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 a system of klein gordon type equations with acoustic boundary conditions we prove the existence uniqueness and uniform stabilization of global solutions for a generalized system of klein gordon type equations with acoustic boundary conditions on a portion of the boundary and the dirichlet boundary condition on the rest

action of the loop group on the self dual yang mills equation recently kac moody symmetry has played an important role in mathematical physics dolan and chau ge and wu discovered an infinitesimal action of the kac moody lie algebra on the space of solutions of sdym we have discovered an action of the loop group on the space of generalized solutions of sdym which exponentiates the kac moody action the group acts by adding a special type of source onto the solution the action is a geometric construction using the twistor picture

invariant recurrence relations for cp n 1 models in this paper we present invariant recurrence relations for the completely integrable cp n 1 euclidean sigma model in two dimensions defined on the riemann sphere s 2 when its action functional is finite we determine the links between successive projection operators wavefunctions of the linear spectral problem and immersion functions of surfaces in the su n algebra together with the outlines of the proofs our formulation preserves the conformal and scaling invariance of these quantities certain geometrical aspects of these relations are described we also discuss the singularities of the meromorphic solutions of the cp n 1 model and show that they do not affect the invariant quantities we illustrate the construction procedure through the examples of the cp 2 and cp 3 models

some nonlinear wave equations with acoustic boundary conditions abstract we prove the existence and uniqueness of global solutions to the mixed problem for the carrier equation u tt m u 2 dx u u t u t 0 with acoustic boundary conditions

inference for a nonlinear counting process regression model martingale and counting process techniques are applied to the problem of inference for general conditional hazard functions this problem was first studied by beran who introduced a class of estimators for the conditional cumulative hazard and survival functions in the special case of time independent covariates here the covariate can be time dependent the classical i i d assumptions are relaxed by replacing them with certain asymptotic stability assumptions and models involving recurrent failures are included this is done within the framework of a general nonparametric counting process regression model important examples of the model include right censored survival data semi markov processes an illness death process with duration dependence and age dependent birth and death processes

backlund transformations and loop group actions we construct a local action of the group of rational maps from 2 to gl n on local solutions of flows of the zs akns sl n hierarchy we show that the actions of simple elements linear fractional transformations give local backlund transformations and we derive a permutability formula from different factorizations of a quadratic element we prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities however the action of the subgroup of the rational maps that satisfy the u n reality condition on the space of global rapidly decaying solutions of the flows in the u n hierarchy is global and the action of a simple element gives a global backlund transformation the actions of certain elements in the rational loop group on the vacuum give rise to explicit time periodic multisolitons multibreathers we show that this theory generalizes the classical backlund theory of the sine gordon equation the group structures of backlund transformations for various hierarchies are determined by their reality conditions we identify the reality conditions the group structures for the sl n u k n k kdv kupershmidt wilson and gel fand dikii hierarchies the actions of linear fractional transformations that satisfy a reality condition modulo the center of the group of rational maps give backlund and darboux transformations for the hierarchy defined by the reality condition since the factorization cannot always be carried out under this reality condition the action is again local and backlund transformations only generate local solutions for these hierarchies unless singular solutions are allowed 2000 john wiley sons inc

the cauchy problem of the ward equation abstract we generalize the results of j villarroel the inverse problem for ward s system stud appl math 83 1990 211 222 a s fokas t a ioannidou the inverse spectral theory for the ward equation and for the 2 1 chiral model comm appl anal 5 2001 235 246 b dai c l terng k uhlenbeck on the space time monopole equation arxiv math dg 0602607 to study the inverse scattering problem of the ward equation with non small data and solve the cauchy problem of the ward equation with a non small purely continuous scattering data

backlund transformations ward solitons and unitons the ward equation also called the modified 2 1 chiral model is obtained by a dimension reduction and a gauge fixing from the self dual yang mills field equation on r it has a lax pair and is an integrable system ward constructed solitons whose extended solutions have distinct simple poles he also used a limiting method to construct 2 solitons whose extended solutions have a double pole ioannidou and zakrzewski and anand constructed more soliton solutions whose extended solutions have a double or triple pole some of the main results of this paper are i we construct algebraic backlund transformations bts that generate new solutions of the ward equation from a given one by an algebraic method ii we use an order k limiting method and algebraic bts to construct explicit ward solitons whose extended solutions have arbitrary poles and multiplicities iii we prove that our construction gives all solitons of the ward equation explicitly and the entries of ward solitons must be rational functions in x y and t iv since stationary ward solitons are unitons our method also gives an explicit construction of all k unitons from finitely many rational maps from c to c

approximation theorems for the propagators of higher order abstract cauchy problems in this paper we present two quite general approximation theorems for the propagators of higher order in time abstract cauchy problems which extend largely the classical trotter kato type approximation theorems for strongly continuous operator semigroups and cosine operator functions then we apply the approximation theorems to deal with the second order dynamical boundary value problems

cumulative regression function tests for regression models for longitudinal data the longitudinal regression model y i j m v i ti j e i j where y i j is the jth measurement of the ith subject at random time i j m is the regression function v i j i is a predictable covariate process observed at time i j and e i j is noise often provides an adequate framework for modeling and comparing groups of data the proposed longitudinal regression model is based on marked point process theory and allows a quite general dependency structure among the observations in this paper we find the asymptotic distribution of the cumulative regression function crf and present a nonparametric test to compare the regression functions for two groups of longitudinal data the proposed test denoted the crf test is based on the cumulative regression function crf and is the regression equivalent of the log rank test in survival analysis we show as a special case that the crf test is valid for groups of independent identically distributed regression data apart from the crf test we also consider a maximal deviation statistic that may be used when the crf test is inefficient

periodic and homoclinic solutions of the modified 2 1 chiral model we use algebraic backlund transformations bts to construct explicit solutions of the modified 2 1 chiral model from t2 r to su n where t2 is a 2 torus algebraic bts are parametrized by z c poles and holomorphic maps from t2 to gr k cn we apply backlund transformations with carefully chosen poles and s to construct infinitely many solutions of the 2 1 chiral model that are i doubly periodic in space variables and periodic in time i e triply periodic ii homoclinic in the sense that the solution u has the same stationary limit u0 as t and is tangent to a stable linear mode of u0 as t and is tangent to an unstable mode of u0 as t

on the wave equation with semilinear porous acoustic boundary conditions the goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary interior sources and a nonlinear boundary interior damping first applying the nonlinear semigroup theory we show the existence and uniqueness of local in time solutions the main difficulty in proving the local existence result is that the neumann boundary conditions experience loss of regularity due to boundary sources using an approximation method involving truncated sources and adapting the ideas in lasiecka and tataru 1993 28 we show that the existence of solutions can still be obtained second we prove that under some restrictions on the source terms then the local solution can be extended to be global in time in addition it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ode and depends on the behavior of the damping terms in several situations the obtained ode can be easily solved and the decay rates can be given explicitly third we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping then the solution ceases to exists and blows up in finite time moreover in either the absence of the interior source or the boundary source then we prove that the solution is unbounded and grows as an exponential function

nonparametric estimation and tests of fit for dose response relations summary various parametric models have been used to describe the dependence of disease risks on continuous explanatory variables the fit of such models is generally tested either against some larger but still parametric class of alternatives or by making the covariates discrete in some arbitrary fashion this communication introduces a more general concept of residuals which may be useful for visualizing the fit of the data subject only to the constraint of monotonicity the maximum likelihood estimate mle of the dose response relation for a single covariate is shown to be a step function with discontinuities only at the covariate values of the failures the mle is found by a constrained newton raphson search and a procedure for reducing the dimensions of the search is suggested applications to the risk of lung cancer in relation to asbestos exposure are given

soliton surfaces associated with sigma models differential and algebraic aspects in this paper we consider both differential and algebraic properties of surfaces associated with sigma models it is shown that surfaces defined by the generalized weierstrass formula for immersion for solutions of the sigma model with finite action defined in the riemann sphere are themselves solutions of the euler lagrange equations for sigma models on the other hand we show that the euler lagrange equations for surfaces immersed in the lie algebra with conformal coordinates that are extremals of the area functional subject to a fixed polynomial identity are exactly the euler lagrange equations for sigma models in addition to these differential constraints the algebraic constraints in the form of eigenvalues of the immersion functions are systematically treated the spectrum of the immersion functions for different dimensions of the model as well as its symmetry properties and its transformation under the action of the ladder operators are discussed another approach to the dynamics is given i e description in terms of the unitary matrix which diagonalizes both the immersion functions and the projectors constituting the model

dispersionless limit of the noncommutative potential kp hierarchy and solutions of the pseudodual chiral model in 2 1 dimensions the usual dispersionless limit of the kp hierarchy does not work in the case where the dependent variable has values in a noncommutative e g matrix algebra passing over to the potential kp hierarchy there is a corresponding scaling limit in the noncommutative case which turns out to be the hierarchy of a pseudodual chiral model in 2 1 dimensions pseudodual to a hierarchy extending ward s modified integrable chiral model applying the scaling procedure to a method generating exact solutions of a matrix potential kp hierarchy from solutions of a matrix linear heat hierarchy leads to a corresponding method that generates exact solutions of the matrix dispersionless potential kp hierarchy i e the pseudodual chiral model hierarchy we use this result to construct classes of exact solutions of the su m pseudodual chiral model in 2 1 dimensions including various multiple lump configurations

on nonlinear wave equations of carrier type abstract this paper is concerned with the initial boundary value problem of the n dimensional carrier equation with an internal damping this damping is a fractional power of the velocity of the points of the material the faedo galerkin method tartar approach and compactness arguments provide the global existence of solutions of the above problem with restriction on the size of the initial data the decay of solutions is obtained by the perturbation method

existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions abstract in this paper we consider a multi dimensional damped semilinear viscoelastic wave equation with variable coefficient and acoustic boundary conditions first we prove a local existence theorem by using the faedo galerkin approximations combined with a contraction mapping theorem second we show that under suitable conditions on the initial data and the relaxation function the solution exists globally in time and we prove the uniform decay rate of the energy

a loop group method for affine harmonic maps into lie groups we generalize the uhlenbeck segal theory for harmonic maps into compact semi simple lie groups to general lie groups equipped with torsion free bi invariant connection

rigidity of holomorphic curves in complex grassmann manifolds in this paper it is proved that the holomorphic curves in complex grassmann manifolds are uniquely determined by its first and second fundamental forms up to rigid motion

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

probability estimates for multiparameter brownian processes soit f une fonction de repartition sur 0 1 d et w f le processus gaussien qui est la limite faible du processus empirique determine par f si g est une fonction sur 0 1 d on obtient des bornes superieures et inferieures pour p sup t 0 1 d w f t g t e

decay rate estimates for wave equations of memory type with acoustic boundary conditions abstract a wave equation of memory type u u 0 t g t s u s d s 0 with acoustic boundary conditions is considered this work is devoted to investigating the influence of the kernel function g and proving general rates of decay of solutions when g does not necessarily decay exponentially

isothermic submanifolds of symmetric r spaces we extend the classical theory of isothermic surfaces in conformal 3 space due to bour christoffel darboux bianchi and others to the more general context of submanifolds of symmetric r spaces with essentially no loss of integrable structure

on a mixed problem with a nonlinear acoustic boundary condition for a non locally reacting boundaries abstract we discuss global solvability to the mixed problem for a nonlinear wave equation in domains with non locally reacting boundaries and nonlinear acoustic boundary condition we prove the existence and uniqueness of solution additionally using the method of nakao we demonstrate the decay of the associated energy stability

su n and so n invariant chiral fields one and two dimensional subspaces applying the harmonic map ansatz we find new classes of su n and so n chiral fields we write the corresponding one dimensional subgroups of su n and so n chiral fields in terms of one and two harmonic maps and the corresponding two dimensional subgroups in terms of su 2 and so 3 chiral fields in other words we reduce the su n and so n chiral equations to harmonic maps in one and two dimensional spaces

the abstract cauchy problem for dissipative operators with respect to metric like functionals abstract this paper is devoted to a characterization of semigroups of lipschitz operators on a closed subset d of a banach space x and the abstract cauchy problem for an operator a in x satisfying the following condition there exists a proper lower semicontinuous functional from x into 0 such that the effective domain of is d a and such that lim n a x n a x in x for any x d a and any sequence x n in d a satisfying two conditions lim n x n x in x and lim sup n x n x the main result asserts that a semigroup of lipschitz operators on d can be generated by an operator a satisfying the above mentioned condition a dissipative condition with respect to a metric like functional and a subtangential condition the kirchhoff equations with acoustic boundary conditions are solved by the method based on the abstract result with the construction of suitable liapunov functionals and the use of a metric like functional on a suitable set

an application of semigroups of locally lipschitz operators to carrier equations with acoustic boundary conditions abstract a generation theorem of semigroups of locally lipschitz operators on a subset of a real banach space is given and applied to the problem of the well posedness of the carrier equation u t t u 2 u u t p 1 u t 0 in 0 with acoustic boundary condition where p 2 and is a bounded domain in an arbitrary dimensional space

an omnibus test for independence of a survival time from a covariate it has been over 60 years since kolmogorov introduced a distribution free omnibus test for the simple null hypothesis that a distribution function coincides with a given distribution function doob subsequently observed that kolmogorov s approach could be simplified by transforming the empirical process to an empirical process based on uniform random variables recent use of more sophisticated transformations has led to the construction of asymptotically distribution free omnibus tests when unknown parameters are present the purpose of the present paper is to use the transformation approach to construct an asymptotically distribution free omnibus test for independence of a survival time from a covariate the test statistic is obtained from a certain test statistic process indexed by time and covariate which is shown to converge in distribution to a brownian sheet a simulation study is carried out to investigate the finite sample properties of the proposed test and an application to data from the british medical research council s 4th myelomatosis trial is given

local likelihood estimation abstract a scatterplot smoother is applied to data of the form x 1 y 1 x 2 y 2 xn yn and uses local fitting to estimate the dependence of y on x a simple example is the running lines smoother which fits a least squares line to the y values falling in a window around each x value the value of the estimated function at x is given by the value of the least squares line at x a smoother generalizes the least squares line which assumes that the dependence of y on x is linear in this article we extend the idea of local fitting to likelihood based regression models one such application is to the class of generalized linear models nelder and wedderburn 1972 we enlarge this class by replacing the covariate form 0 x 1 with an unspecified smooth function s x this function is estimated from the data by a technique we call local likelihood estimation the method consists of maximum likelihood estimation for 0 and 1 applied in a window around each x value multiple covariates are incor

uniform energy decay of a variable coefficient wave equation with nonlinear acoustic boundary conditions abstract a variable coefficient wave equation with nonlinear damped acoustic boundary conditions is considered the riemannian geometry method is applied to deal with the variable coefficients under some checkable conditions on the coefficients the uniform decay of the energy is achieved without any geometrical conditions on the shape of the dissipative portion of the boundary

polynomial stability for wave equations with acoustic boundary conditions and boundary memory damping we study wave equations with acoustic boundary conditions where only one memory damping acts on the acoustic boundary under some conditions on the memory kernel polynomial energy decay rates are established by using higher order energy estimates among some other techniques

identification of nonlinear times series from first order cumulative characteristics abstract we consider the problem of identifying the class of time series model to which a series belongs based on observation of part of the series techniques of nonparametric estimation have been applied to this problem by various authors using kernel estimates of the one step lagged conditional mean and variance functions we study cumulative versions of tukey regressogram estimators of such functions these are more stable than estimates of the mean and variance functions themselves and can be used to construct confidence bands goodness of fit tests for specific parametric models are also developed

well posedness for a variable coefficient wave equation with nonlinear damped acoustic boundary conditions abstract we consider a variable coefficient wave equation with nonlinear damped acoustic boundary conditions well posedness in the hadamard sense for strong and weak solutions is proved by using the theory of nonlinear semigroups

asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions abstract in this paper we consider a multi dimensional wave equation with dynamic boundary conditions related to the kelvin voigt damping global existence and asymptotic stability of solutions starting in a stable set are proved blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained

some permutation tests for survival data summary we introduce two new classes of tests for censored data the first tests for association between survival and a covariate and the second tests for equality of survival distributions between k groups both tests are permutation tests based on nonparametric test statistics and unlike the wald test in the proportional hazards model or the log rank test can detect alternatives of the crossed hazards type the tests are simple to implement and require no theoretical analysis by the user nor an appeal to asymptotic distribution theory simulation results show that the tests are competitive with their classical counterparts under proportional hazards and superior under certain nonproportional hazards alternatives in both the continuous and k sample cases the tests are applied to data from the british medical research council s fourth myelomatosis trial and from a danish study of malignant melanoma

strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions we consider a wave equation with semilinear porous acoustic boundary conditions this is a coupled system of second and first order in time partial differential equations with possibly semilinear boundary conditions on the interface the results obtained are i strong stability for the linear model ii exponential decay rates for the energy of the linear model and iii local exponential decay rates for the energy of the semilinear model this work builds on a previous result showing generation of a well posed dynamical system the main tools used in the proofs are i the stability theorem of arendt batty ii energy methods used in the study of a wave equation with boundary damping and iii an abstract result of i lasiecka applicable to hyperbolic like systems with nonlinearly perturbed boundary conditions

complete second order differential equations in banach spaces with dynamic boundary conditions abstract in this paper we deal with the mixed initial boundary value problem for complete second order in time linear differential equations in banach spaces in which time derivatives occur in the boundary conditions general wellposedness theorems are obtained for the first time which are used to solve the corresponding inhomogeneous problems examples of applications to initial boundary value problems for partial differential equations are also presented

the construction of harmonic maps into complex grassmannians on developpe une technique d analyse des applications harmoniques d une surface de riemann dans un grassmannien complexe en utilisant des diagrammes qui etablissent plusieurs resultats

the cauchy problem of the ward equation with mixed scattering data we solve the cauchy problem of the ward equation with both continuous and discrete scattering data

smoothed cox regression nonparametric regression was shown by beran and mckeague and utikal to provide a flexible method for analysis of censored failure times and more general counting processes models in the presence of covariates we discuss application of kernel smoothing towards estimation in a generalized cox regression model with baseline intensity dependent on a covariate under regularity conditions we show that estimates of the regression parameters are asymptotically normal at rate root n and we also discuss estimation of the baseline cumulative hazard function and related parameters

solutions of the modified chiral model in 2 1 dimensions in this paper we deal with classical solutions of the modified chiral model on r2 1 such solutions are shown to correspond to products of various factors which we call time dependent unitons then the problem of solving the system of second order partial differential equations for the chiral field is reduced to solving a sequence of systems of first order partial differential equations for the unitons

explicit construction of harmonic two spheres into the complex grassmannian we present an explicit description of all harmonic maps of finite uniton number from a riemann surface into a complex grassmannian namely starting from a constant map q and a collection of meromorphic functions and their derivatives we show how to algebraically construct all harmonic maps from the two sphere into a given grassmannian g p mathbb c n in this setting the uniton number depends on q and p and we obtain a sharp estimate for it

stochastic calculus as a tool in survival analysis a review this paper gives a brief survey of the uses of stochastic calculus in survival analysis the role played by martingale central limit theory in deriving asymptotic distributions of estimators and test statistics is described the nelson aalen estimator the kaplan meier estimator cox s proportional hazards model aalen s additive risk model and a goodness of fit test for cox s model are discussed sketches of the proofs of the main results are included

the symmetries of solitons in this article we will retrace one of the great mathematical adventures of this century the discovery of the soliton and the gradual explanation of its remarkable properties in terms of hidden symmetries we will take an historical approach starting with a famous numerical experiment carried out by fermi pasta and ulam on one of the first electronic computers and with zabusky and kruskal s insightful explanation of the surprising results of that experiment and of a follow up experiment of their own in terms of a new concept they called solitons solitons however raised even more questions than they answered in particular the evolution equations that govern solitons were found to be hamiltonian and have infinitely many conserved quantities pointing to the existence of many non obvious symmetries we will cover next the elegant approach to solitons in terms of the inverse scattering transform and lax pairs and finally explain how those ideas led step by step to the discovery that loop groups acting by dressing transformations give a conceptually satisfying explanation of the secret soliton symmetries

invariant recurrence relations for mathbb c p n 1 models in this paper we present invariant recurrence relations for the completely integrable euclidean sigma model in two dimensions defined on the riemann sphere s2 when its action functional is finite we determine the links between successive projection operators wavefunctions of the linear spectral problem and immersion functions of surfaces in the algebra together with the outlines of the proofs our formulation preserves the conformal and scaling invariance of these quantities certain geometrical aspects of these relations are described we also discuss the singularities of the meromorphic solutions of the model and show that they do not affect the invariant quantities we illustrate the construction procedure through the examples of the and models

identifying nonlinear covariate effects in semimartingale regression models letxt be a semimartingale which is either continuous or of counting process type and which satisfies the stochastic differential equationdxt yt t zt dt dmt wherey andz are predictable covariate processes m is a martingale and is an unknown nonrandom function we study inference for by introducing an estimator for a t z int limits 0 z int limits 0 t alpha s x dsdx and deriving a functional central limit theorem for the estimator the asymptotic distribution turns out to be given by a gaussian random field that admits a representation as a stochastic integral with respect to a multiparameter wiener process this result is used to develop a test for independence ofx from the covariatez a test for time homogeneity of and a goodness of fit test for the proportional hazards model t z 1 t a2 z used in survival analysis