uniform homeomorphisms of banach spaces and asymptotic structure we give a general result on the behavior of spreading models in banach spaces which coarse lipschitz embed into asymptotically uniformly convex spaces we use this result to study the uniqueness of the uniform structure in p sums of finite dimensional spaces for 1 p in particular we give some new examples of spaces with unique uniform structure

trees and branches in banach spaces an infinite dimensional notion of asymptotic structure is considered this notion is developed in terms of trees and branches on banach spaces every countably infinite countably branching tree t of a certain type on a space x is presumed to have a branch with some property it is shown that then x can be embedded into a space with an fdd e i so that all normalized sequences in x which are almost a skipped blocking of e i have that property as an application of our work we prove that if x is a separable reflexive banach space and for some 1 0 there exists a subspace of x having finite codimension which c 2 e embeds into the l p sum of finite dimensional spaces

on the geometry of the countably branching diamond graphs abstract in this article the bi lipschitz embeddability of the sequence of countably branching diamond graphs d k k n is investigated in particular it is shown that for every e 0 and k n d k embeds bi lipschiztly with distortion at most 6 1 e into any reflexive banach space with an unconditional asymptotic structure that does not admit an equivalent asymptotically uniformly convex norm on the other hand it is shown that the sequence d k k n does not admit an equi bi lipschitz embedding into any banach space that has an equivalent asymptotically midpoint uniformly convex norm combining these two results one obtains a metric characterization in terms of graph preclusion of the class of asymptotically uniformly convexifiable spaces within the class of reflexive banach spaces with an unconditional asymptotic structure applications to bi lipschitz embeddability into l p spaces and to some problems in renorming theory are also discussed

application of the θ method to a telegraphic model of fluid flow in a dual porosity medium abstract this work focuses mainly on the study of numerical solutions which are obtained using the method of a generalized warren and root model that includes a second order wave like equation in its formulation the solutions approximately describe the single phase hydraulic head in fractures by considering the finite velocity of propagation by means of a cattaneo like equation the corresponding discretized model is obtained by utilizing a non uniform grid and a non uniform time step a simple relationship is proposed to give the time step distribution convergence is analyzed by comparing results from explicit fully implicit and crank nicolson schemes with exact solutions a telegraphic model of fluid flow in a single porosity reservoir with relaxation dynamics the warren and root model and our studied model which is solved with the inverse laplace transform we find that the flux and the hydraulic head have spurious oscillations that most often appear in small time solutions but are attenuated as the solution time progresses furthermore we show that the finite difference method is unable to reproduce the exact flux at time zero obtaining results for oilfield production times which are in the order of months in real units is only feasible using parallel implicit schemes in addition we propose simple parallel algorithms for the memory flux and for the explicit scheme

on the impossibility of dimension reduction in l 1 the johnson lindenstrauss lemma shows that any n points in euclidean space i e n with distances measured under the e 2 norm may be mapped down to o log n e2 dimensions such that no pairwise distance is distorted by more than a 1 p e factor determining whether such dimension reduction is possible in e 1 has been an intriguing open question we show strong lower bounds for general dimension reduction in e 1 we give an explicit family of n points in e 1 such that any embedding with constant distortion d requires n 1 d2 dimensions this proves that there is no analog of the johnson lindenstrauss lemma for e 1 in fact embedding with any constant distortion requires n 1 dimensions further embedding the points into e 1 with 1pe distortion requires nh o e log 1 e dimensions our proof establishes this lower bound for shortest path metrics of series parallel graphs we make extensive use of linear programming and duality in devising our bounds we expect that the tools and techniques we develop will be useful for future investigations of embeddings into e 1

high accuracy wave simulation revised derivation numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation the nearly analytic integration discrete naid method for solving the two dimensional acoustic wave equation has been fully mathematically revised analyzed and tested the naid method is an alternative numerical modeling method for generating synthetic seismograms the acoustic wave equation is first transformed into a system of first order ordinary differential equations odes with respect to time variable t and then directly integrated at a small time interval of tn tn 1 to obtain semi discrete ordinary differential equations the integral kernel is expanded into a truncated taylor series to which the integration operator is explicitly applied the high order temporal derivatives involved in the integral kernel are replaced by high order spatial derivatives which then are approximately calculated as a weighted linear combination of the displacement the particle velocity and their spatial gradients in this article we investigate the theoretical properties of the revised naid method including the discrete error and the stability criteria numerical results for constant and layered velocity models show that comparing to the lax wendroff correction lwc scheme and the staggered grid finite difference method the naid method can effectively suppress the numerical dispersion and source noises caused by the discretization of the acoustic wave equation with too coarse spatial grids or when models have strong velocity contrasts between adjacent layers the proposed naid method has been applied in computing the acoustic wavefields for two heterogeneous models the corner edge model and the marmousi model promising numerical results illustrate that the naid method provides an encouraging tool for large scale and complex wave simulation and inversion problems based on the acoustic equation

equivalent accuracy at a fraction of the cost overcoming temporal dispersion abstractnumerical dispersion in finite difference fd modeling produces coherent artifacts severely constraining the resolution of advanced imaging and inversion techniques conventionally numerical dispersion is reduced by increasing the order of accuracy of the fd operators and we resign ourselves to paying the high computational cost that is incurred assuming no spatial dispersion we have found that fd time dispersion is independent of the medium velocity and the spatial grid for propagation and only depends on the time stepping scheme and the propagation time based on this observation we have devised postpropagation filters to collapse the time dispersion effect of fd modeling our dispersion correction filters are designed by comparing the input waveform with dispersive waveforms obtained by 1d forward modeling these filters are then applied on multidimensional shot records to eliminate the time dispersion by two schemes 1 stationary filtering plus interpolation and 2 nonstationary fil

operators on lp 2 p which factor through xp abstract let t be a bounded linear operator on l p 2 p we show that t factors through x p if and only if there is a change of density of l p so that the induced operator t satisfies an upper l p l 2 tree estimate an operator which factors through x p but fails the upper l p l 2 tree estimate is constructed it is also shown that if t is a closed z p strictly singular operator on l p 2 p then t factors through x p however there exists a z p strictly singular operator which does not factor through x p

removing false images in reverse time migration the concept of de primary abstractprimary reflections like multiples can generate false images when reverse time migration rtm algorithms are used the false images are formed by the zero lag correlation of the source wavefields and primary reflections which are propagated by the migration algorithm along nonphysical paths these paths are generated by strong velocity gradients or reflection interfaces when the two way wave equation is used conceptually this type of artifact can be removed by separating up and downgoing waves but such separation may be impractical because it often requires storing the entire wavefields at all time steps we have developed a de primary rtm method in which such separation can be accomplished without saving the wavefields the computational cost of the proposed method was only approximately 33 higher than that of conventional rtm algorithms using field and synthetic data sets we have demonstrated the existence of this endemic rtm problem and verified the effectiveness of the de primary rt

an optimized wave equation for seismic modeling and reverse time migration the acoustic wave equation has been widely used for the modeling and reverse time migration of seismic data numerical implementation of this equation via finite difference techniques has established itself as a valuable approach and has long been a favored choice in the industry to ensure quality results accurate approximations are required for spatial and time derivatives traditionally they are achieved numerically by using either relatively very fine computation grids or very long finite difference operators otherwise the numerical error known as numerical dispersion is present in the data and contaminates the signals however either approach will result in a considerable increase in the computational cost a simple and computationally low cost modification to the standard acoustic wave equation is presented to suppress numerical dispersion this dispersion attenuator is one analogy of the antialiasing operator widely applied in kirchhoff migration when the new wave equation is solved numerically using finite difference schemes numerical dispersion in the original wave equation is attenuated significantly leading to a much more accurate finite difference scheme with little additional computational cost numerical tests on both synthetic and field data sets in both two and three dimensions demonstrate that the optimized wave equation dramatically improves the image quality by successfully attenuating dispersive noise the adaptive application of this new wave equation only increases the computational cost slightly

a c0 saturated banach space with tight structure it is shown that variants of the hi methods could yield objects closely connected to the classical banach spaces thus we present a new c0 saturated space denoted as x0 with rather tight structure the space x0 is not embedded into a space with an unconditional basis and its complemented subspaces have the following structure everyone is either of type i namely contains an isomorph of x0 itself or else is isomorphic to a subspace of c0 type ii furthermore for any analytic decomposition of x0 into two subspaces one is of type i and the other is of type ii the operators of x0 share common features with those of hi spaces

displacement rank and quasitriangular decomposition for r toeplitz matrices abstract an r toeplitz r t matrix can be regarded as a block toeplitz matrix of order mr from which an l shaped border of k r rows and columns is removed the concept of displacement rank is extended to r t matrices and it is shown that both an r t matrix and its inverse can be expressed as the sum of at most two products of r t matrices this result reduces to a standard lu decomposition in the case when an r t matrix becomes an arbitrary matrix an explicit expression for the inverse is given which contains three triangular matrices and one quasitriangular matrix furthermore the inverse of an r t matrix is completely determined by the elements in its first and last r rows and columns the results presented generalize known properties of toeplitz and block toeplitz matrices

a characterization of subspaces and quotients of reflexive banach spaces with unconditional bases we prove that the dual or any quotient of a separable reflexive banach space with the unconditional tree property has the unconditional tree property this is used to prove that a separable reflexive banach space with the unconditional tree property embeds into a reflexive banach space with an unconditional basis this solves several long standing open problems in particular it yields that a quotient of a reflexive banach space with an unconditional finite dimensional decomposition embeds into a reflexive banach space with an unconditional basis

small subspaces of l_p we prove that if x is a subspace of lp 2 p 1 then either x embeds isomorphically into p 2 or x contains a subspace y which is isomorphic to p 2 we also give an intrinsic characterization of when x embeds into p 2 in terms of weakly null trees in x or equivalently in terms of the innite asymptotic game played in x this solves problems concerning small subspaces of lp originating in the 1970 s the techniques used were developed over several decades the most recent being that of weakly null trees developed in the 2000 s

the uniform structure of banach spaces we explore the existence of uniformly continuous sections for quotient maps using this approach we are able to give a number of new examples in the theory of the uniform structure of banach spaces we show for example that there are two non isomorphic separable mathcal l 1 subspaces of l1 which are uniformly homeomorphic we also prove the existence of two coarsely homeomorphic banach spaces i e with lipschitz isomorphic nets which are not uniformly homeomorphic answering a question of johnson lindenstrauss and schechtman we construct a closed subspace of l1 whose unit ball is not an absolute uniform retract answering a question of the author

a new approach to low distortion embeddings of finite metric spaces into non superreflexive banach spaces abstract the main goal of this paper is to develop a new embedding method which we use to show that some finite metric spaces admit low distortion embeddings into all non superreflexive spaces this method is based on the theory of equal signs additive sequences developed by brunel and sucheston 1975 1976 we also show that some of the low distortion embeddability results obtained using this method cannot be obtained using the method based on the factorization between the summing basis and the unit vector basis of l 1 which was used by bourgain 1986 and johnson and schechtman 2009

sound propagation above an inhomogeneous impedance plane abstract boundary integral equation bie methods are described for the prediction of sound propagation in particular from a line source over a flat plane of inhomogeneous impedance approximate methods which satisfy reciprocity for the calculation of the wave field over a two impedance plane are proposed these approximations applied to propagation from a line source give results agreeing well with those of the bie method when they are applied to propagation from a point source agreement with experiment is shown

banach spaces with separable duals it is proved that every banach space with a separable dual em beds into a space with a shrinking basis it follows that every separable re flexive space can be embedded in a reflexive space with a basis

the determination of companion matrices characterizing toeplitz and r toeplitz matrices abstract it is shown that the unknown elements in the companion matrix used to characterize a nonsingular toeplitz matrix are easily found from the elements of the inverse of the toeplitz matrix similar formulae are obtained for the two companion like matrices characterizing an r toeplitz matrix this case includes the result for a block toeplitz matrix

lenses and asymptotic midpoint uniform convexity abstract we introduce a new isometric geometric property namely asymptotic midpoint uniform convexity and investigate its connection to similar asymptotic properties

simulation of acoustic wavefields in heterogeneous media a robust method for automatic suppression of numerical dispersion numerical dispersion limits the application of numerical simulation methods for solving the acoustic wave equation in largescalecomputation thenearlyanalyticdiscretemethodnadm and its improved version for suppressing numerical dispersion weredevelopedrecently thisnewmethodisarefinementoftwo previousmethodsandfurtherincreasestheabilityofsuppressing numerical dispersion for modeling acoustic wave propagation in 2d heterogeneous media which uses acoustic wave displacement particle velocity and their gradients to restructure the acoustic wavefield via the truncated taylor expansion and the high order interpolation approximate method for the method proposed weinvestigateitsimplementationandcompareitwith the higher order lax wendroff correction lwc scheme the original nearly analytic discrete method nadm and its improved version with regard to numerical dispersion computational costs and computer storage requirements the numerical dispersionrelationsprovidedbytherefinedalgorithmfor1dand 2d cases are analyzed as well as the numerical results obtained bythismethodagainsttheexactsolutionforthe2dacousticcase numerical results show that the refined method gives no visible numericaldispersionforverylargespatialgridincrements itcan simulatehigh frequencywavepropagationforagivengridinterval and automatically suppress the numerical dispersion when the acoustic wave equation is discretized when too few samples per wavelength are used or when models have large velocity contrasts unlike the high order lwc methods our present methodcansavesubstantialcomputationalcostsandmemoryrequirements because very large grid increments can be used the refined method can be used for the simulation of large scale wavefields

the universality of l1 as a dual space let x be a banach space with a separable dual we prove that x embeds isomorphically into a mathcal l infty space z whose dual is isomorphic to l1 if moreover u is a space with separable dual so that u and x are totally incomparable then we construct such a z so that z and u are totally incomparable if x is separable and reflexive we show that z can be made to be somewhat reflexive

elimination of numerical dispersion in finite difference modeling and migration by flux corrected transport finite difference acoustic wave modeling and reverse time depth migration based on the full wave equation are general approaches that can take into account arbitrary variations in velocity and density and can handle turning waves as well however conventional finite difference methods for solving the acoustic or elastic wave equation suffer from numerical dispersion when too few samples per wavelength are used the flux corrected transport fct algorithm adapted from hydrodynamics reduces the numerical dispersion in finite difference wavefield continuation the flux correction procedure endeavors to incorporate diffusion into the wavefield continuation process only where needed to suppress the numerical dispersion incorporating the flux correction procedure in conventional finite difference modeling or reverse time migration can provide finite difference solutions with no numerical dispersion even for impulsive sources the fct correction which can be applied to finite difference approximations of any order in space and time is an efficient alternative to use for finite difference approximations of increasing order through demonstrations of modeling and migration on both synthetic and field data we show the benefits of the fct algorithm as well as its inability to fully recover resolution lost when the spatial sampling becomes too coarse

extended local rytov fourier migration method we develop a novel depth migration method termed the extended local rytov fourier elrf migration method it is based on the scalar wave equation and a local application of the rytov approximation within each extrapolation interval wavefields are fourier transformed back and forth between the frequency space and frequency wavenumber domains during wavefield extrapolation the lateral slowness variations are taken into account in the frequency space domain the method is efficient due to the use of a fast fourier transform algorithm under the small angle approximation the elrf method leads to the split step fourier ssf method that is unconditionally stable the elrf method and the extended local born fourier elbf method that we previously developed can handle wider propagation angles than the ssf method and account for the phase and amplitude changes due to the lateral variations of slowness whereas the ssf method only accounts for the phase changes the stability of the elrf method is controlled

proximity to lp and c0 in banach spaces abstract we construct a class of minimal trees and use these trees to establish a number of coloring theorems on general trees among the applications of these trees and coloring theorems are quantification of the bourgain l p and c 0 indices dualization of the bourgain c 0 index establishing sharp positive and negative results for constant reduction and estimating the bourgain l p index of an arbitrary banach space x in terms of a subspace y and the quotient x y

3d weak dispersion reverse time migration using a stereo modeling operator reliable 3d imaging is a required tool for developing models of complex geologic structures reverse time migration rtm as the most powerful depth imaging method has become the preferred imaging tool because of its ability to handle complex velocity models including steeply dipping interfaces and large velocity contrasts finite difference methods are among the most popular numerical approaches used for rtm however these methods often encounter a serious issue of numerical dispersion which is typically suppressed by reducing the grid interval of the propagation model resulting in large computation and memory requirements in addition even with small grid spacing numerical anisotropy may degrade images or worse provide images that appear to be focused but position events incorrectly recently stereo operators have been developed to approximate the partial differential operator in space these operators have been used to develop several weak dispersion and efficient stereo modeling methods that have been found to be superior to conventional algorithms in suppressing numerical dispersion and numerical anisotropy we generalized one stereo modeling method fourth order nearly analytic central difference nacd from 2d to 3d and applied it to 3d rtm the rtm results for the 3d seg eage phase a classic data set 1 and the seg advanced modeling project model demonstrated that even when using a large grid size the nacd method can handle very complex velocity models and produced better images than can be obtained using the fourth order and eighth order lax wendroff correction lwc schemes we also applied 3d nacd and fourth order lwc to a field data set and illustrated significant improvements in terms of structure imaging horizon layer continuity and positioning we also investigated numerical dispersion and found that not only does the nacd method have superior dispersion characteristics but also that the angular variation of dispersion is significantly less than for lwc

characterisation and properties of r toeplitz matrices abstract an r toeplitz rt matrix can be regarded as a block toeplitz matrix with order some multiple of r from which a bottom border and a right hand border have been removed it is shown that some known properties of toeplitz matrices extend to this new class in particular a necessary and sufficient condition is given in terms of a linear matrix equation for a nonsingular matrix to have rt form from which it follows that the elements of the inverse are completely determined by those in its first and last r rows and columns in addition some results are presented for cases when the inverse of an rt matrix has certain banded forms

third order symplectic integration method with inverse time dispersion transform for long term simulation the symplectic integration method is popular in high accuracy numerical simulations when discretizing temporal derivatives however it still suffers from time dispersion error when the temporal interval is coarse especially for long term simulations and large scale models we employ the inverse time dispersion transform itdt to the third order symplectic integration method to reduce the time dispersion error first we adopt the pseudospectral algorithm for the spatial discretization and the third order symplectic integration method for the temporal discretization then we apply the itdt to eliminate time dispersion error from the synthetic data as a post processing method the itdt can be easily cascaded in traditional numerical simulations we implement the itdt in one typical exiting third order symplectic scheme and compare its performances with the performances of the conventional second order scheme and the rapid expansion method theoretical analyses and numerical experiments show that the itdt can significantly reduce the time dispersion error especially for long travel times the implementation of the itdt requires some additional computations on correcting the time dispersion error but it allows us to use the maximum temporal interval under stability conditions thus its final computational efficiency would be higher than that of the traditional symplectic integration method for long term simulations with the aid of the itdt we can obtain much more accurate simulation results but with a lower computational cost we use the inverse time dispersion transform to the 3rd order symplectic integration it can greatly reduce the time dispersion error especially for long travel times it is superior to the conventional 2nd and high order finite difference methods it allows the maximum temporal interval under stability conditions thus is efficient we can obtain more accurate simulation results but with a lower computational cost

the coarse lipschitz geometry of ell_p oplus ell_q we show that for 1 p with p 2 the space l p 0 1 is not uniformly homeomorphic to ell p oplus ell 2 we also show that if 1 p 2 q the space ell p oplus ell q has unique uniform structure answering a question of johnson lindenstrauss and schechtman geom funct anal 6 430 470 1996

an optimal nearly analytic discrete weighted runge kutta discontinuous galerkin hybrid method for acoustic wavefield modeling abstractthe newly developed optimal nearly analytic discrete onad and the weighted runge kutta discontinuous galerkin wrkdg methods can effectively suppress the numerical dispersion caused by discretizing wave equations but it is difficult for onad to implement on flexible meshes whereas the wrkdg has high computational cost for wavefield simulations we have developed a new hybrid algorithm by combining the onad method with the wrkdg method in this hybrid algorithm the computational domain was split into several subdomains in which the subdomain for the onad method used regular cartesian grids whereas the subdomain for the wrkdg method used triangular grids the hybrid method was at least third order spatially accurate we have applied the proposed method to simulate the scalar wavefields for different models including a homogeneous model a rough topography model a fracture model and a cave model the numerical results found that the hybrid method can deal with complicated geometrical struc