dynamical analysis of schrodinger operators with growing sparse potentials we consider discrete half line schrodinger operators h with potentials of the form v n s n q n here q is any compactly supported real function open image in new window if n l n and s n 0 otherwise where 0 1 and l n is a very fast growing sequence we study in a rather detailed manner the time averaged dynamics exp ith for various initial states in particular for some we calculate explicitly the intermittency function p which turns out to be nonconstant the dynamical results obtained imply that the spectral measure of h has exact hausdorff dimension for all boundary conditions improving the result of jitomirskaya and last

dynamical localization for the random dimer schrodinger operator we study the one dimensional random dimer model with hamiltonian h v where for all x mathbb z v 2x v 2x 1 and where the v 2x are i i d bernoulli random variables taking the values v v 0 we show that for all values of vand with probability one in the spectrum of his pure point if v 1 and v 1 sqrt 2 the lyapunov exponent vanishes only at the two critical energies given by e v for the particular value v 1 sqrt 2 respectively v sqrt 2 we show the existence of new additional critical energies at e 3 sqrt 2 respectively e 0 on any compact interval inot containing the critical energies the eigenfunctions are then shown to be semi uniformly exponentially localized and this implies dynamical localization for all q 0 and for all ell 2 mathbb z with sufficiently rapid decrease mathop sup limits t r psi i left q right kern 1pt left t right mathop sup limits t left langle p i left h omega right psi t left x right q p i left h omega right psi t right rangle infty here psi t e ih omega t psi and pi h is the spectral projector of h onto the interval i in particular if v 1 and v sqrt 2 these results hold on the entire spectrum so that one can take i h

constructing multiple test procedures for partially ordered hypothesis sets a popular method to control multiplicity in confirmatory clinical trials is to use a so called hierarchical or fixed sequence test procedure this requires that the null hypotheses are ordered a priori for example in order of clinical importance the procedure tests the hypotheses in this order using level tests until one is not rejected it then stops so that no subsequent hypotheses are eligible for rejection this procedure strongly controls the familywise error rate fwe that is to say the probability that any true hypotheses are rejected this paper describes a simple generalization of this approach in which the null hypotheses are partially ordered it is convenient to display the partial ordering in a directed acyclic graph dag we consider sequentially rejective procedures based on the partial ordering in which a hypothesis is tested only when all preceding hypotheses have been tested and rejected in general such procedures do not control the fwe but it is shown that when certain intersection hypotheses are added strong control of the fwe is obtained the purpose of the method is to construct inference strategies for confirmatory clinical trials that better reflect the trial objectives copyright 2007 john wiley sons ltd

non monotonic random schrodinger operators the anderson model abstract we prove exponential localization at all energies for one dimensional continuous anderson type models with single site potentials of changing sign a periodic background potential is allowed the main problem arises from non monotonicity i e the operator does not depend monotonically in the form sense on the random parameters we show that the method of two parameter spectral averaging recently devised by buschmann and stolz to prove localization for poisson and random displacement models can be modified to work for the type of anderson model considered here

power law subordinacy and singular spectra i half line operators we study hausdorff dimensional spectral properties of certain whole line quasiperiodic discrete schrodinger operators by using the extension of the gilbert pearson subordinacy theory that we previously developed in 19

modified sequentially rejective multiple test procedures abstract suppose that n hypotheses h 1 h 2 h n with associated test statistics t 1 t 2 t n are to be tested by a procedure with experimentwise significance level the probability of rejecting one or more true hypotheses smaller than or equal to some specified value a commonly used procedure satisfying this condition is the bonferroni b procedure which consists of rejecting h i for any i iff the associated test statistic t i is significant at the level n holm 1979 introduced a modified bonferroni procedure with greater power than the b procedure under holm s sequentially rejective bonferroni srb procedure if any hypothesis is rejected at the level n the denominator of for the next test is n 1 and the criterion continues to be modified in a stagewise manner with the denominator of reduced by 1 each time a hypothesis is rejected so that tests can be conducted at successively higher significance levels holm proved that the experimentwise significance level

stochastic approximation with averaging of the iterates optimal asymptotic rate of convergence for general processes consider the stochastic approximation algorithm x n 1 x n a n g x n xi n in an important paper polyak and juditsky siam j control optim 30 1992 pp 838 855 showed that loosely speaking if the coefficients a n go to zero slower than o 1 n then the averaged sequence sum nolimits i 1 n x i n converged to its limit at an optimum rate for any coefficient sequence the conditions were rather special and direct constructions were used here a rather simple proof is given that results of this type are generic to stochastic approximation and essentially hold any time that the classical asymptotic normality of the normalized and centered iterates holds considerable intuitive insight is provided into the procedure simulations have well borne out the importance of the method

dynamical upper bounds on wavepacket spreading we derive a general upper bound on the spreading rate of wavepackets in the framework of schrodinger time evolution our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time where the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time independent schrodinger equation we also derive a new lower bound on the spreading rate which is strongly connected with our upper bound we apply these new bounds to the fibonacci hamiltonian the most studied one dimensional model of quasicrystals as a result we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization the bounds have the same qualitative behavior in the limit of large coupling

critical energies in random palindrome models we investigate the occurrence of critical energies where the lyapunov exponent vanishes in random schrodinger operators when the potentials have some local order which we call random palindrome models we give necessary and sufficient conditions for the presence of such critical energies the commutativity of finite word elliptic transfer matrices finally we perform some numerical calculations of the lyapunov exponents showing their behavior near the critical energies and the respective time evolution of an initially localized wave packet obtaining the exponent ruling the algebraic growth of the second momentum we also consider special random palindrome models with one letter bounded gap property the transport effects of such long range order are showed numerically

dynamical localization for discrete anderson dirac operators we establish dynamical localization for random dirac operators on the d dimensional lattice with d in left 1 2 3 right in the three usual regimes large disorder band edge and 1d these operators are discrete versions of the continuous dirac operators and consist in the sum of a discrete free dirac operator with a random potential the potential is a diagonal matrix formed by different scalar potentials which are sequences of independent and identically distributed random variables according to an absolutely continuous probability measure with bounded density and of compact support we prove the exponential decay of fractional moments of the green function for such models in each of the above regimes i e j throughout the spectrum at larger disorder jj for energies near the band edges at arbitrary disorder and jjj in dimension one for all energies in the spectrum and arbitrary disorder dynamical localization in theses regimes follows from the fractional moments method the result in the one dimensional regime contrast with one that was previously obtained for 1d dirac model with bernoulli potential

dynamical localization for discrete and continuous random schrodinger operators we show for a large class of random schrodinger operators h on and on that dynamical localization holds i e that with probability one for a suitable energy interval i and for q a positive real here is a function of sufficiently rapid decrease and p i h is the spectral projector of h corresponding to the interval i the result is obtained through the control of the decay of the eigenfunctions of h and covers in the discrete case the anderson tight binding model with bernoulli potential dimension 1 or singular potential 1 and in the continuous case anderson as well as random landau hamiltonians

delocalization in random polymer models a random polymer model is a one dimensional jacobi matrix randomly composed of two finite building blocks if the two associated transfer matrices commute the corresponding energy is called critical such critical energies appear in physical models an example being the widely studied random dimer model it is proven that the lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport showing that it is almost surely overdiffusive even though the models are known to have pure point spectrum with exponentially localized eigenstates for almost every configuration of the polymers furthermore the level spacing is shown to be regular at the critical energy

stepwise confidence intervals without multiplicity adjustment for dose response and toxicity studies abstract not all simultaneous inferences need multiplicity adjustment if the sequence of individual inferences is predefined and failure to achieve the desired inference at any step renders subsequent inferences unnecessary then multiplicity adjustment is not needed this can be justified using the closed testing principle to test appropriate hypotheses that are nested in sequence starting with the most restrictive one but what hypotheses are appropriate may not be obvious in some problems we give a fundamentally different confidence set based justification by partitioning the parameter space naturally and using the principle that exactly one member of the partition contains the true parameter in dose response studies designed to show superiority of treatments over a placebo negative control or a drug known to be efficacious active control the confidence set approach generates methods with meaningful guarantee against incorrect decision whereas previous applications of the closed testing app

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

localization for one dimensional continuum bernoulli anderson models we use scattering theoretic methods to prove strong dynamical and exponential localization for one dimensional continuum anderson type models with singular distributions in particular the case of a bernoulli dis tribution is covered the operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms our main tools are the reflection and transmission coefficients for compactly supported single site perturbations of a periodic background which we use to verify the necessary hypotheses of multi scale analysis we show that non reflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs

dynamical delocalization for the 1d bernoulli discrete dirac operator a 1d tight binding version of the dirac equation is considered after checking that it recovers the usual discrete schrodinger equation in the nonrelativistic limit it is found that for two valued bernoulli potentials the zero mass case presents the absence of dynamical localization for some specific values of the energy albeit it has no continuous spectrum for the other energy values again excluding some very specific ones the bernoulli dirac system is localized independently of the mass

the analysis of multiple endpoints in clinical trials treatment comparisons in randomized clinical trials usually involve several endpoints such that conventional significance testing can seriously inflate the overall type i error rate one option is to select a single primary endpoint for formal statistical inference but this is not always feasible another approach is to apply bonferroni correction i e multiply each p value by the total number of endpoints its conservatism for correlated endpoints is examined for multivariate normal data a third approach is to derive an appropriate global test statistic and this paper explores one such test applicable to any set of asymptotically normal test statistics quantitative binary and survival endpoints are all considered within this general framework two examples are presented and the relative merits of the proposed strategies are discussed

multiple testing in clinical trials the basic ideas of multiple testing are outlined and the problem of how to control the probability of erroneous decisions is discussed the main emphasis is on the concept of the multiple level of significance controlling the experiment or family error in the strong sense which can be achieved by applying the principle of closed tests various practical situations encountered in multiple testing in clinical trials are considered more than one end point more than two treatments such as comparisons with a single control comparisons using ordered alternatives all pairwise comparisons and contrast methods and more than one trial tests based on global statistics the union intersection principle and other criteria are discussed the application of the multiple test concept in sequential sampling is investigated finally some comments are made on multiple power multiple confidence intervals and directed decisions

lower transport bounds for one dimensional continuum schrodinger operators we prove quantum dynamical lower bounds for one dimensional continuum schrodinger operators that possess critical energies for which there is slow growth of transfer matrix norms and a large class of compactly supported initial states this general result is applied to a number of models including the bernoulli anderson model with a constant single site potential

on subordinacy and analysis of the spectrum of one dimensional schrodinger operators abstract by considering the behaviour as n of the ratio of l2 0 n norms of solutions of d 2 dr 2 v r u xu 0 r x r a characterisation of the absolutely continuous and singular spectra of one dimensional schrodinger operators is deduced the analysis is applicable to all operators for which l d 2 dr 2 v r is regular at 0 and in the limit point case at infinity with v r locally integrable

analysis of stochastic gradient algorithms for linear regression problems parameter estimation problems that can be formulated as linear regressions are quite common in many applications recursive on line sequential estimation of such parameters can be performed using the recursive least squares rls algorithm or a stochastic gradient version of this algorithm in this paper the convergence properties of the gradient algorithm are analyzed under the assumption that the gain tends to zero the technique is the same as the so called ordinary differential equation approach but the treatment here is self contained and includes a proof of the boundedness of the estimates a main result is that the convergence conditions for the gradient algorithm are the same as those for the recursive least squares algorithm

quantum hamiltonians with quasi ballistic dynamics and point spectrum abstract consider the family of schrodinger operators and also its dirac version on l 2 z or l 2 n h s w f s n w where s is a transformation on compact metric f is a real lipschitz function and w is a sufficiently fast power decaying perturbation under certain conditions it is shown that h s w presents quasi ballistic dynamics for in a dense g set applications include potentials generated by rotations of the torus with analytic condition on f doubling map axiom a dynamical systems and the anderson model if w is a rank one perturbation examples of h s w with quasi ballistic dynamics and point spectrum are also presented

power law growth for the resistance in the fibonacci model many one dimensional quasiperiodic systems based on the fibonacci rule such as the tight binding hamiltonianh n n 1 n 1 v n n ne el2 e wherev n n 1 n x denoting the integer part ofx and the golden mean sqrt 5 1 2 give rise to the same recursion relation for the transfer matrices it is proved that the wave functions and the norm of transfer matrices are polynomially bounded critical regime if and only if the energy is in the spectrum of the hamiltonian this solves a conjecture of kohmoto and sutherland on the power law growth of the resistance in a one dimensional quasicrystal

the sequential rejection principle of familywise error control closed testing and partitioning are recognized as fundamental principles of familywise error control in this paper we argue that sequential rejection can be considered equally fundamental as a general principle of multiple testing we present a general sequentially rejective multiple testing procedure and show that many well known familywise error controlling methods can be constructed as special cases of this procedure among which are the procedures of holm shaffer and hochberg parallel and serial gatekeeping procedures modern procedures for multiple testing in graphs resampling based multiple testing procedures and even the closed testing and partitioning procedures themselves we also give a general proof that sequentially rejective multiple testing procedures strongly control the familywise error if they fulfill simple criteria of monotonicity of the critical values and a limited form of weak familywise error control in each single step the sequential rejection principle gives a novel theoretical perspective on many well known multiple testing procedures emphasizing the sequential aspect its main practical usefulness is for the development of multiple testing procedures for null hypotheses possibly logically related that are structured in a graph we illustrate this by presenting a uniform improvement of a recently published procedure

power law bounds on transfer matrices and quantum dynamics in one dimension we present an approach to quantum dynamical lower bounds for discrete one dimensional schrodinger operators which is based on power law bounds on transfer matrices it suffices to have such bounds for a nonempty set of energies we apply this result to various models including the fibonacci hamiltonian

dynamical lower bounds for 1d dirac operators quantum dynamical lower bounds for continuous and discrete one dimensional dirac operators are established in terms of transfer matrices then such results are applied to various models including the bernoulli dirac one and in contrast to the discrete case critical energies are also found for the continuous dirac case with positive mass

convergence of an adaptive linear estimation algorithm in this work we prove the almost sure convergence of an adaptive linear estimator governed by a stochastic gradient algorithm with decreasing step size in the presence of correlated observations two complementary contributions are added to the famous 1977 ljung theorem first we drop the condition of nondivergence of the algorithm assumed by ljung while that condition can be ensured by adding a barrier the convergence of the suitably bounded algorithm itself is not established even on the basis of ljung theorem here the barrier problem is overcome by proving that it is not necessary for the convergence our second contribution is to generalize the model describing the correlated observations no state space model is used and no linear relationship between the observations and the signal to be estimated needs to be assumed instead we use a decreasing covariance model that agrees with a very wide class of practical applications

acceleration of stochastic approximation by averaging a new recursive algorithm of stochastic approximation type with the averaging of trajectories is investigated convergence with probability one is proved for a variety of classical optimization and identification problems it is also demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence

power law bounds on transfer matrices and quantum dynamics in one dimension ii abstract we establish quantum dynamical lower bounds for a number of discrete one dimensional schrodinger operators these dynamical bounds are derived from power law upper bounds on the norms of transfer matrices we develop further the approach from part i and study many examples particular focus is put on models with finitely or at most countably many exceptional energies for which one can prove power law bounds on transfer matrices the models discussed in this paper include substitution models sturmian models a hierarchical model the prime model and a class of moderately sparse potentials

sparse 1d discrete dirac operators i quantum transport abstract some dynamical lower bounds for one dimensional discrete dirac operators with different classes of sparse potentials are presented and the particular role of the particle mass is emphasized

resistance of one dimensional quasicrystals abstract one dimensional quasiperiodic systems are often based on an inflation rule giving rise to a recursion formula on the transfer matrices it is shown in a general setting that the norm growth of these matrices can be evaluated a consequence is for instance a purely critical regime namely polynomial bounds in function of the sample size for the electrical resistance when the energy is in the spectrum this solves a conjecture by sutherland and kohmoto

a graphical approach to sequentially rejective multiple test procedures for clinical trials with multiple treatment arms or endpoints a variety of sequentially rejective weighted bonferroni type tests have been proposed such as gatekeeping procedures fixed sequence tests and fallback procedures they allow to map the difference in importance as well as the relationship between the various research questions onto an adequate multiple test procedure since these procedures rely on the closed test principle they usually require the explicit specification of a large number of intersection hypotheses tests the underlying test strategy may therefore be difficult to communicate we propose a simple iterative graphical approach to construct and perform such bonferroni type tests the resulting multiple test procedures are represented by directed weighted graphs where each node corresponds to an elementary hypothesis together with a simple algorithm to generate such graphs while sequentially testing the individual hypotheses the approach is illustrated with the visualization of several common gatekeeping strategies a case study is used to illustrate how the methods from this article can be used to tailor a multiple test procedure to given study objectives

rates of convergence of adaptive step size of stochastic approximation algorithms we propose a new adaptive algorithm with decreasing step size for stochastic approximations the use of adaptive algorithms in various applications is widely spread across fields such as system identification and adaptive control we analyze the rate of convergence of the proposed algorithms an averaging algorithm on its optimality of the rate of convergence is used to control the step sizes our proofs are based on recent results in stochastic approximations and gaussian approximation theorem