latent structure analysis of a set of multidimensional contingency tables abstract statistical methods are introduced for latent structure analysis of a set of two or more multidimensional contingency tables three basic classes of models are considered a models that assume complete homogeneity across tables b models that allow partial homogeneity across tables and c models that allow complete heterogeneity methods are presented for testing whether these models are congruent with the data in the tables and for assessing the significance of differences among the tables in the estimated parameters to illustrate the wide applicability of these models and methods we present analyses of two quite different sets of data

an approach to n mode components analysis as an extension of lastovicka s four mode components analysis ann mode components analysis is developed using a convenient notation both a canonical and a least squares solution are derived the relation between both solutions and their computational aspects are discussed

multiplicity of solutions for a fourth order problem with exponential nonlinearity abstract let b be the unit ball in r n n 5 and n be the exterior unit normal vector on the boundary we consider radial solutions to 2 u e u in b u 0 and u n 0 on b where 0 we show that there exists a unique s 0 such that if s there is a radial singular solution if 5 n 12 then for s there exist infinitely many regular radial solutions and as s the number of such solutions goes to infinity if n 13 we prove uniqueness of smooth radial solutions we derive similar results for the same equation with navier boundary conditions

existence and stability of entire solutions to a semilinear fourth order elliptic problem for a semilinear biharmonic equation with exponential nonlinearity we study the existence and the asymptotic behavior of entire solutions furthermore their stability and stability outside a compact set of rn n 2 is discussed in any space dimension n

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

fixed distance latent class models for the analysis of sets of two way contingency tables this paper develops and describes the application of modified latent class models for analyzing sets of two way contingency tables the proposed fixed distance models differ from traditional latent class models in that the assumption of local stochastic independence is superseded by allowing interactions of the manifest variables within each class which can be represented by a single association parameter as an example two data sets on eye color by hair color collected in caithness n1 5 387 and aberdeen n2 22 361 and fixed distance models with up to six classes three classes per data set are considered finally leading to satisfactory fit and rather simple interpretation

the analysis of three way contingency tables by three mode association models therc m association model goodman 1979 1985 1986 1991 is useful for analyzing the relationship between the variables of a 2 way cross classification the models presented here are generalizations of therc m association model for 3 way tables the family of models proposed here 3 mode association models use tucker s 3 mode components model tucker 1964 1966 kroonenberg 1983 to represent either the three factor interaction or the combined effects of two and three factor interactions an example from a study in developmental psychology kramer gottman 1992 is provided to illustrate the usefulness of the proposed models

some remarks on embedding of weighted sobolev spaces abstract in this paper we generalize the well known caffarelli kohn nirenberg inequality to the fourth order case then by using such embeddings we obtain the existence of positive solutions of navier boundary value problems of a weighted fourth order elliptic equation

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

existence of stable solutions to δ mu eu in rn with m 3 and n 2m abstract we consider the polyharmonic equation m u e u in r n with m 3 and n 2 m we prove the existence of many entire stable solutions this answers some questions raised by farina and ferrero in 7

entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity we investigate entire radial solutions of the semilinear biharmonic equation 2 u exp u in r n n 5 0 being a parameter we show that singular radial solutions of the corresponding dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of r n in particular they cannot be expanded as power series in the natural variable s log x next we prove the existence of infinitely many entire regular radial solutions they all diverge to as x and we specify their asymptotic behaviour as in the case with power type nonlinearities f gazzola h ch grunau radial entire solutions for supercritical biharmonic equations math ann 334 2006 905 936 the entire singular solution x 4l og x plays the role of a separatrix in the bifurcation picture finally a technique for the computer assisted study of a broad class of equations is developed it is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of odes in the case n 5

age patterns of marital fertility revising the coale trussell method abstract this article revises the coale trussell method for analyzing data from the world fertility survey by proposing and testing alternative log linear and log multiplicative models the models in one form or another represent the structural constraint underlying the coale trussell method on the variation in the age pattern of human fertility with a poisson distribution assumption for the number of births several parameters of the models are simultaneously estimated via maximum likelihood it is shown that the new approach can be adopted whenever fertility limitation is compared across multiple populations or subpopulations future users of the coale trussell method for single populations or subpopulations are advised to use the new v a estimates from this study in place of those of coale and trussell

optimal hardy rellich inequalities maximum principle and related eigenvalue problem in this paper we deal with three types of problems concerning the hardy rellich s embedding for a bilaplacian operator first we obtain the hardy rellich inequalities in the critical dimension n 4 then we derive a maximum principle for fourth order operators with singular terms then we study the existence non existence simplicity and asymptotic behavior of the first eigenvalue of the hardy rellich operator 2 n 2 n 4 2 16 q x x 4 under various assumptions on the perturbation q

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

qualitative behavior of global solutions to some nonlinear fourth order differential equations abstract we study global solutions to a fourth order semilinear ordinary differential equation we determine sufficient conditions on the nonlinearity that ensure global continuation of the solutions furthermore we discuss their qualitative behaviors such as oscillations and boundedness

obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory branched covers of orbit cylinders are the basic examples of holomorphic curves studied in symplectic field theory since all curves with fredholm index one can never be regular for any choice of cylindrical almost complex structure we generalize the obstruction bundle technique of taubes for determining multiple cover contributions from gromov witten theory to the case of moduli spaces with boundary our result proves that the differential in rational symplectic field theory and contact homology is strictly decreasing with respect to the natural action filtration

gravitational descendants in symplectic field theory it was pointed out by y eliashberg in his icm 2006 plenary talk that the rich algebraic formalism of symplectic field theory leads to a natural appearance of quantum and classical integrable systems at least in the case when the contact manifold is the prequantization space of a symplectic manifold in this paper we generalize the definition of gravitational descendants in sft from circle bundles in the morse bott case to general contact manifolds after we have shown using the ideas in okounkov and pandharipande ann math 163 2 517 560 2006 that for the basic examples of holomorphic curves in sft that is branched covers of cylinders over closed reeb orbits the gravitational descendants have a geometric interpretation in terms of branching conditions we follow the ideas in cieliebak and latschev http arixiv org abs 0706 3284v2 math s6 2007 to compute the corresponding sequence of poisson commuting functions when the contact manifold is the unit cotangent bundle of a riemannian manifold

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

graphical representations and odds ratios in a distance association model for the analysis of cross classified data although rc m association models have become a generally useful tool for the analysis of cross classified data the graphical representation resulting from such an analysis can at times be misleading the relationships present between row category points and column category points cannot be interpreted by inter point distances but only through projection in order to avoid incorrect interpretation by a distance rule joint plots should be made that either represent the row categories or the column categories as vectors in contrast the present study proposes models in which the distances between row and column points can be interpreted directly with a large small distance corresponding to a small large value for the association the models provide expressions for the odds ratios in terms of distances which is a feature that makes the proposed models attractive reparametrizations to the usual rc m parametrization comparisons to existing data analysis techniques plus an overview of related models and their connections are also provided

a single general method for the analysis of cross classified data reconciliation and synthesis of some methods of pearson yule and fisher and also some methods of correspondence analysis and association analysis 1 2 3 4 5 6 7 8 9 10 contents introduction and summary a general measure of nonindependence for the cells in the table a general method of nonindependence analysis some simple examples some special cases of generalized nonindependence analysis standardized generalized nonindependence analysis overall indices and graphical displays of generalized nonindependence saturated and unsaturated models symmetric nonindependence and generalized quasi symmetry closing comments on 2 x 2 table measures of nonindependence and some brief historical notes

radial symmetry of entire solutions of a bi harmonic equation with exponential nonlinearity abstract we obtain necessary and sufficient conditions for an entire solution u of a bi harmonic equation with exponential nonlinearity e u to be a radially symmetric solution the standard tool to obtain the radial symmetry for a system of equations is the moving plane method mpm in order to apply the mpm we need to know the asymptotic expansions of u and u at we overcome the difficulties due to the fact that e u is supercritical for n 5 e u l n 4 r n and get the right decay rate of u and u at in order to start the moving plane procedure

correspondence analysis used complementary to loglinear analysis loglinear analysis and correspondence analysis provide us with two different methods for the decomposition of contingency tables in this paper we will show that there are cases in which these two techniques can be used complementary to each other more specifically we will show that often correspondence analysis can be viewed as providing a decomposition of the difference between two matrices each following a specific loglinear model therefore in these cases the correspondence analysis solution can be interpreted in terms of the difference between these loglinear models a generalization of correspondence analysis recently proposed by escofier will also be discussed with this decomposition which includes classical correspondence analysis as a special case it is possible to use correspondence analysis complementary to loglinear analysis in more instances than those described for classical correspondence analysis in this context correspondence analysis is used for the decomposition of the residuals of specific restricted loglinear models

the gel fand problem for the biharmonic operator we study stable and finite morse index solutions of the equation delta 2 u e u if the equation is posed in mathbb r n we classify radial stable solutions we then construct nonradial stable solutions and we prove that unlike the corresponding second order problem no liouville type theorem holds unless additional information is available on the asymptotics of solutions at infinity thanks to this analysis we prove that stable solutions of the equation on a smoothly bounded domain supplemented with navier boundary conditions are smooth if and only if n leqq 12 we find an upper bound for the hausdorff dimension of their singular set in higher dimensions and conclude with an a priori estimate for solutions of bounded morse index provided they are controlled in a suitable morrey norm

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

exploratory analysis of contingency tables by loglinear formulation and generalizations of correspondence analysis goodman s 1979 1981 1985 loglinear formulation for bi way contingency tables is extended to tables with or without missing cells and is used for exploratory purposes a similar formulation is done for three way tables and generalizations of correspondence analysis are deduced a generalized version of goodman s algorithm based on newton s elementary unidimensional method is used to estimate the scores in all cases

on the bayesian analysis of population size summary we consider the problem of estimating the total size of a population from a series of incomplete census data we observe that inference is typically highly sensitive to the choice of model and we demonstrate how bayesian model averaging techniques easily overcome this problem we combine and extend the work of madigan york 1997 and dellaportas forster 1999 using reversible jump markov chain monte carlo simulation to calculate posterior model probabilities which can then be used to estimate modelaveraged statistics of interest we provide a detailed description of the simulation procedures involved and consider a wide variety of modelling issues such as the range of models considered their parameterisation both prior choice and sensitivity and computational efficiency we consider a detailed example concerning adolescent injuries in pennsylvania on the basis of medical school and survey data in the context of this example we discuss the relationship between posterior model probabilities and the associated information criteria values for model selection we also discuss cost efficiency issues with particular reference to inclusion and exclusion of sources on the grounds of cost we consider a decision theoretic approach which balances the cost and accuracy of different combinations of data sources to guide future decisions on data collection

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

gee for multinomial responses using a local odds ratios parameterization in this article we propose a generalized estimating equations gee approach for correlated ordinal or nominal multinomial responses using a local odds ratios parameterization our motivation lies upon observing that i modeling the dependence between correlated multinomial responses via the local odds ratios is meaningful both for ordinal and nominal response scales and ii ordinary gee methods might not ensure the joint existence of the estimates of the marginal regression parameters and of the dependence structure to avoid ii we treat the so called working association vector as a nuisance parameter vector that defines the local odds ratios structure at the marginalized contingency tables after tabulating the responses without a covariate adjustment at each time pair to estimate and simultaneously approximate adequately possible underlying dependence structures we employ the family of association models proposed by goodman in simulations the parameter estimators with the proposed gee method for a marginal cumulative probit model appear to be less biased and more efficient than those with the independence working model especially for studies having time varying covariates and strong correlation

coherent orientations in symplectic field theory we study the coherent orientations of the moduli spaces of holomorphic curves in symplectic field theory generalizing a construction due to floer and hofer in particular we examine their behavior at multiple closed reeb orbits under change of the asymptotic direction the orientations are determined by a certain choice of orientation at each closed reeb orbit that is similar to the orientation of the unstable tangent spaces of critical points in finite dimensional morse theory

some additional results on principal components analysis of three mode data by means of alternating least squares algorithms kroonenberg and de leeuw 1980 have developed an alternating least squares method tuckals 3 as a solution for tucker s three way principal components model the present paper offers some additional features of their method starting from a reanalysis of tucker s problem in terms of a rank constrained regression problem it is shown that the fitted sum of squares in tuckals 3 can be partitioned according to elements of each mode of the three way data matrix an upper bound to the total fitted sum of squares is derived finally a special case of tuckals 3 is related to the carroll harshman candecomp parafac model

the analysis of multivariate contingency tables by restricted canonical and restricted association models abstract restricted canonical models and restricted association models are proposed and applied to multiway contingency tables these models have been previously applied to two way contingency tables however multivariate generalization has been impeded in the past since canonical and association models both depend on singular value decompositions that apply only to two way arrays in this article this restriction to two way arrays is overcome by division of the cross classified variables into explanatory and response variables the explanatory variables are treated as a single polytomous variable and the response variables are treated as a second single polytomous variable in this fashion the multiway table is reduced to a two way array to which traditional canonical and association models may be applied use of linear restrictions on parameters in canonical and association models is especially important in multiway tables if useful models are to be constructed the class of models considered in th

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