convergence ball and error analysis of a family of iterative methods with cubic convergence a family of iterative methods with cubic convergence to find approximate solutions of nonlinear equations is proposed the radius of the convergence ball of any iterative method of the family is estimated and quadratic convergence is obtained under the assumption that the first order frechet derivative of the operator involved is lipschitz continuous if the second order frechet derivative of the operator involved is assumed also to be lipschitz continuous cubic convergence of any iterative method of the family is obtained finally some examples are provided to show applications of our theorems

generating a residual intersection let i be an ideal of height g 0 in a commutative noetherian ring r and let 2i be a proper sub ideal of i generated by 3 g elements if the ideal j 21 i has height at least s then j is called an s residual intersection of i see artin and nagata 1 huneke 19 or huneke and ulrich 22 the case s g corresponds to the theory of linkage and is well understood at least from the homolo cal point of view the cornerstone of this theory is that if i is perfect then a free resolution of r j can be produced from a free resolution of r z and the koszul complex on a set of generators of cu very little is known about resolutions of s residual intersections for s g a few special cases have been resolved s 10 301 and the end of the resolution or in other words a set of generators for the canonical module of r j has been described 22 however in general even the generators of j remain unknown here we show that j can be approximated by a sum of links of i if s g 1 and we give conditions under which this approximation in fact gives a full set of generators of j since these generators arise from comparison maps from koszul complexes to known resolutions at least assuming that the resolution of r i is in some sense known it is feasible to count them compute their degrees in the graded case or even to give explicit formulas for them after stating our main results theorem 2 15 and 2 16 and corollary 2 18 precisely in section 2 we lay out the argument in sections 3 4 and 5 in section 6 we are concerned with the question of when a g t 1 residual intersection j is

analytic spread modulo an element and symbolic rees algebras let t n be a d dimensional regular local ring p a prime ideal of t having height equal d 1 and f p within this setting we are interested in computing the analytic spread of p f t we have two reasons for seeking this analytic spread the first is to identify prime ideals in local hypersurface rings whose analytic spreads and heights are the same the second is to aid in a better understanding of the symbolic rees algebra p t of p the ideal pen is the nth symbolic power of p our interest in hypersurface rings essentially originated in hug there we were interested in determining when powers and symbolic powers are equal for a prime ideal in a local hypersurface ring for a prime ideal q adjacent to the maximal ideal i e height m q l the only time such an equality can occur for all large powers is when the analytic spread of q equals the height of q equality does not however necessarily occur under this analytic spread condition cf hu2 example 2 lo in order to force equality for all powers one must make additional assumptions on the associated graded ring of p f t q cf hu2 theorem 2 11 section 2 of the present article contains some general methods for recognizing the analytic spread of p f t and section 3 contains some illustrative examples where the analytic spread of p f t equals the height of p f t the noetherian property of the symbolic rees algebra has created recent interest due to a question of cowsik in connection with his result in c that p t noetherian implies p is a set theoretic complete intersection when t is a regular local ring and dim t p 1 cowsik asked whether p t is noetherian for every prime ideal in a regular local ring an example due to p roberts ro showed the answer is no in general but the question has continued to draw attention hl h2 kr 0 e the

blowing up finitely supported complete ideals in a regular local ring let i be a finitely supported complete m primary ideal of a regular local ring r m we consider singularities of the projective models projr it and projr it over spec r where r it denotes the integral closure of the rees algebra r it a theorem of lipman implies that the ideal i has a unique factorization as a product of special simple complete ideals with possibly negative exponents for some of the factors if projr it is regular we prove that projr it is the regular model obtained by blowing up the finite set of base points of i extending work of lipman and huneke sally in dimension 2 we prove that every local ring s on projr it that is a unique factorization domain is regular moreover if dim s 2 and s dominates r then s is an infinitely near point to r that is s is obtained from r by a finite sequence of local quadratic transforms

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

cohen macaulay rees algebras and degrees of polynomial relations e mail aron sunrnp vfba br 2 department of mathematics michigan state university east lansing mi 48824 usa e mail 21144bfu msu edu 3 department of mathematics rutgers university new brunswick nj 08903 usa e mail vasconce rings rutgers edu received 5 october 1993

factorization of complete ideals in normal birational extensions in dimension two abstract let r m r be a two dimensional regular local ring and let o be a normal birational extension of r we relate the factorizations and semifactorization of complete m o primary ideals in o to the factorizations of some complete m r primary ideals in r to this aim and given a complete m r primary ideal i r we show that we can associate to each complete ideal sheaf with finite cosupport on x b l i r a complete ideal in r from which it can be recovered

mixing and asymptotic properties of markov semigroups on von neumann algebras in the paper we investigate weak mixing and approach to equilibrium of a markov semigroup i e a semigroup alpha t colon t ge 0 of linear normal positive unital mappings on a von neumann algebra in particular we show that weak mixing is equivalent to ergodictity and triviality of the point spectrum of the semigroup and give conditions assuring that each normal state tends to an equilibrium state under the action of the semigroup

rees algebras of modules we study rees algebras of modules within a fairly general framework we introduce an approach through the notion of bourbaki ideals that allows the use of deformation theory one can talk about the essentially unique generic bourbaki ideal i e of a module e which in many situations allows one to reduce the nature of the rees algebra of e to that of its bourbaki ideal i e properties such as cohen macaulayness normality and being of linear type are viewed from this perspective the known numerical invariants such as the analytic spread the reduction number and the analytic deviation of an ideal and its associated algebras are considered in the case of modules corresponding notions of complete intersection almost complete intersection and equimultiple modules are examined in some detail special consideration is given to certain modules which are fairly ubiquitous because interesting vector bundles appear in this way for these modules one is able to estimate the reduction number and other invariants in terms of the buchsbaum rim multiplicity

a note on a paper by d k r babajee and m z dauhoo abstract a counterexample is provided in this short note to show that some of local convergence theorems established in d k r babajee m z dauhoo an analysis of the properties of the variants of newton s method with third order convergence appl math comput 183 2006 659 684 are not always true some mistakes in the proofs of these theorems are pointed out

asymptotic behaviour of arithmetically cohen macaulay blow ups this paper addresses problems related to the existence of arithmetic macaulayfications of projective schemes let y be the blow up of a projective scheme x projr along the ideal sheaf of i r it is known that there are embeddings y projk i e c for c d i e 1 where d i denotes the maximal generating degree of i and that there exists a cohen macaulay ring of the form k i e c if and only if h 0 y oy k h i y oy 0 for i 1 dimy 1 y is equidimensional and cohen macaulay cutkosky and herzog asked when there is a linear bound on c and e ensuring that k i e c is a cohen macaulay ring we obtain a surprising compelte answer to this question namely that under the above conditions there are well determined invariants and e0 such that k i e c is cohen macaulay for all c d i e and e e0 our approach is based on recent results on the asymptotic linearity of the castelnuovo mumford regularity of ideal powers we also investigate the existence of a cohen macaulay rees algebra of the form r i e ct which provides an arithmetic macaulayfication for x if r has negative a invariant we prove that such a cohen macaulay rees algebra exists if and only if oy ox r i oy 0 for i 0 y is equidimensional and cohen macaulay especially these conditions imply the cohen macaulayness of r i e ct for all c d i e and e e0 the above results can be applied to obtain several new classes of cohen macaulay algebras

eigenvalues and eigenspaces of quantum dynamical systems and their tensor products the paper deals with spectral properties of abelian dynamical semigroups on von neumann algebras the notions of an eigenvalue and eigenspace for such semigroups are defined generalizing those known for the classical case and the existence of normal projections on the eigenspaces is proved a more detailed study of properties of these projections is performed also spectral properties of tensor products of completely positive dynamical semigroups are investigated

the theory of d sequences and powers of ideals in this paper we study the question of finding depth r i given an ideal i in a commutative ring this problem is difficult in general even for simple examples depth r z can be difficult to compute brodmann has shown that for any commutative noetherian ring depth r z become stable for large n finding this value however is quite hard this problem originally arose through consideration of the following example suppose k is a field and x xij is an n

asymptotic behavior and eigenvalues of dynamical semi groups on operator algebras positive linear maps are very interesting objects in operator algebra theory and have been studied by many authors cf 1 3 151 in recent years the theory of dynamical semi groups which are defined as certain semi groups of completely positive maps has made remarkable progress cf 6 8 11 the study of such semi groups was initiated by physicists as one of the natural mathematical frameworks for quantum theory of open systems and so the investigations have been mainly made under rather strong restrictions which are physically justified on the other hand we wish to study such semi groups from the mathematical point of view thus we shall consider more genera1 semigroups that is we define a dynamical semi group a a r o as a u weakly continuous one parameter semi group of normal positive maps on a von neumann algebra m in 171 we have shown various ergodic theorems for such a semi group under the assumption that it possesses an invariant faithful normal state p the triplet m a p is called a dynamical system the purpose of this paper is to investigate the relationship between the asymptotic behavior such as ergodicity or mixing properties of m a p and eigenvalues of a in section 2 we present basic notations and terminologies and consider dual dynamical systems in section 3 we define eigenvalues of a and examine their basic properties in particular we show that for any real eigenvalues 1 of a there exists a a weakly continuous norm one projection onto the eigenspace ma under the additional assumption that every a is 2 positive in section 4 eigenspaces of products of dynamical semi groups are computed in section 5 we show that ergodicity of m a p is equivalent to extremality of p in the set of all a invariant states of m and is also equivalent to that p is only a a invariant normal state finally in section 6 mixing properties are characterized by using various conditions especially we show that m a p is weakly mixing if and only if it is 411 0022 247x 82 0404li 14 02 00 0

gorenstein graded rings associated to ideals let a be a noetherian local ring with the maximal ideal m and dim a d let i 6 a be an ideal in a and s hta i this article studies the question of when the associated graded ring g i i 0 i i is gorenstein to state our result we set up some notation let be an integer such that d and let j be a reduction of i generated by elements a1 a2 a we denote by rj i the reduction number of i with respect to j the analytic spread of i is i dim a m a g i then s i we always assume that the generating set a1 a2 a of j is a basic generating set for j in the sense of aberbach huneke and trung aht which means that jiaq is a reduction of iaq for all q v i with i hta q here let v i be a set of prime ideals in a containing i and ji a1 a2 ai for 0 i by aht 7 2 there always exists a basic generating set for j if the field a m is infinite let

asymptotic vanishing of homogeneous components of multigraded modules and its applications abstract in this article we give a condition on the vanishing of finitely many homogeneous components which must imply the asymptotic vanishing for multigraded modules we apply our result to multi rees algebras of ideals as a consequence we obtain a result on normality of monomial ideals which extends and improves the results of reid roberts and vitulli 14 singla 16 and sarkar and verma 15

an analysis of the properties of the variants of newton s method with third order convergence for the last five years the variants of the newton s method with cubic convergence have become popular iterative methods to find approximate solutions to the roots of non linear equations these methods both enjoy cubic convergence at simple roots and do not require the evaluation of second order derivatives in this paper we investigate about the relationship between these methods which are in fact based on the approximation of the second order derivative present in the third order limited taylor expansion we also prove that they are different forms of the halley method and are all contractive iterative methods in a common neighbourhood we extend some of these variants to multivariate cases and prove their respective local cubic convergence from their corresponding linear models

strongly positive semigroups and faithful invariant states let jf9 denote a ff algebra j a semigroup 0 of linear maps of j into j and a faithful invariant normal state over j we assume that is strongly positive in the sense that t a a t a t a for all aej and 0 therefore one can define a contraction semigroup t on t a aej where is the cyclic and separating vector associated with we prove 1 the fixed points ji of are given by j j r t j c e where e is the orthogonal projection onto the subspace of t invariant vectors 2 the state has a unique decomposition into ergodic states if and only if j or jiyje is abelian or equivalently if j is r abelian 3 the state is ergodic if and only if mkje is irreducible or if inf 0 eco o

rees algebras of ideals having small analytic deviation in this article we identify two large families of ideals of a cohen macaulay sometimes gorenstein local ring whose rees algebras are cohen macaulay our main results imply for example that if r m is a regular local ring and p is a prime ideal of r such that p n is unmixed for all n 1 then the rees algebra r pt is cohen macaulay if either dim r p 2 o dim r p 3 r p is cohen macaulay and r p is integrally closed

strongly cohen macaulay schemes and residual intersections this paper studies the local properties of closed subschemes y in cohen macaulay schemes x such that locally the defining ideal of y in x has the property that its koszul homology is cohen macaulay whenever this occurs y is said to be strongly cohen macaulay in x this paper proves several facts about such embeddings chiefly with reference to the residual intersections of y in x the main result states that any residual intersection of y in x is again cohen macaulay introduction our purpose in this paper is to investigate a property of a closed subscheme y in a noetherian cohen macaulay scheme x which we call strong cohen macaulayness this is a local property which we first describe for a local ring let x spec r where r is a cohen macaulay local ring and let y spec r i

cohen macaulay graded rings associated to ideals let g i be the associated graded ring of an ideal i in a cohen macaulay local ring a we give a sufficient condition for g i to be a cohen macaulay ring it is described in terms of the depths of a i n for finitely many n the reduction numbers of i q for certain prime ideals q and the artin nagata property of i in the sense of ulrich contemp math vol 159 pp 373 400 the main theorem unifies several results which are already known in this aspect of the theory although the statement of the main theorem is rather complicated we restate the conditions under special situations so that they are practical and simple

cohen macaulay rees algebras and their specialization the rees algebra of an ideal j in a commutative ring r is by definition the graded algebra where t is an indeterminate in this paper we are concerned with proving that certain rees algebras are cohen macaulay and with answering under strong hypotheses the question if r s n is a factor ring of s and i jr when is i r a specialization of j s that is when is the natural epimorphism r a j s r i r an isomorphism these questions have interest partly because if s and s j s are cohen macaulay then so is gr s s j j j 16 1 and under these hypotheses if n is perfect and r s j s i r then r and 7 1 r are cohen macaulay too thus gr r is cohen macaulay and its torsion freeness and normality for exmple can be characterized in terms of analytic spreads as in 161 see section 3 for details we deal with the specialization question in section 1 it is answered for the case where as above s and 2 j s are cohen macaulay and n is 202

ideals having the expected reduction number 1 introduction let r be a noetherian local ring with infinite residue field k and let be an r ideal when studying algebraic properties of the blow up of spec along v l it is often important to have good upper bounds for the reduction number of the ideal recall that an ideal c is a reduction of if the extension of rees algebras r jt c r it is module finite or equivalently if ir l j r for some r 0 32 the least such r is denoted by rj i a reduction is minimal if it is minimal with respect to inclusion and the reduction number r l of is defined as min o j a minimal reduction of finally the analytic spread i of is the krull dimension of the fiber ring r lt r k or equivalently the minimal number of generators jbi j of any minimal reduction j of 1 32 philosophically speaking j is a simplification of with the reduction number r l measuring how closely the two ideals are related of the numerous recent results about rees algebras and associated graded rings of ideals having small reduction number 18 19 38 35 12 13 14 4 3

necessary and sufficient conditions for the cohen macaulayness of form rings abstract let a be a cohen macaulay local ring with an infinite residue field and let i a be an ideal of height h 0 supposing that depth a i n is high enough for certain values of n we give in this paper necessary and sufficient conditions for the form ring gr a i to be cohen macaulay with a invariant h or h 1

approximation complexes of blowing up rings ii this paper is a sequel to 111 where we studied the relationships that hold between the arithmetical properties of the rees ring 9 i and the symmetric algebra sym l of an ideal i and some of their fibers and the depth properties of the koszul homology modules h i r on a set of generators of i the connection between these objects is realized by certain differential graded algebras the so called approximation complexes that are built out of ordinary koszul complexes these compiexes show however different sensitivities being acyclic in situations much broader than the usual context of regular sequences it has been found that certain extensions thereof d sequences and proper sequences play here that role of acyclic sequence the interplay between sequences and approximation complexes will be a basic point of view here where we give