a proportional hazards model for interval censored failure time data this paper develops a method for fitting the proportional hazards regression model when the data contain left right or interval censored observations results given for testing the hypothesis of a zero regression coefficient lead to a generalization of the log rank test for comparison of several survival curves the method is used to analyze data from an animal tumorigenicity study and also a clinical trial

sharp estimates for solutions of multi bubbles in compact riemann surfaces in this paper we consider a sequence of multibubble solutions u k of the equation 0 1 0 u k k he uk f m he u k d 1 0 in m where h is a c 2 positive function in a compact riemann surface m and k is a constant satisfying lim k k 8m for some positive integer m 1 we prove among other things that k 8m 2 m m j 1h 1 p k j 0 log h p k j 8m 2k p k j k j e k j o e k j where p k j are centers of the bubbles of u k and k j are the local maxima of u k after adding a constant this yields a uniform bound of solutions as k converges to 8m from below provided that 0 log h p k j 8m 2k p k j 0 it generalizes a previous result due to ding jost li and wang 18 and nolasco and tarantello 31 which says that any sequence of minimizers u k is uniformly bounded if k 0 for any maximum point p of the sum of 2 log h and the regular part of the green function where k is the gaussian curvature of m the analytic work of this paper is the first step toward computing the topological degree of 0 1 which was initiated by li 24

taylor collocation method and convergence analysis for the volterra fredholm integral equations in this paper the taylor collocation method is presented for numerically solving the volterra fredholm integral equations in terms of taylor polynomials this method transforms the integral equation to a matrix equation via the collocation points the convergence analysis of the collocation solution on the integral equation is given the method is computationally attractive and applications are demonstrated through illustrative examples

an improved geometric inequality via vanishing moments with applications to singular liouville equations we consider a class of singular liouville equations on compact surfaces motivated by the study of electroweak and self dual chern simons theories the gaussian curvature prescription with conical singularities and onsager s description of turbulence we analyse the problem of existence variationally and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the moser trudinger inequality and in turn to lower bounds on the euler lagrange functional we then discuss existence and non existence results

a taylor collocation method for the approximate solution of general linear fredholm volterra integro difference equations with mixed argument abstract in this paper a taylor collocation method is developed to find an approximate solution of high order linear fredholm volterra integro difference equations with variable coefficients under the mixed conditions the solution is obtained in terms of taylor polynomial near any point also examples are presented which illustrate the partinent features of the method and the results are discussed

regression analysis of interval censored survival data with covariates using log linear models we considered the regression analysis of the event time data with left right or interval censored observations we extended life table techniques for censored survival data using log linear models to incorporate interval censored failures the em algorithm was used to calculate maximum likelihood estimates for the parameters we assumed that the hazard function was a stepwise function over disjoint intervals of time thus the nonparametric model the parametric exponential model and the semiparametric cox proportional hazard model were easily implemented as special cases we adapted the restricted em algorithm to test hypotheses and to construct confidence intervals for the parameters these methods were applied in an analysis of the recurrence time for treated melanoma patients

two dimensional solutions of a mean field equation on flat tori abstract we study the mean field equation on the flat torus t c z z u e u t e u 1 t 0 where is a real parameter for a general flat torus we obtain the existence of two dimensional solutions bifurcating from the trivial solution at each eigenvalue up to a multiplicative constant t of laplace operator on the torus in the space of even symmetric functions we further characterize the subset of all eigenvalues through which only one bifurcating curve passes finally local convexity near bifurcating points of the solution curves are obtained

the approximate solution of high order linear difference equations with variable coefficients in terms of taylor polynomials this paper presents a numerical method for the approximate solution of mth order linear difference equations with variable coefficients under the mixed conditions in terms of taylor polynomials about any point in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

constant q curvature metrics in arbitrary dimension working in a given conformal class we prove existence of constant q curvature metrics on compact manifolds of arbitrary dimension under generic assumptions the problem is equivalent to solving a nth order non linear elliptic differential or integral equation with variational structure where n is the dimension of the manifold since the corresponding euler functional is in general unbounded from above and below we use critical point theory jointly with a compactness result for the above equation

asymptotic estimate for a perturbed scalar curvature equation abstract we consider the equation u v u n 2 n 2 w u with n n 2 n 2 n 2 and we give some minimal conditions on v and w for having a uniform estimate for their solutions when 0

on the solutions of liouville systems for some prescribed m 0 together with the dirichlet boundary condition u 0 on 0 the liouville equation finds applications in various fields of physics and mathematics to name a few it represents the newtonian potential of a cluster of self gravitating mass distribution where a 0 1 6 36 37 the electric potential induced by charge carriers in the theory of electrolytes where a 0 31 and maximal entropy solutions of the incompressible euler equation 23 9 in the field of differential geometry it stands for the problem of finding a metric corresponding to a prescribed gaussian curvature 3 10 the system analog of l takes the form

compactness of solutions to some geometric fourth order equations we prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds the proof is based on a refined bubbling analysis for which the main estimates are given in integral form our result is used in a subsequent paper to find critical points via minimax arguments of some geometric functional which give rise to conformal metrics of constant q curvature as a byproduct of our method we also obtain compactness of such metrics

a flexible computationally efficient method for fitting the proportional hazards model to interval censored data the proportional hazards model ph is currently the most popular regression model for analyzing time to event data despite its popularity the analysis of interval censored data under the ph model can be challenging using many available techniques this article presents a new method for analyzing interval censored data under the ph model the proposed approach uses a monotone spline representation to approximate the unknown nondecreasing cumulative baseline hazard function formulating the ph model in this fashion results in a finite number of parameters to estimate while maintaining substantial modeling flexibility a novel expectation maximization em algorithm is developed for finding the maximum likelihood estimates of the parameters the derivation of the em algorithm relies on a two stage data augmentation involving latent poisson random variables the resulting algorithm is easy to implement robust to initialization enjoys quick convergence and provides closed form variance estimates the performance of the proposed regression methodology is evaluated through a simulation study and is further illustrated using data from a large population based randomized trial designed and sponsored by the united states national cancer institute

chebyshev polynomial solutions of systems of high order linear differential equations with variable coefficients a chebyshev collocation method has been presented for numerically solving systems of high order linear ordinary differential equations with variable coefficients using the chebyshev collocation points this method transforms the ode system and the given conditions to matrix equations with unknown chebyshev coefficients by means of the obtained matrix equations a new system of equations which corresponds to the system of linear algebraic equations is gained hence by finding the chebyshev coefficients easily the finite chebyshev series approach is obtained

ignorability and coarse data we present a general statistical model for data coarsening which includes as special cases rounded heaped censored partially categorized and missing data formally with coarse data observations are made not in the sample space of the random variable of interest but rather in its power set grouping is a special case in which the degree of coarsening is known and nonstochastic we establish simple conditions under which the possible stochastic nature of the coarsening mechanism can be ignored when drawing bayesian and likelihood inferences and thus the data can be validly treated as grouped data the conditions are that the data be coarsened at random a generalization of the condition missing at random and that the parameters of the data and the coarsening process be distinct applications of the general model and the ignorability condition are illustrated in a numerical example and described briefly in a variety of special cases

a weighted logistic regression model for estimation of recurrence of adenomas in a colorectal polyp prevention trial some participants might have their follow up colonoscopy conducted before the scheduled time i e at the end of the trial this results in variable follow up lengths for participants and the data of recurrence status at the end of the trial can be considered as current status data in this paper we use a weighted logistic regression model to estimate recurrence rate of adenoma data at the end of the trial the weights are used to adjust for variable follow up we show that logistic regression tends to underestimate recurrence rate in a simulation study we show that kaplan meier estimator derived from the right endpoint of the current status data tends to overestimate recurrence rate in contrast to logistic regression and the weighted logistic regression method can produce reasonable estimates of recurrence rate even under a high non compliance rate compared to conventional logistic regression and kaplan meier estimator the method described here is illustrated with an example from a colon cancer study

a new polynomial approach for solving difference and fredholm integro difference equations with mixed argument the purpose of this study is to give a taylor polynomial approximation for the solution of high order linear fredholm difference equations with variable coefficients and mixed argument under the mixed conditions about any point for this purpose a taylor matrix method is introduced this method is based on first taking the truncated taylor expansions of the functions in the difference equation and then substituting their matrix forms into the given equation hence the result matrix equation can be solved and the unknown taylor coefficients can be found approximately also examples that illustrate the pertinent features of the method are presented and the results are discussed

a class of estimators of the survival function from interval censored data a model of interval censorship of a failure time t is considered when there is only one inspection time y the observable data are n independent copies of the pair y where t y we construct a class of self consistent estimators of the survival function of t defined implicitly through two equations and show their strong consistency under certain conditions the properties of the nonparametric maximum likelihood estimator are also investigated

supercritical mean field equations on convex domains and the onsager s statistical description of two dimensional turbulence we are motivated by the study of the microcanonical variational principle within onsager s description of two dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails we obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding mean field equation on convex and thin enough domains in the supercritical with respect to the moser trudinger inequality regime this is a brand new achievement since existence results in the supercritical region were previously known only on multiply connected domains we then study the structure of these solutions by the analysis of their linearized problems and we also obtain a new uniqueness result for solutions of the mean field equation on thin domains whose energy is uniformly bounded from above finally we evaluate the asymptotic expansion of those solutions with respect to the thinning parameter and combining it with all the results obtained so far we solve the microcanonical variational principle in a small range of supercritical energies where the entropy is shown to be concave

an estimate on the blowing up solutions of a fourth order equation abstract in this short paper i prove a harnack type inequality of the blowing up solutions for a class of fourth order equations with exponential growth on a compact four manifold the main method i use is the moving plane method

a sup inf type inequality for conformal metrics on riemann surfaces with conical singularities abstract a sup inf type inequality is proved for the regular part of conformal factors for riemannian metrics on surfaces with conical singularities of positive order 0 the proof is based on the explicit representation formula for solutions of the singular liouville equation and generalizes to the singular case an argument by i shafrir

a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities let u be a solution of the equation u v x eu in a domain r2 where 0 a v b and v is lipschitz continuous we prove that sup u can be controlled in terms of inf u more precisely supku inf u c a b k v l for any compact subset k this extends an earlier result of shafrir who obtained a similar conclusion when v 1

on a sharp sobolev type inequality on two dimensional compact manifolds motivated by the asymptotic analysis of double vortex condensates in the chern simons higgs theory we construct a suitable minimizing sequence for a sharp sobolev inequality a la moser for two dimensional compact manifolds as a consequence we first obtain a direct proof of the sharp character of such an inequality secondly and more interestingly we use such minimizing sequence to show that for the flat torus the corresponding extremal problem attains its infimum

application of fibonacci collocation method for solving volterra fredholm integral equations in this paper a new matrix method based on fibonacci polynomials and collocation points is proposed for numerically solving the volterra fredholm integral equations in fact the approximate solution of the problem in the truncated fibonacci series form is obtained by this method also convergence analysis of the proposed method is provided under several mild conditions the reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments

ignorability and coarse data some biomedical examples heitjan and rubin 1991 annals of statistics 19 2244 2253 define data to be coarse when one observes not the exact value of the data but only some set a subset of the sample space that contains the exact value this definition covers a number of incomplete data problems arising in biomedicine including rounded heaped censored and missing data in analyzing coarse data it is common to proceed as though the degree of coarseness is fixed in advance in a word to ignore the randomness in the coarsening mechanism when coarsening is actually stochastic however inferences that ignore this randomness may be seriously misleading heitjan and rubin 1991 have proposed a general model of data coarsening and established conditions under which it is appropriate to ignore the stochastic nature of the coarsening the conditions are that the data be coarsened at random a generalization of missing at random rubin 1976 biometrika 63 581 592 and that the parameters of the data and the coarsening process be distinct this article presents detailed applications of the general model and the ignorability conditions to a variety of coarse data problems arising in biomedical statistics a reanalysis of the stanford heart transplant data crowley and hu 1977 journal of the american statistical association 72 27 36 reveals significant evidence that censoring of pretransplant survival times by transplantation was nonignorable suggesting a greater benefit from cardiac transplantation than previous analyses had found

a quantization property for blow up solutions of singular liouville type equations abstract motivated by the study of self dual vortices in gauge field theory cf vortices and monopoles birkhauser boston 1980 solitons in field theory and nonlinear analysis monographs in mathematics springer new york 2001 we consider a concentrating solution sequence u k satisfying u k z 2 k v k z e u k in b 1 z 2 k v k z e u k z 0 weakly in the sense of measures in b 1 where i k r k ii v k c 0 1 b 1 0 a v k b and v k a in b 1 we prove that necessarily 8 n 8 1 8 z the result is sharp as shown by explicit examples and should be compared with that obtained by li shafrir ind univ math j 43 4 1994 1255 concerning the situation where k 0 k n

topological degree for solutions of fourth order mean field equations we consider the following fourth order mean field equation with navier boundary condition delta 2 u rho frac h x e u int omega h e u rm in omega quad u delta u 0 rm on partial omega quad quad quad quad quad quad quad quad quad quad where h is a c2 positive function is a bounded and smooth domain in mathbb r 4 we prove that for rho in 32m sigma 3 32 m 1 sigma 3 the degree counting formula for is given by d rho left begin array ll frac 1 m chi omega 1 cdot cdot cdot chi omega m rm for m 0 1 rm for m 0 end array right where is the euler characteristic of similar result is also proved for the corresponding dirichlet problem delta 2 u rho frac h x e u int omega h e u quad rm in omega quad u nabla u 0 quad rm on partial omega quad quad quad quad quad quad quad quad quad quad

a sup inf inequality near r 0 let be a domain in r2 not necessarily bounded consider the semi linear elliptic equation u r x eu x we prove that for any compact subset k of there is a constant c such that the inequality 1 supku inf u c holds for all solutions u this type of inequality was first established by brezis li and shafrir h brezis y y li i shafrir a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities j funct anal 115 1993 344 358 under the assumption that r x is positive and bounded away from zero it has become a useful tool in estimating the solutions of semi linear elliptic equations either in euclidean spaces or on riemannian manifolds see h brezis y y li i shafrir a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities j funct anal 115 1993 344 358 c c chen c s lin a sharp sup inf inequality for a nonlinear elliptic equation in r2 comm anal geom 6 1998 1 19 w chen c li gaussian curvature in the negative case proc amer math soc 131 2003 741 744 w chen c li indefinite elliptic problems with critical exponent in advances in non linear pde and related areas world scientific 1998 pp 67 79 y y li i shafrir blow up analysis for solutions of u veu in dimension two indiana univ math j 43 1994 1255 1270 in brezis li and shafrir s result the constant c depends on the lower bound of the function r x in this paper we remove this restriction and extend the inequality to the case where r x is allowed to have zeros so that it can be applied to obtain a priori estimates for a broader class of equations as we will illustrate in the last section the key to prove this inequality is the analysis of asymptotic decay near infinity of solutions for a corresponding limiting equation in r2 which is interesting and useful in its own right

power series cure rate model for spatially correlated interval censored data based on generalized extreme value distribution abstract in this work we propose a flexible cure rate model to allow for spatial correlations by including spatial frailty in the interval censored data setting the proposed model is quite flexible and generalizes the bernoulli geometric poisson and logarithmic models it can be tested for the best fit in a straightforward way our approach enables different underlying activation mechanisms that lead to the event of interest and the number of competing causes that can be responsible for the occurrence of the event of interest follows a flexible exponential discrete power series distribution mcmc methods are used in bayesian inference for the proposed models and bayesian comparison criteria are used for model comparison moreover we conduct influence diagnostics through the diagnostic measures in order to detect possible influential or extreme observations that can cause distortions in the analysis results finally the proposed models are used to analyze a real dataset

remission duration an example of interval censored observations remission duration data from two observational studies the german all aul study on acute lymphoblastic and acute undifferentiated leukaemia and the kiel lymphoma study are used to demonstrate the adequacy of an interval censoring approach the turnbull estimator is contrasted with the conventionally used kaplan meier estimator in addition a parametric model is used for estimation or simulation of the delay times of complete remission diagnosis and relapse diagnosis two possible consequences of the conventional approach are discussed biased estimation for example overestimation of remission duration and underestimation of the true error variance which may lead to false positive results on the other hand the applicability of the interval censoring approach is doubtful when the censoring mechanism examination pattern patient s behaviour is confounded with the endpoints

generalized taylor polynomials for axisymmetric plates and shells this work proposes the use of a mesh free technique derived from the generalized taylor polynomials for the analysis of axisymmetric plates and shells the primary solution variable s is are assumed to take the form of a truncated taylor series around a point c and the unknown coefficients of the expansion are determined using the governing differential equation s and boundary conditions the method is free of shape parameter calibration needed in some other famous mesh free techniques such as the rbf and is quite easy to formulate and program successful application of the method to several benchmark problems of axisymmetric plate and shell structures proves its robustness the results have been verified using the existing rigorous analytical solutions that are in most cases not suited to practical engineering calculations