condition numbers for the outer inverse and constrained singular linear system in this paper we investigate the condition number of the outer inverse at s 2 and outer inverse at s 2 solution of constrained linear system ax b x t where a is a real m n matrix b and x are real vectors t is a subspace let and be two positive real numbers when we consider the weighted frobenius norm a b qp f on the data we get the formula of condition number for the outer inverse at s 2 solution of constrained linear system for the normwise condition number the sensitivity of the relative condition number itself is studied and the componentwise perturbation is also investigated

on a partial order defined by the weighted moore penrose inverse the weighted moore penrose inverse of a matrix can be used to define a partial order on the set of mxn complex matrices and to introduce the concept of weighted ep matrices in this paper we study the weighted star partial order on the set of weighted ep matrices in addition some properties that relate the eigenprojection at zero with the weighted star partial order are obtained

perturbation of the drazin inverse for matrices with equal eigenprojections at zero abstract let a denote the eigenprojection of a matrix a corresponding to the eigenvalue 0 we characterize matrices b such that b a and derive from the results 1 error bounds for the drazin inverse of a perturbation 2 improvement of error bounds given recently by wei 10 wei and wang 11 and castro gonzalez et al 4 and 3 a new characterization of ep matrices

birational extensions in dimension two and integrally closed ideals the aim of this paper is to examine primarily from an algebraic point of view the structure of a 2 dimensional normal local domain s n which birationally dominates a 2 dimensinal regular local ring r m from the geometric point of view the sine qua non is lipman s paper lr on rational singularities for it follows from one of the early results in that paper that s has a rational singularity using elementary algebraic techniques we are able to recover much information concerning the structure of s we have particularly focused on the fact from lipman ll and artin ar that s must have minimal multiplicity we hope this approach will aid in understanding the algebra of rational singularities and in the exploration of many open questions in dimension two and in higher dimensions the framework of the paper is set up as follows in the first section we use basic algebraic techniques such as analytic independence and zariski s main theorem to prove for example that s has a regular height 1 prime if r m is infinite if s is not regular we may assume that r is maximally regular in s and when this is the case we show for instance that the

on stable perturbations for outer inverses of linear operators in banach spaces abstract in this paper we investigate stable perturbations and their characterizations for various types of outer inverses such as generalized 2 3 2 4 2 5 1 2 3 1 2 4 moore penrose group drazin and generalized drazin inverses some known results are improved and extended

the star partial order and the eigenprojection at 0 on ep matrices abstract the space of n n complex matrices with the star partial order is considered in the first part of this paper the class of ep matrices is analyzed and several properties related to this order are given in addition some information about predecessors and successors of a given ep matrix is obtained the second part is dedicated to the study of some properties that relate the eigenprojection at 0 with the star and sharp partial orders

hilbert functions for two ideals let r m denote a noetherian local ring r with maximal ideal m let i and j be ideals contained in m and assume i j is m primary then for all non negative integers r and s the r module r i j s has finite length we denote the length of this r module by l r i j s r s the function r s is called the hilbert function of i and j let denote the integers and for p e let p n e n p in this paper we prove the following theorem suppose for some non negative integer p there exist ordered pairs r 1 s 1 r n s n e p p such that i j s i 1 n i r r i j s s i i r i j sus i i r 1 j s 1 for all r s p then there exists a polynomial f x y e of x y the rational numbers such that r s f r s for all r s 0 furthermore x f dim r j y f dim r i and f l i l j here x f y f denotes the degree of f as a polynomial in x y and f denotes the total degree of f dim s is the krull dimension of the ring s and l u is the analytical spread of the ideal u

on the depth of the fiber cone of filtrations preprint enviat per a la seva publicacio en una revista cientifica journal of algebra 15 december 1997 volume 198 issue 2 pages 428 445 https doi org 10 1006 jabr 1997 7147

integral closures of ideals generated by regular sequences in particular our proposition contains s theorem 1 i as a special case in section 6 we shall make some remarks on theorem 2 in appendix 1 we shall give a proof of lemma 4 see section 1 for the sake of convenience this lemma is basic to our discussion of rees rings and the assertions in the lemma are all found in 9 if the ring a in the lemma is an integral domain in appendix 2 we shall make some remarks on the local cohomology modules of rees rings 390 0021 8693 88 83 00

weighted binary relations involving the drazin inverse the drazin inverse of a matrix has been used in the literature to define a pre order on the set of square complex matrices in this paper we analyze new binary relations defined on the set of rectangular complex matrices and some relationships to the w support idempotent we introduce the class of weighted drazin equal projectors and analyze the pre orders on this class moreover adjacent matrices are studied under the considered relations finally some observations on weighted partial orders are given

on the structure of the fiber cone of ideals with analytic spread one for a given a local ring a m we study the fiber cone of ideals in a with analytic spread one in this case the fiber cone has a structure as a module over its noether normalization which is a polynomial ring in one variable over the residue field one may then apply the structure theorem for modules over a principal domain to get a complete description of the fiber cone as a module we analyze this structure in order to study and characterize in terms of the ideal itself the arithmetical properties and other numerical invariants of the fiber cone as multiplicity reduction number or castelnuovo mumford regularity

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

three notes on the order of ideals defining hypersurfaces let r n be a regular local ring and let i be a proper ideal in r the order of i is defined to be ord z max it i zc tp it is clear that for f en n l the samuel multiplicity of r fr is e r fr ord fr n in this paper we give sufficient conditions on rji and i so that the converse is true more precisely we describe three situations where e r z ord z n implies i fr for some f e n n in the first situation we consider following an idea of ikeda buchsbaum rings r z the second situation is based in a more general frame on the equimultiplicity of i in the third situation we discuss the case of a graded polynomial ring r over an algebraically closed field and a homogeneous ideal zc r here the methods of the proof were outlined by s ikeda and influenced by j l vicente in the light of these results we ask the following question is it true that if r i satisfies serre s condition s then r i is a hypersurface if and only if e r z ord l this question seems to be open even in the special case that r i is a domain satisfying s with dim r i 2 and e r z ord i 2

operators with equal projections related to their generalized inverses in this article we characterize operators on banach spaces which have the same projections related to their outer or inner generalized inverses as corollaries we obtain well known results for the drazin inverse of bounded operators

bivariate hilbert functions for the torsion functor abstract let r p be a commutative local noetherian ring i j ideals m and n finitely generated r modules suppose j ann r m ann r n is p primary the main result of this paper is theorem 6 which gives necessary and sufficient conditions for the length of tori m inm n jmn to agree with a polynomial for m n 0 as a corollary it is shown that the length of tori m inm n inn always agrees with a polynomial in n for n 0 provided i ann r m ann r n is p primary

singular integral operators on function spaces we study the singular integral operator fx t y f x ty defined on all test functions f w hereb is a bounded function 0 is suitable distribution on the unit sphere s n 1 satisfying some cancellation conditions we prove certain boundedness properties of t on the triebel lizorkin spaces and on the besov spaces we also use our results to study the littlewood paley functions these results improve and extend some well known results

on the associated graded modules of canonical modules the canonical modules of cohen macaulay rings give a useful technical tool in commutative algebra algebraic geometry and singularity theory via various duality theorems but not much is known about the canonical modules themselves in this paper we investigate the structure of the canonical moduies expecially their associated graded modules we ask the following two questions let k be a canonical module over a cohenmacaulay local ring r then

perturbation and expression for inner inverses in banach spaces and its applications abstract it is well known that if t is invertible and t 1 is its inverse then t 1 i t t t 1 1 is the inverse of t for t t t 1 1 the main purpose of this paper is to study the following problem if t is an inner inverse of t and t t t 1 is t i t t t 1 an inner inverse of t if the answer is no when is t i t t t 1 an inner inverse of t in this paper we give a complete answer to these problems and provide some characterizations for t i t t t 1 to be an inner inverse of t in banach spaces utilizing these results we investigate similar problems for the generalized inverse 1 3 inverse 1 4 inverse 1 5 inverse 1 2 3 inverse 1 2 4 inverse moore penrose inverse group inverse and drazin inverse the results obtained in this paper extend and improve many recent results in this area

on the perturbation and subproper splittings for the generalized inverse at s 2 of rectangular matrix a abstract in this paper the perturbation and subproper splittings for the generalized inverse at s 2 the unique matrix x such that xax x r x t and n x s are considered we present lower and upper bounds for the perturbation of at s 2 convergence of subproper splittings for computing the special solution at s 2 b of restricted rectangular linear system ax b x t are studied for the solution at s 2 b we develop a characterization therefore we give a unified treatment of the related problems considered in literature by ben israel berman hanke neumann plemmons etc

hilbert series of fiber cones of ideals with almost minimal mixed multiplicity abstract let r m be a cohen macaulay local ring and let i be an m primary ideal we introduce ideals of almost minimal mixed multiplicity which are analogues of ideals studied by j sally compositio math 40 1980 167 175 the main theorem describes the hilbert series of fiber cones of these ideals

on the perturbation of the group generalized inverse for a class of bounded operators in banach spaces abstract given a bounded operator a on a banach space x with drazin inverse a d and index r we study the class of group invertible bounded operators b such that i a d b a is invertible and r b n a r 0 we show that they can be written with respect to the decomposition x r a r n a r as a matrix operator b b 1 b 12 b 21 b 21 b 1 1 b 12 where b 1 and b 1 2 b 12 b 21 are invertible several characterizations of the perturbed operators are established extending matrix results we analyze the perturbation of the drazin inverse and we provide explicit upper bounds of b a d and b b a d a we obtain a result on the continuity of the group inverse for operators on banach spaces

normal cones of monomial primes we explicitly calculate the normal cones of all monomial primes which define the curves of the form tl tl 1 tl n where n 4 all of these normal cones are reduced and cohen macaulay and their reduction numbers are independent of the reduction these monomial primes are new examples of integrally closed ideals for which the product with the maximal homogeneous ideal is also integrally closed substantial use was made of the computer algebra packages maple and macaulay2