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

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

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

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

on testing an unspecified function through a linear mixed effects model with multiple variance components summary we examine a generalized f test of a nonparametric function through penalized splines and a linear mixed effects model representation with a mixed effects model representation of penalized splines we imbed the test of an unspecified function into a test of some fixed effects and a variance component in a linear mixed effects model with nuisance variance components under the null the procedure can be used to test a nonparametric function or varying coefficient with clustered data compare two spline functions test the significance of an unspecified function in an additive model with multiple components and test a row or a column effect in a two way analysis of variance model through a spectral decomposition of the residual sum of squares we provide a fast algorithm for computing the null distribution of the test which significantly improves the computational efficiency over bootstrap the spectral representation reveals a connection between the likelihood ratio test lrt in a multiple variance components model and a single component model we examine our methods through simulations where we show that the power of the generalized f test may be higher than the lrt depending on the hypothesis of interest and the true model under the alternative we apply these methods to compute the genome wide critical value and p value of a genetic association test in a genome wide association study gwas where the usual bootstrap is computationally intensive up to 108 simulations and asymptotic approximation may be unreliable and conservative

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

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

a note on the multiplicities of gorenstein algebras abstract in this note we give a formula for the multiplicities of homogenous gorenstein algebras herzog huneke and srinivasan have conjectured bounds for the multiplicities of homogeneous cohen macaulay algebras herzog and srinivasan have proved this conjecture for c m algebras with quasi pure resolutions using this formula for the multiplicities of gorenstein algebras we prove a stronger bound for the multiplicities of goerenstein algebras with quasi pure resolutions

periodic solutions of higher order delay differential equations abstract in this paper we prove existence results for periodic solutions concerning the higher order delay differential equations our method is based upon the coincidence degree theory of mawhin the results obtained are new examples are given to illustrate the main results

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

some loci of rational cubic fourfolds in this paper we investigate the divisor mathcal c 14 inside the moduli space of smooth cubic hypersurfaces in mathbb p 5 whose general element is a smooth cubic containing a smooth quartic rational normal scroll by showing that all degenerations of quartic scrolls in mathbb p 5 contained in a smooth cubic hypersurface are surfaces with one apparent double point we prove that every cubic hypersurface contained in mathcal c 14 is rational combining our proof with the hodge theoretic definition of mathcal c 14 we deduce that on a smooth cubic fourfold every class t in text h 2 2 x mathbb z with t 2 10 and t cdot h 2 4 is represented by a possibly reducible surface of degree four which has one apparent double point as an application of our results and of the construction of some explicit examples we also prove that the pfaffian locus is not open in mathcal c 14

existence and uniqueness of periodic solution for a class of differential systems in this paper sufficient conditions are obtained for the existence of a unique periodic solution for a class of differential systems

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

on the existence and uniqueness for higher order periodic boundary value problems abstract this paper discusses the existence and uniqueness for the n th order periodic boundary value problem l n u t f t u t 0 t 2 u i 0 u i 2 i 0 1 n 1 where l n u t u n t i 0 n 1 a i u i t is an n th order linear differential operator n 2 and f 0 2 r r is continuous in the case that l n has an even order derivative we present some new spectral conditions for the nonlinearity f t u to guarantee the existence and uniqueness these spectral conditions allow f t u to be a superlinear growth and are the extension for the spectral separation condition presented recently in y li existence and uniqueness for higher order periodic boundary value problems under spectral separation conditions j math anal appl 322 2 2006 530 539

on the asymptotic behaviour of the pseudolikelihood ratio test statistic this paper concerns the asymptotic distribution of the likehood ratio statistic t for testing h 0 0 based on the pseudolikelihood l where is a simple estimator of we show that the asymptotic distribution of t under h 0 is a weighted sum of indepepdent x 1 2 variables where the weights involve the asymptotic joint covariance matrix of and the score function for some sufficient conditions are provided for the limiting distribution to be x 2 the result is extended to allow 0 to be a boundary value of the parameter space and to be misspecified in l we also examine the issue of power loss when is misspecified in l several examples including variance component models multivariate survival models genetic linkage analysis and the behrens fisher problem are presented to demonstrate the scope of the problems considered and to illustrate the results

on high order delay differential equation in this paper we consider a general high order delay differential equation in the following form x n t f t x t x t t t x t x n 1 t where t is a continuous periodic function defined on r with period t f is a continuous function defined on r n 2 and periodic to t f t c c 0 0 0 for all c r by applying mawhin s continuation theorem and a new inequality established in this paper we obtain sufficient conditions for the existence of periodic solutions for the equation our results are new and the methods are different from previous works moreover an example to illustrate the results is given

a functional generalized f test for signal detection with applications to event related potentials significance analysis motivated by the analysis of complex dependent functional data such as event related brain potentials erp this paper considers a time varying coefficient multivariate regression model with fixed time covariates for testing global hypotheses about population mean curves based on a reduced rank modeling of the time correlation of the stochastic process of pointwise test statistics a functional generalized f test is proposed and its asymptotic null distribution is derived our analytical results show that the proposed test is more powerful than functional analysis of variance testing methods and competing signal detection procedures for dependent data simulation studies confirm such power gain for data with patterns of dependence similar to those observed in erps the new testing procedure is illustrated with an analysis of the erp data from a study of neural correlates of impulse control

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

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

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

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

a note on the multiplicity of determinantal ideals abstract herzog huneke and srinivasan have conjectured that for any homogeneous k algebra the multiplicity is bounded above by a function of the maximal degrees of the syzygies and below by a function of the minimal degrees of the syzygies the goal of this paper is to establish the multiplicity conjecture of herzog huneke and srinivasan about the multiplicity of graded cohen macaulay algebras over a field k for k algebras k x 1 x n i when i is a determinantal ideal of arbitrary codimension

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

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

periodic solutions for 2kth order ordinary differential equations with resonance in this paper a non variational version of a max min principle is extended and some unique existence results are obtained for the periodic boundary value problem of the higher order ordinary differential equations under a resonant condition

periodic solutions for higher order differential equations with deviating argument abstract by using the coincidence degree theory of mawhin we study the existence of periodic solutions for higher order differential equations with deviating argument x n t i 1 n 1 b i x i t f t x t x t 1 t x t m t p t some new results on the existence of periodic solutions of the equations are obtained meanwhile an example is given to illustrate our results

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

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

simple linear time algorithms to test chordality of graphs test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs chordal graphs arise naturally in the study of gaussian elimination on sparse symmetric matrices acyclic hypergraphs arise in the study of relational data bases rose tarjan and lueker siam j c

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

sharp bounds for higher linear syzygies and classifications of projective varieties in the present paper we consider upper bounds of higher linear syzygies i e graded betti numbers in the first linear strand of the minimal free resolutions of projective varieties in arbitrary characteristic for this purpose we first remind partial elimination ideals peis theory and introduce a new framework in which one can study the syzygies of embedded projective varieties well using peis theory and the reduction method via inner projections next we establish fundamental inequalities which govern the relations between the graded betti numbers in the first linear strand of an algebraic set x and those of its inner projection x q using these results we obtain some natural sharp upper bounds for higher linear syzygies of any nondegenerate projective variety in terms of the codimension with respect to its own embedding and classify what the extremal case and the next to extremal case are this is a generalization of castelnuovo and fano s results on the number of quadrics containing a given variety and another characterization of varieties of minimal degree and del pezzo varieties from the viewpoint of syzygies note that our method could also be applied to get similar results for more general categories e g connected in codimension one algebraic sets

periodic solutions for nonlinear nth order differential equations with delays by applying the continuation theorem of coincidence degree theory we establish the existence of 2 periodic solutions for a class of nonlinear nth order differential equations with delays

odd periodic solutions for 2nth order ordinary differential equations this paper deals with the existence of periodic solutions for the 2nth order ordinary differential equation u 2n t f t u t u t u 2n 2 t where the nonlinear term f r rn r is a continuous odd function and f t x0 x1 xn 1 is 2 periodic in t some existence results for odd 2 periodic solutions are obtained under the condition that f satisfies some linear superlinear or sublinear growth conditions on x0 x1 xn 1

degrees of freedom tests for smoothing splines when using smoothing splines to estimate a function the user faces the problem of choosing the smoothing parameter several techniques are available for selecting this parameter according to certain optimality criteria here we take a different point of view and we propose a technique for choosing between two alternatives for example allowing for two different levels of degrees of freedom the problem is addressed in the framework of a mixed effects model whose assumptions ensure that the resulting estimator is unbiased a likelihood ratio type test statistic is proposed and its exact distribution is derived tests of linearity and overall effect follow directly we then extend this idea to additive models where it provides a more attractive alternative than multi parameter optimisation and where it gives exact distributional results that can be used in an analysis of deviance type approach examples on real data and a simulation study of level and power complete the paper

on periodic solutions for even order differential equations abstract a class of sufficient conditions are obtained for the existence of a unique 2 periodic solution of even order differential equations employing an initial value problem

existence and uniqueness of periodic solutions for 2n 1 th order differential equations applying the schauder fixed point theorem and the continuation theorem of mawhin we give the existence and uniqueness results of periodic solutions for a class of 2n 1 th order nonlinear ordinary differential equations

minimax theorems and applications to the existence and uniqueness of solutions of some differential equations in this paper minimax theorems due to stepan a tersian were generalized the existence and uniqueness of solutions for several semilinear equations were proved by employing our generalized theorems and the existence and uniqueness results of nonresonance problem for these semilinear equations under the asymptotic un uniformity conditions were presented

on higher order fully periodic boundary value problems abstract in this paper we present sufficient conditions for the existence of periodic solutions of the higher order fully differential equation u n x f x u x u x u n 1 x with n 3 x a b and f a b r n r a continuous function verifying a nagumo type growth condition a new type of lower and upper solutions eventually non ordered allows us to obtain not only the existence but also some qualitative properties on the solution the last section contains two examples to stress the application to both cases of n odd and n even

existence and uniqueness for higher order periodic boundary value problems under spectral separation conditions abstract this paper deals with the existence and uniqueness for the nth order periodic boundary value problem l n u t f t u t 0 t 2 u i 0 u i 2 i 0 1 n 1 where l n u t u n t i 0 n 1 a i u i t is an nth order linear differential operator and f 0 2 r r is continuous we present some spectral conditions for the nonlinearity f t u to guarantee the existence and uniqueness these spectral conditions are the generalization for nonresonance condition of duffing equation

conditional akaike information for mixed effects models summary this paper focuses on the akaike information criterion aic for linear mixed effects models in the analysis of clustered data we make the distinction between questions regarding the population and questions regarding the particular clusters in the data we show that the aic in current use is not appropriate for the focus on clusters and we propose instead the conditional akaike information and its corresponding criterion the conditional aic caic the penalty term in caic is related to the effective degrees of freedom p for a linear mixed model proposed by hodges sargent 2001 p reflects an intermediate level of complexity between a fixed effects model with no cluster effect and a corresponding model with fixed cluster effects the caic is defined for both maximum likelihood and residual maximum likelihood estimation a pharmacokinetics data appli cation is used to illuminate the distinction between the two inference settings and to illustrate the use of the conditional aic in model selection

new approaches to bounding the multiplicity of an ideal abstract we use the theory of resolutions for a given hilbert function to investigate the multiplicity conjectures of huneke and srinivasan and herzog and srinivasan to prove the conjectures for all modules with a particular hilbert function we show that it is enough to prove the statements only for elements at the bottom of the partially ordered set of resolutions with that hilbert function this enables us to test the conjectured upper bound for the multiplicity efficiently with the computer algebra system macaulay 2 and we verify the upper bound for many artinian modules in three variables with small socle degree moreover with this approach we show that though numerical techniques have been sufficient in several of the known special cases they are insufficient to prove the conjectures in general finally we apply a result of herzog and srinivasan on ideals with a quasipure resolution to prove the upper bound for cohen macaulay quotients by ideals with generators in high degrees relative to the regularity

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

periodic solutions of 2n 1 th order ordinary differential equations abstract the goal of this paper is to prove the existence and uniqueness of periodic solutions for a class of the odd order differential equations here the leray schauder degree theory and the schwarz inequality are used

anti periodic solutions for a class of nonlinear nth order differential equations with delays in this paper we use the leray schauder degree theory to establish new results on the existence and uniqueness of anti periodic solutions for a class of nonlinear nth order differential equations with delays of the form x n t f t x n 1 t g t x email protected t e t

counting degrees of freedom in hierarchical and other richly parameterised models summary drawing on linear model theory we rigorously extend the notion of degrees of freedom to richly parameterised models including linear hierarchical and random effect models some smoothers and spatial models and combinations of these the number of degrees of freedom is often much smaller than the number of parameters our notion of degrees of freedom is compatible with similar ideas long associated with smoothers but is applicable to new classes of models and can be interpreted using the projection theory of linear models we use an example to illustrate the two applications of setting prior distributions for variances and fixing model complexity by fixing degrees of freedom

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

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

a marginal approach to reduced rank penalized spline smoothing with application to multilevel functional data multilevel functional data are collected in many biomedical studies for example in a study of the effect of nimodipine on patients with subarachnoid hemorrhage sah patients underwent multiple 4 hr treatment cycles within each treatment cycle subjects vital signs were reported every 10 min these data have a natural multilevel structure with treatment cycles nested within subjects and measurements nested within cycles most literature on nonparametric analysis of such multilevel functional data focuses on conditional approaches using functional mixed effects models however parameters obtained from the conditional models do not have direct interpretations as population average effects when population effects are of interest we may employ marginal regression models in this work we propose marginal approaches to fit multilevel functional data through penalized spline generalized estimating equation penalized spline gee the procedure is effective for modeling multilevel correlated generalized

variance components testing in the longitudinal mixed effects model this article discusses the asymptotic behavior of likelihood ratio tests for nonzero variance components in the longitudinal mixed effects linear model described by laird and ware 1982 biometrics 38 963 974 our discussion of the large sample behavior of likelihood ratio tests for nonzero variance components is based on the results for nonstandard testing situations by self and liang 1987 journal of the american statistical association 82 605 610

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

exact likelihood ratio tests for penalised splines summary penalised spline based additive models allow a simple mixed model representation where the variance components control departures from linear models the smoothing parameter is the ratio of the random coefficient and error variances and tests for linear regression reduce to tests for zero random coefficient variances we propose exact likelihood and restricted likelihood ratio tests for testing polynomial regression versus a general alternative modelled by penalised splines their spectral decompositions are used as the basis of fast simulation algorithms we derive the asymptotic local power properties of the tests under weak conditions in particular we characterise the local alternatives that are detected with asymptotic probability one confidence intervals for the smoothing parameter are obtained by inverting the tests for a fixed smoothing parameter versus a general alternative we discuss f and r tests and show that ignoring the variability in the smoothing parameter estimator can have a dramatic effect on their null distributions the powers of several known tests are investigated and a small set of tests with good power properties is identified the restricted likelihood ratio test is among the best in terms of power

analysis on some infinite modules inner projection and applications a projective scheme x is called quadratic if x is scheme theoretically cut out by homogeneous equations of degree 2 furthermore we say that x satisfies property n2 p if it is quadratic and the quadratic ideal has only linear syzygies up to the first p th steps in the present paper we compare the linear syzygies of the inner projections with those of x and obtain a theorem on embedded linear syzygies as one of our main results this is the natural projection analogue of restricting linear syzygies in the linear section case as an immediate corollary we show that the inner projections of x satisfy property n2 p 1 for any reduced scheme x with property n2 p moreover we also obtain the neccessary lower bound codimx p p p 1 2 which is sharp on the number of quadrics vanishing on x in order to satisfy n2 p and show that the arithmetic depths of inner projections are equal to that of the quadratic scheme x these results admit an interesting syzygetic rigidity theorem on property n2 p which leads the classifications of extremal and next to extremal cases for these results we develop the elimination mapping cone theorem for infinitely generated graded modules and improve the partial elimination ideal theory initiated by m green this new method allows us to treat a wider class of projective schemes which cannot be covered by the koszul cohomology techniques because these are not projectively normal in general

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

an application of a global inversion theorem to an existence and uniqueness theorem for a class of nonlinear systems of differential equations abstract in this paper a class of systems of nonlinear differential equations at resonance is considered with the use of a global inversion theorem which is an extended form of a non variational version of a max min principle we prove that this class of equations possesses a unique 2 periodic solution under a rather weaker condition for existence and uniqueness than those given in papers j chen w li periodic solution for 2 k th boundary value problem with resonance j math anal appl 314 2006 661 671 f cong periodic solutions for 2 k th order ordinary differential equations with nonresonance nonlinear anal 32 1998 787 793 f cong periodic solutions for second order differential equations appl math lett 18 2005 957 961 w li periodic solutions for 2 k th order ordinary differential equations with resonance j math anal appl 259 2001 157 167 w li h li a min max theorem and its applications to nonconservative systems int j math math sci 17 2003 1101 1110 w li z shen a constructive proof of existence and uniqueness of 2 periodic solution to duffing equation nonlinear anal 42 2000 1209 1220 this result extends the results known so far

a simple test for random effects in regression models summary testing that random effects are zero is difficult because the null hypothesis restricts the corresponding variance parameter to the edge of the feasible parameter space in the context of generalized linear mixed models this paper exploits the link between random effects and penalized regression to develop a simple test for a zero effect the idea is to treat the variance components not being tested as fixed at their estimates and then to express the likelihood ratio as a readily computed quadratic form in the predicted values of the random effects under the null hypothesis this has the distribution of a weighted sum of squares of independent standardnormalrandomvariables thetestcanbeusedwithgeneralizedlinearmixedmodels including those estimated by penalized quasilikelihood

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

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

the regularity of partial elimination ideals castelnuovo normality and syzygies abstract let x be a reduced closed subscheme in p n x x p n 1 be a projection from a point outside x and z i x x be the closed subscheme defined by the i th partial elimination ideal k i i x which is supported on the i 1 th multiple points of in this paper motivated from projection methods to prove eisenbud goto conjecture on regularity in many cases we describe the syzygetic behaviors and castelnuovo normality of the projection with a viewpoint of the regularity of the partial elimination ideal k i i x i 1 or that of the multiple locus z i x of we also give some applications to the syzygies and castelnuovo normality of successive projections which recover and generalize some known results in 1 3 15 16