some aspects of polya tree distributions for statistical modelling the definition and elementary properties of polya tree distributions are reviewed two theorems are presented showing that polya trees can be constructed to concentrate arbitrarily closely about any desired pdf and that polya tree priors can put positive mass in every relative entropy neighborhood of every positive density with finite entropy thereby satisfying a consistency condition such theorems are false for dirichlet processes models are constructed combining partially specified polya trees with other information such as monotonicity or unimodality it is shown how to compute bounds on posterior expectations over the class of all priors with the given specifications a numerical example is given a theorem of diaconis and freedman about dirichlet processes is generalized to polya trees allowing polya trees to be the models for errors in regression problems finally empirical bayes models using dirichlet processes are generalized to polya trees an example from berry and christensen is reanalyzed with a polya tree model

inference for mixtures of finite polya tree models mixtures of polya tree models provide a flexible alternative when a parametric model may only hold approximately i provide computational strategies for obtaining full semiparametric inference for mixtures of finite polya tree models given a standard parameterization including models that would be troublesome to fit using dirichlet process mixtures recommendations are put forth on choosing the level of a finite polya tree and model comparison is discussed several examples demonstrate the utility of finite polya tree modeling including data fit to generalized linear mixed models and several survival models

integrable modules for affine lie superalgebras irreducible nonzero level modules with finite dimensional weight spaces are discussed for nontwisted affine lie superalgebras a complete classification of such modules is obtained for superalgebras of type a m n and c n using mathieu s classification of cuspidal modules over simple lie algebras in other cases the classification problem is reduced to the classification of cuspidal modules over finite dimensional cuspidal lie superalgebras described by dimitrov mathieu and penkov based on these results a complete classification of irreducible integrable in the sense of kac and wakimoto modules is obtained by showing that any such module is of highest weight in which case the problem was solved by kac and wakimoto

generalized verma modules over sln 2 induced from u hn free sln 1 modules abstract a class of generalized verma modules over sl n 2 are constructed from sl n 1 modules which are u h n free modules of rank 1 the necessary and sufficient conditions for these sl n 2 modules to be simple are determined this leads to a class of new simple sl n 2 modules

coherent families of weight modules of lie superalgebras and an explicit description of the simple admissible sl n 1 1 modules abstract let g be a classical lie superalgebra of type i we introduce coherent families of weight g modules with bounded weight multiplicities and establish a correspondence between cuspidal and highest weight submodules of these families by extending mathieu s work ann inst fourier 50 2000 537 this enables us to reduce the description of the g 0 module structure of arbitrary simple weight g modules with bounded weight multiplicities to the g 0 module structure of highest weight modules with bounded weight multiplicities we then construct tensor coherent families for g sl n 1 1 which yield an explicit description of the g 0 structure of an arbitrary simple weight module with bounded weight multiplicities in particular we show that for g sl n 1 1 the maximal length of an indecomposable g 0 submodule of a simple weight module with bounded multiplicities equals 5

mixtures of polya trees for flexible spatial frailty survival modelling mixtures of polya trees offer a very flexible nonparametric approach for modelling time to event data many such settings also feature spatial association that requires further sophistication either at the point level or at the lattice level in this paper we combine these two aspects within three competing survival models obtaining a data analytic approach that remains computationally feasible in a fully hierarchical bayesian framework using markov chain monte carlo methods we illustrate our proposed methods with an analysis of spatially oriented breast cancer survival data from the surveillance epidemiology and end results program of the national cancer institute our results indicate appreciable advantages for our approach over competing methods that impose unrealistic parametric assumptions ignore spatial association or both

the poincare series of the module of derivations of affine monomial curves abstract let a be a graded k algebra and m be a finitely generated a module the poincare series p m a z is the formal power series i dim k tor i a k m z i we study the poincare series of der k k s the module of derivations of a numerical semigroup ring k s and we relate it to the poincare series of k over k s and to the type of s we then use this in order to determine the poincare series of der k k s or at least its rationality for some classes of examples we finally give an example of a non rational p der k k s k s z

modeling multivariate survival data by a semiparametric random effects proportional odds model summary in this article the focus is on the analysis of multivariate survival time data with various types of dependence structures examples of multivariate survival data include clustered data and repeated measurements from the same subject such as the interrecurrence times of cancer tumors a random effect semiparametric proportional odds model is proposed as an alternative to the proportional hazards model the distribution of the random effects is assumed to be multivariate normal and the random effect is assumed to act additively to the baseline log odds function this class of models which includes the usual shared random effects model the additive variance components model and the dynamic random effects model as special cases is highly flexible and is capable of modeling a wide range of multivariate survival data a unified estimation procedure is proposed to estimate the regression and dependence parameters simultaneously by means of a marginal likelihood approach unlike the fully parametric case the regression parameter estimate is not sensitive to the choice of correlation structure of the random effects the marginal likelihood is approximated by the monte carlo method simulation studies are carried out to investigate the performance of the proposed method the proposed method is applied to two well known data sets including clustered data and recurrent event times data

singular gelfand tsetlin modules of gl n abstract the classical gelfand tsetlin formulas provide a basis in terms of tableaux and an explicit action of the generators of gl n for every irreducible finite dimensional gl n module these formulas can be used to define a gl n module structure on some infinite dimensional modules the so called generic gelfand tsetlin modules the generic gelfand tsetlin modules are convenient to work with since for every generic tableau there exists a unique irreducible generic gelfand tsetlin module containing this tableau as a basis element in this paper we initiate the systematic study of a large class of non generic gelfand tsetlin modules the class of 1 singular gelfand tsetlin modules an explicit tableaux realization and the action of gl n on these modules is provided using a new construction which we call derivative tableaux our construction of 1 singular modules provides a large family of new irreducible gelfand tsetlin modules of gl n and is a part of the classification of all such irreducible modules for n 3

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

classification of the tor algebras of codimension four almost complete intersections let r m k be a local ring in which 2 is a unit assume that every element of k has a square root in k we classify the algebras tor r r j k as j varies over all grade four almost complete intersection ideals in r the analogous classification has already been found when j varies over all grade four gorenstein ideals 21 and when j varies over all ideals of grade at most three 5 30 the present paper makes use of the classification in 21 of the tor algebras of codimension four gorenstein rings as well as the usually nonminimal dg algebra resolution of a codimension four almost complete intersection which is produced in 25 and 26

semiparametric bayesian analysis of survival data abstract this review article investigates the potential of bayes methods for the analysis of survival data using semiparametric models based on either the hazard or the intensity function the nonparametric part of every model is assumed to be a realization of a stochastic process the parametric part which may include a regression parameter or a parameter quantifying the heterogeneity of a population is assumed to have a prior distribution with possibly unknown hyperparameters careful applications of some recently popular computational tools including sampling based algorithms are used to find posterior estimates of several quantities of interest even when dealing with complex models and unusual data structures the methodologies developed herein are motivated and aimed at analyzing some common types of survival data from different medical studies here we focus on univariate survival data in the presence of fixed and time dependent covariates multiple event time data for repeated nonfatal events

on the existence of mathfrak g mathfrak k modules of finite type let g be a reductive lie algebra over an algebraically closed field of characteristic zero and k be a subalgebra reductive in g we prove that g admits an irreducible g k module m which has finite k multiplicities and which is not a g k 0 module for any proper inclusion of reductive subalgebras k k 0 g if and only if k contains its centralizer in g the main point of the proof is a geometric construction of g k modules which is analogous to cohomological induction for g gl n we show that whenever k contains its centralizer there is an irreducible g k module m of finite type over k such that k coincides with the subalgebra of all g 2 g which act locally finitely on m finally for a root subalgebra k gl n we describe all possibilities for the subalgebra l k of all elements acting locally finitely on some m math subject classification 2000 primary 17b10 secondary 22e46

maximum likelihood estimation in the proportional odds model abstract we consider maximum likelihood estimation of the parameters in the proportional odds model with right censored data the estimator of the regression coefficient is shown to be asymptotically normal with efficient variance the maximum likelihood estimator of the unknown monotonic transformation of the survival time converges uniformly at a parametric rate to the true transformation estimates for the standard errors of the estimated regression coefficients are obtained by differentiation of the profile likelihood and are shown to be consistent a likelihood ratio test for the regression coefficient is also considered

cuspidal representations of sl n 1 abstract in this paper we study the subcategory of cuspidal modules of the category of weight modules over the lie algebra sl n 1 our main result is a complete classification and an explicit description of indecomposable cuspidal modules

computing bayes factors by combining simulation and asymptotic approximations abstract the bayes factor is a ratio of two posterior normalizing constants which may be difficult to compute we compare several methods of estimating bayes factors when it is possible to simulate observations from the posterior distributions via markov chain monte carlo or other techniques the methods that we study are all easily applied without consideration of special features of the problem provided that each posterior distribution is well behaved in the sense of having a single dominant mode we consider a simulated version of laplace s method a simulated version of bartlett correction importance sampling and a reciprocal importance sampling technique we also introduce local volume corrections for each of these in addition we apply the bridge sampling method of meng and wong we find that a simulated version of laplace s method with local volume correction furnishes an accurate approximation that is especially useful when likelihood function evaluations are costly a simple bridge sampli

generalized verma modules induced from sl 2 c and associated verma modules abstract with each generalized verma module induced from a well embedded parabolic subalgebra of a lie algebra with triangular decomposition we associate a verma module over the same algebra in a natural way in the case when the semisimple part of the levi factor of the parabolic subalgebra is isomorphic to sl 2 c and the generalized verma module is induced from an infinite dimensional simple module we prove that the associated verma module is simple if and only if the original generalized verma module is simple

new singular gelfand tsetlin mathfrak gl n modules of index 2 singular gelfand tsetlin modules of index 2 are modules whose tableaux bases may have singular pairs but no singular triples of entries on each row in this paper we construct singular gelfand tsetlin modules for arbitrary singular character of index 2 explicit bases of derivative tableaux and the action of the generators of mathfrak gl n are given for these modules our construction leads to new families of irreducible gelfand tsetlin modules and also provides tableaux bases for some simple verma modules

on the generic free resolutions the problem we discuss in this paper is to find the arithmetic structure of finite free resolutions it started with hilbert s characterisation of ideals of projective dimension 1 with gcd 1 in a polynomial ring he proved that such ideals are given by n x n minors of an n x n 1 matrix this result was then generalized by several authors notably by burch b the hilbert burch theorem has since found many important applications to deformation theory and algebraic geometry in their paper buchsbaum and eisenbud be1 generalized the hilbertburch theorem they proved several structure theorems concerning various factorizations of the minors of the maps involved for resolutions of arbitrary length hochster h has shown that buchsbaum eisenbud structure theorems are enough to give the full structure of resolutions of length 2 he found in this case the universal pair r if i e a ring r and a resolution 5 over it such that for each resolution g with the same dimensions of modules over the ring s there exists a unique map r s such that g for s he proved that r is noetherian normal and cohenmacaulay if we not have the uniqueness in the above definition we speak of the generic pair r f if r f is a universal generic pair and r is noetherian we speak of the universal noetherian generic noetherian pair

root fernando kac subalgebras of finite type abstract let g be a finite dimensional lie algebra and m be a g module the fernando kac subalgebra of g associated to m is the subset g m g of all elements g g which act locally finitely on m a subalgebra l g for which there exists an irreducible module m with g m l is called a fernando kac subalgebra of g a fernando kac subalgebra of g is of finite type if in addition m can be chosen to have finite jordan holder l multiplicities under the assumption that g is simple i penkov has conjectured an explicit combinatorial criterion describing all fernando kac subalgebras of finite type which contain a cartan subalgebra in the present paper we prove this conjecture for g e 8

on the structure of weight modules given any simple lie superalgebra g we investigate the structure of an arbitrary simple weight 0 module we introduce two invariants of simple weight modules the shadow and the small weyl group generalizing results of fernando and futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a levi subsuperalgebra then we classify the cuspidal levi subsuperalgebras of all simple classical lie superalgebras and of the lie superalgebra w n most of them are simply levi subalgebras of go in which case the classification of all finite cuspidal representations has recently been carried out by one of us mathieu our results reduce the classification of the finite simple weight modules over all classical simple lie superalgebras to classifying the finite cuspidal modules over certain lie superalgebras which we list explicitly

generalized linear models with unknown link functions function by incorporating it as an unknown in the model since the link function is usually taken to be strictly increasing by a strictly increasing transformation of its range to the unit interval we can model it as a strictly increasing cumulative distribution function the transformation results in a domain which is 0 1 we model the cumulative distribution function as a mixture of beta cumulative distribution functions noting that the latter family is dense within the collection of all continuous densities on 0 1 for the fitting of the model we take a bayesian approach encouraging vague priors to focus upon the likelihood we discuss choices of such priors as well as the integrability of the resultant posteriors implementation of the bayesian approach is carried out using sampling based methods in particular a tailored metropolis within gibbs algorithm an illustrative example utilising data involving wave damage to cargo ships is provided

bayesian and conditional frequentist testing of a parametric model versus nonparametric alternatives testing the fit of data to a parametric model can be done by embedding the parametric model in a nonparametric alternative and computing the bayes factor of the parametric model to the nonparametric alternative doing so by specifying the nonparametric alternative via a polya tree process is particularly attractive from both theoretical and methodological perspectives among the benefits is a degree of computational simplicity that even allows for robustness analyses to be implemented default nonsubjective versions of this analysis are developed herein in the sense that recommended choices are provided for the many features of the polya tree process that need to be specified considerable discussion of these features is also provided to assist those who might be interested in subjective choices a variety of examples involving location scale models are studied finally it is shown that the resulting procedure can be viewed as a conditional frequentist test resulting in data dependent reported err

complete reducibility of torsion free cn modules of finite degree abstract we show that every torsion free weight module with finite dimensional weight spaces over a symplectic complex lie algebra which is different from sp 2 c is completely reducible

extensions of the conformal representations for orthogonal lie algebras abstract the conformal transformations with respect to the metric defining o n c give rise to an inhomogeneous polynomial representation of o n 2 c using shen s technique of mixed product we generalize the above representation to an inhomogeneous representation of o n 2 c on the tensor space of any finite dimensional irreducible o n c module with the polynomial space where a hidden central transformation is involved moreover we find a condition on the constant value taken by the central transformation such that the generalized representation is irreducible in our approach pieri s formulas invariant operators and the idea of kostant s characteristic identities play key roles the result could be useful in understanding higher dimensional conformal field theory with the constant value taken by the central transformation as the central charge our representations virtually provide natural extensions of the conformal transformations on a riemannian manifold to its vector bundles

marginal bayesian semiparametric modeling of mismeasured multivariate interval censored data motivated by data gathered in an oral health study we propose a bayesian nonparametric approach for population averaged modeling of correlated time to event data when the responses can only be de

on the structure of free resolutions of length 3 the sources of structure theory of finite free resolutions go back to hilbert in his famous paper on invariant theory he proved that every ideal of projective dimension 1 over a polynomial ring is determinantal this theorem was then extended by burch to arbitrary commutative rings there are various motivations for extending this results one of them is a variety of applications of the hilbert burch theorem in deformation theory and algebraic geometry cf for example s another reason comes from serre s conjecture one would like to understand in terms of resolutions why the lifting property of grothendieck does not hold in 1974 buchsbaum and eisenbud in bel introduced so called structure theorems various factorisations which occur for acyclic complexes of arbitrary length then hochster in h generalized the hilbert burch theorem to arbitrary free complexes of length 2 his results were then improved in br pw still the structure of complexes of length bigger than 2 is not understood the first steps in this direction are the results of b and the construction of a generic ring for complexes of length 3 of type 1 rz 1 1 given in pw in this paper i construct a ring which i believe is the generic ring for complexes of length 3 the construction involves new structure theorems and a new idea the notion of defects of structure theorems the paper is organized as follows section 1 is introductory i give there the background from representation theory and the cohomology groups of a lie algebra section 2 contains the proof of the new structure theorems we need the construction of the generic ring r and of the defect lie algebra l in section 3 i show that genericity of r follows from vanising of the homology of a certain family of complexes over u l the enveloping algebra of l in section 4 i apply the results of section 2 to study multi

heterogeneity in models for bivariate survival the importance of the mixing distribution we consider bivariate survival models in which the dependence between two survival times is by way of stochastically related unobserved components we analyse the role of the mixing or heterogeneity distribution in such models and examine how this distribution affects the correlation between the two survival times we derive explicit expressions for the correlation between the survival times and examine some properties of this correlation furthermore we examine to what extent the choice of this heterogeneity distribution is important for the hazards in the sample and their time paths the time path of the hazard among the survivors is determined by the moments of the distribution describing the heterogeneity among the survivors

simple sl_ n 1 module structures on u h abstract we study the category m consisting of u sl n 1 modules whose restriction to u h is free of rank 1 in particular we classify isomorphism classes of objects in m and determine their submodule structure this leads to new sl n 1 modules for n 1 we also find the central characters and derive an explicit formula for taking tensor product with a simple finite dimensional module

semiparametric inference in the proportional odds regression model abstract for fitting the proportional odds regression model with right censored survival times we introduce some weighted empirical odds functions these functions are solutions of some self consistency equations and have a nice martingale representation from these functions several classes of new regression estimators such as the pseudo maximum likelihood estimator martingale residual based estimators and minimum distance estimators are derived these estimators have desirable properties such as easy computation asymptotic normality via a martingale analysis and reliable asymptotic covariance estimation in closed form extensive numerical studies show that the minimum l 2 distance estimators have very good finite sample behaviors compared to existing methods results of some simulation studies and applications to a real dataset are given the weighted odds function based approach also provides inference on the baseline odds function and some measures for lack of fit analysis

predicting survival probabilities with semiparametric transformation models abstract prediction of survival probabilities for future patients is one of the main goals of fitting survival data with regression models in this article we consider a large class of semiparametric transformation models which includes the well known proportional hazards and proportional odds models for the analysis of failure time data specifically we propose pointwise and simultaneous confidence interval procedures for the survival probability of future patients with specific covariates these procedures can be easily implemented through simulation and are illustrated with the data from two well known clinical studies

supports of weight modules over witt algebras as the first step towards a classification of simple weight modules with finite dimensional weight spaces over witt algebras w n we explicitly describe the supports of such modules we also obtain

semiparametric regression analysis for clustered failure time data inference procedures based on the partial likelihood function for the cox proportional hazards model have been generalised to the case in which the data consist of a large number of independent small groups of correlated failure time observations lee wei amato 1992 liang self chang 1993 cai prentice 1997 however the cox model may not fit the data well a class of linear transformation models which includes the proportional hazards and odds models as special cases has been studied extensively for univariate event times in this paper statistical methods to analyse such correlated observations are proposed for these models we use the data from a recent study of the genetic aetiology of alcoholism to illustrate the new procedures for estimation prediction and model selection

irreducible completely pointed modules of quantum groups of type a abstract we give a classification of all irreducible completely pointed u q sl n 1 modules over a characteristic zero field in which q is not a root of unity this generalizes the classification result of benkart britten and lemire in the non quantum case we also show that any infinite dimensional irreducible completely pointed u q sl n 1 can be obtained from some irreducible completely pointed module over the quantized weyl algebra a n 1 q

finite weight modules over twisted affine lie superalgebras abstract this work provides the first step toward the classification of irreducible finite weight modules over twisted affine lie superalgebras we divide the class of such modules into two subclasses called hybrid and tight we reduce the classification of hybrid irreducible finite weight modules to the classification of cuspidal modules of finite dimensional cuspidal lie superalgebras which is discussed in a work of dimitrov mathieu and penkov

a koszul category of representations of finitary lie algebras abstract we find for each simple finitary lie algebra g a category t g of integrable modules in which the tensor product of copies of the natural and conatural modules is injective the objects in t g can be defined as the finite length absolute weight modules where by absolute weight module we mean a module which is a weight module for every splitting cartan subalgebra of g the category t g is koszul in the sense that it is antiequivalent to the category of locally unitary finite dimensional modules over a certain direct limit of finite dimensional koszul algebras we describe these finite dimensional algebras explicitly we also prove an equivalence of the categories t o and t sp corresponding respectively to the orthogonal and symplectic finitary lie algebras o sp

bayesian semiparametric proportional odds models methodology for implementing the proportional odds regression model for survival data assuming a mixture of finite polya trees mpt prior on baseline survival is presented extensions to frailties and generalized odds rates are discussed although all manner of censoring and truncation can be accommodated we discuss model implementation regression diagnostics and model comparison for right censored data an advantage of the mpt model is the relative ease with which predictive densities survival and hazard curves are generated much discussion is devoted to practical implementation of the proposed models and a novel mcmc algorithm based on an approximating parametric normal model is developed a modest simulation study comparing the small sample behavior of the mpt model to a rank based estimator and a real data example is presented

a combinatorial description of blocks in o p λ associated with sl 2 induction abstract we study the category o p where is an admissible category of dense weight sl 2 modules we give a combinatorial description of projectively stratified algebras arising from o p and prove a double centralizer property moreover we determine the characters of tilting modules in o p and prove that the finite dimensional algebra associated with the principal block of our o p is its own ringel dual

markov chain monte carlo methods for computing bayes factors a comparative review the problem of calculating posterior probabilities for a collection of competing models and associated bayes factors continues to be a formidable challenge for applied bayesian statisticians current approaches that take advantage of modern markov chain monte carlo computing methods include those that attempt to sample over some form of the joint space created by the model indicators and the parameters for each model others that sample over the model space alone and still others that attempt to estimate the marginal likelihood of each model directly because the collection of these is equivalent to the collection of model probabilities themselves we review several methods and compare them in the context of three examples a simple regression example a more challenging hierarchical longitudinal model and a binary data latent variable model we find that the joint model parameter space search methods perform adequately but can be difficult to program and tune whereas the marginal likelihood methods o

maximum likelihood estimation for the proportional odds model with random effects in this article we study the semiparametric proportional odds model with random effects for correlated right censored failure time data we establish that the maximum likelihood estimators for the parameters of this model are consistent and asymptotically gaussian furthermore the limiting variances achieve the semiparametric efficiency bounds and can be consistently estimated simulation studies show that the asymptotic approximations are accurate for practical sample sizes and that the efficiency gains of the proposed estimators over those of cai cheng and wei can be substantial a real example is provided to illustrate the proposed methods

a bayesian semiparametric accelerated failure time model a bayesian semiparametric approach is described for an accelerated failure time model the error distribution is assigned a polya tree prior and the regression parameters a noninformative hierarchical prior two cases are considered the first assumes error terms are exchangeable the second assumes that error terms are partially exchangeable a markov chain monte carlo algorithm is described to obtain a predictive distribution for a future observation given both uncensored and censored data

proportional odds regression and sieve maximum likelihood estimation summary this paper introduces under a censoring mechanism an estimation procedure for the baseline function and the regression parameters in the proportional odds model based on the sieve maximum likelihood method the procedure uses monotone splines of variable orders and knots as approximations for the baseline function and an approximately unbiased estimator of the comparative kullback leibler risk to select the best approximation the procedure adapts to various unknown structures of the baseline function such as spatial structures and smoothness of an unknown degree the proposed procedure is implemented for right censored and so called case 2 interval censored data the estimated regression parameters are shown to be asymptotically normal and efficient the small sample operating characteristics of the proposed procedure are examined via simulations and are illustrated on a dataset from a study of breast cancer patients

hierarchical generalized linear models and frailty models with bayesian nonparametric mixing summary this paper proposes bayesian nonparametric mixing for some well known and popular models the distribution of the observations is assumed to contain an unknown mixed effects term which includes a fixed effects term a function of the observed covariates and an additive or multiplicative random effects term typically these random effects are assumed to be independent of the observed covariates and independent and identically distributed from a distribution from some known parametric family this assumption may be suspect if either there is interaction between observed covariates and unobserved covariates or the fixed effects predictor of observed covariates is misspecified another cause for concern might be simply that the covariates affect more than just the location of the mixed effects distribution as a consequence the distribution of the random effects could be highly irregular in modality and skewness leaving parametric families unable to model the distribution adequately this paper therefore proposes a bayesian nonparametric prior for the random effects to capture possible deviances in modality and skewness and to explore the observed covariates effect on the distribution of the mixed effects

the resolution of the bracket powers of the maximal ideal in a diagonal hypersurface ring abstract let k be a field for each pair of positive integers n n we resolve q r x n y n z n as a module over the ring r k x y z x n y n z n write n in the form n a n r for integers a and r with r between 0 and n 1 if n does not divide n and the characteristic of k is fixed then the value of a determines whether q has finite or infinite projective dimension if q has infinite projective dimension then value of r together with the parity of a determines the periodic part of the infinite resolution when q has infinite projective dimension we give an explicit presentation for the module of first syzygies of q this presentation is quite complicated we also give an explicit presentation for the module of second syzygies for q this presentation is remarkably uncomplicated we use linkage to find an explicit generating set for the grade three gorenstein ideal x n y n z n x n y n z n in the polynomial ring k x y z the question does q have finite projective dimension is intimately connected to the question does k x y z x a y a z a have the weak lefschetz property the second question is connected to the enumeration of plane partitions when the field k has positive characteristic we investigate three questions about the frobenius powers f t q of q when does there exist a pair n n so that q has infinite projective dimension and f q has finite projective dimension is the tail of the resolution of the frobenius power f t q eventually a periodic function of t up to shift in particular we exhibit a situation where the tail of the resolution of f t q after shifting is periodic as a function of t with an arbitrarily large period can one use socle degrees to predict that the tail of the resolution of f t q is a shift of the tail of the resolution of q

classification of irreducible nonzero level modules with finite dimensional weight spaces for affine lie algebras abstract we classify the irreducible weight affine lie algebra modules with finite dimensional weight spaces on which the central element acts nontrivially in particular we show that any such module is a quotient of a generalized verma module the classification of such irreducible modules is reduced to the classification of irreducible torsion free modules for algebras of type a n and c n which is due to o mathieu forthcoming

faces and maximizer subsets of highest weight modules abstract in this paper we study general highest weight modules v over a complex finite dimensional semisimple lie algebra g we present three formulas for the set of weights of a large family of modules v which include but are not restricted to all simple modules and all parabolic verma modules these formulas are direct and do not involve cancellations and were not previously known in the literature our results extend the notion of the weyl polytope to general highest weight g modules v we also show that for all simple modules the convex hull of the weights is a w j invariant polyhedron for some parabolic subgroup w j we compute its vertices faces and symmetries more generally we also do this for all parabolic verma modules and for all modules v with highest weight not on a simple root hyperplane to show our results we extend the notion of convexity to arbitrary additive subgroups a r of coefficients our techniques enable us to completely classify weak a faces of the support sets wt v in the process extending classical results of satake borel tits vinberg and casselman as well as modern variants by chari dolbin ridenour and cellini marietti to general highest weight modules

on modules of bounded multiplicities for the symplectic algebras simple infinite dimensional highest weight modules having bounded weight multipicities are classified as submodules of a tensor product also it is shown that a simple torsion free module of finite degree tensored with a finite dimensional module is completely reducible

modular representations of the special linear groups with small weight multiplicities abstract we classify irreducible representations of the special linear groups in positive characteristic with small weight multiplicities with respect to the group rank and give estimates for the maximal weight multiplicities for the natural embeddings of the classical groups inductive systems of representations with totally bounded weight multiplicities are classified an analogue of the steinberg tensor product theorem for arbitrary indecomposable inductive systems for such embeddings is proved

lie algebra modules with finite dimensional weight spaces i let g denote a reductive lie algebra over an algebraically closed field of characteristic zero and let 4 denote a cartan subalgebra of g in this paper we study finitely generated g modules that decompose into direct sums of finite dimensional 4 weight spaces we show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules those we call torsion free modules we also show that if g is a simple lie algebra that admits a torsion free module then g is of type a or c

cuspidal sln modules and deformations of certain brauer tree algebras we show that the algebras describing blocks of the category of cuspidal weight resp generalized weight sln modules are one parameter resp multi parameter deformations of certain brauer tree algebras we explicitly determine these deformations both graded and ungraded the algebras we deform also appear as special centralizer subalgebras of temperley lieb algebras or as generalized khovanov algebras they show up in the context of highest weight representations of the virasoro algebra in the context of rational representations of the general linear group and schur algebras and in the study of the milnor fiber of kleinian singularities

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

semiparametric bayes proportional odds models for current status data with underreporting current status data are a type of interval censored event time data in which all the individuals are either left or right censored for example our motivation is drawn from a cross sectional study which measured whether or not fibroid onset had occurred by the age of an ultrasound exam for each woman we propose a semiparametric bayesian proportional odds model in which the baseline event time distribution is estimated nonparametrically by using adaptive monotone splines in a logistic regression model and the potential risk factors are included in the parametric part of the mean structure the proposed approach has the advantage of being straightforward to implement using a simple and efficient gibbs sampler whereas alternative semiparametric bayes event time models encounter problems for current status data the model is generalized to allow systematic underreporting in a subset of the data and the methods are applied to an epidemiologic study of uterine fibroids

large annihilator category o for sl o sp abstract we construct a new analogue of the bgg category o for the infinite dimensional lie algebras g sl o sp a main difference with the categories studied in 9 and 2 is that all objects of our category satisfy the large annihilator condition introduced in 5 despite the fact that the splitting borel subalgebras b of g are not conjugate one can eliminate the dependency on the choice of b and introduce a universal highest weight category ola of g modules the letters la coming from large annihilator the subcategory of integrable objects in ola is precisely the category t g studied in 5 we investigate the structure of ola and in particular compute the multiplicities of simple objects in standard objects and the multiplicities of standard objects in indecomposable injectives we also complete the annihilators in u g of simple objects of ola

a classification of simple lie modules having a 1 dimensional weight space let l denote a simple lie algebra over the complex numbers in this paper we classify and construct all simple l modules which may be infinite dimensional but have at least one 1 dimensional weight space this completes the study begun earlier by the authors for the case of l a the approach presented here relies heavily on the results of suren fernando whose dissertation dealt with simple weight modules and their weight systems 0 introduction let l be a finite dimensional simple lie algebra over the complex field c having a cartan subalgebra h and denote by c l the centralizer of h in the universal enveloping algebra u of l if x h c is a weight function of a 1 dimensional weight space mx in a simple l module m then q c l c defined by q c v cv for v e mx and c e c l is an algebra homomorphism called a mass function of m clearly x restricted to h is equal to x conversely given any algebra homomorphism q c l c one can construct a unique simple l module which admits x as a mass function 4 10 in 4 the authors determined all algebra homomorphisms q c l c for the simple lie algebras l of type an and using these classified all pointed an modules where we call a module pointed if it is simple and has at least one 1 dimensional weight space in this paper we complete the classification of all pointed l modules for arbitrary simple lie algebras the collection of all pointed l modules clearly includes the highest weight l modules and is included in the collection of all harish chandra l modules relative to the cartan subalgebra h i e simple l modules having a weight space decomposition and finite dimensional weight spaces relative to h the latter inclusion is strict since there exist examples of harish chandra a2 modules in which every weight space is two dimensional in the special case of a1 modules every harish chandra a1 module is pointed our approach to this problem makes heavy use of the results from fernando s thesis 8 in particular from fernando s results we see that in place of determining all algebra homomorphisms q c l c it suffices to find only those algebra homomorphisms x which are associated with so called pointed torsion free l received by the editors may 23 1984 this paper was presented at the 1984 canadian mathematical society s summer seminar on lie algebras and related topics held at the university of windsor 1980 mathematics subject classification primary 17b10

irreducible representations of _ with a 1 dimensional weight space in this paper we classify all irreducible linear representations of the simple lie algebra an which admit a one dimensional weight space with respect to some cartan subalgebra h of a we first show that the problem is equivalent to determining all algebra homomorphisms from the centralizer of the cartan subalgebra h in the universal enveloping algebra of an to the base field we construct all such algebra homomorphisms and provide conditions under which two such algebra homomorphisms provide inequivalent irreducible representations of an

a structure theorem for type 3 grade 3 perfect ideals abstract in 1977 buchsbaum and eisenbud gave a complete characterization of grade 3 perfect ideals i of type 1 i e gorenstein in a noetherian local ring r exploiting the fact that a minimal free resolution for r i has a structure of associative commutative differential graded algebra they also studied successfully the structure of grade 3 almost complete intersections ideals in 1981 and 1982 kustin and miller studied the structure of gorenstein ideals of grade 4 but they introduced a new variable to the problem they defined the concept of defect of an ideal and they found that for grade 4 gorenstein ideals the structures vary in part according to the value of d i the defect of i later in 1984 ann brown in her ph d thesis gave a structure theorem for grade 3 perfect ideals of type 2 and defect positive she also proved that for grade 3 perfect ideals of type 2 d i can be either 0 or 1 the purpose of this paper is to give a structure theorem for grade 3 perfect ideals of type 3 and defect 2 or greater furthermore this structure theorem permits us to establish that for grade 3 type 3 perfect ideals d i can be either 0 or 1 or 2 or 4

weight modules for current algebras for any finite dimensional simple lie algebra g and commutative associative algebra s of finite type we give a complete classification of the simple weight modules of g s with bounded weight multiplicities the classification of simple weight modules is widely seen as a difficult problem in a spectacular tour de force olivier mathieu gave a complete description of such modules for finite dimensional simple lie algebras ma the crucial part of mathieu s classifi cation was understanding the simple admissible highest weight modules in the present paper we consider this question in the context of generalised current algebras l g k s where g is a finite dimensional simple lie algebra over k and s is a commutative associative and unital k algebra which is of finite type these algebras include the finite dimensional simple lie algebras g and the loop algebras g c c t t 1 of affine kac moody theory our methods work not only for the classification of simple admissible highest weight modules but also describe the simple objects in the much larger category of admissible modules weight modules whose weight multiplicities are uniformly bounded we hope that this paper will contribute to solving the harder problem of classifying all simple weight modules for these algebras the category of admissible modules includes the finite dimensional simple modules all of which are known to be evaluation modules see the introductions of l1 cfk or the more general results in nss lp l2 for details what is surprising is that this larger category of generally infinite dimensional modules can also be described in terms of evaluations we now summarize the content of this paper funding from the natural sciences and engineering research council of canada is gratefully ac knowledged

z graded oscillator representations of symplectic lie algebras abstract in this paper we study two parameter z graded oscillator representations of symplectic lie algebras obtained from the natural oscillator representation by partially swapping differential operators and multiplication operators we prove that the homogeneous subspaces with nonzero degree or unequal parameters are infinite dimensional irreducible weight submodules with finite dimensional weight subspaces when the two parameters are equal the homogeneous subspace of degree zero is exactly a direct sum of two infinite dimensional irreducible weight submodules with finite dimensional weight subspaces in generic case these irreducible modules are non unitary modules without highest weight vectors our results are extensions of howe s oscillator construction of infinite dimensional multiplicity free irreducible representations for special linear lie algebras

analysis of transformation models with censored data summary in this paper we consider a class of semi parametric transformation models under which an unknown transformation of the survival time is linearly related to the covariates with various completely specified error distributions this class of regression models includes the proportional hazards and proportional odds models inference procedures derived from a class of generalised estimating equations are proposed to examine the covariate effects with censored observations numerical studies are conducted to investigate the properties of our proposals for practical sample sizes these transformation models coupled with the new simple inference procedures provide many useful alternatives to the cox regression model in survival analysis

analysis of survival data by the proportional odds model a model is presented for the analysis of lifetime data in which the rates of mortality for separate groups of patients converge with time a non parametric estimate is given for the survivor function the theoretical basis for the model assumes that prognostic factors have a multiplicative effect on the odds against survival beyond any given time the model is fitted to data using maximum likelihood estimation and an example of its use in the analysis of a lung cancer trial is given

modeling regression error with a mixture of polya trees we model the error distribution in the standard linear model as a mixture of absolutely continuous polya trees constrained to have median 0 by considering a mixture we smooth out the partitioning effects of a simple polya tree and the predictive error density has a derivative everywhere except 0 the error distribution is centered around a standard parametric family of distributions and thus may be viewed as a generalization of standard models in which important data driven features such as skewness and multimodality are allowed by marginalizing the polya tree exact inference is possible up to markov chain monte carlo error

schur functors and schur complexes the study of schur functors has a relatively long history its main impetus derived from representation theory originally in characteristic zero over the years however with the development of modular representations and algebraic geometry over fields of positive characteristic the need for a theory of universal polynomial functors increased and since the mid 1960s approaches to a characteristic free treatment of schur functors have been developing see for instance the recent book of green 1 l in which the treatments by carter and lusztig 6 higman 12 and towber 20 among others are discussed our own interest in such a treatment was awakened by the work of lascoux 141 on resolutions of determinantal ideals although his thesis treated only the characteristic zero case it suggested that a general and elementary theory of schur functors could be developed using only the rudiments of multilinear algebra involving the hopf algebra structures of the symmetric exterior and divided power algebras moreover this elementary development admitted of a natural generalization to the idea of schur complexes whose usefulness was demonstrated for instance in our construction of a universal minimal