radiation conditions and scattering theory for n particle hamiltonians the correct form of the angular part of radiation conditions is found in scattering problem forn particle quantum systems the estimates obtained allow us to give an elementary proof of asymptotic completeness for such systems in the framework of the theory of smooth perturbations

a probabilistic approach to one dimensional schrodinger operators with sparse potentials we consider the one dimensional schrodinger equation with sparse potential v i e mainly v 0 it is shown that the asymptotics of the solutions corresponding to positive energies e can be approximately described by an infinite sum of independent random variables using results from probability theory we can then determine the spectral properties of the operators under consideration we prove absolute continuity for a general class of potentials and we also have examples with singular continuous spectrum

a determining form for the two dimensional navier stokes equations the fourier modes case the determining modes for the two dimensional incompressible navier stokes equations nse are shown to satisfy an ordinary differential equation ode of the form dv dt f v in the banach space x of all bounded continuous functions of the variable s r with values in certain finite dimensional linear space this new evolution ode named determining form induces an infinite dimensional dynamical system in the space x which is noteworthy for two reasons one is that f is globally lipschitz from x into itself the other is that the long term dynamics of the determining form contains that of the nse the traveling wave solutions of the determining form i e those of the form v t s v0 t s correspond exactly to initial data v0 that are projections of solutions of the global attractor of the nse onto the determining modes the determining form is also shown to be dissipative an estimate for the radius of an absorbing ball is derived in terms of the number of determining modes and the grashof num

a small gain condition for interconnections of iss systems with mixed iss characterizations we consider interconnected nonlinear systems with external inputs where each of the subsystems is assumed to be input to state stable iss sufficient conditions of small gain type are provided guaranteeing that the interconnection is iss with respect to the external input to this end we extend recently obtained small gain theorems to a more general type of interconnections the small gain theorem provided here is applicable to situations where the iss conditions are formulated differently for each subsystem and are either given in the maximization or the summation sense furthermore it is shown that the conditions are compatible in the sense that it is always possible to transform sum formulations to maximum formulations without destroying a given small gain condition an example shows the advantages of our results in comparison with the known ones

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

nodal sets for groundstates of schrodinger operators with zero magnetic field in non simply connected domains we investigate nodal sets of magnetic schrodinger operators with zero magnetic field acting on a non simply connected domain in 2 for the case of circulation 1 2 of the magnetic vector potential around each hole in the region we obtain a characterisation of the nodal set and use this to obtain bounds on the multiplicity of the groundstate for the case of one hole and a fixed electric potential we show that the first eigenvalue takes its highest value for circulation 1 2

a new proof of localization in the anderson tight binding model we give a new proof of exponential localization in the anderson tight binding model which uses many ideas of the frohlich martinelli scoppola and spencer proof but is technically simpler particularly the probabilistic estimates

determining modes for the surface quasi geostrophic equation abstract we introduce a determining wavenumber for the surface quasi geostrophic sqg equation defined for each individual trajectory and then study its dependence on the force while in the subcritical and critical cases this wavenumber has a uniform upper bound it may blow up when the equation is supercritical a bound on the determining wavenumber provides determining modes and measures the number of degrees of freedom of the flow or resolution needed to describe a solution to the sqg equation

distribution of poles for scattering on the real line abstract the density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential in the case of a potential with finite singularities at the endpoints of the support asymptotic formulae for the poles are given while in the c 0 case an example of a potential with infinitely many scattering poles on i r is constructed the scattering amplitude of a compactly supported potential is also characterized

new criteria on p th moment input to state stability of impulsive stochastic delayed differential systems by employing the razumikhin technique the uniformly exponentially stable function and the average dwell time adt approach some razumikhin type theorems on input to state stability iss for time varying impulsive stochastic delay differential systems isdds are obtained it is shown that if the time varying continuous stochastic delayed dynamics is iss and the impulsive effects are destabilizing then the isdds is input to state stable iss with respect to a lower bound of the average dwell time adt the existing results on iss of impulsive systems in the literature require the coefficients of the estimated upper bound for the diffusion operator of a lyapunov function to be constant numbers while the results in this note allow us the coefficients of the estimated upper bound of the diffusion operator to be sign changing time varying function an example is given to illustrate the effectiveness of our results from the example one can see the existing results fail to be used to determine the iss for some type of isdds

small gain theorems for large scale systems and construction of iss lyapunov functions we consider interconnections of n nonlinear subsystems in the input to state stability iss framework for each subsystem an iss lyapunov function is given that treats the other subsystems as independent inputs a gain matrix is used to encode the mutual dependencies of the systems in the network under a small gain assumption on the monotone operator induced by the gain matrix a locally lipschitz continuous iss lyapunov function is obtained constructively for the entire network by appropriately scaling the individual lyapunov functions for the subsystems the results are obtained in a general formulation of iss the cases of summation maximization and separation with respect to external gains are obtained as corollaries

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

stability theory of hybrid dynamical systems with time delay by defining the bounded time lag space and other related concepts this note formulates the notion of hybrid dynamical systems with time delay which provides a unified framework for describing a large class of systems and is suitable for qualitative analysis it also investigates stability properties of invariant sets by the method of lyapunov functions and the razumihin techniques criteria on uniform stability and uniform asymptotic stability are established for hybrid dynamical systems with time delay those criteria are then applied to conduct stability analysis for impulsive delay differential systems and nonlinear sampled data feedback control systems with time delay

discrete data assimilation in the lorenz and 2d navier stokes equations consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing in this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time this paper studies discrete data assimilation for the lorenz equations and the incompressible two dimensional navier stokes equations in both cases we obtain bounds on the time interval h between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

stability analysis of linear time varying time delay systems by non quadratic lyapunov functions with indefinite derivatives abstract this paper presents a stability analysis of linear time varying ltv time delay systems by using non quadratic lyapunov functions and functionals two types of sufficient conditions are proposed for testing several different kinds of stability such as asymptotic stability exponential stability and uniformly exponential stability for ltv systems without delay then by constructing suitable non quadratic lyapunov functions functionals that are respectively the time varying weighted l 1 and l norms of the state variables and by using properties of the uniformly asymptotically stable uas function and the recently established improved razumikhin and krasovskii stability theorems both delay dependent and delay independent conditions are proposed for testing uniformly exponential stability of a class of ltv time delay systems the time derivatives of the non quadratic lyapunov functions functionals along the solutions are allowed to be indefinite namely to take both negative and positive values numerical examples show that the non quadratic lyapunov functions functionals based methods are more efficient than the existing ones that are based on quadratic functions functionals

operators with singular continuous spectrum i general operators the baire category theorem implies that the family f of dense sets g in fixed metric space x is a candidate for generic sets since it is closed under countable intersections and if x is perfect has no isolated point then a f has uncountable intersections with any open ball in x

razumikhin type theorem for stochastic functional differential systems via vector lyapunov function abstract this paper is concerned with input to state stability of sfdss by using stochastic analysis techniques razumikhin techniques and vector lyapunov function method vector razumikhin type theorem has been established on input to state stability for sfdss novel sufficient criteria on the pth moment exponential input to state stability are obtained by the established vector razumikhin type theorem when input is zero an improved criterion on exponential stability is obtained two examples are provided to demonstrate validity of the obtained results

input to state stability of impulsive and switching hybrid systems with time delay this paper investigates input to state stability iss and integral input to state stability iiss of impulsive and switching hybrid systems with time delay using the method of multiple lyapunov krasovskii functionals it is shown that even if all the subsystems governing the continuous dynamics in the absence of impulses are not iss iiss impulses can successfully stabilize the system in the iss iiss sense provided that there are no overly long intervals between impulses i e the impulsive and switching signal satisfies a dwell time upper bound condition moreover these impulsive iss iiss stabilization results can be applied to systems with arbitrarily large time delays conversely in the case when all the subsystems governing the continuous dynamics are iss iiss in the absence of impulses the iss iiss properties can be retained if the impulses and switching do not occur too frequently i e the impulsive and switching signal satisfies a dwell time lower bound condition several illustrative examples are presented with their numerical simulations to demonstrate the main results

generalized lyapunov approach for functional differential inclusions abstract this paper is concerned with the generalized lyapunov approach for functional differential inclusions fdi at first we prove a class of generalized halanay s inequalities then by applying the generalization of halanay s inequalities we investigate the uniformly ultimate boundedness and uniformly asymptotic stability of fdi

determining modes for continuous data assimilation in 2d turbulence we study the number of determining modes necessary for continuous data assimilation in the two dimensional incompressible navier stokes equations our focus is on how the spatial structure of the body forcing affects the rate of continuous data assimilation and the number of determining modes we treat this problem analytically by proving a convergence result depending on the h 1 norm of f and computationally by considering a family of forcing functions with identical grashof numbers that are supported on different annuli in fourier space the rate of continuous data assimilation and the number of determining modes is shown to depend strongly on the length scales present in the forcing

estimates on enstrophy palinstrophy and invariant measures for 2 d turbulence we construct semi integral curves which bound the projection of the global attractor of the 2 d navier stokes equations in the plane spanned by enstrophy and palinstrophy of particular interest are certain regions of the plane where palinstrophy dominates enstrophy previous work shows that if solutions on the global attractor spend a significant amount of time in such a region then there is a cascade of enstrophy to smaller length scales one of the main features of 2 d turbulence theory the semi integral curves divide the plane into regions having limited ranges for the direction of the flow this allows us to estimate the average time it would take for an intermittent solution to burst into a region of large palinstrophy we also derive a sharp universal upper bound on the average palinstrophy and show that it is achieved only for forces that admit statistical steady states where the nonlinear term is zero

some jacobi matrices with decaying potential and dense point spectrum we discuss doubly infinite matrices of the formm ij i j 1 i j 1 v i ij as operators on l2 we present for each 0 examples of potentialsv n with v n o n 1 2 and wherem has only point spectrum our discussion should be viewed as a remark on the recent work of delyon kunz and souillard

on positive solutions of second order elliptic equations stability results and classification soit p un operateur uniformement elliptique d ordre 2 defini dans un domaine cr n soit p le cone convexe de toutes les solutions positives de l equation pu 0 dans on etudie les proprietes generales de p pour p et generaux

on the leading correction of the thomas fermi model lower bound we prove that the quantum mechanical ground state energy of an atom with nuclear chargez can be bounded from below by the sum of the thomas fermi energy of the problem plusq 8z2 plus terms of ordero z 2

construction of strict lyapunov krasovskii functionals for time varying time delay systems abstract for stability analysis of time varying time delay systems it is known that time derivatives time shifts in the discrete time setting of the lyapunov krasovskii functionals lkfs for both internal stability and input to state stability iss can be indefinite which can improve the resulting stability conditions in some cases however the non strict lkfs may be insufficient and inconvenient to use in practice in this paper based on the non strict iss lkfs for time varying time delay systems and with the help of uniformly asymptotically stable uas and uniformly exponentially bounded ueb scalar functions three classes of strict iss lkfs are constructed such that their time derivatives are strictly negative definite along the trajectories of the considered system a general construction of strict iss lkfs by using the positive definite and uniformly bounded solution to a scalar lyapunov differential equation is established which includes the proposed three classes of strict iss lkfs as special cases the approaches are also extended to deal with a lyapunov inequality whose right hand side is indefinite and nonlinear both continuous time and discrete time systems are considered the effectiveness of the proposed methods is illustrated by some examples borrowed from the literature

determination of the solutions of the navier stokes equations by a set of nodal values we consider the navier stokes equations of a viscous incompresible fluid and we want to see to what extent these solutions can be determined by a discrete set of nodal values of these solutions the results presented here are exact results and not approximate ones we show in several cases that the solutions are entirely determined by their values on a discrete set provided this set contains enough points and these points are sufficiently densely distributed in a sense described in the article two typical results are the following ones two stationary solutions coincide if they coincide on a set sufficiently dense but finite similarly if the large time behavior of the solutions to the navier stokes equations is known on an appropriate discrete set then the large time behavior of the solution itself is totally determined

input to state stability of nonlinear systems based on an indefinite lyapunov function abstract this study investigated the input to state stability iss and integral iss iiss of nonlinear systems and discovered new sufficient conditions for them these conditions are more relaxed than others in that they employ an indefinite lyapunov function rather than a negative definite one thus a nonlinear time varying system satisfying them has uniform asymptotic stability two numerical examples show their effectiveness and advantages

indefinite derivative lyapunov krasovskii functional method for input to state stability of nonlinear systems with time delay in this paper the problem of the input to state stability iss for nonlinear systems with time delay is investigated a continuously differentiable lyapunov krasovskii functional with indefinite derivative is introduced to derive the iss of the systems which generalizes the classic lyapunov krasovskii functional with positive definite derivative as a result the uniform stability and uniform asymptotic stability criteria of nonlinear systems with time delay are also established by employing the proposed lyapunov krasovskii functional

integral input to state stability for hybrid delayed systems with unstable continuous dynamics this paper considers integral input to state stability iiss for a class of hybrid time delay systems discrete dynamics includes impulsive and switching signals and continuous dynamics is not necessarily stable based on multiple lyapunov krasovskii functionals a dwell time bound is explicitly given to guarantee iiss of the hybrid delayed system compared with existing results on related problems the obtained stability criteria can be applied to a larger class of hybrid delayed systems moreover the obtained dwell time bound is less conservative than existing ones at last an example related to networked control systems ncss is provided to illustrate the effectiveness of the proposed result

operators with singular continuous spectrum iii almost periodic schrodinger operators we prove that one dimensional schrodinger operators with even almost periodic potential have no point spectrum for a dense g in the hull this implies purely singular continuous spectrum for the almost mathieu equation for coupling larger than 2 and a dense g in even if the frequency is an irrational with good diophantine properties

on a criterion for locating stable stationary solutions to the navier stokes equations on donne une estimation superieure uniforme explicite a la borne de constantin foias ternam et on donne une forme applicable explicite aux conditions suffisantes on considere seulement le cas a 2 dimensions

improved razumikhin and krasovskii stability criteria for time varying stochastic time delay systems the problem of pth moment stability for time varying stochastic time delay systems with markovian switching is investigated in this paper some novel stability criteria are obtained by applying the generalized razumikhin and krasovskii stability approaches both pth moment asymptotic stability and integral input to state stability are considered based on the notion and properties of uniformly stable functions and the improved comparison lemmas the established results show that time derivatives of the constructed razumikhin functions and krasovskii functionals are allowed to be indefinite which improve the existing results on this topic by applying the obtained results for stochastic systems we also analyze briefly the stability of time varying deterministic time delay systems finally examples are provided to illustrate the effectiveness of the proposed results

a determining form for the damped driven nonlinear schrodinger equation fourier modes case in this paper we show that the global attractor of the 1d damped driven nonlinear schrodinger equation nls is embedded in the long time dynamics of a determining form the determining form is an ordinary differential equation in a space of trajectories x cb1 r pmh2 where pm is the l2 projector onto the span of the first m fourier modes there is a one to one identification with the trajectories in the global attractor of the nls and the steady states of the determining form we also give an improved estimate for the number of the determining modes

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

a unified razumikhin type criterion on input to state stability of time varying impulsive delayed systems abstract this paper discusses the problem of input to state stability iss of time varying impulsive delayed systems by introducing a switching parameter and using the notion of average impulsive interval a unified razumikhin type criterion on iss which is simultaneously effective for stabilizing impulses and destabilizing impulses is derived the condition which requires the coefficient of the estimated upper bound of the derivative of a lyapunov function to be constant in the existing results on iss of impulsive systems is weakened the results in this paper allow the coefficient of the derivative of a lyapunov function to be time varying function which can be both positive and negative and may even be unbounded furthermore the impulsive intervals of an impulsive sequence are allowed to have arbitrarily small lower bound and large enough upper bound simultaneously as a by product a unified criterion on iss for time varying impulsive delay free systems is also presented two examples are presented to illustrate the effectiveness of our results

a note on equivalence between two integral inequalities for time delay systems jensen s inequality and extended jensen s inequality are two important integral inequalities when problems of stability analysis and controller synthesis for time delay systems are considered the extended jensen s inequality introduces two additional free matrices and is generally regarded to be less conservative than jensen s inequality the equivalence between jensen s inequality and extended jensen s inequality in bounding the quadratic term h t h t x t s z x s d s in lyapunov functional of time delay systems is presented and theoretically proved it is shown that the extended jensen s inequality does not decrease the lower bound of this quadratic term obtained using jensen s inequality and then it does not reduce the conservativeness though two additional free matrices m 1 and m 2 are involved

improved razumikhin type theorem for input to state stability of nonlinear time delay systems this note describes a lyapunov razumikhin function with an indefinite derivative that can be used to determine the input to state stability iss of a nonlinear time delay system first an extended comparison principle for estimating the solution of a differential inequality with an upper bound that is not semi negative definite is presented then that principle and a lyapunov razumikhin function with an indefinite derivative are employed to derive an iss criterion finally two examples show the effectiveness of our method and its superiority over others

extensions of razumikhin type stability theorems for nonlinear time delay systems on time scales this paper focuses on the stability analysis of nonlinear time delay systems on time scales which are of generality as they can include not only the traditional continuous and discrete ones but a

statistical estimates for the navier stokes equations and the kraichnan theory of 2 d fully developed turbulence a mathematical formulation of the kraichnan theory for 2 d fully developed turbulence is given in terms of ensemble averages of solutions to the navier stokes equations a simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber up to a logarithmic correction this is followed by partial rigorous support for kraichnan s eddy breakup mechanism a rigorous estimate for the total energy is found to be consistent with kraichnan s theory finally it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case

connections between razumikhin type theorems and the iss nonlinear small gain theorem a razumikhin type theorem that guarantees input to state stability for functional differential equations with disturbances is established using the nonlinear small gain theorem the result is used to show that input to state stabilizability for nonlinear finite dimensional control systems is robust in an appropriate sense to small time delays at the input also relaxed razumikhin type conditions guaranteeing global asymptotic stability for differential difference equations are given

on the existence of eigenvalues of the schrodinger operatorh λw in a gap of σ h the authors prove a number of results asserting the existence of eigenvalues of the schrodinger operatorh w in a gap of h whereh v andv andw are bounded the existence of these eigenvalues is an important element in the theory of the colour of crystals the basic theorems are proved in v stronger results forv 1 are also presented

asymptotic analysis of the navier stokes equations abstract new bounds are established on the number of modes which determine the solutions of the navier stokes equations in two dimensions the best bound available at present is nearly proportional to the generalized grashof number defined in the paper and less than logarithmically dependent on the spatial structure or the shape of the force driving the flow to the extent than for the case of 2 dimensional rayleigh benard convection the generalized grashof number may be identified with the usual grashof number the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed

estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems we show that the long time behavior of the projection of the exact solutions to the navier stokes equations and other dissipative evolution equations on the finite dimensional space of interpolant polynomials determines the long time behavior of the solution itself provided that the spatial mesh is fine enough we also provide an explicit estimate on the size of the mesh moreover we show that if the evolution equation has an inertial manifold then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite dimensional space spanned by the approximating polynomials our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation

local properties of coulombic wave functions we investigate the local behaviour of solutions of a nonrelativistic schrodinger equation which describe coulombic systems firstly we give a representation theorem for such solutions in the neighbourhood of coulombic singularities generalizing previous results cusp conditions due to kato and others secondly we investigate the influence of fermi statistics on the local behaviour of many fermionic wave functions showing that e g ann electron wave function must have zeros of order at leastn4 3 for largen