the minimal period problem for nonconvex even second order hamiltonian systems abstract in this paper we study the minimal period problem for even autonomous second order hamiltonian systems defined on r nwithout any convexity assumption by using the variational methods we obtain estimates on the minimal period of the corresponding nonconstant periodic solution of the superquadratic and asymptotically linear hamiltonian systems

nontrivial periodic solutions for the asymptotically linear hamiltonian systems with resonance at infinity abstract in this paper we are concerned with the existence of nontrivial periodic solutions of an asymptotically linear hamiltonian systems when a kind of resonance condition is satisfied we prove several new existence results via the morse theory the garlerkin methods and the moslov index theory

morse theory and asymptotic linear hamiltonian system on considere l existence de solutions 2 periodiques des systemes hamiltoniens asymptotiquement quadratiques jz h z t ou j 0 i i 0 i in la matrice identite de n n

infinite dimensional cohomology groups and periodic solutions of asymptotically linear hamiltonian systems abstract in this paper we study the existence of nontrivial 2 periodic solutions of asymptotically linear hamiltonian systems we consider the case of resonance both at zero and at infinity and we permit time dependent asymptotic matrices our main tools are an infinite dimensional cohomology theory and a corresponding morse theory recently constructed by w kryszewski and the first author we develop a method to compute the new critical groups

sequential monitoring of randomization tests stratified randomization in many clinical trials it is desirable to establish a sequential monitoring plan whereby the test statistic is computed at an interim point or points in the trial and a decision is made whether to stop early due to evidence of treatment efficacy in this article we will set up a sequential monitoring plan for randomization based inference under the permuted block design stratified block design and stratified urn design we will also propose a definition of information fraction in these settings and discuss its calculation under these different designs

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

periodic solutions for asymptotically linear hamiltonian system with resonance both at infinity and origin via computation of critical groups abstract in this paper we study the existence of nontrivial periodic solutions for asymptotically linear hamiltonian system which has the same linear term both at infinity and origin by computation of critical groups at origin and infinity associated with maslov type index we establish the existence for nontrivial periodic solution of the system moreover some conditions employed for the computation of critical groups are weakened

periodic solutions of classical hamiltonian systems without palais smale condition abstract in this paper we study the existence of periodic solutions for classical hamiltonian systems without the palais smale condition we prove that the information of the potential function contained in a finite domain is sufficient for the existence of periodic solutions moreover we establish the existence of infinitely many periodic solutions without any symmetric condition on the potential function v

periodic solutions of autonomous newtonian systems partially asymptotically linear at infinity abstract the goal of this article is to study the existence continuation and global bifurcation of autonomous newtonian systems partially asymptotically linear at infinity the main idea is to prove a kind of splitting lemma and apply the degree for s 1 equivariant gradient maps defined by rybicki 1994 in 16

multiple periodic solutions for asymptotically linear duffing equations with resonance we investigate multiple periodic solutions of asymptotically linear duffing equation with resonance using index theory and morse theory and obtain a new result

k treatment comparisons with restricted randomization rules in clinical trials in the course of conducting a clinical trial to compare k 2 treatments it is often desirable to balance the trial with respect to the assignments of patients to treatments on the other hand some form of randomization of treatment assignments is essential for reducing experimental bias in this article the large sample approximation to the null distribution of k sample randomization tests generated from a broad class of restricted randomization rules is derived the implication of this result for conditional inference is also discussed

spectral flow and bifurcation of critical points of strongly indefinite functionals abstract our main results here are as follows let x be a family of 2 periodic hamiltonian vectorfields that depend smoothly on a real parameter in a b and has a known trivial branch s of 2 periodic solutions let p be the poincare map of the linearization of x at s if the conley zehnder index of the path p does not vanish then any neighborhood of the trivial branch of periodic solutions contains 2 periodic solutions not on the branch moreover if each solution s is constant and each linearization a of x at s is time independent then bifurcation of 2 periodic orbits from the branch of equilibria arises whenever i a b i a b where i a is the index of the linear hamiltonian system ju au

the cohomological span of ls conley index abstract in this paper we introduce a new homotopy invariant the cohomological span of ls conley index we prove the theorems on the existence of critical points for a class of strongly indefinite functionals with the gradient of the form l x k x where l is bounded linear and k is completely continuous we give examples of hamiltonian systems for which our methods give better results than the morse inequalities we also give a formula for the ls index of an isolated critical point which is an extension of the classical dancer theorem for the case of ls index

infinitely many periodic solutions for second order hamiltonian systems in this paper we study the existence of infinitely many periodic solutions for second order hamiltonian systems u uv t u 0 where v t u is either asymptotically quadratic or superquadratic as u

numerical computation of rank one convex envelopes we describe an algorithm for the numerical computation of the rank one convex envelope of a function f mm m times n rightarrow rr we prove its convergence and an error estimate in l infty

nontrivial periodic solutions for strong resonance hamiltonian systems via a local linking theorem in this paper we first establish an existence result of critical points for a class of functionals defined on hilbert spaces by using a local linking idea then as an application of the existence result we obtain the existence of periodic solutions of strong resonance hamiltonian systems which are asymptotically linear both at infinity and at origin

an asymmetric superlinear elliptic problem at resonance abstract in this article we study the existence and multiplicity of solutions to the problem u g x u in u 0 on where is a bounded domain in r n n 2 with smooth boundary and g r r is a differentiable function we will assume that g x s has a resonant behavior for large negative values of s and that a landesman lazer type condition is satisfied we also assume that g x s is superlinear but subcritical for large positive values of s we prove the existence and multiplicity of solutions for problem 1 1 by using minimax methods and infinite dimensional morse theory

discrete sequential boundaries for clinical trials summary pocock 1977 o brien fleming 1979 and slud wei 1982 have proposed different methods to construct discrete sequential boundaries for clinical trials these methods require that the total number of decision times be specified in advance in the present paper we propose a more flexible way to construct discrete sequential boundaries the method is based on the choice of a function a t which characterizes the rate at which the error level ac is spent the boundary at a decision time is determined by a t and by past and current decision times but does not depend on the future decision times or the total number of decision times

significance tests with restricted randomization design in an experiment comparing the effectiveness of two treatments a and b subjects arrive at an experimental site sequentially and must be assigned immediately to one of the two treatment groups the complete randomization scheme tossing a fair coin each time independently to decide which treatment will be used for the next subject has been widely employed because this scheme is free of different kinds of experimental bias efron 1971 and it also provides a basis for statistical inference lehmann 1975 chapter 1 but in a small experiment complete randomization may result in a severe imbalance between the numbers of subjects assigned to the different treatment groups a simple way to attain a perfectly balanced experiment is to use the systematic design abab however this plan can easily bias the results of the experiment in a number of ways for example if the experimenter is predisposed toward treatment a and if he knows or guesses that treatment a will be the next assignment then he may tend to choose a subject with a high expected response because of this bias at the end of the experiment treatment a might appear to be significantly better than treatment b even if there is in fact no difference between the two obviously the systematic design maximizes this kind of bias several compromises between a perfectly balanced assignment scheme and a completely randomized one have been proposed for example the permuted block design the biased coin design efron 1971 the urn design wei 1977 1978a the adaptive biased coin design wei 1978b and the random allocation design stigler 1969 wei 1978c simon 1979 has given an excellent review in particular the urn and adaptive biased coin designs force a small experiment to be balanced but they tend to the complete randomization scheme as the size of the experiment increases recently a specific urn design has been shown to be d optimal by atkinson 1982 sometimes it may be inappropriate to postulate a population from which the subjects in the experiment have been obtained by random sampling instead a randomization model lehmann 1975 chapter 1 is utilized when the data are analysed with this approach the significance tests used for testing the null hypothesis ho that there is no difference between a and b must be only those tests generated by the experimental randomization design actually employed for any given sequence of responses from the subjects one can tabulate all possible patterns of treatment assignments to subjects using the restricted randomization rule and calculate the corresponding treatment

optimum biased coin designs for sequential clinical trials with prognostic factors patients in a clinical trial arrive sequentially and are assigned to one of t treatments this assignment should maintain a balance between the numbers receiving each treatment yet should be sufficiently random to avoid any suspicion of conscious or unconscious cheating to balance these requirements efron 1971 introduced biased coin designs for the comparison of two treatments in which allocation of the treatment is determined probabilistically but with a bias towards the underrepresented treatment one disadvantage of efron s scheme is that it does not include balance over covariates or prognostic factors which may affect the response of the patient to the treatment biased coin schemes which do force balance over both treatments and prognostic factors are given by pocock simon 1975 and efron 1980 the properties of the designs have been elucidated by numerical studies and they are now being increasingly used in clinical trials reviews of the literature and discussions of the practical implications of the designs are given by pocock 1979 and simon 1979 however the designs suffer from the disadvantage that they rely on arbitrary functions to achieve the desired balance the procedures thus lack a firm theoretical framework an alternative approach in the presence of prognostic factors begg iglewicz 1980 uses optimum design theory to suggest a deterministic design criterion which is then modified for computational convenience in this paper i use optimum design theory to provide a procedure of the biased coin type for an arbitrary number of treatments in the presence or absence of prognostic factors this has the theoretical advantage of obviating dependence on a series of arbitrary functions the necessary optimum design theory is presented in 2 and in 3 applied to biased coin experiments the simplest case that of two treatments in the absence of prognostic factors is studied in 4 the extension to three or more treatments is in 5 followed in 6 by the allowance for prognostic factors although in the biased coin designs allocation of treatments is made probabilistically the results of the paper can be used for the sequential construction of da optimum designs in which randomization is not introduced by the experimenter all the designs mentioned so far are sequential but not adaptive in the useful distinction made by pericchi 1981 they are data dependent but not outcomedependent each allocation depends on the previous allocations and prognostic factors

multiple periodic solutions for asymptotically linear ordinary differential equations with double resonance abstract in this paper using index theory and morse theory we investigate multiple periodic solutions of asymptotically linear ordinary differential equation with double resonance at infinity and with resonance at origin and obtain new results

multiple periodic orbits of asymptotically linear autonomous convex hamiltonian systems we investigate multiple periodic solutions of convex asymptotically linear autonomous hamiltonian systems our theorem generalizes a result by mawhin and willem

global bifurcations of critical orbits of g invariant strongly indefinite functionals abstract let h be a separable hilbert space which is an orthogonal representation of a compact lie group g and let h r be a g invariant strongly indefinite functional of the class c 1 to study critical orbits of we have defined the degree for g invariant strongly indefinite functionals which is an element of the euler ring u g using this degree we have formulated the rabinowitz alternative for g invariant strongly indefinite functionals the abstract results are applied to the study of global bifurcations of weak solutions of non cooperative systems of elliptic differential equations

periodic solutions of asymptotically linear autonomous hamiltonian systems abstract in this paper we study the existence and the continuation of periodic solutions of autonomous hamiltonian systems we apply the degree for g invariant strongly indefinite functionals defined in gol biewska and rybicki 2011 2 we consider resonant as well as nonresonant case at stationary solutions

minimal period estimates of periodic solutions for superquadratic hamiltonian systems in this paper we study the minimal period problem for the first order hamiltonian systems which may not be strictly convex by using variational methods and the iteration formula of the maslov type index theory we obtain estimates of the minimal period of the corresponding nonconstant periodic solutions

a twist condition and periodic solutions of hamiltonian systems abstract in this paper we investigate existence of nontrivial periodic solutions to the hamiltonian system hs j z h t z z r 2 n under a general twist condition for the hamiltonian function in terms of the difference of the conley zehnder index at the origin and at infinity we establish existence of nontrivial periodic solutions compared with the existing work in the literature our results do not require the hamiltonian function to have linearization at infinity our results allow interactions at infinity between the hamiltonian and the linear spectra the general twist condition raised here seems to resemble more the spirit of poincare s last geometric theorem

computations of cohomology groups and nontrivial periodic solutions of hamiltonian systems by computing the e critical groups at and infinity of the corresponding functional of hamiltonian systems we proved the existence of nontrivial periodic solutions for the systems which may be resonant at and infinity under some new conditions some results in the literature are extended and some new type of theorems are proved the main tool is the e morse theory developed by kryszewski and szulkin

existence of nontrivial periodic solutions for a class of resonance hamiltonian systems abstract in this paper we study the asymptotically linear hamiltonian systems jz h t z with resonance at infinity via the morse theory and the galerkin methods and obtain some new existence results about the nontrivial periodic solutions of this system

nontrivial solutions of operator equations and morse indices of critical points of min max type on montre que si les hypotheses du theoreme de point selle sont satisfaites si dim y 2 et si f 1 c contient seulement des points critiques non degeneres alors f 1 c contient au moins un point critique u 0 tel que 2 indice de morse u 0 dim y

spectral flow and bifurcation of critical points of strongly indefinite functionals part i general theory abstract spectral flow is a well known homotopy invariant of paths of selfadjoint fredholm operators we describe here a new construction of this invariant and prove the following theorem let i u rbe ac2function on the product of a real intervali a b with a neighborhood u of the origin of a real separable hilbert space h and such that for each in i 0is a critical point of the functional assume that the hessian l of at0is fredholm and moreover that laand lbare nonsingular if the spectral flow of the path l idoes not vanish then the interval i contains at least one bifurcation point from the trivial branch for solutions of the equation x x 0 equivalently every neighborhood of i 0 contains points of the form x wherex 0 is a critical point of

a new index theory for gl 2 paths with applications to asymptotically linear systems abstract in this article we establish a new index theory defined for the general non degenerate matrix paths in gl 2 this is done by the complete homotopy classification for such paths the parity relation theorem is established for relating this index to the morse index of the corresponding differential operator as an application we establish the existence result of a non trivial periodic solution of the asymptotically linear differential equation systems under an index twist condition

on resonance hamiltonian systems without the palais smale condition abstract in this paper we study periodic solutions of resonance hamiltonian systems by combining conley index theory and galerkin approximation method we prove the existence of periodic solutions of hamiltonain systems where both the palais smale condition and the strong resonance method fail

multiple solutions for asymptotically linear duffing equations with neumann boundary value conditions ii abstract we study multiple solutions of asymptotically linear duffing equations satisfying neumann boundary value conditions with resonance by using index theory and lusternik schnirelmann theory and obtain some new results

existence of solutions for a class of semilinear elliptic equations with the robin boundary value condition abstract in this paper we show the existence of at least one nontrivial solution for a class of semilinear elliptic equations with the robin boundary value condition by using linking methods