log linear non uniform association models for agreement between two ratings on an ordinal scale in agreement studies when objects are rated independently by two raters or twice by the same rater an association between their ratings on two categories arises reflecting the distinguishability of these two categories for these raters when ratings are performed on an ordinal scale this association between ratings on two categories increases when the distance between these categories increases on the ordinal scale goodman s log linear models derived for the analysis of agreement between two raters on an ordinal scale assume that distinguishabilities between adjacent categories are either constant or a priori fixed log non linear models that allow variations of the distinguishabilities between adjacent categories along the scale may lead to difficulties in parameter estimation this paper describes a new class of log linear non uniform association models these models extend the log linear uniform association model by allowing variations of distinguishability between adjacent categories along the scale these new models are used to analyse ordinal agreement between dermatologists when assessing the severity of different cutaneous signs of ageing on women faces

association models and the bivariate normal for contingency tables with ordered categories summary association models considered by goodman 1979 for contingency tables with ordered categories are presented here in a somewhat different form to facilitate comparison with the bivariate normal association models can be applied when an underlying bivariate normal is assumed and also under more general conditions and they provide alternatives to tetrachoric and polychoric correlation these models agree more closely with the bivariate normal than does plackett s model 1965 when the row and column classifications arise from underlying univariate normal distributions the general utility of the association models is illustrated here

a necessary and sufficient criterion for the stability of a convex set of matrices a hurwitz stability criterion for the convex hull of two matrices due to c r johnson 1976 is generalized to the case of any compact convex set of matrices it is shown that the same criterion is not available in the case of schur stability

the extreme eigenvalues and stability of real symmetric interval matrices a novel khoritonov like algorithm for computing the minimal and maximal eigenvalues of n n dimensional symmetric interval matrices is presented it is proved that the maximal eigenvalue of a given set of interval matrices coincides with the maximal eigenvalue of a special set of 2 sup n 1 symmetric vertex matrices whereas its minimal eigenvalue coincides with the minimal of another special set of 2 sup n 1 symmetric vertex matrices as immediate corollaries of this algorithm weak necessary and sufficient conditions for testing the hurwitz and schur stability of symmetric interval matrices where one has to test the stability of 2 sup n 1 and 2 sup n symmetric vertex matrices respectively are obtained

on the rigidity theorems of witten in this paper we prove the rigidity theorems predicted by witten in 1986 about the index of certain elliptic operators on manifolds with an s1 action w witten s insight was the culmination of an interesting interchange of ideas between him and hopkins landweber ochanine and stong for the detailed history we refer the reader to la the present account is essentially a reinterpretation of the second author s taubes t original proof of the theorem the senior author s contribution was solely to notice that the rather densely written arguments of the original manuscript could be formulated in terms of the well known fixed point formulae of equivariant index theory and equivariant cohomology in this context the final proof then appears in direct lineage of the atiyah hirzebruch theorem concerning the vanishing of the a genus of spin manifolds admitting a circle action a h and of the even older idea of lusztig concerning circle actions in the complex case there remains however the beautiful physics inspired novelty of connecting these techniques with elliptic function theory

convex combinations of matrices full rank characterization abstract let a 1 a k be complex n by m matrices using the notion of a block p matrix introduced recently by elsner and the second author a characterization of the full rank property of all convex combinations of a 1 a k is derived

s 1 actions and elliptic genera a proof is given of witten s conjectures for the rigidity of the index of the dirac ramond operator on the loop space of a spin manifold which admits ans1 symmetry

stability of interval matrices the real eigenvalue case c v hollot and a c bartlett 1987 showed that testing at most 2 to the n sup 2 certain matrices for stability is sufficient for verifying stability of an n n interval matrix with real eigenvalues it is proven that this upper bound can be reduced to 2 sup 2n 1 and a special case where testing only two matrices is needed is considered

elliptic genera of singular varieties the notions of orbifold elliptic genus and elliptic genus of singular varieties are introduced and the relation between them is studied the elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of calabi yau hypersurfaces in fano gorenstein toric varieties introduced earlier the orbifold elliptic genus is given in terms of the fixed point sets of the action we show that the generating function for the orbifold elliptic genus ellorb x 6n p for symmetric groups 6n acting on n fold products coincides with the one proposed by r dijkgraaf g moore e verlinde and h verlinde the two notions of elliptic genera are conjectured to coincide

assessing rater agreement using marginal association models new models that are useful in the assessment of rater agreement particularly when the rating scale is ordered or partially ordered are presented the models are parameterized to address two important aspects of rater agreement i agreement in terms of the overall frequency in which raters assign categories and ii the extent to which raters agree on the category assigned to individual subjects or items we present methodology for the simultaneous modelling of univariate marginal responses and bivariate marginal associations in the k way contingency table representing the joint distribution of k rater responses the univariate marginal responses provide information for evaluating agreement in terms of the overall frequency of responses and the bivariate marginal associations provide information on category wise agreement among pairs of raters in addition estimated scores within a generalized log non linear model for bivariate associations facilitate the assessment of category distinguishability

analysis of sets of two way contingency tables using association models abstract a class of models is introduced for the analysis of group differences in the association between two discrete variables the rc m association model for two way tables is reviewed and alternative weighting systems for identifying interaction parameters are presented this model is generalized for the setting where a two way contingency table is available for two or more groups various restricted models can be used to examine possible sources of intergroup heterogeneity in the association these sources pertain to heterogeneity in the intrinsic association and or in the scores for the row and column variables the importance of weights used to identify the row and column scores is emphasized a classical set of data previously analyzed by many authors is used to illustrate the advantages of the models and methods developed here

a latent trait finite mixture model for the analysis of rating agreement this article presents a latent distribution model for the analysis of agreement on dichotomous or ordered category ratings the model includes parameters that characterize bias category definitions and measurement error for each rater or test parameter estimates can be used to evaluate rater performance and to improve classification or measurement with use of multiple ratings a simple maximum likelihood estimation procedure is described two examples illustrate the approach although considered in the context of analyzing rater agreement the model provides a general approach for mixture analysis using two or more ordered caregory measures

on cones and stability abstract lyapunov while studying the asymptotic stability of solutions of differential systems proved a theorem which yields a necessary and sufficient condition for stability of a matrix lyapunov s theorem was a breakthrough in the research of stability among others it links cones and stability in various directions in this survey we review some of the links between convexity and matrix stability mainly such links that are derived from lyapunov s theorem the article does not cover the wide variety of all such links but rather gives some of the flavor of this rich theory

hodge genera of algebraic varieties ii we study the behavior of hodge genera under algebraic maps we prove thatthemotivic c y genussatisfiesthe stratifiedmultiplicativeproperty whichshows how to compute the invariant of the source of a morphism from its values on varieties arising from the singularities of the map by considering morphisms to a curve we obtain a hodge theoretic version of the riemann hurwitz formula we also study the monodromy contributions to the y genus of a family of compact complex manifolds and prove an atiyah meyer type formula in the algebraic and analytic contexts this formula measures the deviation from multiplicativity of the y genus and expresses the correction terms as higher genera associated to the period map these higher gen era are hodge theoretic extensions of novikov higher signatures to analytic and alge braic settings characteristic class formulae of atiyah meyer type are also obtained by making use of saito s theory of mixed hodge modules

on certain multi variable rational identities derived from the rigidity of signature of manifolds abstract song derives certain multi variable rational identities by studying torus actions on some homogeneous manifolds and applying the atiyah bott segal singer lefschetz fixed point theorem in this paper we give a direct proof of these rational identities by using the q lucas theorem moreover we also give a similar new rational identity

a pair of matrices sharing common lyapunov solutions a closer look abstract let a b be a pair of matrices with regular inertia if ha a h and hb b h are both positive definite for some hermitian matrix h then all matrices in conv a a 1 b b 1 have identical regular inertia this in turn implies that both conv a b and conv a b 1 consist of non singular matrices in general neither of the converse implications holds in this paper we seek situations where they do hold in particular when a and b are real 2 2 matrices several aspects of the above statements for n n matrices are discussed a connection to the characterization of the convex hull of matrices with regular inertia is introduced differences between the real and the complex case are indicated

ordinal association in contingency tables some interpretive aspects abstract two families of models for ordered contingency tables goodman s association models and canonical correlation models are investigated and compared with respect to the interpretation of their parameters we show that the two families of models actually refer to different kinds of ordinal association stochastic order extremity for correlation models and stochastic order entropy for association models this difference is related to the way the two models scale interaction the scale difference is proven to be of substantial consequence especially under strong association

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

convex invertible cones of matrices a unified framework for the equations of sylvester lyapunov and riccati abstract convex cones of matrices which are closed under matrix inversion were defined and various connections with the algebraic lyapunov equation of general inertia were studied in n cohen i lewkowicz linear algebra and its appl 250 1997 105 131 here in the price of doubling the size of the matrices involved we introduce a unified framework for the equations of sylvester lyapunov and riccati this enables one to extend the convex invertible cone structure to all these three equations and explore related properties in particular the use of the matrix sign function for solving these equations is examined

a characterization of convex cones of matrices with constant regular inertia abstract let a be a convex cone of n n matrices in this paper we present a necessary and sufficient condition for a to contain matrices with a constant regular inertia based on a version of the lyapunov equation the condition involves only the normalized extreme points of a this extends a previous paper by the authors where a robust stability criterion for a was obtained

canonical analysis of contingency tables by maximum likelihood abstract canonical analysis has often been employed instead of log linear models to analyze the relationship of two polytomous random variables however until the last few years analysis has been informal in this article models are examined that place nontrivial restrictions on the values of the canonical parameters so that a parsimonious description of association is obtained maximum likelihood is used to obtain parameter estimates for these restricted models approximate confidence intervals are derived for parameters and chi squared tests are used to check adequacy of models the resulting models may be used to determine the appropriateness of latent class analysis or to determine whether a set of canonical scores has specified patterns results are illustrated through analysis of two tables previously analyzed in the statistical literature comparisons are made with alternate methods of analysis based on a log linear parameterization of cell probabilities it is shown that canonical analysis wh

chern numbers for singular varieties and elliptic homology a fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones intersection homology for example directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety minimal model theory suggests the possibility of working more indirectly by relating any singular variety to a variety which is smooth or nearly so here we use ideas from minimal model theory to define some characteristic numbers for singular varieties generalizing the chern numbers of a smooth variety this was suggested by goresky and macpherson as a next natural problem after the definition of intersection homology 11 we find that only a subspace of the chern numbers can be defined for singular varieties a convenient way to describe this subspace is to say that a smooth variety has a fundamental class in complex bordism whereas a singular variety can at most have a fundamental class in a weaker homology theory elliptic homology we use this idea to give an algebro geometric definition of elliptic homology complex bordism modulo flops equals elliptic homology this paper was inspired by some questions asked by jack morava the descriptions of elliptic homology given by gerald hohn 13 were also an important influence thanks to dave bayer mike stillman john stembridge sheldon katz and stein stromme for their computer algebra programs macaulay sf and schubert which helped in guessing the right answer

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

chern classes of automorphic vector bundles 1 1 suppose x is a compact n dimensional complex manifold each partition i i1 i2 ir of n corresponds to a chern number c x ci1 x ci2 x cir x x z where c x h x z are the chern classes of the tangent bundle x h2n x z is the fundamental class and h0 x z z is the augmentation many invariants of x such as its complex cobordism class may be expressed in terms of its chern numbers mi st during the last 25 years characteristic classes of singular spaces have been defined in a variety of contexts whitney classes of euler spaces su h t ak todd classes of singular varieties bfm chern classes of singular algebraic varieties mac l classes of stratified spaces with even codimension strata gm1 wu classes of singular spaces go2 gp to name a few however these characteristic classes are invariably homology classes and as such they cannot be multiplied with each other in some cases it has been found possible to lift these classes from homology to intersection homology where some characteristic numbers may be formed bbf bw go2 gp t the case of locally symmetric spaces is particularly interesting suppose is a torsionfree arithmetic group acting on a complex n dimensional hermitian symmetric domain d g k where g is the group of real points of a semisimple algebraic group g defined over q with g q and where k g is a maximal compact subgroup then x d is a hermitian locally symmetric space when x is compact hirzebruch s proportionality theorem hr1 says that there is a number v q so that for every partition i i1 i2 ir of n the chern number satisfies c x v c where gu k is the compact dual symmetric space and gu is a compact real form of g c containing k if x d is noncompact it has a canonical baily borel satake compactification x this is a usually highly singular complex projective algebraic variety to formulate a proportionality theorem in the noncompact case one might hope that the tangent bundle tx extends as a complex vector bundle over x but this is false in mu1 d mumford showed that tx has a particular extension e x over any toroidal resolution x x of the baily borel compactification and that for any partition i of n the resulting chern numbers c e c e c e c e x 1 1 1

elliptic genera and quantum field theory it is shown that in elliptic cohomology as recently formulated in the mathematical literature the supercharge of the supersymmetric nonlinear signa model plays a role similar to the role of the dirac operator ink theory this leads to several insights concerning both elliptic cohomology and string theory some of the relevant calculations have been done previously by schellekens and warner in a different context

generalized bilinear models generalized bilinear models are presented for the statistical analysis of two way arrays these models combine bilinear models and generalized linear modeling and yield a family of models that includes many existing models as well as suggest other potentially useful ones this approach both unifies and extends models for two way arrays including the ability to treat response and explanatory variables differently in the models and the incorporation of external information about the variables directly into the analysis a unifying framework for the generalized bilinear models is provided by considering four particular cases which have been proposed and used in the existing statistical literature a three step procedure is proposed to analyze data sets by generalized bilinear models two data sets of different nature are analyzed

convex invertible cones and the lyapunov equation abstract convex cones of matrices which are closed under matrix inversion are defined and their structure is studied various connections with the algebraic lyapunov equation of general inertia are explored

quasi symmetric latent class models with application to rater agreement suppose we observe responses on several categorical variables having the same scale we consider latent class models for the joint classification that satisfy quasi symmetry the models apply when subject specific response distributions are such that i for a given subject responses on different variables are independent and ii odds ratios comparing marginal distributions of the variables are identical for each subject these assumptions are often reasonable in modeling multirater agreement when a sample of subjects is rated independently by different observers in this application the model parameters describe two components of agreement strength of association between classifications by pairs of observers and degree of heterogeneity among the observers marginal distributions we illustrate the models by analyzing a data set in which seven pathologists classified 118 subjects in terms of presence or absence of carcinoma yielding seven categorical classifications with the same binary scale a good fitting model has a latent classification that differentiates between subjects on whom there is agreement and subjects on whom there is disagreement

invariance and factorial models summary two factors having the same set of levels are said to be homologous this paper aims to extend the domain of factorial models to designs that include homologous factors in doing so it is necessary first to identify the characteristic property of those vector spaces that constitute the standard factorial models we argue here that essentially every interesting statistical model specified by a vector space is necessarily a representation of some algebraic category logical consistency of the sort associated with the standard marginality conditions is guaranteed by category representations but not by group representations marginality is thus interpreted as invariance under selection of factor levels z representations and invariance under replication of levels s representations for designs in which each factor occurs once the representations of the product category coincide with the standard factorial models for designs that include homologous factors the set of srepresentations is a subset of the z representations it is shown that symmetry and quasi symmetry are representations in both senses but that not all representations include the constant functions intercept the beginnings of an extended algebra for constructing general z representations is described and illustrated by a diallel cross design

stringy chern classes of singular varieties motivic integration m kontsevich motivic integration lecture at orsay 1995 and macpherson s transformation r macpherson chern classes for singular varieties ann of math 100 1974 423 432 are combined in this paper to construct a theory of stringy chern classes for singular varieties these classes enjoy strong birational invariance properties and their definition encodes data coming from resolution of singularities the singularities allowed in the theory are those typical of the minimal model program examples are given by quotients of manifolds by finite groups for the latter an explicit formula is proven assuming that the canonical line bundle of the manifold descends to the quotient this gives an expression of the stringy chern class of the quotient in terms of chern schwartz macpherson classes of the fixed point set data

on the stability of a convex set of matrices in this paper we give an alternative proof of the constant inertia theorem for convex compact sets of complex matrices it is shown that the companion matrix whose non trivial column is negative satisfies the directional lyapunov condition inclusion for real multiplier vectors an example of a real matrix polytope that satisfies the directional lyapunov condition for real multiplier vectors and which has nonconstant inertia is given a new stability criterion for convex compact sets of real z matrices is given this criterion uses only real vectors and positive definite diagonal matrices

convex invertible cones and positive real analytic functions in this paper we study classical spaces of analytic functions which are convex cones and closed under the involution f 1 f these include the spaces of positive real positive real odd and strictly positive real functions these spaces are associated in the engineering literature with energy dissipation in the sense of lyapunov we discuss the geometric aspects of these spaces in analogy with similar spaces in matrix theory which are also related to the matrix lyapunov equation stability and in view of a general theory of convex invertible cones

using association models to analyse agreement data two examples two examples demonstrate how one can use association models to analyse agreement data the first example concerns intra rater variability in the classification of sputum cytology slides and the second deals with variability associated with the reporting of passive smoking histories the paper emphasizes models in which one estimates category scores from the data that is models that are not in the log linear family of models such models have use in assessment of category distinguishability and provide insights not easily obtained with log linear models

necessary and sufficient conditions for the hurwitz and schur stability of interval matrices establishes a set of new sufficient conditions for the hurwitz and schur stability of interval matrices the authors use these results to establish necessary and sufficient conditions for the hurwitz and schur stability of interval matrices the authors relate the above results to the existence of quadratic lyapunov functions for linear time invariant systems with interval valued coefficient matrices using the above results the authors develop an algorithm to determine the hurwitz and the schur stability properties of interval matrices the authors demonstrate the applicability of their results by means of two specific examples

convex invertible sets and matrix sign function it is proved that in two dimensional unital real algebras a step by step construction of minimal convex invertible sets terminates in two steps an analogous construction of minimal convex conditionally invertible sets of the complex field also terminates in two steps and the complex field is the only finite dimensional complex algebra with this property connections are made with the matrix sign function the matrix sign function of a matrix belongs to the convex invertible set generated by the matrix an example is given to show that the matrix sign function cannot always be obtained in two steps by following the step by step construction

algebraic elliptic cohomology theory and flops i we define the algebraic elliptic cohomology theory coming from krichever s elliptic genus as an oriented cohomology theory on smooth varieties over an arbitrary perfect field we show that in the algebraic cobordism ring with rational coefficients the ideal generated by differences of classical flops coincides with the kernel of krichever s elliptic genus this generalizes a theorem of b totaro in the complex analytic setting

hodge genera of algebraic varieties i the aim of this paper is to study the behavior of hodge theoretic intersection homology genera and their associated characteristic classes under proper morphisms of complex algebraic varieties we obtain formulae that relate parametrized families of global invariants of a complex algebraic variety x to such invariants of singularities of proper algebraic maps defined on x such formulae severely constrain both topologically and analytically the singularities of complex maps even between smooth varieties similar results were announced by the first and third author in 13 32 2007 wiley periodicals inc

geometric representations of the formal affine hecke algebra abstract for any formal group law there is a formal affine hecke algebra defined by hoffnung malagon lopez savage zainoulline coming from this formal group law there is also an oriented cohomology theory we identify the formal affine hecke algebra with a convolution algebra coming from the oriented cohomology theory applied to the steinberg variety as a consequence this algebra acts on the corresponding cohomology of the springer fibers this generalizes the action of classical affine hecke algebra on the k theory of the springer fibers constructed by lusztig we also give a residue interpretation of the formal affine hecke algebra which generalizes the residue construction of ginzburg kapranov vasserot when the formal group law comes from a 1 dimensional algebraic group

mckay correspondence for elliptic genera we establish a correspondence between orbifold and singular elliptic genera of a global quotient while the former is defined in terms of the fixed point set of the action the latter is defined in terms of the resolution of singularities as a byproduct the second quantization formula of dijkgraaf moore verlinde and verlinde is extended to arbitrary kawamata log terminal pairs

dualization invariance and a new complex elliptic genus we define a new elliptic genus on the complex bordism ring with co efficients in z 1 2 we prove that it induces an isomorphism of the complex bordism ring modulo the ideal which is generated by all differences p e p e of projective bundles and their duals onto a polynomial ring on 4 generators in degrees 2 4 6 and 8 as an alternative geometric description of we prove that it is the universal genus which is multiplicative in projective bundles over calabi yau 3 folds with the help of the q expansion of modular forms we will see that for a complex manifold m the value m is a holomorphic euler characteristic we also compare with krichever h complex elliptic genus and see that their only common specializations are ochanine s elliptic genus and the y genus

quasisymmetric models for the analysis of square contingency tables summary a parametric model is presented for the analysis of square contingency tables where there is a one to one correspondence between the categories of the row variable and the categories of the column variable the model is applicable to cross classifications where the categories are unordered partially ordered or completely ordered but the interpretation of the model and associated parameters is most straightforward when the categories are at least partially ordered connections with the mover stayer model and the usual model of quasisymmetry are described two data sets are used to illustrate the utility of the model