energy supercritical nonlinear schrodinger equations quasiperiodic solutions we construct time quasiperiodic solutions to the energy supercritical nonlinear schrodinger equations on the torus in arbitrary dimensions this introduces a new approach which could have general applicability

evolving cellular automata to perform computations mechanisms and impediments abstract we present results from experiments in which a genetic algorithm ga was used to evolve cellular automata cas to perform a particular computational task one dimensional density classification we look in detail at the evolutionary mechanisms producing the ga s behavior on this task and the impediments faced by the ga in particular we identify four epochs of innovation in which new ca strategies for solving the problem are discovered by the ga describe how these strategies are implemented in ca rule tables and identify the ga mechanisms underlying their discovery the epochs are characterized by a breaking of the task s symmetries on the part of the ga the symmetry breaking results in a short term fitness gain but ultimately prevents the discovery of the most highly fit strategies we discuss the extent to which symmetry breaking and other impediments are general phenomena in any ga search

symmetric second derivative integration methods abstract the purpose of this paper is the derivation of symmetric second derivative general linear methods sglms for general time reversible differential equations for this purpose we find algebraic conditions on the coefficients matrices of the methods which are sufficient for the method to be symmetric some symmetric methods of order 4 are constructed and implemented on the famous reversible problems numerical experiments show that the constructed symmetric sglms approximately conserve the invariants of motion over long time intervals for reversible hamiltonian systems

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

role of finite populations in determining evolutionary dynamics the connection between the finite size of an evolving population and its dynamical behavior is examined through analytical and computational studies of a simple model of evolution the infinite population limit of the model is shown to be governed by a special case of the quasispecies equations a flat fitness landscape yields identical results for the dynamics of infinite and finite populations on the other hand a monotonically increasing fitness landscape shows epochs in the dynamics of finite populations that become more pronounced as the rate of mutation decreases the details of the dynamics are profoundly different for any two simulation runs in that events arising from the stochastic noise in the pseudorandom number sequence are amplified as the population size is increased or equivalently the mutation rate is increased these epochs become smaller but do not entirely disappear

a composite integration scheme for the numerical solution of systems of parabolic pdes in one space dimension abstract carroll j a composite integration scheme for the numerical solution of systems of parabolic pdes in one space dimension journal of computational and applied mathematics 46 1993 327 343 we apply the numerical method of lines to approximate the solution of a system of parabolic partial differential equations in one space dimension by approximating the spatial derivatives by discrete values of the solution at a set of mesh points in space we use these approximations to represent the partial differential equations at each mesh point giving a semidiscrete system of ordinary differential equations in the time direction the resulting ode system is integrated numerically using a second order l stable composite integration scheme the author this journal 1989 using variable stepsize integration an implementation of the composite scheme is applied to approximate the solution of some test problems selected from the literature using a uniform spatial mesh we present the results of a number of numerical experiments which allow an appraisal of the proposed approach with published works

an application of single term haar wavelet series in the solution of nonlinear oscillator equations in this paper a novel single term haar wavelet series sthws method is implemented for the solution of the duffing equation and painleve s transcendents pi and pii the results in the form of a block pulse and a discrete solution are presented unlike classical numerical schemes the sthws method has no restrictions on the coefficients of the duffing equation as regards its solution pi and pii are analysed as regards their solutions up to nearest singularities poles using the sthws also an efficient computational implementation shows the remarkable features of wavelet based techniques

growth of sobolev norms and controllability of the schrodinger equation in this paper we obtain a stabilization result for both linear and nonlinear schrodinger equations under generic assumptions on the potential then we consider the schrodinger equations with a potential which has a random time dependent amplitude we show that if the distribution of the amplitude is sufficiently non degenerate then any trajectory of the system is almost surely non bounded in sobolev spaces

a higher order local linearization method for solving ordinary differential equations the local linearization ll method for the integration of ordinary differential equations is an explicit one step method that has a number of suitable dynamical properties however a major drawback of the ll integrator is that its order of convergence is only two the present paper overcomes this limitation by introducing a new class of numerical integrators called the llt method that is based on the addition of a correction term to the ll approximation in this way an arbitrary order of convergence can be achieved while retaining the dynamic properties of the ll method in particular it is proved that the llt method reproduces correctly the phase portrait of a dynamical system near hyperbolic stationary points to the order of convergence the performance of the introduced method is further illustrated through computer simulations

a composite integration scheme for the numerical solution of systems of ordinary differential equations abstract a generalization of a composite linear multistep method 2 is developed and applied to the approximate integration of systems of ordinary differential equations the proposed scheme is second order accurate and l stable an algorithm based on the integration formula derived in this paper is applied to approximate the solutions of a number of standard test problems the numerical results indicate that the method is competitive with other fixed order methods particularly in terms of computational overhead and could provide the basis for efficient temporal integration in the semidiscretization of time dependent partial differential equations

integration of large deviation kernels and applications to large deviations for evolutionary games we investigate large deviation properties for systems of probability kernels and these kernels when integrated with respect to a system of measures that also satisfy the large deviation principle the results of our analysis are then used to demonstrate the large deviation property and derive rate functions for systems of measures derived from the distributions of evolutionary game processes

statistical dynamics of the royal road genetic algorithm abstract metastability is a common phenomenon many evolutionary processes both natural and artificial alternate between periods of stasis and brief periods of rapid change in their behavior in this paper an analytical model for the dynamics of a mutation only genetic algorithm ga is introduced that identifies a new and general mechanism causing metastability in evolutionary dynamics the ga s population dynamics is described in terms of flows in the space of fitness distributions the trajectories through fitness distribution space are derived in closed form in the limit of infinite populations we then show how finite populations induce metastability even in regions where fitness does not exhibit a local optimum in particular the model predicts the occurrence of fitness epochs periods of stasis in population fitness distributions at finite population size and identifies the locations of these fitness epochs with the flow s hyperbolic fixed points this enables exact predictions of the metastable fitness distributions during the fitness epochs as well as giving insight into the nature of the periods of stasis and the innovations between them all these results are obtained as closed form expressions in terms of the ga s parameters an analysis of the jacobian matrices in the neighborhood of an epoch s metastable fitness distribution allows for the calculation of its stable and unstable manifold dimensions and so reveals the state space s topological structure more general quantitative features of the dynamics fitness fluctuation amplitudes epoch stability and speed of the innovations are also determined from the jacobian eigenvalues the analysis shows how quantitative predictions for a range of dynamical behaviors that are specific to the finite population dynamics can be derived from the solution of the infinite population dynamics the theoretical predictions are shown to agree very well with statistics from ga simulations we also discuss the connections of our results with those from population genetics and molecular evolution theory

adaptive linear barycentric rational finite differences method for stiff odes abstract it is our purpose to introduce a simple multistep method based on linear barycentric rational interpolation for solving ordinary differential equations also we design an adaptive version having one free parameter which is used to improve the stability properties numerical experiments of the constructed methods on some well known stiff problems indicate efficiency and capability of the methods in solving stiff problems

an improvement on the positivity results for 2 stage explicit runge kutta methods in this paper we investigate the positivity property for a class of 2 stage explicit runge kutta rk2 methods of order two when applied to the numerical solution of special nonlinear initial value problems ivps for ordinary differential equations odes we also pay particular attention to monotonicity property we obtain new results for positivity which are important in practical applications we provide some numerical examples to illustrate our results

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

the integration of stiff initial value problems in odes using modified extended backward differentiation formulae abstract a class of modified extended backward differentiation formulae mebdf suitable for the approximate numerical integration of first order systems of stiff ordinary differential equations is introduced the computational aspects of this new class of formulae are examined in detail in particular algorithms for varying both order and stepsize are given and this leads to a variable step variable order process using highly stable formulae of order 2 8 extensive numerical results for the well known detest set are given and on this basis a comparison is made between a code incorporating mebdf a code due to hindmarsh based on conventional backward differentiation formulae and a code due to skeel and kong based on blended linear multistep methods it is shown that on this test set the mebdf code is very reliable and is competitive with the other two codes on a significant class of problems a computer listing of the mebdf code used to obtain the results presented in this paper is available from the author

a bdf a generalization of the backward differentiation formulae a one parameter family of generalized bdf the so called a bdf is proposed and analyzed formulae of this new family are a0 stable up to order 7 especially a stable in orders 1 and 2 and a alpha stable in orders 3 to 7 respectively the nonexistence result of a0 stable formulae of order larger than 7 given by c w cryer bit 12 1972 pp 17 25 d m creedon and j j h miller bit 15 1975 pp 244 249 is extended to the a bdf after implementing these formulae in lsode a considerably larger set of stiff problems with oscillatory modes can be solved efficiently and accurately

sharp asymptotics for fixation times in stochastic population genetics models at low mutation probabilities this article describes new models in population genetics that extend the neutral wright fisher model by including strong selection and mutation fixation times are studied in the limit of small mutation probabilities within the framework of markov chains with rare transitions the main result outlines the role of the discrete geometry of the fitness landscape and provides a mean for estimating the expected number of generations for an individual with better fitness value to appear some connections to evolutionary algorithms are discussed as well

efficient variable stiffness methods for cooling of hot rolled steel sections abstract an industrial finite element package for the simulation of alternative cooling strategies for hot rolled steel sections has been enhanced by the incorporation of a variable stiffness second order time integration scheme based on a specially developed family of extended stability explicit runge kutta methods and an l stable semi implicit formula the integration scheme uses local error estimation to vary step size and runge kutta method selection is achieved by monitoring the numerical stiffness through the solution period using a computationally inexpensive estimate for the spectral radius of the system jacobian numerical tests on a range of section cooling problems indicate that the variable stiffness rk code developed is considerably more efficient than standard first order integration methods for the levels of accuracy required for industrial simulations the efficiency compares favorably with other high efficiency codes such as the variable order backward differentiation formulae vode and the nearly stiff rkc explicit solver

stability and diversity in collective adaptation we derive a class of macroscopic differential equations that describe collective adaptation starting from a discrete time stochastic microscopic model the behavior of each agent is a dynamic balance between adaptation that locally achieves the best action and memory loss that leads to randomized behavior we show that although individual agents interact with their environment and other agents in a purely self interested way macroscopic behavior can be interpreted as game dynamics application to several familiar explicit game interactions shows that the adaptation dynamics exhibits a diversity of collective behaviors the simplicity of the assumptions underlying the macroscopic equations suggests that these behaviors should be expected broadly in collective adaptation we also analyze the adaptation dynamics from an information theoretic viewpoint and discuss self organization induced by the dynamics of uncertainty giving a novel view of collective adaptation

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

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

the dynamics of a genetic algorithm for a simple learning problem a formalism for describing the dynamics of genetic algorithms gas using method s from statistical mechanics is applied to the problem of generalization in a perceptron with binary weights the dynamics are solved for the case where a new batch of training patterns is presented to each population member each generation which considerably simplifies the calculation the theory is shown to agree closely to simulations of a real ga averaged over many runs accurately predicting the mean best solution found for weak selection and large problem size the difference equations describing the dynamics can be expressed analytically and we find that the effects of noise due to the finite size of each training batch can be removed by increasing the population size appropriately if this population resizing is used one can deduce the most computationally efficient size of training batch each generation for independent patterns this choice also gives the minimum total number of training patterns used although using independent patterns is a very inefficient use of training patterns in general this work may also prove useful for determining the optimum batch size in the case where patterns are recycled

the quasispecies regime for the simple genetic algorithm with ranking selection we study the simple genetic algorithm with a ranking selection mechanism linear ranking or tournament we denote by the length of the chromosomes by m the population size by pc the crossover probability and by pm the mutation probability we introduce a parameter called the selection drift which measures the selection intensity of the fittest chromosome we show that the dynamics of the genetic algorithm depend in a critical way on the parameter 1 pc 1 pm if 1 then the genetic algorithm operates in a quasispecies regime an advantageous mutant invades a positive fraction of the population with probability larger than a constant p which does not depend on m we estimate next the probability of the occurrence of a catastrophe the whole population falls below a fitness level which was previously reached by a positive fraction of the population the asymptotic results suggest the following rules 1 pc 1 pm should be slightly larger than 1 pm should be of order 1 m should be larger than ln the running time should be of exponential order in m the first condition requires that pm pc ln these conclusions must be taken with great care they come from an asymptotic regime and it is a formidable task to understand the relevance of this regime for a real world problem at least we hope that these conclusions provide interesting guidelines for the practical implementation of the simple genetic algorithm

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

drift analysis and average time complexity of evolutionary algorithms the computational time complexity is an important topic in the theory of evolutionary algorithms eas this paper reports some new results on the average time complexity of eas based on drift analysis some useful drift conditions for deriving the time complexity of eas are studied including conditions under which an ea will take no more than polynomial time in problem size to solve a problem and conditions under which an ea will take at least exponential time in problem size to solve a problem the paper first presents the general results and then uses several problems as examples to illustrate how these general results can be applied to concrete problems in analyzing the average time complexity of eas while previous work only considered 1 1 eas without any crossover the eas considered in this paper are fairly general which use a finite population crossover mutation and selection 2001 elsevier science b v all rights reserved

on reducibility of schrodinger equations with quasiperiodic in time potentials we prove that a linear d dimensional schrodinger equation with an x periodic and t quasiperiodic potential reduces to an autonomous equation for most values of the frequency vector the reduction is made by means of a non autonomous linear transformation of the space of x periodic functions this transformation is a quasiperiodic function of t

the calculi of emergence computation dynamics and induction abstract defining structure and detecting the emergence of complexity in nature are inherently subjective though essential scientific activities despite the difficulties these problems can be analyzed in terms of how model building observers infer from measurements the computational capabilities embedded in nonlinear processes an observer s notion of what is ordered what is random and what is complex in its environment depends directly on its computational resources the amount of raw measurement data of memory and of time available for estimation and inference the discovery of structure in an environment depends more critically and subtlely though on how those resources are organized the descriptive power of the observer s chosen or implicit computational model class for example can be an overwhelming determinant in finding regularity in data this paper presents an overview of an inductive framework hierarchical machine reconstruction in which the emergence of complexity is associated with the innovation of new computational model classes complexity metrics for detecting structure and quantifying emergence along with an analysis of the constraints on the dynamics of innovation are outlined illustrative examples are drawn from the onset of unpredictability in nonlinear systems finitary nondeterministic processes and cellular automata pattern recognition they demonstrate how finite inference resources drive the innovation of new structures and so lead to the emergence of complexity

a matricial exponentially fitted scheme for the numerical solution of stiff initial value problems abstract we present an exponentially fitted scheme for the numerical integration of stiff systems of initial value problems the basic algorithm consists of an explicit one step method which requires the computation of the exponential of a matrix this task is achieved using ward s scaling and squaring algorithm 1 the results of some numerical experiments are presented which allow an appraisal of the proposed approach with other methods

bounded sobolev norms for klein gordon equations under non resonant perturbation in this paper we prove anderson localization for the klein gordon operator on the circle x1d54b under non resonant perturbations furthermore using the result we show that the sobolev norms of solutions to the corresponding klein gordon equations remain bounded for all time

solving stiff differential equations with the method of patches we introduce a new method for solving very stiff sets of ordinary differential equations the basic idea is to replace the original nonlinear equations with a set of equally stiff equations that are piecewise linear and therefore can be solved exactly we demonstrate the value of the method on small systems of equations for which some other methods are inefficient or produce spurious solutions estimate error bounds and discuss extensions of the method to larger systems of equations and to partial differential equations

stepsize conditions for boundedness in numerical initial value problems for runge kutta methods rkms linear multistep methods lmms and classes of general linear methods glms much attention has been paid in the literature to special nonlinear stability requirements indicated by the terms total variation diminishing strong stability preserving and monotonicity stepsize conditions guaranteeing these properties were derived by shu osher j comput phys 77 1988 pp 439 471 and in numerous subsequent papers these special stability requirements imply essential boundedness properties for the numerical methods among which the property of being total variation bounded unfortunately for many well known methods the above special requirements are violated so that one cannot conclude in this way that the methods are total variation bounded in this paper we focus on stepsize conditions for boundedness directly rather than via the detour of the above special stability properties we present a generic framework for deriving best possible stepsize conditions which guarantee boundedness of actual rkms lmms and glms thereby generalizing results on the special stability properties mentioned above

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

bounded sobolev norms for linear schrodinger equations under resonant perturbations abstract we prove that the sobolev norms for a 1 d periodic schrodinger equation remain bounded under small resonant perturbations

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

perturbed mebdf methods we investigate mebdf methods of cash from general linear methods point of view some perturbations of these methods are constructed which preserve the order of these formulas and improve their stability properties

a class of contractive second derivatives multistep methods in this paper we study some properties of contractivity regions of second derivative multistep methods these properties lead to the design of classes of a o and a contractive methods

optimal adaptive performance and delocalization in nk fitness landscapes abstract we investigate the evolutionary dynamics of a finite population of sequences adapting to nk fitness landscapes we find that unlike in the case of an infinite population the average fitness in a finite population is maximized at a small but finite rather than vanishing mutation rate the highest local maxima in the landscape are visited for even larger mutation rates close to a transition point at which the population delocalizes i e leaves the fitness peak at which it was localized and starts traversing the sequence space if the mutation rate is increased even further the population undergoes a second transition and loses all sensitivity to fitness peaks this second transition corresponds to the standard error threshold transition first described by eigen we discuss the implications of our results for biological evolution and for evolutionary optimization techniques

the potential for parallelism in runge kutta methods part 1 rk formulas in standard form the authors examine the potential for parallelism in runge kutta rk methods based on formulas in standard one step form both negative and positive results are presented many of the negative results are based on a theorem that bounds the order of an rk formula in terms of the minimal polynomial associated with its coefficient matrix the positive results are largely examples of prototypical formulas that offer a potential for effective coarse grain parallelism on machines with a few processors

the linear barycentric rational quadrature method for volterra integral equations we introduce two direct quadrature methods based on linear rational interpolation for solving general volterra integral equations of the second kind the first deduced by a direct application of linear barycentric rational quadrature given in former work is shown to converge at the same rate as the rational quadrature rule but is costly on long integration intervals the second based on a composite version of this quadrature rule loses one order of convergence but is much cheaper both require only a sample of the involved functions at equispaced nodes and yield an infinitely smooth solution of most classical examples with machine precision

on growth of sobolev norms in time dependent linear schrodinger equations on spheres abstract we show that solutions of linear schrodinger equations with a time dependent gevrey potential on spheres have at most logarithmically growing sobolev norms we do not use any specific knowledge of the eigenfunctions of the laplacian beltrami operator which is the main difference with our previous paper

implicit schemes for differential equations abstract implicit high order accurate temporal integration schemes are developed to reduce partial differential equations of evolution type to a sequence of ordinary differential equations involving the spatial directions only the ordinary differential equations are solved using various accurate integration methods the schemes are free from extraneous boundary and or initial conditions and are stable for large courant number

when a genetic algorithm outperforms hill climbing a toy optimisation problem is introduced which consists of a fitness gradient broken up by a series of hurdles the performance of a hill climber and a stochastic hill climber are computed these are compared with the empirically observed performance of a genetic algorithm ga with and without the hill climber with a sufficiently large neighbourhood outperforms the stochastic hill climber but is outperformed by a ga both with and without crossover the ga with crossover substantially outperforms all the other heuristics considered here the relevance of this result to real world problems is discussed

l stable parallel one block methods for ordinary differential equations in this contribution the author considers the one block methods designed by sommeijer couzy and van der houwen for the purpose of solving ordinary differential equations odes on a parallel computer the author also derives a new set of order conditions studies the stability and exhibits a new class of parallel methods which contains l stable schemes up to order eleven

a new approach to estimating the expected first hitting time of evolutionary algorithms evolutionary algorithms ea have been shown to be very effective in solving practical problems yet many important theoretical issues of them are not clear the expected first hitting time is one of the most important theoretical issues of evolutionary algorithms since it implies the average computational time complexity in this paper we establish a bridge between the expected first hitting time and another important theoretical issue i e convergence rate through this bridge we propose a new general approach to estimating the expected first hitting time using this approach we analyze eas with different configurations including three mutation operators with without population a recombination operator and a time variant mutation operator on a hard problem the results show that the proposed approach is helpful for analyzing a broad range of evolutionary algorithms moreover we give an explanation of what makes a problem hard to eas and based on the recognition we prove the hardness of a general problem

the positivity of low order explicit runge kutta schemes applied in splitting methods splitting methods are frequently used for the solution of large stiff initial value problems of ordinary differential equations with an additively split right hand side function such systems arise for instance as method of lines discretizations of evolutionary partial differential equations in many applications we consider the choice of explicit runge kutta rk schemes in implicit explicit splitting methods our main objective is the preservation of positivity in the numerical solution of linear and nonlinear positive problems while maintaining a sufficient degree of accuracy and computational efficiency a three stage second order explicit rk method is proposed which has optimized positivity properties this method compares well with standard s stage explicit rk schemes of order s s 2 3 it has advantages in the low accuracy range and this range is interesting for an application in splitting methods numerical results are presented

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

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

growth of sobolev norms in linear schrodinger equations with quasi periodic potential in this paper we consider the following problem let iu t u v x t u 0 be a linear schrodinger equation periodic boundary conditions where v is a real bounded real analytic potential which is periodic in x and quasi periodic in t with diophantine frequency vector denote s t the corresponding flow map thus s t preserves the l 2 norm and our aim is to study its behaviour on h s t d s 0 our main result is the growth in time is at most logarithmic thus if h s then more precisely is proven in 1d and 2d when v is small we also exhibit examples showing that a growth of higher sobolev norms may occur in this context and is thus essentially best possible

bootstrap multiscale analysis and localization in random media we introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events for quantum and classical waves in random media we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side l with probability higher than 1 e l for any 0 1 the starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent at some sufficiently large scale with probability bigger than 1 d is the dimension note that from the same starting hypothesis we get conclusions that are valid for any 0 1 this is achieved by the repeated use of a bootstrap argument as an application we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the hilbert schmidt norm and better estimates on the behavior of the eigenfunctions