a note on the quadratic exponential binary distribution summary the joint distribution of p binary variables is studied in the quadratic exponential form containing only main effects and two factor interactions in the log probabilities approximate versions of marginalized forms of the distribution are studied based on taylor expansion and a number of conclusions drawn

properly embedded minimal planar domains in 1997 collin proved that any properly embedded minimal surface in r 3 with nite topology and more than one end has

embedded minimal surfaces of finite topology in this paper we prove that any complete embedded minimal surface m in r 3 with finite topology and compact boundary possibly empty is conformally a compact riemann surface m with boundary punctured in a finite number of interior points and that m can be represented in terms of meromorphic data on its conformal completion m in particular we demonstrate that m is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior ofm

embedded minimal surfaces with an infinite number of ends we prove the existence of an infinite family of periodic properly embedded minimal surfaces with an infinite number of annular ends 25 refs 29 figs

on singly periodic minimal surfaces with planar ends the spaces of nondegenerate properly embedded minimal surfaces in quotients of r3 by nontrivial translations or by screw motions with nontrivial rotational part fixed finite topology and planar type ends are endowed with natural structures of finite dimensional real analytic manifolds this nondegeneracy is defined in terms of jacobi functions riemann s minimal examples are characterized as the only nondegenerate surfaces with genus one in their corresponding spaces we also give natural immersions of these spaces into certain complex euclidean spaces which turn out to be lagrangian immersions with respect to the standard symplectic structures

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

degree theory on oriented infinite dimensional varieties and the morse number of minimal surfaces spanning a curve in ℝn the algebraic number of disc minimal surfaces spanning a wire in 3 is defined and shown to be equal to one

uniqueness of the riemann minimal examples we prove that a properly embedded minimal surface in r 3 of genus zero with infinite symmetry group is a plane a catenoid a helicoid or a riemann minimal example we introduce the language of hurwitz schemes to understand the underlying moduli space of surfaces in our setting

tests of hypotheses in overdispersed poisson regression and other quasi likelihood models abstract test statistics for evaluating the significance of added variables in a regression equation are developed for mixed poisson models where the structural parameter that determines the mean variance relationship var 2 is estimated by the method of moments and the regression coefficients are estimated by quasi likelihood the formulas presented for test statistics and related estimating equations are applicable generally to quasi likelihood models specified by an arbitrary mean value function x together with a variance function v that contains one or more unknown parameters two versions of the wald and score tests are investigated one calculated from the usual model based covariance matrix whose validity depends on correct specification of the variance function and another using an empirical covariance matrix that has a more general asymptotic justification monte carlo simulations demonstrate that the quasi likelihood method of moments ql m procedures yield ap

two end solutions to the allen cahn equation in r3 abstract in this paper we study axially symmetric solutions of allen cahn equation in the three dimensional euclidean space using a sophisticated continuation method we show the existence of a complete family of axially symmetric solutions with a range of logarithmic growth rates which may be regarded as the analogue of the family of catenoids and hence called two end solutions nonexistence of two end solution with a small growth rate is also shown which differs from the theory of minimal surfaces

the structure of singly periodic minimal surfaces the only properly embedded minimal surface with more than one topological end and a cont inuous g roup of symmetries is the catenoid considering periodic minimal surfaces with more than one end a surface in n 3 is periodic if it is invariant under an infinite discrete g roup of isometrics that act freely on 1r3 the only classical examples are those discovered by riemann the riemann examples possess a quite special set of properties 1 they have an infinite number of flat annular ends 1 2 they are invariant under a nontrivial t ranslat ion t 3 the surfaces t h a v e genus 1 and 2 ends 4 the surfaces t have total curvature equal to 8 zr mot iva ted by the riemann examples we established the existence of an infinite family of proper ly embedded periodic minimal surfaces d k each with an infinite number of ends 1 these surfaces which are discussed below see sect 1 2 on rigidity also have an infinite number of flat annular ends and are invariant under a t ranslat ion t the quot ient of these surfaces by t is a surface of genus 2 k 1 with two ends and total curvature 8 r e k 1 k l we have been able to twis t each of these new surfaces that is we construct surfaces with the same properties that are invariant under screw motions t r where r is a rota t ion a round an axis in the direction of t see sect 1 3 below together with the catenoid and the examples of riemann these new surfaces comprise all the known properly embedded minimal surfaces with more than one topological end whose symmetry g roup contains an infinite discrete subgroup these examples share m a n y of the properties of the riemann examples a major goal of this paper is to show that this is not accidental

product integration methods for solving a system of nonlinear volterra integral equations in this paper the technique of subtracting out singularities is used to derive explicit and implicit product euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of volterra integral equations with a weakly singular kernel the convergence proofs of the numerical schemes are presented these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global lipschitz condition

on monotone abel volterra integral equations on the half line in this paper we reanalyze the trapezoidal method for the solution of nonlinear abel volterra integral equations on the half line we prove the convergence of the method in the uniform norm provided the nonlinearity is lipschitz continuous and strictly monotone

stability analysis of product t methods for abel integral equations of the second kind stability analysis of product methods for abel integral equations of the second kind based on the test equation y t 1 lambda int limits 0 1 frac y s t s alpha ds t geqq 0 0 0 and we investigate whether this property is inherited by numerical approximations toy

variable transformations in the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels extensions to fredholm integral equations abstract a number of techniques that use variable transformations in numerical integration have been developed recently cf sidi numerical integration iv h brass g hammerlin eds birkhauser basel 1993 pp 359 373 laurie j comput appl math 66 1996 337 344 the use of these transformations resulted in increasing the order of convergence of the trapezoidal and the midpoint quadrature rule in this paper the application of variable transformation techniques of sidi and laurie type to the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels is considered since the transformations are such that the end points of integration need not be used as mesh points the methods introduced can be used for vie with both continuous and weakly singular kernel in a uniform way the methods have also the advantages of simplicity of application and of achieving high order of convergence the application of the idea to fredholm integral equations with continuous and weakly singular equations is also considered numerical results are included and they verify the expected increased order of convergence they were obtained by using the trapezoidal formula for the evaluation of the transformed integrals

mathematical programming methods in the numerical solution of volterra integral and integro differential equations with weakly singular kernel abstract nonlinear volterra integral and integro differential equations with weakly singular kernel are considered and solved numerically using nonlinear mathematical programming methods based on minimax approximations in both cases polynomial and multiquadric approximation are used

exponential mixtures and quadratic exponential families summary conditions are derived under which quadratic and polynomial exponential models can be generated as mixtures of linear exponential models the conditions are highly restrictive for continuous sample spaces but less restrictive in the discrete case some properties of binary quadratic exponential models are explored with a view towards finding models that have properties suitable for epidemiological applications

testing covariance structure in multivariate models application to family disease data abstract recent interest in modeling multivariate responses for members of groups has emphasized the need for testing goodness of fit here we describe a way to test the covariance structure of a multivariate distribution parameterized by a vector the idea is to extend this distribution the null distribution to a more general distribution that depends on an additional scalar and a specific quadratic function of the response vector chosen to capture features of an alternative covariance structure when 0 the more general distribution reduces to the null one standard likelihood theory yields a score test for 0 that is a test of fit of the null distribution the score statistic is the standardized difference between observed and expected values of the quadratic function where the expectation is taken with respect to the null distribution with replaced by its maximum likelihood estimate applying the methods to case control data on familial cancers of the ovary and breast we illu

statistical testing of genetic linkage under heterogeneity summary recent advances in human genetics have led to a renewed interest in statistical methods for the detection of linkage from family data for example between marker loci and disease traits statistical analysis of linkage between two loci is carried out almost exclusively by means of the lod log odds score test equivalent to a likelihood ratio test the current practice is to declare genetic linkage between loci when the maximum lod score exceeds 3 as will be discussed here the lod score approach is not appropriate for the detection of linkage from heterogeneous data e g when families consist of a mixture of linked and unlinked types heterogeneity may arise for example when rare mutations at different genetic loci are responsible for the same disease trait as an alternative approach to account for possible heterogeneity in the detection of linkage we propose the application of largesample test statistics that are members of neyman s class of c av or partial score tests the convergence of the proposed test statistics to their asymptotic distributions is investigated via monte carlo simulation for typical study designs applicable in human genetics

segregation analysis of case control data using generalized estimating equations generalized estimating equations gees liang and zeger 1986 biometrika 73 13 22 are used to fit genetic models to binary disease data for families of subjects in case control studies the gees include model specification of both the disease probabilities and the two way and possibly three way correlation coefficients of the family disease data these quantities are modelled as nonlinear functions of unobserved genotypes observed environmental covariates and the unknown parameters the functions reflect the method used to ascertain the family data goodness of fit is tested by allowing more flexible forms for the correlation coefficients regressing them against covariates specific to the relevant pair or triple of family members the approach is applied to family data obtained from simulated and real case control studies this semiparametric approach is less dependent on unverifiable assumptions and more computationally tractable than other methods for segregation analysis

piecewise continuous collocation for integral equations the collocation method for integral equations of the second kind is surveyed and analyzed for the case in which the approximate solutions are only piecewise continuous difficulties with the usual function space setting of l infty d are discussed and a satisfactory sense of point evaluation is given for elements of l infty d other approaches which are discussed include i reformulation as a degenerate kernel method ii the prolongation restriction framework of noble iii other function space settings and iv reformulation as a continuous approximation problem by iterating the piecewise continuous approximate solution in the original integral equation

stability results for collocation methods for volterra integral equations abstract stability behavior is investigated for polynomial spline collocation applied to volterra integral equations with special emphasis on a weakly singular test equation the characterization of the corresponding stability domain is related to stability results for solutions of finite recursion relations as they occur for methods applied to ordinary differential equations estimates of the stability domains are established and as a consequence conditions for a 2 stability are obtained exploiting these estimates collocation approximations with constant linear and continuous splines are considered in more depth and several numerical illustrations are presented

a mathematical treatment of the fluorescence capillary fill device a mathematical model in the form of two coupled diffusion equations is provided for a competitive chemical reaction between an antigen and a labeled antigen for antibody sites on a cell wall boundary conditions are such that the problem is both nonlinear and nonlocal this is then recharacterized first as a pair of coupled singular integro differential equations and then as a system of four volterra integral equations the latter permits a proof of existence and uniqueness of the solution of the original problem small and large time asymptotic solutions are derived and from the first characterization a regular perturbation solution is obtained numerical schemes are briefly discussed and graphical results are presented for human immunoglobulin

the strong halfspace theorem for minimal surfaces d hoffman and w h meeks iii department of mathematics university of massachusetts amherst ma 01003 usa in 6 jorge and xavier constructed examples of complete nonplanar minimal surfaces contained between two parallel planes in 3 recently rosenberg and toubiana have found complete minimally immersed annuli that are proper in an open slab 1 12 on the basis of these and other results the question has arisen as to whether or not there could exist a properly immersed minimal surface in 3 that was contained between two parallel planes the answer is negative an immediate consequence of the following theorem

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

the numerical solution of weakly singular volterra integral equations by collocation on graded meshes since the solution of a second kind volterra integral equation with weakly singular kernel has in general unbounded derivatives at the left endpoint of the interval of integration its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates in this paper we investigate the convergence rates with respect to graded meshes and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation

the classification of doubly periodic minimal tori with parallel ends let k be the space of properly embedded minimal tori in quotients of r 3 by two independent translations with any fixed even number of parallel ends after an appropriate normalization we prove that k is a 3 dimensional real analytic manifold that reduces to the finite coverings of the examples defined by karcher meeks and rosenberg in 9 10 15 the degenerate limits of surfaces in k are the catenoid the helicoid and three 1 parameter families of surfaces the simply and doubly periodic scherk minimal surfaces and the riemann minimal examples

riemann bilinear relations on minimal surfaces from a physical point of view minimal surfaces in r are objects submitted to a balanced force system consisting in the forces associated to non zero onedimensional homology classes in the surface several geometric properties can be studied in terms of those forces as embeddedness symmetries and deformations leading up to uniqueness and non existence results see 11 13 more precisely each closed curve in a minimal surface s r carries a force that expresses the stress produced by an unit conormal vector field along on the whole surface the action of this conormal field provides a tendency of translation or linear momentum f and another one of rotation around an axis or angular momentum m the first one is given by the force vector

goodness of fit tests for censored survival data the problem of testing the underlying distribution in a general regression model is considered when the data are right censored it is proposed that tests be based on efficient scores from a certain class of parametric alternatives the connection between tests based on the scores and more traditional approaches using empirical distribution functions is explored neyman s smooth test is extended to allow for censored data

martingale based residuals for survival models summary graphical methods based on the analysis of residuals are considered for the setting of the highly used cox 1972 regression model and for the andersen gill 1982 generalization of that model we start with a class of martingale based residuals as proposed by barlow prentice 1988 these residuals and or their transforms are useful for investigating the functional form of a covariate the proportional hazards assumption the leverage of each subject upon the estimates of 13 and the lack of model fit to a given subject 1 1 model consider a set of n independent subjects such that the counting process ni ni t t o for the ith subject in the set indicates the number of observed events experienced over time t the sample paths of the ni are step functions with jumps of size 1 and with ni 0 0 we assume that the intensity function for ni t is given by yi t da t zi t yi t ep z t dao t 1 where yi t is a 0 1 process indicating whether the ith subject is a risk at time t 13 is a vector of regression coefficients zi t is a p dimensional vector of covariate processes and ao is the baseline cumulative hazard function several familar survival models fit into this framework the andersen gill 1982 generalization of the cox 1972 model arises when ao t is completely unspecified the further restriction that yi t 1 until the first event or censoring and is 0 thereafter yields the cox model the parametric form ao t t yields a poisson model or an exponential if restricted to a single event per subject and ao t tp a weibull model our attention will focus primarily on the andersen gill and cox models however the

some uniqueness and nonexistence theorems for embedded minimal surfaces thanks to the works of callahan costa hoffman karcher meeks rosenberg wei etc see for example 1 2 3 5 8 11 12 16 20 we dispose now of a large number of properly embedded minimal surfaces in the euclidean space ir other than the classical examples all those surfaces can be viewed as minimal surfaces with finite total curvature properly embedded in complete flat three manifolds the most basic invariants associated to a surface of this type are its topology and its periodicity it is an interesting problem to decide if the simplest examples like the catenoid the helicoid the scherk s surfaces or the riemann example can be characterized in terms of some of these invariants in the non periodic case there are two important uniqueness theorems in this direction the first one obtained by schoen 17 characterizes the catenoid as the only complete minimal surface embedded

uniform error estimates of galerkin methods for monotone abel volterra integral equations on the half line we consider galerkin methods for monotone abel volterra integral equations of the second kind on the half line the l2 theory follows from kolodner s theory of monotone hammerstein equations we derive the loo theory from the l2 theory by relating the l2and loo spectra of operators of the form x b ax to one another here denotes convolution and b e l1 and a e loo as an extra condition we need b t o t a 1 with a 0 we also prove the discrete analogue in particular we verify that the galerkin matrix satisfies the discrete conditions

stability and robustness of collocation methods for abel type integral equations we study stability aspects of collocation methods for abel type integral equations of the first kind using piecewise polynomials these collocation methods may be formulated as projection methods stability is defined as the boundedness of the sequence of projectors in their natural setting robustness is essentially the optimal asymptotic insensitivity to perturbations in the data we show that stability and robustness are equivalent for the above collocation methods this allows us to obtain optimal error estimates for some methods that are well known to be robust we also present numerical results for some methods which appear to be robust

a family of methods for abel integral equations of the second kind abstract a class of methods depending on some parameters are introduced for the numerical solution of the abel integral equations of the second kind some bounds on the parameters are determined so that the corresponding methods have infinite stability intervals

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

galerkin methods for two point boundary value problems for first order systems galerkin methods for two point boundary value problems for first order systems are studied for a model problem rigorous upper and lower estimates upon the error are derived these estimates show the previous work to be sharp in some cases and pessimistic in others a nonstandard least squares type of method is then analyzed this method gives the optimal accuracy of the usual least squares method when the system does not satisfy a regularity assumption the nonstandard method gives slightly more accurate answers measured in l 2 the linear system arising from the nonstandard method has condition number o h 1 contrasted with o h 2 for the usual least squares method

the space of complete minimal surfaces with finite total curvature as lagrangian submanifold the space m of nondegenerate properly embedded minimal surfaces in r3 with finite total curvature and fixed topology is an analytic lagrangian submanifold of cn where n is the number of ends of the surface in this paper we give two expressions one integral and the other pointwise for the second fundamental form of this submanifold we also consider the compact boundary case and we show that the space of stable nonflat minimal annuli that bound a fixed convex curve in a horizontal plane having a horizontal end of finite total curvature is a locally convex curve in the plane c mathematics subject classification 53a10 53c42

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