on the minimal free resolution of n 1 general forms o roig abstract let r k x1 xn and let i be the ideal of n 1 generically chosen forms of degrees d1 dn 1 we give the precise graded betti numbers of r i in the following cases n 3 n 4 and p5 1 di is even n 4 p 5 1 di is odd and d2 d3 d4 d1 d5 4 n is even and all generators have the same degree a which is even pn 1 i 1 di i n is even and d2 dn d1 dn 1 n p n 1 i 1 di i n is odd n 6 is even d2 dn d1 dn 1 n and d1 dn i dn 1 i n 0 we give very good bounds on the graded betti numbers in many other cases we also extend a result of m boij by giving the graded betti numbers for a generic compressed gorenstein algebra i e one for which the hilbert function is maximal given n and the socle degree when n is even and the socle degree is large a recurring theme is to examine when and why the minimal free resolution may be forced to have redundant summands we conjecture that if the forms all have the same degree then there are no redundant summands and we present some evidence for this conjecture

local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

information geometry on hierarchy of probability distributions an exponential family or mixture family of probability distributions has a natural hierarchical structure this paper gives an orthogonal decomposition of such a system based on information geometry a typical example is the decomposition of stochastic dependency among a number of random variables in general they have a complex structure of dependencies pairwise dependency is easily represented by correlation but it is more difficult to measure effects of pure triplewise or higher order interactions dependencies among these variables stochastic dependency is decomposed quantitatively into an orthogonal sum of pairwise triplewise and further higher order dependencies this gives a new invariant decomposition of joint entropy this problem is important for extracting intrinsic interactions in firing patterns of an ensemble of neurons and for estimating its functional connections the orthogonal decomposition is given in a wide class of hierarchical structures including both exponential and mixture families as an example we decompose the dependency in a higher order markov chain into a sum of those in various lower order markov chains

free resolutions of artinian compressed algebras abstract we construct free resolutions of compressed therefore generic artinian graded algebra quotients of polynomial rings and give a method to reduce them to a minimal resolutions this yields results on the form of the resolution and the degrees of the forms in the matrices of the differentials but not precise betti numbers in addition we discuss how this construction is related to a conjecture of boij last we treat the noncompressed case

the absolutely koszul and backelin roos properties for spaces of quadrics of small codimension abstract let k be a field r a standard graded quadratic k algebra with dim k r 2 3 and let k denote an algebraic closure of k we construct a graded surjective golod homomorphism p r k k such that p is a complete intersection of codimension at most 3 furthermore we show that r is absolutely koszul that is every finitely generated r module has finite linearity defect if and only if r is koszul if and only if r is not a trivial fiber extension of a standard graded k algebra with hilbert series 1 2 t 2 t 3 1 t 1 in particular we recover earlier results on the koszul property of backelin 4 conca 7 and d ali 11

on automatic boundary corrections many popular curve estimators based on smoothing have difficulties caused by boundary effects these effects are visually disturbing in practice and can play a dominant role in theoretical analysis local polynomial regression smoothers are known to correct boundary effects automatically some analogs are implemented for density estimation and the resulting estimators also achieve automatic boundary corrections in both settings of density and regression estimation we investigate best weight functions for local polynomial fitting at the endpoints and find a simple solution the solution is universal for general degree of local polynomial fitting and general order of estimated derivative furthermore such local polynomial estimators are best among all linear estimators in a weak minimax sense and they are highly efficient even in the usual linear minimax sense

isomorphism classes of short gorenstein local rings via macaulay s inverse system in this paper we study the isomorphism classes of artinian gorenstein local rings with socle degree three by means of macaulay s inverse system we prove that their classification is equivalent to the projective classification of the hypersurfaces of p n of degree three this is an unexpected result because it reduces the study of this class of local rings to the homogeneous case the result has applications in problems concerning the punctual hilbert scheme hilbd p n and

synchronized clusters in coupled map networks ii stability analysis we study self organized and driven synchronization in some simple coupled map networks namely globally coupled networks and complete bipartite networks using both linear stability analysis and lyapunov function approach and determine stability conditions for synchronization the phase diagrams for the networks studied here have features very similar to the different kinds of structurally similar networks studied in part i lyapunov function approach shows that when any two nodes are in driven synchronization all the coupling terms in the difference between the variables of these two nodes cancel out whereas when they are in self organized synchronization the direct coupling term between the two nodes adds an extra term while the other couplings cancel out we also discuss the conditions for the occurrence of a floating node and suggest that the fluctuations of the conditional lyapunov exponent about zero can be a criterion for its occurrence

on the dimension of bivariate spline spaces of smoothnessr and degreed 3r 1 we consider the well known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally to date dimension formulae for such spaces have been established only whend 3r 2 except for the special case wherer 1 andd 4 in this paper we establish dimension formulae for allr 1 andd 3r 1 for almost all triangulations

poincare series of compressed local artinian rings with odd top socle degree abstract we define a notion of compressed local artinian ring that does not require the ring to contain a field let r m be a compressed local artinian ring with odd top socle degree s at least five and socle r m s 1 m s we prove that the poincare series of all finitely generated modules over r are rational sharing a common denominator and that there is a golod homomorphism from a complete intersection onto r

improved estimators in nonparametric regression problems abstract linear estimators of multivariate means are considered generalizations of some well known theorems about admissibility of linear estimators are given the results then are applied to show that commonly used kernel type estimators in nonparametric regression problems can be constructively improved in a simple way an asymptotic result is described that gives a quantitative measure of the maximum improvement to be gained in certain situations a theoretical bound shows that gains are achievable in the relative risk of up to 58 6 rectangular kernel or 29 2 epanechnikov kernel some examples of smaller sample size are also investigated and these show relative risk gains ranging up to 18 in realistic settings

the explicit minimal resolution constructed from a macaulay inverse system abstract let a be a standard graded artinian gorenstein algebra of embedding codimension three over a field k in the generic case the minimal homogeneous resolution g of a by free sym k a 1 modules is gorenstein linear fix a basis x y z for the k vector space a 1 if g is gorenstein linear then the socle degree of a is necessarily even and if n is the least index with dim k a n less than dim k sym n k a 1 then the socle degree of a is 2 n 2 let m m as m roams over the monomials in x y z of degree 2 n 2 with m k be an arbitrary homogeneous element of degree 2 n 2 in the divided power module d k a 1 the annihilator of denoted ann is the ideal of elements f in sym k a 1 with f 0 the element of d k a 1 is the macaulay inverse system for the ring sym k a 1 ann which is necessarily gorenstein and artinian consider the matrix m m as m and m roam over the monomials in x y z of degree n 1 the ring sym k a 1 ann has a gorenstein linear resolution if and only if det m m 0 if det m m 0 then we give explicit formulas for the minimal homogeneous resolution of sym k a 1 ann in terms the m s and x y z

finding inverse systems from coordinates abstract let i be a homogeneous ideal in r k x 0 x n such that r i is an artinian gorenstein ring a famous theorem of macaulay says that in this instance i is the ideal of polynomial differential operators with constant coefficients that cancel the same homogeneous polynomial f a major question related to this result is to be able to describe f in terms of the ideal i in this note we give a partial answer to this question by analyzing the case when i is the artinian reduction of the ideal of a reduced arithmetically gorenstein zero dimensional scheme p n we obtain f from the coordinates of the points of

zonotopal algebra and forward exchange matroids abstract zonotopal algebra is the study of a family of pairs of dual vector spaces of multivariate polynomials that can be associated with a list of vectors x it connects objects from combinatorics geometry and approximation theory the origin of zonotopal algebra is the pair d x p x where d x denotes the dahmen micchelli space that is spanned by the local pieces of the box spline and p x is a space spanned by products of linear forms the first main result of this paper is the construction of a canonical basis for d x we show that it is dual to the canonical basis for p x that is already known the second main result of this paper is the construction of a new family of zonotopal spaces that is far more general than the ones that were recently studied by ardila postnikov holtz ron holtz ron xu li ron and others we call the underlying combinatorial structure of those spaces forward exchange matroid a forward exchange matroid is an ordered matroid together with a subset of its set of bases that satisfies a weak version of the basis exchange axiom

compressed algebras artin algebras having given socle degrees and maximal length j emsalem and the author showed in 18 that a general polynomialf of degree j in the ring sp k yl yr has jr rl 1 linearly independent partial derivates of order i for i 0 1 t j 2 here we generalize the proof to show that the various partial derivates of s polynomials of specified degrees are as independent as possible given the room available using this result we construct and describe the varieties g e and z e parametrizing the graded and nongraded compressed algebra quotients a r i of the power series ring r k xl sxr having given socle type e these algebras are artin algebras having maximal length dimk a possible given the embedding degree r and given the socle type sequence e els s es where ei is the number of generators of the dual module a of a having degree i the variety z e is locally closed irreducible and is a bundle over g e fibred by affine spaces an whose dimension is known we consider the compressed algebras a new class of interesting algebras and a source of examples many of them are nonsmoothable have no deformation to k k for dimension reasons for some choices of the sequence e d buchsbaum d eisenbud and the author have shown that the graded compressed algebras of socle type e have almost linear minimal resolutions over r with ranks and degrees determined by e other examples have given type e dimk socle a and are defined by an ideal i with certain given numbers of generators in r

complexity measures from interaction structures we evaluate information theoretic quantities that quantify complexity in terms of k text th order statistical dependences that cannot be reduced to interactions among k ensuremath 1 random variables using symbolic dynamics of coupled maps and cellular automata as model systems we demonstrate that these measures are able to identify complex dynamical regimes

synchronization of coupled systems with spatiotemporal chaos we argue that the synchronization transition of stochastically coupled cellular automata discovered recently by l g morelli it et al phys rev bf 58 e r8 1998 is generically in the directed percolation universality class in particular this holds numerically for the specific example studied by these authors in contrast to their claim for real valued systems with spatiotemporal chaos such as coupled map lattices we claim that the synchronization transition is generically in the universality class of the kardar parisi zhang equation with a nonlinear growth limiting term

bivariate measure of redundant information we define a measure of redundant information based on projections in the space of probability distributions redundant information between random variables is information that is shared between those variables but in contrast to mutual information redundant information denotes information that is shared about the outcome of a third variable formalizing this concept and being able to measure it is required for the non negative decomposition of mutual information into redundant and synergistic information previous attempts to formalize redundant or synergistic information struggle to capture some desired properties we introduce a new formalism for redundant information and prove that it satisfies all the properties necessary outlined in earlier work as well as an additional criterion that we propose to be necessary to capture redundancy we also demonstrate the behavior of this new measure for several examples compare it to previous measures and apply it to the decomposition of transfer entropy

the structure of gorenstein linear resolutions of artinian algebras abstract let k be a field a a standard graded artinian gorenstein k algebra s the standard graded polynomial ring sym k a 1 i the kernel of the natural map d the vector space dimension dim k a 1 and n the least index with i n 0 assume that 3 d and 2 n in this paper we give the structure of the minimal homogeneous resolution b of a by free s modules provided b is gorenstein linear keep in mind that if a has even socle degree and is generic then a has a gorenstein linear minimal resolution our description of b depends on a fixed but arbitrary decomposition of a 1 of the form k x 1 v 0 for some non zero element x 1 and some d 1 dimensional subspace v 0 of a 1 much information about b is already contained in the complex b b x 1 b which we call the skeleton of b one striking feature of b is the fact that the skeleton of b is completely determined by the data d n no other information about a is used in the construction of b the skeleton b is the mapping cone of zero k l where l is a well known resolution of buchsbaum and eisenbud k is the dual of l and l and k are comprised of schur and weyl modules associated to hooks respectively the decomposition of b into schur and weyl modules lifts to a decomposition of b furthermore b inherits the natural self duality of b the differentials of b are explicitly given in a polynomial manner in terms of the coefficients of a macaulay inverse system for a in light of the properties of b the description of the differentials of b amounts to giving a minimal generating set of i and for the interior differentials giving the coefficients of x 1 as an application we observe that every non zero element of a 1 is a weak lefschetz element for a

fat points inverse systems and piecewise polynomial functions abstract we explore the connection between ideals of fat points which correspond to subschemes of p nobtained by intersecting mixed powers of ideals of points and piecewise polynomial functions splines on ad dimensional simplicial complex embedded inrd using the inverse system approach introduced by macaulay 11 we give a complete characterization of the free resolutions possible for ideals ink x y generated by powers of homogeneous linear forms we allow the powers to differ we show how ideals generated by powers of homogeneous linear forms are related to the question of determining for some fixed the dimension of the vector space of splines on of degree less than or equal tok we use this relationship and the results above to derive a formula which gives the number of planar mixed splines in sufficiently high degree

reducible family of height three level algebras let r k x 1 x r be the polynomial ring in r variables over an infinite field k and let m be the maximal ideal of r here a level algebra will be a graded artinian quotient a of r having socle soc a 0 m in a single degree j the hilbert function h a h 0 h 1 h j gives the dimension h i dim k a i of each degree i graded piece of a for 0 i j the embedding dimension of a is h 1 and the type of a is dim k soc a here h j the family levalg h of level algebra quotients of r having hilbert function h forms an open subscheme of the family of graded algebras or via macaulay duality of a grassmannian we show that for each of the hilbert functions h 1 3 4 4 and h 2 1 3 6 8 9 3 the family levalg h has several irreducible components theorems 2 3 a 2 4 we show also that these examples each lift to points however in the first example an irreducible betti stratum for artinian algebras becomes reducible when lifted to points theorem 2 3 b we show that the second example is the first in an infinite sequence of examples of type three hilbert functions h c in which also the number of components gets arbitrarily large theorem 2 10 the first case where the phenomenon of multiple components can occur i e the lowest embedding dimension and then the lowest type is that of dimension three and type two examples of this first case have been obtained by the authors unpublished and also by j o kleppe

minimax linear estimation in a white noise problem linear estimation of f x at a point in a white noise model is considered the exact linear minimax estimator of f 0 is found for the family of f x in which f x is lip m the resulting estimator is then used to verify a conjecture of sacks and ylvisaker concerning the near optimality of the epanechnikov kernel

long range effects in an elementary cellular automaton we present evidence that one of the elementary one dimensional cellular automata in the sense of wolfram rule 22 in wolfram s notation involves very complex long range effects similar to a critical phenomenon this is in contrast to superficial evidence that would suggest that this rule leads to fairly simple behavior

renormalizing upper and lower bounds for integrated risk in the white noise model where wt is brownian motion many results which might be difficult to prove in the density estimation or nonparametric regression context take on a more transparent form in this white noise model a sample size of n in the density estimation and nonparametric regression problems corresponds to o 0 o xn in 1 when ois suitably chosen in particular the tools of rescaling developed in low 1992 and donoho and low 1992 and the hardest one dimensional subfamily arguments of donoho and liu 1987 1991 have yielded a fairly complete picture of how to estimate both bounded and unbounded linear functionals on the basis of observations generated by 1 a separate literature is developing to show how to replace density estimation and regression problems by the corresponding white noise problems see for example low 1992 brown and low 1990 and donoho and low 1990 in this paper we focus attention on estimating the entire function f on the basis of the observation scheme given by 1 using integrated squared error as a measure of loss in particular we shall let r f o denote the minimax risk under this loss function that is

inverse system of a symbolic power i abstract the authors determine macaulay s inverse system to a symbolic power of the graded ideal of functions vanishing at a union of irreducible varieties in projective space

free resolutions over short gorenstein local rings let r be a local ring with maximal ideal mathfrak m admitting a non zero element a in mathfrak m for which the ideal 0 a is isomorphic to r ar we study minimal free resolutions of finitely generated r modules m with particular attention to the case when mathfrak m 4 0 let e denote the minimal number of generators of mathfrak m if r is gorenstein with mathfrak m 4 0 and e 3 we show that rm p m r t is rational with denominator h r t 1 et et 2 t 3 for each finitely generated r module m in particular this conclusion applies to generic gorenstein algebras of socle degree 3

quantum algorithm for multivariate polynomial interpolation how many quantum queries are required to determine the coefficients of a degree d polynomial in n variables we present and analyse quantum algorithms for this multivariate polynomial interpolation

dynamical properties of the synchronization transition we use spreading dynamics to study the synchronization transition st of one dimensional coupled map lattices cml s recently baroni et al phys rev e 63 036226 2001 have shown that the st belongs to the directed percolation dp universality class for discontinuous cml s this was confirmed by accurate numerical simulations for the bernoulli map by ahlers and pikovsky phys rev lett 88 254101 2002 spreading dynamics confirms such an identification only for random synchronized states for homogeneous synchronized states the spreading exponents eta and delta are different from the dp exponents but their sum equals the corresponding sum of the dp exponents such a relation is typical of models with infinitely many absorbing states moreover we calculate the spreading exponents for the tent map for which the st belongs to the bounded kardar parisi zhang bkpz universality class the estimation of spreading exponents for random synchronized states is consistent with the hyperscaling relation while it is inconsistent for the homogeneous ones finally we examine the asymmetric tent map for small asymmetry the st remains of the bkpz type however for large asymmetry a different critical behavior appears with exponents being relatively close to those for dp

waring like decompositions of polynomials 1 abstract let f be a homogeneous form of degree d in n variables a waring decomposition of f is a way to express f as a sum of d th powers of linear forms in this paper we consider the decompositions of a form as a sum of expressions each of which is a fixed monomial evaluated at linear forms

inverse limits of macaulay s inverse systems abstract generalizing a result of masuti and the second author we describe inverse limits of macaulay s inverse systems for cohen macaulay factor algebras of formal power series or polynomial rings over an infinite field on the way we find a strictness result for filtrations defined by regular sequences it generalizes both a lemma of uli walther and the rees isomorphism

families of artinian and one dimensional algebras abstract the purpose of this paper is to study families of artinian or one dimensional quotients of a polynomial ring r with a special look to level algebras let gradalg h r be the scheme parametrizing graded quotients of r with hilbert function h let b a be any graded surjection of quotients of r with hilbert function h b 1 h 1 h j and h a respectively if dim a 0 respectively dim a depth a 1 and a is a truncation of b in the sense that h a 1 h 1 h j 1 0 0 respectively h a 1 h 1 h j 1 for some h j then we show there is a close relationship between gradalg h a r and gradalg h b r concerning e g smoothness and dimension at the points a and b respectively provided b is a complete intersection or provided the castelnuovo mumford regularity of a is at least 3 sometimes 2 larger than the regularity of b in the complete intersection case we generalize this relationship to non truncated artinian algebras a which are compressed or close to being compressed for more general artinian algebras we describe the dual of the tangent and obstruction space of graded deformations in a manageable form which we make rather explicit for level algebras of cohen macaulay type 2 this description and a linkage theorem for families allow us to prove a conjecture of iarrobino on the existence of at least two irreducible components of gradalg h r h 1 3 6 10 14 10 6 2 whose general elements are artinian level algebras of type 2

information capacity and pattern formation in a tent map network featuring statistical periodicity we provide quantitative support to the observation that lattices of coupled maps are efficient information coding devices it has been suggested recently that lattices of coupled maps may provide a model of information coding in the nervous system because of their ability to create structured and stimulus dependent activity patterns which have the potential to be used for storing information in this paper we give an upper bound to the effective number of patterns that can be used to store information in the lattice by evaluating numerically its information capacity or information rate as a function of the coupling strength between the maps we also estimate the time taken by the lattice to establish a limiting activity pattern

statistical estimation and optimal recovery new formulas are given for the minimax linear risk in estimating a linear functional of an unknown object from indirect data contaminated with random gaussian noise the formulas cover a variety of loss functions and do not require the symmetry of the convex a priori class it is shown that affine minimax rules are within a few percent of minimax even among nonlinear rules for a variety of loss functions it is also shown that difficulty of estimation is measured by the modulus of continuity of the functional to be estimated the method of proof exposes a correspondence between minimax affine estimates in the statistical estimation problem and optimal algorithms in the theory of optimal recovery

dimension of families of determinantal schemes a scheme x p n c of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous t x t c 1 matrix and x is said to be good determinantal if it is standard determinantal and a generic complete intersection given integers a 0 a 1 a t c 2 and b 1 b t we denote by w b a c hilb p p n c resp w s b a the locus of good resp standard determinantal schemes x c p n c of codimension c defined by the maximal minors of a t x t c 1 matrix f ij i 1 t j 0 t c 2 where f ij k x 0 x 1 x n c is a homogeneous polynomial of degree a j b i in this paper we address the following three fundamental problems to determine 1 the dimension of w b a resp w s b a in terms of a j and b i 2 whether the closure of w b a is an irreducible component of hilb p p n c and 3 when hilb p p n c is generically smooth along w b a concerning question 1 we give an upper bound for the dimension of w b a resp w s b a which works for all integers a 0 a 1 a t c 2 and b 1 b t and we conjecture that this bound is sharp the conjecture is proved for 2 2 and for c 5 under certain numerical assumptions

nonexistence of informative unbiased estimators in singular problems in many nonparametric problems such as density estimation nonparametric regression and so on all the existing informative estimators are biased asymptotic or finite sample there has long been a suspicion that either informative unbiased estimators do not exist for such problems or they must be quite complicated in this paper we clarify the nonexistence of informative unbiased estimators in all singular problems both for fixed sample size and asymptotically this includes most problems with optimal rate of convergence slower than n1 2 we also discuss situations in regular problems where such nonexistences can occur

homology of smooth splines generic triangulations and a conjecture of strang for a a triangulated d dimensional region in rd let sr a denote the vector space of all cr functions f on a that restricted to any simplex in a are given by polynomials of degree at most m we consider the problem of computing the dimension of such spaces we develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds a in the plane getting lower bounds on the dimension of sr a for all r for r 1 we prove a conjecture of strang concerning the generic dimension of the space of cl splines over a triangulated manifold in r2 finally we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region

maximal families of gorenstein algebras the purpose of this paper is to study maximal irreducible families of gorenstein quotients of a polynomial ring r let gradalg h r be the scheme parametrizing graded quotients of r with hilbert function h we prove there is a close relationship between the irreducible components of gradalg h r whose general member is a gorenstein codimension c 1 quotient and the irreducible components of gradalg h r whose general member b is a codimension c cohen macaulay algebra of hilbert function h related to h if the castelnuovo mumford regularity of the gorenstein quotient is large compared to the castelnuovo mumford regularity of b this relationship actually determines a well defined injective mapping from such cohen macaulay components of gradalg h r to gorenstein components of gradalg h r in which generically smooth components correspond moreover the dimension of the gorenstein components is computed in terms of the dimension of the corresponding cohen macaulay component and a sum of two invariants of b using linkage by a complete intersection we show how to compute these invariants linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main theorem

variance estimation in nonparametric regression via the difference sequence method consider a gaussian nonparametric regression problem having both an unknown mean function and unknown variance function this article presents a class of difference based kernel estimators for the variance function optimal convergence rates that are uniform over broad functional classes and bandwidths are fully characterized and asymptotic normality is also established we also show that for suitable asymptotic formulations our estimators achieve the minimax rate

inverse systems of zero dimensional schemes in p n abstract the authors construct the global macaulay inverse system l z for a zero dimensional subscheme z of projective n space p n over an algebraically closed field k from the local inverse systems of the irreducible components of z they show that when z is locally gorenstein a generic element f of degree d apolar to z determines z if d is larger than an invariant z as a consequence of this globalization they show that a natural upper bound for the hilbert function of gorenstein artin quotients of the coordinate ring of z is achieved for large socle degree they also show the uniqueness of generalized additive decompositions of a homogeneous form into powers of linear forms under suitable hypotheses the main tools are elementary but delicate they involve a careful study of how to homogenize a local inverse system and of the behavior of the homogenization under a change of coordinates

combinatorics and geometry of power ideals we investigate ideals in a polynomial ring which are generated by powers of linear forms such ideals are closely related to the theories of fat point ideals cox rings and box splines we pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement a we prove that their hilbert series are determined by the combinatorics of a and can be computed from its tutte polynomial we also obtain formulas for the hilbert series of certain closely related fat point ideals and zonotopal cox rings our work unifies and generalizes results due to dahmen micchelli holtz ron postnikov shapiro shapiro and sturmfels xu among others it also settles a conjecture of holtz ron on the spline interpolation of functions on the lattice points of a zonotope

free and non free multiplicities on the a3 arrangement abstract we give a complete classification of free and non free multiplicities on the a 3 braid arrangement namely we show that all free multiplicities on a 3 fall into two families that have been identified by abe terao wakefield 2007 and abe nuida numata 2009 the main tool is a new homological obstruction to freeness derived via a connection to multivariate spline theory

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

a lower bound for the risk in estimating the value of a probability density abstract this article considers estimation of f 0 of a density function satisfying a lipshitz condition in a neighborhood of 0 a nonstandard use of the cramer rao inequality yields numerical lower bounds on the minimax squared error risk of any estimator these bounds are then compared with the minimax risk of the asymptotically optimal kernel type estimator the asymptotic bounds obtained as the sample size n are not quite as good as those in donoho and liu in press a b but bounds are presented here also for finite values of n and that paper contains no such bounds for this problem the numerical results reported in table 1 show that the asymptotically optimal kernel estimator performs within a factor of 3 of the minimax bound even for sample size n 30 as n increases the relative performance improves to its limiting value although the convergence is fairly slow

an asymptotically efficient solution to the bandwidth problem of kernel density estimation a data driven method of choosing the bandwidth h of a kernel density estimator is heuristically motivated by considering modifications of the kullback leibler or pseudo likelihood cross validation function it is seen that this means of choosing h is asymptotically equivalent to taking the h that minimizes some compelling error criteria such as the average squared error and the integrated squared error thus for a given kernel function the bandwidth can be chosen optimally without making precise smoothness assumptions on the underlying density

some more exact enumeration results for 1d cellular automata the author presents exact enumeration results for the elementary cellular automation rules 22 30 45 and 120 in wolfram s notation obtained by a new algorithm they improve previous results considerably they confirm the existence of hidden power behaved long range correlations in the patterns created by these rules

poincare series of modules over local rings of small embedding codepth or small linking number abstract let s n k be a commutative noetherian local ring and m be a finitely generated s module then the poincare series of m is rational if any of the following holds edim s depth s 3 edim s depth s 4 and s is gorenstein s is one link from a complete intersection s is two links from a complete intersection and s is gorenstein to prove this result we assume without loss of generality that s r i with r m regular local and the key to the argument is to produce a factorization of the canonical map r s as a composition of a complete intersection r c with a golod map c s this is accomplished by invoking a theorem of avramov and backelin once one has existence of a dg algebra structure on a minimal r free resolution of s together with detailed knowledge of the structure of the induced homology algebra tor r s k h k s linkage theory provides the main technical tool for this analysis

artinian level algebras of socle degree 4 abstract in this paper we study the o sequences of local or graded k algebras of socle degree 4 more precisely we prove that an o sequence h 1 3 h 2 h 3 h 4 where h 4 2 is the h vector of a local level k algebra if and only if h 3 3 h 4 a characterization is also presented for gorenstein o sequences in each of these cases we give an effective method to construct a local level k algebra with a given h vector moreover we refine a result of elias and rossi by showing that if h 1 h 1 h 2 h 3 1 is a unimodal gorenstein o sequence then h forces the corresponding gorenstein k algebra to be canonically graded if and only if h 1 h 3 and h 2 h 1 1 2 that is the h vector is maximal we discuss analogue problems for higher socle degrees

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

homological properties of powers of the maximal ideal of a local ring abstract it is known that the powers m n of the maximal ideal of a local noetherian ring share certain homological properties for all sufficiently large integers n for example the natural homomorphisms r r m n are golod respectively small for all large n we give effective bounds on the smallest integers n for which such properties begin to hold

the structure of the inverse system of gorenstein k algebras abstract macaulay s inverse system gives an effective method to construct artinian gorenstein k algebras to date a general structure for gorenstein k algebras of any dimension and codimension is not understood in this paper we extend macaulay s correspondence characterizing the submodules of the divided power ring in one to one correspondence with gorenstein d dimensional k algebras we discuss effective methods for constructing gorenstein graded rings several examples illustrating our results are given

sharp adaptive estimation of linear functionals we consider estimation of a linear functional t f where f is an unknown function observed in gaussian white noise we find asymptotically sharp adaptive estimators on various scales of smoothness classes in multidimensional situations the results allow evaluating explicitly the effect of dimension and treating general scales of classes furthermore we establish a connection between sharp adaptation and optimal recovery namely we propose a scheme that reduces the construction of sharp adaptive estimators on a scale of functional classes to a solution of the corresponding optimization problem

numerical invariants of liaison classes 302 c huneke intersection what properties are then enjoyed by v for instance buchweitz was able to show that all such v are strongly nonobstructed see 5 for details the two questions above have been completely answered if x spec r with r a regular local ring and if codim v 2 in this case

poincare series of modules over compressed gorenstein local rings abstract given two positive integers e and s we consider gorenstein artinian local rings r whose maximal ideal m satisfies m s 0 m s 1 and rank r m m m 2 e we say that r is a compressed gorenstein local ring when it has maximal length among such rings it is known that generic gorenstein artinian algebras are compressed if s 3 we prove that the poincare series of all finitely generated modules over a compressed gorenstein local ring are rational sharing a common denominator a formula for the denominator is given when s is even this formula depends only on the integers e and s note that for s 3 examples of compressed gorenstein local rings with transcendental poincare series exist due to bogvad

clustering coding switching hierarchical ordering and control in a network of chaotic elements abstract a network of chaotic elements is investigated with the use of globally coupled maps a simple coding of many attractors with clustering is shown through the coding the attractors are organized so that their change exhibits bifurcation like phenomena a precision dependent tree is constructed which leads to the similarity of our attractor with those of spin glasses hierarchical dynamics is constructed on the tree which leads to the dynamical change of trees and the temporal change of effective degrees of freedom by a simple input on a site we can switch among attractors and tune the strength of chaos a threshold on a cluster size is found beyond which a peculiar posi nega switch occurs possible application to biological information processing is discussed with the emphasis on the fuzzy switch chaotic search and hierarchical code categorization

the jacobian module the milnor fiber and the d module generated by f s for a germ f on a complex manifold x we introduce a complex derived from the liouville form acting on logarithmic differential forms and give an exactness criterion we use this liouville complex to connect properties of the d module generated by f s to homological data of the jacobian ideal specifically we show that for a large class of germs the annihilator of f s is generated by derivations through local cohomology we connect the cohomology of the milnor fiber to the jacobian module via logarithmic differentials in particular we consider not necessarily reduced hyperplane arrangements we prove a conjecture of terao on the annihilator of 1 f we confirm in many cases a corresponding conjecture on the annihilator of f s but we disprove it in general we show that the bernstein sato polynomial of an arrangement is not determined by its intersection lattice we prove that arrangements for which the annihilator of f s is generated by derivations fulfill the strong monodromy conjecture and that this includes as very special cases all arrangements of coxeter and of crystallographic type and all multi arrangements in dimension 3

extending sdl and lmc complexity measures to quantum states an extension of sdl shiner davison landsberg and lmc lopez ruiz mancini calbet complexity measures is proposed for the quantum information context considering that von neumann entropy is a natural disorder quantifier for quantum states as a first example of application the simple qubit was studied presenting results similar to that obtained by applying sdl and lmc measures to a classical probability distribution then for the werner state a mixture of bell states sdl and lmc measures were calculated depending on the mixing factor providing some conjectures concerning quantum systems

on the rate of convergence of orthogonal series density estimators on montre que si l optimalite est evaluee en termes de comportement des coefficients de fourier alors plusieurs estimateurs de series orthogonales tres simples sont asymptotiquement optimaux

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

certain reflexive sheaves on ⁿ_ and a problem in approximation theory this paper establishes a link between certain local problems in the theory of splines and properties of vector bundles and reflexive sheaves on complex projective spaces