frequency domain elastic wave modeling by finite differences a tool for crosshole seismic imaging the migration imaging or inversion of wide aperture cross hole data depends on the ability to model wave propagation in complex media for multiple source positions computational costs can be considerably reduced in frequency domain imaging by modeling the frequency domain steady state equations rather than the time domain equations of motion i develop a frequency domain approach in this note that is competitive with time domain modeling when solutions for multiple sources are required or when only a limited number of frequency components of the solution are required

simultaneous multifrequency inversion of full waveform seismic data we present a simultaneous multifrequency inversion approach for seismic data interpretation this algorithm inverts all frequency data components simultaneously a data weighting scheme balances the contributions from different frequency data components so the inversion process does not become dominated by high frequency data components which produce a velocity image with many artifacts a gauss newton minimization approach achieves a high convergence rate and an accurate reconstructed velocity image by introducing a modified adjoint formulation we can calculate the jacobian matrix efficiently allowing the material properties in the perfectly matched layers pmls to be updated automatically during the inversion process this feature ensures the correct behavior of the inversion and implies that the algorithm is appropriate for realistic applications where a priori information of the background medium is unavailable two different regularization schemes an l2 norm and a weighted l2 norm function a

an optimal 9 point finite difference frequency space 2 d scalar wave extrapolator in this study a new finite difference technique is designed to reduce the number of grid points needed in frequency space domain modeling the new algorithm uses optimal nine point operators for the approximation of the laplacian and the mass acceleration terms the coefficients can be found by using the steepest descent method so that the best normalized phase curves can be obtained this method reduces the number of grid points per wavelength to 4 or less with consequent reductions of computer memory and cpu time that are factors of tens less than those involved in the conventional second order approximation formula when a band type solver is used on a scalar machine

double hopf bifurcation for van der pol duffing oscillator with parametric delay feedback control the stability and bifurcation of a van der pol duffing oscillator with the delay feedback are investigated in which the strength of feedback control is a nonlinear function of delay a geometrical method in conjunction with an analytical method is developed to identify the critical values for stability switches and hopf bifurcations the hopf bifurcation curves and multi stable regions are obtained as two parameters vary some weak resonant and non resonant double hopf bifurcation phenomena are observed due to the vanishing of the real parts of two pairs of characteristic roots on the margins of the death island regions simultaneously by applying the center manifold theory the normal forms near the double hopf bifurcation points as well as classifications of local dynamics are analyzed furthermore some quasi periodic and chaotic motions are verified in both theoretical and numerical ways

restrictions and unfolding of double hopf bifurcation in functional differential equations abstract the normal form of a vector field generated by scalar delay differential equations at nonresonant double hopf bifurcation points is investigated using the methods developed by faria and magalhaes j differential equations 122 1995 181 we show that 1 there exists linearly independent unfolding parameters of classes of delay differential equations for a double hopf point which generically map to linearly independent unfolding parameters of the normal form equations ordinary differential equations 2 there are generically no restrictions on the possible flows near a double hopf point for both general and z 2 symmetric first order scalar equations with two delays in the nonlinearity and 3 there always are restrictions on the possible flows near a double hopf point for first order scalar delay differential equations with one delay in the nonlinearity and in nth order scalar delay differential equations n 2 with one delay feedback

van der pol s oscillator under delayed feedback the effect of delayed feedback on oscillatory behaviour is investigated for the classical van der pol oscillator it is shown how the presence of delay can change the amplitude of limit cycle oscillations or suppress them altogether the result is compared to the conventional proportional and derivative feedback the derivative like effect of delay is also demonstrated in a modified equation where a delayed term provides the damping

compact gravity inversion we present a new criterion for the inversion of gravity data the principle employed is to minimize the volume of the causative body which is equivalent to maximizing its compactness the anomalous density distribution is obtained using an iterative technique which is numerically stable and rapidly convergent the principle can also be adapted to include modeling of gravity anomalies by single density sources the advantage of this approach is that desirable geologic characteristics are automatically incorporated into the model with a minimum of subjective judgments on the part of the interpreter the treatment of noise in the data fits naturally into the formulation of the inversion procedure the method is illustrated by the inversion of noise free and noisy data generated from a two dimensional model consisting of a regular array of identical rectangular blocks whose densities can be individually specified in every case the algorithm successfully recovers the correct density distribution from the dat

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

efficient calculation of a partial derivative wavefield using reciprocity for seismic imaging and inversion linearized inversion of surface seismic data for a model of the earth s subsurface requires estimating the sensitivity of the seismic response to perturbations in the earth s subsurface this sensitivity or jacobian matrix is usually quite expensive to estimate for all but the simplest model parameterizations we exploit the numerical structure of the finite element method modern sparse matrix technology and source receiver reciprocity to develop an algorithm that explicitly calculates the jacobian matrix at only the cost of a forward model solution furthermore we show that we can achieve improved subsurface images using only one inversion iteration through proper scaling of the image by a diagonal approximation of the hessian matrix as predicted by the classical gauss newton method our method is applicable to the full suite of wave scattering problems amenable to finite element forward modeling we demonstrate our method through some simple 2 d synthetic examples

some asymptotic properties of kernel estimators of a density function in case of censored data on considere l adaptation de l estimateur du noyau a des donnees censurees en utilisant l estimateur de kaplan meier

stability and bifurcation analysis in van der pol s oscillator with delayed feedback abstract the classical van der pol equation with delayed feedback and a modified equation where a delayed term provides the damping are considered linear stability of the equations is investigated by analyzing the associated characteristic equations it is found that there exist the stability switches when delay varies and the hopf bifurcation occurs when the delay passes through a sequence of critical values the bifurcation diagram is drawn in e k plane and the stability and direction of the hopf bifurcation are determined by applying the normal form theory and the center manifold theorem

nonresonant hopf hopf bifurcation and a chaotic attractor in neutral functional differential equations abstract an algorithm for calculating the third order normal form of a nonresonant hopf hopf singularity in a neutral functional differential equation nfde is established the van der pol equation with extended delay feedback is investigated as an nfde of second order the existence of hopf hopf bifurcation is studied and the unfolding near these critical points is given by applying this algorithm periodic solutions and quasi periodic solutions are found with the aid of the bifurcation diagram and corresponding numerical illustrations are presented with the breaking down of the 3 torus a chaotic attractor appears in this nfde of second order following the ruelle takens newhouse scenario which usually arises for an ordinary differential equation of order at least 4 this transition is shown via both theoretical and numerical approaches

hopf pitchfork bifurcation in van der pol s oscillator with nonlinear delayed feedback abstract first we identify the critical values for hopf pitchfork bifurcation second we derive the normal forms up to third order and their unfolding with original parameters in the system near the bifurcation point by the normal form method and center manifold theory then we give a complete bifurcation diagram for original parameters of the system and obtain complete classifications of dynamics for the system furthermore we find some interesting phenomena such as the coexistence of two asymptotically stable states two stable periodic orbits and two attractive quasi periodic motions which are verified both theoretically and numerically

multiscale seismic waveform inversion iterative inversion methods have been unsuccessful at inverting seismic data obtained from complicated earth models e g the marmousi model the primary difficulty being the presence of numerous local minima in the objective function the presence of local minima at all scales in the seismic inversion problem prevent iterative methods of inversion from attaining a reasonable degree of convergence to the neighborhood of the global minimum the multigrid method is a technique that improves the performance of iterative inversion by decomposing the problem by scale at long scales there are fewer local minima and those that remain are further apart from each other thus at long scales iterative methods can get closer to the neighborhood of the global minimum we apply the multigrid method to a subsampled low frequency version of the marmousi data set although issues of source estimation source bandwidth and noise are not treated results show that iterative inversion methods perform much better when employed with a decomposition by scale furthermore the method greatly reduces the computational burden of the inversion that will be of importance for 3 d extensions to the method

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

multifrequency viscoacoustic modeling and inversion modeling and inversion for seismic wavefields that include the attenuation and phase dispersion effects of q can be implemented in the space frequency domain the viscoacoustic wave equation is solved by the moment method absorbing boundary conditions are implemented by reducing q and adjusting the complex velocity to reduce q dependent reflectivity in a zone around the edges of the model grid nonlinear inversion is performed using iterative linearized inversions the residual wavefield at a single frequency is back projected using an anticausal green s function along the viscoacoustic wavepath in an estimate of the model to get updated velocity and q distributions the model obtained from data at one frequency becomes input to inversion at the next higher frequency velocity and q are inverted simultaneously as they are interdependent both modeling and inversion algorithms are successfully tested with synthetic examples data at two or three frequencies are sufficient to produce reliable images from noise free synthetic data

normal forms for retarded functional differential equations with parameters and applications to hopf bifurcation abstract the paper addresses for retarded functional differential equations fdes with parameters the computation of normal forms associated with the flow on a finite dimensional invariant manifold tangent to an invariant space for the infinitesimal generator of the linearized equation at a singularity the analysis is based on the theory previously developed for fdes without parameters by considering an extension of a phase space appropriate to the computation of these normal forms as an application the generic hopf bifurcation for scalar retarded fdes is considered

a note on the variable kernel estimator of the hazard function from randomly censored data in a recent paper tanner and wong 1983b a family of data based nonparametric hazard estimators was introduced several of these estimators were studied in an extensive simulation experiment the estimator which allows for variable bandwidth was found to have a superior performance in this note sufficient conditions for the variable kernel estimator to be strongly consistent are presented

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

nonlinear conjugate gradients algorithm for 2 d magnetotelluric inversion we investigate a new algorithm for computing regularized solutions of the 2 d magnetotelluric inverse problem the algorithm employs a nonlinear conjugate gradients nlcg scheme to minimize an objective function that penalizes data residuals and second spatial derivatives of resistivity we compare this algorithm theoretically and numerically to two previous algorithms for constructing such minimum structure models the gauss newton method which solves a sequence of linearized inverse problems and has been the standard approach to nonlinear inversion in geophysics and an algorithm due to mackie and madden which solves a sequence of linearized inverse problems incompletely using a linear conjugate gradients technique numerical experiments involving synthetic and field data indicate that the two algorithms based on conjugate gradients nlcg and mackie madden are more efficient than the gauss newton algorithm in terms of both computer memory requirements and cpu time needed to find accurate solutio

waveform inversion using a logarithmic wavefield although waveform inversion has been studied extensively since its beginning 20 years ago applications to seismic field data have been limited and most of those applications have been for global seismology or engineering seismology scale problems not for exploration scale data as an alternative to classical waveform inversion we propose the use of a new objective function constructed by taking the logarithm of wavefields allowing consideration of three types of objective function namely amplitude only phase only or both in our waveform inversion we estimate the source signature as well as the velocity structure by including functions of amplitudes and phases of the source signature in the objective function we compute the steepest descent directions by using a matrix formalism derived from a frequency domain finite element finite difference modeling technique our numerical algorithms are similar to those of reverse time migration and waveform inversion based on the adjoint state of the wave equation in order to demonstrate the practical applicability of our algorithm we use a synthetic data set from the marmousi model and seismic data collected from the korean continental shelf for noisefree synthetic data the velocity structure produced by our inversion algorithm is closer to the true velocity structure than that obtained with conventional waveform inversion when random noise is added the inverted velocity model is also close to the true marmousi model but when frequencies below 5 hz are removed from the data the velocity structure is not as good as those for the noise free and noisy data for field data we compare the time domain synthetic seismograms generated for the velocity model inverted by our algorithm with real seismograms and find that the results show that our inversion algorithm reveals short period features of the subsurface although we use wrapped phases in our examples we still obtain reasonable results we expect that if we were to use correctly unwrapped phases in the inversion algorithm we would obtain better results

local convergence of empirical measures in the random censorship situation with application to density and rate estimators on etudie les deviations locales de la mesure empirique definie par l estimateur de kaplan meier 1958 pour la fonction de survie

two dimensional nonlinear inversion of seismic waveforms numerical results the nonlinear problem of inversion of seismic waveforms can be set up using least squares methods the inverse problem is then reduced to the problem of minimizing a lp nonquadratic function in a space of many 104to106 variables using gradient methods leads to iterative algorithms each iteration implying a forward propagation generated by the actual sources a backward propagation generated by the data residuals acting as if they were sources and a correlation at each point of the space of the two fields thus obtained which gives the updated model the quality of the results of any inverse method depends heavily on the realism of the forward modeling finite difference schemes are a good choice relative to realism because although they are time consuming they give excellent results numerical tests performed with multioffset synthetic data from a two dimensional model prove the feasibility of the approach if only surface recorded reflections are used the high spatial frequency content of the

accurate finite difference methods for time harmonic wave propagation finite difference methods for solving problems of time harmonic acoustics are developed and analyzed multi dimensional inhomogeneous problems with variable possibly discontinuous coefficients are considered accounting for the effects of employing non uniform grids a weighted average representation is less sensitive to transition in wave resolution due to variable wave numbers or non uniform grids than the standard pointwise representation further enhancement in method performance is obtained by basing the stencils on generalizations of pade approximation or generalized definitions of the derivative reducing spurious dispersion anisotropy and reflection and by improving the representation of source terms the resulting schemes have fourth order accurate local truncation error on uniform grids and third order in the non uniform case guidelines for discretization pertaining to grid orientation and resolution are presented

ck centre unstable manifolds we consider the existence and smoothness of global centre unstable manifolds for finite and infinite dimensional flows or maps we show that every global centre unstable manifold can be expressed as a graph of a c k map provided that the nonlinearities are c k smooth the proofs are based on a lemma by d henry on a necessary and sufficient condition for a lipschitz map to be continuously differentiable

chaotic attractors on a 3 torus and torus break up abstract two coupled driven van der pol oscillators can have three frequency quasiperiodic attractors which lie on a 3 torus the evidence presented in this paper indicates that the torus is destroyed when the stable and unstable manifolds of an unstable orbit become tangent furthermore no chaotic orbits lying on a torus were observed suggesting that in most cases at least in the case of this system orbits do not become chaotic before their tori are destroyed to expedite the calculations a method was developed which can be used to determine if an orbit is on a torus without actually displaying that orbit the method also described in this paper was designed specifically for our system the basic idea however could be used for studying attractors of other systems very few modifications of the method if any would be necessary when studying systems with the number of degrees of freedom equal to that of our van der pol system

application of the nearly perfectly matched layer in acoustic wave modeling in this work we successfully applied an alternative formulation of the perfectly matched layer pml the so called nearly pml npml to acoustic wave propagation modeling the npml formulation shows great advantages over the standard complex stretched coordinate pml the npml formulation deviates from the standard pml through an inexact variable change but this fact only affects the wave behavior in the npml layer which is outside the region of interest the equivalence of the wave absorbing performance between these two pml formulations the standard complex stretched coordinate pml formulation and the npml formulation in 3d cartesian coordinates for acoustic wave propagation modeling is proved mathematically in this work in time domain methods the advantages of the npml over the standard pml were explained by both the analytical analysis and the numerical simulations in terms of implementation simplicity and computational efficiency the computation time saving is up to 17 for the 2d example used in this work for 3d problems this computational saving is more significant after theoretically analyzing the numerical reflections from the npml and the standard pml we concluded that these two pml formulations have exactly the same performance even after spatial discretization this conclusion is validated by numerical experiment finally we tested the npml in the marmousi velocity model and found its wave absorbing rate is high enough even for this realistic structure

density and hazard rate estimation for censored data via strong representation of the kaplan meier estimator we study the estimation of a density and a hazard rate function based on censored data by the kernel smoothing method our technique is facilitated by a recent result of lo and singh 1986 which establishes a strong uniform approximation of the kaplan meier estimator by an average of independent random variables note that the approximation is carried out on the original probability space which should be distinguished from the hungarian embedding approach pointwise strong consistency and a law of iterated logarithm are derived as well as the mean squared error expression and asymptotic normality which is obtain using a more traditional method as compared with the hajek projection employed by tanner and wong 1983

seismic waveform inversion in the frequency domain part 1 theory and verification in a physical scale model seismic waveforms contain much information that is ignored under standard processing schemes seismic waveform inversion seeks to use the full information content of the recorded wavefield in this paper i present apply and evaluate a frequency space domain approach to waveform inversion the method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data the model data are computed using a full wave equation viscoacoustic frequency domain finite difference method ray asymptotics are avoided and higher order effects such as diffractions and multiple scattering are accounted for automatically the theory of frequency domain waveform wavefield inversion can be expressed compactly using a matrix formalism that uses finite difference finite element frequency domain modeling equations expressions for fast local descent inversion using back propagation techniques then follow naturally implementation of

a strategy for nonlinear elastic inversion of seismic reflection data the problem of interpretation of seismic reflection data can be posed with sufficient generality using the concepts of inverse theory in its roughest formulation the inverse problem consists of obtaining the earth model for which the predicted data best fit the observed data if an adequate forward model is used this best model will give the best images of the earth s interior three parameters are needed for describing a perfectly elastic isotropic earth the density x and the lame parameters x and x or the density x and the p wave and s wave velocities x and x the choice of parameters is not neutral in the sense that although theoretically equivalent if they are not adequately chosen the numerical algorithms in the inversion can be inefficient in the long spatial wavelengths of the model adequate parameters are the p wave and s wave velocities while in the short spatial wavelengths p wave impedance s wave impedance and density are adequate the problem of inversion o

estimating regression parameters using linear rank tests for censored data a class of estimates for regression parameters in a linear model with right censored data is proposed these estimates are derived by using linear rank tests for right censored data as estimating equations they are shown to be consistent and asymptotically normal with covariance matrix for which estimates are proposed efficient estimates within this class are derived together with conditions when they are fully efficient 0 introduction the problem of estimating regression coefficients using ranks has received much attention in the literature for uncensored data and the results are summarized very nicely in hettmansperger 1984 there has also been some work with censored data for the two sample problem specifically louis 1981 considered estimation of a scale change between two distributions with the use of a log rank test whereas wei and gail 1983 generalize to other linear rank tests as well we shall consider the general problem of multiple linear regression and show how linear rank tests can be used as estimating equations the resulting linear rank estimates will be shown to be asymptotically normal with variances that can be estimated consistently we shall also look at the efficiency of these estimates under various situations

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

hazard function estimators a simulation study kernel based methods for the smooth non parametric estimation of the hazard function have received considerable attention in the statistical literature although the mathematical properties of the kernel based hazard estimators have been carefully studied their statistical properties have not we reviewed various kernel based methods for hazard function estimation from right censored data and compared the statistical properties of these estimators through computer simulations our simulations covered seven distributions three levels of random censoring four types of bandwidth functions two sample sizes and three types of boundary correction we conducted a total of 504 simulation experiments with 500 independent samples each our results confirmed the advantages of two recent innovations in kernel estimation boundary correction and locally optimal bandwidths the median relative improvement decrease in mean square error over fixed bandwidth estimators without boundary correction was 3 per cent for fixed bandwidth estimators with left boundary correction 52 per cent locally optimal bandwidths without boundary correction and 66 per cent for locally optimal bandwidths with left boundary correction the locally optimal bandwidth estimators with left boundary correction also outperformed three previously published and publicly available algorithms with median relative improvements in mean square error of 31 per cent 77 per cent and 80 per cent

bogdanov takens singularity in van der pol s oscillator with delayed feedback abstract in this paper we study the classical van der pol s equation with delayed feedback we investigate the bogdanov takens singularity and bifurcation occurring in the system with the variation of the original parameters by using the normal form method and predict the suitable choice in the feedback control in order to obtain the asymptotic stable state

occam s inversion a practical algorithm for generating smooth models from electromagnetic sounding data the inversion of electromagnetic sounding data does not yield a unique solution but inevitably a single model to interpret the observations is sought we recommend that this model be as simple or smooth as possible in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models to obtain smooth models the nonlinear forward problem is linearized about a starting model in the usual way but it is then solved explicitly for the desired model rather than for a model correction by parameterizing the model in terms of its first or second derivative with depth the minimum norm solution yields the smoothest possible model rather than fitting the experimental data as well as possible which maximizes the roughness of the model the smoothest model which fits the data to within an expected tolerance is sought a practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models resulting in a stable and rapidly convergent algorithm the inversion of both magnetotelluric and schlumberger sounding field data and a joint magnetotelluric resistivity inversion demonstrate the method and show it to have practical application

the estimation of the hazard function from randomly censored data by the kernel method by convolution smoothing of the empirical hazards a kernel estimate of the hazard function from censored data is obtained small and large sample expressions for the mean and the variance of the estimator are given conditions for asymptotic normality are investigated using the hajek projection method

linear regression analysis for multivariate failure time observations abstract in this article we consider the case that each patient in a longitudinal study may experience two or more distinct failures the corresponding failure times which are possible censored are recorded for each patient the logarithm of each marginal failure time is assumed to be linearly related to its covariates however the distributional form of the error term in the model does not have to be specified in the analysis furthermore no specific structure of dependence among the distinct failure times on each subject has to be imposed various linear regression methods for analyzing multivariate failure time observations are proposed our procedures do not involve the unstable nonparametric hazard function estimation extensive numerical studies are conducted to evaluate the new proposals recommendations are also made for their practical usage

normal forms of non resonance and weak resonance double hopf bifurcation in the retarded functional differential equations and applications the norm forms of the double hopf bifurcation for a general dde is established the van der pol duffing oscillator with delayed position and velocity feedback is investigated the dynamical classification near the double hopf bifurcation point is studied the stable periodic doubling solutions and unstable three dimensional torus are found in this paper we firstly present the general framework of calculation of normal forms of non resonance and weak resonance double hopf bifurcation for the general retarded functional differential equations by using the normal form theory of delay differential equations due to faria and magalh a es then the dynamical behavior of van der pol duffing oscillator with delayed position and velocity feedback is considered specifically the dynamical classification near the double hopf bifurcation point is investigated by analyzing the obtained normal form finally the numerical simulations support the theoretical results and present some interesting phenomena