the extended euler lagrange condition for nonconvex variational problems this paper provides necessary conditions of optimality for a general variational problem for which the dynamic constraint is a differential inclusion with a possibly nonconvex right side they take the form of an euler lagrange inclusion involving convexification in only one coordinate supplemented by the transversality and weierstrass conditions it is also shown that for time invariant free time problems the adjoint arc can be chosen so that the hamiltonian function is constant along the minimizing state arc the methods used here based on simple finite dimensional nonsmooth calculus clarke decoupling and a rudimentary version of the maximum principle offer an alternative and somewhat simpler derivation of such results to those used by ioffe and rockafellar in concurrent research

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

integral closedness of complete intersection ideals the aim of this paper is to give some structure theorem of certain integrally closed ideals in noetherian rings let a be a noetherian ring and i an ideal of a then an element x of a is said to be integral over z if x satisfies an equation x c x c 0 in a with ci e i the integral closure i of z is by definition the set of elements in a which are integral over z and the ideal z is called integrally closed if f i let ua z resp ht i denote the smallest number of elements in systems of generators for z resp the height of i let min a z be the set of minimal prime divisors of i in this paper we shall study the problem when the ideals z of a such that ua z ht z are integrally closed and our result is stated as follows

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

the adjoint arc in nonsmooth optimization we extend the theory of necessary conditions for nonsmooth problems of bolza in three ways first we incorporate state constraints of the intrinsic type x t e x t for all t second we make no assumption of calmness or normality and third we show that a single adjoint function of bounded variation simultaneously satisfies the hamiltonian inclusion the euler lagrange inclusion and the weierstrass pontryagin maximum condition along with the usual transversality relations

asymptotic prime divisors of torsion free symmetric powers of modules let r be a noetherian ring f rr and m f a submodule of rank r let a m denote the stable value of ass fn mn for n large where fn is the nth symmetric power of fn and mn is the image of the nth symmetric power of m in fn we provide a number of characterizations for a prime ideal to belong to a m we also show that a m a m where a m denotes the stable value of ass fn mn

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

on the finite generation of symbolic blow ups symbolic blow up p n s a noetherian ring if s is noetherian and dim n 0 rip l cowsik observed that p must be a set theoretic complete intersection if r is not regular there is no hope in general of s being noetherian in fact if r is any normal two dimensional local ring and p is a height one prime ideal of r which is not torsion in the class group of r then s is never noetherian this idea is behind the rees counterexample 6 to the generalized zariski question as far as the author knows there are no known counter examples to the question of cowsik the most trivial case is the following if r is nagata and

integrally closed modules over two dimensional regular local rings this paper is based on work of rees on integral closures of modules and initiates the study of integrally closed modules over two dimensional regular local rings in analogy with the classical theory of complete ideals of zariski the main results can be regarded as generalizations of zariski s product theorem they assert that the tensor product mod torsion of integrally closed modules is integrally closed that the symmetric algebra mod torsion of an integrally closed module is a normal domain and that the first fitting ideal of an integrally closed module is an integrally closed ideal a construction of indecomposable integrally closed modules is also given the primary technical tool is a study of the buchsbaum rim multiplicity

singular schemes of hypersurfaces let y be the singular locus of a hypersurface x in a smooth variety m with the scheme structure defined by the jacobian ideal of x we will say then that y is the singular scheme of x to emphasize that the scheme structure of y is important for our considerations in this note we consider a class in the chow group of y which arises naturally in this setup and which captures much intersection theoretic information about the situation the guiding question we have in mind is which schemes y can arise as singular schemes of hypersurfaces we will obtain strong constraints showing for instance that the only hypersurfaces in p whose singular schemes are positive dimensional linear subspaces of p are quadrics and that no reduced nodal curve can be the singular scheme of a hypersurface in a non singular variety many more statements of this sort can be found in section 3 a different type of application is in section 2 we show how our class relates to other invariants of the singularity of a hypersurface the class can be used to recover results of holme and parusi ski on degree and multiplicity of dual varieties and leads naturally to a generalization of the notion of ranks of a smooth projective variety also we obtain a strengthening of landman s parity result and a new proof of a result of zak on the dimension of the dual of a smooth variety the duality results follow by applying the framework to hyperplane sections of m the singular scheme of a section is supported on the locus of contact of the hyperplane with m and the class can be used to measure this contact for example the class measures how general a given section is we show corollary 2 6 that if the contact scheme is a linear subspace pr 1 then the corresponding hyperplane is a smooth point of the dual variety of m and the dual variety has codimension r the main general results are in section 1 where we prove corollary 1 7 that the class we introduce depends in fact only on y and on the line bundle l o x y and not on the ambient variety m provided that y is the singular scheme of a

on the cohen macaulayness of the conormal module of an ideal abstract in the present paper we investigate a question stemming from a long standing conjecture of vasconcelos given a generically complete intersection perfect ideal i in a regular local ring r when is it true that the cohen macaulayness of i i 2 or r i 2 implies that r i is gorenstein this property is known to hold for licci ideals and essentially squarefree monomial ideals we show that a positive answer actually holds for every monomial ideal we then give a positive answer for several special classes of ideals and provide application to algebroid curves with low multiplicity we also exhibit prime ideals in regular local rings and homogeneous level ideals in polynomial rings for which the answer is negative and use them to show the sharpness of our main result as they lie in the first class of ideals not covered by it the homogeneous examples have been found thanks to the help of j c migliore as a by product we exhibit several classes of cohen macaulay ideals whose square is not cohen macaulay our methods work both in the homogeneous and in the local settings

discrete approximations and refined euler lagrange conditions fornonconvex differential inclusions this paper deals with the bolza problem p for differential inclusions subject to general endpoint constraints we pursue a twofold goal first we develop a finite difference method for studying p and construct a discrete approximation to p that ensures a strong convergence of optimal solutions second we use this direct method to obtain necessary optimality conditions in a refined euler lagrange form without standard convexity assumptions in general we prove necessary conditions for the so called intermediate relaxed local minimum that takes an intermediate place between the classical concepts of strong and weak minima in the case of a mayer cost functional or boundary solutions to differential inclusions this euler lagrange form holds without any relaxation the results obtained are expressed in terms of nonconvex valued generalized differentiation constructions for nonsmooth mappings and sets

reductions and special parts of closures abstract we provide an axiomatic framework for working with a wide variety of closure operations on ideals and submodules in commutative algebra including notions of reduction independence spread and special parts of closures this framework is applied to tight frobenius and integral closures applications are given to evolutions and special briancon skoda theorems

integral closedness of mi and the formula of hoskin and deligne for finitely supported complete ideals abstract we obtain necessary and sufficient conditions for a finitely supported monomial ideal i in a polynomial ring of dimension at least three for mi to be integrally closed this is obtained via the higher dimensional analogue of the formula of hoskin and deligne for the length of a finitely supported ideal in a regular local ring

uniform multi agent deployment on a ring we consider two variants of the task of spreading a swarm of agents uniformly on a ring graph ant like oblivious agents having limited capabilities are considered the agents are assumed to have little memory they all execute the same algorithm and no direct communication is allowed between them furthermore the agents do not possess any global information in particular the size of the ring n and the number of agents in the swarm k are unknown to them the agents are assumed to operate on an unweighted ring graph every agent can measure the distance to his two neighbors on the ring up to a limited range of v edges the first task considered is dynamical i e in motion uniform deployment on the ring we show that if either the ring is unoriented or the visibility range is less than n email protected this is an impossible mission for the agents then for an oriented ring and v email protected n email protected we propose an algorithm which achieves the deployment task in optimal time the second task discussed called quiescent spread requires the agents to spread uniformly over the ring and stop moving we prove that under our model in which every agent can measure the distance only to his two neighbors this task is impossible subsequently we propose an algorithm which achieves quiescent but only almost uniform spread the algorithms we present are scalable and robust in case the environment the size of the ring or the number of agents changes during the run the swarm adapts and re deploys without requiring any outside interference

integrally closed and componentwise linear ideals in a two dimensional regular local ring integrally closed ideals have a unique factorization property and their associated graded ring is cohen macaulay in higher dimension these properties do not hold and the goal of the paper is to identify a subclass of integrally closed ideals for which they do we restrict our attention to 0 dimensional homogeneous ideals in polynomial rings r of arbitrary dimension we identify a class of integrally closed ideals the goto class mathcal g which is closed under product and it has a suitable unique factorization property ideals in mathcal g have a cohen macaulay associated graded ring if either they are monomial or dim r 3 our approach is based on the study of the relationship between the notions of integrally closed contracted full and componentwise linear ideals

on linear complete intersections or some of its localizations becomes a complete intersection is a major factor in section 1 we describe a jacobian formulation of serre s normality criterion for symmetric algebras it is used to provide several classes of examples of normal and non normal rees algebras of prime ideals in regular domains adding to the catalog of cohen macaulay domains that are symmetric algebras we examine almost complete intersections these bear a close relationship to certain height three ideals of

a characterization of two dimensional rational singularities via core of ideals abstract the notion of p g ideals for normal surface singularities has been proved to be very useful on the other hand the core of ideals has been proved to be very important concept and also very mysterious one however the computation of the core of an ideal seems to be given only for very special cases in this paper we will give an explicit description of the core of p g ideals of normal surface singularities as a consequence we give a characterization of rational singularities using the inclusion of the core of integrally closed ideals

linkage and the koszul homology of ideals 0 introduction let r be a local ring and i an ideal of r associated to i are several graded algebras the symmetric algebra of the module i sym i the rees algebra of i defined to be the algebra r i gn o inf and the associated graded algebra of i gr1 r rii g ii 2 in in 1 i i i our purpose in this paper is to study the behavior of these graded algebras when we change i through linkage recall the definition of linkage found in the paper of peskine and szpiro 22

finitely supported simple complete ideals and multiplicities in a regular local ring let r m and s n be regular local rings of dim s dim r 2 such that s birationally dominates r and let v be the order valuation ring of s with corresponding valuation ords assume that is s and reessis let u t with is is where s then v w q r with w r it q s isu q where q min mr it and q min ns isu let p p be the center of w on r it and s isu respectively we prove that if sn rm 1 then r it p s isu p let i be a finitely supported complete m primary ideal of a regular local ring r m of dimension d 2 let t be a terminal base point of i and v be the mt adic order valuation of t with corresponding valuation v ordt let n 1 be an integer assume that it mtn and tmt rm 1 let p min mr it such that p q r it with v r it q q r where q min mr it we prove that the quotient ring r it p is d dimensional normal cohen macaulay standard graded domain over k with the multiplicity nd 1 in particular r it p is regular if and only if n 1 we prove that k rm is relatively algebraically closed in kv vmv also we determine the multiplicity of r it p and we prove that if it mt then r it p is regular if and only if tmt rm 1

on unique and almost unique factorization of complete ideals ii the theory of complete ideals was begun by zariski in 1938 z 1 and further developed in appendix 5 of zs zariski developed a theory of factorization of complete ideals in regular two dimensional local rings in this paper we develop the theory of factorization of complete ideals in complete normal two dimensional local rings let r m be a complete normal two dimensional local ring and let re r be the semi group of complete m primary ideals of r we show how the divisor class group of r ci r is related to factorization in re r we generalize results obtained in c2 where it is assumed that r m is algebraically closed assume that r is any regular local ring of dimension 2 zariski proved that any complete ideal i of r has unique factorization into products of simple complete ideals the distinct simple complete ideals occuring in the factorization of i are in 1 1 correspondence with the irreducible components of the exceptional divisor of the blowing up of i this shows that any proper birational morphism x spec r with x normal can be decomposed into a sequence of blowups with irreducible exceptional divisors now assume that r is a complete normal local domain of dimension 2 it is natural to ask if the condition that m r has unique factorization is equivalent to r being a ufd we show theorem 5 that if re r has unique factorization then r is a ufd we also give an example example 1 to show that the converse does not hold however lipman has shown in l that if r is a ufd and if r m is algebraically closed then re r has unique factorization recall that r is a ufd if and only if ci r 0 g6hner g introduced the notion of a semigroup being semi factorial this definition is reproduced in definition 3 of this paper we prove g6hner s conjecture 2 3 of theorem 4 that re r is semifactorial if and only if ci r is a torsion group as a corollary corollary to theorem 4 we obtain that if rim is algebraically closed of characteristic zero then re r is semi factorial if and only if r has a rational singularity another corollary is that ci r is torsion if and only if r satisfies the condition n of muhly and sakuma ms 1r of theorem 4 r satisfies condition n if any proper birational morphism x spec r factors as a sequence of blowups of height 2 ideals with irreducible exceptional divisors

complete ideals and multiplicities in two dimensional regular local rings let i be a complete m primary ideal of a two dimensional regular local ring r m the beautiful theory developed by zariski about complete ideals of r implies that the rees valuation rings v of i are in a natural one to one correspondence with the minimal primes p of the ideal mr it in the rees algebra r it in the previous work of huneke sally and the authors the structure of r it p is considered in the case where the residue field r m k is relatively algebraically closed in the residue field kv of v in this paper we consider the structure of r it p without the assumption that k is relatively algebraically closed in kv and obtain the following results we give necessary and sufficient conditions for r it p to be normal we determine the multiplicity of r it p we examine the cohen macaulay property of r it p and we describe implications for affine components of the blowup of i

on synchronous robotic networks part ii time complexity of rendezvous and deployment algorithms this paper analyzes a number of basic coordination algorithms running on synchronous robotic networks we provide upper and lower bounds on the time complexity of the move toward average and circumcenter laws both achieving rendezvous and of the centroid law achieving deployment over a region of interest the results are derived via novel analysis methods including a set of results on the convergence rates of linear dynamical systems defined by tridiagonal toeplitz and circulant matrices

symbolic squares in regular local rings in this paper we prove for several classes of ideals in regular local rings of equicharacteristic 0 that the symbolic square of the ideal contains no minimal generator of the ideal our techniques come from residual intersections integral closures of ideals and differentials

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

a note on complete intersections let r be a regular local ring and let r t be a polynomial algebra in one variable over r in this paper the author proves that every maximal ideal of r t is complete intersection in each of the following cases 1 r is a local ring of an affine algebra over an infinite perfect field 2 r is a power series ring over a field introduction let f be a regular local ring let r t be a polynomial algebra in one variable over r in d g the following question has been asked question is every maximal ideal of r t complete intersection in this paper we prove that the answer to the above question is affirmative in each of the following cases 1 f is a local ring of an affine algebra over an infinite perfect field 2 f is a power series ring over a field this paper is divided into three sections in 1 we fix notations and state a theorem without proof which is used in 2 and 3 in 2 we prove some lemmas and propositions which are used in proving the result when f is a local ring of an affine algebra 3 deals with the power series case 1 throughout this paper we consider commutative noetherian rings with 1 for a ring r dim r denotes its krull dimension which we always assume to be finite if r is a local ring then 911 f will always denote its unique maximal ideal if m is a finitely generated f module then ii m will denote the minimal number of genera tors of m for an ideal i of r i i denotes the height of definition let be an unmixed ideal of r of height r then i is said to be complete intersection in r if i 2lri xrai where ax a2 ar is a regular r sequence remark if r is cohen macaulay then i is complete intersection if and only if p ht 7 let r and s be two local rings definition r is said to be a local extension of s if s is a subring of r and 91t s 911 f d s r is said to be unramified over s if gjt s r 911 f and f 91t f is separable over 5 91l s let l k be a finite separable extension of k then l is a simple extension of k by a minimal polynomial of l over k we always mean an irreducible monic polynomial over k satisfied by a generator of l over k now we state a theorem which has been proved in d g theorem 3

dynamic optimization problems with free time and active state constraints free time dynamic optimization problems are treated with state constraints in which the data is permitted to be measurable with respect to the time variable necessary conditions for such problems have previously been derived only under the assumption that the state constraint is inactive at the optimal endtimes a more detailed analysis than has yet been undertaken of the interaction between the optimal free endtimes and the state constraint permits the removal of this assumption an example clarifies the nature of the new necessary condition

weak subintegral closure of ideals we describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex analytic settings we focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ideal we start by giving a new geometric interpretation of the reid roberts singh criterion for when an element is weakly subintegral over a subring we give new characterizations of the weak subintegral closure of an ideal we associate with an ideal i of a ring a an ideal i which consists of all elements of a such that v a v i for all rees valuations v of i the ideal i plays an important role in conditions from stratification theory such as whitney s condition a and thom s condition af and is contained in every reduction of i we close with a valuative criterion for when an element is in the weak subintegral closure of an ideal for this we introduce a new closure operation for a pair of modules which we call relative weak closure we illustrate the usefulness of our valuative criterion

finitely supported simple complete ideals in a regular local ring abstract let i be a finitely supported complete m primary ideal of a regular local ring r m a theorem of lipman implies that i has a unique factorization as a product of special simple complete ideals with possibly negative exponents for some of the factors the existence of negative exponents occurs if dim r 3 because of the existence of finitely supported simple ideals that are not special we consider properties of special simple complete ideals such as their rees valuations and point basis let r m be a d dimensional equicharacteristic regular local ring with m x 1 x d r we define monomial quadratic transforms of r and consider transforms and inverse transforms of monomial ideals for a large class of monomial ideals i that includes complete inverse transforms we prove that the minimal number of generators of i is completely determined by the order of i we give necessary and sufficient conditions for the complete inverse transform of a product of monomial ideals to be the product of the complete inverse transforms of the factors this yields examples of finitely supported simple monomial ideals that are not special we prove that a finitely supported simple monomial ideal with linearly ordered base points is special simple

koszul homology and extremal properties of gin and lex for every homogeneous ideal i in a polynomial ring r and for every p dim r we consider the koszul homology h i p r i with respect to a sequence of p of generic linear forms the koszul betti number ijp r i is by definition the dimension of the degree j part of h i p r i in characteristic 0 we show that the koszul betti numbers of any ideal i are bounded above by those of the gin revlex gin i of i and also by those of the lex segment lex i of i we show that ijp r i ijp r gin i iff i is componentwise linear and that and ijp r i ijp r lex i iff i is gotzmann we also investigate the set gins i of all the gin of i and show that the koszul betti numbers of any ideal in gins i are bounded below by those of the gin revlex of i on the other hand we present examples showing that in general there is no j is gins i such that the koszul betti numbers of any ideal in gins i are bounded above by those of j

on the self stabilization of mobile oblivious robots in uniform rings we investigate self stabilizing ss algorithms for oblivious robots in rings we propose probabilistic ss algorithms for the gathering and orientation problems our algorithms assume the ssync model and the global strong multiplicity detection we show these assumptions are weakest to realize probabilistic ss algorithms we investigate self stabilizing algorithms for anonymous and oblivious robots in uniform ring networks that is we focus on algorithms that can start from any initial configuration including those with multiplicity points first we show that no probabilistic self stabilizing gathering algorithm exists in the asynchronous async model or if only global weak and local strong multiplicity detection is available this impossibility result implies that a common assumption about initial configurations no two robots share a node initially is a very strong one on the positive side we give a probabilistic self stabilizing algorithm for the gathering and orientation problems in the semi synchronous ssync model with global strong multiplicity detection with respect to impossibility results those are the weakest system hypotheses in addition as an application of the previous algorithm we provide a self stabilizing algorithm for the set formation problem our results imply that any static set formation can be realized in a self stabilizing manner in this model

the gorensteinness of the symbolic blow ups for certain space monomial curves let p p n 1 n 2 n 3 denote the prime ideal in the formal power series ring a k x y z over a field k defining the space monomial curve x t n1 y t n2 and z t n3 with gcd n 1 n 2 n 3 1 then the symbolic rees algebras r s p n 0 p n are gorenstein rings for the prime ideals p p n 1 n 2 n 3 with min n 1 n 2 n 3 4 and p p m m 1 m 4 with m 9 13 the rings r s p for p p 9 10 13 and p p 13 14 17 are noetherian but non cohen macaulay if ch k 3

local spreading algorithms for autonomous robot systems this paper studies local algorithms for autonomous robot systems namely algorithms that use only information of the positions of a bounded number of their nearest neighbors the paper focuses on the spreading problem it defines measures for the quality of spreading presents a local algorithm for the one dimensional spreading problem proves its convergence to the equally spaced configuration and discusses its convergence rate in the synchronous and semi synchronous settings it then presents a local algorithm achieving the exact equally spaced configuration in finite time in the synchronous setting and proves it is time optimal for local algorithms finally the paper also proposes a possible algorithm for the two dimensional case and presents partial simulation results of its effectiveness

on the occurrence of intermediate local minimizers that are not strong local minimizers abstract recently it has been shown by mordukhovich that certain necessary conditions of optimality in control theory are satisfied at intermediate local minimizers in addition to strong local minimizers it is of interest to know then whether the class of intermediate local minimizers differs from that of strong local minimizers we give examples illustrating that the two concepts of minimizers are indeed distinct and also distinguish intermediate minimizers of different rank we also extend the class of necessary conditions which are known to be satisfied at an intermediate local minimizer

poincare series of resolutions of surface singularities let x spec r be a resolution of singularities of a normal surface singularity spec r with integral exceptional divisors e 1 e r we consider the poincare series g h n t n where h n r x o x n 1 e 1 n r e r we show that if r m has characteristic zero and pic 0 x is a semi abelian variety then the poincare series g is rational however we give examples to show that this series can be irrational if either of these conditions fails

variational analysis of evolution inclusions the paper is devoted to optimization problems of the bolza and mayer types for evolution systems governed by nonconvex lipschitzian differential inclusions in banach spaces under endpoint constraints described by finitely many equalities and inequalities with generally nonsmooth functions we develop a variational analysis of such problems mainly based on their discrete approximations and the usage of advanced tools of generalized differentiation satisfying comprehensive calculus rules in the framework of asplund and hence any reflexive banach spaces in this way we establish extended results on stability of discrete approximations with the strong w 1 2 convergence of optimal solutions under consistent perturbations of endpoint constraints and derive necessary optimality conditions for nonconvex discrete time and continuous time systems in the refined euler lagrange and weierstrass pontryagin forms accompanied by the appropriate transversality inclusions in contrast to the case of geometric endpoint constraints in infinite dimensions the necessary optimality conditions obtained in this paper do not impose any nonempty interiority finite codimension normal compactness assumptions the approach and results developed in the paper make a bridge between optimal control dynamic optimization and constrained mathematical programming problems in infinite dimensional spaces

depths of multiplier ideals and integral closure in this note we study how the depths of multiplier ideals behave under restriction we also study possible values of the depths of multiplier ideals in the filtrations induced from maximal ideal sheaves we then use it to give a sufficient condition for the integral closedness of the product of a multiplier ideal and a power of maximal ideal sheaf in the spirit of huneke

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

hilbert coefficients of hilbert filtrations adic filtrations see 7 11 18 21 wx in the first section of this paper we extend most of the techniques of 4 w5 to the case of hilbert filtrations and by doing so we are also able toxoffer alternative simple proofs of some results of 7 wxin the second section thanks to the setting just described if

linear equivalence of ideal topologies it is proved that whenever p is a prime ideal in a commutative noetherian ring such that the p adic and thep symbolic topologies are equivalent then the two topologies are equivalent linearly several explicit examples are calculated in particular for all prime ideals corresponding to non torsion points on nonsingular elliptic cubic curves

variational problems with nonconvex noncoercive highly discontinuous integrands characterization and existence of minimizers we consider the functional f v int a b f t v prime t dt in cal h p v in w 1 p v a 0 v b d p in 1 infty under only the assumption that the integrand is cal l otimes cal b n measurable we prove characterizations of strong and weak minimizers both in terms of the minimizers of the relaxed functional and by means of the euler lagrange inclusion as an application we provide necessary and sufficient conditions for the existence of the minimum expressed in terms of a limitation on the width of the slope d

symmetric powers of complete modules over a two dimensional regular local ring this is the published version also available here http dx doi org 10 1090 s0002 9947 97 01819 9 first published in trans amer math soc in 1997 published by the american mathematical society

graded rings associated with contracted ideals the study of the ideals in a regular local ring r m of dimension 2 has a long and important tradition dating back to the fundamental work of zariski zs more recent contributions are due to several authors including cutkosky huneke lipman rees sally and teissier among others see c1 c2 h hs l lt r one of the main result in this setting is the unique factorization theorem for complete i e integrally closed ideals proved originally by zariski zs theorem 3 appendix 5 it asserts that any complete ideal can be factorized as a product of simple complete ideals in a unique way up to the order of the factors by definition an ideal is simple if it cannot be written as a product of two proper ideals another important property of a complete ideal i is that its reduction number is 1 which in turns implies that the associated graded ring gri r is cohen macaulay and its hilbert series is well understood this is due to lipman and teissier lt see also hs the class of contracted ideals plays an important role in the original work of zariski as well as in the work of huneke h an ideal i of r is contracted if i r ir m x for some x m m2 any complete ideal is contracted but not the other way round the associated graded ring gri r of a contracted ideal i need not be cohen macaulay and its hilbert series can be very complicated

arcs and wedges on sandwiched surface singularities a wedge on a surface singularity s p is a formal parametrization of s by power series in two variables locally at p a genericity condition which is expected to be enough to guarantee that a wedge lifts to the minimal desingularization of the surface is proved to be so if the singularity is sandwiched

sandwiched singularities and desingularization of surfaces by normalized nash transformations we prove that a succession of normalized nash transformations resolves the singularities of a complex surface we also give a classification of sandwiched surface singularities