superconvergent explicit two step peer methods we consider explicit two step peer methods for the solution of nonstiff differential systems by an additional condition a subclass of optimally zero stable methods is identified that is superconvergent of order p s 1 where s is the number of stages the new condition allows us to reduce the number of coefficients in a numerical search for good methods we present methods with 4 7 stages which are tested in fortran90 and compared with dopri5 and dop853 the results confirm the high potential of the new class of methods

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

strong stability preserving explicit peer methods in this paper we study explicit peer methods up to order p 13 which have the strong stability preserving ssp property this class of general linear methods has the favourable property of a high stage order the effective ssp coefficient is maximized by solving a nonlinear constraint optimization problem numerically to high precision the coefficient matrices of the optimized methods are sparse in a very structured way linear multistep methods are obtained as a special case of only one stage

a flexible approach to the crossing hazards problem we propose a simple and flexible framework for the crossing hazards problem the method is not confined to two sample problems but may also work with continuous exposure variables whose effect changes its sign at some time point of the observed follow up time penalized partial likelihood estimation relies upon the assumption of a smooth hazard ratio via low rank basis splines with a conventional difference penalty to ensure smoothness and additional ad hoc penalties to obtain restricted estimates useful in the context of crossing hazards the framework naturally also leads to a statistical test that has good power for revealing a global effect under several alternatives including crossing hazards we provide the results from a real data analysis and from some simulations to illustrate empirically the performance of the proposed approach as compared with the possible alternatives

highly efficient strong stability preserving runge kutta methods with low storage implementations strong stability preserving ssp runge kutta methods were developed for time integration of semidiscretizations of partial differential equations ssp methods preserve stability properties satisfied by forward euler time integration under a modified time step restriction we consider the problem of finding explicit runge kutta methods with optimal ssp time step restrictions first for the case of linear autonomous ordinary differential equations and then for nonlinear or nonautonomous equations by using alternate formulations of the associated optimization problems and introducing a new more general class of low storage implementations of runge kutta methods new optimal low storage methods and new low storage implementations of known optimal methods are found the results include families of low storage second and third order methods that achieve the maximum theoretically achievable effective ssp coefficient independent of stage number as well as low storage fourth order methods that are more efficient than current full storage methods the theoretical properties of these methods are confirmed by numerical experiment

tests for comparing estimated survival functions we describe a class of statistical tests for the comparison of two or more survival curves typically estimated using the kaplan meier method the class is based on the construction of o quigley 2003 and some special cases are of particular interest underlying the inferential development are the arguments of efron hinkley 1978 leading to a theoretical sampling model that is in some sense closer to the observed data the log rank and weighted log rank tests arise as special members of the class in practice the log rank test will often be a suboptimal even poor test due to the presence of non proportional hazards the proposed test maintains good power and in all the cases considered has greater power than the log rank test under non proportional hazards the power will depend on the alternatives being considered and under reasonable assumptions on the alternatives we conclude that the proposed test is more powerful than the log rank test simulations support these conclusions an example is given as an illustration

the control of parasitism in g symplectic methods g symplectic general linear methods are designed to approximately preserve symplectic invariants for hamiltonian systems in this paper the properties of g symplectic methods are explored computationally and theoretically good preservation properties are observed over long times for many parameter ranges but for other parameter values the parasitic behavior to which multivalue methods are prone corrupts the numerical solution by the growth of small perturbations two approaches for alleviating this effect are considered first compositions of methods with growth parameters of opposite signs can be used to cancel the long term effect of parasitism second methods can be constructed for which the growth parameters are zero by design each of these remedies is found to be successful in eliminating parasitic behavior in long term simulations using a variety of test problems

suppressing or inducing chaos in a model of robot arms and mechanical manipulators we study the problems of suppressing or inducing chaotic dynamics in a simple model of robot arms and mechanical manipulators assuming that the unperturbed systems possess multiple non transverse homoclinic and or heteroclinic orbits depending on the model parameters based on the melnikov method and numerical computations for melnikov integrals fixed points and turning points we obtain conditions for chaos suppression and generation we prove that the initial phase difference plays an important role in suppressing or inducing chaos in complex systems our results indicate that these methods of controlling or inducing chaos can be easily applied to many systems in natural science and engineering

a comparison of sample size methods for the logrank statistic several methods are available for sample size calculation for clinical trials when survival curves are to be compared using the logrank statistic we discuss advantages and disadvantages of some of these methods and present simulation results under exponential proportional hazards and non proportional hazard situations

a sequential testing approach to detecting multiple change points in the proportional hazards model the semi parametric proportional hazards model has been widely adopted in clinical trials with time to event outcomes a key assumption in the model is that the hazard ratio function is a constant over time which is frequently violated as there is often a lag period before an experimental treatment reaches its full effect one existing approach uses maximal score tests and monte carlo sampling to identify multiple change points in the hazard ratio function which requires the number of change points that exist in the model to be known we propose a sequential testing approach to detecting multiple change points in the hazard ratio function using likelihood ratio tests and the distributions of the likelihood ratio statistics under the null hypothesis are evaluated via resampling an important feature of the proposed approach is that the number of change points in the model is inferred from the data and does not need to be specified numerical results based on simulated clinical trials and a real time to event study show that the proposed approach can accurately detect the change points in the hazard ratio function

construction of high order quadratically stable second derivative general linear methods for the numerical integration of stiff odes theory of general linear methods glms for the numerical solution of autonomous system of ordinary differential equations of the form y f y is extended to include the second derivative y g y f y f y this extension of glms is called second derivative general linear methods sglms in this paper we will construct two stage a and l stable sglms of order p and stage order q p up to six with low error constants we will show the efficiency of the proposed methods by implementing on some well known stiff problems

on the zero stability of variable stepsize multistep methods the spectral radius approach in this paper we illustrate a novel approach for studying the asymptotic behaviour of the solutions of linear difference equations with variable coefficients in particular we deal with the zero stability of the 3 step bdf method on grids with variable stepsize for the numerical solution of ivps for odes our approach is based on the theory of the spectral radius of a family of matrices and yields almost optimal results which give a slight improvement to the best results already known from the literature the success got on the chosen example suggests that our approach has a good potential for more general and harder stability analyses of numerical methods

estimating a treatment effect with repeated measurements accounting for varying effectiveness duration to assess treatment efficacy in clinical trials certain clinical outcomes are repeatedly measured over time for the same subject the difference in their means may characterize a treatment effect since treatment effectiveness lag and saturation times may exist erosion of treatment effect often occurs during the observation period instead of using models based on ad hoc parametric or purely nonparametric time varying coefficients we model the treatment effectiveness durations which are the time intervals between the lag and saturation times then we use some mean response models to include such treatment effectiveness durations our methodology is demonstrated by simulations and analysis of a landmark hiv aids clinical trial of short course nevirapine against mother to child hiv vertical transmission during labour and delivery

construction of highly stable two step w methods for ordinary differential equations we describe the construction of linearly implicit two step w methods of high stage order and desirable stability properties for the numerical solution of stiff differential systems the presented methods are demonstrated to be a and l stable when the stepsize is held constant they also preserve good stability properties in a variable stepsize environment under quite demanding conditions imposed on the stepsize pattern

the use of maximin efficiency robust tests in combining contingency tables and survival analysis abstract a test is the maximin efficiency robust test for a family of possible models underlying the data if no other test has higher minimum efficiency relative to the asymptotically optimum test for each model methods used to examine the efficiency robustness of rank tests for the two sample problem are adapted to obtain maximin efficiency robust procedures for testing the equality of proportions across several 2 2 tables for combining the results of tests for trend in several 2 x j tables in which the dose response function is one of a set of possible monotone functions and to analyze censored survival data when either the wilcoxon or log rank may be appropriate in the survival setting the robust test has maximin efficiency 93 3 relative to the wilcoxon or long rank when each is optimum in contrast to the 75 relative efficiency each statistic has when the other is optimum

on the asymptotic properties of a family of matrices in this paper we consider bounded families f of complex n n matrices after introducing the concept of asymptotic order we investigate how the norm of products of matrices behaves as the number of factors goes to infinity in the case of defective families f u sing the asymptotic order allows us to get a deeper knowledge of the asymptotic behaviour than just considering the so called generalized spectral radius with reference to the well known finiteness conjecture for finite families we also introduce the concepts of spectrum maximizing product and limit spectrum maximizing product showing that for finite families of 2 2matrices defectivity is equivalent to the existence of defective such limit products 2001 elsevier science inc all rights reserved

explicit nordsieck methods with extended stability regions abstract we describe the construction of explicit nordsieck methods of order p and stage order q p with large regions of absolute stability we also discuss error propagation and estimation of local discretization errors the error estimators are derived for examples of general linear methods constructed in this paper some numerical experiments are presented which illustrate the effectiveness of proposed methods

a monte carlo approach for change point detection in the cox proportional hazards model detecting a time lag of treatment effect or identifying change points in a hazard function is of great interest and importance in survival analysis the testing procedures hereto are primarily based on analytical approximations for the asymptotic null distribution of either the likelihood ratio test or the score test in the presence of random censoring and or covariates however the justification for the limiting distribution often requires some technical assumptions and conditions that are difficult to verify in practice moreover a satisfactory asymptotic theory for testing the existence of multiple change points in hazard function has not emerged in this paper we consider maximal score tests for detecting change point s in the cox proportional hazards model with censored data we propose to use a simple monte carlo approach for assessing the statistical significance of tests the proposed approach is applicable for testing a single change point in the cox model with covariates and sample stratifications over various types of candidate regions including discrete time point sets or disjoint intervals we also show that the proposed test statistics and the monte carlo procedure are well applicable under situations with multiple change points simulation studies and an analysis of a real data from a randomized cancer trial are conducted to demonstrate the finite sample performance of the proposed approach

stepsize restrictions for the total variation diminishing property in general runge kutta methods much attention has been paid in the literature to total variation diminishing tvd numerical processes in the solution of nonlinear hyperbolic differential equations for special runge kutta methods conditions on the stepsize were derived that are sufficient for the tvd property see e g shu and osher j comput phys 77 1988 pp 439 471 and gottlieb and shu math comp 67 1998 pp 73 85 various basic questions are still open regarding the following issues 1 the extension of the above conditions to more general runge kutta methods 2 simple restrictions on the stepsize which are not only sufficient but at the same time necessary for the tvd property and 3 the determination of optimal runge kutta methods with the tvd property in this paper we propose a theory by means of which we are able to clarify the above questions moreover by applying our theory we settle analogous questions regarding the related strong stability preserving ssp property see e g gottlieb shu and

numerical treatment of o d es the theory ofa methods almost all commonly used methods for o d es and their most miscellaneous compositions area methods i e they can be reduced toz o z j az j 1 h x j 1 z j 1 z j h z j s a s s j 1 1 m this paper presents a general theory fora methods and discusses its practical consequences an analysis of local discretization error l d e accumulation results in a general order criterium and reveals which part of the l d e effectively influences the global error this facilitates the comparison of methods and generalizes considerably the concept of error constants it is shown as a consequence that the global error cannot be safely controlled by the size of the l d e and that the conventional error control may fail in important cases furthermore butcher s effective order methods the concept of nordsieck forms and gear s interpretation of lineark step schemes as relaxation methods are generalized the stability of step changing is shortly discussed

implementation of two step runge kutta methods for ordinary differential equations we investigate the potential for efficient implementation of two step runge kutta methods tsrk a new class of methods introduced recently by jackiewicz and tracogna for numerical integration of ordinary differential equations the implementation issues addressed are the local error estimation changing stepsize using nordsieck technique and construction of interpolants the numerical experiments indicate that the constructed error estimates are very reliable in a fixed and variable stepsize environment

likelihood ratio tests for a changepoint with survival data summary we consider the likelihood ratio test for a changepoint or lag of the effect of some covariates e g treatment in regression modelling of survival data subject to right censoring asymptotic distributions under null hypotheses are derived numerical examples are used to illustrate the implementation of the procedure finally we apply the theory to data from a large randomised cancer prevention trial

construction of algebraically stable dimsims the class of general linear methods for ordinary differential equations combines the advantages of linear multistep methods high efficiency and runge kutta methods good stability properties such as a l or algebraic stability while at the same time avoiding the disadvantages of these methods poor stability of linear multistep methods high cost for runge kutta methods in this paper we describe the construction of algebraically stable general linear methods based on the criteria proposed recently by hewitt and hill we also introduce the new concept of e algebraic stability and investigate its consequences examples of e algebraically stable methods are given up to order p 4

the efficiency of a weighed log rank test under a percent error misspecification model for the log hazard ratio for comparison of two survival distributions it is natural to use a weighted log rank test with weight function given by the log hazard ratio function that is anticipated a priori this paper investigates the efficiency of this test when the a priori estimate of the log hazard ratio is subject to a specified percentage error the test is shown to be the maximum efficiency robust test over the class of alternatives in question and a simple expression for the maximum efficiency is established

stepsize conditions for general monotonicity in numerical initial value problems for runge kutta methods and linear multistep methods much attention has been paid in the literature to special nonlinear stability properties indicated by the terms total variation diminishing tvd strong stability preserving ssp and monotonicity stepsize conditions guaranteeing these properties were studied e g by shu and osher j comput phys 77 1988 pp 439 471 gottlieb shu and tadmor siam rev 43 2001 pp 89 112 hundsdorfer and ruuth monotonicity for time discretizations dundee conference report na 217 2003 university of dundee dundee uk 2003 pp 85 94 higueras j sci comput 21 2004 pp 193 223 and siam j numer anal 43 2005 pp 924 948 spiteri and ruuth siam j numer anal 40 2002 pp 469 491 gottlieb j sci comput 25 2005 pp 105 128 and ferracina and spijker siam j numer anal 42 2004 pp 1073 1093 and math comp 74 2005 pp 201 219 in the present paper we obtain a special stepsize condition guaranteeing the above properties for a generic numerical process this condition is best possible in a well defined and natural sense it is applicable to the important class of general linear methods and it can also be used to answer some open questions for methods of which the above stability properties were studied earlier

two step runge kutta methods implicit two step runge kutta methods are studied it will be shown that these methods require fewer stages to achieve the same order as one step runge kutta methods which means the two step methods are potentially more efficient than one step methods order conditions are derived and examples of two step one stage methods of order 2 and two step two stage methods of order 4 are presented stability properties of these methods with respect to y ay are studied and a stable two step methods of order 2 are characterized two step two stage methods of order 4 which are a stable are found by an extensive computer search semi implicit two stage methods of order 4 were also constructed this is in contrast to the situation encountered in the runge kutta theory where the unique two stage method of order 4 is not semi implicit

an extension and analysis of the shu osher representation of runge kutta methods in the context of solving nonlinear partial differential equations shu and osher introduced representations of explicit runge kutta methods which lead to stepsize conditions under which the numerical process is total variation diminishing tvd much attention has been paid to these representations in the literature in general a shu osher representation of a given runge kutta method is not unique therefore of special importance are representations of a given method which are best possible with regard to the stepsize condition that can be derived from them several basic questions are still open notably regarding the following issues 1 the formulation of a simple and general strategy for finding a best possible shu osher representation for any given runge kutta method 2 the question of whether the tvd property of a given runge kutta method can still be guaranteed when the stepsize condition corresponding to a best possible shu osher representation of the method is violated 3 the generalization of the shu osher approach to general possibly implicit runge kutta methods in this paper we give an extension and analysis of the original shu osher representation by means of which the above questions can be settled moreover we clarify analogous questions regarding properties which are referred to in the literature by the terms monotonicity and strong stability preserving ssp

sample sizes based on the log rank statistic in complex clinical trials the log rank test is frequently used to compare survival curves while sample size estimation for comparison of binomial proportions has been adapted to typical clinical trial conditions such as noncompliance lag time and staggered entry the estimation of sample size when the log rank statistic is to be used has not been generalized to these types of clinical trial conditions this paper presents a method of estimating sample sizes for the comparison of survival curves by the log rank statistic in the presence of unrestricted rates of noncompliance lag time and so forth the method applies to stratified trials in which the above conditions may vary across the different strata and does not assume proportional hazards power and duration as well as sample sizes can be estimated the method also produces estimates for binomial proportions and the tarone ware class of statistics

efficient a stable peer two step methods abstract a stability is a desirable property for stiff integrators however it may be difficult to prove or implement in an automated search for multi stage multi step methods requiring eigenvalue computations on complex sets of parameters in this paper we apply a purely algebraic criterion for a stability of peer two step methods requiring only the solution of algebraic equations and combine it with a new criterion for zero stability of peer methods on general time grids both criteria lead to formulations solvable by standard numerical methods and are easily verifiable by anyone if sufficient data are provided several 4 stage methods of order 3 and 4 are constructed and compared with existing methods on a few standard test problems

representations of runge kutta methods and strong stability preserving methods over the last few years a great effort has been made to develop monotone high order explicit runge kutta methods by means of their shu osher representations in this context the stepsize restriction to obtain numerical monotonicity is normally computed using the optimal representation in this paper we extend the shu osher representations for any runge kutta method giving sufficient conditions for monotonicity we show how optimal shu osher representations can be constructed from the butcher tableau of a runge kutta method the optimum stepsize restriction for monotonicity is given by the radius of absolute monotonicity of the runge kutta method l ferracina and m n spijker siam j numer anal 42 2004 pp 1073 1093 and hence if this radius is zero the method is not monotone in the shu osher representation methods with zero radius require negative coefficients and to deal with them an extra associate problem is considered in this paper we interpret these schemes as representations of perturbed runge kutta methods we extend the concept of radius of absolute monotonicity and give sufficient conditions for monotonicity optimal representations can be constructed from the butcher tableau of a perturbed runge kutta method

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

additive hazards models with latent treatment effectiveness lag time summary in many clinical trials for evaluating treatment efficacy it is believed that there may exist latent treatment effectiveness lag times after which medical treatment procedure or chemical compound would be in full effect in this paper semiparametric regression models are proposed and studied for estimating the treatment effect accounting for such latent lag times the new models take advantage of the invariant property of the additive hazards model in marginalising over an additive latent variable parameters in the models are thus easily estimated and interpreted while the flexibility of not having to specify the baseline hazard function is preserved monte carlo simulation studies demonstrate the appropriateness of the proposed semiparametric estimation procedure the methodology is applied to data collected in a randomised clinical trial which evaluates the efficacy of biodegradable carmustine polymers for treatment of recurrent brain tumours

the runge kutta theory in a nutshell in an earlier paper albrecht siam j numer anal 24 1987 pp 391 406 runge kutta methods were treated as composite linear methods the resulting linear theory is elementary yielding the order conditions as orthogonal relations from a recursion this paper further extends this approach and discusses its implications it is extended to fehlberg forms and the global as well as the stage errors are given in great detail including a new presentation of the principal error function that simplifies error minimization the orthogonal structure of the order conditions has many advantages it provides in particular a powerful strategy for existence proofs and facilitates the calculation of rk methods the recursion which originally was designed to generate the order conditions is considerably generalized by mappings and becomes a major tool also for the classical theory it can be used for a recursive generation of rooted trees also butcher s elementary differentials and weights as well as his order conditions can be obtained recursively to this purpose the elementary weights are reformulated and the rooted trees are complemented by blossoms the whole approach extends to rosenbrock methods

a new theoretical approach to runge kutta methods runge kutta rk methods are treated here as linear multistage methods within the author s concept of a methods as a consequence the traditional difference in treatment of rk methods and linear multistep methods vanishes in particular it is not necessary to consider elementary differentials it is shown how the order of a rk method depends on the error constants of its stages and the conditions nonlinear equations for order p are generated from a very simple recursion

on non parametric and generalized tests for the two sample problem with location and scale change alternatives various tests have been proposed for the two sample problem when the alternative is more general than a simple shift in location non parametric tests o brien s generalized t and rank sum tests and other tests related to the t we show that the generalized tests are directly related to non parametric tests proposed by lepage as a result we obtain a wider more flexible class of o brien type procedures which inherit the level robustness property of non parametric tests we have also computed the tests empirical sizes and powers under several models the non parametric procedures and the related o brien type tests are valid and yield good power in the settings investigated they are preferable to the t test and related procedures whose type i errors differ noticeably from nominal size for skewed and long tailed distributions

high order linear multistep methods with general monotonicity and boundedness properties we consider linear multistep methods that possess general monotonicity and boundedness properties strict monotonicity in terms of arbitrary starting values for the multistep schemes is only valid for a small class of methods under very stringent step size restrictions this makes them uncompetitive with the strong stability preserving ssp runge kutta methods by relaxing these strict monotonicity requirements a larger class of methods can be considered including many methods of practical interest in this paper we construct linear multistep methods of high order up to six that possess relaxed monotonicity or boundedness properties with optimal step size conditions numerical experiments show that the new schemes perform much better than the classical monotonicity preserving multistep schemes moreover there is a substantial gain in efficiency compared to recently constructed ssp runge kutta ssprk methods

dichotomous noise induced chaos in a generalized duffing type oscillator with fractional order deflection abstract the chaotic behavior of a generalized duffing type oscillator with fractional order deflection under dichotomous noise excitation is discussed in detail using the melnikov method together with mean square criterion necessary conditions for the onsets of chaos are derived it is shown that with the increase of noise transition rate the threshold of noise amplitude for chaos firstly decreases to a minimum and then increases the effects of dichotomous noise on the duffing type oscillator are also obtained by vanishing the mean largest lyapunov exponent of the oscillator this is further verified by phase portraits and time histories with numerical simulations

numerical search for algebraically stable two step almost collocation methods we investigate algebraic stability of the new class of two step almost collocation methods for ordinary differential equations these continuous methods are obtained by relaxing some of the interpolation and collocation conditions to achieve strong stability properties together with uniform order of convergence on the whole interval of integration we describe the search for algebraically stable methods using the criterion based on the nyquist stability function proposed recently by hill this criterion leads to a minimization problem in one variable which is solved using the subroutine fminsearch from matlab examples of algebraically stable methods in this class are also presented

second derivative extended backward differentiation formulas for the numerical integration of stiff systems a class of second derivative extended backward differentiation formulas suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is examine

strong stability preserving transformed dimsims abstract in this paper we investigate the strong stability preserving ssp property of transformed diagonally implicit multistage integration methods dimsims within this class examples of ssp methods of order p 1 2 3 and 4 and stage order q p have been determined and also suitable starting and finishing procedure have been constructed the numerical experiments performed on a set of test problems have shown that ssp transformed dimsims can be more accurate and competitive with ssp runge kutta methods of the same order

a new approach to error estimation for general linear methods we present a simple accurate and reliable approach to the estimation of the local discretization error for general linear methods for ordinary differential equations in this approach the input vector for the next step from x n to x n 1 x n h is rescaled and modified accordingly to compensate for the change of stepsize from h to h r h

a class of rank test procedures for censored survival data summary a class of linear rank statistics is proposed for the k sample problem with rightcensored survival data the class contains as special cases the log rank test mantel 1966 cox 1972 and a test essentially equivalent to peto peto s 1972 generalization of the wilcoxon test martingale theory is used to establish asymptotic normality of test statistics under the null hypotheses considered and to derive expressions for asymptotic relative efficiencies under contiguous sequences of alternative hypotheses a class of distributions is presented which corresponds to the class of rank statistics in the sense that for each distribution there is a statistic with some optimal properties for detecting location alternatives from that distribution some monte carlo results are displayed which present small sample behaviour

criteria for chaos of a three well potential oscillator with homoclinic and heteroclinic orbits abstract criteria for chaotic vibrations of a non linear elastic beam having three stable and two unstable equilibrium positions are compared the responses of the system exhibit chaotic behavior in certain regions in the driving frequency and amplitude plane with the use of the melnikov technique necessary conditions for chaos are derived based on homoclinic and heteroclinic bifurcations of the system numerical simulations of the stable and unstable manifolds poincare maps lyapunov exponents and fractal basin boundaries are performed the results indicate that both homoclinic and heteroclinic bifurcations are necessary conditions for chaos thus at least two necessary conditions must be satisfied for any chaotic responses however the final steady state of motion for many initial conditions becomes unpredictable in the sense of fractal basin boundaries if at least one homoclinic or heteroclinic bifurcation occurs finally an experimental criterion for chaotic motion and an heuristic criterion based on classical methods in non linear dynamics are also obtained and compared with the bifurcation criteria as a function of forcing amplitude and frequency

convergence and order reduction of runge kutta schemes applied to evolutionary problems in partial differential equations we address the question of convergence of fully discrete runge kutta approximations we prove that under certain conditions the order in time of the fully discrete scheme equals the conventional order of the runge kutta formula being used however these conditions which are necessary for the result to hold are not natural as a result in many problems the order in time will be strictly smaller than the conventional one a phenomenon called order reduction this phenomenon is extensively discussed both analytically and numerically as distinct from earlier contributions we here treat explicit runge kutta schemes although our results are valid for both parabolic and hyperbolic problems the examples we present are therefore taken from the hyperbolic field as it is in this area that explicit discretizations are most appealing

optimal explicit strong stability preserving general linear methods this paper constructs strong stability preserving general linear time stepping methods that are well suited for hyperbolic pdes discretized by the method of lines these methods generalize both runge kutta rk and linear multistep schemes they have high stage orders and hence are less susceptible than rk methods to order reduction from source terms or nonhomogeneous boundary conditions a global optimization strategy is used to find the most efficient schemes that have low storage requirements numerical results illustrate the theoretical findings

exploiting structure in the construction of dimsims we describe the structure of the nonlinear system for the coefficients of diagonally implicit multistage integration methods for ordinary differential equations this structure is then utilized in the search for methods of high order and stage order with a given stability function new methods were obtained by solving the nonlinear systems by state of the art software based on least squares minimization

partially implicit peer methods for the compressible euler equations when cut cells are used for the representation of orography in numerical weather prediction models this leads to very small cells on one hand this results in very harsh time step restrictions for explicit methods due to the cfl criterion on the other hand cut cells only appear in a small region of the domain therefore we consider a partially implicit method in cut cells the jacobian incorporates advection diffusion and acoustics while in the full cells of the free atmosphere only the acoustic part is used i e the method is linearly implicit in the cut cell regions and semi explicit in the free regions and computes with time step sizes restricted only by the cfl condition in the free atmosphere furthermore we use a simplified jacobian in the cut cell regions in order to save storage and gain computational efficiency while the method retains the order independently of the jacobian we present a linear stability theory which takes the effects of the simplifications of the jacobian on stability into account the presented method is as stable and accurate as the underlying split explicit method but furthermore it can compute with cut cells with nearly no additional effort

implementation of diagonally implicit multistage integration methods for ordinary differential equations we investigate the implementation of diagonally implicit multistage integration methods dimsims the implementation issues addressed are the local error estimation changing stepsize using the nordsieck technique and the construction of continuous interpolants numerical experiments with a method of order three indicate that the error estimates that have been constructed are very reliable in both a fixed and a variable stepsize environment

on quasi consistent integration by nordsieck methods in this paper we study advantages of numerical integration by quasi consistent nordsieck formulas all quasi consistent numerical methods possess at least one important property for practical use which has not attracted attention yet i e the global error of a quasi consistent method has the same order as its local error this means that the usual local error control will produce a numerical solution for the prescribed accuracy requirement if the principal term of the local error dominates strongly over remaining terms in other words the global error control can be as cheap as the local error control in the methods under discussion here we apply the above mentioned idea to nordsieck adams moulton methods which are known to be quasi consistent moreover some nordsieck adams moulton methods are even super quasi consistent the latter property means that their propagation matrices annihilate two leading terms in the defect expansion of such methods in turn this can impose a strong relation between the local and global errors of the numerical solution and allow the global error to be controlled effectively by a local error control we also introduce implicitly extended nordsieck methods such that in some sense they form pairs of embedded formulas with their source nordsieck methods this facilitates the local error control in quasi consistent nordsieck schemes numerical examples presented in this paper confirm clearly the power of quasi consistent integration in practice

supremum versions of the log rank and generalized wilcoxon statistics abstract considerable progress has been made in generalizing to censored data hypothesis tests based on rank statistics the most commonly used generalizations are the log rank test statistic which extends the savage exponential scores statistic cox 1972 mantel 1966 and the generalized wilcoxon statistics gehan 1965 peto and peto 1972 gill 1980 showed that in the two sample problem the asymptotic efficiency properties of the savage and wilcoxon statistics are maintained in censored data by the log rank and peto and peto wilcoxon statistics respectively when censoring is the same in both samples heuristics and simulations have implied that some recently proposed supremum statistics closely related to these two popular linear rank statistics can be used to construct tests that provide good power against a much wider range of nonlocal alternatives see e g fleming and harrington 1981 fleming o brien o fallon and harrington 1980 gill 1980 schumacher 1984 in this article we consider n

construction of two step runge kutta methods with large regions of absolute stability we describe the construction of explicit two step runge kutta methods of order p and stage order q p 1 or q p with large regions of absolute stability this process is illustrated for the methods of order p 2 and 3 and leads to new methods which are competitive with explicit runge kutta methods of the same order

regularity properties of multistage integration methods abstract the numerical method for ordinary differential equations is regular if it has the same set of finite asymptotic values as the underlying differential system this paper examines the regularity and strong regularity properties of diagonally implicit multistage integration methods dimsims introduced recently by j c butcher a sufficient condition for regularity and strong regularity of such methods of any order is given and it is proved that this condition is also necessary for two step two stage dimsims of order greater than or equal to two it is also demonstrated that there exist regular schemes in the class of explicit dimsims this is in contrast to explicit runge kutta methods with more than one stage which are always irregular

weighted log rank type statistics for comparing survival curves when there is a time lag in the effectiveness of treatment summary in certain long term treatment trials one expects some lag period before the treatment is fully effective this paper presents two weighted log rank type statistics designed to have good efficiency over a wide range of lags a maximin efficiency robust statistic and a simplified version of this statistic both statistics can be computed as easily as the log rank statistic asymptotic efficiency calculations supported by small sample simulations show that the two proposed statistics are substantially more efficient than the conventional log rank statistic in certain lag situations with comparatively little efficiency loss relative to the log rank statistic when no lag exists we recommend the maximin statistic for situations where a lag is expected but cannot be specified precisely in advance

nordsieck methods with computationally verified algebraic stability abstract we describe the search for algebraically stable nordsieck methods of order p s and stage order q p where s is the number of stages this search is based on the theoretical criteria for algebraic stability proposed recently by hill and hewitt and hill for general linear methods for ordinary differential equations these criteria which are expressed in terms of the non negativity of the eigenvalues of a hermitian matrix on the unit circle are then verified computationally for the derived nordsieck methods of order p 2

explicit two step peer methods we present a new class of explicit two step peer methods for the solution of nonstiff differential systems a construction principle for methods of order p s s the number of stages with optimal zero stability is given two methods of order p 6 found by numerical search are tested in matlab on several representative nonstiff problems the comparison with ode45 confirms the high potential of the new class of methods

the mel nikov technique for highly dissipative systems we present explicit calculations that extend the applicability of the mel nikov technique to include a general class of highly dissipative systems in particular the dissipation may be in the form of large positive or negative damping the only required assumption is that each system of this class possesses a homoclinic or a heteroclinic orbit we also show that sufficiently small time sinusoidal perturbation of these systems results in nonempty transversal intersection of stable and unstable manifolds for all but at most discretely many frequencies the results are then demonstrated via computer simulation of the highly damped pendulum with constant plus small time sinusoidal forcing

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

a new semiparametric estimation method for accelerated hazard model summary the accelerated hazard model has been proposed for more than a decade however its application is still very limited partly due to the complexity of the existing semiparametric estimation method we propose a new semiparametric estimation method based on a kernel smoothed approximation to the limit of a profile likelihood function of the model the method leads to smooth estimating equations and is easy to use the estimates from the method are proved to be consistent and asymptotically normal our numerical study shows that the new method is more efficient than the existing method the proposed method is employed to reanalyze the data from a brain tumor treatment study

weighted logrank tests to detect a transient improvement in survivorship summary in investigating whether an intervention has an effect on survival we exploit the flexibility of weighted logrank tests for constructing optimal tests sensitive to specified patterns of changes in the ratios or differences of the hazards as a function of time the tests presented seem appropriate if either the intervention on trial imposes changes in life style such as a diet modification or an exercise regimen or if the effect of therapy itself is limited in duration such as in the case of drugs for aids that may induce drug resistant viruses for such problems we postulate no difference between the hazards of the two groups initially increased discrepancies as time goes on and an eventual leveling off of the discrepancies in hazards perhaps due to compliance problems or to adoption of the behavior change under study by individuals in the control group we suggest a quadratic weight function for such problems and show how to evaluate the power of the proposed tests as well as their efficiency relative to other logrank tests other uses of quadratic weight functions for example for evaluating effects with lag times and for some parametric procedures are also described additive and multiplicative models are presented and both discrete and continuous times are considered

models for survival data from cancer prevention studies summary we propose a new parametric survival model for cancer prevention studies the formulation of the model is in the spirit of stochastic modelling of the occurrences of tumours through two stages initiation of an undetected tumour and promotion of the tumour to a detectable cancer several novel properties of the model proposed are derived in addition we examine the relationship of our model with the existing lagged regression model of zucker and lakatos also we bridge the difference between two distinct stochastic modelling methods for cancer data one used primarily for cancer therapeutic trials and the other used for cancer prevention trials

elements of a general theory of composite integration methods this contribution is a survey of results on general linear methods after an introduction into the concept of a methods which is a slight generalization of general linear methods and its main results rk methods are presented in the a method notation which offers the following advantages a the order conditions are obtained from a simple recursion without relying on rooted trees or elementary differentials b they appear in a transparent form as orthogonality relations in dependence of the error constants of the stages of the rk method c their orthogonal structure permits elegant existence proofs and yields new auxiliary conditions d the error accumulation within the stages of the method is completely understood the proof of the main result on rk methods theorem 5 is simplified

order conditions for general linear methods we describe the derivation of order conditions without restrictions on stage order for general linear methods for ordinary differential equations this derivation is based on the extension of the albrecht approach proposed in the context of runge kutta and composite and linear cyclic methods this approach was generalized by jackiewicz and tracogna to two step runge kutta methods by jackiewicz and vermiglio to general linear methods with external stages of different orders and by garrappa to some classes of runge kutta methods for volterra integral equations with weakly singular kernels this leads to general order conditions for many special cases of general linear methods such as diagonally implicit multistage integration methods nordsieck methods and general linear methods with inherent runge kutta stability exact coefficients for several low order methods with some desirable stability properties are presented for illustration

accelerated hazards model based on parametric families generalized with bernstein polynomials a transformed bernstein polynomial that is centered at standard parametric families such as weibull or log logistic is proposed for use in the accelerated hazards model this class provides a convenient way towards creating a bayesian nonparametric prior for smooth densities blending the merits of parametric and nonparametric methods that is amenable to standard estimation approaches for example optimization methods in sas or r can yield the posterior mode and asymptotic covariance matrix this novel nonparametric prior is employed in the accelerated hazards model which is further generalized to time dependent covariates the proposed approach fares considerably better than previous approaches in simulations data on the effectiveness of biodegradable carmustine polymers on recurrent brain malignant gliomas is investigated

on the inversion of the vandermonde matrix abstract the inversion of the vandermonde matrix has received much attention for its role in the solution of some problems of numerical analysis and control theory this work deals with the problem of getting an explicit formula for the generic element of the inverse we derive two algorithms in o n2 and o n3 and compare them with the parker traub and the bjorck pereyra algorithms

a general class of two step runge kutta methods for ordinary differential equations a general class of two step runge kutta methods that depend on stage values at two consecutive steps is studied these methods are special cases of general linear methods introduced by butcher and are quite efficient with respect to the number of function evaluations required for a given order general order conditions are derived using the approach proposed recently by albrecht and examples of methods are given up to the order 5 these methods can be divided into four classes that are appropriate for the numerical solution of nonstiff or stiff differential equations in sequential or parallel computing environments

a versatile test for equality of two survival functions based on weighted differences of kaplan meier curves with censored event time observations the logrank test is the most popular tool for testing the equality of two underlying survival distributions although this test is asymptotically distribution free it may not be powerful when the proportional hazards assumption is violated various other novel testing procedures have been proposed which generally are derived by assuming a class of specific alternative hypotheses with respect to the hazard functions the test considered by pepe and fleming 1989 is based on a linear combination of weighted differences of the two kaplan meier curves over time and is a natural tool to assess the difference of two survival functions directly in this article we take a similar approach but choose weights that are proportional to the observed standardized difference of the estimated survival curves at each time point the new proposal automatically makes weighting adjustments empirically the new test statistic is aimed at a one sided general alternative hypothesis and is distributed with a short right tail under the null hypothesis but with a heavy tail under the alternative the results from extensive numerical studies demonstrate that the new procedure performs well under various general alternatives with a caution of a minor inflation of the type i error rate when the sample size is small or the number of observed events is small the survival data from a recent cancer comparative study are utilized for illustrating the implementation of the process copyright 2015 john wiley sons ltd

an approximate algorithm for estimating treatment lags from right censored data abstract the problem of likelihood estimate of a function given limited data and a set of constraints finds many practical applications one of them is the lag function estimation for treatment effects in clinical trials however searching for the global optimum in such problems often turns out to be prohibitively complex in this paper we present an algorithm that can be effectively used to approximate such solutions this algorithm modifies standard genetic algorithms which model the natural processes of information inheritance and selective pressure the most important modifications deal with the complex strong constraints imposed on the sought solution we present few experiments with a simplified version of the problem and discuss possible future extensions that relax the restrictions

designing cancer immunotherapy trials with random treatment time lag effect in some clinical settings such as the cancer immunotherapy trials a treatment time lag effect may be present and the lag duration possibly vary from subject to subject an efficient study design and analysis procedure should not only take into account the time lag effect but also consider the individual heterogeneity in the lag duration in this paper we present a generalized piecewise weighted logrank gpw logrank test designed to account for the random time lag effect while maximizing the study power with respect to the weights based on the proposed test both analytic and numeric approaches are developed for the sample size and power calculation asymptotic properties are derived and finite sample efficiency is evaluated in simulations compared with the standard practice ignoring the delayed effect the proposed design and analysis procedures are substantially more efficient when a random lag is expected further compared with the existing methods by xu et al considering the fixed time lag effect the proposed approaches are significantly more robust when the lag model is misspecified an r package delayedeffect design is developed for implementation

p stable general nystrom methods for y f y t we focus our attention on the family of general linear methods glms for the numerical solution of second order ordinary differential equations odes these are multivalue methods introduced in d ambrosio et al 2012 3 with the aim to provide an unifying approach for the analysis of the properties of accuracy of numerical methods for second order odes our investigation is addressed to providing the building blocks useful to analyze the linear stability properties of glms for second order odes thus we present the extension of the classical notions of stability matrix stability polynomial stability and periodicity interval a stability and p stability to the family of glms special attention will be given to the derivation of highly stable glms whose stability properties depend on the stability polynomial of indirect runge kutta nystrom methods based on gauss legendre collocation points which are known to be p stable in this way we are able to provide p stable glms whose order of convergence is greater than that of the corresponding rkn method without heightening the computational cost we finally provide and discuss examples of p stable irreducible glms satisfying the mentioned features

on monotonicity and boundedness properties of linear multistep methods in this paper an analysis is provided of nonlinear monotonicity and boundedness properties for linear multistep methods instead of strict monotonicity for arbitrary starting values we shall focus on generalized monotonicity or boundedness with runge kutta starting procedures this allows many multistep methods of practical interest to be included in the theory in a related manner we also consider contractivity and stability in arbitrary norms

search for efficient general linear methods for ordinary differential equations we analyze implicit general linear methods with s internal stages and r s 1 external stages of order p s 1 and stage order q s or q s 1 these methods might eventually lead to more efficient formulas than the class of dimsims and the class of general linear methods with inherent runge kutta stability we analyze also error propagation and estimation of local discretization errors examples of such methods which are a and l stable are derived up to the stage order q 3 or q 4 and order p 4

a nonlinear optimization approach to the construction of general linear methods of high order we describe the construction of diagonally implicit multistage integration methods of order and stage order p q 7 and p q 8 for ordinary differential equations these methods were obtained using state of the art optimization methods particularly variable model trust region least squares algorithms

monotonicity preserving linear multistep methods in this paper we provide an analysis of monotonicity properties for linear multistep methods these monotonicity properties include positivity and the diminishing of total variation we also pay particular attention to related boundedness properties such as the total variation bounded tvb property in the analysis the multistep methods are considered in combination with suitable starting procedures this allows for monotonicity statements for classes of methods which are important and often used in practice but which were thus far not covered by theoretical results

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

computation of optimal monotonicity preserving general linear methods monotonicity preserving numerical methods for ordinary differential equations prevent the growth of propagated errors and preserve convex boundedness properties of the solution we formulate the problem of finding optimal monotonicity preserving general linear methods for linear autonomous equations and propose an efficient algorithm for its solution this algorithm reliably finds optimal methods even among classes involving very high order accuracy and that use many steps and or stages the optimality of some recently proposed methods is verified and many more efficient methods are found we use similar algorithms to find optimal strong stability preserving linear multistep methods of both explicit and implicit type including methods for hyperbolic pdes that use downwind biased operators

strong stability preserving explicit runge kutta methods of maximal effective order we apply the concept of effective order to strong stability preserving ssp explicit runge kutta methods relative to classical runge kutta methods methods with an effective order of accuracy are designed to satisfy a relaxed set of order conditions but yield higher order accuracy when composed with special starting and stopping methods we show that this allows the construc tion of four stage ssp methods with effective order four such methods cannot have classical order four however we also prove that effective order five methods like classical order five methods require the use of nonpositive weights and so cannot be ssp by numerical optimization we construct explicit ssp runge kutta methods up to effective order four and establish the optimality of many of them numerical experiments demonstrate the validity of these methods in practice