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

partially monotone tensor spline estimation of the joint distribution function with bivariate current status data the analysis of the joint cumulative distribution function cdf with bivariate event time data is a challenging problem both theoretically and numerically this paper develops a tensor spline based sieve maximum likelihood estimation method to estimate the joint cdf with bivariate current status data the i splines are used to approximate the joint cdf in order to simplify the numerical computation of a constrained maximum likelihood estimation problem the generalized gradient projection algorithm is used to compute the constrained optimization problem based on the properties of b spline basis functions it is shown that the proposed tensor spline based nonparametric sieve maximum likelihood estimator is consistent with a rate of convergence potentially better than n 1 3 under some mild regularity conditions the simulation studies with moderate sample sizes are carried out to demonstrate that the finite sample performance of the proposed estimator is generally satisfactory

the classification of homogeneous cohen macaulay rings of finite representation type in this paper we classify the homogeneous cohen macaulay rings which are of finite representation type that is the cohen macaulay rings which are positively graded and generated in degree 1 with an algebraically closed field of characteristic 0 in degree 0 and which have up to isomorphisms and shifts in the grading only a finite number of indecomposable maximal cohen macaulay modules mcm modules our main contribution is to show that a homogeneous cohen macaulay ring of finite representation type and dimension 2 must have minimal multiplicity putting this together with previous results of auslander auslander reiten buchweitz greuel schreyer greuel kn6rrer and solberg we obtain

gl v invariant ulrich bundles on grassmannians in this paper we give a full classification of all homogeneous ulrich bundles on a grassmannian mathrm gr k n of k planes on mathbb p n

locally efficient estimation with bivariate right censored data estimation of the survival curve for independently right censored bivariate failure time data is a problem that has been studied extensively over the past 20 years in this article we propose a new class of estimators for the bivariate survivor function based on locally efficient le estimation the le estimator takes bivariate estimators fn and gn of the distributions of the time variables t1 t2 and the censoring variables c1 c2 and maps them to the resulting estimator le if fn and gn are appropriate consistent estimators of f and g then le will be nonparametrically efficient thus the term locally efficient however if either fn or gn but not both is not a consistent estimator of f or g then le will still be consistent and asymptotically normally distributed we propose a locally efficient estimator that uses a consistent nonparametric estimator for g and allows the user to supply lower dimensional semiparametric or parametric working model for f because the estimator that we cho

determinantal representations of singular hypersurfaces in pn abstract a global determinantal representation of projective hypersurface x p n is a matrix whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface we study the properties of such representations for singular possibly reducible or non reduced hypersurfaces in particular we obtain the decomposability criteria for determinantal representations of globally reducible hypersurfaces further we classify the determinantal representations in terms of the corresponding kernel sheaves on x finally we extend the results to the case of symmetric self adjoint representations with implications to hyperbolic polynomials and the generalized lax conjecture

acm sheaves of pure rank two on reducible hyperquadrics abstract we classify a special type of arithmetically cohen macaulay sheaves of rank two on reducible and reduced quadric hypersurfaces as a consequence we show that a reducible and reduced quadric surface is of wild type

ulrich modules over cyclic quotient surface singularities abstract in this paper we characterize ulrich modules over cyclic quotient surface singularities using the notion of special cohen macaulay modules we also investigate the number of indecomposable ulrich modules for a given cyclic quotient surface singularity and show that the number of exceptional curves in the minimal resolution determines a boundary on the number of indecomposable ulrich modules

multiplicative update rules for concurrent nonnegative matrix factorization and maximum margin classification the state of the art classification methods which employ nonnegative matrix factorization nmf employ two consecutive independent steps the first one performs data transformation dimensionality reduction and the second one classifies the transformed data using classification methods such as nearest neighbor centroid or support vector machines svms in the following we focus on using nmf factorization followed by svm classification typically the parameters of these two steps e g the nmf bases coefficients and the support vectors are optimized independently thus leading to suboptimal classification performance in this paper we merge these two steps into one by incorporating maximum margin classification constraints into the standard nmf optimization the notion behind the proposed framework is to perform nmf while ensuring that the margin between the projected data of the two classes is maximal the concurrent nmf factorization and support vector optimization are performed through a set of multiplicative update rules in the same context the maximum margin classification constraints are imposed on the nmf problem with additional discriminant constraints and respective multiplicative update rules are extracted the impact of the maximum margin classification constraints on the nmf factorization problem is addressed in section vi experimental results in several databases indicate that the incorporation of the maximum margin classification constraints into the nmf and discriminant nmf objective functions improves the accuracy of the classification

rank two acm bundles on the del pezzo threefold with picard number 3 abstract a del pezzo threefold f with maximal picard number is isomorphic to p 1 p 1 p 1 in the present paper we completely classify locally free sheaves e of rank 2 such that h i f e t 0 for i 1 2 and t z such a classification extends similar results proved by e arrondo and l costa regarding del pezzo threefolds with picard number 1

the geometry of ulrich bundles on del pezzo surfaces abstract given a smooth del pezzo surface x d p d of degree d we isolate the essential geometric obstruction to a vector bundle on x d being an ulrich bundle by showing that an irreducible curve d of degree dr on x d represents the first chern class of a rank r ulrich bundle on x d if and only if the kernel bundle of the general smooth element of d admits a generalized theta divisor moreover we show that any smooth arithmetically gorenstein surface whose ulrich bundles admit such a characterization is necessarily del pezzo this result is applied to produce new examples of complete intersection curves with semistable kernel bundle and also combined with work of farkas musta and popa to relate the existence of ulrich bundles on x d to the minimal resolution conjecture for curves lying on x d in particular we show that a smooth irreducible curve d of degree 3r lying on a smooth cubic surface x 3 represents the first chern class of an ulrich bundle on x 3 if and only if the minimal resolution conjecture holds for the general smooth element of d

the representation type of segre varieties abstract the goal of this work is to prove that all segre varieties n 1 n s p n n i 1 s n i 1 1 unless the quadric surface in p 3 support families of arbitrarily large dimension and rank of simple ulrich and hence acm vector bundles therefore they are of wild representation type

on profile likelihood abstract we show that semiparametric profile likelihoods where the nuisance parameter has been profiled out behave like ordinary likelihoods in that they have a quadratic expansion in this expansion the score function and the fisher information are replaced by the efficient score function and efficient fisher information the expansion may be used among others to prove the asymptotic normality of the maximum likelihood estimator to derive the asymptotic chi squared distribution of the log likelihood ratio statistic and to prove the consistency of the observed information as an estimator of the inverse of the asymptotic variance

on the growth of the betti sequence of the canonical module we study the growth of the betti sequence of the canonical module of a cohen macaulay local ring it is an open question whether this sequence grows exponentially whenever the ring is not gorenstein we answer the question of exponential growth affirmatively for a large class of rings and prove that the growth is in general not extremal as an application of growth we give criteria for a cohen macaulay ring possessing a canonical module to be gorenstein

multiplicative update rules for incremental training of multiclass support vector machines we present a new method for the incremental training of multiclass support vector machines that can simultaneously modify each class separating hyperplane and provide computational efficiency for training tasks where the training data collection is sequentially enriched and dynamic adaptation of the classifier is required over time an auxiliary function has been designed that incorporates some desired characteristics in order to provide an upper bound for the objective function which summarizes the multiclass classification task a novel set of multiplicative update rules is proposed which is independent from any kind of learning rate parameter provides computational efficiency compared to the conventional batch training approach and is easy to implement convergence to the global minimum is guaranteed since the optimization problem is convex and the global minimizer for the enriched dataset is found using a warm start algorithm experimental evidence on various data collections verified that our method is faster than retraining the classifier from scratch while the achieved classification accuracy rate is maintained at the same level

locally efficient estimation of a multivariate survival function in longitudinal studies we consider estimation of the joint distribution of multivariate survival times t t1 tk which are subject to right censoring by a common censoring variablec two estimators are proposed an initial inverse probability of censoring weighted ipcw estimator and a one step estimator both estimators incorporate information on available time independent and time dependent prognostic factor covariate data the ipcw estimator is consistent and asymptotically normal can under coarsening at random car and a correct specification of a model for the hazard of censoring given the past covariate and failure data the one step estimator is a locally efficient doubly robust estimator that is i it is can under the assumption of car and either but not necessarily both correct specification of a model for the hazard of censoring given the past or correct specification of a model for the conditional distribution of t given past failure and covariate information and ii it is efficient when both these

gorenstein biliaison and acm sheaves abstract let x be a normal arithmetically gorenstein scheme in p n we give a criterion for all codimension two acm subschemes of x to be in the same gorenstein biliaison class on x in terms of the category of acm sheaves on x these are sheaves that correspond to the graded maximal cohen macaulay modules on the homogeneous coordinate ring of x using known results on mcm modules we are able to determine the gorenstein biliaison classes of codimension two subschemes of certain varieties including the nonsingular quadric surface in p 3 and the cone over it in p 4 as an application we obtain a new proof of some theorems of lesperance about curves in p 4 and answer some questions he raised

notes on the hilbert kunz function abstract we note certain properties of the hilbert kunz function and hilbert kunz multiplicity including a strengthened inequality between hilbert kunz and hilbert samuel multiplicities and a characterization of a mixed hilbert function incorporating both ordinary and frobenius powers of ideals

deformations of modules of maximal grade and the hilbert scheme at determinantal schemes let r be a polynomial ring and m a finitely generated graded r module of maximal grade which means that the ideal it a generated by the maximal minors of a homogeneous presentation matrix a of m has maximal codimension in r suppose x proj r it a is smooth in a sufficiently large open subset and dimx 1 then we prove that the local graded deformation functor of m is isomorphic to the local hilbert scheme functor at x proj r under a weak assumption which holds if dimx 2 under this assumption we get that the hilbert scheme is smooth at x and we give an explicit formula for the dimension of its local ring as a corollary we prove a conjecture of r m miro roig and the author that the closure of the locus of standard determinantal schemes with fixed degrees of the entries in a presentation matrix is a generically smooth component v of the hilbert scheme also their conjecture on the dimension of v is proved for dimx 1 the cohomology h i nx of the normal sheaf of x in proj r is shown to vanish for 1 i dimx 2 finally the mentioned results slightly adapted remain true replacing r by any cohen macaulay quotient of a polynomial ring

on the cohen macaulay modules of graded subrings we give several characterizations for the linearity property for a maximal cohen macaulay module over a local or graded ring as well as proofs of existence in some new cases in particular we prove that the existence of such modules is preserved when taking segre products as well as when passing to veronese subrings in low dimensions the former result even yields new results on the existence of finitely generated maximal cohen macaulay modules over non cohen macaulay rings

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

linkage and the koszul homology of ideals 0 introduction let r be a local ring and i an ideal of r associated to i are several graded algebras the symmetric algebra of the module i sym i the rees algebra of i defined to be the algebra r i gn o inf and the associated graded algebra of i gr1 r rii g ii 2 in in 1 i i i our purpose in this paper is to study the behavior of these graded algebras when we change i through linkage recall the definition of linkage found in the paper of peskine and szpiro 22

covariance and survivor function estimation using censored multivariate failure time data summary the covariance between counting process martingales is used to characterize the dependence between two failure time variates a representation of the bivariate survivor function is obtained in terms of the marginal survivor functions and this covariance function a closely related representation expresses the bivariate survivor function in terms of marginal survivor functions and a conditional covariance function leading to a new nonparametric survivor function estimator generalizations to higher dimensional failure time variates are also given simulation evaluations of the survivor function estimator are presented and generalizations to regression problems are outlined

warm start of the primal dual method applied in the cutting plane scheme a practical warm start procedure is described for the infeasible primal dual interior point method ipm employed to solve the restricted master problem within the cutting plane method in contrast to the theoretical developments in this field the approach presented in this paper does not make the unrealistic assumption that the new cuts are shallow moreover it treats systematically the case when a large number of cuts are added at one time the technique proposed in this paper has been implemented in the context of hopdm the state of the art yet public domain interior point code numerical results confirm a high degree of efficiency of this approach regardless of the number of cuts added at one time can be thousands in the largest examples and regardless of the depth of the new cuts reoptimizations are usually done with a few additional iterations

on the algorithmic implementation of multiclass kernel based vector machines in this paper we describe the algorithmic implementation of multiclass kernel based vector machines our starting point is a generalized notion of the margin to multiclass problems using this notion we cast multiclass categorization problems as a constrained optimization problem with a quadratic objective function unlike most of previous approaches which typically decompose a multiclass problem into multiple independent binary classification tasks our notion of margin yields a direct method for training multiclass predictors by using the dual of the optimization problem we are able to incorporate kernels with a compact set of constraints and decompose the dual problem into multiple optimization problems of reduced size we describe an efficient fixed point algorithm for solving the reduced optimization problems and prove its convergence we then discuss technical details that yield significant running time improvements for large datasets finally we describe various experiments with our approach comparing it to previously studied kernel based methods our experiments indicate that for multiclass problems we attain state of the art accuracy

kaplan meier estimate on the plane estimation of the bivariate survival function from censored data is considered the product integral representation of univariate survival functions is generalized to the bivariate case and used to determine identifiability of the survival function of the partially observed data a bivariate analogue of the kaplan meier estimate is introduced and its almost sure consistency is studied extensions to the general multivariate case are sketched

on the normal sheaf of determinantal varieties o roig abstract let x be a standard determinantal scheme xp n of codimension c i e a scheme defined by the maximal minors of a t t c 1 homogeneous polynomial matrix a in this paper we study the main features of its normal sheaf nx we prove that under some mild restrictions 1 there exists a line bundle l on x sing x such that nx l is arithmetically cohen macaulay and even more it is ulrich whenever the entries of a are linear forms 2 nx is simple hence indecomposable and finally 3 nx is semi stable provided the entries of a are linear forms

linkage theory for algebras with pure resolutions abstract let s be a graded cohen macaulay quotient r i of a polynomial ring r k x 1 x n over an infinite field k if s is in the linkage class of a complete intersection then its koszul homology and twisted conormal module are cohen macaulay by passing to zero and one dimensional specializations of s one can convert these properties into numerical test conditions for a graded algebra to be in the linkage class of a complete intersection we prove that if s is generically a complete intersection and the minimal free solution of s is pure and almost linear then s is not in the linkage class of a complete intersection apart from a few obvious exceptions for algebras of small codimension the results are stronger if s has codimension three and has a pure resolution then s is in the linkage class of a complete intersection if and only if s is gorenstein if s is a codimension four gorenstein algebra with a pure resolution then s is in the linkage class of a complete intersection if and only if s is a complete intersection

learning algorithms with optimal stability in neural networks to ensure large basins of attraction in spin glass like neural networks of two state elements xi imu or 1 the authors propose to study learning rules with optimal stability delta where delta is the largest number satisfying delta or sigma j jij xi jmu xi imu mu 1 p i 1 n where n is the number of neurons and p is the number of patterns they motivate this proposal and provide optimal stability learning rules for two different choices of normalisation for the synaptic matrix jij in addition numerical work is presented which gives the value of the optimal stability for random uncorrelated patterns

resultants and chow forms via exterior syzygies let w be a vector space of dimension n 1 over a field k the chow divisor of a k dimensional variety x in p p w is the hypersurface in the grassmannian gk 1 of planes of codimension k 1 in p whose points over the algebraic closure of k are the planes that meet x the chow form of x is the defining equation of the chow divisor for example the resultant of k 1 forms of degree e in k 1 variables is the chow form of p embedded by the e th veronese mapping in p with n k e k 1 more generally the chow divisor of a k cycle i ni vi on projective space is defined to be i nidi where di is the chow divisor of vi the chow divisor of a sheaf f with k dimensional support is the chow divisor of the associated k cycle of f in this paper we will give a new expression for the chow divisor and apply it to give explicit formulas in many new cases starting with a sheaf f on p we use exterior algebra methods to define a canonical and effectively computable chow complex of f on each grassmannian of planes in p if f has k dimensional support we show that the chow form of f is the determinant of the chow complex of f on the grassmannian of planes of codimension k 1 the beilinson monad of f beilinson 1978 is the chow complex of f on the grassmannian of 0 planes that is on p itself in particular we are able to give explicit determinantal and pfaffian formulas for resultants in some cases where no polynomial formulas were known for example the horrocks mumford bundle gives rise to polynomial formulas for the resultant of five homogeneous forms of degrees 4 6 or 8 in five variables the easiest of our new formulas to write down is for the resultant of 3 quadratic forms in three variables the chow form of the veronese surface in p using the tangent bundle of p conclude that it can be written in bezout form described below as the pfaffian of the matrix 0 245 345 135 045 035 145 235

an introduction to kernel based learning algorithms this paper provides an introduction to support vector machines kernel fisher discriminant analysis and kernel principal component analysis as examples for successful kernel based learning methods we first give a short background about vapnik chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and unsupervised scenarios including practical and algorithmic considerations we illustrate the usefulness of kernel algorithms by discussing applications such as optical character recognition and dna analysis

multiplicative updates for nonnegative quadratic programming many problems in neural computation and statistical learning involve optimizations with nonnegativity constraints in this article we study convex problems in quadratic programming where the optimization is confined to an axis aligned region in the nonnegative orthant for these problems we derive multiplicative updates that improve the value of the objective function at each iteration and converge monotonically to the global minimum the updates have a simple closed form and do not involve any heuristics or free parameters that must be tuned to ensure convergence despite their simplicity they differ strikingly in form from other multiplicative updates used in machine learning we provide complete proofs of convergence for these updates and describe their application to problems in signal processing and pattern recognition

reoptimization with the primal dual interior point method reoptimization techniques for an interior point method applied to solving a sequence of linear programming problems are discussed conditions are given for problem perturbations that can be absorbed in merely one newton step the analysis is performed for both short step and long step feasible path following methods a practical procedure is then derived for an infeasible path following method it is applied in the context of crash start for several large scale structured linear programs numerical results with oops a new object oriented parallel solver demonstrate the efficiency of the approach for large structured linear programs crash start leads to about 40 reduction in the number of iterations and translates into a 25 reduction of the solution time the crash procedure parallelizes well and speed ups between 3 1 3 8 on four processors are achieved

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

tame and wild projective curves and classification of vector bundles abstract we propose a new method of classifying vector bundles on projective curves especially singular ones according to their representation type in particular we prove that the classification problem of vector bundles respectively of torsion free sheaves on projective curves is always finite tame or wild we completely classify curves which are of finite respectively tame vector bundle type by their dual graph moreover our methods yield a geometric description of all indecomposable vector bundles and torsion free sheaves on finite and tame curves

nonparametric estimation of a bivariate survival function in the presence of censoring a new family of estimators of a bivariate survival function based on censored vectors is obtained from a decomposition of the bivariate survival function these estimators are uniformly consistent under bivariate censoring and are self consistent under univariate censoring

efficient estimation in the bivariate censoring model and repairing npmle the npmle in the bivariate censoring model is not consistent the problem is caused by the singly censored observations in this paper we prove that the npmle based on interval censoring the singly censored observations is ecient for this reduced data and moreover if we let the width of the interval converge to zero slowly enough then the npmle is also ecient for the original data we are able to determine a lower bound for the rate at which the bandwidth should converge to zero the eciency proof uses the general identity which holds for npmle of a linear parameter in convex models as proved in van der laan 1993a

input space versus feature space in kernel based methods this paper collects some ideas targeted at advancing our understanding of the feature spaces associated with support vector sv kernel functions we first discuss the geometry of feature space in particular we review what is known about the shape of the image of input space under the feature space map and how this influences the capacity of sv methods following this we describe how the metric governing the intrinsic geometry of the mapped surface can be computed in terms of the kernel using the example of the class of inhomogeneous polynomial kernels which are often used in sv pattern recognition we then discuss the connection between feature space and input space by dealing with the question of how one can given some vector in feature space find a preimage exact or approximate in input space we describe algorithms to tackle this issue and show their utility in two applications of kernel methods first we use it to reduce the computational complexity of sv decision functions second we combine it with the kernel pca algorithm thereby constructing a nonlinear statistical denoising technique which is shown to perform well on real world data