the tight integral closure of a set of ideals abstract this paper celebrates the enormous contributions of david buchsbaum to commutative algebra homological algebra and representation theory with grateful appreciation for the inspiration he has provided for myself and so many others

the canonical sheaf of du bois singularities we prove that a cohen macaulay normal variety x has du bois singularities if and only if x g x for a log resolution x x where g is the reduced exceptional divisor of many basic theorems about du bois singularities become transparent using this characterization including the fact that cohen macaulay log canonical singularities are du bois we also give a straightforward and self contained proof that generalizations of semi log canonical singularities are du bois in the cohen macaulay case it also follows that the kodaira vanishing theorem holds for semi log canonical varieties and that cohen macaulay semi log canonical singularities are cohomologically insignificant in the sense of dolgachev

subdifferential regularities of perturbed distance functions outside the target set in banach spaces abstract this paper is devoted to study the frechet and proximal regularities at points outside the target set of perturbed distance functions d s j determined by a closed subset s and a lipschitz function j also we provide some important results on the frechet proximal and clarke subdifferentials of d s j at those points in banach spaces

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

compactness conditions in the geometric theory of banach spaces in the last few years a lot of papers have appeared devoted to the branch of geometric theory of banach spaces involving compactness conditions the notions being fundamental in the classical geometry of banach spaces such as strict convexity uniform convexity smoothness and uniform smoothness 141 have been translated in terms of a measure of noncompactness by huff 13 goebel and sekowski 111 sekowski and stachura 25 rolewicz 22 231 montesinos 18 and the present author 2 31 these new concepts turn out to be useful tools both in the geometric theory of banach spaces and for many applications cf 3 6 10 11 13 18 20 21 241 the purpose of this paper is to discuss several properties connected with the concepts mentioned above firstly we define the modulus of near convexity which corresponds to the concept introduced by huff 13 it will be shown that this modulus is in a certain sense the inverse function to the modulus of noncompact convexity introduced by goebel and sekowski 111 secondly using the concept of a nearly uniformly smooth banach space 3 we introduce the notion of a modulus of near smoothness and we describe a certain relationship between this modulus and the modulus of near convexity this relationship is an analogue of the classical lindenstrauss dual formula 161 based on this relationship and on properties of the mentioned moduli we can calculate several formulas for the moduli of noncompact convexity near convexity and near smoothness in some classical banach spaces we also extend the classical results due to day 7 81 concerning the classification of the product spaces namely we shall show that some product spaces which are not uniformly convex are nearly uniformly convex or have the so called drop property similar results concerning the concept of smoothness will be also obtained the results of this paper form a continuation of the study from a previous article 3

log canonical singularities are du bois a recurring difficulty in the minimal model program mmp is that while log terminal singularities are quite well behaved for instance they are rational log canonical singularities are much more complicated they need not even be cohenmacaulay the aim of this paper is to prove that as conjectured in kol92 1 13 log canonical singularities are du bois the concept of du bois singularities abbreviated as db was introduced by steenbrink in ste83 as a weakening of rationality it is not clear how to define du bois singularities in positive characteristic so we work over a field of characteristic 0 throughout the paper the precise definition is rather involved see 1 10 but our main applications rely only on the following consequence

some geometric properties related to smoothness of banach spaces abstract in this paper we introduce a new class of banach spaces locally k nearly uniformly smooth spaces which is the dual notion to locally k nearly uniformly convex spaces introduced by denka kutzarova the relationships between locally k nearly uniformly smooth spaces locally k uniformly smooth banach spaces and locally fully k smooth banach spaces are discussed we also introduce the two new smooth properties s k i and s k ii which are the extensions of smooth properties s i and s ii that were introduced by suyalatu and discuss the relations between s k i s k ii and other smooth properties at last the locally fully k smoothness compact locally fully k smoothness smooth properties s k i and s k ii in product space l p x i are discussed

f purity and rational singularity we investigate singularities which are f pure respectively f pure type a ring r of characteristic p is f pure if for every r module m 0 m r m r is exact where 1r denotes the r algebra structure induced on r via the frobenius map if r e r and s e 1r then r s rps in 1r f pure type is defined in characteristic 0 by reducing to characteristicp it is proven that when r s i is the quotient of a regular local ring s r is f pure at the prime ideal q if and only if i pi i q p here j p denotes the ideal ap i a e j several theorems result from this criterion if f is a quasihomogeneous hypersurface having weights r1 r and an isolated singularity at the origin 1 17 1ri i implies k x1 x f has f pure type at m x1 x 2 1n i ri i implies k x1 xn f does not have f pure type at m 3 1 1ri i remains unsolved but does connect with a problem that number theorists have studied for many years this theorem parallels known results about rational singularities it is also proven that classifying f pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces and that the f pure locus in the maximal spectrum of k x1 xn i where k is a perfect field of characteristic p is zariski open an important conjecture is that r fr is f pure type should imply r is f pure type whenever r is a cohen macauley normal local ring it is proven that ext1 1r r 0 is a sufficient though not necessary condition a local ring r m of characteristic p is f injective if the frobenius map induces an injection on the local cohomology modules h r h 1r an example is constructed which is f injective but not f pure from this a counterexample to the conjecture that r fr is f pure implies r is f pure is constructed however it is not a domain much less normal moreover it does not lead to a counterexample to the characteristic 0 version of the conjecture 0 introduction let r be a ring of characteristic p and let r denote the ring r viewed as an r module via the frobenius map f r rp r is f pure if for every r module m 0 r m 4 r m is exact a notion of f pure type is then defined in characteristic 0 by reduction to characteristicp f pure rings are connected with invariant theory and appear in the proof that the ring of invariants of a linearly reductive affine linear group acting on a regular ring is cohen macaulay 3 it has also been demonstrated that f purity measures good singularities in the sense that it implies a great deal of simplification in the computation of local cohomology 1 received by the editors march 30 1982 and in revised form august 3 1982 1980 mathematics subject classification primary 13h99 1983 american mathematical society 0002 9947 82 0000 0829 05 25

another look at cesaro sequence spaces abstract we consider the cesaro sequence space ces p as a closed subspace of the infinite l p sum of finite dimensional spaces we easily obtain many known results for example ces p has property of rolewicz uniform opial property and weak uniform normal structure we also consider some generalized cesaro sequence spaces finally we compute the von neumann jordan and james constants of the two dimensional cesaro sequence space ces p 2 when 1 p 2

mixed hodge complexes on algebraic varieties using the theory of mixed hodge modules we introduce the notion of mixed hodge complex on an algebraic variety and establish the relation between the filtered complex of du bois and the corresponding complex of mixed hodge modules some application to the du bois singularity is given

a generalization of tight closure and multiplier ideals we introduce a new variant of tight closure associated to any fixed ideal a which we call tight closure and study various properties thereof in our theory the annihilator ideal r a of all tight closure relations which is a generalization of the test ideal in the usual tight closure theory plays a particularly important role we prove the correspondence of the ideal a and the multiplier ideal associated to a or the adjoint of a in lipman s sense in normal q gorenstein rings reduced from characteristic zero to characteristic p 0 also in fixed prime characteristic we establish some properties of r a similar to those of multiplier ideals e g a briancon skoda type theorem subadditivity etc with considerably simple proofs and study the relationship between the ideal and the f rationality of rees algebras

f injective singularities are du bois in this paper we prove that singularities of f injective type are du bois this extends the correspondence between singularities associated to the minimal model program and singularities defined by the action of frobenius in positive characteristic

f injectivity and buchsbaum singularities let r mathfrak m k be a local ring that contains a field we show that when r has equal characteristic p 0 and when h mathfrak m i r has finite length for all i dim r then r is f injective if and only if every ideal generated by a system of parameters is frobenius closed as a corollary we show that such an r is in fact a buchsbaum ring this answers positively a question of s takagi that f injective singularities with isolated non cohen macaulay locus are buchsbaum we also study the characteristic 0 analogue of this question and we show that du bois singularities with isolated non cohen macaulay locus are buchsbaum in the graded case

asymptotically nonexpansive mappings in modular function spaces abstract in this paper we prove that if is a convex finite modular function satisfying a 2 type condition c a convex bounded a e compact subset of l and t c c a asymptotically nonexpansive mapping then t has a fixed point in particular any asymptotically nonexpansive self map defined on a convex subset of l 1 which is compact for the topology of local convergence in measure has a fixed point

property β of rolewicz and orthogonal convexities of calderon lozanovskiĭ spaces abstract we find criteria for property of rolewicz and orthogonal convexities in calderon lozanovski spaces in order to present characterization for property in the sequence case we introduce a new respective monotonicity property the particular case of orlicz lorentz spaces will also be discussed

three space property for asymptotically uniformly smooth renormings abstract we prove that if y is a closed subspace of a banach space x such that y and x y admit an equivalent asymptotically uniformly smooth norm then x also admits an equivalent asymptotically uniformly smooth norm the proof is based on the use of the szlenk index and yields a few other applications to renorming theory

square free grobner degenerations let i be a homogeneous ideal of s k x 1 ldots x n and let j be an initial ideal of i with respect to a term order we prove that if j is radical then the hilbert functions of the local cohomology modules supported at the homogeneous maximal ideal of s i and s j coincide in particular text depth s i text depth s j and text reg s i text reg s j

conditions for equivalence between the hu washizu and related formulations and computational behavior in the incompressible limit the relationship of the hu washizu mixed formulation to other mixed and enhanced formulations is examined in detail in the context of linear elasticity with a view to presenting a unified framework for such formulations the hu washizu formulation is considered in both its classical form and in a modified form that is suited to establishing well posedness in the incompressible limit recently established theoretical results on uniform well posedness are studied computationally by considering a range of numerical examples which illustrate where appropriate the good behavior predicted by the theory as well as the locking behavior that is evident in those problems in which the conditions for stability and convergence are not satisfied 2005 elsevier b v all rights reserved

generic solutions for some perturbed optimization problem in non reflexive banach spaces abstract let z be a closed boundedly relatively weakly compact nonempty subset of a banach space x and j z r a lower semicontinuous function bounded from below if x 0 is a convex subset in x and x 0 has approximatively z property k then the set of all points x in x 0 z for which there exists z 0 z such that j z 0 x z 0 x and every sequence z n z satisfying lim n j z n x z n x for x contains a subsequence strongly convergent to an element of z is a dense g subset of x 0 z moreover under the assumption that x 0 is approximatively z strictly convex we show more namely that the set of all points x in x 0 z for which there exists a unique point z 0 z such that j z 0 x z 0 x and every sequence z n z satisfying lim n j z n x z n x for x converges strongly to z 0 is a dense g subset of x 0 z here x inf j z x z z z these extend s cobzas s result j math anal appl 243 2000 344 356

a fully convex banach space which does not have the banach saks property let k 2 2 be an integer a banach space x is said to be fully k convex kr if for every sequence xn in x lim n o l k ii l x j 1 implies that xn is a cauchy sequence in x it is known s that every uniformly convex space is 2r and that every kr space is reflexive it is also known 3 51 that there are 2r spaces which are not superreflexive milman 10 p 973 has raised the question whether every reflexive space is isomorphic to a 2r space a banach space x is said to have the banach saks property bs if for every bounded sequence x in x there exists a subsequence y of xn such that the cesaro means l n y y of yn is convergent in x it is known that 7 all superreflexive spaces have the bs and that 1 l every banach space with bs is reflexive however there exist e g 1 reflexive banach spaces which do not have the bs it is also known 12 that there are banach spaces with the bs which are not superreflexive since the bs is invariant under isomorphism it is natural to ask whether every kr space has the bs an affirmative answer would yield a negative solution to milman s question the purpose of the paper is to renorm baernstein s space 1 so that it is 2r therefore kr spaces do not necessarily have the bs we shall need the following characterization of kr spaces for the proof when k 2 see s 6 s

global parameter test ideals abstract this paper shows the existence of ideals whose localizations and completions at prime ideals are parameter test ideals of the localized and completed rings we do this for cohen macaulay localizations resp completions of non local rings for generalized cohen macaulay rings and for non local rings with isolated non cohen macaulay points each being an isolated non f rational point the tools used to prove these results are constructive in nature and as a consequence our results yield algorithms for the computation of these global parameter test ideals finally we illustrate the power of our methods by analyzing the hsl numbers of local cohomology modules with support at any prime ideal

uniform bounds and symbolic powers on smooth varieties the purpose of this note is to show how one can use multiplier ideals toestablish effective uniform bounds on the multiplicative behavior of certainfamilies of ideal sheaves on a smooth algebraic variety in particular weprove a quick but rather surprising result concerning the symbolic powersof radical ideals on such a variety let

nilpotence of frobenius action and the hodge filtration on local cohomology abstract an f nilpotent local ring is a local ring r m of prime characteristic defined by the nilpotence of the frobenius action on its local cohomology modules h m i r a singularity in characteristic zero is said to be of f nilpotent type if its modulo p reduction is f nilpotent for almost all p in this paper we give a hodge theoretic interpretation of three dimensional normal isolated singularities of f nilpotent type in the graded case this yields a characterization of these singularities in terms of divisor class groups and brauer groups

k β and k nearly uniformly convex banach spaces vol 162 no 2 1991 k and k nearly uniformly convex banach spaces denka kutzarova different uniform geometrical properties have been defined between the uniform convexity and the reflexivity of banach spaces in the present paper we introduce other properties of this type namely k and k nearly uniform convexity the definitions as well as some of the results presented here are announced in 7 sullivan 24 has defined k uniformly rotund banach spaces which by a suggestion of davis are all superreflexive fan and glicksberg 1 have introduced fully k convex banach spaces let x1 xk 1 be vectors in a banach space x the k dimensional volume enclosed by x1 xk 1 is given by v x1 x2 xk 1

weak f regularity deforms in q gorenstein rings weakly f regular inother words does the property of weak f regularity deform it has beenshown by singh that in general weak f regularity does not deform si2 wxit is known that f rationality deforms hh2 4 2 and the proof iswxstraightforward however the same difficulties that arise when trying to

well posedness of a class of perturbed optimization problems in banach spaces abstract let x be a banach space and z a nonempty subset of x let j z r be a lower semicontinuous function bounded from below and p 1 this paper is concerned with the perturbed optimization problem of finding z 0 z such that x z 0 p j z 0 inf z z x z p j z which is denoted by min j x z the notions of the j strictly convex with respect to z and of the kadec with respect to z are introduced and used in the present paper it is proved that if x is a kadec banach space with respect to z and z is a closed relatively boundedly weakly compact subset then the set of all x x for which every minimizing sequence of the problem min j x z has a converging subsequence is a dense g subset of x z 0 where z 0 is the set of all points z z such that z is a solution of the problem min j z z if additionally p 1 and x is j strictly convex with respect to z then the set of all x x for which the problem min j x z is well posed is a dense g subset of x z 0

parametrized variational principles encompassing compressible and incompressible elasticity abstract a parametrized five field variational principle that can accommodate both compressible and incompressible hyperelasticity is presented the primary variables are mean and deviatoric stresses mean and deviatoric strains and displacements through appropriate selection of parameters the functional of this general principle specializes to those previously presented by atluri reissner herrmann and franca

proximinality in banach spaces in this paper we study approximatively compact and strongly chebyshev sets where is the norm or the weak topology we show that the metric projection onto strongly chebyshev sets are norm continuous we characterize approximatively compact and strongly chebyshev hyperplanes and use them to characterize factor reflexive proximinal subspaces in almost locally uniformly rotund spaces we also prove some stability results on approximatively compact and strongly chebyshev subspaces

centers of f purity in this paper we study a positive characteristic analogue of the centers of log canonicity of a pair r we call these analogues centers of f purity we prove positive characteristic analogues of subadjunction like results prove new stronger subadjunction like results and in some cases lift these new results to characteristic zero using a generalization of centers of f purity which we call uniformly f compatible ideals we give a characterization of the test ideal which unifies several previous characterizations finally in the case that 0 we show that uniformly f compatible ideals coincide with the annihilators of the mathcal f e r k submodules of e r k as defined by lyubeznik and smith

k uniform rotundity is equivalent to k uniform convexity abstract r geremia and f sullivan proved that 2 uniform rotundity is equivalent to 2 uniform convexity this paper extends this result to k uniformly rotund spaces

error analysis of some galerkin least squares methods for the elasticity equations the recent technique of stabilizing mixed finite element methods by augmenting the galerkin formulation with least squares terms calculated separately on each element is considered the error analysis is performed in a unified manner yielding improved results for some methods introduced earlier in addition a new formulation is introduced and analyzed method is used within two different contexts the first is in connection with the plain strain or three dimensional problems near the incompressible limit for which the poisson ratio equals one half it is well known that for these problems a standard lower order displacement finite element method will break down by now it is also well known how a working method can be obtained a new unknown the pressure is introduced and discretized with a finite element space different from that used for the displacements the finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by brezzi and babuska 5 7 the application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition for a particular choice of displacement and pressure approximations it can be a difficult task to verify this babuska brezzi or inf sup condition during the last decade this problem has been intensively studied and the performances of many combinations are now known cf 18 and 23 26 for some recent extensions in addition there exists a general technique utilizing bubble functions by which a stable method can

f injectivity and frobenius closure of ideals in noetherian rings of characteristic p 0 abstract the main aim of this article is to study the relation between f injective singularity and the frobenius closure of parameter ideals in noetherian rings of positive characteristic the paper consists of the following themes including many other topics 1 we prove that if every parameter ideal of a noetherian local ring of prime characteristic p 0 is frobenius closed then it is f injective 2 we prove a necessary and sufficient condition for the injectivity of the frobenius action on h m i r for all i f m r where f m r is the finiteness dimension of r as applications we prove the following results a if the ring is f injective then every ideal generated by a filter regular sequence whose length is equal to the finiteness dimension of the ring is frobenius closed it is a generalization of a recent result of ma and which is stated for generalized cohen macaulay local rings b let r m k be a generalized cohen macaulay ring of characteristic p 0 if the frobenius action is injective on the local cohomology h m i r for all i dim r then r is buchsbaum this gives an answer to a question of takagi 3 we consider the problem when the union of two f injective closed subschemes of a noetherian f p scheme is f injective using this idea we construct an f injective local ring r such that r has a parameter ideal that is not frobenius closed this result adds a new member to the family of f singularities 4 we give the first ideal theoretic characterization of f injectivity in terms the frobenius closure and the limit closure we also give an answer to the question about when the frobenius action on the top local cohomology is injective

an asymptotic property of schachermayer s space under renorming let x be a banach space with closed unit ball b given k n x is said to be k respectively k 1 nearly uniformly convex k 1 nuc if for every 0 there exists 0 1 so that for every x b and every separated sequence xn b there are indices ni ki 1 respectively ni k 1i 1 such that 1 k 1 x ki 1 xni 1 respectively 1 k 1 k 1i 1 xni 1 it is shown that a banach space constructed by schachermayer is 2 but is not isomorphic to any 2 nuc banach space modifying this example we also show that there is a 2 nuc banach space which cannot be equivalently renormed to be 1

controllability for a class of time varying controlled switching impulsive systems with time delays this paper investigates two types of controllability for time varying switching impulsive control systems with time delays and the relation between the two concepts by characterizing the solution of the system we establish necessary and sufficient conditions of the controllability of delayed switching impulsive systems with respect to a given switching time sequence for some special cases linear time varying switching impulsive systems with or without time delay the corresponding criteria are obtained compared with some existing results the results presented here are more general and less conservative numerical examples are provided to show the effectiveness of the proposed methods

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

analysis and finite element approximation of compressible and incompressible linear isotropic elasticity based upon a variational principle abstract we propose a variational principle for compressible and incompressible linear isotropic elasticity involving four dependent variables the strain tensor the augmented stress tensor pressure and displacement the variational formulation derived from the principle is analyzed employing a general result due to brezzi finite element methods using the galerkin and sbb methods are shown to converge for a wide family of finite element interpolations

f singularities of pairs and inversion of adjunction of arbitrary codimension we generalize the notions of f regular and f pure rings to pairs r mathfrak a t of rings r and ideals mathfrak a subset r with real exponent t 0 and investigate these properties these f singularities of pairs correspond to singularities of pairs of arbitrary codimension in birational geometry via this correspondence we prove a sort of inversion of adjunction of arbitrary codimension which states that for a pair x y of a smooth variety x and a closed subscheme y subsetneq x if the restriction z y z to a normal gorenstein closed subvariety z subsetneq x is klt resp lc then the pair x y z is plt resp lc near z

a new family of stable elements for nearly incompressible elasticity based on a mixed petrov galerkin finite element formulation adding to the classical hellinger reissner formulation another residual form of the equilibrium equation a new petrov galerkin finite element method is derived it fits within the framework of a mixed finite element method and is proved to be stable for rather general combinations of stress and displacement interpolations including equal order discontinuous stress and continuous displacement interpolations which are unstable within the galerkin approach error estimates are presented using the babuska brezzi theory and numerical results confirm these estimates as well as the good accuracy and stability of the method

local cohomology and f stability abstract we study the relationship between the frobenius stability of an artinian module over an f injective ring and its stable part

some generalizations of locally and weakly locally uniformly convex space abstract in this paper we prove that strongly convex space and almost locally uniformly rotund space very convex space and weakly almost locally uniformly rotund space are respectively equivalent we also investigate a few properties of k strongly convex space and k very convex space and discuss the applications of strongly convex space and very convex space in approximation theory

limiting subdifferentials of perturbed distance functions in banach spaces abstract we explore in arbitrary banach spaces the frechet type e subdifferentials and the limiting subdifferentials for the perturbed distance function d s j determined by a closed subset s and a lower semicontinuous function j defined on s in particular upper and lower estimates for the frechet type e subdifferentials and for the limiting subdifferentials are provided in terms of the corresponding subdifferentials of the sum of the associated functions j and s

an extended hu washizu formulation for elasticity a class of new mixed formulations for elasticity is developed and analysed the formulations are based on the discrete evss method introduced in the context of incompressible viscoelastic flows by a fortin m fortin and co workers a key feature is a stabilization term that renders coercive a problem that might not otherwise be so the focus in this work is on behaviour in the incompressible limit and the goal is that of obtaining formulations that are uniformly stable and convergent concrete examples are presented of element choices that lead to unstable formulations in the classical formulation and which are stable for the formulations introduced here a selection of numerical results illustrates in a comparative way the behaviour of the elements introduced

lenses and asymptotic midpoint uniform convexity abstract we introduce a new isometric geometric property namely asymptotic midpoint uniform convexity and investigate its connection to similar asymptotic properties

simple hironaka resolution in characteristic zero building upon work of villamayor and bierstone milman we give a proof of the canonical hironaka principalization and desingularization the idea of homogenized ideals introduced in the paper gives a priori the canonicity of algorithm and radically simplifies the proof

multiplier ideal sheaves and kahler einstein metrics of positive scalar curvature we present a method for proving the existence of kiihler einstein metrics of positive scalar curvature on certain compact complex manifolds and use the method to produce a large class of examples of compact kiihler einstein manifolds of positive scalar curvature suppose that m is a compact complex manifold of positive first chern class as is well known the existence of a kiihler einstein metric on m is equivalent to the existence of a solution to a certain complex monge ampere equation on m to solve this complex mongeampere equation by the method of continuity one needs only to establish the appropriate zeroth order a priori estimate suppose now that m does not admit a kiihler einstein metric so that the zeroth order a priori estimate fails to hold from this lack of an estimate we extract various global algebro geometric properties of m by introducing a coherent sheaf of ideals j on m called the multiplier ideal sheaf which carefully measures the extent to which the estimate fails the sheaf y is analogous to the subelliptic multiplier ideal sheaf that j j kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d neumann problem now j is a global algebro geometric object on m and it so happens that j satisfies a number of highly nontrivial global algebro geometric conditions including a cohomology vanishing theorem in particular the complex analytic subspace v c m cut out by j is nonempty connected and has arithmetic genus zero if v is zero dimensional then it is a single reduced point while if v is one dimensional then its support is a tree of smooth rational curves the logarithmic geometric genus of m v always vanishes these considerations place nontrivial global algebro geometric restric

porosity of perturbed optimization problems in banach spaces let x be a banach space and z a nonempty closed subset of x l etj z r be a lower semicontinuous function bounded from below this paper is concerned with the perturbed optimization problem infz z j z x z denoted by x j inf for x x in the case when x is compactly fully 2 convex it is proved in the present paper that the set of all points x in x for which there does not exist z0 z such that j z0 x z0 infz z j z x z is a porous set in x f urthermore ifx is assumed additionally to be compactly locally uniformly convex we verify that the set of all points x x z 0 such that the problem x j inf fails to be approximately compact is a porous set in x z 0 w herez 0 denotes the set of all z z such that z pz z moreover a counterexample to which some results of ni r x ni generic solutions for some perturbed optimization problem in nonreflexive banach space j math anal appl 302 2005 417 424 fail is provided

a characteristic p analogue of plt singularities and adjoint ideals we introduce a new variant of tight closure and give an interpretation of adjoint ideals via this tight closure as a corollary we prove that a log pair x is plt if and only if the modulo p reduction of x is divisorially f regular for all large p 0 here divisorially f regular pairs are a class of singularities in positive characteristic introduced by hara and watanabe j algebra geom 11 363 392 2002 in terms of frobenius splitting

on f pure thresholds using the frobenius map we introduce a new invariant for a pair r a of a ring r and an ideal a r which we call the f pure threshold c a of a and study its properties we see that the f pure threshold characterizes several ring theoretic properties by virtue of hara and yoshida s result trans amer math soc 355 2003 3143 the f pure threshold c a in characteristic zero corresponds to the log canonical threshold lc a which is an important invariant in birational geometry using the f pure threshold we prove some ring theoretic properties of three dimensional terminal singularities of characteristic zero also in fixed prime characteristic we establish several properties of the f pure threshold similar to those of the log canonical threshold with quite simple proofs

non commutative resolutions of toric varieties abstract let r be the coordinate ring of an affine toric variety we prove using direct elementary methods that the endomorphism ring end r a where a is the finite direct sum of all isomorphism classes of conic r modules has finite global dimension equal to the dimension of r this gives a precise version and an elementary proof of a theorem of spenko and van den bergh implying that end r a has finite global dimension furthermore we show that end r a is a non commutative crepant resolution if and only if the toric variety is simplicial for toric varieties over a perfect field k of prime characteristic we show that the ring of differential operators d k r has finite global dimension

seminormality and local cohomology of toric face rings abstract we characterize the toric face rings that are normal respectively seminormal extending results about local cohomology of brun bruns ichim li and romer of seminormal monoid rings and stanley toric face rings we prove the vanishing of certain graded parts of local cohomology of seminormal toric face rings the combinatorial formula we obtain generalizes hochster s formula we also characterize all necessarily seminormal toric face rings that are f pure or f split over a field of characteristic p 0 an example is given to show that f injectivity does not behave well with respect to face projections of toric face rings finally it is shown that weakly f regular toric face rings are normal affine monoid rings

compactly uniformly convex spaces and property β of rolewicz abstract we study different facets of property of rolewicz we remark that the notions of compact uniform convexity and property are isometrically equivalent and present new examples of spaces with that property an observation is made that the property can be formulated in terms of graphs and an estimate of the modulus is also given

rees algebras of f regular type abstract we study the f regularity of rees algebras r i a it in terms of the global f regularity of the blowing up x proj r i of spec a as it reads global f regularity is a global analog of strong f regularity defined via splitting of frobenius maps in prime characteristic and these notions are extended to characteristic zero by reduction modulo p 0 we study in detail the case where a m is a two dimensional local ring and i is an m primary ideal in characteristic zero the condition for r i to have f regular type is described in terms of the dual graph of a resolution x on which i o x is invertible we also prove some miscellaneous results concerning singularities of rees algebras and extended rees algebras of higher dimension

k super strongly convex and k super strongly smooth banach spaces abstract in this paper we introduce two types of new banach spaces k super strongly convex spaces and k super strongly smooth spaces it is proved that these two notions are dual we also prove that the class of k super strongly convexifiable spaces is strictly between locally k uniformly rotund spaces and k strongly convex spaces and obtain some necessary and sufficient conditions of k super strongly convex space respectively k super strongly smooth space also for each k 2 it is shown that there exists a k super strongly convex respectively k super strongly smooth space which is not k 1 super strongly convex respectively k 1 super strongly smooth space

existence and porosity for a class of perturbed optimization problems in banach spaces abstract let x be a banach space and z a nonempty closed subset of x let j z r be an upper semicontinuous function bounded from above this paper is concerned with the perturbed optimization problem sup z z j z x z which is denoted by x j sup we shall prove in the present paper that if z is a closed boundedly relatively weakly compact nonempty subset then the set of all x x for which the problem x j sup has a solution is a dense g subset of x in the case when x is uniformly convex and j is bounded we will show that the set of all points x in x for which there does not exist z 0 z such that j z 0 x z 0 sup z z j z x z is a porous subset of x and the set of all points x x z 0 such that there exists a maximizing sequence of the problem x j sup which has no convergent subsequence is a porous subset of x z 0 where z 0 denotes the set of all z z such that z is in the solution set of z j sup

a canonical linear system associated to adjoint divisors in characteristic p 0 suppose that x is a projective variety over an algebraically closed field of characteristic p 0 further suppose that l is an ample or more generally in some sense positive divisor we study a natural linear system in h 0 x ox kx l we further generalize this to incorporate a boundary divisor we show that these subsystems behave like the global sections associated to multiplier ideals h 0 x j x l in characteristic zero in particular we show that these systems are in many cases base point free while the original proof utilized kawamata viehweg vanishing and variants of multiplier ideals our proof uses test ideals

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

on the k uniform rotund and the fully convex banach spaces let x be a real banach space if x is uniformly convex then it is known that x is 2 uniformly rotund s and fully 2 convex 2 furthermore if a banach space x is either k uniformly rotund or fully k convex for some k then x is reflexive in this paper we show that if x is strictly convex and kuniformly rotund then x is fully k 1 convex however there exists a superreflexive space which is fully 2 convex but is not 2 uniformly rotund and for each k 2 there exists a strictly convex space which is k uniformly rotund but is not fully k convex thus for each k 2 there exist fully kconvex banach space which is not fully k 1 convex for x x2 xk in x let

numerical analysis of the modified evss method abstract in this paper we present a proof of the stability of a new mixed finite element method introduced recently by the authors 9 within the context of viscoelastic fluids the mixed formulation is related to the evss elastic viscous split stress method proposed by rajagopalan et al 14 and is based on the introduction of the rate of deformation tensor as an additional unknown the proof applies to the stokes flow and some linearized constitutive equations of slow viscoelastic flows existence and uniqueness of the continuous and the discrete problems are derived from a generalized brezzi babuska theory it is shown that no additional compatibility condition is required between the various variables except the usual one for the velocity and the pressure fields this result allows to choose low order finite element for the stress several numerical experiments on the 4 1 contraction stokes flow will be presented which will confirm the improved stability obtained with this new formulation

generic existence of solutions for some perturbed optimization problems abstract let x be a real banach space z a closed nonvoid subset of x and j z r a lower semicontinuous function bounded from below if x is reflexive and has the kadets property then the set of all x x for which there exists z0 z such that j z0 x z0 inf j z x z z z is g and dense in x this extends a result of k s lau 1978 indiana univ math j 27 791 795 on nearest points the case j 0

f rationality of rees algebras in this paper using the notion of the tight integral closure we will give a criterion for f rationality of rees algebras of m primary ideals in a cohen macaulay local ring as its application we prove the following results 1 in dimension two if a is f rational and i is integrally closed then the rees algebra r i is f rational on the other hand in higher dimensions we construct many examples of cohen macaulay normal rees algebras which are not f rational 2 if both a and r i are f rational then so is the extended rees algebra r i 3 if r i is f rational and a g i 1 then a is f rational on the other hand using resolution of singularities we will prove that a two dimensional rational singularity always admits f rational rees algebras in particular this theorem gives another way than that devised by watanabe 1997 j pure appl algebra122 323 328 to construct counterexamples to the boutot type theorem for f rational rings