integral equation theory for fluids ordered by an external field separable interactions the structural and thermodynamical properties of classical fluids orientationally ordered by an external field are investigated by means of integral equation theories a general theoretical framework for handling these theories is developed and detailed for the particular case of separable interactions between fluid particles this approach is then illustrated for the case of two off lattice models the ferromagnetic heisenberg model and a simple liquid crystal model for which the numerical solution of integral equations such as the percus yevick the hypernetted chain and the reference hypernetted chain closure equations are presented and compared with monte carlo simulation results and the analytical solution of the mean spherical approximation the zero field case is also examined and the spontaneous ordering is analyzed in detail mainly in what concerns the appearance of infinite wavelength singularity in the ornstein zernike equation and the relation with the one body closure equations and the long range orientational ordering that occurs in particular it is shown that the wertheim one body closure equation appears as a sum rule compatible with the ornstein zernike equation the relation between the elastic constant and the long range tail of the pair correlation function is made explicit in particular the long range behavior of the various terms in the expansion of the pair correlation function is depicted the numerical investigation of the two models shows that it is not possible to discriminate between the four integral equations as to which one would be the most accurate in all cases the general trends in the thermodynamical and structural properties seem to indicate that the percus yevick approximation is generally better in the strong ordering case whereas the reference hypernetted chain approximation might be better suited to the study of the isotropic phase and the low ordering regimes

symbolic powers of sums of ideals let i and j be nonzero ideals in two noetherian algebras a and b over a field k let i j denote the ideal generated by i and j in a otimes k b we prove the following expansion for the symbolic powers begin aligned i j n sum i j n i i j j end aligned if a and b are polynomial rings and if mathrm char k 0 or if i and j are monomial ideals we give exact formulas for the depth and the castelnuovo mumford regularity of i j n which depend on the interplay between the symbolic powers of i and j the proof involves a result of independent interest which states that the induced map mathrm tor i a k i n rightarrow mathrm tor i r k i n 1 is zero for any homogeneous ideal i and i ge 0 n ge 0 we also investigate other properties and invariants of i j n such as the equality between ordinary and symbolic powers the waldschmidt constant and the cohen macaulayness

phase diagram in a continuum model of the classical heisenberg ferromagnet mean spherical approximation abstract the continuum heisenberg model of spin fluid is considered the pair correlation functions and the single particle distribution function of ferromagnetic state are obtained by the self consistent solution of the ornstein zernike and the lovett equations in the mean spherical approximation on the basis of the analytical solution the phase diagram is calculated and compared with the available monte carlo results for the same model the agreement between theory and simulation is quite satisfactory especially for the liquid gas coexistence line the msa suggests that only ordered states are involved in the gas liquid transition near the gas liquid critical point this result confirms the monte carlo prediction about the existence of a magnetic critical point

ordinary and symbolic powers are golod abstract let s be a positively graded polynomial ring over a field of characteristic 0 and i s a proper graded ideal in this note it is shown that s i is golod if i 2 i here i denotes the ideal generated by all the partial derivatives of elements of i we apply this result to find large classes of golod ideals including powers symbolic powers and saturations of ideals

containment problem for points on a reducible conic in p2 abstract given an ideal i in a noetherian ring one can ask the containment question for which m and r is the symbolic power i m contained in the ordinary power i r c bocci and b harbourne study the containment question in a geometric setting where the ideal i is in a polynomial ring over a field like them we will consider special geometric constructs in particular we obtain a complete solution in two extreme cases of ideals of points on a pair of lines in p 2 in one case the number of points on each line is the same while in the other all the points but one are on one of the lines

on multiplicities of graded sequences of ideals abstract we generalize a result from l ein et al math ag 0202303 proving that for an arbitrary graded sequence of zero dimensional ideals the multiplicity of the sequence is equal to its volume this is done using a deformation to monomial ideals as a consequence of our result we obtain a formula which computes the multiplicity of an ideal i in terms of the multiplicities of the initial monomial ideals of the powers i m we use this to give a new proof of the inequality between multiplicity and the log canonical threshold from de fernex et al j alg geom to appear

a vanishing theorem and asymptotic regularity of powers of ideal sheaves abstract let i be an ideal sheaf on p n in the first part of this paper we bound the asymptotic regularity of powers of i as s p reg i p s p e where e is a constant and s is the s invariant of i we also give the same upper bound for the asymptotic regularity of symbolic powers of i under some conditions in the second part by using multiplier ideal sheaves we give a vanishing theorem of powers of i when it defines a local complete intersection subvariety with log canonical singularities

analytic approximations of plurisubharmonic singularities we study several classes of isolated singularities of plurisubharmonic functions that can be approximated by analytic singularities with control over their residual monge ampere masses they are characterized in terms of green functions for demailly s approximations relative types and valuations furthermore the classes are shown to appear when studying graded families of ideals of analytic functions and the corresponding asymptotic multiplier ideals

linear subspaces symbolic powers and nagata type conjectures abstract inspired by results of guardo van tuyl and the second author for lines in p 3 we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r dimensional planes in p n for n 2 r 1 these considerations lead to new conjectures that suggest that the well known conjecture of nagata for points in p 2 is not an exotic statement but rather a manifestation of a much more general phenomenon which seems to have been overlooked so far

depth functions of symbolic powers of homogeneous ideals this paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal our investigation is focused on the behavior of the function mathrm depth r i t dim r mathrm pd i t 1 where i t denotes the t th symbolic power of a homogeneous ideal i in a noetherian polynomial ring r and mathrm pd denotes the projective dimension it has been an open question whether the function mathrm depth r i t is non increasing if i is a squarefree monomial ideal we show that mathrm depth r i t is almost non increasing in the sense that mathrm depth r i s ge mathrm depth r i t for all s ge 1 and t in e s where begin aligned e s bigcup i ge 1 t in mathbb n i s 1 1 le t le is end aligned which contains all integers t ge s 1 2 1 the range e s is the best possible since we can find squarefree monomial ideals i such that mathrm depth r i s mathrm depth r i t for t not in e s which gives a negative answer to the above question another open question asks whether the function mathrm depth r i t is always constant for t gg 0 we are able to construct counter examples to this question by monomial ideals on the other hand we show that if i is a monomial ideal such that i t is integrally closed for t gg 0 e g if i is a squarefree monomial ideal then mathrm depth r i t is constant for t gg 0 with begin aligned lim t rightarrow infty mathrm depth r i t dim r dim oplus t ge 0 i t mathfrak m i t end aligned our last result which is the main contribution of this paper shows that for any positive numerical function phi t which is periodic for t gg 0 there exist a polynomial ring r and a homogeneous ideal i such that mathrm depth r i t phi t for all t ge 1 as a consequence for any non negative numerical function psi t which is periodic for t gg 0 there is a homogeneous ideal i and a number c such that mathrm pd i t psi t c for all t ge 1

on the postulation of s d fat points in p d in connection with his counter example to the fourteenth problem of hilbert nagata formulated a conjecture concerning the postulation of r fat points of the same multiplicity in p2 and proved it when r is a square iarrobino formulated a similar conjecture in pd we prove iarrobino s conjecture when r is a dth power as a corollary we obtain new counter examples modeled on those by nagata

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

comparison of symbolic and ordinary powers of ideals all given rings in this paper are commutative associative with identity and noetherian recently l ein r lazarsfeld and k smith els discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0 if h is the largest height of an associated prime of i then i hn i n for all n 0 here if w is the complement of the union of the associated primes of i i t denotes the contraction of i t rw to r where rw is the localization of r at the multiplicative system w their proof depends on the theory of multiplier ideals including an asymptotic version and in particular requires resolution of singularities as well as vanishing theorems we want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist our objective here is to give stronger results that can be proved by methods that are in some ways more elementary our results are valid in both equal characteristic 0 and in positive prime characteristic p but depend on reduction to characteristic p we use tight closure methods and in consequence we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic for the most basic form of the result all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring every ideal is tightly closed we note that the main argument here is closely related to a proof given in hu 5 14 16 p 45 that regular local rings in characteristic p are ufds

renormalisation group study of a lattice model of the isotropic nematic transition the isotropic nematic transition is studied using the lebwohl lasher lattice model of liquid crystal behaviour the details of the transition are investigated using the migdal kadanoff potential moving version of the real space renormalisation group the method gives good results when applied to the closely related heisenberg model of ferromagnetism a second order transition is predicted in disagreement with mean field and some computer simulation results

a geometric effective nullstellensatz the purpose of this paper is to present a geometric theorem which clarifies and extends in several directions work of brownawell kollar and others on the effective nullstellensatz specifically we work on an arbitrary smooth complex projective variety x with the previous classical results corresponding to the case when x is projective space in this setting we prove a local effective nullstellensatz for ideal sheaves and a corresponding global division theorem for adjoint type bundles we also make explicit the connection with the intersection theory of fulton and macpherson finally constructions involving products of prime ideals that appear in earlier work are replaced by geometrically more natural conditions involving orders of vanishing along subvarieties much of the previous activity in this area has been algebraic in nature and seems perhaps not well known in detail among geometers therefore we have felt it worthwhile to include here a rather extended introduction we start with an overview of the questions and earlier work on them then we present the set up and statement of our main theorem we conclude with a series of examples which can be read before the general result of what it yields in special cases

on a non vanishing conjecture of kawamata and the core of an ideal we show under suitable hypothesis which are sharp in a certain sense that the core of an m primary ideal in a regular local ring of dimension d is equal to the adjoint or multiplier ideal of its d th power this generalizes the fundamental formula for the core of an integrally closed ideal in a two dimensional regular local ring due to huneke and swanson we also find a generalization of this result to singular non regular settings which we show to be intimately related to the problem of finding non zero sections of ample line bundles on projective varieties in particular we show that a graded analog of our formula for core would imply a remarkable conjecture of kawamata predicting that every adjoint ample line bundle on a smooth variety admits a non zero section

deformations of canonical singularities m reid defined the concept of canonical singularities as those which appear on canonical models of algebraic varieties whose canonical rings are finitely generated from the view point of the classification theory of algebraic varieties it is expected that small deformations of a variety which has only canonical singularities have only canonical singularities the main result of this paper states that this is the case

seshadri constants at very general points we study the local positivity of an ample line bundle at a very general point of a smooth projective variety we obtain a slight improvement of the result of ein kuchle and lazarsfeld

cohomology of projective schemes from annihilators to vanishing we provide bounds on the castelnuovo mumford regularity in terms of defining equations by using elements that annihilates some cohomology modules inspired by works of miyazaki nagel schenzel and vogel the elements in these annihilators are provided either by liaison or by tight closure theories our results hold in any characteristic

homology and some homotopy decompositions for the james filtration on spheres the filtrations on the james construction on spheres jk s2n have played a major role in the study of the double suspension s2n 1 f f2s2n l and have been used to get information about the homotopy groups of spheres and moore spaces and to construct product decompositions of related spaces in this paper we calculate h qjk s2n z pz for odd primes p when k has the form pt 1 the result is well known but these are exceptional cases in which the homology has polynomial growth we find that in general the homology has exponential growth and in some cases also has higher p torsion the calculations are applied to construct a p local product decomposition of qjk s2n for k p2 p which demonstrates a mod p homotopy exponent in these cases 0 introduction one of the major problems in homotopy theory is to determine the properties of spaces of the form qx where x is a simply connected finite complex a dichotomy in the rational homotopy of such spaces has been observed by felix halperin and thomas fht who showed that for such x either irn x 0 q is 0 for all sufficiently large n or else its dimensions grow exponentially with n expressed in terms of homology rather than homotopy groups this translates to saying that either dim hn qx q has polynomial growth or it has exponential growth they introduced the terminology elliptic for the first type of space and hyperbolic for the second independently john moore observed that this distinction between elliptic and hyperbolic appears to be reflected in the properties of the torsion homotopy groups of the space as well he conjectured that an elliptic complex should have a homotopy exponent for each prime while a hyperbolic complex should not have a homotopy exponent for any prime cf s3 that is given a prime p there should exist an r such that pr p torsion irn x 0 for all n if and only if x is elliptic given a prime p spaces can be further differentiated according to whether or not the mod p homology of their loop space has polynomial growth or not we say that x is mod p elliptic if hn qx z pz has polynomial growth and mod p hyperbolic otherwise it is not known whether the mod p homology of mod p hyperbolic spaces must have exponential growth of course if x is hyperbolic then it is mod p hyberbolic for each prime p but an elliptic space may be mod p hyperbolic as illustrated by the mod p moore space sk l up ek which we will write as pk p received by the editors july 21 1994 1991 mathematics subject classification primary 55p99 55p10 research partially supported by a grant from nserc 1996 american mathematical society 3549 this content downloaded from 207 46 13 149 on mon 03 oct 2016 06 09 35 utc all use subject to http about jstor org terms

on nagata s conjecture abstract this paper gives an improved lower bound on the degrees d such that for general points p 1 p n p 2 and m 0 there is a plane curve of degree d vanishing at each p i with multiplicity at least m

the uniform symbolic topology property for diagonally f regular algebras abstract let k be a field of positive characteristic building on the work of the second named author we define a new class of k algebras called diagonally f regular algebras for which the so called uniform symbolic topology property ustp holds effectively we show that this class contains all essentially smooth k algebras we also show that this class contains certain singular algebras such as the affine cone over p k r p k s when k is perfect by reduction to positive characteristic it follows that ustp holds effectively for the affine cone over p c r p c s and more generally for complex varieties of diagonal f regular type

constructing product fibrations by means of a generalization of a theorem of ganea a theorem of ganea shows that for the principal homotopy fibration qb 4 f 4 e induced from a fibration f 4 e 4 b there is a product decomposition q e f 2 qb x q f qb we will determine the conditions for a fibration x 4 y 4 z to yield a product decomposition q z y x x q x y and generalize it to pushouts using this approach we recover some decompositions originally proved by very computational methods the results are then applied to produce after localization at an odd prime p homotopy decompositions for qjk s2 for some k which include the cases k pt the factors of qjpt s2 consist of the homotopy fibre of the attaching map s2pt 1 j4pt1 s2n for jpt s2n and combinations of spaces occurring in the snaith stable decomposition of q2s2n l

resurgences for ideals of special point configurations in bf p n coming from hyperplane arrangements abstract symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years with a notable recent focus on containment relations between symbolic powers and ordinary powers see for example 3 7 13 16 18 19 20 to cite just a few several invariants have been introduced and studied in the latter context including the resurgence and asymptotic resurgence 3 15 there have been exciting new developments in this area recently it had been expected for several years that i n r n 1 i r should hold for the ideal i of any finite set of points in p n for all r 0 but in the last year various counterexamples have now been constructed see 11 17 8 all involving point sets coming from hyperplane arrangements in the present work we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal

on the jordan holder property for geometric derived categories abstract we prove that the semiorthogonal decompositions of the derived category of the classical godeaux surface x do not satisfy the jordan holder property more precisely there are two maximal exceptional sequences in this category one of length 11 the other of length 9 assuming the noetherian property for semiorthogonal decompositions one can define following kuznetsov the clemens griffiths component cg d for each fixed maximal decomposition d we then show that d b x has two different maximal decompositions for which the clemens griffiths components differ moreover we produce examples of rational fourfolds whose derived categories also violate the jordan holder property

invariance of plurigenera in this paper we give a proof of the following long conjectured result on the invariance of the plurigenera main theorem let p x d be a smooth projective family of compact complex manifolds parametrized by the open unit 1 disk d assume that the bers xt py1 t t 2 d are of general type then for every positive integer m the plurigenus dimc c xt mkxt is independent of t 2 d where kxt is the canonical line bundle of xt

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

asymptotic linear bounds for the castelnuovo mumford regularity we prove asymptotic linear bounds for the castelnuovo mumford regularity of certain filtrations of homogeneous ideals whose rees algebras need not be noetherian

uniform harbourne huneke bounds via flat extensions abstract over an arbitrary field f harbourne 3 conjectured that the symbolic power i n r 1 1 i r for all r 0 and all homogeneous ideals i in s f p n f x 0 x n the conjecture has been disproven for select values of n 2 first by dumnicki szemberg and tutaj gasinska in characteristic zero 7 and then by harbourne and seceleanu in positive characteristic 13 however the ideal containments above do hold when e g i is a monomial ideal in s 3 ex 8 4 5 as a sequel to 21 we present criteria for containments of type i n r 1 1 i r for all r 0 and certain classes of ideals i in a prodigious class of normal rings of particular interest is a result for monomial primes in tensor products of affine semigroup rings indeed we explain how to give effective multipliers n in several cases including the d th veronese subring of any polynomial ring f x 1 x n n 1 and the extension ring f x 1 x n z z d x 1 x n of f x 1 x n

formulas for multiplier ideals on singular varieties we prove a generalization of demailly ein lazarsfeld s subadditivity formula and musta s summation formula for multiplier ideals to the case of singular varieties using characteristic p methods as an application of our formula we improve hochster huneke s result on the growth of symbolic powers of ideals in singular affine algebras

f signature of pairs and the asymptotic behavior of frobenius splittings abstract we generalize f signature to pairs r d where d is a cartier subalgebra on r as defined by the first two authors in particular we show the existence and positivity of the f signature for any strongly f regular pair in one application we answer an open question of aberbach and enescu by showing that the f splitting ratio of an arbitrary f pure local ring is strictly positive furthermore we derive effective methods for computing the f signature and the f splitting ratio in the spirit of the work of r fedder

local syzygies of multiplier ideals the purpose of this note is to prove an elementary but somewhat surprising result concerning the syzygies of multiplier ideals it follows in particular that in dimensions three or higher multiplier ideals are very special among all integrally closed ideals let x be a smooth complex algebraic variety of dimension d and let b ox be an ideal sheaf given a rational or real number c 0 one can construct the multiplier ideal

geometric nullstellensatz and symbolic powers on arbitrary varieties in recent years a multiplier ideal defined on arbitrary varieties so called mather jacobian multiplier ideal has been developed independently by ein ishii mustata and de fernex docampo with this new tool we have a chance of extending some classical results proved in nonsingular case to arbitrary varieties to establish their general forms in this paper we first extend a result of geometric nullstellensatz due to ein lazarsfeld in nonsingular case to any projective varieties then we prove a result on comparison of symbolic powers with ordinary powers on any varieties which extends results of ein lazarsfeld smith and hochster huneke

a relation between one point and multi point seshadri constants abstract t szemberg proposed in 2001 a generalization to arbitrary varieties of m nagata s 1959 open conjecture which claims that the seshadri constant of r 9 very general points of the projective plane is maximal here we prove that nagata s original conjecture implies szemberg s for all smooth surfaces x with an ample divisor l generating ns x and such that l2 is a square more generally we prove the inequality e n 1 l r e n 1 l 1 e n 1 o p n 1 r where en 1 l r stands for the n 1 dimensional seshadri constant of the ample divisor l at r very general points of a normal projective variety x and n dimx

counterexamples to the i 3 i2 containment abstract the purpose of this short note is to show that there is in general no containment i 3 i 2 for an ideal i of points in p 2 this answers in the negative a question asked by huneke and generalized by harbourne the sets of points constituting counterexamples come from the dual of the hesse configuration and more generally from fermat arrangements

fat lines in p3 powers versus symbolic powers abstract we study the symbolic and regular powers of ideals i for a family of special configurations of lines in p 3 for this family we show that i m i m for all integers m if and only if i 3 i 3 we use these configurations to answer a question of huneke that asks whether i m i m for all m if equality holds when m equals the big height of the ideal i

uniform approximation of abhyankar valuation ideals in smooth function fields fix a rank one valuation centered at a smooth point x on an algebraic variety over a field of characteristic zero assume that is abhyankar that is that its rational rank plus its transcendence degree equal the dimension of the variety let a m denote the ideal of elements in the local ring of x whose valuations are at least m our main theorem is that there exists k 0 such that a mn is contained in a m k n for all m and n this can be viewed as a greatly strengthened form of izumi s theorem for abhyankar valuations centered on smooth complex varieties the proof uses the theory of asymptotic multiplier ideals

chudnovsky s conjecture for very general points in pkn abstract we prove a long standing conjecture of chudnovsky for very general and generic points in p k n where k is an algebraically closed field of characteristic zero and for any finite set of points lying on a quadric without any assumptions on k we also prove that for any homogeneous ideal i in the homogeneous coordinate ring r k x 0 x n chudnovsky s conjecture holds for large enough symbolic powers of i

bridging the gap between phenomenology and microscopic theory asymptotes in nematic colloids the ornstein zernike equation is applied to nematic colloids with up down symmetry to determine how the electrostatic analogy and other phenomenological results appear in molecular theory in contrast to phenomenological approaches the molecular theory does not assume particular boundary conditions anchoring at colloidal surfaces for our molecular parameters the resulting anchoring appears to be realistic neither rigid nor infinitely weak for this case the effective force between a colloidal pair at large separation remains essentially constant over the entire region of nematic stability we show that a simple van der waals approximation gives a potential of mean force that in some important aspects is similar to the phenomenological results obtained in the limit of weak anchoring at large separations the potential varies as sigma8 where sigma is the colloidal diameter in contrast the more sophisticated mean spherical approximation yields a sigma6 dependence consistent with phenomenological calculations employing rigid boundary conditions we show that taking proper account of the correlation or magnetic coherence length xi inherent in the nematic sample is essential in an analysis of the sigma dependence at infinite xi the leading sigma dependence is sigma6 but this shifts to sigma8 when xi is finite the correlation length also influences the orientational behavior of the effective interaction the so called quadrupole interaction that determines the long range behavior at infinite xi transforms into a superposition of screened multipoles when xi is finite the basic approach employed in this paper can be readily applied to a broad range of physically interesting systems these include patterned and nonspherical colloids colloids trapped at interfaces and nematic fluids in confined geometries such as droplets

the alpha problem line count configurations abstract motivated by the work of chudnovsky and the eisenbud mazur conjecture on evolutions harbourne and huneke give a series of conjectures that relate symbolic and regular powers of ideals of fat points in projective space p n the conjectures involve both containment statements and bounds for the initial degree in which there is a non zero form in an ideal working with initial degrees we verify two of these conjectures for special line count configurations in projective 2 space over an algebraically closed field of characteristic 0

prime ideals in ultraproducts of commutative rings abstract we describe classes of prime ideals in ultraproducts of commutative rings we consider in particular prime ideals in ultraproducts of noetherian rings krull domains finite character rings and qr domains

uniform symbolic topologies in normal toric rings abstract given a normal toric algebra r we compute a uniform integer d d r 0 such that the symbolic power p d n p n for all n 0 and all monomial primes p we compute the multiplier d explicitly in terms of the polyhedral cone data defining r illustrating the output for segre veronese algebras

star configurations in pn abstract star configurations are certain unions of linear subspaces of projective space they have appeared in several different contexts the study of extremal hilbert functions for fat point schemes in the plane the study of secant varieties of some classical algebraic varieties the study of the resurgence of projective schemes in this paper we study some algebraic properties of the ideals defining star configurations including getting partial results about hilbert functions generators and minimal free resolutions of the ideals and their symbolic powers we also show that their symbolic powers define arithmetically cohen macaulay subschemes and we obtain results about the primary decompositions of the powers of the ideals as an application we compute the resurgence for the ideal of the codimension n 1 star configuration in p n in the monomial case i e when the number of hyperplanes is n 1

molecular field theory approach to the landau theory of liquid crystals uniaxial and biaxial nematics nematic liquid crystal phase diagrams in temperature biaxiality space are usually complex we construct a landau theory based on the analogous molecular field theory for orthorhombic biaxial nematic fluids a formal procedure yields coefficients some of which unusually can be tensorial in this landau expansion correctly predicts the complete set of invariants formed from the ordering tensors and avoids ad hoc parametrization of the molecular biaxiality by regularizing the landau expansion to avoid unwanted order parameter divergences at low temperatures we predict phase behavior over the whole range of biaxiality the resulting phase diagrams have the same topology as those of molecular field theory

minimizing normalized volumes of valuations for any mathbb q gorenstein klt singularity x o we introduce a normalized volume function widehat mathrm vol that is defined on the space of real valuations centered at o and consider the problem of minimizing widehat mathrm vol we prove that the normalized volume has a uniform positive lower bound by proving an izumi type estimate for any mathbb q gorenstein klt singularity furthermore by proving a properness estimate we show that the set of real valuations with uniformly bounded normalized volumes is compact and hence reduce the existence of minimizers for widehat mathrm vol to a conjectural lower semicontinuity property for a smooth point we use an inequality of de fernex ein musta to show that the divisorial valuation associated to the exceptional divisor of the standard blow up is a minimizer of widehat mathrm vol finally the relation to fujita s work on divisorial stability is pointed out

associated points and integral closure of modules abstract let x spec r be an affine noetherian scheme and m n be a pair of finitely generated r modules denote their rees algebras by r m and r n let n n be the nth homogeneous component of r n and let m n be the image of the nth homogeneous component of r m in n n denote by m n be the integral closure of m n in n n we prove that ass x n n m n and ass x n n m n are asymptotically stable generalizing known results for the case where m is an ideal or where n is a free module suppose either that m and n are free at the generic point of each irreducible component of x or n is contained in a free r module when x is universally catenary we prove a generalization of a classical result due to mcadam and obtain a geometric classification of the points appearing in ass x n n m n notably we show that if x ass x n n m n for some n then x is the generic point of a codimension one component of the nonfree locus of n m or x is a generic point of an irreducible set in x where the fiber dimension proj r m x jumps we prove a converse to this result without requiring x to be universally catenary our approach is geometric in spirit also we recover strengthen and prove a sort of converse of an important result of kleiman and thorup about integral dependence of modules

ordinary and symbolic rees algebras for ideals of fermat point configurations abstract fermat ideals define planar point configurations that are closely related to the intersection locus of the members of a specific pencil of curves these ideals have gained recent popularity as counterexamples to some proposed containments between symbolic and ordinary powers 6 we give a systematic treatment of the family of fermat ideals describing explicitly the minimal generators and the minimal free resolutions of all their ordinary powers as well as many symbolic powers we use these to study the ordinary and the symbolic rees algebra of fermat ideals specifically we show that the symbolic rees algebras of fermat ideals are noetherian along the way we give formulas for the castelnuovo mumford regularity of powers of fermat ideals and we determine their reduction ideals

perfectoid multiplier test ideals in regular rings and bounds on symbolic powers using perfectoid algebras we introduce a mixed characteristic analog of the multiplier ideal respectively test ideal from characteristic zero respectively p 0 in the case of a regular ambient ring we prove several properties about this ideal such as subadditivity we then use these techniques to derive a uniform bound on the growth of symbolic powers of radical ideals in all excellent regular rings the analogous result was shown in equal characteristic by ein lazarsfeld smith and hochster huneke

when does the subadditivity theorem for multiplier ideals hold demailly ein and lazarsfeld proved the subadditivity theorem for multiplier ideals on nonsingular varieties which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals we prove that in the two dimensional case the subadditivity theorem holds on log terminal singularities however in the higher dimensional case we have several counterexamples we consider the subadditivity theorem for monomial ideals on toric rings and construct a counterexample on a three dimensional toric ring

xy spin fluids in an external magnetic field an integral equation approach the phase behavior of ising spin fluids is studied in the presence of an external magnetic field with the integral equation method the calculations are performed on the basis of a soft mean spherical approximation using an efficient algorithm for solving the coupled set of the ornstein zernike equations the closure relations and the external field constraint the phase diagrams are obtained in the whole thermodynamic space including the magnetic field h for a wide class of ising fluid models with various ratios r of the strengths of magnetic to nonmagnetic yukawa like interactions the influence of varying the inverse screening lengths z 1 and z 2 corresponding to the magnetic and nonmagnetic yukawa parts of the potential is investigated too it is shown that changes in r as well as in z 1 and z 2 can lead to different topologies of the phase diagrams in particular depending on the value of r the critical temperature of the liquid gas transition either decreases monotonically behaves nonmonotonically or increases monotonically with increasing h the para ferro magnetic transition is also affected by changes in r and the screening lengths at h 0 the ising fluid maps onto a simple model of a symmetric nonmagnetic binary mixture for h infinity it reduces to a pure nonmagnetic fluid the results are compared with available simulations and the predictions of other theoretical methods it is demonstrated that the mean spherical approximation appears to be more accurate compared with mean field theory especially for systems with short ranged attraction potentials when z 1 and z 2 are large in the kac limit z 1 z 2 0 both approaches tend to nearly the same results

uniform annihilators of local cohomology abstract in this paper we study the properties of noetherian rings containing uniform local cohomological annihilators it turns out that all such rings should be universally catenary and locally equidimensional we will prove a necessary and sufficient condition for such rings which enables us to show that if a locally equidimensional ring r is the image of a cohen macaulay ring then r has a uniform local cohomological annihilator moreover we will give a positive answer to a conjecture of huneke c huneke uniform bounds in noetherian rings invent math 107 1992 203 223 conjecture 2 13 about excellent rings with dimension no more than 5

asymptotic resurgences for ideals of positive dimensional subschemes of projective space abstract recent work of ein lazarsfeld smith and hochster huneke raised the containment problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal bocci harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings with a focus on zero dimensional subschemes of projective space here we take the first steps toward extending this work to higher dimensional subschemes we introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals i of smooth subschemes generalizing what is done in bocci and harbourne 2010 5 we apply these bounds to ideals of unions of general lines in p n we also pose a nagata type conjecture for symbolic powers of ideals of lines in p 3

quantifying singularities with differential operators abstract the f signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties for example this invariant detects non singularity and strong f regularity however it is very difficult to compute motivated by different aspects of the f signature we define a numerical invariant for rings of characteristic zero or p 0 that exhibits many of the useful properties of the f signature we also compute many examples of this invariant including cases where the f signature is not known we also obtain a number of results on symbolic powers and bernstein sato polynomials

integral equation procedure based on tailored orthogonal functions for the xy spin fluid in an external magnetic field the classical xy model describes particles in three dimensional space that carry magnetic moments or spins whose motion is restricted to rotations in a plane introduction of an external magnetic field lying in the same plane then generates a system that is anisotropic in the azimuthal angle phi we use numerical simulations and integral equation techniques to study this system producing in the latter case a formalism that is identical to that of the simpler isotropic version having no external field the basis for this simplification is a generalization em phi of the ordinary exponential basis set eimphi that restores orthogonality in the presence of the external field we display results of sample calculations obtained with two integral equation closures reference hypernetted chain and soft mean spherical approximation both coupled to the lovett mou buff wertheim relation along with results from the numerical simulations for comparison construction of the em phi is described in an appendix

uniform bounds on symbolic powers abstract we study the question of obtaining uniform bounds for the growth of symbolic powers of ideals in noetherian rings first we use a generalized version of artin s strong approximation theorem to obtain uniform bounds for prime ideals in the integral closure of a finite galois field extension of the fraction field of an excellent henselian regular local ring as a corollary we obtain a similar result for a complete noetherian local domains we also obtain linear uniform bounds for the growth of symbolic powers in certain normal subrings of equicharacteristic regular rings under mild conditions

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

the multiplier ideals of a sum of ideals we prove that if a b o x are nonzero sheaves of ideals on a complex smooth variety x then for every q we have the following relation between the multiplier ideals of a b and a b i x a b i x a i x b a similar formula holds for the asymptotic multiplier ideals of the sum of two graded systems of ideals we use this result to approximate at a given point arbitrary multiplier ideals by multiplier ideals associated to zero dimensional ideals this is applied to compare the multiplier ideals associated to a scheme in different embeddings

molecular field theory of nematics density functional approach i bulk effects the molecular field theory of nematics is rephrased in density functional language the free energy in the nematic state is expressed as a functional expansion around the isotropic state in powers of the nematic order parameters pl and the fractional density change eta at the isotropic nematic transition the crucial quantities in the expansion are spherical harmonics of the direct correlation functions in the isotropic state sensible approximations to these direct correlation functions lead to other well known formulations of nematic field theory an exact in the authors approximation result for eta is obtained in terms of the potential parameters ul and the order parameters pl at the phase transition the authors discuss the relationship of their approach to other microscopic theories of nematics the theory can be extended to discuss inhomogeneous systems

ideal containments under flat extensions abstract let s k y 0 y n r k y 0 y n be given by y i f i where f 0 f n is an r regular sequence of homogeneous elements of the same degree a recent paper shows for ideals i s of matroids that i m i r if and only if i m i r where i is the ideal generated in r by i we prove this result for saturated homogeneous ideals i of configurations of points in p n and use it to obtain many new counterexamples to i r n n 1 i r from previously known counterexamples