a large sample study of cox s regression model strong consistency and asymptotic normality are established for the maximum partial likelihood estimate of the regression parameter in cox s regression model estimates are also derived for the underlying cumulative hazard function and survival distribution we establish the asymptotic normality of these estimates and calculate the limiting variances

on inconsistency of breslow s estimator as an estimator of the hazard rate in the cox model in the cox proportional hazards model breslow s estimator of the cumulative hazard function is well known this estimator is sometimes presented in such a way that it appears to offer a method of estimating the hazard rate function as well we state and prove an asymptotic result for this estimator of the hazard rate which shows that the estimator is inconsistent and that it is asymptotically unbiased

intermittent incommensurate chaos abstract near the onset of intermittent chaos from quasiperiodic motion lying on an attracting 2d torus with rotation number 2 1 5 1 2 the power spectrum of the cartesian coordinate of the intersection point on the poincare section is studied the poincare section is distorted from the ellipse near the onset of chaos then a sequence of spectral lines are excited at frequencies i i 2 i 1 2 their intensities are found to obey the power law 4 i or 2 i for i 1 according as the poincare section has a sharp wrinkle or not a similar spectrum is obtained also in the chaotic regime e 0 the mean value of time intervals of quasiperiodic states between two consecutive bursts and the square root of their variance are found to be inversely proportional to e near the onset point g 3 0

bifurcation phenomena in coupled chemical oscillators normal form analysis and numerical simulations abstract a class of diffusively coupled chemical oscillators is mapped into a problem of two interacting hopf bifurcations the normal form analysis predicts a cascade of steady state limit cycle 2 torus 3 torus bifurcations as well as the coexistence of two stable limit cycles numerical simulations of the original system confirm these predictions and in particular show that this system provides an example of bifurcation leading to a stable quasiperiodic regime with three incommensurate frequencies

local em estimation of the hazard function for interval censored data summary we propose a smooth hazard estimator for interval censored survival data using the method of local likelihood the model is fit using a local em algorithm the estimator is more descriptive than traditional empirical estimates in regions of concentrated information and takes on a parametric flavor in regions of sparse information we derive two different standard error estimates for the smooth curve one based on asymptotic theory and the other on the bootstrap we illustrate the local em method for times to breast cosmesis deterioration finkelstein 1986 biometrics42 845 854 and for times to hiv 1 infection for individuals with hemophilia kroner et al 1994 journal of aids7 279 286 our hazard estimates for each of these data sets show interesting structures that would not be found using a standard parametric hazard model or empirical survivorship estimates

quality adjusted survival estimation with periodic observations quality adjusted survival is a measure that integrates both longevity and quality of life information the analysis of quality adjusted survival in a clinical study with data collected at periodic intervals encounters difficulties due to incomplete information based on observed time points the time axis is partitioned into a set of disjoint time intervals and under a markovian assumption on patient s health status the expected quality adjusted survival is estimated as the summed product of the quality of life and its mean sojourn time of each health state within partitioned intervals it is shown that the estimator is asymptotically normal with a simple variance calculation a simulation study is conducted to investigate the behavior of the estimator and a stroke study illustrates the use of the estimator

network induced chaos in integrate and fire neuronal ensembles it has been shown that a single standard linear integrate and fire if neuron under a general time dependent stimulus cannot possess chaotic dynamics despite the firing reset discontinuity here we address the issue of whether conductance based pulsed coupled network interactions can induce chaos in an if neuronal ensemble using numerical methods we demonstrate that all to all homogeneously pulse coupled if neuronal networks can indeed give rise to chaotic dynamics under an external periodic current drive we also provide a precise characterization of the largest lyapunov exponent for these high dimensional nonsmooth dynamical systems in addition we present a stable and accurate numerical algorithm for evaluating the largest lyapunov exponent which can overcome difficulties encountered by traditional methods for these nonsmooth dynamical systems with degeneracy induced by e g refractoriness of neurons

circle map behaviour of a modulated relaxation oscillator abstract the relaxation oscillations of sinusoidally and triangularly driven inductively shunted josephson junctions are studied by reducing the description of the dynamics to one dimensional circle maps the phase diagrams show two critical lines the first one marks the transition to complete phase locking without chaos involved contrary to expectations the second one does not mark the transition to chaos this is found to be due to a point of discontinuity in the corresponding maps the hausdorff dimension of the set of quasiperiodic windings at the first critical lines are computed for the sinusoidally and the triangularly driven oscillator to be 0 875 0 003 and 0 71 0 02 in the case of the second critical lines we give numerical evidence for zero dimension

estimation of the mean function with panel count data using monotone polynomial splines we study nonparametric likelihood based estimators of the mean function of counting processes with panel count data using monotone polynomial splines the generalized rosen algorithm proposed by zhang jamshidian 2004 is used to compute the estimators we show that the proposed spline likelihood based estimators are consistent and that their rate of convergence can be faster than n 1 3 simulation studies with moderate samples show that the estimators have smaller variances and mean squared errors than their alternatives proposed by wellner zhang 2000 a real example from a bladder tumour clinical trial is used to illustrate this method

farey level separation in mode locking structure of circle mappings abstract we consider various characteristic scales both on the x axis and in the parameter space for critical and near critical circle mappings juxtaposition of different scales for the set of all rational mode lockings exposes the effect of separation into layers this layered structure is shown to reproduce farey tree levels of the corresponding rotation numbers the separation is studied numerically for maps with various orders of singularity and interpreted in terms of coefficients of rotation function expansion

hazard function estimation using b splines summary a flexible parametric procedure is given to model the hazard function as a linear combination of cubic b splines and to obtain maximum likelihood estimates from censored survival data the approach yields smooth estimates of the hazard and suivivorship functions that are intermediate in structure between strongly parametric and non parametric models a simple method is described for selecting the number and location of knots simulation results show favorable root mean square error compared to non parametric estimates for both the hazard and survivorship functions three methods are given to calculate confidence intervals based on the delta method profile likelihood and bootstrap respectively the procedure is applied to estimate hazard rates for acquired immunodeficiency syndrome aids following infection with human immunodeficiency virus hiv spline methods can accommodate complex censoring mechanisms such as those that arise in the aids setting to illustrate hiv infection incidence is estimated for a cohort of hemophiliacs in which the dates of hiv infection are interval censored and some subjects were born after the onset of the hiv epidemic this report develops flexible parametric methods based on splines to estimate the hazard function hi t that describes the rate of disease onset among persons still susceptible at time t the observed data derive from a random sample yi i 1 n n of disease onset times with survivorship function s t py t the yi are right censored by independent random variables ci drawn from h c p c c one observes xi min yi ci and 3i i y ci the indicator for the event that yi is censored the hazard function

rational rotation numbers for maps of the circle we consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms that is maps like x to x t frac c 2 pi sin 2 pi x bmod 1 withc 1 we prove that the set of parameter values corresponding to irrational rotation numbers has lebesgue measure 0 in other words the intervals on which frequency locking occurs fill up the set of full measure

classical representation of the one dimensional anderson model a new approach is applied to the one dimensional anderson model by using a two dimensional hamiltonian map for a weak disorder this approach allows for a simple derivation of correct expressions for the localization length both at the centre and at the edge of the energy band where standard perturbation theory fails approximate analytical expressions for strong disorder are also obtained

map dependence of the fractal dimension deduced from iterations of circle maps every orientation preserving circle mapg with inflection points including the maps proposed to describe the transition to chaos in phase locking systems gives occasion for a canonical fractal dimensiond namely that of the associated set of for whichf g has irrational rotation number we discuss how this dimension depends on the orderr of the inflection points in particular in the smooth case we find numerically thatd r d r 1 r 1 8

persistent properties of bifurcations abstract this paper reviews results about bifurcations which occur in families of differential equations persistent properties are defined to be those which remain when the family of equations is perturbed we provide a list of such properties which is relevant for numerical studies of dynamical systems

universal transition in a dynamical system forced at two incommensurate frequencies abstract the various routes to chaos are explored for a nonlinear mechanical system mode locked to one of two incommensurate external frequencies two types of transitions are seen in our model system a saddle node transition with its associated intermittency can occur when the mode locking is lost while the attracting two torus remains smooth a less trivial transition can occur in which the attracting torus roughens and the power spectrum of the time series develops singular low frequency components after the breakdown mode locking persists with noisy small scale motions about the former torus at the multicritical point where these two transition lines meet scaling and universal low frequency power spectra are observed a renormalization group treatment is proposed the analysis might also be applicable to transitions in rotationally invariant systems

some remarks concerning stability for nonstationary quantum systems the problem of characterizing stability and instability for general nonstationary quantum systems is investigated some characterizations are reported and some elementary properties of a topological characterization are established then it is proven by considering a simple example that there are nonperiodic driven systems whose orbits are neither precompact nor leave on average any compact set autocorrelation measures are computed and the possible roles of the generalizes quasienergy operator and energy growth are briefly discussed

fidelity and entanglement fidelity for infinite dimensional quantum systems instead of unitary freedom for finite dimensional cases bi contractive freedom in the operator sum representation for quantum channels of infinite dimensional systems is established specifically if the channel sends every pure state to a finite rank state then the isometric freedom feature holds then a method of computing entanglement fidelity and a relation between quantum fidelity and entanglement fidelity for infinite dimensional systems are obtained in addition upper and lower bounds of the quantum fidelity and their connection to the trace distance are also provided

continuity of the von neumann entropy a general method for proving continuity of the von neumann entropy on subsets of positive trace class operators is considered this makes it possible to re derive the known conditions for continuity of the entropy in more general forms and to obtain several new conditions the method is based on a particular approximation of the von neumann entropy by an increasing sequence of concave continuous unitary invariant functions defined using decompositions into finite rank operators the existence of this approximation is a corollary of a general property of the set of quantum states as a convex topological space called the strong stability property this is considered in the first part of the paper

exact eigenfunctions for a quantised map exact eigenfunctions for quantised linear maps on a torus are constructed a connection with the number of periodic orbits of the classical map is established it is shown that in the semiclassical limit the eigenfunctions become more and more random i e their correlation function approaches a delta function

bifurcations of tori and phase locking in a dissipative system of differential equations abstract in a system of ordinary differential equations obtained through a seven mode truncation of the plane incompressible navier stokes equations a two dimensional torus undergoes first two period doubling bifurcations and then a transition to a strange attractor this strange attractor of liapunov dimension larger than three in a wide parameter interval is characterized by a power spectrum which retains the two fundamental frequencies of the original torus superimposed on a broad jagged background as the liapunov dimension goes down towards two an interesting phenomenon of phase locking occurs which gives rise to an alternation of chaotic and periodic behavior

period doubling bifurcations chaos phase locking and devil s staircase in a bonhoeffer van der pol oscillator abstract the effect of a periodic input current a 1 cos t in the bonhoeffer van der pol oscillator along with a bias a 0 is investigated numerically as the parameter a 1 is varied in the absence of bias by holding the other parameters at constant values typical period doubling bifurcation sequence is found to occur leading to chaotic motion in agreement with the feigenbaum scenario when the bias is switched on at the transition to chaos frequency locking is observed in the system the frequency locked intervals exhibit complete devil s staircase similar to the one observed in belousov zhabotinsky reaction

time correlations in mode hopping of coupled oscillators we study the dynamics in a system of coupled oscillators when arnold tongues overlap by varying the initial conditions the deterministic system can be attracted to different limit cycles adding noise the mode hopping between different states become a dominating part of the dynamics we simplify the system through a poincare section and derive a 1d model to describe the dynamics we explain that for some parameter values of the external oscillator the time distribution of occupancy in a state is exponential and thus memoryless in the general case on the other hand it is a sum of exponential distributions characteristic of a system with time correlations

fractal dimensions and homeomorphic conjugacies we investigate the behavior of the spectrum of singularities associated with the invariant measure of some dynamical systems under nonsmooth coordinate changes when the homeomorphic conjugacy is not lipschitz continuous we discuss how its singularities can affect the whole set of generalized fractal dimensions we give applications to homeomorphisms that conjugate critical circle maps with irrational golden mean winding numbers we present numerical studies corroborating the theoretical predictions

what price kaplan meier summary the asymptotic efficiency of the kaplan meier product limit estimator relative to the maximum likelihood estimator of a parametric survival function is examined under a random censoring model

devil s staircase in an optomechanical cavity we study self excited oscillations seos in an on fiber optomechanical cavity while the phase of seos randomly diffuses in time when the laser power injected into the cavity is kept constant phase locking may occur when the laser power is periodically modulated in time we investigate the dependence of phase locking on the amplitude and frequency of the laser power modulation we find that phase locking can be induced with a relatively low modulation amplitude provided that the ratio between the modulation frequency and the frequency of seos is tuned close to a rational number of relatively low hierarchy in the farey tree to account for the experimental results a one dimensional map which allows evaluating the time evolution of the phase of seos is theoretically derived by calculating the winding number of the one dimensional map the regions of phase locking can be mapped in the plane of modulation amplitude and modulation frequency comparison between the theoretical predictions and the experimental findings yields a partial agreement

a numerical study of the likelihood of phase locking abstract we report the results of numerical computation of the size of the set of s for which the hemeomorphism of the circle f k x x k 2 sin 2 x mod 1 is phased locked for values of k ranging from 0 1 to 1 the results suggest that for moderate values of k phase locking is unlikely except for small periods but that for k 1 almost all s give phase locking with large periods accounting for a surprisingly large portion of the total

universality and self similarity in the bifurcations of circle maps abstract the bifurcation structure in a two parameter family of circle maps is considered these maps have a topological degree that may be different from one a generalization of the rotation number is given and symmetries of the bifurcations in parameter space are described continuity arguments are used to establish the existence of periodic orbits by plotting the locus of parameter values associated with superstable cycles self similar bifurcations are found these bifurcations are a generalization of the familiar period doubling cascade in maps with one extrema to two parameter maps with two extrema finally a scheme for the global organization of bifurcation in these maps is proposed

the calculation of lyapunov spectra abstract linearization around the chaotic orbits of dynamical systems offers good prospects for understanding complex systems the naturally available information that can be obtained from such studies are the magnitudes and directions in phase space of stretching and shrinking of an initial small sphere of realizations this information is a property of a time displacement matrix that relates the initial bundle of realizations to its later configuration in this paper we extract information from this matrix through its singular values and the associated orthogonal matrices this given an algebraic foundation to the geometric methods that have been used previously the equations of the orthogonal principal stretching and contraction directions show that asymptotically after a long time they depend only on the location in phase space and not on the orbit as an example we show that these directions sketch out the attractor of the lorenz model

small sample results for the kaplan meier estimator abstract whereas much is known about the asymptotic properties of the kaplan meier 1958 estimator kme of a survival function exact results for small samples have been difficult to obtain in this article we obtain an exact expression for the th moment 0 of the kme under a model of proportional hazards this enables us under proportional hazards to a study the bias of the kme and b compare the exact variance of the kme to its asymptotic variance

properties of tango s index for detecting clustering in time tango proposed an index for detecting disease clustering in time applicable to grouped data obtained from a population that remains fairly stable over the study period in this paper we show that tango s index is a two dimensional u statistic having an asymptotic normal distribution to apply this result in the finite sampling situation an edgeworth expansion is used and is shown to be at least as accurate as tango s best result in approximating the tails of his test statistic under the null hypothesis this is extended to show that the edgeworth expansion can be used to approximate the power of tango s test statistic under selected alternatives to randomness a power study based on simulations is conducted to compare the power of tango s index to that of three of its competitors

a local likelihood proportional hazards model for interval censored data we discuss the use of local likelihood methods to fit proportional hazards regression models to right and interval censored data the assumed model allows for an arbitrary smoothed baseline hazard on which a vector of covariates operates in a proportional manner and thus produces an interpretable baseline hazard function along with estimates of global covariate effects for estimation we extend the modified em algorithm suggested by betensky lindsey ryan and wand we illustrate the method with data on times to deterioration of breast cosmeses and hiv 1 infection rates among haemophiliacs

renormalization in a circle map with two inflection points abstract an extended version of the sine circle map becomes non invertible by developing simultaneously two critical points in part of parameter space universality in the corresponding transition from quasiperiodicity to chaos manifests itself in the organisation of doubly superstable orbits four operators are needed to generate all such orbits the existence of non trivial equivalence relations for operation sequences makes the present theory different from other multi operator renormalisation approaches possible implications of the theory in higher dimensional systems are discussed

complete devil s staircase fractal dimension and universality of mode locking structure in the circle map it is shown numerically that the stability intervals for limit cycles of the circle map form a complete devil s staircase at the onset of chaos the complementary set to the stability intervals is a cantor set of fractal dimension d 0 87 this exponent is found to be universal for a large class of functions

computation continuation and bifurcation of torus solutions for dissipative maps and ordinary differential equations abstract a periodic solution can be considered as a fixed point of a poincare map we shall demonstrate how we can generalize this fixed point equivalence to quasi periodic solutions with two periods thus the quasi periodic solution or bi periodic solution will be a fixed point of a more general poincare map the stability of the bi periodic solution is then equivalent to the stability of the fixed point we demonstrate how we can follow the fixed point in dependence of any control parameter in the system furthermore we demonstrate both how the bi periodic solution may bifurcate to more complicated solutions and how it may arise as a bifurcation from a periodic solution our work is algorithmic in spirit and our statements are based on numerical evidence

nonlinear normal modes for the toda chain abstract the toda chain is a nonlinear mass spring chain which can in principle be integrated analytically by the action angle theory of classical mechanics we show that the transformation from physical variables to action variables can be implemented very simply on a computer we study the correspondence between action variables and the motions of the chain and find the role of the action variables to be very similar to the role of normal mode amplitudes in the harmonic chain this similarity leads to a partly rigorous partly heuristic normal mode analysis for the toda chain new results are obtained when this normal mode analysis is applied to certain perturbations of the toda chain such as the fermi pasta ulam chain or the toda chain with a mass impurity

particle propagation in a random and quasi periodic potential abstract we numerically investigate the anderson transition in an effective dimension d 3 d 11 for one particle propagation in a model random and quasi periodic potential the found critical exponents are different from the standard scaling picture we discuss possible reasons for this difference

an algorithmic view of pseudochaos abstract the controversial concept of quantum chaos the dynamical chaos in bounded mesoscopic quantum systems is presented as the most important and universal instance of a new generic dynamical phenomenon pseudochaos the latter characterizes the irregular behaviour of dynamical systems with discrete energy and or frequency spectrum which include classical systems with discrete phase space the question of randomness is addressed in terms of the algorithmic theory of dynamical systems using the arnold cat map for illustration the relationship between the algorithmic theory and ergodic theory is discussed

bifurcations of circle maps arnol d tongues bistability and rotation intervals we study the bifurcations of two parameter families of circle maps that are similar tofb w x x w b 2 sin 2 x mod1 the bifurcation diagram is constructed in terms of setstr wheretr is the set of parameter values b w for whichfb w has an orbit with rotation numberr we show that the known structure whenb 1 wherefb w is no longer injective and can have an interval of rotation numbers specifically the tongues overlap in a uniform monotonic manner and forr irrational tr opens into a tongue our other theorems give information about the dynamics offb w e g bistability or aperiodicity for b w in various subsets of parameter space

multiple imputation for simple estimation of the hazard function based on interval censored data a data augmentation algorithm is presented for estimating the hazard function and pointwise variability intervals based on interval censored data the algorithm extends that proposed by tanner and wong for grouped right censored data to interval censored data it applies multiple imputation and local likelihood methods to obtain smooth non parametric estimates for the hazard function this approach considerably simplifies the problem of estimation for interval censored data as it transforms it into the more tractable problem of estimation for right censored data the method is illustrated for two real data sets times to breast cosmesis deterioration and times to hiv 1 infection for individuals with haemophilia simulations are presented to assess the effects of various parameters on the estimates and their variances

a characterization of positive linear maps and criteria of entanglement for quantum states let h and k be finite or infinite dimensional complex hilbert spaces a characterization of positive completely bounded normal linear maps from into is given which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras this characterization is then applied to give a representation of quantum channels operations between infinite dimensional systems a necessary and sufficient criterion of separability is given which shows that a state on h k is separable if and only if i 0 for all positive finite rank elementary operators examples of ncp and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the ppt criterion and the realignment criterion

the scaling of arnol d tongues for differentiable homeomorphisms of the circle we show both for diffeomorphisms of the circle and for differentiable homeomorphisms that are not diffeomorphisms that the widths of the arnol d tongues in a one parameter family scale asq 3 whenq is the denominator of the rotation number

the kernel estimate of a regression function in likelihood based models abstract smoothing splines have a penalized likelihood motivation good and gaskins 1971 allowing direct application to nonparametric regression in likelihood based models the notion of a weighted likelihood for the nonparametric kernel estimation of a regression function is proposed generalizing the local likelihood theory of tibshirani and hastie 1987 let the data be of the form xi yi i 1 n where xi 0 1 d are lattice points and the yi are independent random variables from a family of distributions with parameter i g xi with g having continuous partial derivatives of order k 2 the goal is to arrive at a nonparametric estimate o of o g xo for a fixed point xo 0 1 d we consider the estimator o that maximizes the weighted likelihood function w n i 1 w xo xi b log f yi with f the density of yi w a symmetric kernel with compact support and b the bandwidth that controls the degree of smoothing sufficient conditions for consistency and asympt

exact treatment of mode locking for a piecewise linear map a piecewise linear map with one discontinuity is studied by analytic means in the two dimensional parameter space when the slope of the map is less than unity periodic orbits are present and we give the precise symbolic dynamic classification of these the localization of the periodic domains in parameter space is given by closed expressions the winding number forms a devil s terrace a two dimensional function whose cross sections are complete devils s staircases in such a cross section the complementary set to the periodic intervals is a cantor set with dimensiond 0

erosion of regression effect in a survival study lack of persistence or erosion of the regression effect is an alternative to proportional hazards of particular interest in many medical applications such a departure from proportional hazards is often the most likely direction in which the model may be inadequate questions such as is the effect of treatment only transitory or to what extent does an initially measured prognostic variable maintain its impact frequently arise in the context of a simple changepoint model we propose a test of the null hypothesis of proportional hazards against the specific alternative of erosion of the regression effect the particular changepoint model used can be viewed as a first approximation to a more complex reality an approximation that enables us to avoid specifically modeling the functional form that any erosion might take practical guidelines for carrying out the test are provided the approach is illustrated in the context of a study on risk factors for breast cancer survival

spectra of scaling indices for fractal measures theory and experiment abstract we propose a new formalism to characterize fractal measures such as strange attractors of dynamical systems which uses a function f here is the scaling index of the measure about a point on the fractal and f is the dimension of the set of points on the fractal with the same value of the spectrum describes the global scaling structure of the measure the spectrum includes as special points the hausdorff dimension of the fractal and the scaling indices of the most rarefied and the most concentrated regions of the measure we apply this formalism to two well known transitions to chaos the route via period doubling and the route via quasiperiodicity in both cases we find a universal spectrum which includes previously discovered scaling numbers both spectra have been measured experimentally on a forced rayleigh benard system

smooth estimation of survival functions and hazard ratios from interval censored data using bayesian penalized b splines we discuss the use of bayesian p spline and of the composite link model to estimate survival functions and hazard ratios from interval censored data if one further assumes proportionality of the hazards the proposed strategy provides a smoothed estimate of the baseline hazard along with estimates of global covariate effects the frequentist properties of our bayesian estimators are assessed by an extensive simulation study we further illustrate the methodology by three examples showing that the proportionality of the hazards might also be found inappropriate from interval censored data

the dynamics of phase locking and points of resonance in a forced magnetic oscillator abstract we report data on an experimental system a forced symmetric oscillator containing a saturable inductor with magnetic hysteresis it displays a hopf bifurcation to quasiperiodicity entrainment horns and chaos we study in detail the bifurcations and hysteresis occurring near points of resonance particularly strong resonance and show how the observed behavior can be understood using arnold s theory much of the behavior relating to the entrainment horns is explored period doubling and symmetry breaking bifurcations homoclinic bifurcations and crises and other bifurcations taking place at the horn boundaries important features of the behavior related to symmetry properties of the oscillator are studied and explained through the concept of a half cycle map the system is shown to exhibit a hopf bifurcation from a phase locked state to periodic islands similar to those found in hamiltonian systems an initialization technique is used to observe the manifolds of saddle orbits and other hidden structure an unusual differential equation model is developed which is irreversible and generates a noninvertible poincare map of the plane noninvertibility of this planar map has important effects on the behavior observed the poincare map may also be approximated through experimental measurements resulting in a planar map with parameter dependence this model gives good correspondence with the system in a region of the parameter space

quasiperiodically forced dynamical systems with strange nonchaotic attractors abstract we discuss the existence and properties of strange nonchaotic attractors of quasiperiodically forced nonlinear dynamical systems we do this by examining a particular model differential equation cos e cos 2 t where is a two frequency quasiperiodic function of t when 0 the analysis of the equation is facilitated since then it can be related to the schrodinger equation with quasiperiodic potential we show that the equation does indeed exhibit strange nonchaotic attractors and we consider the important question of whether these attractors are typical in the sense that they exist on a set of positive lebesgue measure in parameter space the equation also exhibits two and three frequency quasiperiodic behavior we also show that the strange nonchaotic attractors have distinctive frequency spectrum this property might make them experimentally observable

confidence bands for the median survival time as a function of the covariates in the cox model abstract let p x be the pth quantile of the distribution of the life length of an individual with covariate vector x in the cox model we introduce an estimator p x of p x and develop several families of confidence bands for x as a function of x to construct one type of band we proceed as follows we show that as n where n is the number of individuals in the study converges weakly to a gaussian process w x with a complicated covariance structure we then estimate this covariance structure from the data and simulate many gaussian processes with this estimated covariance structure the critical constants required for the construction of the confidence bands are obtained from the simulated processes another type of bands is obtained by bootstrapping we obtain an asymptotic theory for both types of bands simulation studies are used to compare the two types of bands the methods are illustrated on the stanford heart transplant data

fourier analysis of multi frequency dynamical systems abstract the fourier spectrum of a multiply periodic system appears complex due to resonances at integer combinations of the fundamental frequencies we show how to reorder the peaks in the fourier spectrum to yield a simpler structure which reveals important aspects of the time series which are not clearly seen in the ordinary fourier transform as examples we study time series from circle and torus maps and from experimental data by gollub and benson

arnold cat map ulam method and time reversal we study the properties of the arnold cap map on a torus with a several periodic sections using the ulam method this approach generates a markov chain with the ulam matrix approximant we study numerically the spectrum and eigenstates of this matrix showing their relation with the fokker plank relaxation and the kolmogorov sinai entropy we show that in the frame of the ulam method the time reversal property of the map is preserved only on a short ulam time which grows only logarithmically with the matrix size parallels with the evolution in a regime of quantum chaos are also discussed

irregular wakes in reaction diffusion waves abstract reaction diffusion equations have proved to be highly successful models for a wide range of biological and chemical systems but chaotic solutions have been very rarely documented we present a new mechanism for generating apparently chaotic spatiotemporal irregularity in such systems by analysing in detail the bifurcation structure of a particular set of reaction diffusion equations on an infinite one dimensional domain with particular initial conditions we show that possible solutions include travelling fronts which leave behind either regular or irregular spatiotemporal oscillations using a combination of analytical and numerical analysis we show that the irregular behaviour arises from the instability of oscillations induced by the passage of the front finally we discuss the generality of this mechanism as a way in which spatiotemporal irregularities can arise naturally in reaction diffusion systems

transition to topological chaos for circle maps abstract many biperiodic flows can be modelled by maps of a circle to itself for such maps the transition from zero to positive topological entropy can be achieved in several ways we describe all the possible routes for smooth circle maps and discuss the relevance of our results to the transition to chaos for two frequency systems

complete devil s staircase in an ising model with competing interactions an ising model on a regular cayley tree with competing ferro and antiferromagnetic nearest neighbour interactions can be formulated as a discrete two dimensional mapping the authors use this mapping to obtain sequences of modulated phases at low temperatures associated with complete devil s staircases the fractal dimensionality of the staircases increase with temperature at high temperatures the incommensurate phases may occupy regions of finite measure of the phase diagram

the nonlinear dynamics of the damped and driven toda chain 1 energy bifurcation diagrams abstract the dynamics of a periodically driven damped toda chain with periodic boundary conditions is investigated with the help of energy cross sections it can be shown that besides periodic motions of different periodicity quasiperiodic and aperiodic chaotic motions exist the method of energy bifurcation diagrams is introduced which allows a survey of the resonance properties of the chain

quantum chaos and operator fidelity metric we show that the recently introduced operator fidelity metric provides a natural tool to investigate the crossover to quantum chaotic behavior this metric is an information theoretic measure of the global stability of a unitary evolution against perturbations we use random matrix theory arguments to conjecture that the operator fidelity metric can be used to discriminate phases with regular behavior from quantum chaotic ones a numerical study of the onset of chaotic in the dicke model is given in order to support the conjecture

on quantum fidelities and channel capacities for the discrete memoryless quantum channel we show the equivalence of two different notions of quantum channel capacity that which uses the entanglement fidelity as its criterion for success in transmission and that which uses the minimum fidelity of pure states in a subspace of the input hilbert space as its criterion as a corollary any source with entropy rate less than the capacity may be transmitted with high entanglement fidelity we also show that a restricted class of encodings is sufficient to transmit any quantum source which may be transmitted on a given channel this enables us to simplify a known upper bound for the channel capacity it also enables us to show that the availability of an auxiliary classical channel from encoder to decoder does not increase the quantum capacity

stability and lyapunov stability of dynamical systems a differential approach and a numerical method abstract a set of differential equations for the eigenvalues and eigenvectors of the stability matrix of a dynamical system as well as for the lyapunov exponents and the corresponding eigenvectors is derived the rate of convergence of the lyapunov eigenvectors is shown to be exponential the eigenvectors of the stability matrix can be grouped into sets each spanning a subspace which converges at an exponential rate it is demonstrated that generically the real parts of the eigenvalues of the stability matrix equal the corresponding lyapunov exponents this statement has been tested numerically the values of the lyapunov exponents i are shown to be related to the corresponding finite time values of the lyapunov exponents e g those deduced from a finite time numerical simulation i t by i t i b i i t t the b i s are constants and i t are noise terms of zero mean this observation leads to a method of extrapolation which has been used to predict lyapunov exponents from a finite amount of data it is shown that the use of the standard numerical methods to compute lyapunov exponents introduces an error a i t in the value of i t where the a i s are constants thus the standard method has a rate of convergence which is the same as that of the exact i t s finally we have shown how one can compute the eigenvectors associated with each of the eigenvalues of the stability matrix as well as the lyapunov eigenvectors

computation of quasi periodic solutions of forced dissipative systems ii abstract the problem of finding a t periodic solution of a forced dissipative system of ordinary differential equations is most conveniently reformulated as a fixed point problem of a poincare map mapping the phase space at t 0 into the phase space at t t and the stability of the periodic response is equivalent to the stability of the fixed point it is shown how the problem of determining quasi periodic solutions of forced systems with two forcing frequencies may be reformulated as a fixed point problem of a new type of poincare map that also opens the possibility of treating quasi periodic solutions with more than two frequencies as well as treating bifurcating quasi periodic solutions of autonomous equations three examples are considered one in which the exact solution is known and two others where the quasi periodic solutions have been determined by other means

winding numbers for the supercritical sine circle map abstract the dependence of the winding number on the initial value and on the nonlinearity for the supercritical sine circle map is discussed analytical investigations and numerical results show that some scaling behavior for the winding number can be extracted

the nonlinear dynamics of the damped and driven toda chain ii fourier and lyapunov analysis of tori abstract the dynamics of a periodically damped and driven toda chain with periodic boundary conditions is investigated with the help of spectral energy bifurcation diagrams and lyapunov diagrams it can be shown that quasiperiodic motions on tori tk of dimension k 3 exist the technical problems concerning the fourier and lyapunov analysis of the motions on tori tk k 2 are addressed

the problem of a covariate time qualitative interaction in a survival study we introduce a test for the equality of two survival distributions against the specific alternative of crossing hazards although this kind of alternative is somewhat rare designing a test specifically aimed at detecting such departures from the null hypothesis in this direction leads to powerful procedures upon which we can call in those few cases where such departures are suspected furthermore the proposed test and an approximate version of the test are seen to suffer only moderate losses in power when compared with their optimal counterparts should the alternative be one of proportional hazards our interest in the problem is motivated by clinical studies on the role of acute graft versus host disease as a risk factor in leukemic children and we discuss the analysis of this study in detail the model we use in this work is a special case of the one introduced by anderson and senthilselvan 1982 applied statistics 31 44 51 we propose overcoming an inferential problem stemming from their model by using the methods of davies 1977 biometrika 64 247 254 1987 biometrika 74 33 43 backed up by resampling techniques we also look at an approach relying directly on resampling techniques the distributional aspects of this approach under the null hypothesis are interesting but practically its behaviour is such that its use cannot be generally recommended outlines of the necessary asymptotic theory are presented and for this we use the tools of martingale theory

on the global structure of period doubling flows abstract for a wide class of period doubling flows on r3 we analyze the global structure of the invariant manifolds and the topology of the bifurcating periodic orbits we emphasize aspects of the dynamics which are not visible in an analysis of the associated poincare return map the global manifold structure implies constraints for the subsequent bifurcational behavior of the flow the period doubled orbits are classified using the theory of iterated torus knots this classification reveals an infinite number of topologically distinct period doubling flows and raises the important question as to which sequences of torus knots appear in the asymptotic limit of a period doubling cascade if the asymptotic sequence of torus knots produced is not unique then the self similarity of the poincare map does not imply a self similar structure for the flow in particular at the accumulation point for the cascade one would not expect to observe a universal power spectrum for the flow this possibility is of some experimental interest

attractors on an n torus quasiperiodicity versus chaos the occurrence of quasiperiodic motions in nonconservative dynamical systems is of great fundamental importance however current understanding concerning the question of how prevalent such motions should be is incomplete with this in mind the types of attractors which can exist for flows on an n torus are studied numerically for n 3 and 4 specifically nonlinear perturbations are applied to maps representing n frequency quasiperiodic attractors these perturbations can cause the original n frequency quasiperiodic attractors to bifurcate to other types of attractors our results show that for small and moderate nonlinearity the frequency of occurrence of quasiperiodic motions is as follows n frequency quasiperiodic attractors are the most common followed by n 1 frequency quasiperiodic attractors followed by periodic attractors however as the nonlinearity is further increased n frequency quasiperiodicity becomes less common ceasing to occur when the map becomes noninvertible chaotic attractors are very rare for n 3 for small to moderate nonlinearity but are somewhat more common for n 4 examination of the types of chaotic attractors that occur for n 3 reveals a rich variety of structure and dynamics in particular we see that there are chaotic attractors which apparently fill the entire n torus i e limit sets of orbits on these attractors are the entire torus furthermore these are the most common types of chaotic attractors at moderate nonlinearities

survival estimation using splines a nonparametric maximum likelihood procedure is given for estimating the survivor function from right censored data it approximates the hazard rate by a simple function such as a spline with different approximations yielding different estimators a special case is that proposed by nelson 1969 journal of quality technology 1 27 52 and altshuler 1970 mathematical biosciences 6 1 11 the estimators are uniformly consistent and have the same asymptotic weak convergence properties as the kaplan meier 1958 journal of the american statistical association 53 457 481 estimator however in small and in heavily censored samples the simplest spline estimators have uniformly smaller mean squared error than do the kaplan meier and nelson altshuler estimators the procedure is extended to estimate the baseline hazard rate and regression coefficients in the cox 1972 journal of the royal statistical society series b 34 187 220 proportional hazards model and is illustrated using experimental carcinogenesis data

some statistical properties of simple classically stochastic quantum systems numerical studies are made of simple one and two dimensional quantum models which are stochastic in the classical limit it is shown that the correlation properties of the quantum and corresponding classical motions are only similar for very short time intervals ts and that the evolution of the quantum system unlike the classical one is stable the diffusive excitation of the quantum system under a periodic perturbation is limited to a specific time interval t ts during which the diffusion rate is similar to the corresponding classical diffusion rate for the two dimensional model a continuous component in the correlation spectrum survives for an indefinite period tw t it is shown that when the perturbation is quasiperiodic the interval t increases sharply

multiple temporal cluster detection summary this article proposes a simple method to determine single or multiple temporal clustering on a variable size population by a transformation of the data set the method based on a regression model allows consideration of a variable population size during the time of study a model selection procedure and a resampling method are used to select the number of clusters the results have applications in epidemiological studies of rare diseases

local birth of homoclinic chaos abstract local generation of chaotic behaviours associated with a sil nikov homoclinic orbit is studied near the codimension two bifurcation at the double stationary oscillatory instability of eigenvalues 0 we develop a perturbative calculation of the sil nikov homoclinic orbit and of its bifurcation locus starting from a conservative vector field flat nonanalytic behaviours due to non axisymmetric perturbations of the normal form are discussed and we show that the codimension one locus of sil nikov homoclinicity is unavoidable near bicriticality under certain conditions the results are applied to rossler s model quasiperiodic regimes born at the double instability are also studied and illustrated in the rossler model

fractal bifurcation sets renormalization strange sets and their universal invariants the transition structure of the most common routes to chaos are organized by fractal bifurcation sets examples include the quasiperiodic transitions to chaos and the period doubling structure found in arnol d tongues in this paper i discuss the universality of such fractal bifurcation sets and their relation to strange invariant sets of renormalization transformations an important result is that fractal bifurcation sets from within the same universality chaos are lipeomorphic this implies that they have the same fractal structure and in particular the same hausdorff dimension and scaling spectra some other invariants are introduced

reconstructing bifurcation diagrams only from time waveforms abstract a simple algorithm is proposed for reconstructing bifurcation diagrams only from time waveforms of a dynamical system without knowing an explicit form of the dynamics and its dependence on the parameters reconstruction means that the obtained bifurcation diagram is qualitatively similar to that of the original system the algorithm consists of two steps first a nonlinear predictor is sought using a particular class of functions with many parameters second the karhunen loeve transform is used to find only significant parameters contributing the bifurcations the algorithm is tested against two families of dynamical system the henon family and the coupled logistic delayed logistic family

symbolic dynamics of order preserving orbits abstract the maps we consider are roughly those that can be obtained by truncating non invertible maps to weakly monotonic maps they have a flat piece the binary sequences that correspond to order preserving orbits are shown to satisfy a minimax principle which was already known for order preserving orbits with rational rotation number the converse is also proven all minimax orbits are order preserving with respect to some rotation number for certain families of such circle maps one can solve exactly for the parameter values for which the map has a specified rotation number rho for rho rational we obtain the endpoints of the resonance intervals recursively these parameter values can be organized in a natural way as the nodes of a farey tree we give some applications of the ideas discussed

link based models for survival data with interval and continuous time censoring we present a flexible family of link based regression models with time independent baseline covariates particular choices of the link function yield a proportional hazard or a proportional odds model among others the background hazard function in the model is estimated using b splines this yields a full likelihood under both continuous time and random interval censoring residuals for influence and goodness of fit are described two examples are shown one in continuous time and the other under interval censoring

on the continuability of periodic orbits of parametrized three dimensional differential equations on considere un systeme autonome a n dimensions d equations differentielles ordinaires a 1 parametre x f x on suppose que f est definie et continue sur un sous ensemble ouvert o de r r n et que f x 0 sur o on etudie les orbites periodiques du systeme contenues dans o

the restricted cubic spline as baseline hazard in the proportional hazards model with step function time dependent covariables we incorporate a cubic spline function where the tails are linearly constrained as the baseline hazard into the proportional hazards model we show estimation of covariable coefficients and survival probabilities with this model to be as efficient statistically as with the cox proportional hazards model when covariables are fixed examples show that the inclusion of time dependent covariables defined as step functions into the restricted cubic spline proportional hazards model reduces computation time by a factor of 213 over the cox model advantages of the spline model also include flexibility of the hazard smooth survival curves and confidence limits for the survival and hazard estimates when there are time dependent covariables present

graphical exploration of covariate effects on survival data through nonparametric quantile curves kaplan meier curves provide an effective means of presenting the distributional pattern in a sample of survival data however in order to assess the effect of a covariate a standard scatterplot is often difficult to interpret because of the presence of censored observations several authors have proposed a running median as an effective way of indicating the effect of a covariate this article proposes a form of kernel estimation employing double smoothing that can be applied in a simple and efficient manner to construct an estimator of a percentile of the survival distribution as a function of one or two covariates permutations and bootstrap samples can be used to construct reference bands that help identify whether particular features of the estimates indicate real features of the underlying curve or whether this may be due simply to random variation the techniques are illustrated on data from a study of kidney transplant patients