nonlinear elliptic boundary value problems at strong resonance let a be a linear elliptic partial differential operator on a domain r n such that the boundary value problem 1 1 au u in bu 0 on is selfadjoint and has a discrete set of isolated eigenvalues bounded from below and no other spectrum

asymptotic properties of moment estimators for overdispersed counts and proportions summary consistency and asymptotic normality of moment estimates of regression parameters and an overdispersion parameter are shown for data with extra binomial and extrapoisson variation the asymptotic covariance of the regression parameters is found to be unaffected by estimation of the overdispersion parameter the moment estimates may be obtained using iterated weighted least squares

perturbation expansions for invariant subspaces abstract this paper applies a technique previously described by the author for obtaining perturbation expansions and k th order perturbation estimations for invariant subspaces of a matrix

on the sensitivity of multiple eigenvalues of nonsymmetric matrix pencils this paper considers the sensitivity of semisimple multiple eigenvalues and corresponding generalized eigenvector matrices of a nonsymmetric matrix pencil analytically dependent on several parameters the directional derivatives of the multiple eigenvalues are obtained and the average of eigenvalues and corresponding generalized eigenvector matrices are proved to be analytic the results can be used to define the sensitivity of the semisimple multiple eigenvalues and corresponding generalized eigenvector matrices these are useful for investigating structural optimal design model updating and structural damage detection

nonlinear multipoint boundary value problems on considere l existence et le nombre de solutions d un probleme a n points limites pour une equation differentielle non lineaire d ordre 2 ou n 3 est un entier

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

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

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

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

nonexistence results for a class of nonlinear elliptic equations involving critical sobolev exponents abstract in this paper we prove nonexistence results for some classes of nonlinear elliptic equations with critical growth of the form u g x u u 2 2 u x u 0 x u 0 x where 2 2 n n 2 g x u is a lower order perturbation of u 2 1 and is a bounded strictly star shaped domain in r n n 3 combining pohozaev s identity with classical and weak interpolation inequality we are able to exhibit examples of nonlinear problems without any nontrivial solution bifurcating from infinity in 0 for n 3 and n 4

goodness of fit tests for log linear models in sparse contingency tables abstract the asymptotic normality of the likelihood ratio goodness of fit statistic is demonstrated for testing the fit of log linear models with closed form maximum likelihood estimates in sparse contingency tables unlike the traditional chi squared theory the number of categories in the table increases as the sample size increases but not all of the expected frequencies are required to become large some results of a small monte carlo study are presented the traditional chi squared approximation is reasonably accurate for the pearson statistic for many sparse tables but cases are presented for which it fails the normal approximation can be much more accurate than the chi squared approximation for the likelihood ratio statistic but the bias of estimated moments is a potential problem for very sparse tables

a bounded mountain pass lemma without the ps condition and applications we present a version of the mountain pass lemma which does not require the ps condition we apply this version to problems where the ps condition is not satisfied

multiple eigenvalue sensitivity analysis abstract this paper surveys the development of the sensitivity analysis of multiple eigenvalues and the associated invariant subspaces of two kinds of matrix eigenvalue problems in which the matrices are analytically dependent on several parameters

profile likelihood based confidence interval of the intraclass correlation for binary outcome data sampled from clusters the intraclass correlation in binary outcome data sampled from clusters is an important and versatile measure in many biological and biomedical investigations properties of the different estimators of the intraclass correlation based on the parametric semi parametric and nonparametric approaches have been studied extensively mainly in terms of bias and efficiency see for example ridout et al biometrics 1999 55 137 148 paul et al journal of statistical computation and simulation 2003 73 507 523 and lee statistical modelling 2004 4 113 126 but little attention has been paid to extending these results to the problem of the confidence intervals in this article we generalize the results of the four point estimators by constructing asymptotic confidence intervals obtaining closed form asymptotic and sandwich variance expressions of those four point estimators it appears from simulation results that the asymptotic confidence intervals based on these four estimators have serious under coverage to remedy this we introduce the fisher s z transformation approach on the intraclass correlation coefficient the profile likelihood approach based on the beta binomial model and the hybrid profile variance approach based on the quadratic estimating equation for constructing the confidence intervals of the intraclass correlation for binary outcome data as assessed by monte carlo simulations these confidence interval approaches show significant improvement in the coverage probabilities moreover the profile likelihood approach performs quite well by providing coverage levels close to nominal over a wide range of parameter combinations we provide applications to biological data to illustrate the methods copyright 2012 john wiley sons ltd

on the semi monotone operator theory and applications abstract in this paper eis a real banach space a e e e is a semi monotone operator we first study the variational inequality problem findu k such that a u u v u 0 wherek e is a closed convex subset then the operator equationa u u p and we also construct a degree theory for demi continuous semi monotone operators in reflexive banach spaces

unfolding bifurcations of an elliptic boundary value problem on considere la famille a 2 parametres d equations elliptiques sur un domaine r n a frontiere c donnee par u f u g dans et u uo sur m r 2 f 1 g r r lisses f 0 0 et o 0 lisse on montre que sous certaines restrictions sur f une bifurcation autour de la solution u 0 proche 0 m 0 ou 0 est une valeur propre simple de l equation linearisee w 0 f 0 w 0 dans w 0 sur et 0 0 peut se decrire par la catastrophe en cusp de thom

a new variational principle and duality for periodic solutions of hamilton s equations we are dealing with a classical boundary value problem concerning periodic solutions of hamiltonian equations p f k f 4th p f i f h x t p t 1 where h r x r x r r is a given function called the hamiltonian in mechanics 1 and also in mathematics 171 with a given hamiltonian the lagrangian l defined by the legendre transform l 4 x a sup k p wc x pi d 2

on assessing goodness of fit of generalized linear models to sparse data summary approximations to the first three moments of pearson s statistic are obtained for noncanonical generalized linear models extending the results of mccullagh a first order modification to pearson s statistic is proposed which induces local orthogonality with the regression parameters resulting in substantial simplifications and increased power accurate and easily computed approximations to the moments of the modified pearson statistic conditional on the estimated regression parameters are obtained for testing goodness of fit to sparse data both the pearson statistic and its modification are shown to be asymptotically independent of the regression parameters simulation studies and examples are given

generalization of the bauer fike theorem the bauer fike theorem for diagonalizable matrices is generalized to cases where i non diagonalizable matrices or ii only part of the spectrum is considered

estimating overdispersion in sparse multinomial data multinomial data arise in many areas of the life sciences such as mark recapture studies and phylogenetics and will often by overdispersed with the variance being higher than predicted by a multinomial model the quasi likelihood approach to modeling this overdispersion involves the assumption that the variance is proportional to that specified by the multinomial model as this approach does not require specification of the full distribution of the response variable it can be more robust than fitting a dirichlet multinomial model or adding a random effect to the linear predictor estimation of the amount of overdispersion is often based on pearson s statistic x2 or the deviance d for many types of study such as mark recapture the data will be sparse the estimator based on x2 can then be highly variable and that based on d can have a large negative bias we derive a new estimator which has a smaller asymptotic variance than that based on x2 the difference being most marked for sparse data we illustrate the numerical difference between the three estimators using a mark recapture study of swifts and compare their performance via a simulation study the new estimator has the lowest root mean squared error across a range of scenarios especially when the data are very sparse

infinite dimensional linking let n be a closed separable subspace of a hilbert space e we can define a new norm v w satisfying v w v v n and such that the topology induced by this norm is equivalent to the weak topology of n on bounded subsets of n this can be done as follows let ek be an orthonormal basis for n define displaystyle u v w sum k 1 infty frac u e k v e k 2 k quad u v in n

on multiple eigenvalues of matrices depending on several parameters the eigenvalues and eigenvectors as well as their derivatives of matrices depending on one or several parameters have been studied by sun j comput math 3 1985 pp 351 364 using the implicit function theorems and others but not for the more general multiple eigenvalues in this paper the mean of the multiple eigenvalues in terms of the trace operator tr the corresponding invariant subspaces and their derivatives are studied using a generalization of sun s approach the numerical aspects concerning the computation of such derivatives by direct and iterative methods e g simultaneous and inverse simultaneous iterations will be discussed briefly the main results in this paper apply to clusters of nonmultiple eigenvalues as well the implications e g on simultaneous iteration will be explored

sensitivity analysis of all eigenvalues of a symmetric matrix given a x a ij x a n by n symmetric matrix depending smoothly on a parameter x we study the first order sensitivity of all the eigenvalues lambda m x of a x 1 le m le n under some smoothness assumption like the a ij be c 1 we prove that the directional derivatives d mapsto lambda prime m x d lim t to o lambda m x td lambda m x t do exist and give an explicit expression of them in terms of the data of the parametrized matrix the key idea to circumvent the difficulties inherent to the study of each lambda m taken separately is to consider the functions f m x 1 le m le n defined as the sums of the m largest eigenvalues of a x based on ky fan s variational formulation of f m and some chain rule from nonsmooth analysis we derive an explicit formula for the generalized gradient of f m and a computationally useful formula for the directional derivative of f m using these formulas and the relation lambda m f m f m 1 we then derive the directional derivative of lambda m some properties of this directional derivative as well as an illustrative example are presented

morse conley theory for minimal surfaces of varying topological type after the plateau problem had been solved by douglas and rad6 by finding a possibly branched disk type minimal immersion bounded by a given jordan curve in n soon more general problems were proposed on one hand the existence of minimal surfaces of higher topological structure was investigated by douglas i d courant c and shiffman sh 1 on the other hand the problem of finding unstable minimal surfaces and more generally developing a morse theory for minimal surfaces of disk type was attacked by morse tompkins mt1 and shiffman sh2 a natural question then was to develop a morse theory for minimal surfaces of arbitrary topological structure while morse tompkins in their paper mt2 only treated a very special case namely annulus type surfaces of what their title unstable minimal surfaces of higher topological structure claims shiffman confronted the case of genus 0 and arbitrary connectivity sh3 these classical papers were not in every respect satisfactory the investigations of douglas and courant on the higher genus problem were recently critisized by tromba t 1 however luckhaus i l was able to carry out a systematic reworking of the arguments of courant and shiffman the original approaches of morse tompkins shiffman moreover severely suffer from the fact that these authors work in the c instead of the more natural h l 2 topology in the c dirichlet s integral whose critical points parametrize the sought after minimal surfaces is not differentiable and no intrinsic notion i e depending only on the surface of morse index can be defined this makes it impossible to decide in the work of morse tompkins and shiffman whether the morse relations for minimal surfaces reflect a property of the surfaces

the generalized levinger transformation in this paper we present new results relating the numerical range of a matrix a with the generalized levinger transformation l a a b email protected email protected a where h a and s a are respectively the hermitian and skew hermitian parts of a using these results we then derive expressions for eigenvalues and eigenvectors of the perturbed matrix a l e a b for a fixed matrix e and a b are real parameters

snakes oriented families of periodic orbits their sources sinks and continuation abstract poincare observed that for a differential equation x x depending on a parameter each periodic orbit generally lies in a connected family of orbits in x space in order to investigate certain large connected sets denoted q of orbits containing a given orbit we introduce two indices an orbit index and a center index defined at certain stationary points we show that genetically there are two types of hopf bifurcation those we call sources 1 and sinks 1 generically if the set q is bounded in x space and if there is an upper bound for periods of the orbits in q then q must have as many source hopf bifurcations as sink hopf bifurcations and each source is connected to a sink by an oriented one parameter snake of orbits a snake is a maximal path of orbits that contains no orbits whose orbit index is 0 see fig 1 1

saddle point techniques from the beginning of the calculus of variations over 300 years ago scientists were interested in finding extrema of expressions now called functionals involving unknown functions it was discovered that functions producing extrema are solutions of differential equations called euler equations in one dimentional problems these were ordinary differential equations and the easiest procedure was to solve the corresponding euler equation and then check if the solution produces the desired maximum or minimum however as researchers began to study higher dimensional problems it became obvious that it was more difficult to solve the euler equations than obtain an extremum of the functional moreover many partial differential equations and systems that were being studied by various scientists happened to be the euler equations of functionals in this case any critical point of the corresponding functional will provide a solution of the given equation this led to the search for critical points of functionals in this paper we describe several methods of finding critical points we present applications

renormalized area and properly embedded minimal surfaces in hyperbolic 3 manifolds we study the renormalized area functional mathcal a in the ads cft correspondence specifically for properly embedded minimal surfaces in convex cocompact hyperbolic 3 manifolds and somewhat more broadly poincare einstein spaces our main results include an explicit formula for the renormalized area of such a minimal surface y as an integral of local geometric quantities as well as formulae for the first and second variations of mathcal a which are given by integrals of global quantities over the asymptotic boundary loop of y all of these formulae are also obtained for a broader class of nonminimal surfaces the proper setting for the study of this functional when the ambient space is hyperbolic requires an understanding of the moduli space of all properly embedded minimal surfaces with smoothly embedded asymptotic boundary we show that this moduli space is a smooth banach manifold and develop a mathbb z valued degree theory for the natural map taking a minimal surface to its boundary curve we characterize the nondegenerate critical points of mathcal a for minimal surfaces in mathbb h 3 and finally discuss the relationship of mathcal a to the willmore functional

a mountain pass theorem 143 in other words the mountain separating 0 and e may even be assumed of zero altitude and the conclusion is still true moreover as we shall show if c a the critical point can be chosen with llx ll r it is interesting to ask whether this extension of the ambrosetti rabinowitz theorem remains true in the infinite dimensional case we prove here that if 1 is strengthened a little to the form 1 there exist real numbers u r r such that 0 a for every x e a x e x y a moreover if c a the critical point can be chosen with r i x0 i r roughly speaking in brief the mountain pass theorem continues to hold for a mountain of zero altitude provided it also has non zero thickness in addition if c a the pass itself occurs precisely on the mountain i e satisfying r llroll r there are two interesting and immediate corollaries the first one says that a c function which has iwo local minimum points also has a third critical point the second states that a u periodic c function with a local minimum e has a critical point x e ku k 0 1 2 the precise statements of the above results will be given in the next sec tions the proof depends upon a lemma of clark 3 formulated here in a version suitable to our purpose in section 3 some applications are presen ted for the forced pendulum equation 2

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

the structure of the critical set in the mountain pass theorem we show that the critical set generated by the mountain pass theorem of ambrosetti and rabinowitz must have a well defined structure in particular if the underlying banach space is infinite dimensional then either the critical set contains a saddle point of mountain pass type or the set of local minima intersects at least two components of the set of saddle points related conclusions are also established for the finite dimensional case and when other special conditions are assumed throughout the paper no hypotheses of nondegeneracy are required on the critical set

calculation of eigenpair derivatives for asymmetric damped systems with distinct and repeated eigenvalues summary an algorithm is derived for the computation of eigenpair derivatives of asymmetric quadratic eigenvalue problem with distinct and repeated eigenvalues in the proposed method the eigenvector derivatives of the damped systems are divided into a particular solution and a homogeneous solution by introducing an additional normalization condition we construct two extended systems of linear equations with nonsingular coefficient matrices to calculate the particular solution the method is numerically stable and the homogeneous solutions are computed by the second order derivatives of the eigenequations two numerical examples are used to illustrate the validity of the proposed method copyright 2015 john wiley sons ltd

constructive variational methods for differential equations on considere des methodes constructives pour resoudre des systemes d equations aux derivees partielles non lineaires

unstable solutions of the euler equations of certain variational problems soit cr n un domaine borne avec une frontiere appartenant a c 2 pour 0 1 soit f 0 f 0 x z p appartenant a c loc 2 r r n et soit f 0 x z r n r convexe pour tout x z r alors pour u c 1 on definit i 0 t u f 0 x u u dx on considere le probleme variationnel engendre en restreignant i 0 a l ensemble u c 1 u avec c 2

sequential estimation of structural models with a fixed point constraint this paper considers the estimation problem of structural models for which empirical restrictions are characterized by a fixed point constraint such as structural dynamic discrete choice models or models of dynamic games we analyze a local condition under which the nested pseudo likelihood npl algorithm converges to a consistent estimator and derive its convergence rate we find that the npl algorithm may not necessarily converge to a consistent estimator when the fixed point mapping does not have a local contraction property to address the issue of divergence we propose alternative sequential estimation procedures that can converge to a consistent estimator even when the npl algorithm does not

the conditional distribution of goodness of fit statistics for discrete data abstract i consider the distribution of pearson s statistic and of the likelihood ratio goodness of fit statistic for discrete data in the important case where the data are extensive but sparse it is argued that the appropriate reference distribution is conditional on the sufficient statistic for the unknown regression parameters the first three conditional asymptotic cumulants are derived by edgeworth expansion and these are used for the computation of tail probabilities the principal advantage of the limit considered here as opposed to the more usual x 2 limit is that the cell counts need not be large

derivatives of eigenvalues and eigenvectors of a general complex matrix a survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non hermitian matrices is presented in addition a modification of one method based on a better normalizing condition is proposed methods are classified as direct or adjoint and are evaluated for efficiency operation counts are presented in terms of matrix size number of design variables and number of eigenvalues and eigenvectors of interest the effect of the sparsity of the matrix and its derivatives is also considered and typical solution times are given general guidelines are established for the selection of the most efficient method

semilinear problems in unbounded domains abstract as formulated by silva e a de b e silva linking theorems and applications to semilinear elliptic problems at resonance nonlinear anal 16 1991 455 477 and schechter m schechter a generalization of the saddle point method with applications ann polon math 57 3 1992 269 281 m schechter new saddle point theorems in generalized functions and their applications varanasi 1991 plenum new york 1993 pp 213 219 the sandwich theorem has become a very useful tool in finding critical points of functionals leading to solutions of partial differential equations in the present paper this theorem is strengthened to apply to more general situations we present some applications

le probleme de dirichlet pour des equations de monge ampere complexes modifiees resume let x 2 m g be a strongly pseudoconvex hermitian manifold with c boundary if belongs to c 2 x let us consider the changes of hermitian metric defined by g 1 g i and g 2 g i we solve problems of the form detg i det g 1 f x in x and u on x where f c tx r is an everywhere strictly positive function satisfying some assumptions and u c x

boundary value problems of the ginzburg landau equations for a domain in the euclidean space r d d 2 3 existence of weak solutions for both interior and exterior dirichlet boundary value problems of the ginzburg landau equations are established without any restriction on the range of the coupling constant the size of or the boundary data for the critical choice 1 we prove the existence of confined multivortices in a bounded domain by a constructive monotone iteration method

imperfect transcritical and pitchfork bifurcations imperfect bifurcation phenomena are formulated in framework of analytical bifurcation theory on banach spaces in particular the perturbations of transcritical and pitchfork bifurcations at a simple eigenvalue are examined and two parameter unfoldings of singularities are rigorously established applications include semilinear elliptic equations imperfect euler buckling beam problem and perturbed diffusive logistic equation

an existence theorem for some semilinear elliptic systems abstract an existence theorem is obtained for a class of semilinear second order uniformly elliptic systems obtained formally from a variational principle and modeled on nonlinear helmholtz systems superlinear growth of the nonlinear term precludes application of standard methods to these systems indeed we permit very rapid growth of the nonlinear term so the underlying functional is not defined on the hilbert space within which a solution is naturally sought mollification of the nonlinear term nonetheless results in the resulting functional satisfying the palais smale condition critical points are determined by solution of a dynamical system the limit of vanishing mollification then produces a weak solution of the original problem

computation of derivatives of repeated eigenvalues and corresponding eigenvectors by simultaneous iteration an earlier simultaneous iteration method for computing derivatives of eigenvalues and the corresponding eigenvectors of nonsymmetric matrices depending smoothly on some real parameters is extended to the case of repeated eigenvalues

goodness of fit of generalized linear models to sparse data we derive approximations to the first three moments of the conditional distribution of the deviance statistic for testing the goodness of fit of generalized linear models with non canonical links by using an estimating equations approach for data that are extensive but sparse a supplementary estimating equation is proposed from which the modified deviance statistic is obtained an application of a modified deviance statistic is shown to binomial and poisson data we also conduct a performance study of the modified pearson statistic derived by farrington and the modified deviance statistic derived in this paper in terms of size and power through a small scale simulation experiment both statistics are shown to perform well in terms of size the deviance statistic however shows an advantage of power two examples are given