recursive combination tests we present a method that extends the flexibility of adaptive designs to the number of interim analyses and to the choice of decision boundaries at each stage of the trial the design of the next stage can be determined using all of the information gathered so far additionally one can specify the next stage as the final stage or can plan a further interim analysis the method is based on a recursive application of the two stage combination tests for p values the crucial point is the appropriate definition of a p value function that combines p values from two separate stages to a single p value formally we start with a two stage combination test however the p value of the second stage can be replaced by the p value of a further combination test this applies if the experimenter decides in the first interim analysis to perform another interim analysis thereby extending the design to at least three stages obviously this can be also done in a recursive way in the following interim analyses the test

irregular dynamics and homoclinic orbits to hamiltonian saddle centers we consider 4 dimensional real analytic hamiltonian systems with a saddle center equilibrium related to a pair of real and a pair of imaginary eigenvalues and a homoclinic orbit to it we find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium we also consider one parameter families of reversible 4 dimensional hamiltonian systems we prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points we also present similar results for systems with heteroclinic orbits to saddle center equilibria 1997 john wiley sons inc

homoclinic orbits to a saddle center in a fourth order differential equation abstract we study homoclinic orbits to a saddle center of a fourth order ordinary differential equation which is invariant under the transformation x x involving an eigenvalue parameter q and an odd piece wise cubic type nonlinearity it is found that for a sequence of eigenvalues which tends to infinity homoclinic orbits exist whose complexity increases as the eigenvalue becomes larger these orbits are found to be embedded in branches of homoclinic orbits to periodic orbits as x

periodic solutions for a class of fourth order nonlinear differential equations in this paper by using inequality techniques and coincidence degree theory some sufficient conditions are established for ensuring the existence of t periodic solutions for a class of fourth order nonlinear differential equations two illustrative examples are provided to demonstrate that the results of this paper hold under weaker conditions than the existing ones and are more effective

existence of multiple solutions to nonlinear elliptic equations of nondivergence form abstract in this paper by means of a priori estimates and topological degree theory we prove the existence of two or more solutions to nonlinear elliptic equations of nondivergence form

periodic solutions of some fourth order nonlinear differential equations abstract in this paper using mawhin s continuation theorem we obtain existence results concerning t periodic solutions of fourth order nonlinear differential equations of the type u p u g t u e t where p r g r 2 r is continuous t periodic in t and e r r is continuous t periodic with mean value zero applications are given for the extended fisher kolmogorov and swift hohenberg equations with forcing term

on the periodic orbits of the fourth order differential equation u qu u ef u u u u we provide sufficient conditions for the existence of periodic solutions of the fourth order differential equation u qu u ef u u u u where q and e are real parameters e is small and f is a nonlinear function

on the limit cycles for a class of fourth order differential equations we provide sufficient conditions for the existence of periodic solutions of the fourth order differential equation where p is a rational number different from 0 e is small and f is a nonlinear function

optimal conditional error functions for the control of conditional power ethical considerations and the competitive environment of clinical trials usually require that any given trial have sufficient power to detect a treatment advance if at an interim analysis the available data are used to decide whether the trial is promising enough to be continued investigators and sponsors often wish to have a high conditional power which is the probability to reject the null hypothesis given the interim data and the alternative of interest under this requirement a design with interim sample size recalculation which keeps the overall and conditional power at a prespecified value and preserves the overall type i error rate is a reasonable alternative to a classical group sequential design in which the conditional power is often too small in this article two stage designs with control of overall and conditional power are constructed that minimize the expected sample size either for a simple point alternative or for a random mixture of alternatives given by a prior density for the efficacy parameter the presented optimality result applies to trials with and without an interim hypothesis test in addition one can account for constraints such as a minimal sample size for the second stage the optimal designs will be illustrated with an example and will be compared to the frequently considered method of using the conditional type i error level of a group sequential design

periodic solutions for extended fisher kolmogorov and swift hohenberg equations obtained using a continuation theorem abstract this paper is concerned with an existence result for periodic solutions for a class of fourth order ordinary differential equations involving extended fisher kolmogorov and swift hohenberg equations where under a suitable growth condition on the nonlinear term one proves an existence result by applying mawhin s continuation theorem

controlling the type i error rate in two stage sequential adaptive designs when testing for average bioequivalence in a 2 2 crossover trial for establishing average bioequivalence abe of a generic agent and a currently marketed drug the recommended approach to hypothesis testing is the two one sided test tost procedure which depends among other things on the estimated within subject variability the power of this procedure and therefore the sample size required to achieve a minimum power depends on having a good estimate of this variability when there is uncertainty it is advisable to plan the design in two stages with an interim sample size reestimation after the first stage using an interim estimate of the within subject variability one method and 3 variations of doing this were proposed by potvin et al using simulation the operating characteristics including the empirical type i error rate of the 4 variations called methods a b c and d were assessed by potvin et al and methods b and c were recommended however none of these 4 variations formally controls the type i error rate of falsely claiming abe even though the amount of inflation produced by method c was considered acceptable a major disadvantage of assessing type i error rate inflation using simulation is that unless all possible scenarios for the intended design and analysis are investigated it is impossible to be sure that the type i error rate is controlled here we propose an alternative principled method of sample size reestimation that is guaranteed to control the type i error rate at any given significance level this method uses a new version of the inverse normal combination of p values test in conjunction with standard group sequential techniques that is more robust to large deviations in initial assumptions regarding the variability of the pharmacokinetic endpoints the sample size reestimation step is based on significance levels and power requirements that are conditional on the first stage results this necessitates a discussion and exploitation of the peculiar properties of the power curve of the tost testing procedure we illustrate our approach with an example based on a real abe study and compare the operating characteristics of our proposed method with those of method b of povin et al

periodic and homoclinic solutions of extended fisher kolmogorov equations in this paper we study the existence of periodic solutions of the fourth order equations uiv pu a x u b x u3 0 and uiv pu a x u b x u3 0 where p is a positive constant and a x and b x are continuous positive 2l periodic functions the boundary value problems p1 and p2 for these equations are considered respectively with the boundary conditions u 0 u l u 0 u l 0 existence of nontrivial solutions for p1 is proved using a minimization theorem and a multiplicity result using clark s theorem existence of nontrivial solutions for p2 is proved using the symmetric mountain pass theorem we study also the homoclinic solutions for the fourth order equation uiv pu a x u b x u2 c x u3 0 where p is a constant and a x b x and c x are periodic functions the mountain pass theorem of brezis and nirenberg and concentration compactness arguments are used

periodic solutions for extended fisher kolmogorov and swift hohenberg equations by truncature techniques abstract combining truncature techniques with a variational approach we establish an existence result for nontrivial periodic solutions for a class of fourth order ordinary differential equations involving extended fisher kolmogorov and swift hohenberg equations

homoclinic solutions for swift hohenberg and suspension bridge type equations we establish the existence of homoclinic solutions for a class of fourth order equations which includes the swift hohenberg model and the suspension bridge equation in the first case the nonlinearity has three zeros corresponding to a double well potential while in the second case the nonlinearity is asymptotically constant on one side the swift hohenberg model is a higher order extension of the classical fisher kolmogorov model its more complicated dynamics give rise to further possibilities of pattern formation the suspension bridge equation as studied by chen and mckenna j differential equations 136 1997 325 355 we give a positive answer to an open question raised by the authors c 2002 elsevier science usa

positive and negative solutions of a quasi linear elliptic equation by a mountain pass method and truncature techniques abstract the existence of a positive and a negative solution of a problem of the type u f x u u 0 in u 0 on is proved when f has a growth at infinity depending both on u and on u the techniques are based on an iterative scheme of mountain pass approximated solutions and the use of a suitable truncature method

adaptive sample size calculations in group sequential trials a method for group sequential trials that is based on the inverse normal method for combining the results of the separate stages is proposed without exaggerating the type i error rate this method enables data driven sample size reassessments during the course of the study it uses the stopping boundaries of the classical group sequential tests furthermore exact test procedures may be derived for a wide range of applications the procedure is compared with the classical designs in terms of power and expected sample size

sm 2 hbox 4 kappa fermionic characters and restricted jagged partitions a derivation of the basis of states for the superconformal minimal models is presented it relies on a general hypothesis concerning the role of the null field of dimension 2 1 2 the basis is expressed solely in terms of gr modes and it takes the form of simple exclusion conditions being thus a quasi particle type basis its elements are in correspondence with 2 1 restricted jagged partitions the generating functions of the latter provide novel fermionic forms for the characters of the irreducible representations in both ramond and neveu schwarz sectors

periodic orbits of the fourth order non autonomous differential equation u qu pu ef t u u u u abstract we provide sufficient conditions for the existence of periodic solutions of the fourth order differential equation u qu pu e f t u u u u where q p and e are real parameters e is small and f is a nonlinear non autonomous periodic function with respect to t moreover we provide some applications

existence and numerical approximations of periodic solutions of semilinear fourth order differential equations abstract a multiplicity result of existence of periodic solutions with prescribed wavelength for a class of fourth order nonautonomous differential equations related either to the extended fisher kolmogorov or to the swift hohenberg equation is proved variational approach is used some numerical solutions are calculated via the finite element method

homotopy classes for stable connections between hamiltonian saddle focus equilibria for a class of hamiltonian systems in 4 the set of homoclinic and heteroclinic orbits which connect saddle focus equilibria is studied using a variational approach the oscillatory properties of a saddle focus equilibrium and the variational nature of the problem give rise to connections in many homotopy classes of the configuration plane punctured at the saddle foci this variational approach does not require any assumptions on the intersections of stable and unstable manifolds such as transversality moreover these connections are shown to be local minimizers of an associated action functional this result has applications to spatial pattern formation in a class of fourth order bistable evolution equations

pattern selection of solutions of the swift hohenberg equation abstract we study the large time behaviour of solutions u x t of the swift hohenberg equation on a one dimensional domain 0 l focusing in particular on the role of the eigenvalue parameter and the length l of the domain on the selection of the limiting profiles v we show by means of numerical simulations how different values of these parameters may lead to qualitatively different final profiles and prove the existence of a collection n of disjoint intervals which depend on the value of such that if l n then solutions all converge to the trivial solution v 0 we show that branches of nontrivial solutions bifurcate from the trivial solution at the endpoints of these intervals and we study the local behaviour of these branches we establish global bounds for the solutions u x t and identify the global attractor finally we derive an estimate about the shape of the final state

constructive proof of existence for a class of fourth order nonlinear bvps a new existence proof of solutions for a class of fourth order nonlinear boundary value problems is proposed the proof of the main results is based on the reproducing kernel theorem it is worthwhile to point out that the method presented in this paper can be applied for the existence proof of diverse kinds of boundary conditions

a unified theory of two stage adaptive designs adaptation of a clinical trial occurs when there is a change from the planned course of action in design sample size or method of analysis there are many proposed adaptations in the literature but no comprehensive mathematical theory to support them the standard neyman pearson theory is not suitable for the setting and its use may result in invalid inference this article provides a general formulation of adaptation under which its validity can be broadly and rigorously established the basic theory on distribution stochastic independence point estimation confidence intervals hypothesis testing and asymptotics is developed various examples are given to illustrate applications of the theory the goal is to provide a solid theoretical foundation to aid the development of adaptive methods for clinical investigations