sobolev estimates for operators given by averages over cones we prove a result related to work by a greenleaf and g uhlmann concerning sobolev estimates for operators given by averages over cones this is done using the almost orthogonality lemma of cotlar and stein and the van der corput lemma on oscillatory integrals 0 introduction this paper is inspired by gu1 theorem 3 23 to state the result we need to introduce some notions following gu1 let m g be an n dimensional riemannian manifold we consider a c map ty m x sm 1 x ilt s tm with s tm the unitary tangent bundle on m such that for any x e iltn the map sm 1 3 s ey x s 0 e s txm is an embedding with image eyx c s tzm we define a m x sm 1 x 2 m by x r expx r x r let r x x r as the variables x r vary in the support of the distribution k defined below consider now k x r e zd m x sm 1 x r a distributiorl with very small support whith the following properties first of all we fix a number u with 0 u m then k x x r is a smooth function for r 7 0 satisfying the inequalities 0 1 a l0 x 0l k x a r i cog i r 8 l moreover if u m for any pair of small positive nurnbers o e ry 0 1 b a az g x x u r drl ca

local tomography with nonsmooth attenuation ii we continue investigation of local tomography with nonsmooth attenuation which was proposed earlier by the author our main result is a new and simple characterization of the wave front of the local tomography function tilde f lambda left phi right which holds under fairly general conditions on the density function f and the attenuation coefficient

microlocal analysis of some isospectral deformations we study the microlocal structure of the examples of isospectral deformations of riemannian manifolds given by d deturck and c gordon in det gi the schwartz kernel of the intertwining operators considered by them may be written as an oscillatory integral with a singular phase function and product type amplitude in certain instances we identify them as belonging to the space of fourier integral operators associated with various pairwise intersecting lagrangians after formulating a class of operators incorporating the most relevant features of the operators above we establish a composition calculus for this class and show that is not necessary to introduce new lagrangians in the composition

inverse backscattering in even dimensions this paper investigates the inverse backscattering problem for a potential in even dimensions our results are obtained through the study of a modified backscattering map we use the wave equation approach here and extend melrose and uhlmann s method 6 to even dimensions

models of degenerate fourier integral operators and radon transforms fourier integral operators whose lagrangians project with singularities arise frequently in many areas of analysis and geometry 3 5 9 10 18 20 however there are as yet few analytic tools available for their study and even their simplest regularity properties with respect to the singularities of the lagrangian are still obscure 4 5 13 16 in this paper we study a class of oscillatory integrals ta and fourier integral operators r which can be expected to model the higher order singularities of the lagrangian they have homogeneous polynomial phases in two variables of order n and indeed the case n 3 is the model for lagrangians which project with whitney folds 3 10 11 the main difficulty which sets the higher order degeneracy cases n 4 apart from the the lower ones n 2 3 is that the critical varieties which arise there are usually not smooth manifolds a systematic study of these oscillatory integrals was begun in 14 the main idea in that work was to treat the critical varieties as smooth manifolds away from a lower dimensional subvariety and to keep track of the distance to this lower dimensional subvariety to achieve this we introduced a method of stationary phase which exhibited clearly the dependence on the distance between critical points the method gave sharp bounds on the size of the kernel k x y of tat but no information on its phase and the resulting cancellations it does not seem possible to refine this approach to the phase level and this suggests looking instead for decompositions of the operators ta which can incorporate indirectly the required cancellations the main goal of this paper is to introduce such decompositions these decompositions which we will try to describe momentarily reflect the singular nature of the critical points and are powerful enough to yield sharp bounds for the above oscillatory integrals and radon transforms although the models under consideration are very

nonbinary quantum codes we present several results on quantum codes over general alphabets that is in which the fundamental units may have more than two states in particular we consider codes derived from finite symplectic geometry assumed to have additional global symmetries from this standpoint the analogs of calderbank shor steane codes and of gf 4 linear codes turn out to be special cases of the same construction this allows us to construct families of quantum codes from certain codes over number fields in particular we get analogs of quadratic residue codes including a single error correcting code encoding one letter in five for any alphabet size we also consider the problem of fault tolerant computation through such codes generalizing ideas of gottesman see phys rev a vol 57 no 1 p127 37 1998

reconstruction of singularities from full scattering data by new estimates of bilinear fourier multipliers we prove that the singularities of a complex valued potential q in the schrodinger hamiltonian q can be reconstructed from the linear born approximation for full scattering data by averaging in the extra variables we prove that with this procedure the accuracy in the reconstruction improves the previously known accuracy obtained from fixed angle or backscattering data in particular for q in w alpha 2 for 0 in 2d we recover the main singularity of q with an accuracy of one derivative in 3d the accuracy is epsilon 1 2 increasing with this gives a mathematical basis for diffraction tomography the proof is based on some new estimates for multidimensional bilinear fourier multipliers of independent interest

inverse problems for lorentzian manifolds and non linear hyperbolic equations we study two inverse problems on a globally hyperbolic lorentzian manifold m g the problems are 1 passive observations in spacetime consider observations in an open set v subset m the light observation set corresponding to a point source at q in m is the intersection of v and the light cone emanating from the point q let w subset m be an unknown open relatively compact set we show that under natural causality conditions the family of light observation sets corresponding to point sources at points q in w determine uniquely the conformal type of w 2 active measurements in spacetime we develop a new method for inverse problems for non linear hyperbolic equations that utilizes the non linearity as a tool this enables us to solve inverse problems for non linear equations for which the corresponding problems for linear equations are still unsolved to illustrate this method we solve an inverse problem for semilinear wave equations with quadratic non linearities we assume that we are given the neighborhood v of the time like path mu and the source to solution operator that maps the source supported on v to the restriction of the solution of the wave equation to v when m is 4 dimensional we show that these data determine the topological differentiable and conformal structures of the spacetime in the maximal set where waves can propagate from mu and return back to mu

the inverse scattering problem on a fixed energy level for the two dimensional schrodinger operator this paper contains a solution with complete proofs of the main problems of the inverse scattering at fixed energy e for the schrodinger two dimensional operator with decreasing coefficients uniqueness of the reconstruction a reconstruction procedure characterization of the scattering data most completed results are obtained under the condition of smallness of coefficients in comparison with the number e this paper develops and generalizes several preceded papers

permittivity recovery from multiple characteristic electric waves we use the calculus of fourier integral operators with singular symbols to reconstruct electrical permittivity singularities from remote measurements of scattered electric waves the reconstruction is microlocal and is uniformly valid in the neighborhood of conically refracted waves

spectral and computational properties of band symmetric toeplitz matrices abstract we are investigating spectral properties of band symmetric toeplitz matrices bst matrices by giving a suitable representation of a bst matrix we achieve separation results and multiplicity conditions for the eigenvalues of a 7 r 5 diagonal bst matrix and also structural properties of the eigenvectors we give eigenvalue bounds for a 2k 1 diagonal bst matrix and also necessary and sufficient conditions for positive definiteness which are easy to check the same conditions apply in the case of block bst matrices either with full blocks or with bst blocks we exhibit fast computational methods for the evaluation of the determinant and the characteristic polynomial of a bst matrix either for sequential or for parallel computations two algorithms based on the bisection technique and newton s method are shown to be very fast for computing the eigenvalues of a 7 or 5 diagonal bst matrix

scattering by a potential using hyperbolic methods etude du probleme de diffusion inverse developpement d un algorithme correspondant pour deux variables

inverse backscattering problem for the acoustic equation in even dimensions abstract we show that the sound speed c x of the acoustic wave equation in any even dimension can be uniquely determined by the backscattering data provided that it is close to a constant in the three dimensional case p stefanov and g uhlmann siam j math anal 28 1997 1191 1204 have proved a similar result their method takes advantage of the inversion formula for the radon transform in odd dimensions being a local operator this is not true in even dimensions moreover the odd dimensional lax and phillips modified radon transform fails to work in even dimensions in this paper we overcome these difficulties and prove an even dimensional version of stefanov and uhlmann s result

sobolev estimates for fractional and singular radon transforms abstract we prove sobolev inequalities for singular and fractional radon transforms which are defined as in a paper of phong and stein under certain hypothesis on the corresponding lagrangian n c which does not necessarily have to be a canonical graph in the proof we use oscillatory integrals the cotlar stein almost orthogonality theorem a sort of littlewood paley decomposition for a certain operator some basic facts about fourier integral operators and pseudodifferential operators the main ideas come from papers by phong and stein acta math 157 1986 99 157 and sogge and stein j analyse math 54 1990 165 188

scattering in the presence of fold caustics we consider the linearized scattering operator cal f of acoustics which maps singularities in the sound speed in the subsurface of the earth to singularities in the measured pressure at the earth s surface the source of the acoustic signal is concentrated at a point on the surface application ofthe adjoint scattering operator cal f to the measured pressure is a standard technique of locating sound speed singularities the resulting image is equivalent to application of cal f cal f to the sound speed singularities provided caustics do not occur cal f cal f is a pseudodifferential operator meaning sound speed singularities do not move when it is applied and one obtains a reliable image however if caustics are present cal f cal f is not well understood our main result is that for the case of a simple fold caustic the most common kind cal f cal f is a singular fourier integral operator meaning that some sound speed singularities will be propagated t

averages on the unitary group and applications to the problem of disordered conductors in a recent approach to the study of disordered conductors a maximum entropy criterion is used to perform statistical calculations directly on the transfer matrix for the full physical system quantities of great importance in that approach are averages of products of the u and u matrix elements u being a unitary matrix evaluated with the invariant measure of the unitary group the authors evaluate those averages that contain up to four u and four u matrix elements since they are the ones needed in the calculation of the average and covariance of transmission and reflection coefficients the evaluation involves no direct integration at all but makes use solely of the invariance of the measure the fact that the matrix elements ua alpha are commuting c numbers and the unitarity of u the physical consequences for the average and covariance of transmission and reflection coefficients are briefly discussed

rigidity of broken geodesic flow and inverse problems consider broken geodesics alpha 0 l on a compact riemannian manifold m g with boundary of dimension n geq 3 the broken geodesics are unions of two geodesics with the property that they have a common end point assume that for every broken geodesic alpha 0 l starting at and ending to the boundary partial m we know the starting point and direction alpha 0 alpha 0 the end point and direction alpha l alpha l and the length l we show that this data determines uniquely up to an isometry the manifold m g this result has applications in inverse problems on very heterogeneous media for situations where there are many scattering points in the medium and arises in several applications including geophysics and medical imaging as an example we consider the inverse problem for the radiative transfer equation or the linear transport equation with a nonconstant wave speed assuming that the scattering kernel is everywhere positive we show that the boundary measurements determine the wave speed inside the domain up to an isometry

cone beam local tomography methods systems and processes for providing efficient image reconstruction using local cone beam tomography which provide a reduced level of artifacts without suppressing the strength of the useful features and in a dynamic case provide reconstruction of objects that are undergoing a change during the scan an embodiment provides a method of reconstructing an image from cone beam data provided by at least one detector the method includes collecting cb projection data of an object storing the cb projection data in a memory and reconstructing the image from the local cb projection data in the reconstructing step a combination of derivatives of the cb projection data that will result in suppressing the artifacts are found the combination of derivatives includes collecting cone beam data that represents a collection of integrals that represent the object

inverse obstacle scattering using reduced data the classical result of schiffer in acoustic scattering is that a knowledge of the far field pattern for all observation directions and all incident directions at a fixed wave number k uniquely determines the sound soft scattering obstacle d this is widely believed to be far greater information than is required for uniqueness and more recent results have been obtained that considerably weaken the amount of data needed provided certain restrictions are placed on the scatterer most of this work has concentrated on reducing the number of incident waves required for a unique determination this paper takes another approach and seeks to determine sufficient information to recover the obstacle from measurements of the far field at isolated points our approach will be constructive and some numerical reconstructions will be presented

a kronecker product variant of the facr method for solving the generalized poisson equation we present a fast direct method for the solution of a linear system mx y where m is a block tridiagonal toeplitzmatrix with a on the diagonal and t on the two subdiagonals a and t commute such matrices are obtained from a finite difference approximation to poisson s equation with nonconstant coefficients in one direction among others the new method is called kpcr l method and begins with l steps of cyclic reduction after which the remaining system is solved by a kronecker product method for an appropriate choice of l the asymptotic operation count for an n n grid is o n2 log2 log2 n which is faster than either cyclic reduction or the kronecker product method itself the algorithm is similar to and has the same complexity as the facr l algorithm which is a combination of cyclic reduction and fourier analysis or matrix decomposition however the facr l algorithm only reaches this complexity if a and t can be diagonalized by a fast transformation where the new method is fast for every banded a and t moreover the kpcr l method can be easily generalized to the case where a and t do not commute

boundedness of singular radon transforms on l p spaces under a finite type condition in this paper a new method is presented for proving boundedness of singular radon transforms on l p for 1 p two closely related results generalizing the corresponding theorem of christ nagel stein and wainger are proven 1 introduction in this paper a new method is presented for proving boundedness of singular radon transforms on l p for 1 p two closely related results generalizing the corresponding theorem of christ nagel stein and wainger in 3 are proved singular radon transforms are a type of singular integral operator a prototype for which is the hilbert transform along curves let 1 1 r n be a curve with 0 0 and let f be a schwarz function on r n 1

recovery of singularities for formally determined inverse problems in this paper we considered several formally determined problems in two dimensions there are no global identifiability results for these problems however we can recover an important feature of these functions namely their singularities more precisely we prove that one can determine the location and strength of singularities of anl compactly supported potential by knowing the associated scattering amplitude at a fixed energy also we prove that one can determine the location and strength of the singularities of the sound speed of a medium by making measurements just on the boundary of the medium

singular and maximal radon transforms analysis and geometry part 2 geometric theory 8 curvature introduction 8 1 three notions of curvature 8 2 theorems 8 3 examples 9 curvature some details 9 1 the exponential representation 9 2 diffeomorphism invariance 9 3 curvature condition cy 9 4 two lemmas 9 5 double fibration formulation 10 equivalence of curvature conditions 10 1 invariant submanifolds and deficient lie algebras 10 2 vanishing jacobians 10 3 construction of invariant submanifolds

recovering singularities of a potential from singularities of scattering data in this paper we show that the leading singularities of certain potentials can be determined from the leading singularities of the backscattering as well as other determined sets of scattering data the potentials in question are conormal with respect to smooth surfaces of arbitrary dimension the restrictions on their orders allow for unbounded potentials in all dimension greater than or equal to three

singular radon transforms and oscillatory integrals on considere une reduction des transformees de radon singulieres a certaines integrales oscillatoires

an fio calculus for marine seismic imaging ii sobolev estimates we establish sharp l 2 sobolev estimates for classes of pseudodifferential operators with singular symbols guillemin and uhlmann duke math j 48 251 267 1981 melrose and uhlmann commun pure appl math 32 483 519 1979 whose non pseudodifferential fourier integral operator parts exhibit two sided fold singularities the operators considered include both singular integral operators along curves in mathbb r 2 with simple inflection points and normal operators arising in linearized seismic imaging in the presence of fold caustics felea comm pde 30 1717 1740 2005 felea and greenleaf comm pde 33 45 77 2008 nolan siam j appl math 61 659 672 2000

nonlinear interaction of waves in elastodynamics and an inverse problem we consider nonlinear elastic wave equations generalizing gol dberg s five constants model we analyze the nonlinear interaction of two distorted plane waves and characterize the possible nonlinear responses using the boundary measurements of the nonlinear responses we solve the inverse problem of determining elastic parameters from the displacement to traction map

inverse scattering problem for a two dimensional random potential we study an inverse problem for the two dimensional random schrodinger equation q k 2 u 0 the potential q x is assumed to be a gaussian random function whose covariance operator is a classical pseudodifferential operator we show that the backscattered field obtained from a single realization of the random potential q determines uniquely the principal symbol of the covariance operator of q the analysis is carried out by combining harmonic and microlocal analysis with stochastic methods

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

inverse backscattering problem for the generalized nonlinear schrodinger operator in two dimensions the inverse born approximation is studied for the generalized nonlinear two dimensional schrodinger equation where the real valued unknown potentials l belong to for some 2 p and their fourier transforms belong to the space for some 1 s 2 we prove that the leading order singularities of the sum of the unknown potentials can be recovered from the inverse born approximation we also prove that the approximation is equal to the true function up to sum of two functions one of them from the sobolev space ht while the other being a continuous function

hilbert integrals singular integrals and radon transforms i on introduit deux nouvelles classes d operateurs on etudie leur relation et on montre comment on peut les appliquer a l etude des problemes aux valeurs limites

algebras of semiclassical pseudodifferential operators associated with zoll type domains in cotangent bundles abstract we consider domains in cotangent bundles with the property that the null foliation of their boundary is fibrating and the leaves satisfy a bohr sommerfeld condition for example the unit disk bundle of a zoll metric given such a domain we construct an algebra of associated semiclassical pseudodifferential operators with singular symbols the schwartz kernels of the operators have frequency set contained in the union of the diagonal and the flow out of the null foliation of the boundary of the domain we develop a symbolic calculus prove the existence of projectors under a mild additional assumption whose range can be thought of as quantizing the domain give a symbolic proof of a szego limit theorem and study associated propagators

contribution of periodic diffractive geodesics abstract on an euclidean surface with conical singularities the wave trace is expected to be singular at l where l is the length of some diffractive periodic geodesic in this paper we compute the leading term of the singularity brought to the trace by a regular isolated diffractive geodesic and by a regular family of periodic non diffractive geodesic these results can be applied to polygons

distribution of resonances for open quantum maps we analyze a simple model of quantum chaotic scattering system namely the quantized open baker s map this model provides a numerical confirmation of the fractal weyl law for the semiclassical density of quantum resonances the fractal exponent is related to the dimension of the classical repeller we also consider a variant of this model for which the full resonance spectrum can be rigorously computed and satisfies the fractal weyl law for that model we also compute the shot noise of the conductance through the system and obtain a value close to the prediction of random matrix theory

integration with respect to the haar measure on unitary orthogonal and symplectic group we revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the haar measure on the unitary group u d the previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d also we consider the integrals of polynomial functions on the orthogonal group o d and the symplectic group sp d we obtain an exact character expansion and the asymptotic behavior for large d thus we can show the asymptotic freeness of haar distributed orthogonal and symplectic random matrices as well as the convergence of integrals of the itzykson zuber type

oscillatory singular integrals and harmonic analysis on nilpotent groups abstract several related classes of operators on nilpotent lie groups are considered these operators involve the following features i oscillatory factors that are exponentials of imaginary polynomials ii convolutions with singular kernels supported on lower dimensional submanifolds iii validity in the general context not requiring the existence of dilations that are automorphisms

concerning nikodym type sets in 3 dimensional curved spaces in this paper we shall give some natural partial extensions to the curved space setting of results of bourgain 1 and wolff 21 concerning lower bounds for the dimension of compliments of nikodym sets in euclidean space recall that a classical nikodym set is a subset of 1 1 x 1 1 of lebesgue measure one which has the property that for each x e f there is a line yx so that x nf x because of this the relative compliment q 1 1 x 1 1 f must be a set of measure zero with the property that if q is the set of points x for which there is a line segment y through x with iq n axf a then iq ji 0 for every 0 7 3 if iq i 0 here as before tyx denotes a line segment through x of length ae 0 this lower bound was later improved by wolff 21 to dim q 5 2 if iq i o in both works the lower bounds on the dimension were obtained for somewhat more general sets lower bounds for analogous sets in higher dimensions were also obtained in 1 and 21 the strongest to date are those of wolff 21 showing that if q c ri and iq i 0 for some ae 0 then dim q n 2 2 let us now consider extensions of this result to the curved 3 dimensional setting to this end we shall let m3 g denote a paracompact 3 dimensional riemannian manifold with metric ds2 zgjk x dxjdxk given x e m3 we let ya denote the set of all geodesics containing x with arclength ae that is i y a abusing the classical terminology somewhat we now define nikodym type subsets of m3

semiclassical spectra of gauge fields abstract we study the asymptotic behavior of the eigenvalues of the schrodinger operator with a vector potential on a compact manifold as planck s constant tends to zero we obtain estimates in terms of periodic trajectories of wong s flow which are uniform in the charge parameter

recovery of material parameters in transversely isotropic media in this paper we show that in anisotropic elasticity in the particular case of transversely isotropic media under appropriate convexity conditions knowledge of the qsh wave travel times determines the tilt of the axis of isotropy as well as some of the elastic material parameters and the knowledge of qp and qsv travel times conditionally determines a subset of the remaining parameters in the sense that if some of the remaining parameters are known the rest are determined or if the remaining parameters satisfy a suitable relation they are all determined under certain non degeneracy conditions furthermore we give a precise description of the additional issues which are a subject of ongoing work that need to be resolved for a full treatment

nonlocal inversion formulas for the x ray transform etude de la reconstruction d une fonction f x sur r n a partir de ses integrales sur des plans k dimensionnels cas de k 1

a special class of structured matrices constructed with the kronecker product and its use for difference equations abstract starting from a sequence of specific orthogonal matrices and using the matricial kronecker product we build classes of matrices sk k 1 2 which are shown to possess a certain affiliation with toeplitz and hankel matrices being completely determined by the first row elements their eigenvalues as well as an inversion formula are easily derived therefrom s matrices as they are termed in general may arise through the application of finite difference methods to boundary value problems

near peak scattering and the corrected kirchhoff approximation for a convex obstacle abstract the uniform asympotic behavior of the scattering amplitude near the forward peak in the case of classical scattering of waves by a convex obstacle is derived a microlocal model is obtained for the scattering operator this is achieved by use of a parametrix for diffractive boundary problems and by a new study of a class of fourier integral operators those with folding canonical relations a crucial ingredient consists of putting a fourier integral operator with folding canonical relation into a normal form the analysis also gives the asymptotic behavior of the normal derivative of the scattered wave on a neighbourhood of the shadow boundary thus providing a corrected version of the kirchhoff approximation

singular integrals related to the radon transform and boundary value problems abstract two classes of integral operators are introduced that combine features of calderon zygmund singular integrals the hilbert integral and the radon transform examples and applications arising from integral geometry and several complex variables are discussed

on some properties of orthogonal weingarten functions we give a fourier type formula for computing the orthogonal weingarten formula the weingarten calculus was introduced as a systematic method to compute integrals of polynomials with respect to haar measure over classical groups although a fourier type formula was known in the unitary case the orthogonal counterpart was not known it relies on the jack polynomial generalization of both schur and zonal polynomials this formula substantially reduces the complexity involved in the computation of weingarten formulas we also describe a few more new properties of the weingarten formula state a conjecture and give a table of values

characterization and source receiver continuation of seismic reflection data in reflection seismology one places sources and receivers on the earth s surface the source generates elastic waves in the subsurface that are reflected where the medium properties stiffness and density vary discontinuously in the field often there are obstructions to collect seismic data for all source receiver pairs desirable or needed for data processing and application of inverse scattering methods typically data are measured on the earth s surface we employ the term data continuation to describe the act of computing data that have not been collected in the field seismic data are commonly modeled by a scattering operator developed in a high frequency single scattering approximation we initially focus on the determination of the range of the forward scattering operator that models the singular part of the data in the mentioned approximation this encompasses the analysis of the properties of and the construction of a minimal elliptic projector that projects a space of distributions on the data acquisition manifold to the range of the mentioned scattering operator this projector can be directly used for the purpose of seismic data continuation and is derived from the global parametrix of a homogeneous pseudodifferential equation the solution of which coincides with the range of the scattering operator we illustrate the data continuation by a numerical example

inverse backscattering in two dimensions this article extends the authors previous results commun math phys 124 169 215 1989 to inverse scattering in two space dimensions the new problem in two dimensions is the behavior of the backscattering amplitude near zero energy generically this has the form a xi mathord left vphantom xi left xi right right kern nulldelimiterspace left xi right xi mathord left vphantom xi left xi right left xi right right kern nulldelimiterspace left xi right left xi right 2 pi 2 pi beta ln left xi right 1 b xi whereb 0 0 andb is holder continuous in order to work in weighted holder spaces as before the constant and the functionb must now be interpreted as coordinates on the space of backscattering data in this setting the mapping to backscattering data is again a local diffeomorphism at a dense open set in the real valued potentials

bounds for singular fractional integrals and related fourier integral operators abstract we prove sharp lp lq endpoint bounds for singular fractional integral operators and related fourier integral operators under the nonvanishing rotational curvature hypothesis

global uniqueness for a two dimensional inverse boundary value problem we show that the coefficient y x of the elliptic equation vie qyvu 0 in a two dimensional domain is uniquely determined by the corresponding dirichlet to neumann map on the boundary and give a reconstruction pro

a class of singular fourier integral operators in synthetic aperture radar imaging in this article we analyze the microlocal properties of the linearized forward scattering operator f and the normal operator f f where f is the l 2 adjoint of f which arises in synthetic aperture radar imaging for the common midpoint acquisition geometry when f is applied to the scattered data artifacts appear we show that f f can be decomposed as a sum of four operators each belonging to a class of distributions associated to two cleanly intersecting lagrangians i p l 0 1 thereby explaining the latter artifacts

three dimensional scattering and the heisenberg inequality abstract we show that it is possible to concentrate a radially symmetric potential around the origin in three dimensional space without having its backscattering amplitude spread out as a function of the wave number this result is impossible within the born approximation in fact this result is a three dimensional version of some recent work showing that the nonlinear scattering transform allows for better localization properties than its linearized version namely the fourier transform

recovery of the singularities of a potential from backscattering data in general dimension abstract we prove that in dimension n 2 the main singularities of a complex potential q having a certain a priori regularity are contained in the born approximation q b constructed from backscattering data this is archived using a new explicit formula for the multiple dispersion operators in the fourier transform side we also show that q q b can be up to one derivative more regular than q in the sobolev scale on the other hand we construct counterexamples showing that in general it is not possible to have more than one derivative gain sometimes even strictly less depending on the a priori regularity of q

completeness of products of solutions and some inverse problems for pde abstract in this paper we give conditions which guarantee that products of solutions of partial differential equations pu au 0 are complete in l 2 here p is a linear partial differential operator with constant coefficients and a is a function in l we check these conditions for elliptic parabolic and hyperbolic equations of second order and give applications to inverse problems for related equations where a is to be found

singular solutions of elliptic equations and the determination of conductivity by boundary measurements abstract we improve some results on uniqueness and stability in the determination of the coefficient a in the equation i div a grad u 0 in when all possible pairs of dirichlet and neumann data on are known we also treat cases of anisotropic equations our method relies on the construction of solutions to i having an isolated singularity with prescribed asymptotic behaviour

microlocal analysis of scattering data for nested conormal potentials consider the potential scattering problem for the wave equation begin array rcl partial t 2 bigtriangleup q u 0 text in mathbb r n times mathbb r u delta t x cdot omega text for t rho end array 1 where q is a compactly supported potential and omega in s n 1 is fixed here is the value such that supp q subset vert x vert leq rho

a parametrix for the fundamental solution of the klein gordon equation on asymptotically de sitter spaces abstract in this paper we construct a parametrix for the forward fundamental solution of the wave and klein gordon equations on asymptotically de sitter spaces without caustics we use this parametrix to obtain asymptotic expansions for solutions of u f and to obtain a uniform l p estimate for a family of bump functions traveling to infinity

estimates for singular convolution operators on the heisenberg group soit h n le groupe de heisenberg l operateur a co c h n donne par af f k s etend a un operateur borne de l p dans l p pour 1 p les operateurs ae co c donnes par aef f ke s etendent aux operateurs bornes sur l p pour 1 p avec des normes independantes de e

the inverse backscattering problem in three dimensions this article is a study of the mapping from a potentialq x onr3 to the backscattering amplitude associated with the hamiltonian q x the backscattering amplitude is the restriction of the scattering amplitudea k k es2 s2 toa k we show that in suitable complex banach spaces the map fromq x toa x x x x x is usually a local diffeomorphism hence in contrast to the overdetermined problem of recoveringq from the full scattering amplitude the inverse backscattering problem is well posed