bifurcations of traveling wave solutions for an integrable equation this paper deals with the following equation mt 1 2 1 mk xxx 1 2 1 mk x which is proposed by z j qiao j math phys 48 082701 2007 and qiao and liu chaos solitons fractals 41 587 2009 by adopting the phase analysis method of planar dynamical systems and the theory of the singular traveling wave systems to the traveling wave solutions of the equation it is shown that for different k the equation may have infinitely many solitary wave solutions periodic wave solutions kink antikink wave solutions cusped solitary wave solutions and breaking loop solutions we discuss in a detail the cases of k 2 12 12 2 and parametric representations of all possible bounded traveling wave solutions are given in the different c g parameter regions

cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

on the development of a high order compact scheme for exhibiting the switching and dissipative solution natures in the camassa holm equation in this paper a two step iterative solution algorithm for solving the camassa holm equation which involves only the first order derivative term is presented in each set of the u p and u m differential equations one is governed by the inviscid nonlinear convection reaction equation for the time evolving fluid velocity component along the horizontal direction the other equation is known as the inhomogeneous helmholtz equation the resulting reduction of differential order facilitates us to develop the flux discretization scheme in a stencil with comparatively fewer points for accurately predicting the unidirectional propagation of the shallow water wave the modified equation analysis for eliminating several leading discretization error terms and the fourier analysis for minimizing a particular type of wave like error are employed in this study the fifth order spatially accurate combined compact upwind scheme is developed in a three point stencil for approximating the first order derivative term for the purpose of retaining a long term accurate hamiltonian and multi symplectic geometric structures in camassa holm equation the time integrator or time stepping scheme chosen in this study should conserve symplecticity another main emphasis of conducting the present calculation of camassa holm equation is to shed light on the conservation of hamiltonians up to the time before wave breaking we also intended to elucidate the switching scenario by virtue of the peakon peakon interaction problem and the dissipative scenario after the time of head on collision in the peakon antipeakon interaction problem

new exact travelling wave solutions for the k 2 2 equation with osmosis dispersion abstract in this paper by using bifurcation method we successfully find the k 2 2 equation with osmosis dispersion u t u 2 x u 2 xxx 0 possess two new types of travelling wave solutions called kink like wave solutions and antikink like wave solutions they are defined on some semifinal bounded domains and possess properties of kink waves and anti kink waves their implicit expressions are obtained for some concrete data the graphs of the implicit functions are displayed and the numerical simulation of travelling wave system is made by maple the results show that our theoretical analysis agrees with the numerical simulation

bivariate survival models induced by frailties abstract when two observed survival times depend via a proportional hazards model on the same unobserved variable called a frailty this common dependence induces an association between the observed times this article considers the class of bivariate survival distributions that can arise in this way extends it to allow negative association and shows that the observable bivariate distribution determines the unobserved frailty distribution up to a scale parameter a cross ratio function easily estimated even from censored data is the key to both the characterization results and inferential procedures and diagnostic plots that are introduced and illustrated by real examples the models considered are the natural counterpart for survival data of the latent variable models that have long been used in factor analysis for continuous data or in latent structure analysis for binary data the main distinguishing feature of survival data is the possibility of censoring which makes the standard methods difficu

a class of weighted dependence measures for bivariate failure time data this paper considers a class of summary measures of the dependence between a pair of failure time variables over a finite follow up region the class consists of measures that are weighted averages of local dependence measures and includes the cross ratio measure and finite region version of kendall s recently proposed by the authors two new special cases are identified that can avoid the need to estimate the bivariate survivor function and that admit explicit variance estimators nonparametric estimators of such dependence measures are proposed and are shown to be consistent and asymptotically normal with variances that can be consistently estimated properties of selected estimators are evaluated in a simulation study and the method is illustrated through an analysis of australian twin study data

traveling wave solutions of the camassa holm equation all weak traveling wave solutions of the camassa holm equation are classified we show that in addition to smooth solutions there are a multitude of traveling waves with singularities peakons cuspons stumpons and composite waves

exact solitary and periodic wave solutions of the k 2 2 equation defocusing branch an auxiliary elliptic equation method is presented for constructing exact solitary and periodic travelling wave solutions of the k 2 2 equation defocusing branch some known results in the literature are recovered more efficiently and some new exact travelling wave solutions are obtained also new stationary wave solutions are obtained

compacton peakon cuspons loop solutions and smooth solitons for the generalized kp mew equation in this paper we investigate all possible single traveling solitary wave solutions of the generalized kp mew equation u t u 2 x u 2 x x t x u y y 0 under boundary condition lim u a regular compacton solution of the generalized kp mew equation correspond to the case of a 0 in the case of a 0 we find new exact soliton solutions including loop solution cusped soliton peaked soliton and smooth soliton solutions the parametric conditions of existence of the loop solution cusped soliton peaked soliton smooth soliton and compacton solutions are given by using the phase portrait analytical technique mathematical analysis and numerical simulations are provided for these soliton solutions of the generalized kp mew equation we show some graphs to explain these solutions

single peak solitary wave solutions for the osmosis k 2 2 equation under inhomogeneous boundary condition abstract the qualitative theory of differential equations is applied to the k 2 2 equation with osmosis dispersion smooth peaked and cusped solitary wave solutions of the osmosis k 2 2 equation under inhomogeneous boundary condition are obtained the parametric conditions of existence of the smooth peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique asymptotic analysis and numerical simulations are provided for smooth peaked and cusped solitary wave solutions of the osmosis k 2 2 equation

regression analysis of multivariate incomplete failure time data by modeling marginal distributions abstract many survival studies record the times to two or more distinct failures on each subject the failures may be events of different natures or may be repetitions of the same kind of event in this article we consider the regression analysis of such multivariate failure time observations each marginal distribution of the failure times is formulated by a cox proportional hazards model no specific structure of dependence among the distinct failure times on each subject is imposed the regression parameters in the cox models are estimated by maximizing the failure specific partial likelihoods the resulting estimators are shown to be asymptotically jointly normal with a covariance matrix that can be consistently estimated simultaneous inferential procedures are then proposed extensive monte carlo studies indicate that the normal approximation is adequate for practical use the new methods allow time dependent covariates missing observations and arbitrary patterns of censorship they are illustrat

inferences on the association parameter in copula models for bivariate survival data we investigate two stage parametric and two stage semi parametric estimation procedures for the association parameter in copula models for bivariate survival data where censoring in either or both components is allowed we derive asymptotic properties of the estimators and compare their performance by simulations both parametric and semi parametric estimators of the association parameter are efficient at independence and the parameter estimates in the margins have high efficiency and are robust to misspecification of dependency structures in addition we propose a consistent variance estimator for the semi parametric estimator of the association parameter we apply the proposed methods to an aids data set for illustration

asymptotically optimal definite quadrature formulae of 4th order we construct several sequences of asymptotically optimal definite quadrature formulae of fourth order and evaluate their error constants besides the asymptotical optimality an advantage of our quadrature formulae is the explicit form of their weights and nodes for the remainders of our quadrature formulae monotonicity properties are established when the integrand has a non negative or non positive fourth derivative and a posteriori error estimates are proven

semiparametric estimation of marginal hazard function from case control family studies estimating marginal hazard function from the correlated failure time data arising from case control family studies is complicated by noncohort study design and risk heterogeneity due to unmeasured shared risk factors among the family members accounting for both factors in this article we propose a two stage estimation procedure at the first stage we estimate the dependence parameter in the distribution for the risk heterogeneity without obtaining the marginal distribution first or simultaneously assuming that the dependence parameter is known at the second stage we estimate the marginal hazard function by iterating between estimation of the risk heterogeneity frailty for each family and maximization of the partial likelihood function with an offset to account for the risk heterogeneity we also propose an iterative procedure to improve the efficiency of the dependence parameter estimate the simulation study shows that both methods perform well under finite sample sizes we illustrate the method with a case control family study of early onset breast cancer

explicit peakon and solitary wave solutions for the modified fornberg whitham equation in this paper the modified fornberg whitham equation is studied by using the bifurcation theory and the method of phase portraits analysis in some parametric conditions some peakons and solitary waves are found and their exact parametric representations in explicit form are obtained

marginal hazards regression for retrospective studies within cohort with possibly correlated failure time data a retrospective dental study was conducted to evaluate the degree to which pulpal involvement affects tooth survival due to the clustering of teeth the survival times within each subject could be correlated and thus the conventional method for the case control studies cannot be directly applied in this article we propose a marginal model approach for this type of correlated case control within cohort data weighted estimating equations are proposed for the estimation of the regression parameters different types of weights are also considered for improving the efficiency asymptotic properties of the proposed estimators are investigated and their finite sample properties are assessed via simulations studies the proposed method is applied to the aforementioned dental study

nilpotent singular points and compactons abstract in this paper dynamical system theory is applied to several types of fully nonlinear wave equations these equations can be reduced to planar polynomial differential systems by transformation of variables we treat these polynomial differential systems by phase space analytical technique the results of our study demonstrate that there exist close connection between nilpotent singular points and compactons moreover we find some new elliptic function compactons instead of well known trigonometric function compactons by analyzing nilpotent points two new compactons induced by singular elliptic are also obtained

single peak solitary wave solutions for the generalized kp mew 2 2 equation under boundary condition the qualitative theory of differential equations is applied to the kp mew 2 2 equation q t q 2 x q 2 x x t x q y y 0 our procedure shows that the kp mew 2 2 equation either has the regular peakon soliton cuspon soliton and smooth soliton solutions when sitting on the non zero constant pedestal lim x q x a 0 or possesses compacton solutions only when lim x q x a 0 in particular we present mathematical analysis and numerical simulations are provided for those peakon cuspon compacton loop soliton and smooth soliton solutions

special exact soliton solutions for the k 2 2 equation with non zero constant pedestal abstract special exact solutions of the k 2 2 equation u t u 2 x u 2 xxx 0 are investigated by employing the qualitative theory of differential equations our procedure shows that the k 2 2 equation either has loop soliton cusped soliton and smooth soliton solutions when sitting on the non zero constant pedestal lim x u a 0 or possesses compacton solutions only when lim x u 0 mathematical analysis and numerical simulations are provided for these soliton solutions of the k 2 2 equation

on robustness of marginal regression coefficient estimates and hazard functions in multivariate survival analysis of family data when the frailty distribution is mis specified the shared frailty model is an extension of the cox model to correlated failure times and essentially a random effects model for failure time outcomes in this model the latent frailty shared by individual members in a cluster acts multiplicatively as a factor on the hazard function and is typically modelled parametrically one commonly used distribution is gamma where both shape and scale parameters are set to be the same to allow for unique identification of baseline hazard function it is popular because it is a conjugate prior and the posterior distribution possesses the same form as gamma in addition the parameter can be interpreted as a time independent cross ratio function a natural extension of odds ratio to failure time outcomes in this paper we study the effect of frailty distribution mis specification on the marginal regression estimates and hazard functions under assumed gamma distribution with an application to family studies the simulation results show that the biases are generally 10 and lower even when the true frailty distribution deviates substantially from the assumed gamma distribution this suggests that the gamma frailty model can be a practical choice in real data analyses if the regression parameters and marginal hazard function are of primary interest and individual cluster members are exchangeable with respect to their dependencies

on the monotonicity of sequences of quadrature formulae leti f l f k 0 r 0 vk 1 a k f x k be a quadrature formula and let s n f n 1 be successive approximations of the definite integrali f 0 1 f x dx obtained by the composition ofl i e s n f l n where varphi n x frac 1 n sum nolimits k 0 n 1 f left frac k x n right we prove sufficient conditions for monotonicity of the sequence s n f n 1 as particular cases the monotonicity of well known newton cotes and gauss quadratures is shown finally a recovery theorem based on the monotonicity results is presented

applications of propagation of long wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized kadomtsev petviashvili modified equal width dynamical equation abstract in this research work we constructed the solitary wave solutions of generalized kadomtsev petviashvili modified equal width kp mew equation with the help of new technique which is modification form of extended auxiliary equation mapping method the generalized kp mew equation is the nonlinear pdes which described the propagation of long wave with dissipation and dispersion in nonlinear media as a result families of solitary wave solutions are obtained in different form of solitons bright dark solitons and traveling wave solutions the physical structure of these new solutions is shown graphically in two and three dimensions with the aid of computer software mathematica these obtained new solutions show the power and effectiveness of this new method we can also solve other nonlinear system of pdes which are involved in mathematical physics and many other branches of physical sciences with the help of this new method

complexiton solutions to integrable equations abstract complexiton solutions or complexitons for short are exact solutions newly introduced to integrable equations are newly introduced starting with the solution classification for a linear differential equation the korteweg de vries equation and the toda lattice equation are considered as examples to exhibit complexiton structures of nonlinear integrable equations the crucial step in the solution process is to apply the wronskian and casoratian techniques for hirota s bilinear equations correspondence between complexitons of the korteweg de vries equation and complexitons of the toda lattice equation is provided

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

dynamics of the generalized kp mew burgers equation with external periodic perturbation the generalized kadomtsevpetviashvili modified equal width burgers kp mew burgers equation is introduced for the first time the qualitative change of the traveling wave solutions of the kp mew burgers equation is studied using numerical simulations considering an external periodic perturbation the periodic and chaotic motions of the perturbed kp mew burgers equation are investigated by using the phase projection analysis time series analysis poincare section and bifurcation diagram the parameter a nonlinear coefficient plays a crucial role in the periodic motions and chaotic motions through period doubling route to chaos of the perturbed kp mew burgers equation

proportional hazards tests and diagnostics based on weighted residuals summary nonproportional hazards can often be expressed by extending the cox model to include time varying coefficients e g for a single covariate the hazard function for subject i is modelled as exp fl t zi t a common example is a treatment effect that decreases with time we show that the function 3 t can be directly visualized by smoothing an appropriate residual plot also many tests of proportional hazards including those of cox 1972 gill schumacher 1987 harrell 1986 lin 1991 moreau o quigley mesbah 1985 nagelkerke oosting hart 1984 o quigley pessione 1989 schoenfeld 1980 and wei 1984 are related to time weighted score tests of the proportional hazards hypothesis and can be visualized as a weighted least squares line fitted to the residual plot

frank s family of bivariate distributions summary this paper examines the properties of a new class of bivariate distributions whose members are stochastically ordered and likelihood ratio dependent the proposed class can be used to construct bivariate families of distributions whose marginals are arbitrary and which include the frechet bounds as well as the distribution corresponding to independent variables three nonparametric estimators of the association parameter are suggested and monte carlo experiments are used to compare their small sample behaviour to that of the maximum likelihood estimate

logistic regression of family data from case control studies summary multivariate regression models are applied to binary disease data in families identified from case control studies attention is restricted to marginal or reproducible models i e those whose parameters have the same interpretations in the marginal distributions for all subsets of a family with a logistic specification of each individual s marginal disease probability for such models it is shown that the case control family data can be analysed as if they were obtained from a prospective study with the baseline disease probabilities of case and control probands differing from that of their relatives this result extends that of anderson 1972 and prentice pyke 1979 for the probands data to include disease outcomes and covariates for their families it contrasts with inconsistent estimates of parameters in nonreproducible models that result when the case control sampling design is ignored tosteson rosner redline 1991 the contrast underscores the need to check the plausibility of the reproducibility assumption which requires that the covariates be independent of any unmeasured factors responsible for the correlation of familial disease occurrence before analysing case control data prospectively

exit criteria and monotonicity in compound quadrature of gaussian type in this short note for compound quadrature rules of gaussian type we prove stopping rules and monotonicity results based on peano kernel methods

on dependence estimation using correlated failure time data from case control family studies correlated failure time data arise routinely in case control family studies to measure the dependence between bivariate failure times a cross ratio i e conditional hazard ratio function has been used many approaches ranging from estimating equations to likelihood score equations have been proposed for the estimation of parameters in the cross ratio function from prospective family data however the adaptation of these approaches to case control family data appears too complicated and computationally intensive to be practical in this paper a connection is established between the cross ratio function and the relative risk function in a stratified proportional hazards model leading to the development of a set of partial likelihood based estimating equations a simulation study is conducted to evaluate the performance of this estimation procedure and two datasets are analysed to illustrate this method

the camassa holm hierarchy n dimensional integrable systems and algebro geometric solution on a symplectic submanifold this paper shows that the camassa holm ch spectral problem yields two different integrable hierarchies of nonlinear evolution equations nlees one is of negative order ch hierachy while the other one is of positive order ch hierarchy the two ch hierarchies possess the zero curvature representations through solving a key matrix equation we see that the well known ch equation is included in the negative order ch hierarchy while the dym type equation is included in the positive order ch hierarchy furthermore under two constraint conditions between the potentials and the eigenfunctions the ch spectral problem is cast in 1 a new neumann like n dimensional system when it is restricted into a symplectic submanifold of 2n which is proven to be integrable by using the dirac poisson bracket and the r matrix process and 2 a new bargmann like n dimensional system when it is considered in the whole 2n which is proven to be integrable by using the standard poisson bracket and the r matrix process

a new integrable equation with cuspons and w m shape peaks solitons in this paper we propose a new completely integrable wave equation mt mx u2 ux2 2m2ux 0 m u uxx the equation is derived from the two dimensional euler equation and is proven to have lax pair and bi hamiltonian structures this equation possesses new cusp solitons cuspons instead of regular peakons ce x ct with speed c through investigating the equation we develop a new kind of soliton solutions w m shape peaks solitons there exist no smooth solitons for this integrable water wave equation