a survey on oscillation of impulsive delay differential equations this paper presents a systematic development of the oscillatory results for the linear and nonlinear impulsive delay differential equations we also discuss linearized oscillation theory and give some applications to mathematical biology

splittable ideals and the resolutions of monomial ideals abstract we provide a new combinatorial approach to study the minimal free resolutions of edge ideals that is quadratic square free monomial ideals with this method we can recover most of the known results on resolutions of edge ideals with fuller generality and at the same time obtain new results past investigations on the resolutions of edge ideals usually reduced the problem to computing the dimensions of reduced homology or koszul homology groups our approach circumvents the highly nontrivial problem of computing the dimensions of these groups and turns the problem into combinatorial questions about the associated simple graph we also show that our technique extends successfully to the study of graded betti numbers of arbitrary square free monomial ideals viewed as facet ideals of simplicial complexes

grobner bases via linkage in this paper we give a sufficient condition for a set g of polynomials to be a groebner basis with respect to a given term order for the ideal i that it generates our criterion depends on the linkage pattern of the ideal i and of the ideal generated by the initial terms of the elements of g we then apply this criterion to ideals generated by minors and pfaffians more precisely we consider large families of ideals generated by minors or pfaffians in a matrix or a ladder where the size of the minors or pfaffians is allowed to vary in different regions of the matrix or the ladder we use the sufficient condition that we established to prove that the minors or pfaffians form a reduced groebner basis for the ideal that they generate with respect to any diagonal or anti diagonal term order we also show that the corresponding initial ideal is cohen macaulay our proof relies on known results in liaison theory combined with a simple hilbert function computation in particular our arguments are completely algebraic

oscillation and linearized oscillation of a logistic equation with several delays for a scalar delay logistic equation y t y t k 1mrk t 1 y hk t k hk t t a connection between oscillating properties of this equation the corresponding differential inequalities and the linear equation x t k 1mrk t x hk t 0 is established explicit nonoscillation and oscillation conditions are presented when hk t t k k 1 2 m we also investigate the linearized oscillation of eq

on the ideal theory of graphs we study algebras defined by finite sets g m1 mq of monomials of a polynomial ring r there are two basic algebras i k g k m1 mq the k subalgebra of r spanned by the mi and ii the quotient ring r i g where i g m1 mq they come together in the construction of the rees algebra r i g of the ideal i g the emphasis is almost entirely on sets of squarefree monomials of degree two and their attached graphs the main results are assertions about the cohen macaulay behaviour of the koszul homology of i g and how normality or cohen macaulayness of one of the algebras can be read off the properties of the graph or in the other algebra

cellular resolutions of cointerval ideals minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed this class of d uniform hypergraphs coincides with the complements of interval graphs for the case d 2 and strictly contains the class of strongly stable hypergraphs corresponding to pure shifted simplicial complexes the polyhedral complexes supporting the resolutions are described as certain spaces of directed graph homomorphisms and are realized as subcomplexes of mixed subdivisions of the minkowski sums of simplices resolutions of more general hypergraphs are obtained by considering decompositions into cointerval hypergraphs

factoring the adjoint and maximal cohen macaulay modules over the generic determinant a question of bergman asks whether the adjoint of the generic square matrix over a field can be factored nontrivially as a product of square matrices we show that such factorizations indeed exist over any coefficient ring when the matrix has even size establishing a correspondence between such factorizations and extensions of maximal cohen macaulay modules over the generic determinant we exhibit all factorizations where one of the factors has determinant equal to the generic determinant the classification shows not only that the cohen macaulay representation theory of the generic determinant is wild in the tame wild dichotomy but that it is quite wild even in rank two the isomorphism classes cannot be parametrized by a finite dimensional variety over the coefficients we further relate the factorization problem to the multiplicative structure of the mathop rm ext nolimits algebra of the two nontrivial rank one maximal cohen macaulay modules and determine it completely

m sequences graph ideals and ladder ideals of linear type in this paper we study monomial ideals and ladder determinantal ideals of linear type and their blow up algebras our main tools are grobner bases and sagbi bases deformations and the notion of m sequence of monomials w x let r s k x be a polynomial ring over a field k equipped with a monomial order t let i be an ideal of r generated by polynomials f f consider the presentation 1 s w x w x w x c r t s r t t a r i s r f t f t 1 s i s

extensions of the multiplicity conjecture the multiplicity conjecture of herzog huneke and srinivasan states an upper bound for the multiplicity of any graded k algebra as well as a lower bound for cohen macaulay algebras in this note we extend this conjecture in several directions we discuss when these bounds are sharp find a sharp lower bound in the case of not necessarily arithmetically cohen macaulay one dimensional schemes of 3 space and propose an upper bound for finitely generated graded torsion modules we establish this bound for torsion modules whose codimension is at most two

grobner geometry of schubert polynomials given a permutation w sn we consider a determinantal ideal iw whose generators are certain minors in the generic n n matrix filled with independent variables using emultidegrees f as simple algebraic substitutes for torus equivariant cohomology classes on vector spaces our main theorems describe for each ideal iw

on delay differential equations with impulses abstract sufficient conditions are obtained respectively for the asymptotic stability of the trivial solution of x t ax t j 1 bjx tj t tj t tj and for the existence of a nonoscillatory solution conditions are also obtained for all solutions to be oscillatory the asymptotic behaviour of an impulsively perturbed delay logistic equation is investigated as an extension to a nonlinear equation

strict stability of impulsive functional differential equations strict stability is the kind of stability that can give us some information about the rate of decay of the solutions there are some results about strict stability of differential equations in the present paper we shall extend the strict stability to impulsive functional differential equations by using lyapunov functions and razumikhin technique we shall get some criteria for the strict stability of impulsive functional differential equations and we can see that impulses do contribute to the system s strict stability behavior

a formula for the core of certain strongly stable ideals abstract the core of an ideal is the intersection of all of its reductions the core is a natural object to study from the point of view of blow up algebras and integral closure and it also has geometric significance coming for example from its connection to adjoint and multiplier ideals in general the core is difficult to describe explicitly in this paper we investigate a particular family of strongly stable ideals we prove that ideals in this family satisfy an artin nagata property yet fail to satisfy other stronger standard depth conditions we then show that there is a surprisingly simple explicit formula for the core of these ideals

small schemes and varieties of minimal degree we prove that if x p r is any 2 regular scheme in the sense of castelnuovo mumford then x is small this means that if l is a linear space and y l x is finite then y is linearly independent in the sense that the dimension of the linear span of y is deg y 1 the converse is true and well known for finite schemes but false in general the main result of this paper is that the converse small implies 2 regular is also true for reduced schemes algebraic sets this is proven by means of a delicate geometric analysis leading to a complete classification we show that the components of a small algebraic set are varieties of minimal degree meeting in a particularly simple way from the classification one can show that if x p r is 2 regular then so is xred and so also is the projection of x from any point of x our results extend the del pezzo bertini classification of varieties of minimal degree the charac terization of these as the varieties of regularity 2 by eisenbud goto and the construction of 2 regular square free monomial ideals by froberg 0 introduction throughout this paper we will work with projective schemes x p r over an algebraically closed field k the castelnuovo mumford regularity of x p r is a basic homological measure of the complexity of x and its embedding in p r that gives a bound for the degrees of the generators of the defining ideal ix of x and for many other invariants the only schemes of regular ity 1 are the linear spaces but no classification is known for projective schemes of regularity 2 in this paper we prove a structure theorem for reduced 2 regular schemes showing that their irreducible components are varieties of minimal degree and characterizing how these components can meet we also show that the reduced structure on any 2 regular scheme is 2 regular and thus we obtain a complete description of the reduced structures on 2 regular schemes since a high veronese re embedding of any zero dimensional scheme is 2 regular one cannot hope to characterize the isomorphism types of all 2 regular nonreduced schemes before stating our results we review some basic notions for any subscheme x p r we write span x for the smallest linear subspace of p r containing x recall that every variety reduced irreducible scheme x p r satisfies the condition deg x 1 codim x span x

hilbert functions of points on schubert varieties in the symplectic grassmannian we give an explicit combinatorial description of the multiplicity as well as the hilbert function of the tangent cone at any point on a schubert variety in the symplectic grassmannian

singular loci of ladder determinantal varieties and schubert varieties abstract we relate certain ladder determinantal varieties associated to one sided ladders to certain schubert varieties in sl n q for a suitable n and a suitable parabolic subgroup q and we determine the singular loci of these varieties we state a conjecture on the irreducible components of the singular locus of a schubert variety in the flag variety which is a refinement of the conjecture of lakshmibai and sandhya proc indian acad sci math sci 100 1990 45 52 we prove the conjecture for a certain class of schubert varieties

the core of zero dimensional monomial ideals the core of an ideal is the intersection of all its reductions we describe the core of a zero dimensional monomial ideal i as the largest monomial ideal contained in a general reduction of i this provides a new interpretation of the core in the monomial case as well as an efficient algorithm for computing it we relate the core to adjoints and first coefficient ideals and in dimension two and three we give explicit formulas

linearized oscillation theory for a nonlinear delay impulsive equation for a scalar nonlinear impulsive delay differential equation y t k 1m rk t fk y hk t 0 t j y j ij y j with rk t 0 hk t t limj j such an auxiliary linear impulsive delay differential equation x t k 1m rk t ak t x hk t 0 x j bjx j is constructed that oscillation nonoscillation of the nonlinear equation can be deduced from the corresponding properties of the linear equation coefficients rk t and delays are not assumed to be continuous explicit oscillation and nonoscillation conditions are established for some nonlinear impulsive models of population dynamics such as the impulsive logistic equation and the impulsive generalized lasota wazewska equation which describes the survival of red blood cells it is noted that unlike nonimpulsive delay logistic equations a solution of a delay impulsive logistic equation may become negative

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

cellular resolutions of monomial modules we construct a canonical free resolution for arbitrary monomial modules and lattice ideals this includes monomial ideals and defining ideals of toric varieties and it generalizes our joint results with irena peeva for generic ideals bps ps introduction given a field k we consider the laurent polynomial ring t k x 1 1 x 1 n as a module over the polynomial ring s k x1 xn the module structure comes from the natural inclusion of semigroup algebras s k n k z t a monomial module is an s submodule of t which is generated by monomials x x1 1 xan n a z of special interest are the two cases when m has a minimal monomial generating set which is either finite or forms a group under multiplication in the first case m is isomorphic to a monomial ideal in s in the second case m coincides with the lattice module ml s x a l k x b n l t for some sublattice l z whose intersection with n is the origin 0 0 0 we shall derive free resolutions of m from regular cell complexes whose vertices are the generators of m and whose faces are labeled by the least common multiples of their vertices the basic theory of such cellular resolutions is developed in section 1 our main result is the construction of the hull resolution in section 2 we rescale the exponents of the monomials in m so that their convex hull in r is a polyhedron pt whose bounded faces support a free resolution of m this resolution is new and interesting even for monomial ideals it need not be minimal but unlike minimal resolutions it respects symmetry and is free from arbitrary choices in section 3 we relate the lattice module ml to the z l graded lattice ideal il x x a b l s this class of ideals includes ideals defining toric varieties we express the cyclic smodule s il as the quotient of the infinitely generated s module ml by the action of l in fact we like to think of ml as the universal cover of il many questions about il can thus be reduced to questions about ml in particular we obtain the hull resolution of a lattice ideal il by taking the hull resolution of ml modulo l this paper is inspired by the work of barany howe and scarf bhs who introduced the polyhedron pt in the context of integer programming the hull resolution generalizes results in bps for generic monomial ideals and in ps for generic lattice ideals in these generic cases the hull resolution is minimal

simplicial ideals 2 linear ideals and arithmetical rank abstract in the first part of this paper we study scrollers and linearly joined varieties scrollers were introduced in barile and morales 2004 bm4 linearly joined varieties are an extension of scrollers and were defined in eisenbud et al 2005 eghp there they proved that scrollers are defined by homogeneous ideals having a 2 linear resolution a particular class of varieties of important interest in classical geometry are cohen macaulay varieties of minimal degree they were classified geometrically by the successive contribution of del pezzo 1885 dp bertini 1907 b and xambo 1981 x and algebraically in barile and morales 2000 bm2 they appear naturally studying the fiber cone of a codimension two toric ideals morales 1995 m gimenez et al 1993 1999 gms1 gms2 barile and morales 1998 bm1 ha 2006 h ha and morales 2009 hm let s be a polynomial ring and i s a homogeneous ideal defining a sequence of linearly joined varieties we compute depth s i we prove that c v depth s i 1 where c v is the connectedness dimension of the algebraic set defined by i we characterize sets of generators of i and give an effective algorithm to find equations as an application we prove that ara i projdim s i in the case where v is a union of linear spaces in particular this applies to any square free monomial ideal having a 2 linear resolution in the case where v is a union of linear spaces the ideal i can be characterized by a tableau which is an extension of a ferrer or young tableau all these results are new and extend results in barile and morales 2004 bm4 eisenbud et al 2005 eghp

many toric ideals generated by quadratic binomials possess no quadratic grobner bases abstract let g be a finite connected simple graph and i g the toric ideal of the edge ring k g of g in the present paper we study finite graphs g with the property that i g is generated by quadratic binomials and i g possesses no quadratic grobner basis first we give a nontrivial infinite series of finite graphs with the above property second we implement a combinatorial characterization for i g to be generated by quadratic binomials and by means of the computer search we classify the finite graphs g with the above property up to 8 vertices

oscillation properties of nonlinear impulsive delay differential equations and applications to population models comparison theorem and explicit sufficient conditions are obtained for oscillation and nonoscillation of solutions of nonlinear impulsive delay differential equations which can be utilized to population dynamic models our results in this paper generalize and improve several known results

resolutions and hilbert series of determinantal varieties and unitary highest weight modules abstract let g k be a hermitian symmetric pair and let g and k denote the corresponding complexified lie algebras let g k p p be the usual decomposition of g as a k module there is a natural correspondence between k c orbits in p and a distinguished family of unitarizable highest weight modules for g called the wallach representations we denote by yk the closure of the k c orbit in p that is associated to the kth wallach representation in this article we give explicit formulas for the numerator polynomials of the hilbert series of the varieties yk by using bgg resolutions of unitarizable highest weight modules a preliminary result gives a new branching formula for a certain two parameter family of finite dimensional representations of the even orthogonal groups our work is an extension of previous work by enright and willenbring

hilbert function of powers of ideals of low codimension we study the hilbert function of the powers of homogeneous ideals which are either cohen macaulay of codimension 2 or gorenstein of codimension 3 we show that that if i is an ideal in one of these classes and it is of linear type then for all k the hilbert function of i k depends only on the hilbert function of i in other words if i andj are ideals in one of the above mentioned classes which are both of linear type and they have the same hilbert function then also i k and j k have the same hilbert function for all k

the cell structure the brauer tree structure and extensions of cell modules for hecke orders of dihedral groups we shall show that the hecke order hdn of the dihedral group of order 2 pn over z q q 1 for an odd prime p is a projectively cellular order we describe the corresponding cell ideals and compute the extension groups between the corresponding cell modules some are z torsion free some are fp q q 1 torsion free moreover we show that hdn is a brauer tree order to the tree with central exceptional vertex of multiplicity pn 1 p 1 2

on hilbertian ideals abhyankar defined the index of a monomial in a matrix of indeterminates x to be the maximal size of any minor of x whose principal diagonal divides the given monomial using this concept he characterized a free basis for general type of determinantal ideals formed by the minors coming from a saturated subset of x in this paper to a monomial in x of index p we associate a combinatorial object called a superskeleton of latitude p which can loosely be described as a p tuple of almost nonintersecting paths in a rectangular lattice of points using this map we prove that the ideal generated by the p by p minors of a saturated set in x is hilbertian i e the hilbert polynomial of this ideal coincides with its hilbert function for all nonnegative integers

hilbert polynomials of non standard bigraded algebras this paper investigates hilbert polynomials of bigraded algebras which are generated by elements of bidegrees 1 0 d 1 1 ldots d r 1 where d 1 ldots d r are non negative integers the obtained results can be applied to study rees algebras of homogeneous ideals and their diagonal subalgebras

global attractivity in a food limited population model with impulsive effects abstract sufficient conditions are obtained for the global attractivity of the positive equilibrium of the delay logistic equation with impulsive effect n t r t n t k n p t k c t n p t n k n k exp i k ln n k k p where r t c t are continuous functions p k are positive constants and k k 1 denotes the sequence of impulsive time which assumed to be strictly increasing some special cases are also considered

the persistence of nonoscillatory solutions of delay differential equations under impulsive perturbations abstract we obtain a sufficient condition for the persistence of nonoscillatory solutions of the delay differential equation under impulsive perturbations

oscillation criteria for impulsive delay differential equations abstract some new results on oscillation and nonoscillation for delay differential equations with impulses are established a counterexample is given which shows that a known oscillation criterion is incorrect

oscillation and stability of linear impulsive delay differential equations the main results of this paper are that the oscillation and the stability of a linear impulsive delay differential equation are equivalent respectively to the oscillation and the stability of a corresponding linear delay differential equation without impulses the results of this paper can be used to improve and derive some of the known results in the literature

algebras of minors let i be an ideal of a noetherian ring r the rees algebra of i is the graded r algebra k 0ik the study of the properties of rees algebras and of the other blow up algebras has attracted the attention of many researchers in the last three decades for a detailed account of the theories that have been developed and of the results that have been proved the reader should consult the monograph of vasconcelos v in this paper we treat a special and interesting case the rees algebras of determinantal ideals and their special fibers let x xij be a generic matrix of size m n over a field k and let s be the polynomial ring k xij let it be the ideal of s generated by the minors of size t of x finally let t be the rees algebra of it and let at be the subalgebra of s generated by the t minors of x in the case of maximal minors i e t min m n the algebra at is nothing but the homogeneous coordinate ring of the grassmann variety which is known to be a

existence and global attractivity of positive periodic solutions of functional differential equations with impulses abstract sufficient and realistic conditions are obtained for the existence and global attractivity of positive periodic solutions of the delay differential system with impulses d y t d t y t f t y t 1 t y t n t a e t 0 t t k y t k y t k b k y t k k 1 2 the method involves the application of the gaines and mawhin s coincidence degree theory the constructing suitable lyapunov functionals and estimations of uniform upper bounds on solutions when these results are applied to some special delay population models some new results are obtained and some known results are generalized in particular our results indicate that under the appropriate linear periodic impulsive perturbations the above impulsive delay differential equation preserves the original periodicity and global attractivity of the nonimpulsive delay differential equation d y t d t y t f t y t 1 t y t n t

semidualizing modules of 2 2 ladder determinantal rings abstract we continue our study of ladder determinantal rings over a field k from the perspective of semidualizing modules in particular given a ladder of variables y we show that the associated ladder determinantal ring k y i 2 y admits exactly 2 n non isomorphic semidualizing modules where n is determined from the combinatorics of the ladder y the number n is essentially the number of non gorenstein factors in a certain decomposition of y from this for each n we show explicitly how to find ladders y such that k y i 2 y admits exactly 2 n non isomorphic semidualizing modules this is in contrast to our previous work which demonstrates that large classes of ladders have exactly 2 non isomorphic semidualizing modules

mixed ladder determinantal varieties abstract we investigate ladder determinantal varieties defined by ideals of minors of possibly different sizes in the different regions the steps of one sided ladders l these varieties are an important generalization of the classical ladder determinantal varieties i e with equal size minors since they are very closely related to schubert varieties this being the first main result of this paper we show that they correspond to opposite cells in schubert varieties in flag varieties of type a n in consequence one deduces the normality and the cohen macaulayness of these one sided ladder determinantal varieties with ideals of mixed size minors as well as the fact that they have rational singularities next we show that up to products by affine spaces each of these varieties is a basic open set in a classical ladder determinantal variety i e with equal size minors and that it contains as a basic open set another classical ladder determinantal variety this result along with a general localisation lemma used to show it enables us to compute the divisor class group and singular locus of the coordinate rings of these varieties as well as to determine when they are gorenstein

combinatorial interpretations of some boij soderberg decompositions boij soderberg theory shows that the betti table of a graded module can be written as a linear combination of pure diagrams with integer coefficients using ferrers hypergraphs and simplicial polytopes we provide interpretations of these coefficients for ideals with a d linear resolution their quotient rings and for gorenstein rings whose resolution has essentially at most two linear strands we also establish a structural result on the decomposition in the case of quasi gorenstein modules

oscillatory properties of the first order nonlinear advanced and delayed differential inequalities on donne des conditions suffisantes pour que des inegalites differentielles retardees ou avancees aient des solutions oscillatoires

convergence rate of the fuzzy generalized nearest neighbor rule tizzy k nearest neighbor rule k nnr has been applied in a variety of substantive areas yang and chen l described a fuzzy generalized k nn algorithm which is a unified approach to a variety of fuzzy k nnr s they created the strong consistency of posterior risk of the fuzzy generalized nnr in this paper we give their convergence rate that is the convergence rate of posterior risk of the fuzzy generalized nnr is exponentially fast

ideals of adjacent minors abstract we give a description of the minimal primes of the ideal generated by the 2 2 adjacent minors of a generic matrix we also compute the complete prime decomposition of the ideal of adjacent m m minors of an m n generic matrix when the characteristic of the ground field is zero a key intermediate result is the proof that the ideals which appear as minimal primes are in fact prime ideals this introduces a large new class of mixed determinantal ideals that are prime

linear resolutions of quadratic monomial ideals abstract we study the minimal free resolution of a quadratic monomial ideal in the case where the resolution is linear first we focus on the squarefree case namely that of an edge ideal we provide an explicit minimal free resolution under the assumption that the graph associated with the edge ideal satisfies specific combinatorial conditions in addition we construct a regular cellular structure on the resolution finally we extend our results to non squarefree ideals by means of polarization

on oscillation of a logistic equation with several delays abstract for a scalar delay logistic equation y t y t k 1 m r k t 1 y h k t k h k t t a connection between oscillating properties of this equation the corresponding differential inequalities and the linear equation x t k 1 m r k t x h k t 0 is established explicit nonoscillation and oscillation conditions are presented

the h vector of a ladder determinantal ring cogenerated by 2 2 minors is log concave we show that the h vector of a ladder determinantal ring cogenerated by m u1 v1 is log concave thus we prove an instance of a conjecture of stanley respectively conca and herzog

linear balls and the multiplicity conjecture abstract a linear ball is a simplicial complex whose geometric realization is homeomorphic to a ball and whose stanley reisner ring has a linear resolution it turns out that the stanley reisner ring of the sphere which is the boundary complex of a linear ball satisfies the multiplicity conjecture a class of shellable spheres arising naturally from commutative algebra whose stanley reisner rings satisfy the multiplicity conjecture will be presented

oscillation of nonlinear delay impulsive differential equations and inequalities the oscillatory and asymptotic behavior of first order nonlinear delay impulsive differential equations and inequalities is studied some new sharp sufficient conditions for oscillation and nonoscillation of solutions of the equations and inequalities are obtained many known results are improved in particular the conditions are necessary and sufficient if the equations and inequalities are autonomous

on positivity of singular regular linear time delay time invariant systems subject to multiple internal and external incommensurate point delays this paper deals with the positivity properties of singular regular linear time delay time invariant systems subject to multiple internal and external incommensurate constant point delays the main idea behind the investigation is that its main body is performed based on the construction of the whole state space trajectory solution without using as usual equivalence or similarity transformations on the matrix of dynamics in order to split the state trajectory solution into two parts one being typically associated with a nilpotent matrix in that way the whole state trajectory solution contains impulsive terms associated with the initial conditions and inputs some extensions concerning positivity aspects are given for a special canonical form which separates the dynamics associated with the nilpotent matrix obtained from an equivalence transformation on the singular matrix of the dynamic system

straightening law and powers of determinantal ideals of hankel matrices in this paper we establish a standard monomial theory for generic hankel matrices by a generic hankel matrix we mean a matrix y yij with yij x i j 1 where the x i are indeterminates over a field k we use this structure to determine the symbolic powers and the primary decomposition of the powers of the determinantal ideals it of y further we prove that the symbolic and ordinary rees algebras of it are cohen macaulay normal domains the first standard monomial theory was developed by hodge h to study the homogeneous coordinate ring of the grassmannian variety later standard monomial theories were established for generic matrices by doubilet rota and stein drs and for generic symmetric and generic skew symmetric matrices by de concini and procesi dp these are all examples of algebras with straightening law asl for short over a poset or over a doset the abstract notion of asl was introduced and developed by eisenbud e1 and by de concini eisenbud and procesi dep2 see also bv these structures turned out to be an extremely powerful tool in studying determinantal rings and ideals arising from the above mentioned generic matrices let now x1 xn be indeterminates over an arbitrary field k for j 1 n we denote by xj the j n 1 j hankel matrix with entries x1 xn that is

hankel determinantal rings have rational singularities abstract hankel determinantal rings i e determinantal rings defined by minors of hankel matrices of indeterminates arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves they may also be viewed as linear specializations of generic determinantal rings we prove that over fields of characteristic zero hankel determinantal rings have rational singularities in the case of positive prime characteristic we prove that they are f pure independent of the characteristic we give a complete description of the divisor class groups of these rings and show that each divisor class group element is the class of a maximal cohen macaulay module

monomial and toric ideals associated to ferrers graphs each partition a 1 2 an determines a so called ferrers tableau or equivalently a ferrers bipartite graph its edge ideal dubbed a ferrers ideal is a squarefree monomial ideal that is generated by quadrics we show that such an ideal has a 2 linear minimal free resolution i e it defines a small subscheme in fact we prove that this property characterizes ferrers graphs among bipartite graphs furthermore using a method of bayer and sturmfels we provide an explicit description of the maps in its minimal free resolution this is obtained by associating a suitable polyhedral cell complex to the ideal graph along the way we also determine the irredundant primary decomposition of any ferrers ideal we conclude our analysis by studying several features of toric rings of ferrers graphs in particular we recover establish formulae for the hilbert series the castelnuovo mumford regularity and the multiplicity of these rings while most of the previous works in this highly investigated area of research involve path counting arguments we offer here a new and self contained approach based on results from gorenstein liaison theory

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

some degenerations of kazhdan lusztig ideals and multiplicities of schubert varieties abstract we study hilbert samuel multiplicity for points of schubert varieties in the complete flag variety by grobner degenerations of the kazhdan lusztig ideal in the covexillary case we give a manifestly positive combinatorial rule for multiplicity by establishing with a grobner basis a reduced limit whose stanley reisner simplicial complex is homeomorphic to a shellable ball or sphere we show that multiplicity counts the number of facets of this complex we also obtain a formula for the hilbert series of the local ring in particular our work gives a multiplicity rule for grassmannian schubert varieties providing alternative statements and proofs to formulae of lakshmibai and weyman 1990 26 rosenthal and zelevinsky 2001 37 krattenthaler 2001 22 kodiyalam and raghavan 2003 21 kreiman and lakshmibai 2004 24 ikeda and naruse 2009 13 and woo and yong 2009 40 we suggest extensions of our methodology to the general case

on oscillation of a food limited population model with impulse and delay in this paper we consider a food limited population model with impulsive effect several explicit sufficient conditions are established for oscillation and nonoscillation of solutions of the equations

koszul blowup algebras associated to three dimensional ferrers diagrams abstract we investigate the rees algebra and the toric ring of the squarefree monomial ideal associated to the three dimensional ferrers diagram under the projection property condition we describe explicitly the presentation ideals of the rees algebra and the toric ring we show that the toric ring is a koszul cohen macaulay normal domain while the rees algebra is koszul and the defining ideal is of fiber type

generic graph construction ideals and greene s theorem let x be an m n matrix of indeterminates m n and t a new indeterminate consider the polynomial rings r0 k x and r r0 t for a given positive integer t m consider the ideal it it x generated by the t minors i e the determinants of the t t submatrices of x using all these determinantal ideals we define a new ideal j in r r0 t which we call the generic graph construction ideal as follows

oscillation properties of a logistic equation with several delays abstract for a scalar delay logistic equation y t y t k 1 m r k t 1 y h k t k h k t t the oscillation properties are established which are well known for a linear delay differential equation such as comparison theorems explicit nonoscillation and oscillation conditions dependence of the solution sign on the initial function and the initial value

on ideals of minors fixing a submatrix abstract let m be a n m matrix with entries in a noetherian ring r and let m be the submatrix of m consisting of the first r columns r n 1 consider the ideal jr m of n n minors of m involving the columns of m we obtain the primary decomposition and the homological dimension of jr m in the generic case the proofs rely heavily on the methods and the theory of weak d sequences and straightening laws as a byproduct we obtain exact conditions under which jr m is generated by a d sequence and also a complete picture of the blowing up algebras of jr m in that case the latter proofs rely on recent methods developed by several authors such as those of sliding depth approximation complexes cohen macaulay residual intersections to close the discussion we construct a free resolution of jr m when m n 1 the case r n 1 had been treated before by the present authors a side curiosity herein obtained is an example of a nonperfect radical 3 generated ideal of codimension 2 whose associated graded ring is a cohen macaulay reduced ring that is not gorenstein examples of this sort do not seem to abound