a discrete schrodinger equation via optimal transport on graphs abstract in 1966 edward nelson presented an interesting derivation of the schrodinger equation using brownian motion recently this derivation is linked to the theory of optimal transport which shows that the schrodinger equation is a hamiltonian system on the probability density manifold equipped with the wasserstein metric in this paper we consider similar matters on a finite graph by using discrete optimal transport and its corresponding nelson s approach we derive a discrete schrodinger equation on a finite graph the proposed system is quite different from the commonly referred discretized schrodinger equations it is a system of nonlinear ordinary differential equations odes with many desirable properties several numerical examples are presented to illustrate the properties

periodic nonlinear schrodinger equation and invariant measures in this paper we continue some investigations on the periodic nlseiu u iu xx u u p 2 p 6 started in lrs we prove that the equation is globally wellposed for a set of data of full normalized gibbs measrue e beta h phi hd phi x h phi tfrac 1 2 int left phi right 2 tfrac 1 p int left phi right p after suitablel 2 truncation the set and the measure are invariant under the flow the proof of a similar result for the kdv and modified kdv equations is outlined the main ingredients used are some estimates from b1 on periodic nls and kdv type equations

simple statistical explanation for the localization of energy in nonlinear lattices with two conserved quantities the localization of energy in the discrete nonlinear schrodinger equation is explained with statistical methods the partition function and the entropy of the system are computed for low amplitude initial conditions detailed predictions for the long time solution are derived localized high amplitude excitations absorb a surplus of energy when they emerge as a by product of the production of entropy in the small fluctuations the thermodynamic interpretation of this process applies to many dynamical systems with two conserved quantities

transport in simple networks described by an integrable discrete nonlinear schrodinger equation we elucidate the case in which the ablowitz ladik al type discrete nonlinear schrodinger equation nlse on simple networks e g star graphs and tree graphs becomes completely integrable just as in the case of a simple one dimensional 1d discrete chain the strength of cubic nonlinearity is different from bond to bond and networks are assumed to have at least two semi infinite bonds with one of them working as an incoming bond the present work is a nontrivial extension of our preceding one sobirov et al phys rev e 81 066602 2010 on the continuum nlse to the discrete case we find 1 the solution on each bond is a part of the universal bond independent al soliton solution on the 1d discrete chain but it is multiplied by the inverse of the square root of bond dependent nonlinearity 2 nonlinearities at individual bonds around each vertex must satisfy a sum rule and 3 under findings 1 and 2 there exist an infinite number of constants of motion as a practical issue with the use of an al soliton injected through the incoming bond we obtain transmission probabilities inversely proportional to the strength of nonlinearity on the outgoing bonds

oscillations of networks the role of soft nodes to describe the flow of a miscible quantity on a network we consider the graph wave equation where the standard continuous laplacian is replaced by the graph laplacian the structure of the graph influences strongly the dynamics assuming the graph is forced and damped at specific nodes we derive the amplitude equations using a basis of eigenvectors of the graph laplacian these lead us to introduce the notion of soft nodes we give sufficient conditions for their existence in general graphs they can cause several effects as we show on small graphs for example the ineffectiveness of damping applied to them soft nodes may be of critical importance for complex physical networks and engineering networks like power grids

higher order ostrowski type inequalities over euclidean domains abstract the classical ostrowski inequality for functions on intervals estimates the value of the function minus its average in terms of the maximum of its first derivative this result is extended to functions on general domains using the l norm of its n th partial derivatives for radial functions on balls the inequality is sharp

ostrowski type inequalities on h type groups the main aim of this paper is to establish an ostrowski type inequality on h type groups using the l norm of the horizontal gradient the work has been motivated by the work of anastassiou and goldstein in g a anastassiou j a goldstein higher order ostrowski type inequalities over euclidean domains j math anal appl 337 2008 962 968

invariant measures for nls in infinite volume we study gibbs measures and invariant dynamics for 1d defocusing nls iut uxx u u u p 2 0 when the period l for all p weak solutions are typically holder continuous on bounded space time regions if p 4 uniqueness and continuous dependence on the initial data may be shown

inverse spectral problem for quantum graphs the inverse spectral problem for the laplace operator on a finite metric graph is investigated it is shown that this problem has a unique solution for graphs with rationally independent edges and without vertices having valence 2 to prove the result a trace formula connecting the spectrum of the laplace operator with the set of periodic orbits for the metric graph is established

a complexity approach to the soliton resolution conjecture the soliton resolution conjecture is one of the most interesting open problems in the theory of nonlinear dispersive equations roughly speaking it asserts that a solution with generic initial condition converges to a finite number of solitons plus a radiative term in this paper we use the complexity of a finite object a notion introduced in algorithmic information theory to show that the soliton resolution conjecture is equivalent to the analogous of the second law of thermodynamics for the complexity of a solution of a dispersive equation

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

derivation of the gross pitaevskii equation for the dynamics of bose einstein condensate consider a system of n bosons in three dimensions interacting via a repulsive short range pair potential n 2 v n xi xj where x x1 xn denotes the positions of the particles let hn denote the hamiltonian of the system and let n t be the solution to the schrodinger equation suppose that the initial data n 0 satisfies the energy condition h n 0 h k n n 0i c k n k for k 1 2 we also assume that the k particle density matrices of the initial state are asymptotically factorized as n we prove that the k particle density matrices of n t are also asymptotically factorized and the one particle orbital wave function solves the gross pitaevskii equation a cubic non linear schrodinger equation with the coupling constant given by the scattering length of the potential v we also prove the same conclusion if the energy condition holds only for k 1 but the factorization of n 0 is assumed in a stronger sense ams subject classification number 81v70 81t18 35q55

parabolic approximations of the convection diffusion equation we propose an approximation of the convection diffusion operator which consists in the product of two parabolic operators this approximation is much easier to solve than the full convection diffusion equation which is elliptic in space we prove that this approximation is of order three in the viscosity and that the classical parabolic approximation is of order one in the viscosity numerical examples are given to demonstrate the effectiveness of our new approximation

statistical mechanics of the nonlinear schrodinger equation we investigate the statistical mechanics of a complex fieldo whose dynamics is governed by the nonlinear schrodinger equation such fields describe in suitable idealizations langmuir waves in a plasma a propagating laser field in a nonlinear medium and other phenomena their hamiltonian h phi int omega frac 1 2 nabla phi 2 1 p phi p dx is unbounded below and the system will under certain conditions develop self focusing singularities in a finite time we show that when is the circle and thel 2 norm of the field which is conserved by the dynamics is bounded byn the gibbs measure obtained is absolutely continuous with respect to wiener measure and normalizable if and only ifp andn are such that classical solutions exist for all time no collapse of the solitons this measure is essentially the same as that of a one dimensional version of the more realisitc zakharov model of coupled langmuir and ion acoustic waves in a plasma we also obtain some properties of the gibbs state by both analytic and numerical methods asn and the temperature are varied

ostrowski type inequalities for functions whose higher order derivatives have a single point of non differentiability ostrowski type inequalities for the class of functions whose n 1 th order derivatives are continuous of bounded variation and have a single point of non differentiability are derived special attention is given to functions whose first derivative has a single point of non differentiability improvements of some previously obtained results are provided

bistable reaction diffusion on a network we study analytically and numerically a bistable reaction diffusion equation on an arbitrary finite network we prove that stable fixed points multi fronts exist for any configuration as long as the diffusion is small we also study fold bifurcations leading to depinning and give a simple depinning criterion these results are confirmed by using continuation techniques from bifurcation theory and by solving the time dependent problem near the threshold a qualitative comparison principle is proved and verified for time dependent solutions and for some related models

l2 concentration of blow up solutions for the nonlinear schrodinger equation with critical power nonlinearity abstract we consider the solution of the nonlinear schrodinger equation i u t u f u and u 0 where u is defined on 0 t rn and f is a nonlinear complex valued function we consider the case of the critical power that is f z z 4 n z as z in a sense which will be made precise in addition we suppose that u t blows up in the norm h1 at time t we prove that u t has no limit in l2 as t t in the particular case where has a spherical symmetry we further show a phenomenon of l2 concentration at the origin

overlapping schwarz methods for a singularly perturbed semilinear elliptic problem and their parallel implementation domain decomposition algorithms based on the schwarz alternating method are applied to the numerical solution of a singularly perturbed semilinear elliptic problem the emphasis is given to the description of the algorithms multidomain schwarz alternating method and two level schwarz method as well as their computer realization on a distributed memory multiprocessor computer numerical experiments related to a model problem illustrate the performance of the algorithms on parallel environment as well as their behavior with respect to the critical parameters such as the perturbation parameter the amount of overlapping and sizes of subdomains application of the two level schwarz method to a model semiconductor problem is also discussed

hardy type inequalities and pohozaev type identities for a class of p degenerate subelliptic operators and applications abstract the purpose of this paper is to consider a class of p degenerate subelliptic operators l p constructed by generalizing greiner s vector fields their fundamental solutions at the origin are established with the aid of the properties of radial functions a picone type identity and a hardy type inequality with respect to vector fields are proved some pohozaev type identities and applications to nonlinear equations are given finally a carleman type estimate and uniqueness of the operator l 2 are discussed

soliton propagation on chains with simple nonlocal defects we study the propagation of solitons on complex chains built by inserting finite graphs at two sites of an unbranched chain we compare numerical findings with the results of an analytical linear approximation scheme describing the interaction of large and fast solitons with nonlocal topological defects on a chain we show that the transmission properties of the solitons strongly depend on the structure of the inserted graphs giving a tool for controlling the soliton propagation through the choice of pertinent graphs to attach to the chain

physically motivated domain decomposition for singularly perturbed equations a domain decomposition algorithm suitable for the efficient and accurate solution of a parabolic reaction convection diffusion equation with small parameter multiplying the diffusion term is presented convergence is established via maximum principle arguments the equation arises in the modeling of laminar transonic flow decomposition into subdomains is accomplished via singular perturbation analysis which dictates regions where certain reduced equations may be solved in place of the full equation effectively preconditioning the problem this paper concentrates on the theoretical basis of the method establishing local and global a priori error bounds

on the volume limit of the focusing cubic schrodinger equation we revisit the question of an invariant measure for the focusing cubic schrodinger equation on the line for the periodic problem the appropriate ensemble was introduced by lebowitz rose and speer 3 and proved to be invariant under the flow by mckean 5 these parties and others have also discussed the thermodynamic limit though without consensus simulations carried out in 3 indicated the possibility of a phase transition similar experiments in 1 appeared to contradict that interpretation later a proof was put forward in 6 that the full thermodynamic limit did not exist suggesting a possible explanation for the disparate conclusions drawn from the numerics unfortunately the latter contains an error the main result here is that in the infinite volume the ensemble collapses onto the unit mass on the trivial path along the way sharp estimates for the partition function are established 2002 wiley periodicals inc

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

a domain decomposition method for generating orthogonal polynomials for a gaussian weight on a finite interval abstract a domain decomposition method has been developed for generating orthogonal polynomials for a gaussian weight on 1 1 the method takes advantage of the underlying asymptotic structure of the orthogonal polynomials and hence it is effective in the sense that it makes maximal use of the analytic properties of the solution to increase accuracy and efficiency these polynomials are necessary for constructing gaussian quadrature formulas that are encountered in large quantum chemistry computational packages and in calculating the compton scattering kernel and its associated angular moments

probabilistic methods for discrete nonlinear schrodinger equations we show that the thermodynamics of the focusing cubic discrete nonlinear schrodinger equation are exactly solvable in dimension 3 and higher a number of explicit formulas are derived

statistical mechanics of the 2 dimensional focusing nonlinear schrodinger equation we study a natural construction of an invariant measure for the 2 dimensional periodic focusing nonlinear schrodinger equation with the critical cubic nonlinearity we find that a phase transition occurs as the coupling constant defining the strength of the nonlinearity is increased but that the natural construction successful for the 1 dimensional case and for the 2 dimensional defocusing case cannot produce an invariant measure our methods rely on an analysis of a statistical mechanical model closely related to the spherical model of berlin and kac

general euler ostrowski formulae and applications to quadratures the aim of this paper is to generalize inequality f x 1 b a ab f s1 ds1 f b f a b a x a b 2 f b f a 2 b a x2 a b x a2 b2 4ab 6 f b a 3 6i x a b a obtained in a aglic aljinovic m matic j pecaric improvements of some ostrowski type inequalities j comput anal appl in press and therefore obtain a generalization and improvement of inequality f x 1 b a ab f s1 ds1 f b f a b a x a b 2 f b f a 2 b a x2 a b x a2 b2 4ab 6 f a x b a 3 obtained in g a anastassiou univariate ostrowski inequalities revisited monatsh math 135 2002 175 189 to do this first we derive general euler ostrowski formulae which generalize extended euler formulae obtained in lj dedic m matic j pecaric on generalizations of ostrowski inequality via some euler type identities math inequal appl 3 3 2000 337 353 the main novelty is that a remainder is expressed in terms of bn x mt which enables us to obtain a vide variety of quadrature formulae such as trapezoid midpoint bitrapezoid twopoint formulae and their multipoint generalizations

random data cauchy theory for supercritical wave equations ii a global existence result we prove that the subquartic wave equation on the three dimensional ball with dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in bigcap s 1 2 h s theta times h s 1 theta we obtain this result as a consequence of a general random data cauchy theory for supercritical wave equations developed in our previous work 6 and invariant measure considerations inspired by earlier works by bourgain 2 3 on the non linear schrodinger equation which allow us to obtain also precise large time dynamical informations on our solutions

random data cauchy theory for supercritical wave equations i local theory we study the local existence of strong solutions for the cubic nonlinear wave equation with data in hs m s 1 2 where m is a three dimensional compact riemannian manifold this problem is supercritical and can be shown to be strongly ill posed in the hadamard sense however after a suitable randomization we are able to construct local strong solution for a large set of initial data in hs m where s 1 4 in the case of a boundary less manifold and s 8 21 in the case of a manifold with boundary

invariant measures and the soliton resolution conjecture the soliton resolution conjecture for the focusing nonlinear schrodinger equation nls is the vaguely worded claim that a global solution of the nls for generic initial data will eventually resolve into a radiation component that disperses like a linear solution plus a localized component that behaves like a soliton or multisoliton solution considered to be one of the fundamental open problems in the area of nonlinear dispersive equations this conjecture has eluded a proof or even a precise formulation to date this paper proves a statistical version of this conjecture at mass subcritical nonlinearity in the following sense the uniform probability distribution on the set of all functions with a given mass and energy if such a thing existed would be a natural invariant measure for the nls flow and would reflect the long term behavior for generic initial data with that mass and energy unfortunately such a probability measure does not exist we circumvent this problem by constructing a sequence of discrete measures that in principle approximate this fictitious probability distribution as the grid size goes to 0 we then show that a continuum limit of this sequence of probability measures does exist in a certain sense and in agreement with the soliton resolution conjecture the limit measure concentrates on the unique ground state soliton combining this with results from ergodic theory we present a tentative formulation and proof of the soliton resolution conjecture in the discrete setting the above results following in the footsteps of a program of studying the long term behavior of nonlinear dispersive equations through their natural invariant measures initiated by lebowitz rose and speer and carried forward by bourgain mckean tzvetkov oh and others are proved using a combination of techniques from large deviations pdes harmonic analysis and bare hands probability theory it is valid in any dimension 2014 wiley periodicals inc

derivation of the cubic non linear schrodinger equation from quantum dynamics of many body systems we prove rigorously that the one particle density matrix of three dimensional interacting bose systems with a short scale repulsive pair interaction converges to the solution of the cubic non linear schrodinger equation in a suitable scaling limit the result is extended to k particle density matrices for all positive integer k

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

two level parallel schwarz methods for singularly perturbed semilinear elliptic problems abstract two level domain decomposition methods based on the schwarz alternating procedure are proposed for the numerical solution of singularly perturbed semilinear elliptic problems convergence properties of the two level schwarz methods are established numerical experiments for a test problem illustrate the behaviour of the methods with respect to the critical parameters such as the perturbation parameter the amount of overlapping and sizes of subdomains application of one of the two level schwarz methods to a model semiconductor problem is also discussed

iterative solution of elliptic problems by approximate factorization an iterative method for the numerical solution of singularly perturbed second order linear elliptic problems is presented it is a defect correction iteration in which the approximate operator is the product of two first order operators which is readily inverted numerically the approximate operator is generated by formal asymptotic factorization of the original operator hence this is a quasi analytic defect correction iteration quad both its continuous and discrete versions are analyzed in one dimension the scheme is extended to a variety of two dimensional operators and it is analyzed for a model advection diffusion equation numerical calculations show the effectiveness of the scheme over a wide range of values of the small parameter

comparison of parallel implementation of some multi level schwarz methods for singularly perturbed parabolic problems abstract parallel multi level algorithms combining a time discretization and an overlapping domain decomposition technique are applied to the numerical solution of singularly perturbed parabolic problems two methods based on the schwarz alternating procedure are considered a two level method with auxiliary correcting subproblems as well as a three level method with auxiliary predicting and correcting subproblems moreover modifications of the methods using time extrapolation on subdomain interfaces are investigated the emphasis is given to the description of the algorithms as well as their computer realization on a distributed memory multiprocessor computer numerical experiments illustrate the performance of the algorithms on parallel environment and their behaviour with respect to the critical parameters such as the perturbation parameter and the size of the auxiliary subdomains

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

iterative and non iterative full and approximate factorization methods for multidimensional reaction diffusion equations a variety of second order accurate in both space and time full and approximate factorization methods for the numerical solution of two dimensional reaction diffusion equations is presented these methods may use time linearization and yield linearly implicit techniques and one dimensional operators in each direction it is shown that if the factorization errors are neglected linearly implicit approximate factorization methods provide uncoupled equations whereas if these errors are considered the equations are coupled and must be solved iteratively it is also shown that the allocation of the reaction and diffusion terms to the one dimensional operators plays a paramount role in determining the accuracy of approximate factorization methods and preserving the symmetry of the original differential problem iterative full and approximate factorization methods that do require iterations are also presented and for the problem considered here these methods are shown to converge in about two iterations and provide solutions in agreement with those obtained with linearly implicit full and approximate factorization techniques

a note on mixed matrix moments for the complex ginibre ensemble we consider the mixed matrix moments for the complex ginibre ensemble these are well known we consider the relation to the expected overlap functions of chalker and mehlig this leads to new asymptotic problems for the overlap we obtain some results but we also state some remaining open problems

derivation of the two dimensional nonlinear schrodinger equation from many body quantum dynamics we derive rigorously for both r 2 and l l 2 the cubic nonlinear schrodinger equation in a suitable scaling limit from the two dimensional many body bose systems with short scale repulsive pair interactions we first prove convergence of the solution of the bbgky hierarchy corresponding to the many body systems to a solution of the infinite gross pitaevskii hierarchy corresponding to the cubic nls and then we prove uniqueness for the infinite hierarchy which requires number theoretical techniques in the periodic case

a hybrid numerical asymptotic method for scattering problems we develop a hybrid numerical asymptotic method for the helmholtz equation the method is a galerkin finite element method in which the space of trial solutions is spanned by asymptotically derived basis functions the basis functions are very efficient in representing the solution because each is the product of a smooth amplitude and an oscillatory phase factor like the asymptotic solution the phase is determined a priori by solving the eiconal equation using the ray method while the smooth amplitude is represented by piecewise polynomials the number of unknowns necessary to achieve a given accuracy with this new basis is dramatically smaller than the number necessary with a standard method and it is virtually independent of the wavenumber k we apply the method to the problems of scattering from a parabola and from a circle and compare the results with those of a standard finite element method

an iterative method for systems of nonlinear hyperbolic equations abstract an iterative algorithm for the efficient solution of systems of nonlinear hyperbolic equations is presented parallelism is evident at several levels in the formation of the iteration the equations are decoupled thereby providing large grain parallelism parallelism may also be exploited within the solves for each equation convergence of the iteration is established via a bounding function argument experimental results in two dimensions are presented

fluctuations in the thermodynamic limit of focussing cubic schrodinger we consider the statistical mechanics of a complex field z whose dynamics are governed by the focussing cubic schrodinger equation here the hamiltonian h int omega left frac 1 2 left nabla z right 2 frac 1 4 left z right 4 right dx is unbounded from below preventing the natural gibbs measure from being normalizable this difficulty may be circumvented 5 by taking the circle of perimeter l and fixing the mean square which is conserved by the dynamics int 0 l left z right 2 dx ld for positive density d the resulting probability measure on paths is absolutely continuous to the two dimensional wiener measure and is known to be invariant under the flow 2 7 one way to extend this picture to the whole line flow is to take the thermodynamic limit l unfortunately the unboundedness of h causes vast local concentration of the field as l increases and leads to collapse at l 11 here we attempt to capture fluctuations away from this collapse by performing a joint continuum and infinite volume limit for an appropriate lattice ensemble the result is that for high density the scaled paths go over into a white noise

ostrowski type inequalities and hardy inequalities with boundary terms related to the generalized greiner operator in this paper we first show a representation formula related to the generalized greiner operator by an improved method then using the representation formula and l norm of the horizontal gradient we establish ostrowski type inequalities in the ball and general bounded domain related to the generalized greiner operator moreover in the same spirit we show the hardy inequalities with boundary terms keywords generalized greiner operator representation formula ostrowski type inequal ity hardy inequality mr 2010 subject classification 35r45 26d10 35j70 clc number o178 document code a article id 1000 0917 2015 01 0091 11

brownian motion with restoring drift the petit and micro canonical ensembles letf q be odd and positive near then the non linear wave equation 2q t2 2q x2 f q 0 considered on the circle 0 x l can be written in hamiltonian formq h p p h q with p q cdot and h tfrac 1 2 mathop smallint limits 0 l q 2 mathop smallint limits 0 l f q tfrac 1 2 mathop smallint limits 0 l p 2 the corresponding flow preserves the suitably interpreted petit ensemble e h d qd p and forl q settles down to the stationary diffusion with infinitesimal operator 1 2 2 q2 m q q m being the logarithmic derivative of the ground state of d2 dq2 f q this diffusion is the brownian motion with restoring drift see mckean vaninsky 1993 1 for reasons suggested by the paper of lebowitz rose speer 1988 on nls it is interesting to study the micro canonical ensemble obtained by restricting to the sphere int limits 0 l q 2 n and makingl with fixedd n l now forf q q2 the same type of diffusion appears but with drift arising from the modified potentialf q cq2 c being chosen so that the mean ofq2 is the assigned numberd the proof employs doblin s method of loops 1937 and steepest descent the same is true forf q m2q2 only now the proof is elementary the outcome is also the same iff q q2 0 providedd is smaller than the petit canonical mean ofq2 ford larger than this mean the matter is more subtle and the outcome is unknown

invariant measures for the 2d defocusing nonlinear schrodinger equation consider the2d defocusing cubic nlsiu t u u u 2 0 with hamiltonian smallint left left nabla phi right 2 tfrac 1 2 left phi right 4 right it is shown that the gibbs measure constructed from the wick ordered hamiltonian i e replacing 4 by 4 is an invariant measure for the appropriately modified equationiu t u u u 2 2 u 2 dx u 0 there is a well defined flow on thesupport of the measure in fact it is shown that for almost all data the solutionu u 0 satisfiesu t e it c hs for somes 0 first a result local in time is established and next measure invariance considerations are used to extend the local result to a global one cf b2

gruss and ostrowski type inequalities abstract in this paper we use a method originated in s s dragomir some gruss type inequalities in inner product spaces j inequal pure appl math 4 2 2003 article 42 to establish some gruss and ostrowski type inequalities