fourier transform inequalities with measure weights abstract fourier transform norm inequalities f q p are proved for measure weights on moment subspaces of lp r n density theorems are established to extend the inequalities to all of lp r n in both cases the conditions for validity are computable for n 2 and are radial and the results are applied to prove spherical restriction theorems which include power weights t t n p 1 p n p 1

the zak transform and sampling theorems for wavelet subspaces the zak transform is used for generalizing a sampling theorem of g waiter see ieee trans informat theory vol 38 p 881 884 1992 for wavelet subspaces cardinal series based on signal samples f a n n in z with a possibly unequal to 0 waiter s case are considered the condition number of the sampling operator and worst case aliasing errors are expressed in terms of zak transforms of scaling function and wavelet this shows that the stability of the resulting interpolation formula depends critically on a

isometries fock spaces and spectral analysis of schrodinger operators on trees abstract we construct conjugate operators for the real part of a completely non unitary isometry and we give applications to the spectral and scattering theory of a class of operators on complete fock spaces natural generalizations of the schrodinger operators on trees we consider c algebras generated by such hamiltonians with certain types of anisotropy at infinity we compute their quotient with respect to the ideal of compact operators and give formulas for the essential spectrum of these hamiltonians

global stabilization of the generalized korteweg de vries equation posed on a finite domain this paper is concerned with the internal stabilization of the generalized korteweg de vries equation on a bounded domain the global well posedness and the exponential stability are investigated when the exponent in the nonlinear term ranges over the interval 1 4 the global exponential stability is obtained whatever the location where the damping is active confirming positively a conjecture of perla menzala vasconcellos and zuazua quart appl math 60 2002 pp 111 129

multilinear estimates for periodic kdv equations and applications we prove an endpoint multilinear estimate for the xs b spaces associated to the periodic airy equation as a consequence we obtain sharp local well posedness results for periodic generalized kdv equations as well as some global well posedness results below the energy norm

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

infinite dimensional symplectic capacities and a squeezing theorem for hamiltonian pde s we study partial differential equations of hamiltonian form and treat them as infinite dimensional hamiltonian systems in a functional phase space ofx dependent functions in this phase space we construct an invariant symplectic capacity and prove a version of gromov s non squeezing theorem we give an interpretation of the theorem in terms of the energy transition to high frequencies problem

indirect estimation via l w for a large class of queueing systems little s law l w helps provide a variety of statistical estimators for the long run time average queue length l and the long run customer average waiting time w we apply central limit theorem versions of little s law to investigate the asymptotic efficiency of these estimators we show that an indirect estimator for l using the natural estimator for w plus the known arrival rate is more efficient than a direct estimator for l provided that the interarrival and waiting times are negatively correlated thus extending a variance reduction principle for the gi g s model due to a m law and j s carson we also introduce a general framework for indirect estimation which can be applied to other problems besides l w we show that the issue of indirect versus direct estimation is related to estimation using nonlinear control variables we also show under mild regularity conditions that any nonlinear control variable scheme is equivalent to a linear control va

autonomous ovsyannikov theorem and applications to nonlocal evolution equations and systems abstract this work presents an ovsyannikov type theorem for an autonomous abstract cauchy problem in a scale of decreasing banach spaces which in addition to existence and uniqueness of solution provides an estimate about the analytic lifespan of the solution then using this theorem it studies the cauchy problem for camassa holm type equations and systems with initial data in spaces of analytic functions on both the circle and the line which is the main goal of this paper finally it studies the continuity of the data to solution map in spaces of analytic functions

approximating the revenue maximization problem with sharp demands we consider the revenue maximization problem with sharp multi demand in which m indivisible items have to be sold to n potential buyers each buyer i is interested in getting exactly d i items and each item j gives a benefit v ij to buyer i we distinguish between unrelated and related valuations in the former case the benefit v ij is completely arbitrary while in the latter each item j has a quality q j each buyer i has a value v i and the benefit v ij is defined as the product v i q j the problem asks to determine a price for each item and an allocation of bundles of items to buyers with the aim of maximizing the total revenue that is the sum of the prices of all the sold items the allocation must be envy free that is each buyer must be happy with her assigned bundle and cannot improve her utility we first prove that for related valuations the problem cannot be approximated to a factor o m 1 e for any e 0 unless p np and that such result is asymptotically tight in fact we provide a simple m approximation algorithm even for unrelated valuations we then focus on an interesting subclass of proper instances that do not contain buyers a priori known not being able to receive any item for such instances we design an interesting 2 approximation algorithm and show that no 2 e approximation is possible for any 0 e 1 unless p np we observe that it is possible to efficiently check if an instance is proper and if discarding useless buyers is allowed an instance can be made proper in polynomial time without worsening the value of its optimal solution

global well posedness in l 2 for the periodic benjamin ono equation we prove that the benjamin ono equation is globally well posed in h s bbb t for s ge 0 moreover we show that the associated flow map is lipschitz on every bounded set of h s 0 bbb t s ge 0 and even real analytic in this space for small times this result is sharp in the sense that the flow map if it can be defined and coincides with the standard flow map on h 0 infty bbb t cannot be of class c 1 alpha alpha 0 from h 0 s bbb t into h 0 s bbb t as soon as s 0

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

combination can be hard approximability of the unique coverage problem we prove semilogarithmic inapproximability for a maximization problem called unique coverage given a collection of sets find a subcollection that maximizes the number of elements covered exactly once specifically assuming that mathrm np not subseteq operatorname bptime 2 n varepsilon for an arbitrary varepsilon 0 we prove o 1 log sigma n inapproximability for some constant sigma sigma varepsilon we also prove o 1 log 1 3 varepsilon n inapproximability for any varepsilon 0 assuming that refuting random instances of 3sat is hard on average and we prove o 1 log n inapproximability under a plausible hypothesis concerning the hardness of another problem balanced bipartite independent set we establish an omega 1 log n approximation algorithm even for a more general budgeted setting and obtain an omega 1 log b approximation algorithm when every set has at most b elements we also show that our inapproximability results extend to envy free pricing an important problem in computational economics we describe how the budgeted unique coverage problem motivated by real world applications has close connections to other theoretical problems including max cut maximum coverage and radio broadcasting

global well posedness for schrodinger equation with derivative in h 1 2 r abstract in this paper we consider the cauchy problem of the cubic nonlinear schrodinger equation with derivative in h s r this equation was known to be the local well posedness for s 1 2 takaoka 1999 27 ill posedness for s 1 2 biagioni and linares 2001 1 etc and global well posedness for s 1 2 i team 2002 10 in this paper we show that it is global well posedness in the endpoint space h 1 2 r which remained open previously the main approach is the third generation i method combined with a new resonant decomposition technique the resonant decomposition is applied to control the singularity coming from the resonant interaction

global well posedness and inviscid limit for the korteweg de vries burgers equation abstract considering the cauchy problem for the korteweg de vries burgers equation u t u x x x x 2 u u 2 x 0 u 0 where 0 1 and u is a real valued function we show that it is globally well posed in h s s s and uniformly globally well posed in h s s 3 4 for all 0 1 moreover we prove that for any t 0 its solution converges in c 0 t h s to that of the kdv equation if tends to 0

a higher dispersion kdv equation on the line abstract the cauchy problem for a korteweg devries equation with dispersion of order m 2 j 1 where j is a positive integer kdvm is studied with data in sobolev and analytic spaces first optimal bilinear estimates in bourgain spaces are proved and using them well posedness in sobolev spaces h s s j 1 4 is established then well posedness in analytic gevrey spaces g s 0 is proved by using an analytic version of the bilinear estimates this implies that the uniform radius of analyticity persist for some time for the later times a lower bound for the radius of spacial analyticity is derived which is given by t c t with 4 3 e for any e 0 when j 1 and 1 when j 2 finally it is shown that the regularity of the solution in the time variable is gevrey of order m and this is optimal

continuity of the flow of kdv with regard to the wasserstein metrics and application to an invariant measure abstract in this paper we prove the continuity of the flow of kdv on spaces of probability measures with respect to a combination of wasserstein distances on h s s 0 and l 2 we are motivated by the existence of an invariant measure belonging to the spaces onto which these distances are defined

on social envy freeness in multi unit markets abstract we consider a market setting in which buyers are individuals of a population whose relationships are represented by an underlying social graph given buyers valuations for the items being sold an outcome consists of a pricing of the objects and an allocation of bundles to the buyers an outcome is social envy free if no buyer strictly prefers the bundles of her neighbors in the social graph we focus on the revenue maximization problem in multi unit markets in which there are multiple copies of the same item being sold and each buyer is assigned a subset of identical items we consider four different cases that arise when considering two different buyers valuations i e single minded or general and by adopting two different forms of pricing that is item or bundle pricing for all the above cases we show the hardness of the revenue maximization problem and give corresponding approximation results all our approximation bounds are optimal or nearly optimal moreover under the assumption of social graphs of bounded treewidth we provide an optimal allocation algorithm for general valuations with item pricing finally we determine optimal bounds on the corresponding price of envy freeness that is on the worst case ratio between the maximum revenue that can be achieved without envy freeness constraints and the one obtainable in case of social relationships some of our results close hardness open questions or improve already known ones in the literature concerning the standard setting without sociality

generalized definition of time delay in scattering theory we advocate for the systematic use of a symmetrized definition of time delay in scattering theory in two body scattering processes we show that the symmetrized time delay exists for arbitrary dilated spatial regions symmetric with respect to the origin it is equal to the usual time delay plus a new contribution which vanishes in the case of spherical spatial regions we also prove that the symmetrized time delay is invariant under an appropriate mapping of time reversal these results are also discussed in the context of classical scattering theory

sampling theorem for wavelet subspaces error estimate and irregular sampling the error estimate is useful in the application of the sampling theorem for the classical shannon sampling theorem various errors are widely studied but for the sampling theorem in general wavelet subspaces only the aliasing error is studied in this paper we study three other errors truncation error amplitude error and time jitter error with the same technique a result on irregular sampling is improved

well posedness for the cauchy problem associated to the hirota satsuma equation periodic case abstract we consider a system of korteweg de vries kdv equations coupled through nonlinear terms called the hirota satsuma system we study the initial value problem ivp associated to this system in the periodic case for given data in sobolev spaces h s h s 1 with regularity below the one given by the conservation laws using the fourier transform restriction norm method we prove local well posedness whenever s 1 2 also with some restriction on the parameters of the system we use the recent technique introduced by colliander et al called i method and almost conserved quantities to prove global well posedness for s 3 14

irregular sampling for spline wavelet subspaces spline wavelets spl psi sub m t are important in time frequency localization due to i spl psi sub m can be arbitrarily close to the optimal case as m is sufficiently large ii spl psi sub m has compact support and simple analytic expression which lead to effective computation although the spline wavelet subspaces are so simple walter s well known sampling theorem does not hold if the order of spline m is even moreover when irregular sampling is considered in these spaces it is hard to determine the sampling density which is a serious problem in applications in this correspondence a general sampling theorem is obtained for m spl ges 3 in the sense of iterative construction and the sampling density spl delta sub m is estimated

numerical integration using wavelets in this paper an algorithm for obtaining approximate value of a definite integral as well as double integral using wavelets will be illustrated this approximation depends on the pure scaling functions expansion of the integrand function

multiwavelet bases with extra approximation properties this paper highlights the differences between traditional wavelet and multiwavelet bases with equal approximation order because multiwavelet bases normally lack important properties that traditional wavelet bases of equal approximation order possess the associated discrete multiwavelet transform is less useful for signal processing unless it is preceded by a preprocessing step prefiltering this paper examines the properties and design of orthogonal multiwavelet bases with approximation order 1 that possess those properties that are normally absent for these balanced bases so named by lebrun and vetterli see proc ieee int conf acoust speech signal process icassp munich germany vol 3 p 2473 76 1997 and ibid vol 46 p 1119 24 1998 prefiltering can be avoided by reorganizing the multiwavelet filter bank the development in this paper draws from results regarding the approximation order of m band wavelet bases the main result thereby obtained is a characterization of balanced multiwavelet bases in terms of the divisibility of certain transfer functions by powers of z sup 2r 1 z sup 1 1 for traditional wavelets r 1 this specializes to the usual factor z 1 sup k

numerical analysis of the non uniform sampling problem we give an overview of recent developments in the problem of reconstructing a band limited signal from nonuniform sampling from a numerical analysis view point it is shown that the appropriate design of the finite dimensional model plays a key role in the numerical solution of the nonuniform sampling problem in the one approach often proposed in the literature the finite dimensional model leads to an ill posed problem even in very simple situations the other approach that we consider leads to a well posed problem that preserves important structural properties of the original infinite dimensional problem and gives rise to efficient numerical algorithms furthermore a fast multilevel algorithm is presented that can reconstruct signals of unknown bandwidth from noisy nonuniformly spaced samples we also discuss the design of efficient regularization methods for ill conditioned reconstruction problems numerical examples from spectroscopy and exploration geophysics demonstrate the performance of the proposed methods

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

the power series method for nonlocal and nonlinear evolution equations abstract the initial value problem for a 4 parameter family of nonlocal and nonlinear evolution equations with data in a space of analytic functions is solved by using a power series method in abstract banach spaces in addition to determining the power series expansion of the solution this method also provides an estimate of the analytic lifespan expressed in terms of the norm of the initial data thus establishing an abstract cauchy kovalevsky type theorem for these equations

local well posedness for the dispersion generalized periodic kdv equation abstract in this paper we set up the local well posedness of the initial value problem for the dispersion generalized periodic kdv equation t u x d x u x u 2 u 0 for 2 s 4 and h s t and we show that the 4 is a lower endpoint to obtain the bilinear estimates 1 2 and 1 3 which are the crucial steps to obtain the local well posedness by picard iteration the case 2 was studied in kenig et al 1996 10

well posedness and ill posedness of kdv equation with higher dispersion abstract first the cauchy problem for kdv equation with 2 n 1 order dispersion is studied and the local well posedness result for the initial data in sobolev spaces h s r with s n 1 4 is established via the fourier restriction norm method second we prove that the kdv equation with 2 n 1 order dispersion is ill posed for the initial data in h s r with s n 1 4 n 2 n n if the flow map is c 2 differentiable at zero form h s r to c 0 t h s r finally we obtain the sharp regularity requirement for the kdv equation with 2 n 1 order dispersion s n 1 4

design of prefilters for discrete multiwavelet transforms the pyramid algorithm for computing single wavelet transform coefficients is well known the pyramid algorithm can be implemented by using tree structured multirate filter banks the authors propose a general algorithm to compute multiwavelet transform coefficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets the proposed algorithm can be thought of as a discrete vector valued wavelet transform for certain discrete time vector valued signals the proposed algorithm can be also thought of as a discrete multiwavelet transform for discrete time signals the authors then present some numerical experiments to illustrate the performance of the algorithm which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms

efficient numerical methods in non uniform sampling theory summary we present a new second generation reconstruction algorithm for irregular sampling i e for the problem of recovering a band limited function from its non uniformly sampled values the efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems the choice of adaptive weights can be seen as a simple but very efficient method of preconditioning further substantial acceleration is achieved by utilizing the toeplitztype structure of the system matrix this new algorithm can handle problems of much larger dimension and condition number than have been accessible so far furthermore if some gaps between samples are large then the algorithm can still be used as a very efficient extrapolation method across the gaps

the m band cardinal orthogonal scaling function the m band symmetric cardinal orthogonal scaling function with compact support is of interest in several applications such as sampling theory signal processing computer graphics and numerical algorithms in this paper we provide a complete mathematical analysis for the m band symmetric cardinal orthogonal scaling function firstly we generalize some results of the cardinal orthogonal scaling function from the special case m 2 to the most general case m 2 also we find some new results secondly we obtain the characterizations of the m band symmetric cardinal orthogonal scaling function and revisit some known examples to prove our theory

local and global well posedness for the 2d zakharov kuznetsov burgers equation in low regularity sobolev space abstract in the present paper we consider the cauchy problem of the 2d zakharov kuznetsov burgers zkb equation which has the dissipative term x 2 u this is known that the 2d zakharov kuznetsov equation is well posed in h s r 2 for s 1 2 and the 2d nonlinear parabolic equation with quadratic derivative nonlinearity is well posed in h s r 2 for s 0 by using the fourier restriction norm with dissipative effect we prove the well posedness for zkb equation in h s r 2 for s 1 2

the cardinal orthogonal scaling function and sampling theorem in the wavelet subspaces in this paper we derive that there is a relation between the lowpass filter coefficient and wavelet s samples in its integer points when a scaling function is a cardinal orthogonal scaling function and we give some examples in which the lowpass filter coefficients are constructed from the wavelets then we discuss the symmetry property of cardinal orthogonal scaling function and give some useful characterizations therefore we construct a family of cardinal orthogonal scaling functions with exponential decay and higher approximation order furthermore some examples with exponential decay and higher approximation order are given and they will testify our results in the end we deduce that there no exists a cardinal orthogonal scaling function with exponential decay high approximation order and symmetry property in this family of cardinal orthogonal scaling functions constructed in our paper

generalized sampling theorems in multiresolution subspaces it is well known that under very mild conditions on the scaling function multiresolution subspaces are reproducing kernel hilbert spaces rkhss this allows for the development of a sampling theory in this paper we extend the existing sampling theory for wavelet subspaces in several directions we consider periodically nonuniform sampling sampling of a function and its derivatives oversampling multiband sampling and reconstruction from local averages all these problems are treated in a unified way using the perfect reconstruction pr filter bank theory we give conditions for stable reconstructions in each of these cases sampling theorems developed in the past do not allow the scaling function and the synthesizing function to be both compactly supported except in trivial cases this restriction no longer applies for the generalizations we study here due to the existence of fir pr banks in fact with nonuniform sampling oversampling and reconstruction from local averages we can guarantee compactly supported synthesizing functions moreover local averaging schemes have additional nice properties robustness to the input noise and compression capabilities we also show that some of the proposed methods can be used for efficient computation of inner products in multiresolution analysis after this we extend the sampling theory to random processes we require autocorrelation functions to belong to some subspace related to wavelet subspaces it turns out that we cannot recover random processes themselves unless they are bandlimited but only their power spectral density functions we consider both uniform and nonuniform sampling

on sampling theorem wavelets and wavelet transforms the classical shannon sampling theorem has resulted in many applications and generalizations from a multiresolution point of view it provides the sine scaling function in this case for a band limited signal its wavelet series transform wst coefficients below a certain resolution level can be exactly obtained from the samples with a sampling rate higher than the nyquist rate the authors study the properties of cardinal orthogonal scaling functions cosf which provide the standard sampling theorem in multiresolution spaces with scaling functions as interpolants they show that cosf with compact support have and only have one possibility which is the haar pulse they present a family of cosf with exponential decay which are generalizations of the haar function with these cosf an application is the computation of wst coefficients of a signal by the mallat 1989 algorithm they present some numerical comparisons for different scaling functions to illustrate the advantage of cosf for signals which are not in multiresolution spaces they estimate the aliasing error in the sampling theorem by using uniform samples

self spreading and strength of singularities for solutions to semilinear wave equations algebres microlocales propagation de la regularite solutions lineaires conduisant a l etalement auto etalement des singularites

traveling wave solutions of the degasperis procesi equation we classify all weak traveling wave solutions of the degasperis procesi equation in addition to smooth and peaked solutions the equation is shown to admit more exotic traveling waves such as cuspons stumpons and composite waves

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

optimal envy free pricing with metric substitutability we study the unit demand envy free pricing problem faced by a profit maximizing seller with unlimited supply when there is metric substitutability among the items consumer i s value for item j is v i c i j and the substitution costs c i j form a metric our model is motivated by the observation that sellers often sell the same product at different prices in different locations and rational consumers optimize the tradeoff between prices and substitution costs while the general envy free pricing problem is hard to approximate we show that the problem of maximizing revenue with metric substitutability among items can be solved exactly in polynomial time we do this by first showing that in any optimal price vector the set of nodes that pay exactly their value uniquely determines which nodes buy an item and what price they pay and therefore the revenue we transform the problem of finding an optimal set of such nodes to an instance of weighted independent set on a perfect graph which can be solved in polynomial time by the strong perfect graph theorem proving the result we then analyze the computational tractability of various extensions to our model we begin with relaxing the metric substitutability requirement and show that when the substitution costs do not form a metric even if a 1 epsilon approximate triangle inequality holds the problem becomes np hard thus the triangle inequality characterizes the threshold at which the problem goes from tractable to hard we then relax assumptions on the supply and demand we consider restricting supplies to a subset of locations or the amount of supplies or allowing buyers to demand more than one unit in all cases the problem becomes np hard in addition the multiunit demand case illustrates an interesting paradoxical nonmonotonicity the optimal revenue the seller can extract can actually decrease when consumers demands increase we show the revenue maximization problem with multiunit demand is apx hard even for the simplest valuations with equal marginal values for all items up to the demand constraint and demands of at most 3

asymptotics frequency modulation and low regularity ill posedness for canonical defocusing equations in a recent paper kenig ponce and vega study the low regularity behavior of the focusing nonlinear schrodinger nls focusing modified korteweg de vries mkdv and complex korteweg de vries kdv equations using soliton and breather solutions they demonstrate the lack of local well posedness for these equations below their respective endpoint regularities in this paper we study the defocusing analogues of these equations namely defocusing nls defocusing mkdv and real kdv all in one spatial dimension for which suitable soliton and breather solutions are unavailable we construct for each of these equations classes of modified scattering solutions which exist globally in time and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts these solutions are used to demonstrate lack of local well posedness in certain sobolev spaces in the sense that the dependence of solutions upon initial data fails to be uniformly continuous in particular we show that the mkdv flow is not uniformly continuous in the l 2 topology despite the existence of global weak solutions at this regularity finally we investigate the kdv equation at the endpoint regularity h and construct solutions for both the real and complex kdv equations the construction provides a nontrivial time interval t t and a locally lipschitz continuous map taking the initial data in h to a distributional solution u c 0 t t h which is uniquely defined for all smooth data the proof uses a generalized miura transform to transfer the existing endpoint regularity theory for mkdv to kdv

periodic fourth order cubic nls local well posedness and non squeezing property abstract in this paper we consider the cubic fourth order nonlinear schrodinger equation 4nls under the periodic boundary condition we prove two results one is the local well posedness in h s t with 1 3 s 0 for the cauchy problem of the wick ordered 4nls the other one is the non squeezing property for the flow map of 4nls in the symplectic phase space l 2 t to prove the former we used the ideas introduced in 36 and 27 and to prove the latter we used the ideas in 8

global well posedness results for the 2d schrodinger debye system abstract in this work we study the well posedness of the schrodinger debye system i t u 1 2 u u v t 0 x r 2 t v v u 2 0 1 u 0 x u 0 x v 0 x v 0 x in the context of sobolev spaces of type h k 1 r 2 h k 2 r 2 where k 1 k 2 are integers we prove some a priori estimates and therefore global well posedness using nonlinear interpolation of operators we prove some global well posedness results when u 0 v 0 h s r 2 h l r 2 where s l r are some convex combinations of k 1 and k 2

global well posedness of the one dimensional cubic nonlinear schrodinger equation in almost critical spaces abstract in this paper we first introduce a new function space m h p whose norm is given by the l p sum of modulated h norms of a given function in particular when 1 2 we show that the space m h p agrees with the modulation space m 2 p r on the real line and the fourier lebesgue space f l p t on the circle we use this equivalence of the norms and the galilean symmetry to adapt the conserved quantities constructed by killip visan zhang to the modulation space and fourier lebesgue space setting by applying the scaling symmetry we then prove global well posedness of the one dimensional cubic nonlinear schrodinger equation nls in almost critical spaces more precisely we show that the cubic nls on r is globally well posed in m 2 p r for any p while the renormalized cubic nls on t is globally well posed in f l p t for any p in appendix we also establish analogous global in time bounds for the modified kdv equation mkdv in the modulation spaces on the real line and in the fourier lebesgue spaces on the circle an additional key ingredient of the proof in this case is a galilean transform which converts the mkdv to the mkdv nls equation

quadratic dispersive generalized benjamin ono equation abstract this paper is concerned with quadratic dispersive generalized benjamin ono equation 1 1 1 2 we prove sharp local well posedness in h s s 1 4 with a solution map that is analytic from h s r to c 0 t h s r and the ill posed result in h s r as soon as s s in the sense that the flow map u 0 u t cannot be continuous

a formula relating sojourn times to the time of arrival in hamiltonian dynamics we consider on a manifold m equipped with a poisson bracket a hamiltonian h with complete flow and a family 1 d of abstract position observables satisfying the condition j h h 0 for each j under these assumptions we prove a new formula relating sojourn times in dilated regions defined in terms of to the time of arrival of classical orbits the correspondence between this formula and a formula established recently in the framework of quantum mechanics is put into evidence among other examples our theory applies to stark hamiltonians homogeneous hamiltonians purely kinetic hamiltonians the repulsive harmonic potential central force systems the poincar ball model the wave equation the nonlinear schr dinger equation the korteweg de vries equation and quantum hamiltonians defined via expectation values

the gardner equation and the l 2 stability of the n soliton solution of the korteweg de vries equation multi soliton solutions of the korteweg de vries equation kdv are shown to be globally l2 stable and asymptotically stable in the sense of martel and merle the proof is surprisingly simple and combines the gardner transform which links the gardner and kdv equations together with the martel merle tsai and martel merle recent results on stability and asymptotic stability in the energy space applied this time to the gardner equation as a by product the results of maddocks sachs and merle vega are improved in several directions

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation

global well posedness and nonsqueezing property for the higher order kdv type flow abstract in this paper we prove that the periodic higher order kdv type equation t u 1 j 1 x 2 j 1 u 1 2 x u 2 0 t x r t u 0 x u 0 x u 0 h s t is globally well posed in h s for s j 2 j 3 the proof is based on i method introduced by colliander et al 4 we also prove the nonsqueezing property of the periodic higher order kdv type equation the proof relies on gromov s nonsqueezing theorem for the finite dimensional hamiltonian system and an approximation argument for the solution flow more precisely after taking the frequency truncation to the solution flow we apply the nonsqueezing theorem by using the approximation argument we extend this result to the infinite dimensional system this argument was introduced by kuksin 14 and made concretely by bourgain 2 for the 1d cubic nls flow and colliander et al 5 for the kdv flow one of our observations is that the higher order kdv type equation has the better modulation effect from the non resonant interaction than that the kdv equation has hence unlike the work of colliander et al 5 we can get the nonsqueezing property for the solution flow without the miura transform