assessing proportionality in the proportional odds model for ordinal logistic regression the proportional odds model for ordinal logistic regression provides a useful extension of the binary logistic model to situations where the response variable takes on values in a set of ordered categories the model may be represented by a series of logistic regressions for dependent binary variables with common regression parameters reflecting the proportional odds assumption key to the valid application of the model is the assessment of the proportionality assumption an approach is described arising from comparisons of the separate correlated fits to the binary logistic models underlying the overall model based on asymptotic distributional results formal goodness of fit measures are constructed to supplement informal comparisons of the different fits a number of proposals including application of bootstrap simulation are discussed and illustrated with a data example

emden fowler equations involving critical exponents on considere le probleme y t k y 2 y 2k 3 0 t 0 pour t 2 0 0 soit t inf t 0 y 0 sur t alors t 0 0 on etudie comment la position t du premier zero de y t varie avec

a note on the radial solutions for the supercritical henon equation we prove the existence of a positive radial solution for the henon equation with arbitrary growth the solution is found by means of a shooting method and turns out to be an increasing function of the radial variable some numerical experiments suggest the existence of many positive oscillating solutions

on the existence of positive solutions for semilinear elliptic equations with neumann boundary conditions we give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in unbounded lipschitz domainsd d d 3 having compact boundary with nonlinear neumann boundary conditions on the boundary ofd for this we use an implicit probabilistic representation schauder s fixed point theorem and a recently proved sobolev inequality forw 1 2 d special cases include equations arising from the study of pattern formation in various models in mathematical biology and from problems in geometry concerning the conformal deformation of metrics

a comparison of test statistics for assessing the effects of concomitant variables in survival analysis in data analysis involving the proportional hazards regression model due to cox 1972 journal of the royal statistical society series b 34 187 220 the test criteria commonly used for assessing the partial contribution to survival of subsets of concomitant variables are the classical likelihood ratio lr and wald statistics this paper presents an investigation of three other test criteria with potentially major computational advantages over the classical tests especially for stepwise variable selection in moderate to large data sets the alternative criteria considered are rao s efficient score statistic and two other score statistics under the cox model the performance of these tests is examined empirically and compared with the performance of the lr and wald statistics rao s test performs comparably to the lr test in all the cases considered the performance of the other criteria is competitive in many cases the use of these statistics is illustrated in a study of coronary artery disease

global branches of non radially symmetric solutions to a semilinear neumann problem in a disk let d r 2 be a disk and let f c 3 we assume that there is a r such that f a 0a ndf a 0 in this article we are concerned with the neumann problem u f u 0i nd u 0o n d we show the following there are unbounded continuums consisting of non radially symmetric solutions emanating from the second and third eigenvalues if f u u u u p 1 a 1 or if f is of bistable type then the unbounded branches emanating from non principal eigenvalues are unbounded in the positive direction of the branch emanating from the second eigenvalue is unique near the bifurcation point up to rotation the main tool of this article is the zero level set the nodal curve of u and ux

analysis on a diffusive sis epidemic model with saturated incidence rate and linear source in a heterogeneous environment abstract in this paper we are concerned with a reaction diffusion sis epidemic model with saturated incidence rate and linear source in spatially heterogeneous environments we derive the uniform bounds of parabolic system and study the extinction and persistence of the infectious disease in terms of the basic reproduction number the existence of the endemic equilibrium ee is established when the basic reproduction number is greater than one we further investigate the effects of diffusion and saturation on asymptotic profiles of the endemic equilibrium our study shows that the linear source can enhance persistence of infectious disease and large saturation can help speed up the elimination of disease

analysis of repeated categorical data using generalized estimating equations moment methods for analysing repeated binary responses have been proposed by liang and zeger and extended by prentice and zhao and prentice in these estimating equations models are proposed for the correlation between the repeated binary responses we extend liang and zeger s method to models for the correlation between repeated nominal or ordinal categorical responses in particular when the repeated responses are binary our methods reduce to liang and zeger s method our method is illustrated with two datasets one dataset contains repeated observations of self assessment of arthritis an ordered variable with three categories collected during a randomized comparative study of alternative treatments of patients with rheumatoid arthritis the second dataset is a longitudinal study of the health effects of air pollution in which the repeated ordered multinomial response is the wheezing status no wheeze wheeze with cold wheeze apart from cold of a child at ages 9 10 11 and 12 years

the lin ni conjecture in negative geometries abstract we prove see theorem 1 1 that the lin ni conjecture for closed manifolds is false in dimensions n 4 5 when the scalar curvature is negative but that it holds true in the bounded energy setting when n 6 see theorem 1 2 as a corollary to theorem 1 1 we prove see theorem 2 2 that contrary to the 3 dimensional case the sup inf inequality does not hold in general with a sole control on the size of the potential for positive solutions of stationary critical schrodinger equations

a survey of models for repeated ordered categorical response data we survey models for analysing repeated observations on an ordered categorical response variable the models presented are univariate models that permit correlation among repeated measurements the models describe simultaneously the dependence of marginal response distributions on values of explanatory variables and on the occasion of response we present models for three transformations of the response distribution cumulative logits adjacent category logits and the mean for scores assigned to response categories we discuss three methods for fitting the models maximum likelihood weighted least squares and semi parametric weighted least squares is easily implemented with sas as illustrated with a study designed to compare a drug with a placebo for the treatment of insomnia

goodness of fit tests for gee modeling with binary responses analysis of data with repeated measures is often accomplished through the use of generalized estimating equations gee methodology although methods exist for assessing the adequacy of the fitted models for uncorrelated data with likelihood methods it is not appropriate to use these methods for models fitted with gee methodology we propose model based and robust empirically corrected goodness of fit tests for gee modeling with binary responses based on partitioning the space of covariates into distinct regions and forming score statistics that are asymptotically distributed as chi square random variables with the appropriate degrees of freedom the null distribution and the statistical power of the proposed goodness of fit tests were assessed using simulated data the proposed goodness of fit tests are illustrated by two examples using data from clinical studies

stationary solutions of chemotaxis systems the keller segel model is a system of partial differential equations modelling a mutual attraction of amoebae caused by releasing a chemical substance chemotaxis this paper analyzes the stationary solutions of the system with general nonlinearities via bifurcation techniques and gives a criterion for bifurcation of stable nonhomogeneous aggregation patterns examples are discussed with various kinds of nonlinearities modelling the sensitivity of the chemotaxis response

phase diagram of the two dimensional t t falicov kimball model the ground state phase diagram of the two dimensional falicov kimball model with nearest neighbour and next nearest neighbour hoppings has been studied in the perturbative regime where hoppings are small compared with the on site coulomb interaction the phase diagram at fourth order exhibits a richer structure than the one of the ordinary falicov kimball model

the analysis of longitudinal polytomous data generalized estimating equations and connections with weighted least squares in recent years methods have been developed for modelling repeated observations of a categorical response obtained over time on the same individual although situations in which the repeated response is binary or poisson have been studied extensively relatively little attention has been given to polytomous categorical response variable in this paper we extend the estimating equations initially developed for clustered discrete data by liang and zeger 1986 biometrika 73 13 22 and subsequently extended by prentice 1988 biometrics 44 1033 1048 to polytomous response variables under certain assumptions we illustrate that these estimating equations simplify to the weighted least squares wls equations formalized by koch et al 1977 biometrics 33 133 158 this connection provides a formal framework for obtaining iterated weighted least squares model parameter estimates cumulative logit models are developed and applied to a representative longitudinal data set simulation results comparing wls an iterative form of wls and independence estimating equations using a robust estimate of the variance are presented

on the neumann problem for some semilinear elliptic equations and systems of activator inhibitor type on deduit des estimations a priori pour des solutions positives du probleme de neumann pour des systemes elliptiques semilineaires ainsi que pour des equations isolees semilineaires reliees a ces systemes

best constants and minimizers for embeddings of second order sobolev spaces by considering the kernels of the first two traces four different second order sobolev spaces may be constructed for these spaces embeddings into lebesgue spaces the best embedding constant and the possible existence of minimizers are studied the euler equation corresponding to some of these minimization problems is a semilinear biharmonic equation with boundary conditions involving third order derivatives it is shown that the complementing condition is satisfied

analyzing bivariate ordinal data using a global odds ratio abstract a moment method is proposed for regression analysis of bivariate ordered categorical data using the global odds ratio as the measure of association for modeling the margins this method utilizes the stochastic ordering implicit in the data this method allows for covariate effects in modeling the association between responses an application of the proposed model is illustrated using the ophthalmological data from the wisconsin epidemiologic study of diabetic retinopathy for identifying risk factors among younger onset diabetics this model is also extended to a longitudinal data setting with more than two repeated measures

nonexistence of positive radial solutions of a quasilinear neumann problem with a critical sobolev exponent note that problem 1 1 always admits the constant solution u0 k ny2 4 in adimurthi yadava 3 and budd knaap peletier 6 it has been proved that there exists a constant k0 0 such that for 0 k k0 1 1 admits a nonconstant solution for n 4 5 6 and does not for n 3 in this paper we study problem 1 1 when n 3 7 in this case the method used in 3 and 6 breaks down due to the fact that n 2 ny 2 2 which leads to a drastic change in the behaviour of the associated pohozhaev s functionals in contrast to the case n 4 5 6 we have

segregation in the falicov kimball model the falicov kimball model is a simple quantum lattice model that describes light and heavy electrons interacting with an on site repulsion alternatively it is a model of itinerant electrons and fixed nuclei it can be seen as a simplification of the hubbard model by neglecting the kinetic hopping energy of the spin up particles one gets the falicov kimball model we show that away from half filling i e if the sum of the densities of both kinds of particles differs from 1 the particles segregate at zero temperature and for large enough repulsion in the language of the hubbard model this means creating two regions with a positive and a negative magnetization our key mathematical results are lower and upper bounds for the sum of the low est eigenvalues of the discrete laplace operator in an arbitrary domain with dirichlet boundary conditions the lower bound consists of a bulk term independent of the shape of the domain and of a term proportional to the boundary therefore one lowers the kinetic energy of the itinerant particles by choosing a domain with a small boundary for the falicov kimball model this corresponds to having a single compact domain that has no heavy particles

estimations of the free energy for the hubbard model a series of upper bounds and two lower bounds for the partition function of the hubbard model have been derived these bounds are expressible by certain properties of the falicov kimball model the upper bounds have been derived mainly by the use of the golden thompson inequality and its generalizations and the holder inequality the lower bounds are based on the bogoliubov peierls inequality the numerical values of bounds have also been calculated for small systems

two dimensional berezin li yau inequalities with a correction term we improve the berezin li yau inequality in dimension two by adding a positive correction term to its right hand side it is also shown that the asymptotical behaviour of the correction term is almost optimal this improves a previous result by melas 11

lower bound for the segregation energy in the falicov kimball model in this work a lower bound for the ground state energy of the falicov kimball model for intermediate densities is derived the explicit derivation is important in the proof of the conjecture of segregation of the two kinds of fermions in the falicov kimball model for sufficiently large interactions this bound is given by a bulk term plus a boundary term of the form 1 n where is the region devoid of classical particles and n is the density of electrons a detailed proof is presented for n 1 2 where the coefficient 1 1 2 10 13 is obtained for the two dimensional case although clearly not optimal in terms of order of magnitude this is the largest explicitly calculated coefficient in the range of intermediate densities with suitable modifications the method can also be used to obtain a coefficient for all densities that is the topic of the last section where a sketch of the proof for n 1 2 is shown

segregation in the asymmetric hubbard model we study the asymmetric hubbard model where hoppings of electrons depend on their spin for strong interactions and sufficiently asymmetric hoppings it is proved that the ground state displays phase separation away from half filling this extends a recent result obtained with freericks and lieb for the falicov kimball model it is based on estimates for the sum of lowest eigenvalues of the discrete laplacian in arbitrary domains

positive clusters for smooth perturbations of a critical elliptic equation in dimensions four and five abstract we construct clustering positive solutions for a perturbed critical elliptic equation on a closed manifold of dimension n 4 5 such a construction is already available in the literature in dimensions n 6 see for instance 10 14 30 32 36 and not possible in dimension 3 by 25 this also provides new patterns for the lin ni 26 problem on closed manifolds and completes results by brezis and li 8 about this problem

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

minimizing the ground state energy of an electron in a randomly deformed lattice we provide a characterization of the spectral minimum for a random schrodinger operator of the form h delta sum i in mathbb z d q x i omega i in l 2 mathbb r d where the single site potential q is reflection symmetric compactly supported in the unit cube centered at 0 and the displacement parameters i are restricted so that adjacent single site potentials do not overlap in particular we show that a minimizing configuration of the displacements is given by a periodic pattern of densest possible 2 d clusters of single site potentials

global existence and finite time blow up of solutions of a gierer meinhardt system abstract we are concerned with the gierer meinhardt system with zero neumann boundary condition u t d 1 u a 1 u u p v q 1 x x t 0 v t d 2 v a 2 v u r v s 2 x x t 0 u x 0 u 0 x v x 0 v 0 x x where p 1 s 1 q r d 1 d 2 a 1 a 2 are positive constants 1 2 u 0 v 0 are nonnegative smooth functions r d d 1 is a bounded smooth domain we obtain new sufficient conditions for global existence and finite time blow up of solutions especially in the critical exponent cases p 1 r and q r p 1 s 1

testing proportionality in the proportional odds model fitted with gee generalized estimating equations gee methodology as proposed by liang and zeger has received widespread use in the analysis of correlated binary data miller et al and lipsitz et al extended gee to correlated nominal and ordinal categorical data in particular they used gee for fitting mccullagh s proportional odds model in this paper we consider robust that is empirically corrected and model based versions of both a score test and a wald test for assessing the assumption of proportional odds in the proportional odds model fitted with gee the wald test is based on fitting separate multiple logistic regression models for each dichotomization of the response variable whereas the score test requires fitting just the proportional odds model we evaluate the proposed tests in small to moderate samples by simulating data from a series of simple models we illustrate the use of the tests on three data sets from medical studies

asymptotic profiles of the positive steady state for an sis epidemic reaction diffusion model part i abstract to capture the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease allen et al in l j s allen b m bolker y lou a l nevai asymptotic profiles of the steady states for an sis epidemic reaction diffusion model discrete contin dyn syst ser a 21 1 2008 1 20 proposed a spatial sis susceptible infected susceptible reaction diffusion model and studied the existence uniqueness and particularly the asymptotic behavior of the endemic equilibrium as the diffusion rate of the susceptible individuals goes to zero in the case where a so called low risk subhabitat is created in this work we shall provide further understanding of the impacts of large and small diffusion rates of the susceptible and infected population on the persistence and extinction of the disease which leads us to determine the asymptotic behaviors of the endemic equilibrium when the diffusion rate of either the susceptible or infected population approaches to infinity or zero in the remaining cases consequently our results reveal that in order to eliminate the infected population at least in low risk area it is necessary that one will have to create a low risk subhabitat and reduce at least one of the diffusion rates to zero in this case our results also show that different strategies of controlling the diffusion rates of individuals may lead to very different spatial distributions of the population moreover once the spatial environment is modified to include a low risk subhabitat the optimal strategy of eradicating the epidemic disease is to restrict the diffusion rate of the susceptible individuals rather than that of the infected ones

proportional odds logistic regression effective means of dealing with limited uncertainty in dichotomizing clinical outcomes classifying a measurable clinical outcome as a dichotomous variable often involves difficulty with borderline cases that could fairly be assigned either of the two binary class memberships in such situations the indicated class membership is often highly subjective and subject to for instance a measurement error in other situations the intermediate level of a three level ordinal factor may sometimes be explicitly reserved for cases which could likely belong to either of the two binary classes such indefinite readings are often eliminated from the statistical analysis in this article we review conceptual and methodological aspects of employing proportional odds logistic regression for a three level ordinal factor as a suitable alternative to ordinary logistic regression when dealing with limited uncertainty in classifying clinical outcome as a binary variable

berezin li yau inequalities on domains on the sphere abstract we prove berezin li yau inequalities for the dirichlet and neumann eigenvalues on domains on the sphere s d 1 a sharp explicit bound for the sums of the neumann eigenvalues is obtained for all dimensions d in the case of s 2 we also obtain sharp lower bounds with correction terms for the vector laplacian and the stokes operator

on a conjecture of lin ni for a semilinear neumann problem let q be a bounded domain in rn n 3 and a 0 we consider au au u n 2 n 2 in q2 u 0 in q au av 0 on i2 and show that for a sufficiently small the minimal energy solutions are only constants

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

goodness of fit tests for ordinal response regression models it is well documented that the commonly used pearson chi square and deviance statistics are not adequate for assessing goodness of fit in logistic regression models when continuous covariates are modelled in recent years several methods have been proposed which address this shortcoming in the binary logistic regression setting or assess model fit differently however these techniques have typically not been extended to the ordinal response setting and few techniques exist to assess model fit in that case we present the modified pearson chi square and deviance tests that are appropriate for assessing goodness of fit in ordinal response models when both categorical and continuous covariates are present the methods have good power to detect omitted interaction terms and reasonable power to detect failure of the proportional odds assumption or modelling the wrong functional form of a continuous covariate these tests also provide immediate information as to where a model may not fit well in addition the methods are simple to understand and implement and are non specific that is they do not require prespecification of a type of lack of fit to detect

existence and nonexistence of least energy solutions of the neumann problem for a semilinear elliptic equation with critical sobolev exponent and a critical lower order perturbation let be a smooth bounded domain in rn with n 5 a 0 0 and 2 2nn 2 we show that the exponent q 2 n 1 n 2 plays a critical role regarding the existence of least energy or ground state solutions of the neumann problem u au u2 1 uq 1in u 0in u 0on namely we prove that when q 2 n 1 n 2 there exists an 0 0 such that the problem has a least energy solution if 0