the sharp hardy uncertainty principle for schrodinger evolutions we give a new proof of hardy s uncertainty principle up to the end point case which is only based on calculus the method allows us to ex tend hardy s uncertainty principle to schrodinger equations with non constant coefficients we also deduce optimal gaussian decay bounds for solutions to these schrodinger equations

horvitz thompson estimators for double platform line transect surveys summary horvitz thompson estimators based on estimated inclusion probabilities are developed for line transect surveys in which some of the central assumptions of conventional line transect theory fail the estimators are designed to accommodate uncertain detection of animals in the observers path simultaneously with dependence of detection probabilities on explanatory variables other than perpendicular distance they require that data be collected by two observer teams simultaneously searching the survey region recording for each detection both positional data and details of any other variables that may affect detectability one of the estimators is designed to correct for responsive animal movement this method requires that the primary observer team searches independently of the secondary team whereas the second estimator requires both observer teams to search independently of each other the methods can be viewed as specific forms of mark recapture experiments in which individual capture probabilities are modelled as functions of observable explanatory variables and in which animals detected by both observer teams correspond to recaptures they differ from existing mark recapture horvitz thompson estimators primarily in that they use an assumption about the distribution of perpendicular distances that is implicit in conventional

carleman inequalities for the dirac and laplace operators and unique continuation on demontre les meilleures inegalites de type carleman possibles dans le cas de l operateur de dirac on obtient la propriete de prolongement unique pour d v ou v l loc r n 3n 2 2 n 3

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

multi phase computations of the semiclassical limit of the schrodinger equation and related problems whitham vs wigner abstract we present and compare two different techniques to obtain the multi phase solutions for the schrodinger equation in the semiclassical limit the first is whitham s averaging method which gives the modulation equations governing the evolution of multi phase solutions the second is the wigner transform a convenient tool to derive the semiclassical limit equation in the phase space the vlasov equation for the linear schrodinger equation motivated by the linear superposition principle we derive and prove the multi phase ansatz for the wigner function by the stationary phase method and then use the ansatz to close the moment equations of the vlasov equation and obtain the multi phase equations in the physical space we show that the multi phase equations so derived agree with those derived by whitham s averaging method which can be proved using different arguments generic way of obtaining and computing the multi phase equations by the wigner function is given and kinetic schemes are introduced to solve the multi phase equations the numerical schemes based on multiphase closure of the liouville equation are purely eulerian and only operate in the physical space several numerical examples are given to explore the validity of this approach similar studies are conducted for the linearized korteweg de vries equation and the linear wave equation

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

l 1 approximation of stationary hamilton jacobi equations we describe a nonlinear finite element technique to approximate the solutions of stationary hamilton jacobi equations in two space dimensions using continuous finite elements of arbitrary degree the method consists of minimizing a functional containing the l 1 norm of the hamiltonian plus a discrete entropy it is shown that the approximate sequence converges to the unique viscosity solution under appropriate hypotheses on the hamiltonian and the mesh family

convergence of a non monotone scheme for hamilton jacobi bellman equations with discontinous initial data we prove the convergence of a non monotonous scheme for a one dimensional first order hamilton jacobi bellman equation of the form v t max f x vx 0 v 0 x v0 x the scheme is related to the hjb ultrabee scheme suggested in bokanowski and zidani j sci comput 30 1 1 33 2007 we show for general discontinuous initial data a first order convergence of the scheme in l1 norm towards the viscosity solution we also illustrate the non diffusive behavior of the scheme on several numerical examples

uniqueness properties of solutions of schrodinger equations under suitable assumptions on the potentials v and a we prove that if u c 0 1 h1 is a solution of the linear schrodinger equation i t x u vu a xu on rd 0 1 and if u 0 in x r 0 1 for some r 0 then u 0 in rd 0 1 as a consequence we obtain uniqueness properties of solutions of nonlinear schrodinger equations of the form i t x u g x t u u xu xu on rd 0 1 where g is a suitable nonlinear term the main ingredient in our proof is a carleman inequality of the form e x1 v lx2lt2 e x1 xv bx 2lt2 c e x1 i t x v bx1 2lt2 for any v c r h1 with v t 0 for t 0 1 in this inequality bx 2 and bx1 2 are banach spaces of functions on rd and e x1 is a suitable weight

size bias in line transect sampling a field test an important problem in line transect sampling is that objects or point clusters of objects of different sizes have different sighting probabilities in a recent paper drummer and mcdonald 1987 biometrics 43 13 21 develop a bivariate sighting function their function is dependent on perpendicular distance and object size one important special case is an extension of the exponential power series sighting function first proposed by pollock 1978 biometrics 34 475 478 in this note empirical evidence is given for this model based on a field test of line transect sampling theory beer cans were used to simulate point clusters of objects with cluster sizes 1 2 4 and 8 to achieve approximately equal precision of parameter estimates equal numbers of each cluster size were taken

implementation of the entropy viscosity method with the discontinuous galerkin method the notion of entropy viscosity method introduced in guermond and pasquetti 21 is extended to the discontinuous galerkin framework for scalar conservation laws and the compressible euler equations

convergence of an antidiffusion lagrange euler scheme for quasilinear equations this paper presents an explicit scheme which may be broken into several steps the first one is a transport phase for a time step with a projection on a lagrangian mesh then a new projection is performed on a fixed eulerian mesh lastly a corrected antidiffusion step of the shasta type occurs a stability condition appears to make the scheme easily computable this condition allows larger timesteps than the well known courant friedrichs lewy condition a more complex version of the scheme is proposed which works without any stability condition convergence towards the entropy solution is proved a few numerical experiments are reported showing the quality of the approximation of shocks or giving some information about the precision

on the convergence rate of operator splitting for hamilton jacobi equations with source terms we establish a rate of convergence for a semidiscrete operator splitting method applied to hamilton jacobi equations with source terms the method is based on sequentially solving a hamilton jacobi equation and an ordinary differential equation the hamilton jacobi equation is solved exactly while the ordinary differential equation is solved exactly or by an explicit euler method we prove that the l infty error associated with the operator splitting method is bounded by mathcal o delta t where delta t is the splitting or time step this error bound is an improvement over the existing mathcal o sqrt delta t bound due to souganidis nonlinear anal 9 1985 pp 217 257 in the one dimensional case we present a fully discrete splitting method based on an unconditionally stable front tracking method for homogeneous hamilton jacobi equations it is proved that this fully discrete splitting method possesses a linear convergence rate moreover numerical results are presented to illustrate the theoretical convergence results

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

high order schemes and entropy condition for nonlinear hyperbolic systems of conservation laws on presente une procedure systematique de construction des approximations explicites des systemes strictement hyperboliques des lois de conservation on etablit la convergence

l 1 minimization methods for hamilton jacobi equations the one dimensional case a new approximation technique based on l 1 minimization is introduced it is proven that the approximate solution converges to the viscosity solution in the case of one dimensional stationary hamilton jacobi equation with convex hamiltonian

a quantitative carleman estimate for second order elliptic operators we prove a carleman estimate for elliptic second order partial differential expressions with lipschitz continuous coefficients the carleman estimate is valid for any complex valued function u w2 2 with support in a punctured ball of arbitrary radius the novelty of this carleman estimate is that we establish an explicit dependence on the lipschitz and ellipticity constants the dimension of the space and the radius of the ball in particular we provide a uniform and quantitative bound on the weight function for a class of elliptic operators given explicitly in terms of ellipticity and lipschitz constant

incorporating covariates into standard line transect analyses summary an implicit assumption of standard line transect methodology is that detection probabilities depend solely on the perpendicular distance of detected objects to the transect line heterogeneity in detection probabilities is commonly minimized using stratification but this may be precluded by small sample sizes we develop a general methodology which allows the effects of multiple covariates to be directly incorporated into the estimation procedure using a conditional likelihood approach small sample size properties of estimators are examined via simulations as an example the method is applied to eastern tropical pacific dolphin sightings data

one sided stability and convergence of the nessyahu tadmor scheme non oscillatory schemes are widely used in numerical approximations of nonlinear conservation laws the nessyahu tadmor nt scheme is an example of a second order scheme that is both robust and simple in this paper we prove a new stability property of the nt scheme based on the standard minmod reconstruction in the case of a scalar strictly convex conservation law this property is similar to the one sided lipschitz condition for first order schemes using this new stability we derive the convergence of the nt scheme to the exact entropy solution without imposing any nonhomogeneous limitations on the method we also derive an error estimate for monotone initial data

size bias in line transect sampling summary line transect sampling is used to estimate the number of individuals in a study area it is usually assumed that the detections of individuals constitute independent events and that the probability of detecting an item is a function of its perpendicular distance from the transect however certain species of plants animals tend to aggregate or cluster thus possibly violating the assumption of independence also larger clusters may tend to have a higher probability of detection thus distorting the observed probability distribution of cluster sizes this can cause overestimation of the true population size a similar problem occurs if the experimenter is using line transect data to estimate the mean and total amount of some attribute of the items and this attribute influences the item s probability of detection bivariate detection functions that incorporate the cluster size or another attribute as a covariate are proposed estimators of total population size and of the moments of the additional variate are derived using the theory of weighted distributions an application of the method to a shipboard survey of minke whales is provided line transect sampling is traditionally used to estimate the total number of items in a study area it is usually assumed that the probability of detecting an item is a function of its perpendicular distance from the transect see burnham anderson and laake 1980 for a comprehensive review of line transect estimation procedures the primary application of the proposed procedure is for the case when the population consists of groups or clusters of individuals in order to estimate the density of individuals per unit area the experimenter will need an estimate of mean group size however if group size influences delectability the observed mean group size is a biased estimator of the mean also an experimenter may desire to use line transect surveys to estimate some attribute of the items other than their density number per unit area for example if the item under study is a particular shrub species the experimenter may require an estimate of the mean height of the plants in the study area if larger plants have a greater chance of being detected from the transect then these data are size biased toward larger plants thus the observed mean height may be a poor estimate of the true mean the line transect estimator of density that is based solely on the observed distances from the transect to the items may also be biased it is important to determine whether the additional variable s of interest influences an item s probability of detection if this is the case then estimates of the characteristics of this variable must be adjusted for the size biased sampling phenomenon

high resolution nonoscillatory central schemes for hamilton jacobi equations in this paper we construct second order central schemes for multidimensional hamilton jacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle thus these schemes provide the first examples of nonoscillatory second order godunov type schemes based on global projection operators numerical experiments are performed l 1 l infty errors and convergence rates are calculated for convex hamiltonians numerical evidence confirms that our central schemes converge with second order rates when measured in the l1 norm advocated in our recent paper numer math to appear the standard l infty norm however fails to detect this second order rate

the cauchy problem for quasi linear schrodinger equations with x rn t r with x1 xn and summation convention one may think of the equation in 1 1 as a non linear schrodinger equation where the operator modeling the dispersion relation is non isotropic and depends also on the unknown function its conjugate and their gradients in the space variables under appropriate assumptions on the function f the method presented here extends to fully non linear schrodinger equations of the form

entropy viscosity method for nonlinear conservation laws a new class of high order numerical methods for approximating nonlinear conservation laws is described entropy viscosity method the novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand this new approach does not use any flux or slope limiters applies to equations or systems supplemented with one or more entropy inequalities and does not depend on the mesh type and polynomial approximation various benchmark problems are solved with finite elements spectral elements and fourier series to illustrate the capability of the proposed method

existence of viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games recently m g crandall and p l lions trans amer math soc 277 1983 1 42 introduced the class of viscosity solutions of these equations and proved uniqueness within this class this paper discusses the existence of these solutions under assumptions closely related to the ones which guarantee the uniqueness

convergence of semi lagrangian approximations to convex hamilton jacobi equations under very large courant numbers we consider a class of semi lagrangian high order approximation schemes for convex hamilton jacobi equations in this framework we prove that under certain restrictions on the relationship between delta x and delta t the sequence of approximate solutions is uniformly lipschitz continuous and hence by consistency that it converges to the exact solution the argument is suitable for most reconstructions of interest including high order polynomials and eno reconstructions

uniqueness properties of solutions to schrodinger equations nx j 1 2 u x 2 x 0 harmonic functions in the unit ball x r n x 2 since harmonic functions are still real analytic in x r n x 1 in fact it is well known that if p x d is a linear elliptic differential operator with real analytic coefficients andp x d u 0 in a open set r n then u is real analytic in hence the s u c p also holds for such solutions through the work of hadamard 28 on the uniqueness of the cauchy problem which is closely related to the strong unique continuation property discussed earlier it became clear for applications in nonlinear problems that it would be desirable to establish the strong unique continuation property for operators whose coefficients are not necessarily real analytic oreven c the first results in this direction were found in the pioneering work of carleman 9 when

hardy uncertainty principle convexity and parabolic evolutions we give a new proof of the l2 version of hardy s uncertainty principle based on calculus and on its dynamical version for the heat equation the reasonings rely on new log convexity properties and the derivation of optimal gaussian decay bounds for solutions to the heat equation with gaussian decay at a future time we extend the result to heat equations with lower order variable coefficient

l 1 stability and error estimates for approximate hamilton jacobi solutions we study the l 1 stability and error estimates of general approximate solutions for the cauchy problem associated with multidimensional hamilton jacobi h j equations for strictly convex hamiltonians we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors we then demonstrate this general theory for two types of approximations approximate solutions constructed by the vanishing viscosity method and by godunov type finite difference methods if we let epsilon denote the small scale of such approximations the viscosity amplitude epsilon the spatial grad size delta x etc then our l 1 error estimates are of cal o epsilon and are sharper than the classical l infty results of order one half cal o sqrt epsilon the main building blocks of our theory are the notions of the semi concave stability condition and l 1 measure of the truncation error the whole theory could be viewed as a multidimensional extension of the lip prime stability theory for one dimensional nonlinear conservation laws developed by tadmor et al 34 24 25 in addition we construct new godunov type schemes for h j equations which consist of an exact evolution operator and a global projection operator here we restrict our attention to linear projection operators first order schemes we note however that our convergence theory applies equally well to nonlinear projections used in the context of modern high resolution conservation laws we prove semi concave stability and obtain l 1 bounds on their associated truncation errors l 1 convergence of order one then follows second order central godunov type schemes are also constructed numerical experiments are performed errors and orders are calculated to confirm our l 1 theory

regularity of solutions to the schrodinger equation soit f dans l espace de schwartz s r n et soit s t f x u x t rn exp ix exp it 2 f d x r n t r f est la transformee de fourier de f on considere u solution de l equation de schrodinger u i u t on pose s f x sup 0 t 1 s t f x x r n on introduit des espaces de sobolev h s en posant h s f s f hd s s r ou f hd s rn 1 2 s f 2 d 1 2 on etudie pour quelles valeurs de s on a b s f x 2 dx 1 2 c b f hd s f s r n pour toutes les boucles b de r n

second order numerical methods for first order hamilton jacobi equations we present practical numerical methods which produce provably second order approximations for a class of stationary first order hamilton jacobi partial differential equations using probabilistic methods we derive high order asymptotic expansions for a first order method and then use those results to design second order methods we prove second order convergence for the solution and for its gradient on a subset of the domain where the solution is smooth although we limit our attention to second order schemes in principle the techniques in this paper can be extended to arbitrarily high order methods examples illustrate the rate of convergence as well as global sharp resolution of discontinuities the hamilton jacobi equations we consider correspond to deterministic optimal control problems and our rate of convergence results are valid for the value functions and for the optimal feedback controls

unconditionally stable methods for hamilton jacobi equations we present new numerical methods for constructing approximate solutions to the cauchy problem for hamilton jacobi equations of the form ut h dxu 0 the methods are based on dimensional splitting and front tracking for solving the associated non strictly hyperbolic system of conservation laws pt dx h p 0 where p dx u in particular our methods depend heavily on a front tracking method for one dimensional scalar conservation laws with discontinuous coefficients the proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as large time step godunov type or front tracking methods we present several numerical examples illustrating the main features of the proposed methods we also compare our methods with several methods from the literature

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation

generalized monotone schemes discrete paths of extrema and discrete entropy conditions solutions of conservation laws satisfy the monotonicity property the number of local extrema is a non increasing function of time and local maximum minimum values decrease increase monotonically in time this paper investigates this property from a numerical standpoint we introduce a class of fully discrete in space and time high order accurate difference schemes called generalized monotone schemes convergence toward the entropy solution is proven via a new technique of proof assuming that the initial data has a finite number of extremum values only and the flux function is strictly convex we define discrete paths of extrema by tracking local extremum values in the approximate solution in the course of the analysis we establish the pointwise convergence of the trace of the solution along a path of extremum as a corollary we obtain a proof of convergence for a muscl type scheme that is second order accurate away from sonic points and extrema

convergence of muscl and filtered schemes for scalar conservation laws and hamilton jacobi equations this paper considers the questions of convergence of i muscl type i e second order tvd finite difference approximations towards the entropic weak solution of scalar one dimensional conservation laws with strictly convex flux and ii higher order schemes filtered to preserve an upper bound on some weak second order finite differences towards the viscosity solution of scalar multi dimensional hamilton jacobi equations with convex hamiltonians

numerical schemes for conservation laws via hamilton jacobi equations we present some difference approximation schemes which converge to the entropy solution of a scalar conservation law having a convex flux the numerical methods described here take their origin from approximation schemes for hamilton jacobi bellman equations related to optimal control problems and exhibit several interesting features the convergence result still holds for quite arbitrary time steps the main assumption for convergence can be interpreted as a discrete analogue of oleinik s entropy condition numerical diffusion around the shocks is very limited some tests are included in order to compare the performances of these methods with other classical methods godunov tvd