irreducible calibrated representations of periplectic brauer algebras and hook representations of the symmetric group abstract we construct an infinite tower of irreducible calibrated representations of periplectic brauer algebras on which the cup cap generators act by nonzero matrices as representations of the symmetric group these are exterior powers of the standard representation i e hook representations our approach uses the recently defined degenerate affine periplectic brauer algebra which plays a role similar to that of the degenerate affine hecke algebra in representation theory of the symmetric group we write formulas for the representing matrices in the basis of jucys murphy eigenvectors and we completely describe the spectrum of these representations the tower formed by these representations provides a new non semisimple categorification of pascal s triangle along the way we also prove some basic results about calibrated representations of the degenerate affine periplectic brauer algebra

exact operator solution of the calogero sutherland model the wave functions of the calogero sutherland model are known to be expressible in terms of jack polynomials a formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived the creation operators that enter in this formula of rodrigues type for the jack polynomials involve dunkl operators

a mixed effects model for multivariate ordinal response data including correlated discrete failure times with ordinal responses the mixed effects model for binary responses due to conaway 1990 a random effects model for binary data is extended to accommodate ordinal responses in general and discrete time survival data with ordinal responses in particular given a multinomial likelihood cumulative complementary log log link function and log gamma random effects distribution the resulting marginal likelihood has a closed form as a result a newton raphson estimation procedure is feasible without resorting to numerical approximation based or monte carlo integration techniques the parameters in the model have a proportional hazards interpretation in terms of multivariate discrete time data with ordinal responses using data from a psychological example the proposed method is compared with other mixed effects approaches as well as population averaged models

a note on the existence of the posterior distribution for a class of mixed models for binomial responses summary necessary and sufficient conditions are given for the existence of the posterior distribution of the variance components in a class of mixed models for binomial responses the implications of our results are illustrated through an example

modelling neuromotor ratings with floor effects associated movements ams are a classical diagnostic tool to assess differences between normal children and children with some motor dysfunction this paper presents a methodology to produce age and gender dependent reference curves for ams of normal children for various tasks of a test battery data available consist of separate ratings of duration and extent of ams which are ordinal quantities with few levels other problems are severe age and gender dependent floor effects as well as some ceiling effects leaving little information for analysis at older ages to get a better scale we combined the two ordinal ratings into one meaningful and quasi continuous quantity referred to as intensity of ams in order to solve problems due to floor effects ceiling effects and discreteness we assumed left right and interval censored values respectively we considered a censored regression problem and postulated a truncated normal distribution for the non censored values after an appropriate transformation of the data using wei and tanner s poor man s data augmentation algorithm together with the technique of linear mixed effects modelling useful reference curves could be produced in contrast to the cumulative probabilities approach for ordinal data our methodology allows the calculation of individual age and gender standardized values which puts us in a position to investigate numerous scientific questions

random effects modeling approaches for estimating roc curves from repeated ordinal tests without a gold standard estimating diagnostic accuracy without a gold standard is an important problem in medical testing although there is a fairly large literature on this problem for the case of repeated binary tests there is substantially less work for the case of ordinal tests a noted exception is the work by zhou castelluccio and zhou 2005 biometrics 61 600 609 which proposed a methodology for estimating receiver operating characteristic roc curves without a gold standard from multiple ordinal tests a key assumption in their work was that the test results are independent conditional on the true test result i propose random effects modeling approaches that incorporate dependence between the ordinal tests and i show through asymptotic results and simulations the importance of correctly accounting for the dependence between tests these modeling approaches along with the importance of accounting for the dependence between tests are illustrated by analyzing the uterine cancer pathology data analyzed by zhou et al 2005

consistency of maximum likelihood estimators in general random effects models for binary data we consider a general random effects model for repeated binary measures assuming a latent linear model with any class of mixing distributions the latent model is assumed to have the laird ware structure but the random effects may be from any specified class of multivariate distributions and the error vector may have any specified continuous distribution elementwise threshold crossing then gives the observed vector of binary outcomes special cases of this model include recently discussed mixed logistic regression and probit models which have had either parametric usually gaussian or nonparametric mixing distributions we give sufficient conditions for identifiability of the mixing distribution and fixed effects and for convergence of maximum likelihood estimators for the mixing distribution and fixed effects as expected the conditions are much stronger for nonparametric mixing than for gaussian mixing we illustrate the conditions by applying them to a practical example

on the category o for rational cherednik algebras we study the category o of representations of the rational cherednik algebra aw attached to a complex reflection group w w e construct an exact functor called knizhnik zamolodchikov functor o hw mod where hw is the finite iwahori hecke algebra associated to w we prove that the knizhnik zamolodchikov functor induces an equiv alence between o otor the quotient of o by the subcategory of aw modules supported on the discriminant and the category of finite dimensional hw modules the standard aw modules go under this equivalence to certain modules arising in kazhdan lusztig theory of cells provided w is a weyl group and the hecke algebra hw has equal parameters we prove that the category o is equivalent to the module category over a finite dimensional algebra a generalized q schur algebra associated to w

testing the correlation for clustered categorical and censored discrete time to event data when covariates are measured without with errors summary in the analysis of clustered categorical data it is of common interest to test for the correlation within clusters and the heterogeneity across different clusters we address this problem by proposing a class of score tests for the null hypothesis that the variance components are zero in random effects models for clustered nominal and ordinal categorical responses we extend the results to accommodate clustered censored discrete time to event data we next consider such tests in the situation where covariates are measured with errors we propose using the simex method to construct the score tests for the null hypothesis that the variance components are zero key advantages of the proposed score tests are that they can be easily implemented by fitting standard polytomous regression models and discrete failure time models and that they are robust in the sense that no assumptions need to be made regarding the distributions of the random effects and the unobserved covariates the asymptotic properties of the proposed tests are studied we illustrate these tests by analyzing two data sets and evaluate their performance with simulations

analysis of variance for repeated measures data a generalized estimating equations approach various techniques are available for the analysis of repeated measures data and the appropriate choice depends on distributional assumptions and study design features a correct analysis must account for potential dependence between repeated observations on the same subject liang and zeger proposed a more unified approach to the analysis of repeated measures data based on the application of generalized estimating equations we examine the application of these methods to several types of data in which one estimates the mean response directly for each combination of discrete covariates and uses an identity link computations for fitting this type of model are exceptionally simple numerical examples suggest that the proposed approach yields estimation and hypothesis testing results consistent with more specialized methods

limit of the smallest eigenvalue of a large dimensional sample covariance matrix in this paper the authors show that the smallest if p leq n or the p n 1 th smallest if p n eigenvalue of a sample covariance matrix of the form 1 n xx tends almost surely to the limit 1 sqrt y 2 as n rightarrow infty and p n rightarrow y in 0 infty where x is a p times n matrix with iid entries with mean zero variance 1 and fourth moment finite also as a by product it is shown that the almost sure limit of the largest eigenvalue is 1 sqrt y 2 a known result obtained by yin bai and krishnaiah the present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices

asymptotic syzygies of algebraic varieties we study the asymptotic behavior of the syzygies of a smooth projective variety as the positivity of the embedding line bundle grows the main result asserts that the syzygy modules are non zero in almost all degrees allowed by castelnuovo mumford regularity we also give an effective statement for veronese varieties that we conjecture to be optimal

maximum likelihood variance components estimation for binary data abstract we consider a class of probit normal models for binary data and describe ml and reml estimation of variance components for that class as well as best prediction for the realized values of the random effects ml estimates are calculated using an em algorithm for complicated models em includes a gibbs step the computations are illustrated through two examples

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

concentration of norms and eigenvalues of random matrices we prove concentration results for lpn operator norms of rectangular random matrices and eigenvalues of self adjoint random matrices the random matrices we consider have bounded entries which are independent up to a possible self adjointness constraint our results are based on an isoperimetric inequality for product spaces due to talagrand

mixed model analysis of a censored normal distribution with reference to animal breeding a mixed model procedure for analysis of censored data assuming a multivariate normal distribution is described a bayesian framework is adopted which allows for estimation of fixed effects and variance components and prediction of random effects when records are left censored the procedure can be extended to right and two tailed censoring the model employed is a generalized linear model and the estimation equations resemble those arising in analysis of multivariate normal or categorical data with threshold models estimates of variance components are obtained using expressions similar to those employed in the em algorithm for restricted maximum likelihood reml estimation under normality

best linear recursive estimation for mixed linear models abstract recursive estimation techniques for fixed and completely random models are extended to mixed linear models the kalman filter is used to obtain recursive estimators for a two part random model where the second random factor obeys a generalized autoregressive process by passing to the limit in an appropriate way recursions for the mixed model are derived

impact of missing data due to drop outs on estimators for rates of change in longitudinal studies a simulation study many cohort studies and clinical trials are designed to compare rates of change over time in one or more disease markers in several groups one major problem in such longitudinal studies is missing data due to patient drop out the bias and efficiency of six different methods to estimate rates of changes in longitudinal studies with incomplete observations were compared generalized estimating equation estimates gee proposed by liang and zeger 1986 unweighted average of ordinary least squares olse of individual rates of change uwls weighted average of olse wls conditional linear model estimates cle a covariate type estimates proposed by wu and bailey 1989 random effect re and joint multivariate re jmre estimates the latter method combines a linear re model for the underlying pattern of the marker with a log normal survival model for informative drop out process the performance of these methods in the presence of missing data completely at random mcar at random mar and non ignorable nim were compared in simulation studies data for the disease marker were generated under the linear random effects model with parameter values derived from realistic examples in hiv infection rates of drop out assumed to increase over time were allowed to be independent of marker values or to depend either only on previous marker values or on both previous and current marker values under macr all six methods yielded unbiased estimates of both group mean rates and between group difference however the cross sectional view of the data in the gee method resulted in seriously biased estimates under mar and nim drop out process the bias in the estimates ranged from 30 per cent to 50 per cent the degree of bias in the gee estimates increases with the severity of non randomness and with the proportion of mar data under mcar and mar all the other five methods performed relatively well re and jmre estimates were more efficient that is had smaller variance than uwls wls and cl estimates under nim wls and particularly re estimates tended to underestimate the average rate of marker change bias approximately 10 per cent under nim uwls cl and jmre performed better in terms of bias 3 5 per cent with the jmre giving the most efficient estimates given that markers are key variables related to disease progression missing marker data are likely to be at least mar thus the gee method may not be appropriate for analysing such longitudinal marker data the potential biases due to incomplete data require greater recognition in reports of longitudinal studies sensitivity analyses to assess the effect of drop outs on inferences about the target parameters are important

pieri resolutions for classical groups abstract we generalize the constructions of eisenbud floystad and weyman for equivariant minimal free resolutions over the general linear group and we construct equivariant resolutions over the orthogonal and symplectic groups we also conjecture and provide some partial results for the existence of an equivariant analogue of boij soderberg decompositions for betti tables which were proven to exist in the non equivariant setting by eisenbud and schreyer many examples are given

affine hecke algebras and generalized standard young tableaux abstract this paper introduces calibrated representations for affine hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations the primary technique is to provide indexing sets for controlling the weight space structure of finite dimensional modules for the affine hecke algebra using these indexing sets we show that 1 irreducible calibrated representations are indexed by skew local regions 2 the dimension of an irreducible calibrated representation is the number of chambers in the local region 3 each irreducible calibrated representation is constructed explicitly by formulas which describe the action of the generators of the affine hecke algebra on a specific basis in the representation space the indexing sets for weight spaces are generalizations of standard young tableaux and the construction of the irreducible calibrated affine hecke algebra modules is a generalization of a young s seminormal construction of the irreducible representations of the symmetric group in this sense young s construction has been generalized to arbitrary lie type

zipping tate resolutions and exterior coalgebras abstract we conjecture what the cone of hypercohomology tables of bounded complexes of coherent sheaves on projective spaces is when we have specified regularity conditions on the cohomology sheaves of this complex and its dual there is an injection from this cone into the cone of homological data sets of squarefree modules over a polynomial ring k x 1 x n and we conjecture that this is an isomorphism the tate resolutions of a complex of coherent sheaves on projective space p w and the exterior coalgebra on x 1 x n may be amalgamated together to form a complex of free sym i x i w modules a procedure introduced by cox and materov via a reduction i x i w i x i k we get a complex of free modules over k x 1 x n the extremal rays in the cone of squarefree complexes are conjecturally given by triplets of pure free squarefree complexes introduced in 15 we describe the corresponding classes of hypercohomology tables a class which generalizes vector bundles with supernatural cohomology we also show how various pure resolutions in the literature like resolutions of modules supported on determinantal varieties and tensor complexes may be obtained by the first part of the procedure

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

on a simple test for neglected heterogeneity in panel studies summary this paper deals with the performance of a simple test for the detection of unmeasured heterogeneity in regression models for panel data it is assumed that the unmeasured heterogeneity can be modeled by an individual specific random variable that does not vary over time conditionally on the individual specific effects and the regressor variables the distribution of the dependent variable is assumed to belong to a linear exponential family this class of distributions includes linear models for continuous dependent variables as well as nonlinear models for discrete outcomes the proposed test does not assume a special form of the distribution of the heterogeneity and the test statistic can be calculated with available computer programs the performance of the test is investigated in a simulation experiment

modelling ordered categorical data recent advances and future challenges this article summarizes recent advances in the modelling of ordered categorical ordinal response variables we begin by reviewing some models for ordinal data introduced in the literature in the past 25 years we then survey recent extensions of these models and related methodology for special types of applications such as for repeated measurement and other forms of clustering we also survey other aspects of ordinal modelling such as small sample analyses power and sample size considerations and availability of software throughout we suggest problem areas for future research and we highlight challenges for statisticians who deal with ordinal data

random matrices nonbacktracking walks and orthogonal polynomials several well known results from the random matrix theory such as wigner s law and the marchenko pastur law can be interpreted and proved in terms of nonbacktracking walks on a certain graph orthogonal polynomials with respect to the limiting spectral measure play a role in this approach

bias reduction of approximate maximum likelihood estimates for heritability in threshold models approximate maximum likelihood estimates such as penalized quasi likelihood and iterated reweighted reml estimates for heritability or intraclass correlation in threshold models for binary data can be seriously biased two approaches to correct for this bias are studied with emphasis on animal breeding models for binary data scope for reduction of bias of heritability estimates is found to be slim because of the commonly large number of fixed effects in animal breeding models minimal dimensions for the data are identified such that bias and root mean squared error are of modest size and useful inference on heritability or intraclass correlation is feasible

a proportional odds model with subject specific effects for repeated ordered categorical responses summary suppose subjects make repeated responses on the same ordered categorical scale we propose a generalization of the rasch model that expresses the cumulative logit of the response distribution using subject parameters and a proportional odds structure for item effects parameters in the model describe subject specific rather than populationaveraged effects consistent estimation of the effects requires eliminating the subject parameters we accomplish this by simultaneous fitting of rasch models conditional on sufficient statistics for those parameters for the possible binary collapsings of the response the fitting process uses an improved newton raphson algorithm for fitting generalized loglinear models by maximum likelihood estimation subject to constraints for the case of two items we give simple expressions for an effect estimate and its standard error and suggest a test of marginal homogeneity for ordinal matched pairs

distribution free fitting of logit models with random effects for repeated categorical responses this article discusses random effects models for within subject comparisons of repeated responses on the same categorical scale the models account for the correlation that normally occurs between repeated responses the standard way of fitting such models maximizes the marginal likelihood after integrating with respect to a distribution for the random effect an alternative non parametric approach does not assume a distributional form for the random effects recent literature shows that for certain simple logit models this approach yields essentially the same model parameter estimates as conditional maximum likelihood moreover these estimates also result from fitting corresponding quasi symmetric log linear models for simple data sets in which primary interest relates to subject specific comparisons of the repeated responses one can easily obtain the estimates with standard software for log linear models examples include data from crossover designs and from comparisons of treatment and control groups regarding the change between baseline and follow up observations

an estimate for the average spectral measure of random band matrices for a class of random band matrices of band width w we prove regularity of the average spectral measure at scales w 0 99 and find its asymptotics at these scales

orthogonal functions generalizing jack polynomials the rational cherednik algebra is a certain algebra of differential reflection operators attached to a complex reflection group w and depending on a set of central parameters each irreducible representation s of w corresponds to a standard module m for this paper deals with the infinite family g r 1 n of complex reflection groups our goal is to study the standard modules using a commutative subalgebra t of h discovered by dunkl and opdam in this case the irreducible w modules are indexed by certain sequences of partitions we first show that t acts in an upper triangular fashion on each standard module m with eigenvalues determined by the combinatorics of the set of standard tableaux on as a consequence we construct a basis for m consisting of orthogonal functions on n with values in the representation s for g 1 1 n with n these functions are the non symmetric jack polynomials we use intertwining operators to deduce a norm formula for our orthogonal functions and give an explicit combinatorial description of the lattice of submodules of m in the case in which the orthogonal functions are all well defined a consequence of our results is the construction of a number of interesting finite dimensional modules with intricate structure finally we show that for a certain choice of parameters there is a cyclic group of automorphisms of so that the rational cherednik algebra for g r p n is the fixed subalgebra our results therefore descend to the rational cherednik algebra for g r p n by clifford theory

joint modeling of multiple ordinal adherence outcomes via generalized estimating equations with flexible correlation structure jiang z reprint author univ pittsburgh grad sch publ hlth dept biostat pittsburgh pa 15261 usa zhen jiang fda hhs gov

latent variable models for clustered dichotomous data with multiple subclusters summary regression models for clustered binary data are derived from nonlinear mixed models in terms of latent normal variables the marginal response probabilities are functions of covariates through generalized linear models within a cluster the pairwise tetrachoric correlations are all equal and are not restricted by marginal probabilities this approach accommodates hierarchically nested binary data an algorithm for estimation using generalized estimating equations is proposed an example illustrates the application of this approach

walk generating functions and spectral measures of infinite graphs let g be a locally finite graph with bounded vertex degrees its adjacency matrix a gives rise to a bounded self adjoint operator on l2 v g let e be the resolution of the identity for a and ev the vector in l2 v g having the v coordinate 1 and all other coordinates 0 the function uv e eu ev is called the spectral measure of g corresponding to vertices u v of g it is related to the generating function wuv z for the number of walks of given length between u and v the spectral measures are used to show the following the pairing theorem characterizing finite bipartite graphs in terms of their spectrum is generalized from finite to infinite graphs the average closed walk generating function can be defined in some cases generalizing the plancherel measure to not necessarily vertex transitive graphs some theorems about the convergence of graph measures are given with applications to the calculation of the expected eigenvalue distribution of a large finite random regular or bipartite semiregular graph and the calculation of spectral measures of several examples

the tracy widom law for some sparse random matrices consider the random matrix obtained from the adjacency matrix of a random d regular graph by multiplying every entry by a random sign the largest eigenvalue converges after proper scaling to the tracy widom distribution

a flexible model for multivariate interval censored survival times with complex correlation structure we focus on the analysis of multivariate survival times with highly structured interdependency and subject to interval censoring such data are common in developmental genetics and genetic epidemiology we propose a flexible mixed probit model that deals naturally with complex but uninformative censoring the recorded ages of onset are treated as possibly censored ordinal outcomes with the interval censoring mechanism seen as arising from a coarsened measurement of a continuous variable observed as falling between subject specific thresholds this bypasses the requirement for the failure times to be observed as falling into non overlapping intervals the assumption of a normal age of onset distribution of the standard probit model is relaxed by embedding within it a multivariate box cox transformation whose parameters are jointly estimated with the other parameters of the model complex decompositions of the underlying multivariate normal covariance matrix of the transformed ages of onset become possible the new methodology is here applied to a multivariate study of the ages of first use of tobacco and first consumption of alcohol without parental permission in twins the proposed model allows estimation of the genetic and environmental effects that are shared by both of these risk behaviours as well as those that are specific

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

shapes of free resolutions over a local ring we classify the possible shapes of minimal free resolutions over a regular local ring this illustrates the existence of free resolutions whose betti numbers behave in surprisingly pathological ways we also give an asymptotic characterization of the possible shapes of minimal free resolutions over hypersurface rings our key new technique uses asymptotic arguments to study formal mathbb q betti sequences

the analysis of binomial data by a generalized linear mixed model summary methods for generalized linear models are extended to provide estimates of location and variance parameters for mixed models fitted to binomial data formed by classifying samples from an underlying normal distribution the method estimates the parameters directly on the underlying scale for a balanced one way random effects model the variance estimator simplifies to the usual analysis of variance one the estimation of variances and the prediction of random effects for binomial traits is required by animal breeders the predictors given are analogous to best linear unbiased predictors henderson 1973 but differ from those presented by harville mee 1984

generalized linear models with random effects a gibbs sampling approach abstract generalized linear models have unified the approach to regression for a wide variety of discrete continuous and censored response variables that can be assumed to be independent across experimental units in applications such as longitudinal studies genetic studies of families and survey sampling observations may be obtained in clusters responses from the same cluster cannot be assumed to be independent with linear models correlation has been effectively modeled by assuming there are cluster specific random effects that derive from an underlying mixing distribution extensions of generalized linear models to include random effects has thus far been hampered by the need for numerical integration to evaluate likelihoods in this article we cast the generalized linear random effects model in a bayesian framework and use a monte carlo method the gibbs sampler to overcome the current computational limitations the resulting algorithm is flexible to easily accommodate changes in the number

rate of convergence to the semi circular law a stochastic bound of order op n 1 2 for the kolmogorov distance between the spectral distribution function of an n n matrix from wigner ensemble and the distribution function of the semi circular law is obtained the result holds assuming that the twelfth moment of the entries of the matrix is uniformly bounded

the spectral edge of some random band matrices we study the asymptotic distribution of the eigenvalues of random hermitian periodic band matrices focusing on the spectral edges the eigenvalues close to the edges converge in distribution to the airy point process if and only if the band is sufficiently wide w n 5 6 otherwise a different limiting distribution appears

maximum likelihood algorithms for generalized linear mixed models abstract maximum likelihood algorithms are described for generalized linear mixed models i show how to construct a monte carlo version of the em algorithm propose a monte carlo newton raphson algorithm and evaluate and improve the use of importance sampling ideas calculation of the maximum likelihood estimates is feasible for a wide variety of problems where they were not previously i also use the newton raphson algorithm as a framework to compare maximum likelihood to the joint maximization or penalized quasi likelihood methods and explain why the latter can perform poorly

a random effects ordinal regression model for multilevel analysis a random effects ordinal regression model is proposed for analysis of clustered or longitudinal ordinal response data this model is developed for both the probit and logistic response functions the threshold concept is used in which it is assumed that the observed ordered category is determined by the value of a latent unobservable continuous response that follows a linear regression model incorporating random effects a maximum marginal likelihood mml solution is described using gauss hermite quadrature to numerically integrate over the distribution of random effects an analysis of a dataset where students are clustered or nested within classrooms is used to illustrate features of random effects analysis of clustered ordinal data while an analysis of a longitudinal dataset where psychiatric patients are repeatedly rated as to their severity is used to illustrate features of the random effects approach for longitudinal ordinal data

syzygies of canonical curves and special linear series 0 introduction 105 1 line bundles on scrolls 109 2 scrolls and pencils 113 3 syzygies of a subvariety of a scroll 115 4 canonical curves 118 5 on the equation of rational surfaces 121 6 d gonal curves for d 5 126 7 proof of the main result 131 references 136

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

random monomial ideals abstract inspired by the study of random graphs and simplicial complexes and motivated by the need to understand average behavior of ideals we propose and study probabilistic models of random monomial ideals we prove theorems about the probability distributions expectations and thresholds for events involving monomial ideals with given hilbert function krull dimension and first graded betti numbers and show that our models generalize several well known random simplicial complex models finally we present several experimentally backed conjectures about regularity projective dimension strong genericity and cohen macaulayness of random monomial ideals

completely splittable representations of symmetric groups and affine hecke algebras abstract we study the class of completely splittable representations of the symmetric group and its associated degenerate affine hecke algebra in positive characteristic using techniques developed by okounkov and vershik for the symmetric group over the complex numbers the representations are classified and constructed and we provide a character formula in the symmetric group case

functional limit theorems for random regular graphs consider d uniformly random permutation matrices on n labels consider the sum of these matrices along with their transposes the total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices we consider limit theorems for various combinatorial and analytical properties of this graph or the matrix as n grows to infinity either when d is kept fixed or grows slowly with n in a suitable weak convergence framework we prove that the finite but growing in length sequences of the number of short cycles and of cyclically non backtracking walks converge to distributional limits we estimate the total variation distance from the limit using stein s method as an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix a key step in this latter derivation is an extension of the kahn szemeredi argument for estimating the second largest eigenvalue for all values of d and n

jack polynomials as fractional quantum hall states and the betti numbers of the k 1 equals ideal we show that for jack parameter k 1 r 1 certain jack polynomials studied by feigin jimbo miwa mukhin vanish to order r when k 1 of the coordinates coincide this result was conjectured by bernevig and haldane who proposed that these jack polynomials are model wavefunctions for fractional quantum hall states special cases of these jack polynomials include the wavefunctions of laughlin and read rezayi in fact along these lines we prove several vanishing theorems known as clustering properties for jack polynomials in the mathematical physics literature special cases of which had previously been conjectured by bernevig and haldane motivated by the method of proof which in the case r 2 identifies the span of the relevant jack polynomials with the sn invariant part of a unitary representation of the rational cherednik algebra we conjecture that unitary representations of the type a cherednik algebra have graded minimal free resolutions of bernstein gelfand gelfand type we prove this for the ideal of the k 1 equals arrangement in the case when the number of coordinates n is at most 2k 1 in general our conjecture predicts the graded sn equivariant betti numbers of the ideal of the k 1 equals arrangement with no restriction on the number of ambient dimensions

a mixed model procedure for analyzing ordered categorical data a mixed model procedure is developed for predicting the value of an ordered categorical response from knowledge of various predictor variables this procedure resembles the best linear unbiased procedure of henderson 1975 biometrics 31 423 447 for predicting the value of a quantitative response the approach is based on a mixed model version of the threshold model in which it is assumed that the observed category is determined by the value of an underlying unobservable continuous response that follows a mixed linear model the results are illustrated by an application to the problem of predicting the degree of difficulty that will be experienced by a dairy cow in the birth of her calf

localization of eigenvector centrality in networks with a cut vertex we show that eigenvector centrality exhibits localization phenomena on networks that can be easily partitioned by the removal of a vertex cut set the most extreme example being networks with a cut vertex three distinct types of localization are identified in these structures one is related to the well established hub node localization phenomenon and the other two are introduced and characterized here we gain insights into these problems by deriving the relationship between eigenvector centrality and katz centrality this leads to an interpretation of the principal eigenvector as an approximation to more robust centrality measures which exist in the full span of an eigenbasis of the adjacency matrix

a mixed effects regression model for three level ordinal response data three level data occur frequently in behaviour and medical sciences for example in a multi centre trial subjects within a given site are randomly assigned to treatments and then studied over time in this example the repeated observations level 1 are nested within subjects level 2 who are nested within sites level 3 similarly in twin studies repeated measurements level 1 are taken on each twin level 2 within each twin pair level 3 a three level mixed effects regression model is described here random effects at the second and third level are included in the model additionally both proportional odds and non proportional odds models are developed the latter allows the effects of explanatory variables to vary across the cumulative logits of the model a maximum marginal likelihood mml solution is described and gauss hermite numerical quadrature is used to integrate over the distribution of random effects the random effects are normally distributed in this instance features of this model are illustrated using data from a school based smoking prevention trial and an alzheimer s disease clinical trial

mixed poisson likelihood regression models for longitudinal interval count data summary in many longitudinal studies it is desired to estimate and test the rate over time of a particular recurrent event often only the event counts corresponding to the elapsed time intervals between each subject s successive observation times and baseline covariate data are available the intervals may vary substantially in length and number between subjects so that the corresponding vectors of counts are not directly comparable a family of poisson likelihood regression models incorporating a mixed random multiplicative component in the rate function of each subject is proposed for this longitudinal data structure a related empirical bayes estimate of random effect parameters is also described these methods are illustrated by an analysis of dyspepsia data from the national cooperative gallstone study

a mixed effects multinomial logistic regression model a mixed effects multinomial logistic regression model is described for analysis of clustered or longitudinal nominal or ordinal response data the model is parameterized to allow flexibility in the choice of contrasts used to represent comparisons across the response categories estimation is achieved using a maximum marginal likelihood mml solution that uses quadrature to numerically integrate over the distribution of random effects an analysis of a psychiatric data set in which homeless adults with serious mental illness are repeatedly classified in terms of their living arrangement is used to illustrate features of the model

a recursion and a combinatorial formula for jack polynomials the jack polynomials j x form a remarkable class of symmetric polynomials they are indexed by a partition and depend on a parameter one of their properties is that several classical families of symmetric functions can be obtained by specializing e g the monomial symmetric functions m the elementary functions e 0 the schur functions s 1 and nally the two classes of zonal polynomials 2 1 2 the jack polynomials can be de ned in various ways e g a as an orthogonal family of functions which is compatible with the canonical ltration of the ring of symmetric functions or b as simultaneous eigenfunctions of certain di erential operators the sekiguchi debiard operators recently opdam o constructed a similar family f x of nonsymmetric polynomials the index runs now through all compositions nn they are de ned in a completely similar fashion e g the sekiguchi debiard operators are being replaced by the cherednik di erential re ection operators see sect 3 it is becoming more and more clear that these polynomials are as important as their symmetric counterparts the purpose of this paper is to add to the existing characterizations of jack polynomials two further ones c a recursion formula among the f together with two formulas to obtain j from them d combinatorial formulas of both j and f in terms of certain generalized tableaux

hilbert series howe duality and branching for classical groups an extension of the littlewood restriction rule is given that covers all pertinent parameters and simplifies to the original under littlewood s hypotheses two formulas are derived for the gelfand kirillov dimension of any unitary highest weight representation occurring in a dual pair setting one in terms of the dual pair index and the other in terms of the highest weight for a fixed dual pair setting all the irreducible highest weight representations which occur have the same gelfand kirillov dimension we define a class of unitary highest weight representations and show that each of these representations l has a hilbert series hl q of the form

on the exponential generating function for non backtracking walks we derive an explicit formula for the exponential generating function associated with non backtracking walks around a graph we study both undirected and directed graphs our results allow us to derive computable expressions for non backtracking versions of network centrality measures based on the matrix exponential we find that eliminating backtracking walks in this context does not significantly increase the computational expense we show how the new measures may be interpreted in terms of standard exponential centrality computation on a certain multilayer network insights from this block matrix interpretation also allow us to characterize centrality measures arising from general matrix functions rigorous analysis on the star graph illustrates the effect of non backtracking and shows that unwanted localization effects can be eliminated when we restrict to non backtracking walks we also investigate the localization issue on synthetic networks

sheaf cohomology and free resolutions over exterior algebras in this paper we derive an explicit version of the bernstein gel fand gel fand bgg correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free reso lutions over its koszul dual exterior algebra among the facts about the bgg correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the linear part of a resolution over the exterior algebra we explore the structure of free resolutions over an exterior algebra for example we show that such resolutions are eventually dominated by their linear parts in the sense that erasing all terms of degree 1 in the complex yields a new complex which is eventually exact as applications we give a construction of the beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms the explicitness of our version allows us to to prove two conjectures about the morphisms in the monad and we get an efficient method for machine computation of the cohomology ofsheaves we also construct all the monads for a sheaf that can be built from sums of line bundles and show that they are often characterized by numerical data let v be a finite dimensional vector space over a field k and let w v be the dual space in this paper we will study complexes and resolutions over the exterior algebra e v and their relation to modules over s sym w and sheaves on projective space p w in this paper we study the bernstein gel fand gel fand bgg correspondence 1978 usually stated as an equivalence between the derived category of bounded complexes of coherent sheaves on p w and the stable category of finitely generated graded modules over e its essential content is a functor r from complexes of graded s modules to complexes of graded e modules and its adjoint l for example if m imi is a graded s module regarded as a complex with just one term then the first and third authors are grateful to the nsf for partial support during the preparation of this paper the third author wishes to thank msri for its hospitality

three themes of syzygies we present three exciting themes of syzygies where major progress was made recently boij soderberg theory stillman s question and syzygies over complete intersections

a syzygetic approach to the smoothability of zero dimensional schemes abstract we consider the question of which zero dimensional schemes deform to a collection of distinct points equivalently we ask which artinian k algebras deform to a product of fields we introduce a syzygetic invariant which sheds light on this question for zero dimensional schemes of regularity two this invariant imposes obstructions for smoothability in general and it completely answers the question of smoothability for certain zero dimensional schemes of low degree the tools of this paper also lead to other results about hilbert schemes of points including a characterization of nonsmoothable zero dimensional schemes of minimal degree in every embedding dimension d 4

unitary representations of cyclotomic rational cherednik algebras abstract we classify the irreducible unitary modules in category o c for the rational cherednik algebras of type g r 1 n and give explicit combinatorial formulas for their graded characters more precisely we produce a combinatorial algorithm determining for each r partition of n the closed semi linear set of parameters c for which the contravariant form on the irreducible representation l c is positive definite we use this algorithm to give a closed form answer for the cherednik algebra of the symmetric group recovering a result of etingof stoica and the author and the weyl groups of classical type

flexible modeling of longitudinal highly skewed outcomes the analysis of data from epidemiologic and environmental studies presents challenges such as skewness of distribution rounding and multiple measurements over time to model trends over time based on repeated measurements we propose a general latent model suitable for highly skewed data the model assumes that the observed outcome is determined by an unobservable outcome that follows a weibull distribution to accommodate correlations among repeated responses over time we introduce a general random effect from the power variance function pvf family of distributions including the gamma distribution often employed in the literature the resulting marginal likelihood has a closed form without resorting to numerical or approximation methods we study estimation and hypothesis testing under these models with different choices of random effect distributions simulation studies are conducted to evaluate their performance finally we apply the proposed method to exposure data collected from the michigan polybrominated biphenyl mipbb study