spectral asymptotics for laplacians on self similar sets abstract given a self similar dirichlet form on a self similar set we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a geometric counting function defined through a family of coverings of the self similar set naturally associated with the dirichlet space secondly under sub gaussian heat kernel upper bound we prove a detailed short time asymptotic behavior of the partition function which is the laplace stieltjes transform of the eigenvalue counting function associated with the dirichlet form this result can be applicable to a class of infinitely ramified self similar sets including generalized sierpinski carpets and is an extension of the result given recently by b m hambly for the brownian motion on generalized sierpinski carpets moreover we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function

corner treatments for high order local absorbing boundary conditions in high frequency acoustic scattering this paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal computational domains for the finite element solution of high frequency acoustic scattering problems while high order absorbing boundary conditions habcs are accurate for smooth fictitious boundaries the precision of the solution drops in the presence of corners if no specific treatment is applied we present and analyze two strategies to preserve the accuracy of pade type habcs at corners first by using compatibility relations derived for right angle corners and second by regularizing the boundary at the corner we show that the former strategy is well adapted to right corners and efficient for nearly right corners while the later is better for very obtuse corners numerical results are proposed to analyze and compare the approaches for two and three dimensional problems

application of unsplit convolutional perfectly matched layer for scalar arbitrarily wide angle wave equation abstracta classical split perfectly matched layer pml method has recently been applied to the scalar arbitrarily wide angle wave equation awwe in terms of displacement however the classical split pml obviously increases computational cost and cannot efficiently absorb waves propagating into the absorbing layer at grazing incidence our goal was to improve the computational efficiency of awwe and to enhance the suppression of edge reflections by applying a convolutional pml cpml we reformulated the original awwe as a first order formulation and incorporated the cpml with a general complex frequency shifted stretching operator into the renewed formulation a staggered grid finite difference fd method was adopted to discretize the first order equation system for wavefield depth continuation the first order awwe with the cpml saved memory compared with the original second order awwe with the conventional split pml with the help of numerical examples we verified the correctness of the staggered

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

exact and operator rational approximate solutions of the helmholtz weyl composition equation in underwater acoustics the quadratic profile the application of phase space and path integral methods to both mathematical and computational direct and inverse wave propagation modeling at the level of the scalar one way helmholtz equation is briefly reviewed the role of operator symbols is stressed and their properties briefly discussed the construction of the operator symbol requires the exact or approximate solution of the helmholtz weyl composition equation in the weyl pseudodifferential operator calculus the exact symbols for several quadratic profiles are presented and briefly analyzed the symbols exactly corresponding to a family of operator rational approximations are also presented for the quadratic case these results are used to illustrate several points pertinent to wide angle propagation modeling and the refractive index profile reconstruction problem in underwater acoustics

local times for brownian motion on the sierpinski carpet jointly continuous local times are constructed for brownian motion on the sierpinski carpet a consequence is that the brownian motion hits points the method used is to analyze a sequence of eigenvalue problems

log periodic asymptotic expansion of the spectral partition function for self similar sets let fancyscript z t be the partition function the trace of the heat semigroup of the canonical laplacian on a post critically finite self similar set with uniform resistance scaling factor and good geometric symmetry or on a generalized sierpinski carpet it is proved that fancyscript z t sum k 0 n t d k d mathrm w g k log t o bigl rm exp ct frac 1 d mathrm w 1 bigr as t downarrow 0 for some continuous periodic functions g k mathbb r to mathbb r and c in 0 infty here d mathrm w in 1 infty denotes the walk dimension n 1 for a post critically finite self similar set n d for a d dimensional generalized sierpinski carpet d k k 0 n subset 0 infty is strictly decreasing with d n 0 g 0 is strictly positive and g 1 is either strictly positive or strictly negative depending on the neumann or dirichlet boundary condition

on the resistance of the sierpinski carpet let f text n be the nth stage in the construction of the sierpinski carpet let r text n be the electrical resistance of f text n when the left and right sides are each short circuited and a voltage is applied between them we prove that there exists a constant rho such that textstyle frac 1 4 rho n leq r n leq 4 rho n the motivation for this result came from the problem of establishing a the existence and b the value of the spectral dimension of the sierpinski carpet in this and a subsequent paper we settle a and give bounds for b

asymptotic analysis of weakly nonlinear bessel gaus beams abstract in this paper we investigate the propagation of conical waves in nonlinear media in particular we are interested in the effects resulting from applying a gaussian apodization to an ideal nondiffracting wave first we present a multiple scales approach to derive amplitude equations for weakly nonlinear conical waves from a governing equation of cubic nonlinear schrodinger type from these equations we obtain asymptotic solutions for the linear and the weakly nonlinear problem for which we state several uniform estimates that describe the deviation from the ideal nondiffracting solution moreover we show numerical simulations based on an implementation of our amplitude equations to support and illustrate our analytical results

a mapping approach for handling sloping interfaces a mapping approach for handling sloping interfaces in parabolic equation solutions is developed and tested at each range the medium is rigidly translated vertically so that a sloping interface becomes horizontal to simplify the approach the slope is assumed to be small and the extra terms that arise in the wave equation under the mapping are neglected the effects of these terms can be approximately accounted for by applying a leading order correction to the phase the mapping introduces variations in topography which are relatively easy to handle for the case of a pressure release boundary condition the accuracy of the approach is demonstrated for problems involving fluid sediments the approach should also be accurate for problems involving elastic sediments and should be useful for solving three dimensional problems involving variable topography

wide angle fd and ffd migration using complex pade approximations seismic migration by downward continuation using the one way wave equation approximations has two shortcomings imaging steep dip reflectors and handling evanescent waves complex pade approximations allow a better treatment of evanescent modes stabilizing finite difference migration without requiring special treatment for the migration domain boundaries imaging of steep dip reflectors can be improved using several terms in the pade expansion we discuss the implementation and evaluation of wide angle complex pade approximations for finite difference and fourier finite difference migration methods the dispersion relation and the impulsive response of the migration operator provide criteria to select the number of terms and coefficients in the pade expansion this ensures stability for a prescribed maximum propagation direction the implementations are validated on the marmousi model data set and seg eage salt model data

coupling and harnack inequalities for sierpinski carpets uniform harnack inequalities for harmonic functions on the preand graphical sierpinski carpets are proved using a probabilistic coupling argument various results follow from this including the construction of brownian motion on sierpinski carpets embedded in rd d 3 estimates on the fundamental solution of the heat equation and sobolev and poincare inequalities the sierpinski carpets scs we will study are generalizations of the cantor set let f0 0 1 be the unit cube in r d 2 centered at z0 1 2 1 2 let k a be integers with 1 a k and a k even divide f0 into k equal subcubes remove a central block of a subcubes and let f1 be what remains thus f1 f0 k a 2k k a 2k now repeat this operation on each of the k a remaining subcubes to obtain f2 iterating we obtain a decreasing sequence of closed sets fn then f n 0 fn is a sierpinski carpet and has hausdorff dimension df df f log k a log k when d 2 k 3 and a 1 we get the usual sierpinski carpet let fn kfn 0 and define the pre sierpinski carpet by f n 1 fn see 10 the graphical sierpinski carpet is the graph g v e with vertex set v z0 z f and edge set e x y v x y 1 thus int f is a domain in r with a large scale structure which mimics the small scale structure of f we are interested in the behavior of solutions of the laplace and heat equations on f f and g one reason for this is applications to transport phenomena in disordered media see 6 another is the new type of behavior of the heat kernel on these spaces let w be brownian motion on f with normal reflection on f and let q t x y be the transition density of w so that q solves the heat equation on f with neumann boundary conditions on f 1991 mathematics subject classification primary 60b99 secondary 60j35

the one way wave equation and its invariance properties the harmonic wave equation in inhomogeneous media is exactly split into coupled first order equations with respect to a principal direction of propagation according to the bremmer scheme the resulting one way wave equation is shown not to conserve energy flux for dimensions two and three against the general belief in one way wave propagation or parabolic equation literature conservation of energy flux is only ensured in the high frequency limit on the other hand a simple invariant is found that may be seen as a generalization of the snell law to arbitrary non stratified media similarly the reciprocity property is not fully ensured in general and the time reversal symmetry is ensured for propagating fields besides in the one way wave equation the additional term to the standard parabolic equation is shown to strengthen mode coupling the analysis encompasses the evanescent waves

entropy based algorithms for best basis selection adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals it permits efficient compression of a variety of signals such as sound and images the predefined libraries of modulated waveforms include orthogonal wavelet packets and localized trigonometric functions and have reasonably well controlled time frequency localization properties the idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost the method relies heavily on the remarkable orthogonality properties of the new libraries all expansions in a given library conserve energy and are thus comparable several cost functionals are useful one of the most attractive is shannon entropy which has a geometric interpretation in this context

a trace theorem for dirichlet forms on fractals abstract we consider a trace theorem for self similar dirichlet forms on self similar sets to self similar subsets in particular we characterize the trace of the domains of dirichlet forms on sierpinski gaskets and sierpinski carpets to their boundaries where the boundaries are represented by triangles and squares that confine the gaskets and the carpets as an application we construct diffusion processes on a collection of fractals called fractal fields these processes behave as an appropriate fractal diffusion within each fractal component of the field

a hybrid finite difference surface green s function method for computing transmission losses in an inhomogeneous atmosphere over irregular terrain a standard parabolic equation spe is used to approximate the helmholtz equation for electromagnetic propagation in an inhomogeneous atmosphere an implicit finite difference ifd scheme to solve the spe is applied between the irregularly shaped ground and an altitude z z sub h below which all inhomogeneities of the medium are assumed localized the boundary condition at z z sub h is obtained by matching the ifd solution to a surface green s function sgf solution within the uniform region above z z sub h for ground slopes above about 1 degrees the ifd implementation of the impedance boundary condition at the ground is shown to maintain the validity of the approximation only for vertically polarized waves predictions using this hybrid finite difference fd sgf method agree well with results obtained using other computational methods

spatial log periodic oscillations of first passage observables in fractals for transport processes in geometrically restricted domains the mean first passage time mfpt admits a general scaling dependence on space parameters for diffusion anomalous diffusion and diffusion in disordered or fractal media for transport in self similar fractal structures we obtain an expression for the source target distance dependence of the mfpt that exhibits both the leading power law behavior depending on the hausdorff and spectral dimension of the fractal as well as small log periodic oscillations that are a clear and definitive signal of the underlying fractal structure we also present refined numerical results for the sierpinski gasket that confirm this oscillatory behavior

perfectly matched absorbing layers for the paraxial equations a new absorbing boundary technique for the paraxial wave equations is proposed and analyzed numerical results show the efficiency of the method

rational square root approximations for parabolic equation algorithms in this article stable pade approximations to the function 1 z are derived by choosing a branch cut in the negative half plane the pade coefficients are complex and may be derived analytically to arbitrary order from the corresponding real coefficients associated with the principal branch defined by z 1 i z 0 i z denotes the imaginary part of z the characteristics of the corresponding square root approximation are illustrated for various segments of the complex plane in particular for waveguide problems it is shown that an increasingly accurate representation may be obtained of both the evanescent part of the mode spectrum for the acoustic case and the complex mode spectrum for the elastic case an elastic parabolic equation algorithm is used to illustrate the application of the new pade approximations to a realistic ocean environment including elasticity in the ocean bottom

an explicit analytical solution for sound propagation in a three dimensional penetrable wedge with small apex angle a problem of sound propagation in a shallow water waveguide with a weakly sloping penetrable bottom is considered the adiabatic mode parabolic equations are used to approximate the solution of the three dimensional 3d helmholtz equation by modal decomposition of the acoustic pressure field the mode amplitudes satisfy parabolic equations that admit analytical solutions in the special case of the 3d wedge using the analytical formula for modal amplitudes an explicit and remarkably simple expression for the acoustic pressure in the wedge is obtained the proposed solution is validated by the comparison with a solution of the 3d penetrable wedge problem obtained using a fully 3d parabolic equation that includes a leading order cross term correction

mode functions for the wide angle approximation to the parabolic equation the parabolic approximation to the wave equation is examined within the context of normal mode theory in a layered waveguide the horizontal propagation constants and modal amplitudes of the field satisfying the standard parabolic equation spe approximation can be mapped exactly into the amplitudes and wave numbers of the normal modes for the field p satisfying the helmholtz wave equation however this is not the case for certain other parabolic approximations such as the wide angle parabolic equation wape approximation approximate mode functions for the wape approximation are developed these mode functions are then used to decompose range independent sound pressure fields computed using the wape approximation the resulting modal coefficients and eigenfunctions obtained using the wape mode functions are compared with those obtained using standard normal mode theory

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

energy and laplacian on hanoi type fractal quantum graphs this article studies potential theory and spectral analysis on compact metric spaces which we refer to as fractal quantum graphs these spaces can be represented as a possibly infinite union of one dimensional intervals and a totally disconnected possibly uncountable compact set which roughly speaking represents the set of junction points classical quantum graphs and fractal spaces such as the hanoi attractor are included among them we begin with proving the existence of a resistance form on the hanoi attractor and go on to establish heat kernel estimates and upper and lower bounds on the eigenvalue counting function of laplacians corresponding to weakly self similar measures on the hanoi attractor these estimates and bounds rely heavily on the relation between the length and volume scaling factors of the fractal we then state and prove a necessary and sufficient condition for the existence of a resistance form on a general fractal quantum graph finally we extend our spectral results to a large class of weakly self similar fractal quantum graphs

discrete non local boundary conditions for split step pade approximations of the one way helmholtz equation this paper deals with the efficient numerical solution of the two dimensional one way helmholtz equation posed on an unbounded domain in this case one has to introduce artificial boundary conditions to confine the computational domain the main topic of this work is the construction of the so called discrete transparent boundary conditions for state of the art parabolic equation methods namely a split step discretization of the high order parabolic approximation and the split step pade algorithm of collins finally several numerical examples arising in optics and underwater acoustics illustrate the efficiency and accuracy of our approach

one way large range step methods for helmholtz waveguides a useful approach for long range computation of the helmholtz equation in a waveguide is to re formulate it as the operator differential riccati equation for the dirichlet to neumann map for waveguides with slow range dependence the piecewise range independent approximation is used to derive a second order range stepping method for this one way re formulation the range marching formulas are exact for each range independent piece and a large range step is possible if the range dependence is gradual based on a fourth order conservative exponential method for linear evolution equations a fourth order method that admits even larger range steps is developed for the one way re formulation numerical examples are used to demonstrate the improved accuracy of the fourth order method

migration with arbitrarily wide angle wave equations we develop a new scalar migration technique that is highly accurate for imaging steep dips in heterogeneous media this method is based on arbitrarily wide angle wave equations awwes that are highly accurate space domain one way wave equations and have a form similar to the 15 equation the accuracy of the proposed method is increased by introducing auxiliary variables as well as adjusting the parameters of the approximation poststack migration is carried out by downward continuation using the awwe for which we have developed a stable explicit double marching scheme up to 80 accuracy is achieved by second order awwe migration with only 2 3 times the computational effort of the 15 equation and requiring almost the same storage we illustrate the performance of awwe migration using impulse response graphs a single dipping reflector and a slice of the seg eage salt model

arbitrarily wide angle wave equations for complex media by combining various ideas related to one way wave equations owwes half space stiffness relation special finite element discretization and complex coordinate stretching a systematic procedure is developed for deriving a series of highly accurate space domain versions of owwes the resulting procedure is applicable to complex media where the governing equation full wave equation is a second order differential system making the procedure applicable for general heterogeneous anisotropic porous viscoelastic media owing to their high accuracy in representing waves propagating in an arbitrarily wide range of angles the resulting equations are named arbitrarily wide angle wave equations awwes in order to illustrate the proposed procedure awwes are derived for one way propagation in acoustic as well as elastic media while acoustic awwes can be considered as modified versions of well known space domain owwes based on rational approximations of the square root operator the elastic awwes are significantly different from the existing elastic owwes unlike the existing elastic owwes elastic awwes are displacement based and are applicable to general anisotropic media furthermore awwes are simple in their form and appear amenable to easy numerical implementation

dispersive waves and pml performance for large stencil differencing in the parabolic equation the discrete reflection absorption performance of perfectly matched layer pml boundary conditions is a function of the field discretization scheme throughout the computational domain and the form of the complex coordinate extension that characterizes the pml region most previous work analyzing pml reflection performance has assumed use of low order second or fourth small stencil three or five point spatial discretizations whose dispersion is poor outside a narrow low wavenumber band large stencil differencing schemes regardless of order are easily created with physically realistic wide band dispersive characteristics but the effect their adoption has on pml performance has not been analyzed previously in the context of a semi discrete approximation to the parabolic paraxial equation this work derives the analytic form of the discrete pml reflection coefficient for an arbitrary order arbitrary stencil interior differencing scheme coupled to an arbitrary order arbitrary stencil pml region this result is then used to show that the absorption performance of the pml regions is largely insensitive to changes in interior differencing as a consequence the benefits of superior dispersion characteristics provided by use of large stencil differencing in the interior physical domain can be had without incurring discrete reflection penalties at the pml

the flattened surface parabolic equation a simple surface flattening transformation is applied to the problem of underwater acoustic scattering from an irregular sea surface the flattened surface elliptic wave equation is factored to obtain a parabolic equation that has the same high angle capability as recent parabolic equation models the flattened surface parabolic equation contains an additional surface slope term but includes forward scattering from the irregular sea surface the transformation creates additional volume irregularities and irregularities in the water sediment interface that the range and depth mesh need to resolve the surface flattening transformation is easy to implement in a parabolic equation model it compares very well with results obtained using boundary integral equations an example is given of scattering at 400 hz from a sea surface with wave heights of 3 m or nearly a wavelength

anisotropic complex pade hybrid finite difference depth migration standard real valued finite difference fd and fourier finite difference ffd migrations cannot handle evanescent waves correctly which can lead to numerical instabilities in the presence of strong velocity variations a possible solution to these problems is the complex pade approximation which avoids problems with evanescent waves by rotating the branch cut of the complex square root we have applied this approximation to the acoustic wave equation for vertical transversely isotropic media to derive more stable fd and hybrid fd ffd migrations for such media our analysis of the dispersion relation of the new method indicates that it should provide more stable migration results with fewer artifacts and higher accuracy at steep dips our studies lead to the conclusion that the rotation angle of the branch cut that should yield the most stable image is 60 for fd migration as confirmed by numerical impulse responses and work with synthetic data

leading order cross term correction of three dimensional parabolic equation models the issue of handling a leading order cross multiplied term in three dimensional 3d parabolic equation pe based models is addressed in particular numerical results obtained incorporating a leading order cross term correction in an existing 3d pe model written in cylindrical coordinates based on higher order pade approximations in both depth and azimuth and a splitting operator technique are reported note that the numerical algorithm proposed in this paper could be used in the future to update any 3d pe codes that neglect cross terms and use a splitting numerical technique the 3d penetrable wedge benchmark problem is chosen to illustrate the accuracy of the now fully wide angle enhanced 3d pe model the comparisons with a 3d reference solution based on the image source clearly show that handling the leading order cross term in the 3d pe computation is sufficient to remove the phase errors inherent to any 3d pe models that neglect cross terms in their formulations

highly accurate absorbing boundary conditions for wide angle wave equations we develop a new class of absorbing boundary conditions abcs to prevent unwanted artifacts and wraparounds associated with aperture truncation in migration modeling using high order one way wave equations the fundamental approach behind the proposed development is the efficient discretization of the half space beyond the boundary of interest using midpoint integrated imaginary finite elements an idea recently utilized in the development of effective one way wave equations the proposed absorbing boundary conditions essentially add absorbing layers at the aperture truncation points we derive the absorbing boundary conditions analyze their properties and develop a stable explicit finite difference scheme to solve the downward continuation problem modified by these boundary conditions with the help of numerical examples we conclude that with as few as three absorbing layers i e two additional gridpoints the waves can be absorbed completely thus preventing associated artifacts

parabolic wave equation approximations in heterogenous media the properties of different variants of parabolic approximations of scalar wave equations are analyzed these equations are of general form which includes those used in seismology underwater acoustics and other applications a new version of the parabolic approximation is derived for heterogeneous media it has optimal properties with respect to wave reflection at material interfaces the amplitudes of the reflected and transmitted waves depend continuously on the interface existence uniqueness and energy estimates are proved

besov class via heat semigroup on dirichlet spaces i sobolev type inequalities abstract we introduce heat semigroup based besov classes in the general framework of dirichlet spaces general properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained as a highlight of the paper we obtain a far reaching l p analogue p 1 of the sobolev inequality that was proved for p 2 by n varopoulos under the assumption of ultracontractivity for the heat semigroup the case p 1 is of special interest since it yields isoperimetric type inequalities

rational operators for filtering the parabolic equation pe method has been applied to approximate the wave equation construct radiation conditions solve scattering problems construct initial conditions and derive energy flux conditions a new application of the pe method filters based on rational approximations of depth operators is described and tested the rational approximation is designed to act as the identity operator on the desired part of the spectrum of the depth separated wave equation and to annihilate other parts of the spectrum this approach does not require explicit knowledge of the spectrum the applications of the filter include directly solving eigenvalue problems annihilating components of the wave number spectrum generating initial conditions at the source range and eliminating gibbs oscillations that arise in energy conserving pe solutions

spectral dimension and bohr s formula for schrodinger operators on unbounded fractal spaces we establish an asymptotic formula for the eigenvalue counting function of the schrodinger operator for some unbounded potentials v on several types of unbounded fractal spaces we give sufficient conditions for bohr s formula to hold on metric measure spaces which admit a cellular decomposition and then verify these conditions for fractafolds and fractal fields based on nested fractals in particular we partially answer a question of fan khandker and strichartz regarding the spectral asymptotics of the harmonic oscillator potential on the infinite blow up of a sierpinski gasket

continuity and estimates of the liouville heat kernel with applications to spectral dimensions the liouville brownian motion lbm recently introduced by garban rhodes and vargas and in a weaker form also by berestycki is a diffusion process evolving in a planar random geometry induced by the liouville measure m gamma formally written as m gamma dz e gamma x z gamma 2 mathbb e x z 2 2 dz gamma in 0 2 for a massive gaussian free field x it is an m gamma symmetric diffusion defined as the time change of the two dimensional brownian motion by the positive continuous additive functional with revuz measure m gamma in this paper we provide a detailed analysis of the heat kernel p t x y of the lbm specifically we prove its joint continuity a locally uniform sub gaussian upper bound of the form p t x y le c 1 t 1 log t 1 exp bigl c 2 x y beta wedge 1 t frac 1 beta 1 bigr for t in bigl 0 frac 1 2 bigr for each beta frac 1 2 gamma 2 2 and an on diagonal lower bound of the form p t x x ge c 3 t 1 bigl log t 1 bigr eta for t in 0 t eta x with t eta x in bigl 0 frac 1 2 bigr heavily dependent on x for each eta 18 for m gamma almost every x as applications we deduce that the pointwise spectral dimension equals 2 m gamma a e and that the global spectral dimension is also 2

transparent boundary conditions for split step pade approximations of the one way helmholtz equation in this paper we generalize the nonlocal discrete transparent boundary condition introduced by f schmidt and p deuflhard 1995 comput math appl 29 53 76 and by f schmidt and d yevick 1997 j comput phys 134 96 107 to propagation methods based on arbitrary pade approximations of the two dimensional one way helmholtz equation our approach leads to a recursive formula for the coefficients appearing in the nonlocal condition which then yields an unconditionally stable propagation method

complex pade approximants for wide angle acoustic propagators procedures for constructing wide angle propagators that attenuate evanescent modes in parabolic equation pe models of underwater sound propagation are examined in this context two new split step pade approximations to exponentiated square root operators are proposed the first method involves a propagator derived from a pade approximant of a square root operator that has been rotated in the complex plane while the propagator of the second method is obtained from a complex coefficient rational approximation of the square root operator numerical evidence confirms that these proposed methods yield improved behavior compared to previously suggested techniques in selected areas of the spectral domain

exact discrete nonlocal boundary conditions for high order pade parabolic equations the nonlocal boundary conditions nlbcs for high order finite difference parabolic equations pes are obtained by z transformation of the discrete pe in a homogeneous medium the considered nlbcs include the free space radiation condition possibly with a density jump at the nlbc interface the nlbc at an arbitrary impedance interface and the nlbcs for sources and the starting field beyond the nlbc interface the derivation is presented for the multiterm pade pe model owwe one way wave equation but the developed technique is applicable to a broad class of finite difference pes the obtained nlbcs are exact for the given finite difference scheme they are not limited by the order of pade approximation or by the pe steps in range and depth the nlbc convolution coefficients are calculated by the numerical inverse z transformation the accuracy and performance of the algorithm are analyzed for several benchmark problems the solution is robust for the range steps over 25 wavelengths and or the approximations up to the tenth order the nlbcs are faster and more accurate for large steps because fewer previous range steps contribute to the convolution precomputation of the nlbc coefficients may be required in time demanding applications the results are compared with earlier proposed nlbcs for high order pes

optimization of the parameters in complex pade fourier finite difference migration abstractcomplex pade fourier finite difference migration is a stable one way wave equation technique that allows for better treatment of evanescent modes than its real counterpart in this way producing fewer artifacts as for real fourier finite difference ffd migration its parameters can be optimized to improve the imaging of steeply dipping reflectors the dip limitation of the ffd operator depends on the variation of the velocity field we have developed a wide angle approximation for the one way continuation operator by means of optimization of the pade coefficients and the most important velocity dependent parameter we have evaluated the achieved quality of the approximate dispersion relation in dependence on the chosen function of the ratio between the model and reference velocities under consideration of the number of terms in the pade approximation and the branch cut rotation angle the optimized parameters are chosen based on the migration results and the computational cost we found that by

higher order paraxial wave equation approximations in heterogenous media a new family of paraxial wave equation approximations is derived these approximations are of higher order accuracy than the parabolic approximation and they can be applied to the same computationa

on the use of higher order azimuthal schemes in 3 d pe modeling in this paper the issue of using higher order finite difference schemes to handle the azimuthal derivative term in a three dimensional parabolic equation based model is addressed the three dimensional penetrable wedge benchmark problem is chosen to illustrate the accuracy and efficiency of the proposed schemes both point source and modal initializations of the pressure field are considered for each higher order finite difference scheme used in azimuth the convergence of the numerical solution with respect to the azimuth is investigated and the cpu times are given some comparisons with solutions obtained from another 3 d model j a fawcett j acoust soc am 93 2627 2632 1993 are presented the numerical simulations show that the use of a higher order scheme in azimuth allows one to reduce the required number of points in the azimuthal direction while still obtaining accurate solutions the higher order schemes have approximately the same efficiency as a fft based approach in fact may outperform it slightly however the finite difference approach has the advantage that it may be more flexible than the fft approach for various pe approximations

dirichlet forms on fractals poincare constant and resistance we study dirichlet forms associated with random walks on fractal like finite grahs we consider related poincare constants and resistance and study their asymptotic behaviour we construct a markov semi group on fractals as a subsequence of random walks and study its properties finally we construct self similar diffusion processes on fractals which have a certain recurrence property and plenty of symmetries

matching pursuit filters applied to face identification we present a face identification algorithm that automatically processes an unknown image by locating and identifying the face the heart of the algorithm is the use of pursuit filters a matching pursuit filter is an adapted wavelet expansion where the expansion is adapted to both the data and the pattern recognition problem being addressed for identification the filters find the features that differentiate among faces whereas for detection the filters encode the similarities among faces the filters are designed though a simultaneous decomposition of a training set into a two dimensional 2 d wavelet expansion this yields a representation that is explicitly 2 d and encodes information locally the algorithm uses coarse to fine processing to locate a small set of key facial features which are restricted to the nose and eye regions of the face the result is an algorithm that is robust to variations in facial expression hair style and the surrounding environment based on the locations of the facial features the identification module searches the data base for the identity of the unknown face using matching pursuit filters to make the identification the algorithm was demonstrated on three sets of images the first set was images from the feret data base the second set was infrared and visible images of the same people this demonstration was done to compare performance on infrared and visible images individually and on fusing the results from both modalities the third set was mugshot data from a law enforcement application

transition densities for brownian motion on the sierpinski carpet upper and lower bounds are obtained for the transition densitiesp t x y of brownian motion on the sierpinski carpet these are of the same form as those which hold for the sierpinski gasket in addition the joint continuity ofp t x y is proved the existence of the spectral dimension is established and the einstein relation connecting the spectral dimension the hausdorff dimension and the resistance exponent is shown to hold

energy conserving and reciprocal solutions for higher order parabolic equations conservation of sound energy in nonabsorbing media and acoustical reciprocity in stationary fluids at rest are the fundamental properties of the acoustic field preserving these properties in parabolic approximations is essential for accurate modeling of sound propagation in shallow water environments recently godin proposed a family of wide angle parabolic equations pe s for motionless fluid which are energy conserving reciprocal and allow improved description of mode coupling without adversely affecting phase accuracy for moving media a generalized claerbout pe was also derived its solutions preserve the acoustic energy in the absence of dissipation and satisfy the flow reversal theorem frt which is an extension of reciprocity principle to moving media the report presents a numerical scheme for solving the proposed pe s the algorithm preserves the conservational properties of the original pseudo differential equations implicit finite difference solutions satisfy discrete analogs of the e

weyl s problem for the spectral distribution of laplacians on p c f self similar fractals we establish an analogue of weyl s classical theorem for the asymptotics of eigenvalues of laplacians on a finitely ramified i e p c f self similar fractalk such as for example the sierpinski gasket we consider both dirichlet and neumann boundary conditions as well as laplacians associated with bernoulli type multifractal measures onk from a physical point of view we study the density of states for diffusions or for wave propagation in fractal media more precisely let x be the number of eigenvalues less thanx then we show that x is of the order ofxds 2 asx where the spectral exponent ds is computed in terms of the geometric as well as analytic structures ofk further we give an effective condition that guarantees the existence of the limit ofx ds 2 x asx this condition is in some sense generic in addition we define in terms of the above spectral exponents and calculate explicitly the spectral dimension ofk

wave number sampling at short range the self starter is an efficient approach for solving geoacoustic inverse problems involving a vertical array of receivers located on the order of ten wavelengths from a source r j cederberg and m d collins j acoust soc am 22 102 109 1997 with this approach accurate solutions can be obtained by sampling the wave number spectrum at approximately ten points there have been unsubstantiated claims that the spectral solution provides similar efficiency when implemented with an approach that involves perturbing the integration contour off the real line f b jensen et al computational ocean acoustics american institute of physics new york 1994 pp 231 240 it is demonstrated that this quadrature scheme breaks down at short ranges and that the self starter solution is about an order of magnitude faster these conclusions are based on comparisons of the rational approximations associated with the self starter and the quadrature scheme comparisons of acoustic fields for particular proble

from projection pursuit and cart to adaptive discriminant analysis while many efforts have been put into the development of nonlinear approximation theory and its applications to signal and image compression encoding and denoising there seems to be very few theoretical developments of adaptive discriminant representations in the area of feature extraction selection and signal classification in this paper we try to advocate the idea that such developments and efforts are worthwhile based on the theoretical study of a data driven discriminant analysis method on a simple yet instructive example we consider the problem of classifying a signal drawn from a mixture of two classes using its projections onto low dimensional subspaces unlike the linear discriminant analysis lda strategy which selects subspaces that do not depend on the observed signal we consider an adaptive sequential selection of projections in the spirit of nonlinear approximation and classification and regression trees cart at each step the subspace is enlarged in a direction that maximizes the mutual information with the unknown class we derive explicit characterizations of this adaptive discriminant analysis ada strategy in two situations when the two classes are gaussian with the same covariance matrix but different means the adaptive subspaces are actually nonadaptive and can be computed with an algorithm similar to orthonormal matching pursuit when the classes are centered gaussians with different covariances the adaptive subspaces are spanned by eigen vectors of an operator given by the covariance matrices just as could be predicted by regular lda however we prove that the order of observation of the components along these eigen vectors actually depends on the observed signal numerical experiments on synthetic data illustrate how data dependent features can be used to outperform lda on a classification task and we discuss how our results could be applied in practice