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

compactified picard stacks over the moduli stack of stable curves with marked points abstract in this paper we give a construction of algebraic artin stacks endowed with a modular map onto the moduli stack of stable curves of genus g with n marked points the stacks we construct are smooth irreducible and have dimension 4 g 3 n yielding a geometrically meaningful compactification of the universal picard stack parametrizing n pointed smooth curves together with a line bundle

models for the analysis of association in multivariate contingency tables abstract the present work describes a family of models that can be used for the analysis of association in multiway cross classifications the models are complementary to the usual i e the so called hierarchical log linear models an advantage of this family of models is that it is frequently possible to give a more parsimonious description of the association than is possible with the usual log linear models approach the models will be especially useful when some or all of the variables are ordinal the models are derived from a reparameterization of the saturated model based on singular value decompositions of interaction terms in the log linear parameterization multivariate association models given in goodman 1979 1986 clogg 1982a b agresti and kezouh 1983 gilula and haberman 1988 and becker and clogg 1989 are obtained as special cases the utility of the models developed in this article is demonstrated with the analysis of two examples the first example a cross classification of

multivariate normal integrals and contingency tables with ordered categories ak dimensional multivariate normal distribution is made discrete by partitioning thek dimensional euclidean space with rectangular grids the collection of probability integrals over the partitioned cubes is ak dimensional contingency table with ordered categories it is shown that loglinear model with main effects plus two way interactions provides an accurate approximation for thek dimensional table the complete multivariate normal integral table is computed via the iterative proportional fitting algorithm from bivariate normal integral tables this approach imposes no restriction on the correlation matrix comparisons with other numerical integration algorithms are reported the approximation suggests association models for discretized multivariate normal distributions and contingency tables with ordered categories

graphs related to held s simple group abstract we analyze the permutation representations of low degree of held s simple group he we also determine its primitive multiplicity free permutation representations and show that there is no graph on which it or its automorphism group acts as a distance transitive group of automorphisms in doing so we supply a computer free construction of he

a computer free construction of j4 define a finite simple group j to be of j4 type or simply j4 provided that j contains an involution z withcjz 21 12 3aut mat22 the purpose of this paper is to give the first computer free construction of a group of j4 type in addition we achieve yet another uniqueness proof for groups of j4 type via the simple connectedness of the 2 local geometry of such a group

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

conditionally specified continuous distributions a distribution is conditionally specified when its model constraints are expressed conditionally for example besag s 1974 spatial model was specified conditioned on the neighbouring states and pseudolikelihood is intended to approximate the likelihood using conditional likelihoods there are three issues of interest existence uniqueness and computation of a joint distribution in the literature most results and proofs are for discrete probabilities here we exclusively study distributions with continuous state space we examine all three issues using the dependence functions derived from decomposition of the conditional densities we show that certain dependence functions of the joint density are shared with its conditional densities therefore two conditional densities involving the same set of variables are compatible if their overlapping dependence functions are identical we prove that the joint density is unique when the set of dependence functions is both compatible and complete in addition a joint density apart from a constant can be computed from the dependence functions in closed form since all of the results are expressed in terms of dependence functions we consider our approach to be dependence based whereas methods in the literature are generally density based applications of the dependence based formulation are discussed

the numerical evaluation of certain multivariate normal integrals abstract we show that a multivariate normal integral with tridiagonal covariance matrix can be computed efficiently using iterated integration

on asymptotic power and efficiency of tests of independence in contingency tables with ordered classifications abstract asymptotic power formulas for tests of independence against simple association models in i j contingency tables with ordered categories are presented yates s score correlation test and spearman s tied rank correlation test are compared with a likelihood ratio test proposed by s j haberman for testing independence in log linear models with ordered classifications asymptotic power and relative efficiencies are derived under local alternatives that need not be included in the likelihood model assumed by haberman s procedure some computations for real and fictitious data are presented

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

towards the geometry of double hurwitz numbers abstract double hurwitz numbers count branched covers of cp 1 with fixed branch points with simple branching required over all but two points 0 and and the branching over 0 and specified by partitions of the degree with m and n parts respectively single hurwitz numbers or more usually hurwitz numbers have a rich structure explored by many authors in fields as diverse as algebraic geometry symplectic geometry combinatorics representation theory and mathematical physics the remarkable elsv formula relates single hurwitz numbers to intersection theory on the moduli space of curves this connection has led to many consequences including okounkov and pandharipande s proof of witten s conjecture in this paper we determine the structure of double hurwitz numbers using techniques from geometry algebra and representation theory our motivation is geometric we give evidence that double hurwitz numbers are top intersections on a moduli space of curves with a line bundle a universal picard variety in particular we prove a piecewise polynomiality result analogous to that implied by the elsv formula in the case m 1 complete branching over one point and n is arbitrary we conjecture an elsv type formula and show it to be true in genus 0 and 1 the corresponding witten type correlation function has a richer structure than that for single hurwitz numbers and we show that it satisfies many geometric properties such as the string and dilaton equations and an itzykson zuber style genus expansion ansatz we give a symmetric function description of the double hurwitz generating series which leads to explicit formulae for double hurwitz numbers with given m and n as a function of genus in the case where m is fixed but not necessarily 1 we prove a topological recursion on the corresponding generating series which leads to closed form expressions for double hurwitz numbers and an analogue of the goulden jackson polynomiality conjecture an early conjectural variant of the elsv formula in a later paper faber s intersection number conjecture and genus 0 double hurwitz numbers 2005 in preparation the formulae in genus 0 will be shown to be equivalent to the formulae for top intersections on the moduli space of smooth curves m g for example three formulae we give there will imply faber s intersection number conjecture in moduli of curves and abelian varieties aspects of mathematics vol e33 vieweg braunschweig 1999 pp 109 129 in arbitrary genus with up to three points

the stability space of compactified universal jacobians in this paper we describe compactified universal jacobians i e compactifications of the moduli space of line bundles on smooth curves obtained as moduli spaces of rank 1 torsion free sheaves on stable curves using an approach due to oda seshadri we focus on the combinatorics of the stability conditions used to define compactified universal jacobians we explicitly describe an affine space the stability space with a decomposition into polytopes such that each polytope corresponds to a proper deligne mumford stack that compactifies the moduli space of line bundles we apply this description to describe the set of isomorphism classes of compactified universal jacobians answering a question of melo and to resolve the indeterminacy of the abel jacobi sections addressing a problem raised by grushevsky zakharov

association models and canonical correlation in the analysis of cross classifications having ordered categories abstract the association models considered in goodman 1979a for the analysis of cross classifications having ordered categories are presented in a somewhat different form in the present article to facilitate comparison of the results obtained using these models with those obtained using the earlier canonical correlation approach both the association models and the canonical correlation approach can provide meaningful scores for the row and column categories and these scores can be used to partition into relevant components the usual chi squared statistic for testing the null hypothesis of statistical independence between the row classification and column classification however while the usual procedure for testing the statistical significance of chi squared components is invalid when these components are based on the canonical correlations a corresponding procedure is valid when these components are obtained with the association models the components of association obtained with the association mo

neron models and compactified picard schemes over the moduli stack of stable curves we construct modular deligne mumford stacks representable over parametrizing neron models of jacobians as follows let b be a smooth curve and k its function field let be a smooth genus g curve over k admitting stable minimal model over b the neron model b is then the base change of via the moduli map b of f i e b moreover is compactified by a deligne mumford stack over giving a completion of neron models naturally stratified in terms of neron models

a computer free construction of the third group of janko abstract we give a new constructive computer free existence proof of the third group of janko j 3 using the 9 dimensional unitary representation of 3 j 3 over gf 4 thereby we embed the c 2 c geometries for 3 j 3 and j 3 into the 9 dimensional unitary space and the 9 dimensional projective space respectively

a compactification over overline _ of the universal moduli space of slope semistable vector bundles 0 introduction 425 1 the quotient construction 428 2 the fiberwise g i t problem 433 3 cohomology bounds 435 4 slope unstable torsion free sheaves 437 5 special torsion bounded sheaves 440 6 slope semistable torsion free sheaves 443 7 two results in geometric invariant theory 449 8 the construction of ug e r 454 9 basic properties of ug e r 458 10 the isomorphism between ug e 1 and pg e 464 references 470

gromov witten gauge theory we introduce a modular completion of the stack of maps from stable marked curves to the quotient stack pt c x and use this stack to construct some gauge theoretic analogues of the gromov witten invariants we also indicate the generalization of these invariants to the quotient stacks x c x where x is a smooth proper complex algebraic variety

the probability integrals of bivariate normal distributions a contingency table approach summary after partitioning the plane into fine rectangular regions the bivariate normal probability integrals over all the rectangular regions can be considered as a population contingency table the row and column marginals of this contingency table are two discretized univariate normal distributions the logarithm of the local cross product ratios are approximately linear given the correlation coefficient of the bivariate normal a contingency table with discretized normal marginals and proper cross product ratios can be constructed which represents a table of probability integrals for the bivariate normal

compactified picard stacks over overline mathcal m _g we study algebraic artin stacks over overline mathcal m g giving a functorial way of compactifying the relative degree d picard variety for families of stable curves we also describe for every d the locus of genus g stable curves over which we get deligne mumford stacks strongly representable over overline mathcal m g

extensions of the universal theta divisor abstract the jacobian varieties of smooth curves fit together to form a family the universal jacobian over the moduli space of smooth pointed curves and the theta divisors of these curves form a divisor in the universal jacobian in this paper we describe how to extend these families over the moduli space of stable pointed curves using a stability parameter we then prove a wall crossing formula describing how the theta divisor varies with this parameter we use this result to analyze divisors on the moduli space of smooth pointed curves that have recently been studied by grushevsky zakharov hain and muller finally we compute the pullback of the theta divisor studied in alexeev s work on stable semiabelic varieties and in caporaso s work on theta divisors of compactified jacobians

hurwitz numbers and intersections on moduli spaces of curves 1 department of mathematics university of stockholm s 10691 stockholm sweden e mail teke matematik su se 2 higher college of mathematics independent university of moscow and institute for system research ras moscow russia e mail lando mccme ru 3 department of mathematics royal institute of technology s 10044 stockholm sweden e mail mshapiro math kth se 4 department of mathematics and department of computer science university of haifa haifa 31905 israel e mail alek mathcs haifa ac il

some models for the analysis of association in multiway cross classifications having ordered categories abstract goodman recently presented a class of models for the analysis of association between two discrete ordinal variables the association was measured in terms of the odds ratios in 2 2 subtables formed from adjacent rows and adjacent columns of the cross classification and models were devised that allowed the odds ratios to depend on an overall effect on row effects on column effects and on other effects this article presents some generalizations of this approach appropriate for multiway cross classifications including a models for the analysis of conditional association b models for the analysis of partial association and c models for the analysis of symmetric association three cross classifications are analyzed with these models and methods and rather simple interpretations of the association in each are provided

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