the 2 1 dimensional sine gordon equation integrability and localized solutions in this paper the 2 1 dimensional sine gordon equation 2dsg introduced by konopelchenko and rogers is investigated and is shown to satisfy the painleve property a variable coefficient hirota bilinear form is constructed by judiciously using the painleve analysis with a non conventional choice of the vacuum solutions first the line kinks are constructed then exact localized coherent structures in the 2dsgi equation are generated by the collision of two non parallel ghost solitons which drive the two non trivial boundaries present in the system also the reason for the difficulty in identifying localized solutions in the 2dsgii equation is indicated we also highlight the significance of the asymptotic values of the boundaries of the system

linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

ideals with stable betti numbers componentwise linear ideals were introduced earlier to generalize the result that the stanley reisner ideal i of a simplicial complex has a linear resolution if and only if its alexander dual is cohen macaulay it turns out that i is componentwise linear if and only if is sequentially cohen macaulay in this paper we discuss betti number properties of componentwise linear ideals let i be a graded ideal of a polynomial ring s and gin i the generic initial ideal of i with respect to the reverse lexicographic term order on s our main result is that i and gin i admit the same graded betti numbers if and only if i is componentwise linear for the proof of this fact we describe some properties of the betti diagram of a generic initial ideal combinatorial implications for shifted complexes will also be discussed

multigraded castelnuovo mumford regularity we develop a multigraded variant of castelnuovo mumford regularity motivated by toric geometry we work with modules over a polynomial ring graded by a finitely generated abelian group as in the standard graded case our definition of multigraded regularity involves the vanishing of graded components of local cohomology we establish the key properties of regularity its connection with the minimal generators of a module and its behavior in exact sequences for an ideal sheaf on a simplicial toric variety x we prove that its multigraded regularity bounds the equations that cut out the associated subvariety we also provide a criterion for testing if an ample line bundle on x gives a projectively normal embedding

castelnuovo mumford regularity by approximation abstract the castelnuovo mumford regularity of a module gives a rough measure of its complexity we bound the regularity of a module given a system of approximating modules whose regularities are known such approximations can arise naturally for modules constructed by inductive combinatorial means we apply these methods to bound the regularity of ideals constructed as combinations of linear ideals and the module of derivations of a hyperplane arrangement as well as to give degree bounds for invariants of finite groups

generalized castelnuovo mumford regularity for affine kac moody algebras abstract the graded modules over noncommutative algebras often have minimal free resolutions of infinite length resulting in infinite castelnuovo mumford regularity in kang et al 2010 6 we introduced a generalized notion of castelnuovo mumford regularity to overcome this difficulty in this paper we compute the generalized castelnuovo mumford regularity for integrable highest weight representations of all affine kac moody algebras it is shown that the generalized regularity depends only on the type and rank of algebras and the level of representations

a diagonal bound for cohomological postulation numbers of projective schemes abstract let x be a projective scheme over a field k and let f be a coherent sheaf of o x modules we show that the cohomological postulation numbers f i of f e g the ultimate places at which the cohomological hilbert functions n dim k h i x f n h f i n start to be polynomial for n 0 are polynomially bounded in terms of the cohomology diagonal h f i i i 0 dim f of f as a consequence we obtain that there are only finitely many different cohomological hilbert functions h f i if f runs through all coherent sheaves of o x modules with fixed cohomology diagonal in order to prove these results we extend the regularity bound of bayer and mumford computational algebraic geometry and commutative algebra proc cortona 1991 cambridge univ press 1993 pp 1 48 from graded ideals to graded modules moreover we prove that the castelnuovo mumford regularity of the dual f of a coherent sheaf of o p k r modules f is polynomially bounded in terms of the cohomology diagonal of f

symmetry group and non propagating solitons in the 2 1 dimensional sine gordon equation starting from the compact version of the 2 1 dimensional sine gordon equation the symmetry group and then the lie symmetries and the related algebra can be obtained via a simple combination of a gauge transformation of the spectral function and the transformations of the space time variables thereout applying the group transformation theorem on a two straight line soliton solution one can derive two types of meaningful non propagating dromion lattice excitations

the information encoded in initial ideals we consider homogeneous ideals i and the initial ideal in i for the revlex order first we give a sequence of invariants computed from i giving better and better approximations to the initial ideal and ending in an equivalent description then we apply this to different settings in algebraic geometry to understand what information is encoded in the generic initial ideal of the ideal of a projective scheme we also consider the higher initial ideals as defined in a paper by fl0ystad in particular we show that giving the generic higher initial ideal of a space curve is equivalent to giving the generic initial ideal of a linked curve

recent improvements in the complexity of the effective nullstellensatz we bring up to date the estimates on the complexity of the effective nullstellensatz and the membership problem

a combinatorial proof of the eisenbud goto conjecture for monomial curves and some simplicial semigroup rings abstract we will give a pure combinatorial proof of the eisenbud goto conjecture for arbitrary monomial curves moreover we will show that the conjecture holds for certain simplicial affine semigroup rings

on the componentwise linearity and the minimal free resolution of a tetrahedral curve abstract a tetrahedral curve is an unmixed usually nonreduced one dimensional subscheme of projective 3 space whose homogeneous ideal is the intersection of powers of the ideals of the six coordinate lines the second and third authors have shown that these curves have very nice combinatorial properties and they have made a careful study of the even liaison classes of these curves we build on this work by showing that they are almost always componentwise linear i e their homogeneous ideals have the property that for any d the degree d component of the ideal generates a new ideal whose minimal free resolution is linear the one type of exception is clearly spelled out and studied as well the main technique is a careful study of the way that basic double linkage behaves on tetrahedral curves and the connection to the tetrahedral curves that are minimal in their even liaison classes with this preparation we also describe the minimal free resolution of a tetrahedral curve and in particular we show that in any fixed even liaison class there are only finitely many tetrahedral curves with linear resolution finally we begin the study of the generic initial ideal gin of a tetrahedral curve we produce the gin for arithmetically cohen macaulay tetrahedral curves and for minimal arithmetically buchsbaum tetrahedral curves and we show how to obtain it for any nonminimal tetrahedral curve in terms of the gin of the minimal curve in that even liaison class

symmetry analysis and exact solutions of the 2 1 dimensional sine gordon system according to the space variable exchange symmetry we change the 2 1 dimensional sine gordon system obtained by konopelchenko and rogers kr to a variant form some types of similarity reductions are obtained by using some lie symmetry analysis from these similarity reductions we find that the soliton structure of the system possesses quite a rich structure the line solitons parallel to the lines x y 0 and x y 0 may have an arbitrary shape in addition to the well known dromion solution which is constructed by two line solitons one may find many other kinds of soliton solutions localized in all directions for instance some kinds of ring type basin like plateau like and bowl like and instanton type soliton solutions can be found directly by selecting the arbitrary functions included in the single soliton solution the dromion solutions may also be constructed by straight line and curved line solitons

on generalized loewner systems novel integrable equations in 2 1 dimensions a reinterpretation and generalization of a class of infinitesimal backlund transformations originally introduced in a gas dynamics context by loewner in 1952 leads to a linear representation for a novel class of 2 1 dimensional nonlinear equations the latter may be parametrized in terms of a triad of eigenfunctions moreover like the well known integrable davey stewartson and nizhnik novikov veselov equations the nonlinear systems typically contain the two spatial variables on an equal footing the dressing method is outlined for these generalized loewner systems it is noted that basic reductions lead to 2 1 dimensional versions of the principal chiral fields model toda lattice system and notably the classical sine gordon equation loewner systems on grassmannian the projective cpn and rpn manifolds are considered in particular the 2 1 dimensional integrable sine gordon system in which x and y occur in a symmetric manner arises naturally in this context a gauge equivalent system likewise eme

comparing castelnuovo mumford regularity and extended degree the borderline cases castelnuovo mumford regularity and any extended degree function can be thought of as complexity measures for the structure of finitely generated graded modules a recent result of doering gunston and vasconcelos shows that both can be compared in the case of a graded algebra we extend this result to modules and analyze when the estimate is in fact an equality a complete classification is obtained if we choose as extended degree the homological or the smallest extended degree the corresponding algebras are characterized in three ways by relations among the algebra generators by using generic initial ideals and by their hilbert series

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

saturation and castelnuovo mumford regularity abstract let i be a homogeneous ideal of the polynomial ring k x 0 x n where k is an arbitrary field avoiding the construction of a minimal graded free resolution of i we provide effective methods for computing the castelnuovo mumford regularity of i that also compute other cohomological invariants of k x 0 x n i

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

the lex plus powers inequality for local cohomology modules we prove an inequality between hilbert functions of local cohomology modules supported in the homogeneous maximal ideal of standard graded algebras over a field within the framework of embeddings of posets of hilbert functions as a main application we prove an analogue for local cohomology of evans lex plus powers conjecture for betti numbers this results implies some cases of the classical lex plus powers conjecture namely an inequality between extremal betti numbers in particular for the classes of ideals for which the eisenbud green harris conjecture is currently known the projective dimension and the castelnuovo mumford regularity of a graded ideal do not decrease by passing to the corresponding lex plus powers ideal

extremal betti numbers and applications to monomial ideals recall that the mumford castelnuovo regularity of m is the least integer such that for each i all free generators of fi lie in degree i that is i j 0 for j i in terms of macaulay mac regularity is the number of rows in the diagram produced by the betti command a betti number i j 0 will be called extremal if l r 0 for all l i r j 1 and r l j i that is if i j is the nonzero top left corner in a block of zeroes in the macaulay betti diagram in other words extremal betti numbers account for notches in the shape of the minimal free resolution and one of them computes the regularity in this sense extremal betti numbers can be seen as a refinement of the notion of mumford castelnuovo regularity in the first part of this note we connect the extremal betti numbers of an arbitrary submodule of a free s module with those of its generic initial module in the second part which can be read independently of the first we relate extremal multigraded betti numbers in the minimal resolution of a square free monomial ideal with those of the monomial ideal corresponding to the alexander dual simplicial complex our techniques give also a simple geometric proof of a more precise version of a recent result of terai te97 see also ft97 for a homological reformulation and related results generalizing eagon and reiner s theorem er96 that a stanleyreisner ring is cohen macaulay if and only if the homogeneous ideal corresponding to the alexander dual simplicial complex has a linear resolution we are grateful to david eisenbud for useful discussions

oscillations and limit cycles in lotka volterra systems with delays abstract in this paper competitive lotka volterra systems are studied that have distributed delays and constant coefficients on interaction terms and have time dependent growth rate vectors with an asymptotically constant average algebraic conditions are found to rule out non vanishing oscillations for such systems and heteroclinic limit cycles for autonomous systems as a supplement to these results simple sufficient conditions are provided for certain components of all solutions to vanish and a criterion is given for partial permanence an outstanding feature of all these results is that the conditions are irrelevant of the size and distribution of the delays

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

the regularity of a toric variety abstract we give a method for computing the degrees of the minimal syzygies of a toric variety by means of combinatorial techniques indeed we complete the explicit description of the minimal free resolution of the associated semigroup algebra using the simplicial representation of koszul homology which appeared in a campillo and c marijuan 1991 sem theor nombres bordeaux3 249 260 as an application we obtain an algorithm for computing the castelnuovo mumford regularity of a projective toric variety this regularity is explicitly bounded by means of the semigroup generators which parametrize the variety

circuit complexity proof complexity and polynomial identity testing the ideal proof system we introduce a new and natural algebraic proof system whose complexity measure is essentially the algebraic circuit size of nullstellensatz certificates this enables us to exhibit close connections between effective nullstellensatze proof complexity and algebraic circuit complexity in particular we show that any super polynomial lower bound on any boolean tautology in our proof system implies that the permanent does not have polynomial size algebraic circuits vnp vp we also show that super polynomial lower bounds on the number of lines in polynomial calculus proofs imply the permanent versus determinant conjecture note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound our proof system helps clarify the relationships between previous algebraic proof systems in doing so we highlight the importance of polynomial identity testing pit in proof complexity in particular we use pit to illuminate ac0 p frege lower bounds which have been open for nearly 30 years with no satisfactory explanation as to their apparent difficulty finally we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity using the algebraic structure of our proof system we propose a novel route to such lower bounds although such lower bounds remain elusive this proposal should be contrasted with the difficulty of extending ac0 p circuit lower bounds to ac0 p frege lower bounds

application of the generalized weierstrass preparation theorem to the study of homogeneous ideals the system of weierstrass polynomiajs defined originally for ideals in convergent power series rings together with its sequence of degrees allows us to analyze a homogeneous ideal directly making use of it we study local cohomology modules syzygies and then graded buchsbaum rings our results give a formula which to some extent clarifies the connection among the matrices appearing in the free resolution starting from a system of weierstrass polynomials a rough classification of graded buchsbaum rings in the general case and a complete classification of graded buchsbaum integral domains of codimension two

almost regular sequences and betti numbers let be a simplicial complex and s the shifted simplicial complex of as defined by kalai it is shown that the stanley reisner ideals i and i s have the same extremal betti numbers this theorem is the squarefree analogue of the corresponding result by bayer charalambous and popescu which asserts that a graded ideal and its generic ideal have the same extremal betti numbers our theorem implies well known results by kalai according to which and s have the same simplicial homology and is cohen macaulay if and only if its shifted complex is cohen macaulay

bounds for the castelnuovo mumford regularity of modules we establish bounds for the castelnuovo mumford regularity of a finitely generated graded module and its symmetric powers in terms of the degrees of the generators of the module and the degrees of their relations we extend to modules and improve the known bounds for homogeneous ideals in a polynomial ring established by galligo giusti caviglia and sbarra

extended multilinear variable separation approach and multivalued localized excitations for some 2 1 dimensional integrable systems the multilinear variable separation approach and the related universal formula have been applied to many 2 1 dimensional nonlinear systems starting from the universal formula abundant 2 1 dimensional localized excitations have been found in this paper the universal formula is extended in two different ways one is obtained for the modified nizhnik novikov veselov equation such that two universal terms can be combined linearly and this type of extension is also valid for the 2 1 dimensional symmetric sine gordon system the other is for the dispersive long wave equation the broer kaup kupershmidt system the higher order broer kaup kupershmidt system and the burgers system where arbitrary number of variable separated functions can be involved because of the existence of the arbitrary functions in both the original universal formula and its extended forms the multivalued functions can be used to construct a new type of localized excitations folded solitary waves fsws and foldons the fsws

castelnuovo mumford regularity postulation numbers and reduction numbers abstract suppose g is a standard graded ring over an infinite field we obtain a sharp lower bound for the regularity of g in terms of the postulation number the depth and the dimension of g we present a class of examples in dimension 1 where the postulation number is 0 and the regularity of g can take on any value between 1 and the embedding codimension of g suppose g gr m r is the associated graded ring of a cohen macaulay local ring r m we compute the regularity the reduction number and the postulation number of g and consider the relationship among these invariants in the case where dim g grade g 1 a precise description is known as to how these integers are related we consider the case where dim g grade g 2 and prove that if dim g grade g 2 then reg g max p g dim g 1 r m where p g is the postulation number of g and r m is the reduction number of m

curves on normal surfaces we study locally cohen macaulay space curves lying on normal surfaces we prove some theorems on the behaviour of the cohomology functions and initial ideals of such space curves which give a basic distinction between those curves and curves lying on non normal surfaces a natural step in the study and classification of space curves is to classify the hilbert functions of space curves or more generally cohomology functions of space curves a closely related approach is to study the somewhat more refined invariants the generic initial ideal and higher initial ideal and classify which of them can occur for smooth integral or locally cohen macaulay space curves given a space curve c on a surface s p one may do descending biliaison on the surface s and end up with a curve x minimal in its biliaison class on the surface s on the other hand one might consider a surface t of minimal degree such that c may be linked s t to a curve y in these cases it will often happen thatx or y will have non reduced components the invariants of c may be deduced from the invariants of x or y hence it is natural to try to get an understanding of the invariants of curves on s with nonreduced components here it turns out that there is a basic distinction between when s is normal and when s is not normal the reason is that if one has a reduced curve d on s then a multiplicity n structure on d in s is uniquely determined if s is normal while if s is not normal there may exist many in fact infinitely many multiplicity n structures on d in s for c a curve on a surface s of degree s there is a basic sequence 0 l os oc 0 1 where l is the ideal sheaf of c in s this also gives rise to a sequence 0 os s 4 l s 4 c 0 2 in both cases l and l s 4 are torsion free rank one sheaves on s with local projective dimension one as op3 modules we prove the following theorem 2 1 let s p be a normal surface of degree s and l a torsion free rank one sheaf on s with local projective dimension one as op3 module let the minimal generators of h l n zh l n occur in degrees n0 n1 nr then nj nj 1 s 2 for j 1 r and the regularity of l is nr s 2 received by the editors july 4 1996 and in revised form september 22 1997 2000 mathematics subject classification primary 14h50

vanishing theorems a theorem of severi and the equations defining projective varieties in 1903 severi s proved that if x x c p are smooth surfaces whose union is the complete intersection of r 2 hypersurfaces of degrees di dr 2 then hypersurfaces of degree k z di r cut out a complete linear series on x cf sr xiii 9 8 the purpose of this paper is first of all to show that elementary arguments using the kodaira vanishing theorem lead to a simple variant of severi s statement theorem 7 below which extends it in serval directions more importantly we hope to convince the reader that this result has a surprising number of applications to questions involving the equations defining projective varieties consider to begin with a smooth complex projective variety x c pr of dimension n and codimension e r n in this setting our theorem asserts the vanishing of the higher cohomology of suitable twists of the ath power of the ideal sheaf yx of x in apr proposition 1 assume that x is cut out scheme theoretically in p by hypersurfaces of degrees d1 d2 d then h p r k 0 for i 1 provided k ad d2 de r note that only the degrees of the first e codim x pr defining equations come into play here when n 2 and a 1 this is a consequence of severi s

localized excitations of the 2 1 dimensional sine gordon system the 2 1 dimensional sine gordon system is investigated via a generalized bilinear operator representation plateau basin bowl and saddle type ring solitons are thereby constructed it is indicated how iterated moutard transformations may be employed to extend the range of the method

conditions for generic initial ideals to be almost reverse lexicographic abstract let i be a homogeneous artinian ideal in a polynomial ring r k x 1 x n over a field k of characteristic 0 we study an equivalent condition for the generic initial ideal gin i with respect to reverse lexicographic order to be almost reverse lexicographic as a result we show that the moreno socias conjecture implies the froberg conjecture and the pardue conjecture and for the case codim i 3 we show that r i has the strong lefschetz property if and only if gin i is almost reverse lexicographic finally we give a positive partial answer to the moreno socias conjecture and hence to the pardue conjecture

a stratification of the hilbert scheme of points in the projective plane this result is analogous to a theorem of iarrobino and brian on on the stratification of the punctual hilbert scheme where a different hilbert function defined by initial forms instead is used however cp i l thm 2 9 p 26 the proof essentially makes use of the smoothness of h a cp f p 517 and a general result of bialynicki birula on the action of the multiplicative group on a smooth projective scheme as already was done in a study of the homology of h a by ellingsrud and stromme this method is applied here to a study of h a around points which are invariant under the borel group in the appendix a method for computing the dimension of h is represented which rests on iarrobino s description of the tangent space to certain points of h a and the theorem of bialynicki birula finally i would like to mention that hirschowitz has outlined a general program for studying stratifications including those defined by hilbert functions cp h p 175 but the result obtained here can at best be regarded only as the most simple perhaps non trivial example

hilbert functions of d regular ideals abstract in the present paper we characterize all possible hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to d where d is a positive integer in addition we prove the following result which is a generalization of bigatti hulett and pardue s result let p 0 and d 0 be integers if the base field is a field of characteristic 0 and there is a graded ideal i whose projective dimension proj dim i is smaller than or equal to p and whose regularity reg i is smaller than or equal to d then there exists a monomial ideal l having the maximal graded betti numbers among graded ideals j which have the same hilbert function as i and which satisfy proj dim j p and reg j d we also prove the same fact for squarefree monomial ideals the main methods for proofs are generic initial ideals and combinatorics on strongly stable ideals

lefschetz properties and hyperplane arrangements abstract in this article we study the weak and strong lefschetz properties and the related notion of almost revlex ideal in the non artinian case proving that several results known in the artinian case hold also in this more general setting we then apply the obtained results to the study of the jacobian algebra of hyperplane arrangements

a generalization of castelnuovo mumford regularity for representations of noncommutative algebras abstract we introduce and generalize the notion of castelnuovo mumford regularity for representations of noncommutative algebras effectively establishing a measure of complexity for such objects the grobner shirshov basis theory for modules over noncommutative algebras is developed by which a noncommutative analogue of schreyer s theorem is proved for computing syzygies by a repeated application of this theorem we construct free resolutions for representations of noncommutative algebras some interesting examples are included in which graded free resolutions and regularities are computed for representations of various algebras in particular using the bernstein gelfand gelfand resolutions for integrable highest weight modules over kac moody algebras we compute the projective dimensions and regularities explicitly for the cases of finite type and affine type a n 1

castelnuovo mumford regularity of seminormal simplicial affine semigroup rings abstract we show that the eisenbud goto conjecture holds for homogeneous seminormal simplicial affine semigroup rings moreover we prove an upper bound for the castelnuovo mumford regularity in terms of the dimension which is similar as in the normal case finally we compute explicitly the regularity of full veronese rings

localized excitations in 2 1 dimensional systems by means of a special variable separation approach a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various 2 1 dimensional models such as the davey stewartson ds model the nizhnik novikov veselov nnv system asymmetric nnv equation asymmetric ds equation dispersive long wave equation broer kaup kupershmidt system long wave short wave interaction model maccari system and a general n m component ablowitz kaup newell segur akns system selecting the arbitrary functions appropriately one may obtain abundant stable localized interesting excitations such as the multidromions lumps ring soliton solutions breathers instantons etc it is shown that some types of lower dimensional chaotic patterns such as the chaotic chaotic patterns periodic chaotic patterns chaotic line soliton patterns chaotic dromion patterns fractal lump patterns and fractal dromion patterns may be found in higher dimensional soliton systems the interactions between the traveling ring type soliton solutions are completely elastic the traveling ring solitons pass through each other and preserve their shapes velocities and phases some types of localized weak solutions peakons are also discussed especially the interactions between two peakons are not completely elastic after the interactions the traveling peakons also pass through each other and preserve their velocities and phases however they completely exchange their shapes

direct methods for primary decomposition leti be an ideal in a polynomial ring over a perfect field we given new methods for computing the equidimensional parts and radical ofi for localizingi with respect to another ideal and thus for finding the primary decomposition ofi our methods rest on modern ideas from commutative algebra and are direct in the sense that they avoid the generic projections used by hermann 1926 and all others until now

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

the sine gordon equation in the finite line a rather complete numerical study of the perturbed and nonperturbed sine gordon equation in one dimensional cartesian coordinates and finite domains is presented the numerical study is based on five three point linearly implicit finite difference methods of different spatial accuracy for the unperturbed sge the accuracy of these five methods is found to be very sensitive to the implicitness parameter less sensitive to the spatial order of accuracy and almost no sensitive to the time step for second order accurate temporal discretizations the largest errors of the five methods were found to occur at the front of the soliton solution of the unperturbed sge but the techniques were very accurate for long time computations of the unperturbed sge in both infinite domains and in finite domains upon many collisions of the kinks and antikinks with the boundaries despite the fact that they do not preserve a discrete energy it is shown that the largest values of the kinetic and total energies and the smallest values of the strain and potential energies occur when solitons and antisolitons strike on the boundaries subject to homogeneous neumann conditions and the four energies remain constant between collisions with the boundaries for the malomed s perturbation one of the fronts of the soliton soliton doublet and soliton antisoliton doublet is trapped at one boundary and the other one undergoes a change in amplitude and its front describes a curved trajectory the soliton antisoliton breather oscillates in a damped manner and slowly drifts towards one boundary the soliton antisoliton soliton soliton doublet and soliton antisoliton doublet of the unperturbed sge are robust under kivshar and malomed s perturbations whereas the soliton antisoliton breather loses its integrity becomes a kink antikink pair and then yields a chaotic behaviour in both space and time small amplifications proportional to the first order temporal derivative of the amplitude do not alter substantially the initial dynamics of the soliton antisoliton soliton soliton doublet and soliton antisoliton doublet but their amplitudes exhibit oscillatory behaviour of increasing amplitude at large times however small amplifications cause first a transition from the soliton antisoliton breather to the kink antikink solution a linearized stability analysis of the spatially homogeneous solutions of the unperturbed sge is performed and analytical solutions are obtained by means of the method of separation of variables for both homogeneous dirichlet and neumann boundary conditions and the results show that the solution corresponding to a zero amplitude is oscillatory in both space and time whereas that corresponding to an amplitude equal to p may exhibit exponential or linear growth depending on the length of the domain if either of these growths is present it is shown that the evolution of small initial perturbations may result in homoclinic crossings and chaotic behaviour in both space and time in the absence of perturbations in the sge and the existence of an attractor when perturbations of the malomed s type are present in either case the solution may exhibit certain spatial symmetries or antisymmetries at large times finally the results presented in this paper indicate that the accurate computation of homoclinic orbits of the sge is not only controlled by the spatial accuracy of the numerical method but also by its energy preserving characteristics and perhaps by the number of grid points used in the numerical technique

finiteness of hilbert functions and bounds for castelnuovo mumford regularity of initial ideals bounds for the castelnuovo mumford regularity and hilbert coefficients are given in terms of the arithmetic degree if the ring is reduced or in terms of the defining degrees from this it follows that there exists only a finite number of hilbert functions associated with reduced algebras over an algebraically closed field with a given arithmetic degree and dimension a good bound is also given for the castelnuovo mumford regularity of initial ideals which depends neither on term orders nor on the coordinates and holds for any field

the ubiquity of smooth hilbert schemes we investigate the geography of hilbert schemes parametrizing closed subschemes of projective space with specified hilbert polynomials we classify hilbert schemes with unique borel fixed points via combinatorial expressions for their hilbert polynomials these expressions naturally lead to an arrangement of nonempty hilbert schemes as the vertices of an infinite full binary tree here we discover regularities in the geometry of hilbert schemes specifically under natural probability distributions on the tree we prove that hilbert schemes are irreducible and nonsingular with probability greater than 0 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

linear normality of general linear sections and some graded betti numbers of 3 regular projective schemes in this paper we study graded betti numbers of any nondegenerate 3 regular algebraic set x in a projective space pn more concretely via generic initial ideals gins method we mainly consider tailing betti numbers whose homological index is at least codim x pn for this purpose we first introduce a key definition nd 1 property which provides a suitable ground where one can generalize the concepts such as being nondegenerate or of minimal degree from the case of varieties to the case of more general closed subschemes and give a clear interpretation on the tailing betti numbers next we recall basic notions and facts on gins theory and we analyze the generation structure of the reverse lexicographic rlex gins of 3 regular nd 1 subschemes as a result we present exact formulae for these tailing betti numbers which connect them with linear normality of general linear sections of x with a linear subspace of dimension at least codim x pn finally we consider some applications and related examples

test multiplier and invariant ideals abstract this paper gives an explicit formula for the multiplier ideals and consequently for the log canonical thresholds of any gl v gl w invariant ideal in s sym v w where v and w are vector spaces over a field of characteristic 0 this characterization is done in terms of a polytope constructed from the set of young diagrams corresponding to the schur modules generating the ideal our approach consists in computing the test ideals of some invariant ideals of s in positive characteristic namely we compute the test ideals and so the f pure thresholds of any sum of products of determinantal ideals not all the invariant ideals are as the latter not even in characteristic 0 but they are up to integral closure and this is enough to reach our goals the results concerning the test ideals are obtained as a consequence of general results holding true in a special situation within such framework fall determinantal objects of a generic matrix as well as of a symmetric matrix and of a skew symmetric one similar results are thus deduced for the gl v invariant ideals in sym sym 2 v and in sym 2 v monomial ideals also fall in this framework thus we recover howald s formula for their multiplier ideals and more generally hara yoshida s formula for their test ideals in the proof we introduce the notion of floating test ideals a property that in a sense is satisfied by ideals defining schemes with the nicest possible singularities as will be shown products of determinantal ideals and by passing to characteristic 0 ideals generated by a single schur module have this property

generic initial ideals of artinian ideals having lefschetz properties or the strong stanley property for a standard artinian k algebra a r i we give equivalent conditions for a to have the weak or strong lefschetz property or the strong stanley property in terms of the minimal system of generators of gin i using the equivalent condition for the weak lefschetz property we show that some graded betti numbers of gin i are determined just by the hilbert function of i if a has the weak lefschetz property furthermore for the case that a is a standard artinian k algebra of codimension 3 we show that every graded betti number of gin i is determined by the graded betti numbers of i if a has the weak lefschetz property and if a has the strong lefschetz respectively stanley property then we show that the minimal system of generators of gin i is determined by the graded betti numbers respectively by the hilbert function of i

borel fixed ideals and reduction number abstract the aim of this paper is to study the relationship between the reduction number and borel fixed ideals in all characteristics especially it is shown that r r i r r i lex where i lex denotes the unique lex segment ideal whose hilbert function is equal to that of i this solves a recent question by conca