algebraic shifting and graded betti numbers let s k x 1 x n denote the polynomial ring in n variables over a field k with each deg x i 1 let be a simplicial complex on n 1 n and i c s its stanley reisner ideal we write e for the exterior algebraic shifted complex of a and c for a combinatorial shifted complex of a let ii j i dim k tor i k i i j denote the graded betti numbers of i in the present paper it will be proved that i ii j i e ii j i c for all i and j where the base field is infinite and ii ii j i ii j i c for all i and j where the base field is arbitrary thus in particular one has ii j i ii j i lex for all i and j where lex is the unique lexsegment simplicial complex with the same f vector as a and where the base field is arbitrary

maximal minimal resolutions that is exact everywhere except at f0 where the cokernel is m such a resolution always exists if r is a noetherian n graded ring with r0 a field and m is a finitely generated z graded r module then a resolution may be constructed in a minimal way in which each fi is graded and each differential is homogeneous of degree 0 the rank of the ith free module in a minimal free resolution of m is an invariant of m called the ith betti number of m and denoted i m likewise r ij m the number of elements of degree j in a minimal set of homogeneous generators of fi is also an invariant of m called the i j th graded betti number of m in this paper we study modules with maximal graded betti numbers more precisely consider a set consisting of pairs r m where r is a noetherian n graded ring with r0 a field and m is a finitely generated graded r module is there some r m such that ij m ij m for every i and j and every r m there are trivial examples in which the answer is yes e g has only one element or no e g consists of all such pairs theorems 1 and 2 below give affirmative answers for certain sets defined by conditions on the hilbert series and the depths of r and m theorem 3 allows one to compute the maximal graded betti numbers for the pairs in the sets considered in theorems 1 and 2 throughout this paper k denotes a field and q the polynomial ring k x1 xn with the usual n grading given by degxi 1 the hilbert series of a finitely generated graded q module m is hm s dimk mis we define d lexicographic ideals and submodules in section 2 the poincare series of an r module m is p m s t ij m s t we write 4 for coefficient wise inequality of laurent series with coefficients in z

poset embeddings of hilbert functions for a standard graded algebra r we consider embeddings of the poset of hilbert functions of r ideals into the poset of r ideals as a way of classification of hilbert functions there are examples of rings for which such embeddings do not exist we describe how the embedding can be lifted to certain ring extensions which is then used in the case of polarization and distraction a version of a theorem of clements lindstrom is proved we exhibit a condition on the embedding that ensures that the classification of hilbert functions is obtained with images of lexicographic segment ideals

distractions of shakin rings abstract we study by means of embeddings of hilbert functions a class of rings which we call shakin rings i e quotients k x 1 x n a of a polynomial ring over a field k by ideals a l p which are the sum of a piecewise lex segment ideal l as defined by shakin and a pure powers ideal p our main results extend abedelfatah s recent work on the eisenbud green harris conjecture shakin s generalization of macaulay and bigatti hulett pardue theorems on betti numbers and when char k 0 mermin murai theorem on the lex plus power inequality from monomial regular sequences to a larger class of ideals we also prove an extremality property of embeddings induced by distractions in terms of hilbert functions of local cohomology modules

betti splitting via componentwise linear ideals abstract a monomial ideal i admits a betti splitting i j k if the betti numbers of i can be determined in terms of the betti numbers of the ideals j k and j k given a monomial ideal i we prove that i j k is a betti splitting of i provided j and k are componentwise linear generalizing a result of francisco ha and van tuyl if i has a linear resolution the converse also holds we apply this result recursively to the alexander dual of vertex decomposable shellable and constructible simplicial complexes moreover we determine the graded betti numbers of the defining ideal of three general fat points in the projective space

cayley bacharach theorems and conjectures a theorem of pappus of alexandria proved in the fourth century a d began a long development in algebraic geometry in its changing expressions one can see reflected the changing concerns of the field from synthetic geometry to projective plane curves to riemann surfaces to the modern development of schemes and duality we survey this development historically and use it to motivate a brief treatment of a part of duality theory we then explain one of the modern developments arising from it a series of conjectures about the linear conditions imposed by a set of points in projective space on the forms that vanish on them we give a proof of the conjectures in a new

applications of mapping cones over clements lindstrom rings abstract we prove that gotzmann s persistence theorem holds over every clements lindstrom ring we also construct the infinite minimal free resolution of a square free borel ideal over such a ring

ideals containing the squares of the variables we study the betti numbers of graded ideals containing the squares of the variables in a polynomial ring we prove the lex plus powers conjecture for such ideals

global qualitative description of a class of nonlinear dynamical systems in this paper we propose a methodology to derive a qualitative description of the behavior of a system from an incompletely known nonlinear dynamical model the model is written as an algebraic structure with unknown parameters and or functions under some hypotheses we obtain a graph describing the possible transitions between regions defined by the trends of the state variables and their relative positions a qualitative simulation of the model can be compared with on line data for fault detection purpose we give the example of a nonlinear biological model in dimension three for the growth of cells in a bioreactor

a generalized modified bessel function and a higher level analogue of the theta transformation formula abstract a new generalization of the modified bessel function of the second kind k z x is studied elegant series and integral representations a differential difference equation and asymptotic expansions are obtained for it thereby anticipating a rich theory that it may possess the motivation behind introducing this generalization is to have a function which gives a new pair of functions reciprocal in the koshliakov kernel cos z m 2 z 4 x sin z j 2 z 4 x and which subsumes the self reciprocal pair involving k z x its application towards finding modular type transformations of the form f z w f z i w where 1 is given as an example we obtain a beautiful generalization of a famous formula of ramanujan and guinand equivalent to the functional equation of a non holomorphic eisenstein series on s l 2 z this generalization can be considered as a higher level analogue of the general theta transformation formula we then use it to evaluate an integral involving the riemann function and consisting of a sum of products of two confluent hypergeometric functions

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

algebra structures on some canonical resolutions the study of algebra structures on finite free resolutions of cyclic modules begins with buchsbaum and eisenbud b el the classes of cyclic modules whose minimal resolutions are known to admit an algebra structure include the residue field gu complete intersections koszul complex modules of homological dimension atmost 3 b el gorenstein of codimension four k m and herzog algebras k m23 avramov al gave examples to show that there are cyclic modules whose minimal resolutions do not admit an algebra structure let r be a noetherian local ring with maximal ideal m in this paper we construct algebra structures on the minimal resolutions of two classes of cyclic modules r i namely when z is of the form jk where j is an ideal generated by a regular sequence and when z is the ideal of maximal minors of a generic n x m matrix provided r contains the rationals in al avramov defined certain obstructions to the existance of an algebra structure on the minimal resolution of a module and then produced modules with non zero obstructions the general question is whether the vanishing of these obstructions is also sufficient for the existence of a minimal algebra resolution of a cyclic module when z is an ideal generated by a regular sequence in r and m r z for any positive integer k then these obstructions are all zero therefore avramov and schlessinger asked whether the minimal resolutions of r z admit an algebra structure corollary 3 6 of this paper provides an affirmative answer to this question

on the uniformity of zero dimensional complete intersections abstract after showing that the general cayley bacharach conjecture formulated by d eisenbud m green and j harris 1996 6 is equivalent to a conjecture about the region of uniformity of a zero dimensional complete intersection we prove this conjecture in a number of special cases in particular after splitting the conjecture into several intervals we prove it for the first the last and part of the penultimate interval moreover we generalize the uniformity results of j hansen 2003 12 and l gold j little and h schenck 2005 9 to level schemes and apply them to obtain bounds for the minimal distance of generalized reed muller codes

pruned cellular free resolutions of monomial ideals abstract using discrete morse theory we give an algorithm that prunes the excess of information in the taylor resolution and constructs a new cellular free resolution for an arbitrary monomial ideal the pruned resolution is not simplicial in general but we can slightly modify our algorithm in order to obtain a simplicial resolution we also show that the lyubeznik resolution fits into our pruning strategy the pruned resolution is not always minimal but it is a lot closer to the minimal resolution than the taylor and the lyubeznik resolutions as we will see in some examples we finally use our methods to give a different approach to the theory of splitting of monomial ideals we deduce from this splitting strategy that the pruned resolution is always minimal in the case of edge ideals of paths and cycles

new pathways and connections in number theory and analysis motivated by two incorrect claims of ramanujan abstract the focus of this paper commences with an examination of three not obviously related pages in ramanujan s lost notebook pages 336 335 and 332 in decreasing order of attention on page 336 ramanujan proposes two identities but the formulas are wrong each is vitiated by divergent series we concentrate on only one of the two incorrect identities which may have been devised to attack the extended divisor problem we prove here a corrected version of ramanujan s claim which contains the convergent series appearing in it the convergent series in ramanujan s faulty claim is similar to one used by g f voronoi g h hardy and others in their study of the classical dirichlet divisor problem this now brings us to page 335 which comprises two formulas featuring doubly infinite series of bessel functions the first being conjoined with the classical circle problem initiated by gauss and the second being associated with the dirichlet divisor problem the first and fourth authors along with sun kim have written several papers providing proofs of these two difficult formulas in different interpretations in this monograph we return to these two formulas and examine them in more general settings the aforementioned convergent series in ramanujan s identity is also similar to one that appears in a curious identity found in chapter 15 in ramanujan s second notebook written in a more elegant equivalent formulation on page 332 in the lost notebook this formula may be regarded as a formula for 1 2 and in 1925 s wigert obtained a generalization giving a formula for 1 k for any even integer k 2 we extend the work of ramanujan and wigert in this paper the voronoi summation formula appears prominently in our study in particular we generalize work of j r wilton and derive an analogue involving the sum of divisors function s n the modified bessel functions k s x arise in several contexts as do lommel functions we establish here new series and integral identities involving modified bessel functions and modified lommel functions among other results we establish a modular transformation for an infinite series involving s n and modified lommel functions we also discuss certain obscure related work of n s koshliakov we define and discuss two new related classes of integral transforms which we call koshliakov transforms because he first found elegant special cases of each

reverse lex ideals we study reverse lex ideals in a polynomial ring and compare their properties to those of lex ideals in particular we provide an analogue of green s theorem for reverse lex ideals we also compare the betti numbers of strongly stable and square free strongly stable monomial ideals to those of reverse lex ideals

free resolutions of lex ideals over a koszul toric ring in this paper we study the minimal free resolution of lex ideals over a koszul toric ring in particular we study in which toric ring r all lex ideals are componentwise linear we give a certain necessity and sufficiency condition for this property and show that lex ideals in a strongly koszul toric ring are componentwise linear in addition it is shown that in the toric ring arising from the segre product p p every hilbert function of a graded ideal is attained by a lex ideal and that lex ideals have the greatest graded betti numbers among all ideals having the same hilbert function

symbolic reachability analysis of genetic regulatory networks using discrete abstractions we use hybrid systems techniques for the analysis of reachability properties of a class of piecewise affine pa differential equations that are particularly suitable for the modeling of genetic regulatory networks more specifically we introduce a hyperrectangular partition of the state space that forms the basis for a discrete abstraction preserving the sign of the derivatives of the state variables the resulting discrete transition system provides a qualitative description of the network dynamics that is well adapted to available experimental data and that can be efficiently computed in a symbolic manner from inequality constraints on the parameters

hilbert schemes and maximal betti numbers over veronese rings macaulay s theorem macaulay in proc lond math soc 26 531 555 1927 characterizes the hilbert functions of graded ideals in a polynomial ring over a field we characterize the hilbert functions of graded ideals in a veronese ring r the coordinate ring of a veronese embedding of pr 1 we also prove that the hilbert scheme which parametrizes all graded ideals in r with a fixed hilbert function is connected this is an analogue of hartshorne s theorem hartshorne in math ihes 29 5 48 1966 that hilbert schemes over a polynomial ring are connected furthermore we prove that each lex ideal in r has the greatest betti numbers among all graded ideals with the same hilbert function

on the resolution of ideals of fat points abstract in this paper we determine the hilbert function and the minimal system of generators of r 1 n 1 general fat points of p n furthermore by looking at the minimal system of generators we can show that the ideal associated to two fat points in p n or the ideal associated to r 1 n general fat points all with the same multiplicities is a splittable ideal and this is the first step in constructing a minimal resolution

a dynamic systems perspective on qualitative simulation abstract this paper presents a simple geometric interpretation of qualitative simulation qs based on the phase space representation of dynamic systems theory qs consists of two steps transition analysis determines the sequence of qualitative states that a system traverses and global interpretation derives its long term behavior i recast transition analysis as a search problem in phase space and replace the assorted transition rules with two algebraic conditions the first condition determines transitions between arbitrarily shaped regions in phase space as opposed to qs which only handles n dimensional rectangles it also provides more accurate results by considering only the boundaries between regions the second condition determines whether nearby trajectories approach a fixed point asymptotically it obtains better results than qs by exploiting local stability properties i recast global interpretation of dissipative systems as a search for attractors in phase space and present a global interpretation strategy for the subset of these systems whose local behavior determines global behavior uniquely although limited in scope this strategy handles many systems that appear in the qs literature

twisted second moments of the riemann zeta function and applications abstract in order to compute a twisted second moment of the riemann zeta function two different mollifiers each being a combination of two different dirichlet polynomials were introduced separately by bui conrey and young and by feng in this article we introduce a mollifier which is a combination of four dirichlet polynomials of different shapes we provide an asymptotic result for the twisted second moment of s for such choice of mollifier a small increment on the percentage of zeros of the riemann zeta function on the critical line is given as an application of our results

resolutions of a stable ideals abstract let a a 1 a 2 a n be a sequence of integers or we introduce a stable ideals in a polynomial ring and study their homological properties our results generalize results on square free monomial ideals by aramova avramov herzog hibi and srinivasan

qualitative stability of certain nonlinear systems to certain nonlinear dynamical systems naturally correspond simplicial complexes this correspondence is a generalization of the familiar relationship between the interaction matrix of a linear dynamical system and the signed digraph of that matrix by defining stability in terms of attractor regions as opposed to attractor trajectories we can specify qualitative linear algebraic conditions involving the simplicial complex which insure stability of the nonlinear system the analysis uses only signs of coefficients in the dynamical system a model of e lorenz 3 is an example of a three dimensional system which fulfills the stability conditions and which is known to have a strange attractor within the attractor region in summary the linear qualitative tests described theorems 2 and 3 can be applied to certain nonlinear dynamical systems to yield preliminary information about the global stability of such systems

automatic qualitative analysis of dynamic systems using piecewise linear approximations abstract this paper explores automating the qualitative analysis of physical systems it describes a program called plr for piecewise linear reasoner that formalizes an analysis strategy employed by experts plr takes parameterized ordinary differential equations as input and produces a qualitative description of the solutions for all initial values it approximates intractable nonlinear systems with piecewise linear ones analyzes the approximations and draws conclusions about the original systems it chooses approximations that are accurate enough to reproduce the essential properties of their nonlinear prototypes yet simple enough to be analyzed completely and efficiently it derives additional properties such as boundedness or periodicity by theoretical methods plr s analysis depends on abstract properties of systems rather than on specific numeric values this makes its conclusions more robust and enables it to handle parameterized equations transparently i demonstrate plr on several common nonlinear systems and on published examples from mechanical engineering

koshliakov kernel and identities involving the riemann zeta function abstract some integral identities involving the riemann zeta function and functions reciprocal in a kernel involving the bessel functions j z x y z x and k z x are studied interesting special cases of these identities are derived one of which is connected to a well known transformation due to ramanujan and guinand

almost complete intersections and the lex plus powers conjecture abstract we prove the almost complete intersection case of the lex plus powers conjecture on graded betti numbers we show that the resolution of a lex plus powers almost complete intersection provides an upper bound for the graded betti numbers of any other ideal with regular sequence in the same degrees and the same hilbert function a key ingredient is finding an explicit comparison map between two koszul complexes finally we obtain bounds on the hilbert function of an almost complete intersection including a special case of a conjecture of eisenbud green harris

box invariance in biologically inspired dynamical systems a dynamical system is box invariant if there exists a box shaped positively invariant region we show that box invariance can be checked in cubic time for linear and affine systems and that it remains decidable for classes of nonlinear systems of interest with polynomial structure we present results on the robustness of box invariance for linear systems using spectral properties of metzler matrices we also present sufficient conditions for establishing box invariance of switched and hybrid systems in general we argue that box invariance is a characteristic of many biologically inspired dynamical models

on the lex plus powers conjecture abstract let s be a polynomial ring over a field and i s a homogeneous ideal containing a regular sequence of forms of degrees d 1 d c in this paper we prove the lex plus powers conjecture when the field has characteristic 0 for all regular sequences such that d i j 1 i 1 d j 1 1 for each i that is we show that the betti table of i is bounded above by the betti table of the lex plus powers ideal of i

zeros of combinations of the riemann ξ function and the confluent hypergeometric function on bounded vertical shifts abstract in 1914 hardy proved that infinitely many non trivial zeros of the riemann zeta function lie on the critical line using the transformation formula of the jacobi theta function recently the first author obtained an integral representation involving the riemann function and the confluent hypergeometric function linked to the general theta transformation using this result we show that a series consisting of bounded vertical shifts of a product of the riemann function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line thereby generalizing a previous result due to the first and the last authors along with roy and robles the latter itself is a generalization of hardy s theorem

structural decomposition of monomial resolutions abstract we express the multigraded betti numbers of an arbitrary monomial ideal in terms of the multigraded betti numbers of two basic classes of ideals this decomposition has multiple applications in some concrete cases we use it to construct minimal resolutions of classes of monomial ideals in other cases we use it to compute projective dimensions to illustrate the effectiveness of the structural decomposition we give a new proof of a classic theorem by charalambous that states the following let k be a field and m an artinian monomial ideal in s k x 1 x n then for all i b i s m n i

poset embeddings of hilbert functions and betti numbers abstract we study inequalities between graded betti numbers of ideals in a standard graded algebra over a field and their images under embedding maps defined earlier by us in caviglia and kummini 2013 5 we show that if graded betti numbers do not decrease when we replace ideals in an algebra by their embedded versions then the same behaviour is carried over to ring extensions as a corollary we give alternative inductive proofs of earlier results of bigatti hulett pardue mermin peeva stillman and murai we extend a hypersurface restriction theorem of herzog and popescu to the situation of embeddings we show that we can obtain the betti table of an ideal in the extension ring from the betti table of its embedded version by a sequence of consecutive cancellations we further show that the lex plus powers conjecture of evans reduces to the artinian situation