on a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients we develop a perturbation theory for stochastic differential equations sdes by which we mean both stochastic ordinary differential equations sodes and stochastic partial differential equations spdes in particular we estimate the l p distance between the solution process of an sde and an arbitrary ito process which we view as a perturbation of the solution process of the sde by the l q distances of the differences of the local characteristics for suitable p q 0 as one application of the developed perturbation theory we establish strong convergence rates for numerical approximations of a class of sodes with nonglobally monotone coefficients as another application of the developed perturbation theory we prove strong convergence rates for spatial spectral galerkin approximations of solutions of semilinear spdes with nonglobally monotone nonlinearities including cahn hilliard cook type equations and stochastic burgers equations further applications of the developed perturbation theory include regularity analyses of solutions of sdes with respect to their initial values as well as small noise analyses for ordinary and partial differential equations

finite transforms of a noetherian ring let r be a noetherian ring and let uc spec r let t be the transform 1 u is in the total quotient ring of r and r u p for all pe u let b r p e spec r i grade p 1 and let 8 r p e spec r 1 the completion r has a depth 1 prime divisor of zero the main result of the first section says that tci is a finite r module if and only if every regular prime in p r is contained in some prime in u and with linitely many exceptions every regular prime in g r is contained in some prime in u we then apply this result to various well known transforms of r a transform which recently has drawn interest is r t for u d r it was known that r is the union of all ideal transforms of r which happen to be finite r modules but it was not known when r itself is finite here we show that r is a finite r module if and only if the set a k pe r i no prime in d r contains p is finite the second section of the paper studies when a r and the larger set jl r y r b r are finite we show that this occurs quite frequently a major result is that if r e 4 c b are noetherian rings with b finitely generated over r and if r is semilocal then m a is finite and so a is a finite a module the third section asks some questions and makes related comments

galerkin approximations for the stochastic burgers equation existence and uniqueness for semilinear stochastic evolution equations with additive noise by means of finite dimensional galerkin approximations is established and the convergence rate of the galerkin approximations to the solution of the stochastic evolution equation is estimated these abstract results are applied to several examples of stochastic partial differential equations spdes of evolutionary type including a stochastic heat equation a stochastic reaction diffusion equation and a stochastic burgers equation the estimated convergence rates are illustrated by numerical simulations the main novelty in this article is the estimation of the difference of the finite dimensional galerkin approximations and of the solution of the infinite dimensional spde uniformly in space i e in the l infty topology instead of the usual hilbert space estimates in the l 2 topology that were shown before

ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

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

absolute continuity and numerical approximation of stochastic cahn hilliard equation with unbounded noise diffusion abstract in this article we consider the absolute continuity and numerical approximation of the solution of the stochastic cahn hilliard equation with unbounded noise diffusion we first obtain the holder continuity and malliavin differentiability of the solution of the stochastic cahn hilliard equation by using the strong convergence of the spectral gakerkin approximation then we prove the existence and strict positivity of the density function of the law of the exact solution for the stochastic cahn hilliard equation with sublinear growth diffusion coefficient which fills a gap for the existed result when the diffusion coefficient satisfies a growth condition of order 1 3 1 to approximate the density function of the exact solution we propose a full discretization based on the spatial spectral galerkin approximation and the temporal drift implicit euler scheme furthermore a general framework for deriving the strong convergence rate of the full discretization is developed based on the variation approach and the factorization method consequently we obtain the sharp mean square convergence rates in both time and space via sobolev interpolation inequalities and semigroup theories to the best of our knowledge this is the first result on the convergence rate of full discretizations for the considered equation

prime divisors asymptotic r sequences and unmixed local rings a current trend in commutative algebra has been to characterize properties of prime divisors in the completion of a local ring in terms of the independence of various elements and sequences in the local ring itself for instance in 2 bruns characterizes the minimal depth of prime divisors of zero in the completion r of a local ring r m as the largest number of elements which are p independent for n sufficiently large more recently the notion of asymptotic sequence has been used to characterize the minimal depth of minimal prime divisors of zero in r this was done by the author in s and l j ratliff jr in 19 elements x xd form an asymptotic sequence if x p u pi pe 2 x x where a z ass r i large n and r is the integral closure of i using this result as a starting point a full blown theory of asymptotic grade has been developed in 19 and 7 in this paper we seek a similar characterization of the minimal depth of all prime divisors of zero in r in particular we introduce the set

full discretization of semilinear stochastic wave equations driven by multiplicative noise a fully discrete approximation of the semilinear stochastic wave equation driven by multiplicative noise is presented a standard linear finite element approximation is used in space and a stochastic trigonometric method is used for the temporal approximation this explicit time integrator allows for mean square error bounds independent of the space discretization and thus does not suffer from a step size restriction as in the often used stormer verlet leapfrog scheme furthermore it satisfies an almost trace formula i e a linear drift of the expected value of the energy of the problem numerical experiments are presented and confirm the theoretical results

approximating stochastic evolution equations with additive white and rough noises in this paper we analyze galerkin approximations for stochastic evolution equations driven by an additive gaussian noise which is temporally white and spatially fractional with hurst index less than or equal to 1 2 first we regularize the noise by the wong zakai approximation and obtain its optimal order of convergence then we apply the galerkin method to discretize the stochastic evolution equations with regularized noises optimal error estimates are obtained for the galerkin approximations in particular our error estimates remove an infinitesimal factor which appears in the error estimates of various numerical methods for stochastic evolution equations in existing literatures

on the kernel of a monadic transformation abstract let j b0 b1 bh a be an ideal in a noetherian ring a such that b0 is nonnilpotent and let b a x1 xh then a complete description of the asymptotic essential and quintessential prime divisors of k b0x1 b1 b0xh bh b is given and a more detailed description of these prime divisors of k and of ker b a b 1 b 0 b h b 0 is given when b0 b1 bh are an asymptotic essential or a sequence also it is shown that there exists a unique essential prime divisor of k if and only if b0 b1 bh are an essential sequence and a has exactly one prime divisor of zero

finite element approximation of the cahn hilliard cook equation we study the nonlinear stochastic cahn hilliard equation perturbed by additive colored noise we show almost sure existence and regularity of solutions we introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to 1 we also prove strong convergence without known rate

vibration of shear deformable beams using a spline function approach b spline functions are used as trial functions in a rayleigh ritz analysis of the free vibration of shear deformable timoshenko beams in a first approach it is demonstrated in numerical applications that when the lateral deflection and the cross sectional rotation are represented by functions of equal order the calculated natural frequencies are of good accuracy for stocky beams but can overestimate the true frequencies very considerably for slender beams this is identified as a shear locking difficulty and consideration of its causes points clearly to the adoption of a new displacement field in which the deflection is represented by a b spline function which is one order higher than that used to represent the rotation numerical results using this new displacement field demonstrate good accuracy for both stocky and slender beams the shear locking difficulty is completely eliminated this has clear significance for the analysis of shear deformable plates and shells when using b spline functions

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

strong convergence rate of finite difference approximations for stochastic cubic schrodinger equations abstract in this paper we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic schrodinger equations driven by a multiplicative q wiener process beyond the uniform boundedness of moments for high order derivatives of the exact solution the key requirement of our approach is the exponential integrability of both the exact and numerical solutions by constructing and analyzing a lyapunov functional and its discrete correspondence we derive the uniform boundedness of moments for high order derivatives of the exact solution and the first order derivative of the numerical solution which immediately yields the well posedness of both the continuous and discrete problems the latter exponential integrability is obtained through a variant of a criterion given by cox hutzenthaler and jentzen arxiv 1309 5595 as a by product of this exponential integrability we prove that the exact and numerical solutions depend continuously on the initial data and obtain a large deviation type result on the dependence of the noise with first order strong convergence rate

an exponential wagner platen type scheme for spdes the strong numerical approximation of semilinear stochastic partial differential equations spdes driven by infinite dimensional wiener processes is investigated there are a number of results in the literature that show that euler type approximation methods converge under suitable assumptions to the exact solutions of such spdes with strong order 1 2 or at least with strong order 1 2 varepsilon where varepsilon 0 is arbitrarily small recent results extend these results and show that milstein type approximation methods converge under suitable assumptions to the exact solutions of such spdes with strong order 1 varepsilon it has also been shown that splitting up approximation methods converge under suitable assumptions with strong order 1 to the exact solutions of such spdes in this article an exponential wagner platen type numerical approximation method for such spdes is proposed and shown to converge under suitable assumptions with strong order 3 2

a new boundary spectral strip method for non periodical geometrical entities based on analytical integrations abstract a method based on expanding the boundary integral equation into a trigonometric series or a high order polynom is proposed depending on the domain geometry it is a non element method which yields solutions to elastostatic and potential problems using a small computer memory yet obtaining more precise results as compared with other common numerical methods when the geometry of the problem contains circles and straight lines all the integrations required for solution of the boundary integral are solved analytically some elastostatic problems are solved here and compared with the boundary element method bem which shows some remarkable advantages of the boundary strip method over the bem

bases of ideals and rees valuation rings let i be a regular proper ideal in a noetherian ring r we prove that there exists a simple free integral extension ring a of r such that the ideal ia has a rees good basis that is a basis c1 cg such that ciw iw for i 1 g and for all rees valuation rings w of ia moreover a may be constructed so that i ia and i have the same rees integers with possibly different cardinalities and ii ap is unramified over rp r for each asymptotic prime divisor p of ia indeed if h is a regular ideal in r such that each asymptotic prime divisor of h is contained in an asymptotic prime divisor of i then ii holds for ha if card reesh card reesi we prove that i also holds for ha and h if i b1 bg r and b1 bg is an asymptotic sequence we prove that b1 bg is a rees good basis of i

asymptotic lech s inequality abstract we explore the classical lech s inequality relating the hilbert samuel multiplicity and colength of an m primary ideal in a noetherian local ring r m we prove optimal versions of lech s inequality for sufficiently deep ideals in characteristic p 0 and we conjecture that they hold in all characteristics our main technical result shows that if r m has characteristic p 0 and r is reduced equidimensional and has an isolated singularity then for any sufficiently deep m primary ideal i the colength and hilbert kunz multiplicity of i are sufficiently close to each other more precisely for all e 0 there exists n 0 such that for any i r with l r i n we have 1 e l r i e hk i 1 e l r i

quintessential modulated ideals abstract let r denote a commutative noetherian ring and i an ideal of r the concept of quintessential sequences over zero ideal was introduced by mcadam and ratliff 1985 8 they showed that these sequences enjoy many of the basic properties of asymptotic sequences over zero ideal it was shown that quintessential sequences over ideals i 0 r are not a good analogue of asymptotic sequences over i 0 r by making use of the new concept of quintessential grade of an ideal over another ideal we show that there exists a class of ideals i for which quintessential sequences over i are an excellent analogue of asymptotic sequences over i also we give more results on quintessential sequences over an ideal and derive generalizations of some mcadam ratliff s results 1985 8 13

projectively full ideals in noetherian rings ii abstract let r be a noetherian commutative ring with unit 1 0 and let i be a regular proper ideal of r the set p i of integrally closed ideals projectively equivalent to i is linearly ordered by inclusion and discrete there is naturally associated to p i a numerical semigroup s i we have s i n if and only if every element of p i is the integral closure of a power of the largest element j of p i if this holds the ideal j and the set p i are said to be projectively full if i is invertible and r is integrally closed we prove that p i is projectively full we investigate the behavior of projectively full ideals in various types of ring extensions we prove that a normal ideal i of a local ring r m is projectively full if i m 2 and both the associated graded ring g m and the fiber cone ring f i are reduced we present examples of normal local domains r m of altitude two for which the maximal ideal m is not projectively full

strong convergence rate of splitting schemes for stochastic nonlinear schrodinger equations abstract in this paper we show that solutions of stochastic nonlinear schrodinger nls equations can be approximated by solutions of coupled splitting systems based on these systems we propose a new kind of fully discrete splitting schemes which possess algebraic strong convergence rates for stochastic nls equations key ingredients of our approach are using the exponential integrability and stability of the corresponding splitting systems and numerical approximations in particular under very mild conditions we derive the optimal strong convergence rate o n 2 1 2 of the spectral splitting crank nicolson scheme where n and denote the dimension of the approximate space and the time step size respectively

quasi one fibered ideals in two dimensional muhly local domains abstract in this paper a two dimensional integrally closed noetherian local domain r m with algebraically closed residue field is said to be a muhly local domain if the associated graded ring is an integrally closed domain using the theory of degree functions developed by rees and sharp and the zariski lipman theory of complete ideals in two dimensional regular local rings we study m primary ideals of r having among their rees valuations exactly one valuation ord r we therefore call these ideals quasi one fibered some consequences are deduced

a class of projectively full ideals in two dimensional muhly local domains abstract let r m be a regular local domain of dimension d 2 and let x 1 x d be a regular system of parameters then ciuperca heinzer ratliff jr and rush have proved that every ideal of the form i x 1 n x 2 x d r with n n is projectively full if r m is a two dimensional muhly local domain i e an integrally closed noetherian local domain with algebraically closed residue field and the associated graded ring an integrally closed domain then we are able to prove a similar result for every minimal ideal basis x 1 x d of m such that x 1 rad x 2 x d

essential sequences and rees rings abstract it is shown that if b 1 b s is a u essential sequence over an ideal i in at noetherian ring r then certain permutations of u tb 1 tb i b i 1 b s are a u esential sequence of ir r b i and over tir r i b i where b i b 1 h b i r 0 i s andr r j is the rees ring of r with respect to its ideal j a number of related results are also given concerning u essential sequences over i and over id or i b k d where d is the monadic transformation ring r b i b k or r i b i b k when r is local the u essential cograde of i a containment relation between the essential prime divisors of b i and the u essential prime divisors of i b i and the fact that every permutation of b 1 b s is a u essential sequence over i and is an essential sequence in r

essential prime divisors and projectively equivalent ideals let z be an ideal in a noetherian ring r we concern ourselves with the essential prime divisors of z an interesting subset of ass r p for all large n we first take z br with b a regular element of r we show that there is a ring t with r c te r such that t is a finite r module and the essential primes of bt are exactly the prime divisors of bt we next consider an arbitrary ideal z and apply our principal arguments to the element u in the kees ring of z we thereby deduce that there is an idcal j projectively equivalent to z such that the set of essential primes of z equals the set u ass r j over ii 1 2 3 notation let z be an ideal in a noetherian ring r we will use w or

essential sequences and projective equivalence it is well known that if r is a noetherian ring and b1 bh are an r sequence then there exist ideals 0 b0 b1 bh in r such that the following hold for i 1 h a bi b1 bi r b ass rbi ass r bi k k 1 c bi 1 k bi r bi 1 k for all k 1 and d if p ass rbi 1 and p bi r r then p is properly contained in some p ass rbi the main theorem in this paper shows that something quite similar to this holds for essential sequences namely if bi bh are an essential sequence in r then there exists ideals 0 j0 j1 jh in r such that the following hold for i 1 h a ji is projectively equivalent to bi bi r b ass rji ass r ji k k 1 c ji 1 k bir ji 1 k for all k 1 and d if p ass rji 1 and p bi r r then p is properly contained in some p ass rji

monoidal extensions of a cohen macaulay unique factorization domain let a be a noetherian cohen macaulay domain b c 1 c g an a sequence j b c 1 c g a and b a j b then b is cohen macaulay there is a natural one to one correspondence between the sets ass b b bb and ass a a j and each q ass a a j has height g 1 if b does not have unique factorization then some height one prime ideals p of b are not principal these primes are identified in terms of j and p a and we consider the question of how far from principal they can be if a is integrally closed necessary and sufficient conditions are given for b to be integrally closed and sufficient conditions are given for b to be a ufd or a krull domain whose class group is torsion finite or finite cyclic it is shown that if p is a height one prime ideal of b then p a also has height one if and only if b p and thus p a has height one for all but finitely many of the height one primes p of b if a has unique factorization a description is given of whether or not such a prime p is a principal prime ideal or has a principal primary ideal in terms of properties of p n a a similar description is also given for the height one prime ideals p of b with p a of height greater than one if the prime factors of b satisfy a mild condition if a is a ufd and b is a power of a prime element then b is a krull domain with torsion class group if and only if j is primary and integrally closed and if this holds then b has finite cyclic class group also if j j is not primary then for each height one prime ideal p contained in at least one but not all prime divisors of j it holds that the height one prime pa 1 b b has no principal primary ideals this applies in particular to the rees ring r a 1 t tj as an application of these results it is shown how to construct for any finitely generated abelian group g a monoidal transform b a j b such that a is a ufd b is cohen macaulay and integrally closed and g cl b the divisor class group of b

a bracket power characterization of analytic spread one ideals the main theorem characterizes in terms of bracket powers analytic spread one ideals in local rings specifically let bl bg x be r egular nonunits in a local noetherian ring r m and assume that i c xr a the integral closure of xr where i b1 bg x r then the main result shows that for all but finitely many units u1 ug in r that are non congruent modulo m and for all large integers n and k it holds that jin ii for j 1 i k and j not divisible by char r m where i i is the j th bracket power bi uix i bg ugx i xi r of i bi ulx bg ugx x r and conversely if there exist positive integers g n and k n g such that i has a basis 31 39 x such that ilkn 3i k 3gk xk n r then i has analytic spread one

numerical simulation of focusing stochastic nonlinear schrodinger equations abstract in this paper we numerically investigate nonlinear schrodinger equations with a stochastic contribution which is of white noise type and acts either as a potential multiplicative noise or as a forcing term additive noise in the subcritical case we recover similar results as in the case of the korteweg de vries equation in the critical or supercritical case we observe that depending on its smoothness the noise may have different effects spatially smooth noises amplify blow up phenomena whereas delta correlated multiplicative noises prevent blow up formation note that in this latter case very few results are known both from a theoretical and a numerical point of view

semigroup splitting and cubature approximations for the stochastic navier stokes equations approximation of the marginal distribution of the solution of the stochastic navier stokes equations on the two dimensional torus by high order numerical methods is considered the corresponding rates of convergence are obtained for a splitting scheme and the method of cubature on wiener space applied to a spectral galerkin discretization of degree n while the estimates exhibit a strong n dependence convergence is obtained for appropriately chosen time step sizes results of numerical simulations are provided and confirm the applicability of the methods

projectively equivalent ideals and rees valuations let r be a noetherian ring two ideals i and j in r are projectively equivalent in case the integral closure of ii is equal to the integral closure of jj for some i j n it is known that if i and j are projectively equivalent then the set reesi of rees valuation rings of i is equal to the set reesj of rees valuation rings of j and the values of i and j with respect to these rees valuation rings are proportional we observe that the converse also holds in particular if the ideal i has only one rees valuation ring v then the ideals j projectively equivalent to i are precisely the ideals j such that reesj v in certain cases such as i dimr 1 or ii r is a two dimensional regular local domain we observe that if i has more than one rees valuation ring then there exist ideals j such that reesi reesj but j is not projectively equivalent to i if i and j are regular ideals of r we prove that reesi reesj reesij with equality holding if dimr 2 but not holding in general if dimr 3 we associate to i and to the set p i of integrally closed ideals projectively equivalent to i a numerical semigroup s i n such that s i n if and only if there exists j p i for which p i jn a n n

numerical approximation of stochastic evolution equations convergence in scale of hilbert spaces the research of hakima bessaih was supported by the nsf united states grant dms 1418838 the research of erika hausenblas was supported by the austrian science fund fwf austria p25968 the research of tsiry randrianasolo was supported by the austrian science fund fwf austria p25968 and the german research council germany as part of the collaborative research center sfb 1283

on the asymptotic cograde of an ideal abstract let i be an ideal in a local ring r m and let s i be the asymptotic cograde of i the maximum length of an asymptotic sequence over i rees has shown that s i altitude r l i where l i is the analytic spread of i we prove several more bounds on s i i s i min little depth p p is an asymptotic prime divisor of i ii s i grade m i the maximum length of an asymptotic sequence contained in m i iii s i grade m grade i iv grade m l i s i and v if r is quasi unmixed then grade r i n s i for all large n also it is shown that the asymptotic prime divisors of i b 1 b s r contain those of i when b 1 b s are an asymptotic sequence over i and that the residue classes modulo i of such a sequence are an asymptotic sequence in r i finally several sufficient conditions for l i to be equal to l i p for some asymptotic prime divisor p of i are given

bounds related to projective equivalence classes of ideals let z be a regular ideal in noetherian ring r and let i denote the integral closure of i a situation which is often of interest is when there is an integer b z 2 0 such that i b a e i for all n 2 1 for instance in an analytically unramified local ring such a b z exists for all i now recall that the ideal h is projectively equivalent to i if there are positive integers r and s with h i in section 1 we make the easy observation that the existence of b z as above is an invariant of the projective equivalence class of i that is if such a b z exists and if the ideal h is projectively equivalent to z then there is a b h with h bch n c h for all n 2 1 of course it may happen that b h b z the main purpose of section 1 is to study the stronger condition that there exist a fixed integer b 3 0 such that for all ideals h projectively equivalent to z h c h for all n 1 we make use of a key result namely that given z there exists an integer d such that for any ideal h projectively equivalent to z one can write hd p for some s let s 2 1 be an integer and let the ideal k be a reduction of i this means there is an integer e 20 depending on s and k with k z z in section 2 we discuss a situation in which e can in fact

well posedness and optimal regularity of stochastic evolution equations with multiplicative noises abstract in this paper we establish the well posedness and optimal trajectory regularity for the solution of stochastic evolution equations with generalized lipschitz type coefficients driven by general multiplicative noises to ensure the well posedness of the problem the linear operator of the equations is only need to be a generator of a c 0 semigroup and the proposed noises are quite general which include space time white noise and rougher noises when the linear operator generates an analytic c 0 semigroup we derive the optimal trajectory regularity of the solution through a generalized criterion of factorization method

effective normality criteria for algebras of linear type abstract the algebras studied here are subalgebras of rings of polynomials generated by 1 forms so called rees algebras with coefficients in a noetherian ring given a normal domain r and a torsionfree module e with a free resolution f 2 f 1 f 0 e 0 we study the role of the matrices of syzygies in the normality of the rees algebra of e when the rees algebra r e and the symmetric algebra s e coincide the main results characterize normality in terms of the ideal ic s e and of the completeness of the first s symmetric powers of e where c rank and s rankf0 ranke it requires that r be a regular domain special results under broader conditions on r are still more effective

normal sally modules of rank one in this paper we explore the structure of the normal sally modules of rank one with respect to an m m primary ideal in a nagata reduced local ring r which is not necessary cohen macaulay as an application of this result when the base ring is cohen macaulay analytically unramified the extremal bound on the first normal hilbert coefficient leads to the depth of the associated graded rings g with respect to a normal filtration is at least dim r 1 dim r 1 and g turns in to cohen macaulay when the third normal hilbert coefficient is vanished

notes on projectively related ideals and residual division abstract let i i i i g and j j i j f be finite collections of ideals of a noetherian ring r then i and j are projectively related in case the integral closures of the ideals i i i i i i g g and j j i i j j f f are equal for some positive integers i 1 i g and j 1 j f some basic properties of this relation are proved and then these are applied when either g 1 or each i i is regular to give several necessary and sufficient conditions for the existence of an ideal k that is projectively related to i such that k n g k n for all n 1 and for all g in a given multiplicatively closed set of nonzero ideals of r it is then shown that if there exists such an ideal k in r then for certain rings a related to r there exist ideals l projectively related to i a such that l n ga l n for all n 1 and for all g

integrally closed almost complete intersection ideals abstract let a m be a local ring we study the existence and structure of the m primary integrally closed ideals generated by dim a 1 elements

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

projective equivalence of ideals in noetherian integral domains abstract let i be a nonzero proper ideal in a noetherian integral domain r in this paper we establish the existence of a finite separable integral extension domain a of r and a positive integer m such that all the rees integers of ia are equal to m moreover if r has altitude one then all the rees integers of j rad i a are equal to one and the ideals j m and ia have the same integral closure thus rad i a j is a projectively full radical ideal that is projectively equivalent to ia in particular if r is dedekind then there exists a dedekind domain a having the following properties i a is a finite separable integral extension of r and ii there exists a radical ideal j of a and a positive integer m such that i a j m in this case the extension a also has the property that for each maximal ideal n of a with i n the canonical inclusion r n r a n is an isomorphism and the integer m is a multiple of a 0 r 0

from rough path estimates to multilevel monte carlo new classes of stochastic differential equations can now be studied using rough path theory see e g t j lyons m caruana and t levy differential equations driven by rough paths springer berlin 2007 or p k friz and m hairer a course on rough paths universitext springer cham 2014 in this paper we investigate from a numerical analysis point of view stochastic differential equations driven by gaussian noise in the aforementioned sense our focus lies on numerical implementations and more specifically on the savings possible via multilevel methods our analysis relies on a subtle combination of pathwise estimates gaussian concentration and multilevel ideas numerical examples are given which both illustrate and confirm our findings

a semi discrete scheme for the stochastic nonlinear schrodinger equation we study the convergence of a semi discretized version of a numerical scheme for a stochastic nonlinear schrodinger equation the nonlinear term is a power law and the noise is multiplicative with a stratonovich product our scheme is implicit in the deterministic part of the equation as is usual for conservative equations we also use an implicit discretization of the noise which is better suited to stratonovich products we consider a subcritical nonlinearity so that the energy can be used to obtain an a priori estimate however in the semi discrete case no ito formula is available and we have to use a discrete form of this tool also in the course of the proof we need to introduce a cut off of the diffusion coefficient which allows to treat the nonlinearity then we prove convergence by a compactness argument due to the presence of noise and to the implicit discretization of the noise this is rather complicated and technical we finally obtain convergence of the discrete solutions in various topologies

projectively full radical ideals in integral extension rings let r be a noetherian commutative ring with unit 1 0 and let i be a regular proper ideal of r the main question considered in this paper is whether there exists a finite integral extension ring a of r for which the nilradical of ia is a projectively full ideal that is projectively equivalent to ia a related and stronger question that we also consider is whether there exists a finite integral extension ring a of r for which the nilradical j of ia is projectively equivalent to ia and for which all the rees integers of j are one the following two results are special cases of the main theorems in the present paper 1 if r is a noetherian integral domain then there exists a finite integral extension ring a of r such that the nilradical of ia is projectively equivalent to ia 2 if also r contains a field of characteristic zero then there exists a finite free integral extension ring a of r for which the nilradical of ia is a projectively full ideal that is projectively equivalent to ia