counter example in three dimension and homogenization of geometric motions in two dimension in this paper we give a counter example to the homogenization of the forced mean curvature motion in a periodic setting in dimension n geqq 3 when the forcing is positive we also prove a general homogenization result for geometric motions in dimension n 2 under the assumption that there exists a constant 0 such that every straight line moving with a normal velocity equal to is a subsolution for the motion we also present a generalization in dimension 2 where we allow sign changing normal velocity and still construct bounded correctors when there exists a subsolution with compact support expanding in all directions

group inverse for the block matrices with an invertible subblock let m abcd a is square be a square block matrix with an invertible subblock over a skew field k in this paper we give the necessary and sufficient conditions for the existence as well as the expressions of the group inverse for m under some conditions

stable norms on complex numbers and quaternions in this paper we study stability properties of norms on the complex numbers and on the quaternions our main findings are that these norms are stable if and only if they majorize the modulus function and that not all stable norms are strongly stable part of the paper is devoted to the standard matrix representations of the above number systems where we show that norms on the corresponding matrix algebras are stable if and only if they are spectrally dominant we conclude by considering proper seminorms observing that none are stable on the complex numbers or on the quaternions

further results on the group inverses and drazin inverses of anti triangular block matrices abstract in this paper the sufficient and necessary conditions for the existence of the group inverse for a class of 2 2 anti triangular block matrices m with some certain conditions satisfied and representations of the group inverse of m are obtained which can be regarded as the extension of bu et al c bu j zhao j zheng group inverse for a class 2 2 block matrices over skew fields appl math comput 204 2008 45 49 and c bu j zhao k zhang some results on group inverses of block matrices over skew fields electron j linear algebra 18 2009 117 125 further we use these results to determine the expressions of the drazin inverse of m finally two numerical examples are given to illustrate our results

on the drazin inverse of the sum of two operators and its application to operator matrices abstract given two bounded linear operators f g on a banach space x such that g 2 f g f 2 0 we derive an explicit expression for the drazin inverse of f g for this purpose firstly we obtain a formula for the resolvent of an auxiliary operator matrix in the form m f i g f g from the provided representation of f g d several special cases are considered in particular we recover the case g f 0 studied by hartwig et al r e hartwig g wang y wei some additive results on drazin inverse linear algebra appl 322 2001 207 217 for matrices and by djordjevic and wei d s djordjevic y wei additive results for the generalized drazin inverse j aust math soc 73 1 2002 115 126 for operators finally we apply our results to obtain representations for the drazin inverse of operator matrices in the form m a b c d which are extensions of some cases given in the literature

a note on the group inverse of some 2 2 block matrices over skew fields abstract let k be a skew field and km n be the set of all m n matrices over k in this paper we give the necessary and sufficient conditions to the existence and the representations of the group inverse for block matrix ax yb a b 0 under some conditions where a y km n x b kn m

weighted l norms for matrices abstract let w w ij be a fixed n n matrix of positive entries and consider the w weighted l norm defined on c n n by a w max i j w ij ij a ij the main purpose of this note is to prove that for this norm multiplicativity strong stability and quadrativity are each equivalent to the condition w 1 2 w 1 where w 1 1 ij is the hadamard inverse of w among other things we also show that if is k bounded for some k 2 then it is stable

a representation for the drazin inverse of block matrices with a singular generalized schur complement abstract consider a 2 2 block complex square matrix m a b c d where a and d are square matrices suppose that i aa d b o and c i aa d o where a d is the drazin inverse of a the representations of the drazin inverse m d have been studied in the case where the generalized schur complement s a ca d b is either zero or nonsingular in this paper we develop a representation under certain conditions for m d when s is singular and group invertible moreover this formula includes the case where s o or nonsingular a numerical example is given to illustrate the result

several expressions for the generalized drazin inverse of a block matrix in a banach algebra we present some new representations for the generalized drazin inverse of a block matrix with generalized schur complement being generalized drazin invertible in a banach algebra under conditions weaker than those used in recent papers on the subject

the 2 2 0 drazin inverse problem abstract we consider the additive drazin problem and we study the existence of the drazin inverse of a two by two matrix with zero 2 2 entry

a note on the representation for the drazin inverse of 2 2 block matrices abstract in this note we consider representations of the drazin inverse of 2 2 block matrices under conditions weaker than those used in recent papers on the subject in particular in d s djordjevic p s stanimirovic on the generalized drazin inverse and generalized resolvent czechoslovak math j 51 126 2001 617 634 r hartwig x li y wei representations for the drazin inverse of 2 2 block matrix siam j matrix anal appl 27 2006 757 771

drazin inverse of partitioned matrices in terms of banachiewicz schur forms abstract let m a b c d be a partitioned matrix where a and d are square matrices denote the drazin inverse of a by a d the purpose of this paper is twofold firstly we develop conditions under which the drazin inverse of m having generalized schur complement s d ca d b group invertible can be expressed in terms of a matrix in the banachiewicz schur form and its powers secondly we deal with partitioned matrices satisfying rank m rank a d rank s d and give conditions under which the group inverse of m exists and a formula for its computation

the group inverse of a triangular matrix block conditions are given for the group inverse of a triangular matrix over a field to exist these involve the zero nonzero schur complement applications are given to idempotent matrices

a note on the representations for the drazin inverse of 2 2 block matrices abstract in 1979 campbell and meyer proposed the problem of finding a formula for the drazin inverse of a 2 2 matrix m a b c d in terms of its various blocks where the blocks a and d are required to be square matrices special cases of the problems have been studied in particular a representation of the drazin inverse of m denoted by md has recently been obtained under the assumptions that c i aad b o and a i aad b o together with the condition that the generalized schur complement d cadb be either nonsingular or zero we derive an alternative representation for md under the same assumptions but with the condition on the schur complement in the hypothesis replaced by the condition that r caa d n b n d where r and n are the range and null space of a matrix

relationships between dmp relation and some partial orders let h be an infinite dimensional complex hilbert space and let b h be the algebra of all bounded linear operators on h the equivalent definition of the dmp relation on b h using operator block matrices is introduced we present a new interpretation of this relation which allows to generalize many known results for matrices to general hilbert spaces regarding the dmp relation and its relationship with the star order and minus partial order also some properties of dmp relation defined with the help of idempotent operators are investigated

a class of seminorms on function algebras abstract let a be a function algebra on a set t in this paper we study seminorms on a of the form sc x cx where c 0 c a is a fixed element and is the sup norm on t we begin by proving that under suitable assumptions elements c d a satisfy c d on t if and only if for some p 0 b of a these results are then used in order to study multiplicativity and quadrativity factors for sc on b i e constants 0 and 0 for which sc xy sc x sc y and sc x2 sc x 2 for all x y b finally for a family t of functions in a we define the seminorm s f x sup sf x f f and provide conditions under which s f has multiplicativity and quadrativity factors by exhibiting an element c a such that s f s c on a

stability condition for difference schemes for parabolic systems recently goldberg studied in m goldberg siam j numer anal 35 1998 pp 478 493 m goldberg siam j numer anal 35 1998 pp 1995 2003 stability conditions of a well known family of difference schemes for the following multidimensional parabolic system of second order frac partial bu bf x t partial t sum 1 le p le q le s a pq frac partial 2 bu bf x t partial x p partial x q sum 1 le p le s b p frac partial bu bf x t partial x p c bu bf x t here we present a modified stability condition for the above problem and prove that this condition is optimal in some sense

some additive results on drazin inverse some additive perturbation results for drazin inverses are given in particular a formula is given for the drazin inverse of a sum of two matrices when one of the products of these matrices vanishes some special applications of this are also considered

periodic travelling wave solutions of a parabolic equation a monotonicity result this paper considers travelling wave solutions of a quasilinear parabolic equation corresponding to the propagation of a front in some striated medium the striations whose geometry play a crucial role on the type of solutions obtained are supposed to be oblique and disposed in a periodic fashion we show that the speed of such travelling waves depends in a monotonous way on the angle of inclination of the striations

representation for the generalized drazin inverse of block matrices in banach algebras abstract several representations of the generalized drazin inverse of a block matrix with a group invertible generalized schur complement in banach algebra are presented

relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices application of perturbation theory for simple invariant subspaces perturbation bounds for the relative error in the eigenvalues of diagonalizable and singular matrices are derived by using perturbation theory for simple invariant subspaces of a matrix and the group inverse of a matrix these upper bounds are supplements to the related perturbation bounds for the eigenvalues of diagonalizable and nonsingular matrices

on the drazin inverse of block matrices and generalized schur complement in this paper we give an additive result for the drazin inverse with its applications we obtain representations for the drazin inverse of a 2x2 complex block matrix having generalized schur complement s d ca db equal to zero or nonsingular several situations are analyzed and recent results are generalized r e hartwig x li y wei representations for the drazin inverse of a 2x2 block matrix siam j matrix anal appl 27 3 2006 757 771

expression of the drazin and mp inverse of partitioned matrix and quotient identity of generalized schur complement in this paper we give representations of the drazin and mp inverse of a 2x2 block matrix and quotient identities for the generalized schur complement of a partitioned 3x3 matrix under conditions different than those used in recent papers on the subject we present numerical examples to illustrate our results

some results on the generalized drazin inverse of operator matrices the generalized drazin inverse m d of a 2 2 operator matrix is considered where a x and d y are generalized drazin invertible expressions for the generalized drazin inverse m d of operator matrix m in terms of the individual blocks a b c d a d and d d are derived under some conditions

multiplicativity factors for orlicz space function norms abstract let be a function norm defined by a young function with respect to a measure space t m and let l be the orlicz space determined by if l is an algebra then a constant 0 is called a multiplicativity factor for if fg f g for all f g l the main objective of this paper is to give conditions under which l is indeed an algebra and to obtain in this case the best least multiplicativity factor for the first of our principal results is that l is an algebra if and only if m inf inf m a 0 a 0 or x sup x 0 x our second main result states that if l is an algebra and t m is free of infinite atoms then the best multiplicativity factor for is 1 1 minf if minf 0 and x if minf 0

generation and propagation of interfaces for reaction diffusion equations abstract this paper is concerned with the asymptotic behavior as e 0 of the solution ue of the reaction diffusion equation in r n r u t 1 2 0 where is the derivative of a bistable potential we show that if the initial data u 0 has values in both domains of attraction of the potential then an interface will develop in a short time o e 2 ln e we also show that if the wells of the potential are of equal depth then this interface will propagate with normal velocity equal to its mean curvature our result is valid as long as the interface remains smooth if the initial interface is compact and n 2 then the interface will disappear in a finite time but not if n 1 in case the depths of the wells are not equal then in order to obtain get reasonable results we must work on the scaled time s t e slower time scale in this scale we show that the interface moves with a constant speed proportional to the difference of the depths of the two wells along the normal towards the domain of the deeper well this result is valid for all s 0 and does not actually depend on the regularity of the interface we also extend the above results to the homogeneous second initial boundary value problem in case the depths of the wells are equal and the initial interface is orthogonal to the boundary we prove that the interface moves with normal velocity equal to its mean curvature provided that there is a family of hypersurfaces which moves according to their mean curvature and intersect the boundary orthogonally

singular limit of a reaction diffusion equation with a spatially inhomogeneous reaction term we study reaction diffusion equations with a spatially inhomogeneous reaction term if the coefficient of these reaction term is much larger than the diffusion coefficient a sharp interface appears between two different phases we show that the equation of motion of such an interface involves a drift term despite the absence of drift in the original diffusion equations in particular we show that the same rich spatial patterns observed for a chemotaxis growth model can be realized by a system without a drift term

generalized transition fronts for one dimensional almost periodic fisher kpp equations this paper investigates the existence of generalized transition fronts for fisher kpp equations in one dimensional almost periodic media assuming that the linearized elliptic operator near the unstable steady state admits an almost periodic eigenfunction we show that such fronts exist if and only if their average speed is above an explicit threshold this hypothesis is satisfied in particular when the reaction term does not depend on x or in some cases is small enough moreover except for the threshold case the fronts we construct and their speeds are almost periodic in a sense when our hypothesis is no longer satisfied such generalized transition fronts still exist for an interval of average speeds with explicit bounds our proof relies on the construction of sub and super solutions based on an accurate analysis of the properties of the generalized principal eigenvalues

a note on the formulas for the drazin inverse of the sum of two matrices abstract in this paper we give the explicit formulas for the drazin inverse of p q under some conditions furthermore we give the representations for the drazin inverse of some block matrices a b c d a and d are square under some conditions

a note on the drazin inverses with banachiewicz schur forms let h be a hilbert space m the closed subspace of h with orthocomplement m according to the orthogonal decomposition email protected m every operator email protected b h can be written in a block form m abcd in this note we give necessary and sufficient conditions for a partitioned operator matrix m to have the drazin inverse with banachiewicz schur form in addition this paper investigates the relations among the drazin inverse the moore penrose inverse and the group inverse when they can be expressed in the banachiewicz schur forms

traveling waves of a curvature flow in almost periodic media abstract in a plane media with almost periodic vertical striations we study a curvature flow and construct two kinds of traveling waves one having a straight line like profile and the other having a v shaped profile for each of the first kind traveling waves its profile is the graph of a function whose derivative is almost periodic for each of the second kind traveling waves its profile is like a pulsating cone whose two tails approach asymptotically the profiles of the first kind traveling waves also we consider a homogenization problem and provide an explicit formula for the homogenized traveling speed

all spectral dominant norms are stable abstract a vector norm on the space of n n complex valued matrices is called stable if for some constant k 0 not depending upon a or m we have am k a m we show that such a norm is stable if and only if it dominates the spectralradius

additive results for the drazin inverse of block matrices and applications in this paper we consider the drazin inverse of a sum of two matrices and derive additive formulas under conditions weaker than those used in some recent papers on the subject as a corollary we get the main results from the paper of yang and liu h yang x liu the drazin inverse of the sum of two matrices and its applications j comput appl math 235 2011 1412 1417 as an application we give some new representations for the drazin inverse of a block matrix

stable difference schemes for parabolic systems a numerical radius approach ii in this paper a numerical radius approach is taken in order to further discuss stability conditions for a well known family of finite difference schemes for the initial value problem associated with the petrowski well posed multispace dimensional parabolic system dst frac partial mathbf u mathbf x t partial t dst sum 1 leq p leq q leq s a pq dst frac partial 2 mathbf u mathbf x t partial x p partial x q sum 1 leq p leq s b p frac partial mathbf u mathbf x t partial x p c mathbf u mathbf x t noindent where a pq b p and c are constant matrices a pq being hermitian

explicit group inverse of an innovative patterned matrix in this paper we present an innovative patterned matrix rfpl toeplitz matrix is neither the extension of toeplitz matrix nor its special case we show that the group inverse of this new patterned matrix can be represented as the sum of products of lower and upper triangular toeplitz matrices first the explicit expression of the group inverse of an rfpl toeplitz matrix is obtained second the decomposition of the group inverse is given finally an example demonstrates availability of the two methods for the group inverse

formulas for the drazin inverse of special block matrices properties of the drazin inverse of the matrix f iie0 with e square are investigated based on this approach it is obtained an explicit formula for the drazin inverse of matrices of the form a ip tquv t where u v p and q are nxk the representation for a d is given in terms of kxk matrices involving the individual blocks under some conditions some special cases are also analyzed

expressions for the g drazin inverse of additive perturbed elements in a banach algebra abstract we study additive properties of the g drazin inverse in a banach algebra a in our development we derive a representation of the resolvent of a 2 2 matrix with entries in a which is then used to find explicit expressions for the g drazin inverse of the sum a b under new conditions on a b a as an application of our results we obtain a representation for the drazin inverse of a 2 2 complex block matrix in terms of the individual blocks under certain conditions

representations for the drazin inverses of 2 2 block matrices abstract let m denote a 2 2 block complex matrix a b c d where a and d are square matrices not necessarily with the same orders in this paper explicit representations for the drazin inverse of m are presented under the condition that bd i c 0 for i 0 1 n 1 where n is the order of d

stable norms from theory to applications and back abstract the purpose of this survey paper is to give a brief review of certain aspects of stability of norms and subnorms acting on algebras over a field f either r or c a norm n on an associative algebra a over f shall be called stable if for some positive constant n a m n a m for all a a m 1 2 3 a norm shall be called strongly stable if the above inequality holds with 1 we begin the paper by discussing several results regarding norm stability including conditions under which norms on certain algebras are stable the second part of the paper is devoted to applications where we employ the notion of norm stability to obtain criteria for the convergence of a well known family of finite difference schemes for the initial value problem associated with the parabolic system u x t t 1 j k s a jk 2 u x t x j x k 1 j s b j u x t x j cu x t where ajk bj and c are constant matrices ajk being hermitian the third and last part of the paper deals with the question of stability for subnorms acting on subsets of power associative algebras that are closed under scalar multiplication and under raising to powers a subnorm f on such a set s is a real valued function satisfying f a 0 for all 0 a s and f a f a for all a s and f

multiplicativity factors for seminorms ii let s be a seminorm on an algebra a in this paper we study multiplicativity and quadrativity factors for s i e constants 0 and 0 for which s xy s x s y and s x2 s x 2 for all x y a we begin by investigating quadrativity factors in terms of the kernel of s we then turn to the question under what conditions does s have multiplicativity factors if it has quadrativity factors we show that if a is commutative then quadrativity factors imply multiplicativity factors we further show that in the noncommutative case there exist both proper seminorms and norms that have quadrativity factors but no multiplicativity factors