nonlinear nonautonomous delay equation x t l x t i s

cramer rule for the unique solution of restricted matrix equations over the quaternion skew field in this paper we establish the determinantal representations of the generalized inverses a r t 1 s 1 2 a l t 2 s 2 2 and a t 1 t 2 s 1 s 2 2 over the quaternion skew field by the theory of the column and row determinants in addition we derive some generalized cramer rules for the unique solution of some restricted quaternion matrix equations the findings of this paper extend some known results in the literature

the representation and characterization of drazin inverses of operators on a hilbert space in this note the representation and characterization of drazin inverses of operators on a hilbert space are established

global attractivity of a nonautonomous logistic difference equation with delay abstract consider the nonautonomous delay logistic difference equation y n 1 y n k n where p n n 0 is a sequence of nonnegative real numbers k n n 0 is a sequence of positive integers n k n is nondecreasing lim n n k n and is a positive constant we obtain new sufficient conditions for the global attractivity of the equilibrium of equation 1 1 which improve some recent results established in 1

global asymptotic behavior of solutions of a forced delay difference equation in this paper we consider the nonautonomous forced delay difference equation xn pnf xn k rn n 0 1 2 where the forcing term rn is a sequence of real numbers pn a sequence of positive real numbers k a nonnegative integer we obtain new sufficient conditions so that every solution of the equation converges to zero when we apply our results to some well known delay difference population models the corollaries improve some recent results

a geometrical approach on generalized inverses by neumann type series the convergence of the neumann type series to 1 2 inverses has been shown by k tanabe linear algebra appl 10 1975 163 in this paper these results indicating conditions characterizing the convergence of this series to different generalized inverses are extended in addition these results for obtaining different generalized inverses from the hyperpower method are applied finally generalized involutory matrices are introduced and characterized using the obtained results

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

partial orders based on outer inverses classes of outer inverses are used to create new partial orders and unify the classification of some of the old partial orders these classes are then compared with those obtained by using g inverses inner inverses

the positive solutions of nonautonomous hyperlogistic delay difference equations consider the nonautonomous hyperlogistic delay difference equation xn pnxn 1 xnkn r n 0 1 2 where pn is a sequence of positive real numbers kn a sequence of nonnegative integers such that n kn is nondecreasing and r a ratio of two odd integers our main results given sufficient conditions that guarantee every solution to be positive the conditions under which for all n 1 either xn 1 or 0 xn 1 are given respectively for the asymptotic behavior we give sufficient conditions that guarantee every positive solution to converge to the equilibrium x 1 of the model or to oscillate about 1 the results improve and generalize some recent results established by chen and yu 1 and zhou and zhang 2 some remarks and examples illustrate our theorems the methods used here are different from those in 3

the reverse order law for 1 3 4 inverse of the product of two matrices the relationship between 1 3 4 inverses of ab and the product of 1 3 4 inverses of a and b have been studied in this paper the necessary and sufficient conditions for b 1 3 4 a 1 3 4 ab 1 3 4 b 1 3 4 a 1 3 4 ab 1 3 4 and b 1 3 4 a 1 3 4 ab 1 3 4 are given

rank equalities related to the generalized inverse at s 2 with applications abstract a variety of rank formulas of some matrix expressions and certain partitioned matrices with respect to the generalized inverse a t s 2 are established some necessary and sufficient conditions are given by using the rank formulas presented in this paper for two three and four ordered matrices to be independent in the generalized inverse a t s 2 as special cases necessary and sufficient conditions are derived for two three and four ordered matrices to be independent in the weighted moore penrose inverse and the drazin inverse some known results can be regarded as the special cases of the results in this paper

on continuity of the generalized inverse at s 2 this paper studies the continuity of the generalized inverse a t s 2 since common important generalized inverses such as a a m n a d a a d w a l 1 and a l etc are all generalized inverse a t s 2 in fact we give a unified continuity of such important generalized inverses

asymptotics of sobolev orthogonal polynomials for hermite coherent pairs abstract let qn be the polynomials orthogonal with respect to the sobolev inner product f g s fg d 0 f g d 1 being 0 1 a coherent pair where one of the measures is the hermite measure the outer relative asymptotics for qn with respect to hermite polynomials are found on the other hand we consider the sobolev scaled polynomials and we obtain the plancherel rotach asymptotics for those as well as a consequence about their zeros

sobolev orthogonal polynomials balance and asymptotics let 0 and 1 be measures supported on an unbounded interval and sn n the extremal varying sobolev polynomial which minimizes

condition number related to the outer inverse of a complex matrix in this paper we obtain the formula for computing the condition number of a complex matrix which is related to the outer generalized inverse of a given matrix we use the schur decomposition of a matrix we characterize the spectral norm and the frobenius norm of the relative condition number of the generalized inverse and the level 2 condition number of the generalized inverse the sensitivity for the generalized drazin inverse solution of linear systems is presented we also present the structured perturbation of the generalized inverse

uniform stability of nonlinear difference systems abstract we offer sufficient conditions for the uniform stability and uniformly asymptotic stability of the zero solution of the nonlinear difference system formula where 0 1 f n s r sis the set of all functions n k 0 rfor somek n andxn sis defined byxn m x n m form n k 0

successive matrix squaring algorithm for computing outer inverses abstract in this paper we derive a successive matrix squaring sms algorithm to approximate an outer generalized inverse with prescribed range and null space of a given matrix a c r m n we generalize the results from the papers l chen e v krishnamurthy i macleod generalized matrix inversion and rank computation by successive matrix powering parallel computing 20 1994 297 311 y wei successive matrix squaring algorithm for computing drazin inverse appl math comput 108 2000 67 75 y wei h wu j wei successive matrix squaring algorithm for parallel computing the weighted generalized inverse a mn appl math comput 116 2000 289 296 and obtain an algorithm for computing various classes of outer generalized inverses of a instead of particular matrices used in these articles we use an appropriate matrix r c s n m s r numerical examples are presented

rank equalities for idempotent and involutary matrices abstract we establish several rank equalities for idempotent and involutary matrices in particular we obtain new formulas for the rank of the difference the sum the product and the commutator of idempotent or involutary matrices extensions to scalar potent matrices are also included our matrices are complex and are not necessarily hermitian

smallest zeros of some types of orthogonal polynomials asymptotics we establish mehler heine type formulas for orthogonal polynomials related to rational modifications of hermite weight on the real line and for hermite sobolev orthogonal polynomials these formulas give us the asymptotic behaviour of the smallest zeros of the corresponding orthogonal polynomials furthermore we solve a conjecture posed in a previous paper about the asymptotics of the smallest zeros of the hermite sobolev polynomials as well as an open problem concerning the asymptotics of these sobolev orthogonal polynomials

global stability and oscillation in nonlinear difference equations of population dynamics in this paper we study the qualitative behavior of solutions of the discrete population modelx n x n 1 px n qx nr x n k m n 0 1 2 where email protected 0 1 q email protected 0 p

the asymptotic behavior of the solutions of a discrete reaction diffusion equation abstract by means of bihari type inequalities we derive sufficient conditions for solutions of a discrete reaction diffusion equation to be bounded or to converge to zero asymptotic representation of solutions are also derived our results yield estimates and explicit attractive regions for the solutions

on the p q outer generalized inverse in banach algebra abstract in this note the p q outer generalized inverse in banach algebra is considered a new representation based on the 1 5 inverse is presented also we construct an iterative method for computing the p q outer generalized inverse in banach algebra

global asymptotic behavior of nonlinear difference equations in this paper the improved 3 2 conditions for the attractivity of the zero solution of nonlinear delay deference equation x n 1 x n f n x g n n 0 1 2 1 1 are obtained some applications are presented to illustrate the advantage of our results

symbolic computation of at s 2 inverses using qdr factorization abstract efficient evaluation of the full rank qdr decomposition is established a method and algorithm for efficient symbolic computation of a t s 2 inverses of a given rational matrix a is defined using the full rank qdr decomposition of an appropriate rational matrix w the algorithm is implemented using mathematica s ability to deal with symbolic expressions as well as with numbers examples including polynomial and rational matrices are presented some comparisons with well known methods for symbolic evaluation of generalized inverses are obtained

preconditionings and splittings for rectangular systems recently eiermann marek and niethammer have shown how to applysemiiterative methods to a fixed point systemx tx c which isinconsistent or in whichthe powers of the fixed point operator t have no limit to obtain iterative methods which converge to some approximate solution to the fixed point system in view of their results we consider here stipulations on apreconditioning qax qb of the systemax b and separately on asplitting a m n which lead to fixed point systems such that with the aid of a semiiterative method the iterative scheme converges to a weighted moore penrose solution to the systemax b we show in several ways that to obtain a meaningful limit point from a semiiterative method requires less restrictions on the splittings or the reconditionings than those which have been required in the classical picard iterative method see e g the works of berman and plemmons berman and neumann and tanabe we pay special attention to the case when the weighted moore penrose solution which is sought is the minimal norm least squares solution toax b

some asymptotics for sobolev orthogonal polynomials involving gegenbauer weights we consider the sobolev inner product email protected 1 1f x g x 1 x 2 a 1 email protected f x g x email protected x a 12 where email protected is a measure involving a gegenbauer weight and with mass points outside the interval 1 1 we study the asymptotic behaviour of the polynomials which are orthogonal with respect to this inner product we obtain the asymptotics of the largest zeros of these polynomials via a mehler heine type formula these results are illustrated with some numerical experiments

the generalized bott duffin inverse and its applications abstract the generalized bott duffin inverse a l of a with respect to a subspace l is defined and by means of it a unified treatment of a class of linear systems and extreme value problems with equality constraints is given

new proofs of two representations and minor of generalized inverse a 2 t s abstract in this paper we prove the limit representation for the generalized inverse a t s 2 of a given complex matrix a in a simpler way then using this representation two known results the generalized inverse a t s 2 as an analogs of convex combination with the minor of adja and the representation of the minor of the generalized inverse a t s 2 were derived anew as a special case of the minor a determinantal representation of the generalized inverse a t s 2 is obtained

generalized inverse a 2 t s and a rank equation in this paper the rank equationrankabcx rank a is considered in the sense that b and c are characterized when a 2 t s is solution to this equation as the special cases similar results for a a m n a d and a g are also derived this contributes to certain recent results in the literature including that obtained by grosz linear algebra appl 289 1999 127 and thome and wei appl math comput 141 2003 471

global attractivity for nonlinear difference equations with delay abstract in this paper the global attractivity of the zero solution of the nonlinear delay defference equation 1 1 x n 1 x n f n x g n n 0 1 2 is studied and the relaxed 3 2 conditions are obtained the results here have improved the known results