on testing the validity of sequential probability forecasts events are observed sequentially and at each stage a probability for the next event is assessed we consider the relationship between the sequence of probability forecasts and the sequence of outcomes we argue that the forecasts may be considered empirically valid when both these sequences are consistent with a common joint distribution for the events to aid in assessing validity we introduce various test statistics that measure in a natural way the empirical performance of the probability forecasts in the light of the outcomes obtained it is shown that under the null hypothesis of forecast validity such statistics will after suitable normalization have a standard normal or chi squared distribution virtually irrespective of the common joint distribution supposed to underlie both sequences

justification of the nonlinear schrodinger approximation for a quasilinear klein gordon equation we consider a nonlinear klein gordon equation with a quasilinear quadratic term the nonlinear schrodinger nls equation can be derived as a formal approximation equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet like solutions to the quasilinear klein gordon equation it is the purpose of this paper to present a method that allows one to prove error estimates in sobolev norms between exact solutions of the quasilinear klein gordon equation and the formal approximation obtained via the nls equation the paper contains the first validity proof of the nls approximation of a nonlinear hyperbolic equation with a quasilinear quadratic term by error estimates in sobolev spaces we expect that the method developed in the present paper will allow an answer to the relevant question of the validity of the nls approximation for other quasilinear hyperbolic systems

some properties of borel summable functions classes of borel summable functions are defined and studied some properties of these functions which are useful for high order calculations pertaining to certain physical theories are proved and discussed they include reciprocal watson like theorems sufficient conditions for membership in the classes and asymptotic behaviors of the expansion coefficients after composition of functions

on the transition from the ginzburg landau equation to the extended fisher kolmogorov equation abstract the ginzburg landau gl equation generically describes the behaviour of small perturbations of a marginally unstable basic state in systems on unbounded domains in this paper we consider the transition from this generic situation to a degenerate co dimension 2 case in which the gl approach is no longer valid instead of studying a general underlying model problem we consider a two dimensional system of coupled reaction diffusion equations in one spatial dimension we show that near the degeneration the behaviour of small perturbations is governed by the extended fisher kolmogorov efk equation at leading order the relation between the gl equation and the efk equation is quite subtle but can be analysed in detail the main goal of this paper is to study this relation which we do asymptotically the asymptotic analysis is compared to numerical simulations of the full reaction diffusion system as one approaches the co dimension 2 point we observe that the stable stationary periodic patterns predicted by the gl equation evolve towards various different families of stable stationary but not necessarily periodic so called multi bump solutions in the literature these multi bump patterns are shown to exist as solutions of the efk equation but there is no proof of the asymptotic stability of these solutions our results suggest that these multi bump patterns can also be asymptotically stable in large classes of model problems

approximation of the korteweg de vries equation by the nonlinear schrodinger equation abstract in 1968 zakharov derived the nonlinear schrodinger equation as an approximation equation for the water wave problem it is still an open problem if there are really solutions of the water wave problem which can be approximated by the solutions of the nonlinear schrodinger equation in answering this question we made some progress for a model problem the approximation of the korteweg de vries equation by the nonlinear schrodinger equation

justification of the nls approximation for a quasilinear water wave model abstract we show how for a quasilinear water wave model the nls approximation can be justified the model presents several new difficulties due to the quadratic terms which have to be eliminated by a normal form transformation due to the quasilinearity of the problem there is some loss of regularity associated with the normal form transformation and there is a nontrivial resonance present in the problem the loss of regularity is dealt with by using a cauchy kowalevskaya like method to treat the initial value problem and the nontrivial resonance is dealt with via a rescaling argument

information processes for semimartingale experiments in this paper we give explicit representations for kullback leibler information numbers between a priori and a posteriori distributions when the observations come from a semimartingale we assume that the distribution of the observed semimartingale is described in terms of the so called triplet of predictable characteristics we end by considering the corresponding notions in a model with a fractional noise

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

on the turing patterns in one dimensional gradient skew gradient dissipative systems in this article fundamental properties concerning the turing patterns are considered in one dimensional dissipative systems with gradient skew gradient structure introduced in m kuwamura and e yanagida phys d 175 2003 pp 185 195 it is a natural extension of free energy which covers reaction diffusion systems of activator inhibitor type the theory based on this concept provides a new perspective on a fundamental problem of what unique turing pattern is to be selected among many

phase dynamics in the complex ginzburg landau equation abstract for 1 stable time periodic solutions a x t aqeiqx i qt are the locally preferred planform for the complex ginzburg landau equation t a a 1 i x 2 a 1 i a a 2 in order to describe the spatial global behavior an evolution equation for the local wave number q can be derived formally the local wave number q satisfies approximately a conservation law q h q it is the purpose of this paper to explain the extent to which the conservation law is valid by proving estimates for this formal approximation

breaking the hidden symmetry in the ginzburg landau equation in this paper we study localised traveling solutions to a ginzburg landau equation to which we have added a small o e 0 e 1 quintic term we consider this term as a model for the higher order nonlinearities which appear in the derivation of the ginzburg landau equation by a combination of a geometrical approach and an explicit perturbation analysis we are able to relate the family of bekki nozaki solutions of the cubic equation n bekki and b nozaki 1985 phys lett a 110 133 135 to a curve of co dimension 2 homoclinic bifurcations in parameter space thus we are able to interpret the hidden symmetry which has been conjectured to explain the existence of the bekki nozaki solutions from the point of view of bifurcation theory we show that the quintic term breaks this hidden symmetry and that the one parameter family of bekki nozaki solutions is embedded in a two parameter family of homoclinic solutions which exist at a co dimension 1 homoclinic bifurcation furthermore we show mainly by geometrical arguments that the addition of the small quintic term can create large families of traveling localised structures that cannot exist in the cubic case these solutions exist in open subsets of the parameter space and correspond to structurally stable multi or n circuit heteroclinic orbits in an ode reduction and have a monotonically decreasing increasing amplitude except for a number n of relatively fast jumps

approximation and attractivity properties of the degenerated ginzburg landau equation abstract we are interested in spatially extended pattern forming systems close to the threshold of the first instability in case when the so called degenerated ginzburg landau equation takes the role of the classical ginzburg landau equation as the amplitude equation of the system this is the case when the relevant nonlinear terms vanish at the bifurcation point here we prove that in this situation every small solution of the pattern forming system develops in such a way that after a certain time it can be approximated by the solutions of the degenerated ginzburg landau equation in this paper we restrict ourselves to a swift hohenberg kuramoto shivashinsky equation as a model for such a pattern forming system

propagating fronts and the center manifold theorem we prove the existence of propagating front solutions for the swift hohenberg equation sh using the center manifold theorem we reduce the problem to a two dimensional system of ordinary differential equations they describe stationary solutions and front solutions of the partial differential equation sh we identify the well known amplitude equation as the lowest order approximation to the equation of motion on the center manifold

bifurcations of poiseuille flow between parallel plates three dimensional solutions with large spanwise wavelength the spatial dynamics approach is applied to the analysis of bifurcations of the three dimensional poiseuille flow between parallel plates in contrast to the classical studies we impose time periodicity as well as spatial periodicity with period 2 in the streamwise direction however we make no assumptions on the behavior in the spanwise direction except the uniform closeness of the bifurcating solution to the basic flow in an abstract setting it is shown how the dimension of the critical eigenspace of the spatial dynamics analysis can be uniquely determined from the classical linear stability problem for the three dimensional poiseuille problem we are able to find all relevant coefficients from the analysis of the purely two dimensional problem moreover we are able to analyze precisely the influence of a spanwise pressure gradient and the associated spanwise mass flux the study of the reduced problem shows that there are two different kinds of solutions spirals and ribbons which are 2 p periodic in the spanwise direction as in the couette taylor problem and both of them bifurcate in the same direction

kawahara dynamics in dispersive media abstract in a previous paper it was shown that in the long wave limit the water wave problem without surface tension can be described approximately by two decoupled kdv equations here we study a model problem and prove that in a degenerate case which occurs for the water wave problem with surface tension and near other codimension two points at which the coefficient in front of the leading order dispersive term in the equation of motion vanishes the long wave limit can be rigorously approximated by two decoupled kawahara equations

on prequential model assessment in life history analysis summary we extend the prequential approach introduced by a p dawid to continuous time marked point processes a prequential forecasting system is expressed by means of the stochastic intensities of the point process the exact independence and 1 exponentiality of the derived compensators are used to assess the predictive performance of the model and the inferential procedure

the non linear stability of front solutions for parabolic partial differential equations for the ginzburg landau equation and similar reaction diffusion equations on the line we show convergence of complex perturbations of front solutions towards the front solutions by exhibiting a coercive functional

functional calculus of the borel transform an explicit form is given of the borel transform b of g f where g f z g f z with f borel summable and g analytic in f 0 and a detailed study of the singularities in b in the borel variable is made examples include g 1 log and e the present paper generalizes results of boenkost et al j math phys 29 1118 1987

justification and failure of the nonlinear schrodinger equation in case of non trivial quadratic resonances abstract the nonlinear schrodinger nls equation can be derived as an amplitude equation describing slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet the purpose of this paper is to prove estimates between the formal approximation obtained via the nls equation and true solutions of the original system in case of non trivial quadratic resonances it turns out that the approximation property app holds if the approximation is stable in the system for the three wave interaction twi associated to the resonance we construct a counterexample showing that the nls equation can fail to approximate the original system if instability occurs for the approximation in the twi system in the unstable case we give some arguments why the validity of the app can be expected for spatially localized solutions and why it cannot be expected for non localized solutions although we restrict ourselves to a nonlinear wave equation as original system we believe that the results hold in more general situations too

modulation equations stochastic bifurcation in large domains we consider the stochastic swift hohenberg equation on a large domain near its change of stability we show that under the appropriate scaling its solutions can be approximated by a periodic wave which is modulated by the solutions to a stochastic ginzburg landau equation we then proceed to show that this approximation also extends to the invariant measures of these equations

residuals for relative risk regression summary several possible definitions of residuals are given for relative risk regression with time varying covariates each such residual has a representation as an estimator of a stochastic integral with respect to the martingale arising from a subject s failure time counting process previously proposed residuals for individual study subjects and for specific time points are shown to be special cases of this definition as are previously derived regression diagnostics an illustration and various generalizations are also given able methods are required to detect various departures from modelling assumptions suitably defined residuals may play an important role in such identification however the nonparametric aspect of the model the possibility that modelled regression variables may be varying with follow up time and most importantly the usual presence of right censorship implies that specialized residual definitions are required the class of residuals considered here is most easily formulated using counting process notation for the failure time data let ni t yi t and zi t represent respectively for the ith subject the values of counting censoring and covariate processes at follow up time t i 1 n while ni u yi u zi u 0c u t specifies the corresponding histories for the ith subject prior to time t thus in a typical univariate failure time application ni with right continuous sample paths will take value zero prior to the time of failure on the ith subject and value one thereafter while yi with left continuous sample paths will take value one at times at which the ith subject is at risk for an observed failure and value zero otherwise the counting process ni can be uniquely decomposed so that for all t i

diffusive mixing of stable states in the ginzburg landau equation the ginzburg landau equation on the real line has spatially periodic steady states of the form with and for we construct solutions which converge for all t 0 to the limiting pattern as these solutions are stable with respect to sufficiently small perturbations and behave asymptotically in time like where is uniquely determined by the boundary conditions this extends a previous result of brk92 by removing the assumption that should be close to zero the existence of the limiting profile is obtained as an application of the theory of monotone operators and the long time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term

the existence of dendritic fronts in this paper we study a fourth order semilinear parabolic equation on the infinite real line we show that in a certain parameter range this equation has propagating front solutions solutions tending to 0 at and advancing to the right with a speedc which leave behind them aperiodic pattern in the laboratory frame this is thus an example of spontaneous pattern formation

annealing markov chain monte carlo with applications to ancestral inference abstract markov chain monte carlo mcmc the metropolis hastings algorithm has been used for many statistical problems including bayesian inference likelihood inference and tests of significance though the method generally works well doubts about convergence often remain here we propose mcmc methods distantly related to simulated annealing our samplers mix rapidly enough to be usable for problems in which other methods would require eons of computing time they simulate realizations from a sequence of distributions allowing the distribution being simulated to vary randomly over time if the sequence of distributions is well chosen then the sampler will mix well and produce accurate answers for all the distributions even when there is only one distribution of interest these annealing like samplers may be the only known way to get a rapidly mixing sampler these methods are essential for attacking very hard problems which arise in areas such as statistical genetics we illustrate the methods wi

the inverse of a borel summable function let b1 be the borel transform of f f 0 0 by means of an integral equation similar to that proposed by t hooft the whys of subnuclear physics edited by a zichichi plenum new york 1979 an explicit representation of the borel transform b2 of 1 f as a series in convolution powers of b1 and its derivative is given it is proved that the singularities of b1 b2 on the circle of convergence coincide and have the same strength when approaching these singularities on a path inside their common analyticity domain but possibly outside the circle of convergence of their power series representation borel circle

martingale based residuals for survival models summary graphical methods based on the analysis of residuals are considered for the setting of the highly used cox 1972 regression model and for the andersen gill 1982 generalization of that model we start with a class of martingale based residuals as proposed by barlow prentice 1988 these residuals and or their transforms are useful for investigating the functional form of a covariate the proportional hazards assumption the leverage of each subject upon the estimates of 13 and the lack of model fit to a given subject 1 1 model consider a set of n independent subjects such that the counting process ni ni t t o for the ith subject in the set indicates the number of observed events experienced over time t the sample paths of the ni are step functions with jumps of size 1 and with ni 0 0 we assume that the intensity function for ni t is given by yi t da t zi t yi t ep z t dao t 1 where yi t is a 0 1 process indicating whether the ith subject is a risk at time t 13 is a vector of regression coefficients zi t is a p dimensional vector of covariate processes and ao is the baseline cumulative hazard function several familar survival models fit into this framework the andersen gill 1982 generalization of the cox 1972 model arises when ao t is completely unspecified the further restriction that yi t 1 until the first event or censoring and is 0 thereafter yields the cox model the parametric form ao t t yields a poisson model or an exponential if restricted to a single event per subject and ao t tp a weibull model our attention will focus primarily on the andersen gill and cox models however the

the validity of phase diffusion equations and of cahn hilliard equations for the modulation of pattern in reaction diffusion systems abstract so called phase diffusion equations and cahn hilliard equations can be derived via multiple scaling analysis in order to describe slow modulations in time and space of stable and slightly unstable spatially periodic pattern it is the purpose of this paper to explain the extent to which these formal approximations are valid by proving estimates between the formal approximations and true solutions of the original system the proof is given for an abstract reaction diffusion system as original system

the time dependent amplitude equation for the swift hohenberg problem precise estimates for the validity of the amplitude approximation for the swift hohenberg equation are given in a fully time dependent framework it is shown that small solutions of ordero which are modulated like stationary solutions have an evolution which is well described in the amplitude approximation for a time of ordero 2 for the proofs we use techniques for nonlinear semigroups and oscillatory integrals

conservation of resonant periodic solutions for the one dimensional nonlinear schrodinger equation we consider the one dimensional nonlinear schrodinger equation with dirichlet boundary conditions in the fully resonant case absence of the mass term we investigate conservation of small amplitude periodic solutions for a large measure set of frequencies in particular we show that there are infinitely many periodic solutions which continue the linear ones involving an arbitrary number of resonant modes provided the corresponding frequencies are large enough say greater than a certain threshold value depending on the number of resonant modes if the frequencies of the latter are close enough to such a threshold then they can not be too distant from each other hence we can interpret such solutions as perturbations of wave packets with large wave number

asymptotics of the cut discontinuity and large order behaviour from the instanton singularity the case of lattice schrodinger operators with exponential disorder we discuss the relation between the singularity structure of the borel transform the asymptotics of the cut discontinuity and the large order behaviour of perturbation theory in an explicit example a tight binding model with exponential disorder we show how to obtain the first instanton singularity from a cluster expansion in the borel variable and as an application we determine the exact decay of the density of states ase the method opens some perspectives for similar problems arising from different models of mathematical physics

characterization of dispersive chaos and related states of binary fluid convection abstract we describe an extensive series of experiments on traveling wave convection in an ethanol water mixture with separation ratio 0 020 in a long narrow annular container as the stress parameter is quasi statically increased through the range 0 2 above onset this system evolves through a series of complex spatiotemporal states before settling into an extended pattern of steady spatially uniform convective rolls much of the dynamical behavior is influenced by strong nonlinear dispersion which causes the repeated and often irregular appearance and sudden collapse of spatially localized bursts of traveling waves small amplitude states seen very close to onset are analyzed quantitatively using a complex ginzburg landau equation complex dynamical states encountered further above onset are analyzed using a number of classical statistical techniques focusing mainly on correlation functions and fourier spectra in addition we use the technique of biorthogonal decomposition in an effort to extract the basic structures of the patterns and to characterize the complexty of their dynamics

justification of the nonlinear schrodinger equation for the evolution of gravity driven 2d surface water waves in a canal of finite depth in 1968 v e zakharov derived the nonlinear schrodinger equation for the two dimensional water wave problem in the absence of surface tension that is for the evolution of gravity driven surface water waves in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet in this paper we give a rigorous proof that the wave packets in the two dimensional water wave problem in a canal of finite depth can be approximated over a physically relevant timespan by solutions of the nonlinear schrodinger equation

testing goodness of fit for proportional hazards model with censored observations abstract a numerical omnibus test of fit for the two sample proportional hazards model is proposed for randomly censored observations the test is derived from cox s partial likelihood and does not need a dummy time dependent covariate or any partition of the time axis consistency of the test is established examples are provided for illustration

asymptotics frequency modulation and low regularity ill posedness for canonical defocusing equations in a recent paper kenig ponce and vega study the low regularity behavior of the focusing nonlinear schrodinger nls focusing modified korteweg de vries mkdv and complex korteweg de vries kdv equations using soliton and breather solutions they demonstrate the lack of local well posedness for these equations below their respective endpoint regularities in this paper we study the defocusing analogues of these equations namely defocusing nls defocusing mkdv and real kdv all in one spatial dimension for which suitable soliton and breather solutions are unavailable we construct for each of these equations classes of modified scattering solutions which exist globally in time and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts these solutions are used to demonstrate lack of local well posedness in certain sobolev spaces in the sense that the dependence of solutions upon initial data fails to be uniformly continuous in particular we show that the mkdv flow is not uniformly continuous in the l 2 topology despite the existence of global weak solutions at this regularity finally we investigate the kdv equation at the endpoint regularity h and construct solutions for both the real and complex kdv equations the construction provides a nontrivial time interval t t and a locally lipschitz continuous map taking the initial data in h to a distributional solution u c 0 t t h which is uniquely defined for all smooth data the proof uses a generalized miura transform to transfer the existing endpoint regularity theory for mkdv to kdv

multi scale expansions in the theory of systems integrable by the inverse scattering transform abstract it is shown that using multi scale expansions conventionally employed in the theory of nonlinear waves one can transform systems integrable by the ist method into other systems of this type

a graphical method for assessing goodness of fit in cox s proportional hazards model abstract suggested here is a simple graphical method for studying the goodness of fit in cox s regression model for survival data the method is easy to use as it does not require the estimation of alternative models and only involves quantities similar to those already appearing in the partial likelihood expression that is needed in the parameter estimation the rationale behind the graphs is very intuitive they make a direct comparison between observed and expected failure frequencies as estimated from the model in a correctly specified model one anticipates an approximate balance between such frequencies otherwise there will typically be groups of individuals for which the expected frequencies are systematically too high or too low to match with the data and this shows in the graphs introduced here in the concrete applications of the method the individuals are stratified in a way that depends on what aspect of the model is being checked against data there is always one graph for each stratum s

attractivity of the ginzburg landau mode distribution for a pattern forming system with marginally stable long modes abstract the mathematical theory of the description of pattern forming systems close to the first instability via the ginzburg landau equation is based on approximation and attractivity results this theory allowed to prove global existence results and upper semicontinuity of attractors for classical hydrodynamical stability problems such as the couette taylor problem recently approximation results for the ginzburg landau approximation for pattern forming systems with marginally stable long modes such as the benard marangoni system have been shown it is the purpose of this paper to prove the second fundamental property namely the attractivity of the ginzburg landau mode distribution for such systems

the validity of modulation equations for extended systems with cubic nonlinearities modulation equations play an essential role in the understanding of complicated systems near the threshold of instability here we show that the modulation equation dominates the dynamics of the full problem locally at least over a long time scale for systems with no quadratic interaction term we develop a method which is much simpler than previous ones it involves a careful bookkeeping of errors and an estimate of gronwall type as an example for the dissipative case we find that the ginzburg landau equation is the modulation equation for the swift hohenberg problem moreover the method also enables us to handle hyperbolic problems the nonlinear schrodinger equation is shown to describe the modulation of wave packets in the sine gordon equation