modular perverse sheaves on flag varieties ii koszul duality and formality building on the theory of parity sheaves due to juteau mautner williamson we develop a formalism of mixed modular perverse sheaves for varieties equipped with a stratification by affine spaces we then give two applications 1 a koszul type derived equivalence relating a given flag variety to the langlands dual flag variety and 2 a formality theorem for the modular derived category of a flag variety extending a previous result of riche soergel williamson

some koszul rings from geometry abstract we give examples of koszul rings that arise naturally in algebraic geometry in the first part we prove a general result on koszul property associated to an ample line bundle on a projective variety specifically we show how koszul property of multiples of a base point free ample line bundle depends on its castelnuovo mumford regularity in the second part we give examples of koszul rings that come from adjoint line bundles on minimal irregular surfaces of general type

parabolic projective functors in type a abstract we classify projective functors on the regular block of rocha caridi s parabolic version of the bgg category o in type a in fact we show that in type a the restriction of an indecomposable projective functor from o to the parabolic category is either indecomposable or zero as a consequence we obtain that projective functors on the parabolic category o in type a are completely determined up to isomorphism by the linear transformations they induce on the level of the grothendieck group which was conjectured by stroppel in 39

equivariant constructible koszul duality for dual toric varieties abstract for affine toric varieties x and x defined by dual cones we define an equivalence of categories between mixed versions of the equivariant derived category d t b x and the derived category of sheaves on x which are locally constant with unipotent monodromy on each orbit this equivalence satisfies the koszul duality formalism of beilinson ginzburg and soergel

representations of semisimple lie algebras in prime characteristic and the noncommutative springer resolution we prove most of lusztig s conjectures on the canonical basis in homology of a springer ber the conjectures predict that this basis controls numerics of representations of the lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic we check this for almost all characteristics to this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the springer resolution on the one hand this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and rumynin on the other hand it is compatible with the t structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the langlands dual group this equivalence established by arkhipov and the rst author ts the framework of local geometric langlands duality the latter compatibility allows one to apply frobenius purity theorem to deduce the desired properties of the basis we expect the noncommutative counterpart of the springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric langlands duality

general fractal discrete scheme for high frequency lung sound production a general scheme is proposed to explain the observed spectral properties of high frequency human respiratory sounds in terms of the interaction between the respiratory flux and a bronchial tree of fractal properties the air flux is treated as composed of discrete decoupled elements while the tree is assumed to have a cantor based geometry according to this model the affine behavior often observed in the high frequency log log spectral range is a direct consequence of the fractal geometry of the bronchial tree in both qualitative and quantitative aspects this strongly indicates that the dynamics underlying the high frequency sound generation must have at most nondominant couplings between the relevant fluid components

correlated binary regression using a quadratic exponential model summary a model of quadratic exponential form is parameterized in terms of marginal means and pairwise correlations for the regression analysis of correlated binary data pseudo maximum likelihood methods using a special case termed the multiplicative model are proposed but are noted to be computationally unattractive if the blocks of correlated responses are at all large on the other hand score estimating functions for mean and correlation parameters are shown to be of a particularly simple form under the quadratic exponential family special cases of such estimating functions having attractive computational properties are identified and illustrated

categorification of induced cell modules and the rough structure of generalised verma modules this paper presents categorifications of right cell modules and induced cell modules for hecke algebras of finite weyl groups fit type a we show that these categorifications depend only on the isomorphism class of the cell module not on the cell itself our main application is multiplicity formulas for parabolically induced modules over a reductive lie algebra of type a which finally determines the so called rough structure of generalised verma modules on the way we present several categorification results and give a positive answer to kostant s problem from a joseph kostant s problem goldie rank and the gelfand kirillov conjecture invent math 56 3 1980 191 213 in many cases we also present a general setup of decategorification precategorification and categorification

higher symplectic reflection algebras and non homogeneous n koszul property abstract we study a class of associative algebras associated to finite groups acting on a vector space these algebras are non homogeneous n koszul algebra generalizations of symplectic reflection algebras p etingof v ginzburg symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism invent math 147 2002 243 348 the koszul property was generalized to homogeneous algebras of degree n 2 in r berger koszulity for nonquadratic algebras j algebra 239 2001 705 734 in the present paper the extension of the koszul property to non homogeneous algebras is realized through a pbw theorem this pbw theorem is the generalization to the n case of a quadratic result obtained by braverman and gaitsgory a braverman d gaitsgory poincare birkhoff witt theorem for quadratic algebras of koszul type j algebra 181 1996 315 328 see also a polishchuk l positselski quadratic algebras univ lecture ser vol 37 amer math soc providence ri 2005 we work in the general setting where the ground ring is an arbitrary von neumann regular ring

sheaves on fibered threefolds and quiver sheaves this paper classifies a class of holomorphic d branes closely related to framed torsion free sheaves on threefolds fibered in resolved ade surfaces over a general curve c in terms of representations with relations of a twisted kronheimer nakajima type quiver in the category coh c of coherent sheaves on c for the local calabi yau case c cong bf a 1 and special choice of framing one recovers the n 1 ade quiver studied by cachazo katz vafa

the impact of litter effects on dose response modeling in teratology the fitting of dose response models to teratology data involving littermates in order to generate estimates of teratogenic risk is receiving increasing attention as a potential alternative to the safety factor approach to risk estimation in this paper we utilize the beta binomial distribution to introduce varying degrees of intralitter correlation and for purposes of illustration consider a logistic dose response model that describes the logit of risk as a straight line function of ln dose the biases and exact and asymptotic variances of the maximum likelihood estimators of the intercept and slope are studied by simulation as a function of the intralitter correlation structure

whittaker coinvariants for gl m n abstract let w m n be the finite w algebra attached to the principal nilpotent orbit in the general linear lie superalgebra gl m n c in this paper we study the whittaker coinvariants functor which is an exact functor from category o for gl m n c to a certain category of finite dimensional modules over w m n we show that this functor has properties similar to soergel s functor v in the setting of category o for a semisimple lie algebra we also use it to compute the center of w m n explicitly and deduce consequences for the classification of blocks of o up to morita derived equivalence

tilting modules in category o and sheaves on moment graphs abstract we describe tilting modules of the deformed category o over a semisimple lie algebra as certain sheaves on a moment graph associated to the corresponding block of o we prove that they map to braden macpherson sheaves constructed along the reversed bruhat order under fiebig s localization functor by this means we get character formulas for tilting modules and explain how soergel s result about the andersen filtration gives a koszul dual proof of the semisimplicity of subquotients of the jantzen filtration

algebraic deformations of toric varieties i general constructions abstract we construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry isospectral deformations and noncommutative geometry in braided monoidal categories our approach utilizes the same fan structure of the variety but deforms the underlying embedded algebraic torus we develop a sheaf theory using techniques from noncommutative algebraic geometry the cases of projective varieties are studied in detail and several explicit examples are worked out including new noncommutative deformations of grassmann and flag varieties our constructions set up the basic ingredients for a thorough study of instantons on noncommutative toric varieties which will be the subject of the sequel to this paper

graded decomposition numbers for the blob algebra abstract we give a closed formula for the graded decomposition numbers of the blob algebra over a field of characteristic zero at a root of unity

a modified pseudolikelihood approach for analysis of longitudinal data we consider the analysis of longitudinal data when the covariance function is modeled by additional parameters to the mean parameters in general inconsistent estimators of the covariance variance correlation parameters will be produced when the working correlation matrix is misspecified which may result in great loss of efficiency of the mean parameter estimators albeit the consistency is preserved we consider using different working correlation models for the variance and the mean parameters in particular we find that an independence working model should be used for estimating the variance parameters to ensure their consistency in case the correlation structure is misspecified the designated working correlation matrices should be used for estimating the mean and the correlation parameters to attain high efficiency for estimating the mean parameters simulation studies indicate that the proposed algorithm performs very well we also applied different estimation procedures to a data set from a clinical trial for illustration

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

lower central series and free resolutions of hyperplane arrangements if m is the complement of a hyperplane arrangement and a h m k is the cohomology ring of m over a field of characteristic 0 then the ranks k of the lower central series quotients of 1 m can be computed from the betti numbers bii dimtor a k k i of the linear strand in a minimal free resolution of k over a we use the cartan eilenberg change of rings spectral sequence to relate these numbers to the graded betti numbers b dimtor e a k j of a minimal resolution of a over the exterior algebra e from this analysis we recover a formula of falk for 3 and obtain a new formula for 4 the exact sequence of low degree terms in the spectral sequence allows us to answer a question of falk on graphic arrangements and also shows that for these arrangements the algebra a is koszul if and only if the arrangement is supersolvable we also give combinatorial lower bounds on the betti numbers b 1 of the linear strand of the free resolution of a over e if the lower bound is attained for i 2 then it is attained for all i 2 for such arrangements we compute the entire linear strand of the resolution and we prove that all components of the first resonance variety of a are local for graphic arrangements which do not attain the lower bound unless they have no braid sub arrangements we show that b 1 is determined by the number of triangles and k4 subgraphs in the graph

estimation bias using the beta binomial distribution in teratology kupper et al 1986 biometrics 42 85 98 considered the fitting of dose response regressions to litter proportions in teratology experiments they found that the estimators which maximise the beta binomial likelihood become biased when the intralitter correlation is incorrectly assumed to be homogeneous this paper considers the nature of this bias and its implications for the analysis of litter proportions in toxicology

affine jacquet functors and harish chandra categories abstract we define an affine jacquet functor and use it to describe the structure of induced affine harish chandra modules at noncritical levels extending the theorem of kac and kazhdan on the structure of verma modules in the bernstein gelfand gelfand categories o for kac moody algebras this is combined with a vanishing result for certain extension groups to construct a block decomposition of the categories of affine harish chandra modules of lian and zuckerman the latter provides an extension of the works of rocha caridi and wallach a rocha caridi n r wallach projective modules over infinite dimensional graded lie algebras math z 180 1982 151 177 and deodhar gabber and kac v deodhar o gabber v kac structure of some categories of representations of infinite dimensional lie algebras adv math 45 1982 92 116 on block decompositions of bgg categories for kac moody algebras we also derive a compatibility relation between the affine jacquet functor and the kazhdan lusztig tensor product and apply it to prove that the affine harish chandra category is stable under fusion tensoring with the kazhdan lusztig category this compatibility will be further applied in studying translation functors for the affine harish chandra category based on the fusion tensor product

highest weight sl_2 categorifications ii structure theory this paper continues the study of highest weight categorical sl2 actions initi ated in part i we start by refining the definition given there and showing that all examples considered in part i are also highest weight categorifications in the refined sense then we prove that any highest weight sl2 categorification can be filtered in such a way that the suc cessive quotients are so called basic highest weight sl2 categorifications for a basic highest weight categorification we determine minimal projective resolutions of standard objects we use this in particular to examine the structure of tilting objects in basic categorifications and to show that the ringel duality is given by the rickard complex we apply some of these structural results to categories o for cyclotomic rational cherednik algebras we finish by discussing open problems

koszul duality for toric varieties we show that certain categories of perverse sheaves on affine toric varieties x and x v defined by dual cones are koszul dual in the sense of beilinson ginzburg and soergel 1996 the functor expressing this duality is constructed explicitly by using a combinatorial model for mixed sheaves on toric varieties

preprojective algebra structure on skew group algebras abstract we give a class of finite subgroups g s l n k for which the skew group algebra k x 1 x n g does not admit the grading structure of a higher preprojective algebra namely we prove that if a finite group g s l n k is conjugate to a subgroup of s l n 1 k s l n 2 k for some n 1 n 2 1 then the skew group algebra k x 1 x n g is not morita equivalent to a higher preprojective algebra this is related to the preprojective algebra structure on the tensor product of two koszul bimodule calabi yau algebras we prove that such an algebra cannot be endowed with a grading structure as required for a higher preprojective algebra moreover we construct explicitly the bound quiver of the higher preprojective algebra over a finite dimensional koszul algebra of finite global dimension we show in addition that preprojective algebras over higher representation infinite koszul algebras are derivation quotient algebras whose relations are given by a superpotential

the classification of blocks in bgg category mathcal o o we classify all equivalences between the indecomposable abelian categories which appear as blocks in bgg category mathcal o for reductive lie algebras our classification implies that a block in category mathcal o only depends on the bruhat order of the relevant parabolic quotient of the weyl group as part of the proof we observe that any finite dimensional algebra with simple preserving duality admits at most one quasi hereditary structure

on trivial extensions and higher preprojective algebras abstract for a kosuzl n properly graded algebra we prove that the n 1 preprojective algebra of its quadratic dual is isomorphic to the quadratic dual of a twisted trivial extension this is applied to the slice algebras of stable n translation algebras and gives a noncommutative version of bernstein gelfand gelfand correspondence for such algebras

analysis of longitudinal data unbalanced over time summary the paper considers modelling estimating and diagnostically verifying the response process generating longitudinal data with emphasis on association between repeated measures from unbalanced longitudinal designs our model is based on separate specifications of the moments for the mean standard deviation and correlation with different components possibly sharing common parameters we propose a general class of correlation structures that comprise random effects measurement errors and a serially correlated process these three elements are combined via flexible time varying weights whereas the serial correlation can depend flexibly on the mean time and lag when the measurement schedule is independent of the response process our estimation procedure yields consistent and asymptotically normal estimates for the mean parameters even when the standard deviation and correlation are misspecified and for the standard deviation parameters even when the correlation is misspecified a generic diagnostic method is developed for verifying the models for the mean standard deviation and in particular the correlation which is applicable even when the data are severely unbalanced the methodology is illustrated by an analysis of data from a longitudinal study that was designed to characterize pulmonary growth in girls

estimating equations for parameters in means and covariances of multivariate discrete and continuous responses generalized estimating equations are introduced in an ad hoc fashion for the covariance matrix of a multivariate response these equations are to be solved jointly with score equations from a generalized linear model for mean parameters a class of quadratic exponential models is used to develop joint estimating equations for mean and covariance parameters in a more systematic fashion and proposals for the use of such equations are developed comments on the relative merits of the ad hoc and model based approaches to estimation are given and a regression illustration with a bivariate response is provided

covariance matrix estimation using repeated measurements when data are incomplete modeling repeated measurements data has been studied extensively lately in the parametric situation however it is significant to study the effect of various treatments over a period of time where the repeated measurements on the same subject are expected to be correlated the correlation among the repeated measurements for all the subjects will be studied via the covariance matrix the positive definite constraint is one of obstacles that encounters modeling the covariance structure however the cholesky decomposition removes this constraint and allows modeling the components of the covariance matrix m pourahmadi joint mean covariance models with applications to longitudinal data unconstrained parameterization biometrika 86 1999 677 690 in this paper we adopt the estimation procedures introduced by diggle and verbyla nonparametric estimation of covariance structure in longitudinal data biometrics 54 1998 401 415 where the variogram cloud as well as the squared residuals are used to estimate the variogram and the variances via the kernel smoothing selecting the appropriate bandwidth value is one of the important steps in the estimation process thus in our data analysis we choose the bandwidth using one of the most simple straight forward methods which is the cross validation method developed by rice and silverman estimating the mean and covariance structure nonparametrically when the data are curves j roy statist soc b 53 1991 233 243 and adapted by others finally we apply these nonparametric techniques as well as a graphical method m pourahmadi joint mean covariance models with applications to longitudinal data unconstrained parameterization biometrika 86 1999 677 690 to a real life data and use the penalized likelihood criterion like aic and bic to compare models of our interest our interest

modular perverse sheaves on flag varieties iii positivity conditions we further develop the general theory of the mixed modular derived category introduced by the authors in a previous paper in this series we then use it to study positivity and q koszulity phenomena on flag varieties

extended generalized estimating equations for clustered data abstract typically analysis of data consisting of multiple observations on a cluster is complicated by within cluster correlation estimating equations for generalized linear modeling of clustered data have recently received much attention this article proposes an extension to the generalized estimating equation method proposed by liang and zeger which treats within cluster correlations as nuisance parameters using ideas from extended quasi likelihood estimating equations for regression and association parameters are provided simultaneously the resulting estimators are proven to be asymptotically normal and consistent under certain conditions the consistency of regression estimators allows incorrect modeling of the correlation among repeated responses the method is illustrated with an analysis of data from a developmental toxicity study

a poincare birkhoff witt theorem for quadratic algebras with group actions braverman gaitsgory polishchuk and positselski gave necessary and sufficient conditions for a nonhomogeneous quadratic algebra to satisfy the poincare birkhoff witt property when its homogeneous version is koszul we widen their viewpoint and consider a quotient of an algebra that is free over some not necessarily semisimple subalgebra we show that their theorem holds under a weaker hypothesis we require the homogeneous version of the nonhomogeneous quadratic algebra to be the skew group algebra semidirect product algebra of a finite group acting on a koszul algebra obtaining conditions for the poincare birkhoff witt property over nonsemisimple group algebras we prove our main results by exploiting a double complex adapted from guccione guccione and valqui formed from a koszul complex and a resolution of the group giving a practical way to analyze hochschild cohomology and deformations of skew group algebras in positive characteristic we apply these conditions to graded hecke algebras and drinfeld orbifold algebras including rational cherednik algebras and symplectic reflection algebras in arbitrary characteristic with special interest in the case when the characteristic of the underlying field divides the order of the acting group

a generalized koszul theory and its relation to the classical theory abstract let a i 0 a i be a graded locally finite k algebra where a 0 is a finite dimensional algebra whose finitistic dimension is 0 in this paper we develop a generalized koszul theory preserving many classical results and show an explicit correspondence between this generalized theory and the classical theory applications in representations of certain categories and extension algebras of standard modules of standardly stratified algebras are described

koszul duality and equivalences of categories let a and a be dual koszul algebras by positselski a filtered algebra u with gr u a is koszul dual to a differential graded algebra a d we relate the module categories of this dual pair by a hom adjunction this descends to give an equivalence of suitable quotient categories and generalizes work of beilinson ginzburg and soergel

some homological properties of category o iii we prove that thick category o associated to a semi simple complex finite dimensional lie algebra is extension full in the category of all modules we also prove the weak alexandru conjecture both for regular blocks of thick category o and the associated categories of harish chandra bimodules but disprove it for singular blocks

coherent sheaves and categorical sl2 actions we introduce the concept of a geometric categorical sl2 action and relate it to that of a strong categorical sl2 action the latter is a special kind of 2 representation in the sense of rouquier the main result is that a geometric categorical sl2 action induces a strong categorical sl2 action this allows one to apply the theory of strong sl2 actions to various geometric situations our main example is the construction of a geometric categorical sl2 action on the derived category of coherent sheaves on cotangent bundles of grassmannians

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

an extended quasi likelihood function summary wedderburn s original definition of quasi likelihood for generalized linear models is extended to allow the comparison of variance functions as well as those of linear predictors and link functions the relationship between generalized linear models and the use of transformations of the response variable is explored and the ideas are illustrated by three examples generalized linear models by allowing the full distributional assumption about the random component in the model to be replaced by a much weaker assumption in which only the first and second moments were defined in making this extension wedderburn widened the scope of generalized linear models in a way very similar to that of gauss when he replaced the assumption of normality in classical linear models by that of equal variance for generalized linear models with distributions in the exponential family likelihood ratio and score tests are used for testing hypotheses concerning nested subsets of covariates in the linear predictor and for assessing hypothesized link functions these methods are also applicable with wedderburn s form of quasi likelihood however neither of these methods is suitable for the comparison of different variance functions in this paper we introduce an extended quasi likelihood function which allows for the comparison of various forms of all the components of a generalized linear model i e the linear predictor the link function and the variance function we then apply the ideas to the analysis of several sets of data

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

noetherianity and gelfand kirillov dimension of components abstract the category of all additive functors mod mod for a finite dimensional algebra were shown to be left noetherian if and only if is of finite representation type by m auslander here we consider the category of all additive graded functors from the category of associated graded category of mod to graded vector spaces this category decomposes into subcategories corresponding to the components of the auslander reiten quiver for a regular component we show that the corresponding graded functor category is left noetherian if and only if the section of the component is extended dynkin or infinite dynkin

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

exponential dispersion models on etudie les proprietes generales de la classe des modeles de dispersion exponentiels qui est la generalisation multivariable des modeles de distribution d erreur de nelder et wedderburn 1972

n cohomology of simple highest weight modules on walls and purity letg p b be respectively a complex connected linear algebraic semisimple group a parabolic subgroup and a borel subgroup the first main result is the following theorem let be a pure complex ong b smooth with respect to bruhat cells then its restriction to anyp orbit is pure as well of the same weight as a consequence we are able to compute then cohomology of simple highest weight modules on walls

fluid flow through ramified structures we investigate the fluid flow through two dimensional ramified structures by direct simulation of the navier stokes equations we show that for trees with n generations the flow distribution strongly depends on the reynolds number re specifically for a tree without loops the flow becomes highly heterogeneous at high re for a tree with loops on the other hand the flow distribution tends to be more uniform at increased re conditions we show that these apparently contradictory behaviors have the same origin namely the effect of inertia on the momentum transport in the channels of the ramified geometry in order to simulate the propagation of the flow imbalance throughout the tree without loops we develop a simple model that incorporates the basic fluid dynamics features of the system for large trees the results of the model indicate that the distribution of flow at the outlet branches can be described by a self affine landscape finally we argue that the nonuniform partitioning of flow found for the structure without loops may contribute to the morphogenesis and functioning of the bronchial tree s1063 651x 99 18210 2

two block springer fibers of types c and d a diagrammatic approach to springer theory we explain an elementary topological construction of the springer representation on the homology of topological springer fibers of types c and d in the case of nilpotent endomorphisms with two jordan blocks the weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types b and d we determine the decomposition of the representations into irreducibles and relate our construction to classical springer theory as an application we obtain presentations of the cohomology rings of all two block springer fibers of types c and d moreover we deduce explicit isomorphisms between the kazhdan lusztig cell modules attached to the induced trivial module and the irreducible specht modules in types c and d

mixed tensors of the general linear supergroup abstract we describe the image of the canonical tensor functor from deligne s interpolating category re p g l m n to r e p g l m n attached to the standard representation this implies explicit tensor product decompositions between any two projective modules and any two kostant modules of g l m n covering the decomposition between any two irreducible g l m 1 representations we also obtain character and dimension formulas for m n we classify the mixed tensors with non vanishing superdimension for m n we characterize the maximally atypical mixed tensors and show some applications regarding tensor products

primary ideals associated to the linear strands of lascoux s resolution and syzygies of the corresponding irreducible representations of the lie superalgebra gl m n abstract we investigate equivariant koszul duality between primary ideals i a b of s s m m n associated to rectangular young diagrams a b and the corresponding atypical irreducible mixed supertensor representations x a b of gl m n in characteristic zero we show i a b to be h 0 s x a b of the koszul dual s x a b and we compute all the higher cohomology of s x a b to be direct sums of i a r b r with multiplicities being given by coefficients of gauss polynomials utilizing this we are able to describe the equivariant syzygies of x a b over s m m n and determine the homological dimension of i a b over s

a likelihood based method for analysing longitudinal binary responses summary in this paper we discuss a likelihood based method for analysing correlated binary responses based on a multivariate model it is related to the pseudo maximum likelihood approach suggested recently by zhao prentice 1990 their parameterization results in a simple pairwise model in which the association between responses is modelled in terms of correlations while the present paper uses conditional log odds ratios with this approach higher order associations can be incorporated in a natural way one important advantage of this parameterization is that the maximum likelihood estimates of the marginal mean parameters are robust to misspecification of the time dependence we describe an iterative two stage procedure for obtaining the maximum likelihood estimates two examples are presented to illustrate this methodology

likelihood quasi likelihood and pseudolikelihood some comparisons summary there is considerable interest in the fitting of models jointly to the mean and dispersion of a response for the mean parameter the wedderburn estimating equations are widely accepted however there is some controversy about estimating the dispersion parameters finite sampling properties of several dispersion estimators are investigated for three models by simulation we compare the maximum extended quasi likelihood estimator the maximum pseudolikelihood estimator and the maximum likelihood estimator if it exists of these estimators the maximum extended quasi likelihood estimator is usually superior in minimizing the mean squared error

effects of variance function misspecification in analysis of longitudinal data the approach of generalized estimating equations gee is based on the framework of generalized linear models but allows for specification of a working matrix for modeling within subject correlations the variance is often assumed to be a known function of the mean this article investigates the impacts of misspecifying the variance function on estimators of the mean parameters for quantitative responses our numerical studies indicate that 1 correct specification of the variance function can improve the estimation efficiency even if the correlation structure is misspecified 2 misspecification of the variance function impacts much more on estimators for within cluster covariates than for cluster level covariates and 3 if the variance function is misspecified correct choice of the correlation structure may not necessarily improve estimation efficiency we illustrate impacts of different variance functions using a real data set from cow growth

group actions on algebras and the graded lie structure of hochschild cohomology hochschild cohomology governs deformations of algebras and its graded lie structure plays a vital role we study this structure for the hochschild cohomology of the skew group algebra formed by a nite group acting on an al gebra by automorphisms we examine the gerstenhaber bracket with a view toward deformations and developing bracket formulas we then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory deformations in this context include graded hecke alge bras and symplectic reection algebras we give some general results describing when brackets are zero for polynomial skew group algebras which allow us in particular to nd noncommutative poisson structures for abelian groups we express the bracket using inner products of group characters lastly we inter pret results for graded hecke algebras

skew group algebras of calabi yau algebras abstract the calabi yau property of skew group algebras is discussed it is shown that the skew group algebra a g of a koszul calabi yau algebra a with a finite subgroup g of automorphisms of a is calabi yau if and only if g is a finite subgroup of the special linear group sl a which is defined by means of the homological determinant using the a algebra structure on the yoneda algebra some results in bocklandt et al 2010 bsw are generalized say every connected graded p koszul calabi yau algebra is derived from a superpotential the superpotential for the skew group algebra a g is also constructed

koszul differential graded algebras and bgg correspondence the concept of koszul differential graded dg for short algebra is introduced in 8 let a be a koszul dg algebra if the ext algebra of a is finite dimensional i e the trivial module a k is a compact object in the derived category of dg a modules then it is shown in 8 that a has many nice properties however if the ext algebra is infinite dimensional little is known about a as shown in 15 see also proposition 2 2 a k is not compact if h a is finite dimensional in this paper it is proved that the koszul duality theorem also holds when h a is finite dimensional by using foxby duality a dg version of the bgg correspondence is deduced from the koszul duality theorem

estimating equations for association structures this paper investigates generalized estimating equations for association parameters which are frequently of interest in family studies with emphasis on covariance estimation separate link functions are used to connect the mean the scale and the correlation to linear predictors involving possibly different sets of covariates and separate estimating equations are proposed for the three sets of parameters simulations show that the robust sandwich variance estimator and the jackknife variance estimator for the correlation parameters are generally close to the empirical variance for the sample size of 50 clusters the results contradict ziegler et al and kastner and ziegler where the sandwich estimator obtained from the software mareg was shown to be unsuitable for practical usage the problem appears to arise because the mareg variance estimator does not account for variability in estimation of the scale parameters but may be valid with fixed scale we also find that the formula for the approximate jackknife variance estimator in ziegler et al is deficient resulting in systematic deviations from the fully iterated jackknife variance estimator a general jackknife formula is provided and performs well in numerical studies data from a study on the genetics of alcoholism is used to illustrate the importance of reliable variance estimation in biomedical applications

generating the fukaya categories of hamiltonian g manifolds abstract let g be a compact lie group and let k be a field of characteristic p geq 0 such that h g has no p torsion if p 0 we show that a free lagrangian orbit of a hamiltonian g action on a compact monotone symplectic manifold x split generates an idempotent summand of the monotone fukaya category mathcal f x k if and only if it represents a nonzero object of that summand slightly more general results are also provided our result is based on an explicit understanding of the wrapped fukaya category mathcal w t g k through koszul twisted complexes involving the zero section and a cotangent fibre and on a functor d b mathcal w t g k to d b mathcal f x times x k canonically associated to the hamiltonian g action on x we explore several examples which can be studied in a uniform manner including toric fano varieties and certain grassmannians

modeling of lung morphogenesis using fractal geometries the fractal dimension d sub f is one measure of the space filling features of a self similar structure additionally since d sub f varies with branching level there may be potential critical locations that are functionality important the authors introduce an algorithm that models lung airway structures and uses computer simulations of growth based on fractal concepts under these conditions limits imposed by simple boundary constraints generate structures that are in good agreement with actual morphometric data

equivariant sheaves on flag varieties we show that the borel equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of differential graded modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves this proves a conjecture of soergel and lunts in the case of flag varieties

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

analogs of renormalization group transformations in random processes abstract we review the properties of a real space renormalization group transformation of the free energy including the existence of oscillatory terms multiplying the non analytic part of the free energy we then construct stochastic processes which incorporate into probability distributions the features of the free energy scaling equation the essential information is obtainable from the scaling equation and a direct solution for a probability is not necessary these random processes are shown to be generated directly from cantor sets in a spatial representation the ensuing random process exhibits a transition between gaussian and fractal behavior in the fractal regime the trajectories will in an average sense form self similar clusters in a temporal representation the random process exhibits a transition between an asymptotically constant renewal rate and fractal behavior the fractal regime represents a frozen state with only transient effects allowed and is related to charge transport in glasses

efficient semiparametric mean association estimation for longitudinal binary responses semiparametric methods for longitudinal data with association within subjects have recently received considerable attention however existing methods for semiparametric longitudinal binary regression modeling i mainly concern mean structures with association parameters treated as nuisance ii generally require a correct specification of the covariance structure for misspecified covariance structure may lead to inefficient mean parameter estimates and iii usually run into computation and estimation problems when the time points are irregularly and possibly subject specific in this article we propose a semiparametric logistic regression model which simultaneously takes into account both the mean and response association structures via conditional log odds ratio for multivariate longitudinal binary outcomes our main interest lies in efficient estimation of both the marginal and association parameters the estimators of the parameters are obtained via the profile kernel approach we evaluate the proposed methodology through simulation studies and apply it to a real dataset both theoretical and empirical results demonstrate that the proposed method yields highly efficient estimators and performs satisfactorily copyright 2012 john wiley sons ltd

consistent model selection and data driven smooth tests for longitudinal data in the estimating equations approach summary model selection for marginal regression analysis of longitudinal data is challenging owing to the presence of correlation and the difficulty of specifying the full likelihood particularly for correlated categorical data the paper introduces a novel bayesian information criterion type model selection procedure based on the quadratic inference function which does not require the full likelihood or quasi likelihood with probability approaching 1 the criterion selects the most parsimonious correct model although a working correlation matrix is assumed there is no need to estimate the nuisance parameters in the working correlation matrix moreover the model selection procedure is robust against the misspecification of the working correlation matrix the criterion proposed can also be used to construct a data driven neyman smooth test for checking the goodness of fit of a postulated model this test is especially useful and often yields much higher power in situations where the classical directional test behaves poorly the finite sample performance of the model selection and model checking procedures is demonstrated through monte carlo studies and analysis of a clinical trial data set

the hypertoric intersection cohomology ring we present a functorial computation of the equivariant intersection cohomology of a hypertoric variety and endow it with a natural ring structure when the hyperplane arrangement associated with the hypertoric variety is unimodular we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category the computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset

on the use of the quasi likelihood method in teratological experiments this paper concerns the impact of litter effects on the inference of dose response relationships in teratological experiments with binary response kupper et al 1986 biometrics 42 85 98 concluded that when intra litter correlations are dose dependent the use of a common intra litter correlation likelihood based on the beta binomial distribution could lead to severe bias in estimation of the parameter beta which characterizes the dose response relationship we show here that the problems of bias and coverage probability for beta could still be substantial when one uses the likelihood with heterogeneous intra litter correlations we then examine through a simulation study the performance of the quasi likelihood method wedderburn 1974 biometrika 61 439 447 and recommend that this method with a common intra litter correlation parameter be used when the number of litters is small or modest

generalized least squares taylor series linearization and fisher s scoring in multivariate nonlinear regression in this article we consider a general multivariate nonlinear regression setting in which the marginal mean and variance covariance structure share a common set of regression parameters estimation is carried out via iteratively reweighted generalized least squares irgls that entails repeated application of taylor series linearization and estimated generalized least squares egls under normality this irgls procedure is equivalent to fisher s method of scoring and hence maximum likelihood estimation mle however estimates from this procedure are also shown to minimize a bias corrected generalized least squares objective function that does not require the assumption of normality under fairly mild regularity conditions the resulting estimates are consistent asymptotically normal and under normality assumptions asymptotically efficient the estimates are compared against those obtained as solutions to the usual generalized estimating equations gee using both simulation and numerical examples

unbiased estimating equations from working correlation models for irregularly timed repeated measures the method of generalized estimating equations gees has been criticized recently for a failure to protect against misspecification of working correlation models which in some cases leads to loss of efficiency or infeasibility of solutions however the feasibility and efficiency of gee methods can be enhanced considerably by using flexible families of working correlation models we propose two ways of constructing unbiased estimating equations from general correlation models for irregularly timed repeated measures to supplement and enhance gee the supplementary estimating equations are obtained by differentiation of the cholesky decomposition of the working correlation or as score equations for decoupled gaussian pseudolikelihood the estimating equations are solved with computational effort equivalent to that required for a first order gee full details and analytic expressions are developed for a generalized markovian model that was evaluated through simulation large sample sandwich standard errors for working correlation parameter estimates are derived and shown to have good performance the proposed estimating functions are further illustrated in an analysis of repeated measures of pulmonary function in children

weighted blowups and mirror symmetry for toric surfaces abstract this paper explores homological mirror symmetry for weighted blowups of toric varietes it will be shown that both the a model and b model categories have natural semi orthogonal decompositions an explicit equivalence of the right orthogonal categories will be shown for the case of toric surfaces

hard lefschetz theorem for nonrational polytopes the hard lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope one can construct the intersection cohomology combinatorially from the polytope hence it is well defined even for nonrational polytopes when there is no variety associated to it we prove the hard lefschetz theorem for the intersection cohomology of a general polytope

multivariate regression analyses for categorical data summary it is common to observe a vector of discrete and or continuous responses in scientific problems where the objective is to characterize the dependence of each response on explanatory variables and to account for the association between the outcomes the response vector can comprise repeated observations on one variable as in longitudinal studies or genetic studies of families or can include observations for different variables this paper discusses a class of models for the marginal expectations of each response and for pairwise associations the marginal models are contrasted with log linear models two generalized estimating equation approaches are compared for parameter estimation the first focuses on the regression parameters the second simultaneously estimates the regression and association parameters the robustness and efficiency of each is discussed the methods are illustrated with analyses of two data sets from public health research

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures