stability of discrete breathers abstract we approach the problem of linear stability of discrete breathers in a rigorous way for networks of not necessarily identical oscillators with general type of coupling in the case of hamiltonian systems using symplectic signature theory we give general conditions on the spectrum of the monodromy map in the uncoupled limit such that the discrete breathers are l 2 linearly stable for weak enough coupling we consider also dissipative networks of oscillators in this case we prove that the discrete breathers are not only stable but also attract a neighbourhood of initial data for any choice of l p topology some examples are considered amongst which an instance of roto breather

efficiency and robustness in the use of repeated measurements efficiencies in repeated measurements experiments are studied by the use of f statistics when appropriate and of hotelling s t2 statistics otherwise bounds on the asymptotic relative efficiencies of these statistics are given and conditions are found under which one test dominates another uniformly similar comparisons are made between t2 and the lawley hotelling t2 0 test in the analysis of repeated vector measurements it is found that the t2 test may be considerably more efficient than its competitors but that its efficiency cannot be substantially less these findings are illustrated numerically by examples from the literature it is shown further that f and t2 tests are exact for all unimodal distributions constant on ellipsoids and that efficiency comparisons among f tests are preserved under scale mixtures of multidimensional gaussian laws

sample size and power for pair matched case control studies a new sample size formula is derived herein for matched pair case control studies this result is based on an unconditional approach to the use of mcnemar s statistic we contrast this method for determining sample size and power with schlesselman s conditional approach and compare the results numerically in a monte carlo study

dispersive estimate for quasi periodic schrodinger operators on 1 d lattices abstract consider the one dimensional discrete schrodinger operator h h q n q n 1 q n 1 v n q n n z with r d diophantine and v a real analytic function on t d r 2 z d for v sufficiently small we prove the dispersive estimate for every l 1 z 1 e i t h l k 0 ln e 0 a ln ln 2 t 2 d t 1 3 l 1 t 1 t 2 with a and k 0 two absolute constants and e 0 an analytic norm of v the estimate holds for every t d

asymptotic stability of ground states in some hamiltonian pdes with symmetry we consider a ground state soliton of a hamiltonian pde we prove that if the soliton is orbitally stable then it is also asymptotically stable the main assumptions are transversal nondegeneracy of the manifold of the ground states linear dispersion in the form of strichartz estimates and nonlinear fermi golden rule we allow the linearization of the equation at the soliton to have an arbitrary number of eigenvalues the theory is tailor made for the application to the translational invariant nls in space dimension 3 the proof is based on the extension of some tools of the theory of hamiltonian systems reduction theory darboux theorem normal form to the case of systems invariant under a symmetry group with unbounded generators

sample size requirement for repeated measurements in continuous data in this paper we extend bloch s discussion on the usefulness and the limitations in the application of repeated measurements per subject in study designs we derive general sample size formulae for any finite number of comparison groups to calculate the required number of subjects with repeated measurements that do not have to be conditionally independent for fixed total cost we discuss the optimal sample allocation for repeated measurements needed to maximize the power and the underestimation when using bloch s sample size formula if in the hypothesis testing procedure the variance parameters are unknown we have also included a quantitative investigation of the effectiveness of taking repeated measurements per subjects to reduced the required number of subjects for a given power at a given alpha level

internal modes of discrete solitons near the anti continuum limit of the dnls equation discrete solitons of the discrete nonlinear schrodinger dnls equation are compactly supported in the anti continuum limit of the zero coupling between lattice sites eigenvalues of the linearization of the dnls equation at the discrete soliton determine its spectral stability small eigenvalues bifurcating from the zero eigenvalue near the anti continuum limit were characterized earlier for this model here we analyze the resolvent operator and prove that it is bounded in the neighborhood of the continuous spectrum if the discrete soliton is simply connected in the anti continuum limit this result rules out the existence of internal modes neutrally stable eigenvalues of the discrete spectrum near the anti continuum limit

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

sample sizes for repeated measurements in dichotomous data when the measurement of outcome varies within studied subjects and the cost of additional subjects is high taking more than one measurement for each subject constitutes a useful alternative to increase the power or to reduce the total cost of the study in this paper i present sample size formulae for repeated measurements in dichotomous data under different situations i also discuss optimal sample allocation for repeated measurements

sample size calculations for two group repeated measures experiments vonesh and schork 1986 biometrics 42 601 610 presented a statistical methodology for computing the minimum sample size required for the within subjects repeated measures design they applied hotelling s t2 analysis and demonstrated the utility of these techniques under general covariance structures in this paper we extend these procedures to the between subjects repeated measures design when there are two treatment groups under consideration the multivariate analysis of variance approach to analyzing repeated measurements is considered and this model also resolves to hotelling s t2 analysis two models are put forward to contend with those situations where the correlation structure among the repeated measures is unknown these include the autoregressive model and the assumption of compound symmetry and they are compared and contrasted throughout tables of the minimum sample sizes required for several hypotheses arising from repeated measures designs are presented

nonlinear resonance in systems with hysteresis a general approach is suggested for the analysis of nonlinear resonance in control systems including an integral linear term and a nonlinear term with hysteresis this approach uses a special property of nonlinear terms which is valid for many important classes of hysteresis operators for systems with such hysteresis nonlinearities we introduce a method to derive conditions for the existence of forced oscillations and for systems with a parameter of nonlinear resonance at infinity

sample size for repeated measures studies with binary responses we consider the sample size required for repeated measures studies when the response variable is binary we propose the use of weighted least squares wls for calculating the minimum sample size required to detect some minimum clinically important treatment effect we provide tabulated values of the estimated sample sizes for a simple example and we discuss some practical considerations in determination of sample size with repeated binary responses

existence of solutions to functional boundary value problem of second order nonlinear differential equation abstract in this paper a functional boundary value problem is studied based on mawhin s coincidence degree theory some existence theorems are obtained in the case of non resonance and the cases of dim ker l 1 and dim ker l 2 at resonance the results not only generalize and improve some known results of multi point and integral boundary value problems but also give some existence theorems for boundary value problems that all their boundary value conditions are relied on both x and x

breathers in nonlinear lattices existence linear stability and quantization abstract it has been proved by the principle of anti integrable limit renamed here anti continuous limit and the implicit function theorem that under rather weak hypotheses there exist spatially localized and time periodic solutions breathers in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong the models can be translationally invariant or not and be of any dimension there are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions multibreathers the whole set of these solutions can be labeled by a discrete coding sequence the condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called phonobreather which correspond to special codes which can be also spatially chaotic and where no oscillator is at rest these results extend almost without change to models with coupled rotors the existence of rotobreathers which are exact solutions consisting of a single rotor rotating while the other rotors oscillate is proved there are also multirotobreather solutions and phono rotobreathers etc it is also shown that the phase of the single breathers constituting a multibreather or multirotobreather solution can be submitted to a phase torsion these new dynamical structures carry a nonzero energy flow in models with two dimensions and more the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow the dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti continuous limit the existence of two kinds of polarobreathers which are spatially exponentially localized dynamical solutions is proved for a small enough electronic transfer integral the polarobreathers of the first kind include the standard static polaron the lattice configuration is static time independent when the electron is in an excited state i e the electronic oscillator oscillates periodically they can be spatially chaotic a polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation the most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site the standard floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the newton operator involved in the implicit function theorem the krein theory of bifurcations is reinterpreted from this point of view the linear stability of the single breather and rotobreather solutions is then proved at least at small enough coupling finally it is shown that in the quantum version of translationally invariant classical models where there cannot exist in principle any strictly localized excitation there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers the bandwidth of these excitations goes exponentially to zero with the number of bound phonons this theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist these numerical applications are currently written elsewhere

sample size determination for constrained longitudinal data analysis the longitudinal data analysis model proposed by liang and zeger sankhy indian j stat ser b 2000 62 134 148 uses the baseline as well as postbaseline values as the dependent variables and the baseline mean responses are constrained to be the same across treatment groups due to randomization compared with the conventional longitudinal analysis of covariance this approach can correctly estimate the variance of within group mean changes and achieve the specified coverage probabilities general results on the sample size and power calculations for this model in the presence of missing data are obtained the sample size relationship between the constrained and unconstrained longitudinal data analysis is established simple expressions for sample size calculation are obtained for the compound symmetry and first order autoregressive correlation structures the sensitivity of the sample size requirement to the configuration of correlation structure and retention pattern is assessed the performance of several ad hoc approximations for longitudinal data analysis sample size calculation is evaluated simulation studies are conducted to assess the validity of the proposed sample size formulas with deviation from normality the sample size formulas are also illustrated in detail using real clinical trial data copyright 2008 john wiley sons ltd

asymptotic stability of breathers in some hamiltonian networks of weakly coupled oscillators we consider a hamiltonian chain of weakly coupled anharmonic oscillators it is well known that if the coupling is weak enough then the system admits families of periodic solutions exponentially localized in space breathers in this paper we prove asymptotic stability in energy space of such solutions the proof is based on two steps first we use canonical perturbation theory to put the system in a suitable normal form in a neighborhood of the breather second we use dispersion in order to prove asymptotic stability the main limitation of the result rests in the fact that the nonlinear part of the on site potential is required to have a zero of order 8 at the origin from a technical point of view the theory differs from that developed for hamiltonian pdes due to the fact that the breather is not a relative equilibrium of the system

double resonance with landesman lazer conditions for planar systems of ordinary differential equations abstract we prove the existence of periodic solutions for first order planar systems at resonance the nonlinearity is indeed allowed to interact with two positively homogeneous hamiltonians both at resonance and some kind of landesman lazer conditions are assumed at both sides we are thus able to obtain as particular cases the existence results proposed in the pioneering papers by lazer and leach 1969 27 and by frederickson and lazer 1969 18 our theorem also applies in the case of asymptotically piecewise linear systems and in particular generalizes fabry s results in fabry 1995 10 for scalar equations with double resonance with respect to the dancer fucik spectrum

adaptive design and estimation in randomized clinical trials with correlated observations summary clinical trial designs involving correlated data often arise in biomedical research the intracluster correlation needs to be taken into account to ensure the validity of sample size and power calculations in contrast to the fixed sample designs we propose a flexible trial design with adaptive monitoring and inference procedures the total sample size is not predetermined but adaptively reestimated using observed data via a systematic mechanism the final inference is based on a weighted average of the block wise test statistics using generalized estimating equations where the weight for each block depends on cumulated data from the ongoing trial when there are no significant treatment effects the devised stopping rule allows for early termination of the trial and acceptance of the null hypothesis the proposed design updates information regarding both the effect size and within cluster correlation based on the cumulated data in order to achieve a desired power estimation of the parameter of interest and its confidence interval are proposed we conduct simulation studies to examine the operating characteristics and illustrate the proposed method with an example

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

demonstration of the stability or instability of multibreathers at low coupling abstract whereas there exists a mathematical proof for one site breathers stability and an unpublished one for two site breathers the methods for determining the stability properties of multibreathers rely on numerical computation of the floquet multipliers or on the weak nonlinearity approximation leading to discrete nonlinear schrodinger equations here we present a set of multibreather stability theorems mst that provides a simple method to determine multibreathers stability in klein gordon systems these theorems are based in the application of degenerate perturbation theory to aubry s band theory we illustrate them with several examples

on dispersion of small energy solutions of the nonlinear klein gordon equation with a potential in this paper we study small amplitude solutions of nonlinear klein gordon equations with a potential under suitable smoothness and decay assumptions on the potential and a genericity assumption on the nonlinearity we prove that all small energy solutions are asymptotically free in cases where the linear system has at most one bound state the result was already proved by soffer and weinstein we obtain here a result valid in the case of an arbitrary number of possibly degenerate bound states the proof is based on a combination of birkhoff normal form techniques and dispersive estimates

dispersive estimate for two periodic discrete one dimensional schrodinger operator abstract for the two periodic discrete one dimensional schrodinger operator h n n 1 n 1 v n n n z with v n 2 v n for every n z and v 0 v 1 r we show the dispersive estimate e i t h l 181 1 v 0 v 1 2 1 4 1 t 1 3 l 1 l 1 z t r

smoothing properties and retarded estimates for some dispersive evolution equations smoothing properties in the form of space time integrability properties play an important role in the study of dispersive evolution equations a number of them follow from a combination of general arguments and specific estimates we present a general formulation which makes the separation between the two types of ingredients as clear as possible and we illustrate it with the examples of the schrodinger equation of the wave equation and of a class of 1 1 dimensional equations related to the benjamin ono equation of special interest for the cauchy problem are retarded estimates expressed in terms of those properties we derive a number of such estimates associated with the last example and we mention briefly an application of those estimates to the cauchy problem for the generalized benjamin ono equation

sample sizes in the multivariate analysis of repeated measurements determination of sample sizes for comparing two or more treatments in repeated measurements experiments is considered multivariate normality of the individual s vector of repeated measures is assumed particular emphasis is placed on applications wherein the error variance covariance matrix is arbitrary positive definite sample size determination is based on power considerations associated with hotelling s t2 test and the desire to detect a specified difference between any pair of treatment means tabulated sample sizes are given in the case of an equal variance unequal covariance structure the utility of these sample sizes is also demonstrated for a more general variance covariance structure applications of these sample sizes are illustrated with two examples

asymptotic stability of lattice solitons in the energy space orbital and asymptotic stability for 1 soliton solutions of the toda lattice equations as well as for small solitary waves of the fpu lattice equations are established in the energy space unlike analogous hamiltonian pdes the lattice equations do not conserve the adjoint momentum in fact the toda lattice equation is a bidirectional model that does not fit in with the existing theory for the hamiltonian systems by grillakis shatah and strauss to prove stability of 1 soliton solutions we split a solution around a 1 soliton into a small solution that moves more slowly than the main solitary wave and an exponentially localized part we apply a decay estimate for solutions to a linearized toda equation which has been recently proved by mizumachi and pego to estimate the localized part we improve the asymptotic stability results for fpu lattices in a weighted space obtained by friesecke and pego

solvability of functional differential equations with multi point boundary value problems at resonance in this paper we discuss the following third order functional differential equations x t f t x t fx t x t gx t x t hx t email protected 0 1 subject to the boundary conditions x 0 0 x 0 0 x 1 email protected email protected ix h i where f 0 1 xr email protected r f g h are three operators a i i 1 m 2 0 0 h 1 h 2 h m 2 1 under some appropriate conditions some existence and multiplicity results are given for the problem at resonance by using a priori estimates and the topological degree theory of mawhin

resonances radiation damping and instabilitym in hamiltonian nonlinear wave equations we consider a class of nonlinear klein gordon equations which are hamiltonian and are perturbations of linear dispersive equations the unperturbed dynamical system has a bound state a spatially localized and time periodic solution we show that for ge neric nonlinear hamiltonian perturbations all small amplitude so lutions decay to zero as time tends to infinity at an anomalously slow rate in particular spatially localized and time periodic solutions of the linear problem are destroyed by generic nonlinear hamiltonian perturbations via slow radiation of energy to infinity these solutions can therefore be thought of as metastable states the main mecha nism is a nonlinear resonant interaction of bound states eigenfunc tions and radiation continuous spectral modes leading to energy transfer from the discrete to continuum modes this is in contrast to the kam theory in which appropriate nonresonance conditions im ply the persistence of invariant tori a hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the fermi golden rule arising in the theory of resonances in quantum mechanics the techniques used involve i a time dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues ii a generalization of the hamiltonian normal form appropriate for infinite dimensional dis persive systems and iii ideas from scattering theory the arguments are quite general and we expect them to apply to a large class of

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

non linear wave and schrodinger equations i instability of periodic and quasiperiodic solutions we investigate stability of periodic and quasiperiodic solutions of linear wave and schrodinger equations under non linear perturbations we show in the case of the wave equations that such solutions are unstable for generic perturbations for the schrodinger equations periodic solutions are stable while the quasiperiodic ones are not we extend these results to periodic solutions of non linear equations

on the spectral theory and dispersive estimates for a discrete schrodinger equation in one dimension based on the recent work komech et al dispersive estimates for 1d discrete schrodinger and klein gordon equations appl anal 85 1487 2006 for compact potentials we develop the spectral theory for the one dimensional discrete schrodinger operator h v n 1 n 1 2 n vn n we show that under appropriate decay conditions on the general potential and a nonresonance condition at the spectral edges the spectrum of h consists of finitely many eigenvalues of finite multiplicities and the essential absolutely continuous spectrum while the resolvent satisfies the limiting absorption principle and the puiseux expansions near the edges these properties imply the dispersive estimates eithpa c h l 2 l 2 t 3 2 for any fixed 52 and any t 0 where pa c h denotes the spectral projection to the absolutely continuous spectrum of h in addition based on the scattering theory for the discrete jost solutions and the previous results by stefanov and kevrekidis asymptotic behaviour of small