on smoothed rank profile tests in eigenstructure methods for directions of arrival estimation we propose a statistical procedure known as the smoothed rank profile srp test that can be applied to an array covariance matrix to determine the source coherency structure and solvability of the directions of arrival estimation problem in the presence of coherent sources the srp is the rank profile of a telescoping series of matrices extracted from the observed array covariance matrix

metastability of the cahn hilliard equation in one space dimension abstract we establish metastability of the one dimensional cahn hilliard equation for initial data that is order one in energy and order one in h 1 away from a point on the so called slow manifold with n well separated layers specifically we show that for such initial data on a system of lengthscale there are three phases of evolution 1 the solution is drawn after a time of order 2 into an algebraically small neighborhood of the n layer branch of the slow manifold 2 the solution is drawn after a time of order 3 into an exponentially small neighborhood of the n layer branch of the slow manifold 3 the solution is trapped for an exponentially long time exponentially close to the n layer branch of the slow manifold the timescale in phase 3 is obtained with the sharp constant in the exponential

slow motion manifolds far from the attractor in multistable reaction diffusion equations abstract we consider a scalar reaction diffusion equation with multistable nonlinearity with a particular symmetry by reduction to a family of transmission problems in r and by contraction arguments a manifold close to an invariant manifold formed by functions exhibiting a pattern of transition layers is constructed an approximation for the associated vector field is also provided this shows that the motion on those manifolds is exponentially slow as in the well known case of the bistable equation however in opposition to the bistable case some of these manifolds are far from the attractor since these manifolds correspond to metastable patterns this shows the importance of the transient motion toward the attractor and the importance of these manifolds in organizing that motion it is also shown that by a suitable perturbation we can obtain new equilibria on those manifolds

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

a comparison of the existence of cross terms in the wigner distribution and the squared magnitude of the wavelet transform and the short time fourier transform it is shown that cross terms comparable to those found in the wigner distribution wd exist for the energy distributions of the wavelet transform wt and the short time fourier transform stft the geometry of the cross terms is described by deriving mathematical expressions for the energy distributions of the stft and the wt of a multicomponent signal from those mathematical expressions it is inferred that the stft and the wt cross terms 1 occur at the intersection of the respective transforms of the two signals under consideration whereas the wd cross terms occur at mid time frequency of the two signals 2 are oscillatory in nature as are the wd cross terms and are modulated by a cosine whose argument is a function of the difference in center times and center frequencies of the signals under consideration and 3 can have a maximum amplitude as large as twice the product of the magnitude of the transforms of the two signals in question like wd cross terms it is shown that the presence of these cross terms could lead to problems in analyzing a multicomponent signal the consequences of this effect with respect to speech applications are discussed

a bootstrap procedure for high resolution velocity analysis a method is given for further improving velocity estimates derived from high resolution velocity analysis in conventional velocity analysis a set of tentative velocities is used to apply a normal moveout nmo correction to a set of spatio temporal windows the coherence measure is evaluated for each velocity and finally the velocity estimate is retrieved from the peak of the coherence measure because analytical expressions for velocity uncertainties are difficult to derive i propose an intensive statistical procedure the bootstrap method to assess the accuracy of the velocity estimate in the bootstrap method i create a data sample by randomly drawing seismic traces with replacement from a window of the common midpoint gather cmp next i calculate the velocity that maximizes the coherence measure for each bootstrap realization the variation of this velocity provides a means to compute standard errors i also use the bootstrap method to construct an average coherence measure and a kernel density estimator of the velocity that maximizes the coherence the average coherence exhibits an important attenuation of spurious events while retaining enough resolution to model reflections properly with similar moveout curves the latter is illustrated with synthetic and field data examples

a general approach for obtaining joint representations in signal analysis i characteristic function operator method a general approach for obtaining joint representations for arbitrary physical quantities is presented the characteristic function operator method of moyal 1949 and ville 1960 and generalized by cohen 1966 1976 and scully and cohen 1987 for the time frequency case is developed for arbitrary variables

spatio temporal spectral analysis by eigenstructure methods this paper presents new algorithms for estimating the spatio temporal spectrum of the signals received by a passive array the algorithms are based on the eigenstructure of the covariance and spectral density matrices of the received signals they allow partial correlation between the sources and thus are applicable to certain kinds of multipath problems simulation results that illustrate the performance of the new algorithms are presented

application of singular value decomposition to vertical seismic profiling an essential part of the interpretation of vertical seismic profiles vsp is the separation of the upgoing and downgoing waves this paper presents a new approach which is based on the decomposition of time shifted vsp sections into eigenimages using singular value decomposition svd the first few eigenimages of the time shifted vsp section contain the contributions of the horizontally aligned downgoing waves the last few eigenimages contain the contribution of uncorrelated noise components the separated upgoing waves are recovered as a partial sum of the eigenimages mportant aspects of this approach are that regular sampling of the recording levels is not required that the first break times need not be measured with extreme accuracy that noise rejection may be automatically included in the processing and that eigenimages or sums of eigenimages which may be computed as part of the approach can provide important additional information

evolutionary periodogram for nonstationary signals presents a novel estimator for the time dependent spectrum of a nonstationary signal by modeling the signal at any given frequency as having a time varying amplitude accurately represented by an orthonormal basis expansion the authors are able to compute a minimum mean squared error estimate of this time varying amplitude repeating the process over all frequencies they obtain a power distribution as a function of time and frequency that is consistent with the wold cramer evolutionary spectrum based on the model assumptions the authors develop the evolutionary periodogram ep for nonstationary signals an estimator analogous to the periodogram used in the stationary case they also derive the time frequency resolution of the new estimator the approach is free of some of the drawbacks of the bilinear distributions and of the short time fourier transform spectral estimates it is guaranteed to produce nonnegative spectra without the cross term behavior of the bilinear distributions and it does not require windowing of data in the time domain examples illustrating the new estimator are given

a high resolution weighted ab semblance for dealing with amplitude variation with offset phenomenon abstractvelocity analysis is an essential step in seismic reflection data processing the conventional and fastest method to estimate how velocity changes with increasing depth is to calculate semblance coefficients traditional semblance has two problems low time and velocity resolution and an inability to handle amplitude variation with offset avo phenomenon although a method known as the ab semblance can arrive at peak velocities in the areas with an avo anomaly it has a lower velocity resolution than conventional semblance we have developed a weighted ab semblance method that can handle both problems simultaneously we have developed two new weighting functions to weight the ab semblance to enhance the resolution of velocity spectra in the time and velocity directions in this way we increase the time and velocity resolution while eliminating the avo problem the first weighting function is defined based on the ratio between the first and the second singular values of the time window to improve

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

phase and amplitude tracking for seismic event separation we have developed a method to decompose seismic records into atomic events each defined by a smooth phase function and a smooth amplitude function this decomposition is intrinsically nonlinear and calls for a nonconvex least squares optimization formulation along the lines of full waveform inversion to overcome the lack of convexity we have developed an iterative refinement expansion scheme to initialize and track the phase and amplitude for each atomic event for short we called the method phase tracking the initialization is carried out by applying multiple signal classification to a few seed traces in which events can be separated and identified by their arrival times and amplitudes we then construct the initial solution at the seed traces using linear phase functions from the arrival times and constant amplitude functions assuming the medium is mostly dispersion free we refine this initial solution to account for dispersion and imperfect knowledge of the wavelet at the seed traces by fitting the observed data using a gradient descent method the resulting phase and amplitude functions are then carefully expanded across the traces in an adequately smooth way to match the whole data record we have evaluated the proposed method on two synthetic records and a field record because the parametrization of the seismic events is physically meaningful it also enables a simple form of bandwidth extension of the observed shot record to unobserved low and high frequencies we tested this procedure on the same shot records bandwidth extension is in principle helpful to initialize full waveform inversion with frequency sweeps and enhanced its resolution

dynamics of localized structures abstract we describe qualitative behaviour of solutions of the gross pitaevskii equation in 2d in terms of motion of vortices and radiation to this end we introduce the notion of the intervortex energy we develop a rather general adiabatic theory of motion of well separated vortices and present the method of effective action which gives a fairly straightforward justification of this theory finally we mention briefly two special situations where we are able to obtain a rather detailed picture of the vortex dynamics our approach is rather general and is applicable to a wide class of evolution nonlinear equation which exhibit localized stable static solutions it yields description of general time dependent solutions in terms of dynamics of those static solutions glued together

a high resolution data adaptive time frequency representation a data adaptive time frequency representation is developed that overcomes some limitations of the short time fourier transform while avoiding the cross terms that make the wigner distribution and other bilinear representations difficult to interpret the adaptive time frequency representations uses gaussian basis functions but varies their time width and chirp rate with time and frequency to achieve high signal concentration everywhere a measure of local signal concentration allows fully automated determination of the optimal basis parameters the adaptive method is computationally expensive but may provide much better performance than any currently known technique

an adaptive optimal kernel time frequency representation time frequency representations with fixed windows or kernels figure prominently in many applications but perform well only for limited classes of signals representations with signal dependent kernels can overcome this limitation however while they often perform well most existing schemes are block oriented techniques unsuitable for on line implementation or for tracking signal components with characteristics that change with time the time frequency representation developed in the present paper based on a signal dependent radially gaussian kernel that adapts over time surmounts these difficulties the method employs a short time ambiguity function both for kernel optimization and as an intermediate step in computing constant time slices of the representation careful algorithm design provides reasonably efficient computation and allows on line implementation certain enhancements such as cone kernel constraints and approximate retention of marginals are easily incorporated with little additional computation while somewhat more expensive than fixed kernel representations this new technique often provides much better performance several examples illustrate its behavior on synthetic and real world signals

rank 1 phenomena for mapping class groups we prove that every element of the mapping class group g has linear growth confirming a conjecture of n ivanov and that g is not boundedly generated we also provide restrictions on linear representations of g and its finite index subgroups

metastability and stability of patterns in a convolution model for phase transitions where e is a small positive parameter and f is bistable having only three zeros 1 a 1 and is balanced i e r 1 1 f u du 1 4 0 see fig 1 for a typical initial datum j which crosses a several times at the beginning the solution evolves as if the diffusion term euxx were not there and in the time scale o ej n ej the solution gets close to 1 or 1 in intervals separated by thin layers of thickness at the order of o ej n ej 17 28 at this moment a spatial pattern is generated it persists on a time scale o ece but is eventually destroyed typically u x t 1 u x t 1 uniformly for x on 1 20 1 as t 1 it is possible that u x t converges to a or a nonconstant steady state as t 1 but this is an unlikely event because all these steady states are unstable 16 27 research supported in part by the national science foundation and bor leqsf

slow dynamics for the cahn hilliard equation in higher space dimensions the motion of bubbles it is known that the van der waals cahn hilliard w c h dynamics can be approximated by a quasi static stefan problem with surface tension it turns out that the stefan problem has a manifold of equilibria equal in dimension to that of the domain any sphere of fixed radius with interface contained in the domain is an equilibrium indistinguishable from the point of view of the perimeter functional we resolve this degeneracy by showing that at the w c h level this manifold is replaced by a quasi invariant stable manifold on which the typical solution preserves its bubble like shape until it reaches the boundary moreover we show that the bubble moves superslowly we also obtain an equation that determines those special spheres that correspond to equilibria at the w c h level our work establishes the phenomenon of superslow motion in higher space dimensions in the class of single interface solutions

slow motion in higher order systems and g convergence in one space dimension the slow motion in higher order systems and convergence in one space dimension was studied some of the theorems and computations that support and proves the stability of the equations are presented

quadratic wavefront and traveltime approximations in inhomogeneous layered media with curved interfaces a quadratic approximation for the square of the traveltime from a source region to a receiver region is given for a three dimensional 3 d medium consisting of inhomogeneous layers with curved interfaces the square of the traveltime being a function of source and receiver coordinates is developed in a taylor series around a reference source and receiver point the relationships of the traveltime parameters to the ray parameters and the wavefront curvature matrices are shown using midpoint half offset coordinates gives a simplified traveltime function compared to using source receiver coordinates only in the case that the reference source point and the reference receiver point coincide zero offset approximation for a medium consisting of homogeneous layers with plane dipping interfaces the traveltime approximation is further simplified the derived traveltime approximation is shown to be exact for a reflection from a plane dipping interface in a homogeneous medium explicit expressions for the tr

window length selection for optimum slowness resolution of the local slant stack transform one of the most critical decisions in the design of a local slant stack transform lsst is the selection of its aperture or more precisely the selection of the appropriate number of traces and their weighting coefficients for each slant stack the challenge is to achieve a good compromise between the slowness and the spatial resolution conventionally the window size is chosen in a more intuitive manner by visual inspection and some limited tests we analyzed the lsst to establish rigorous criteria for the window selection to achieve the optimum slowness and spatial resolution in the transformed domain for a given data set for this purpose we estimated the slowness resolution in the lsst domain as a function of the spatial window bandwidth and of the spectral characteristics of the waves for a wave with a given bandpass spectrum the slowness resolution the stopband attenuation and the wavefront tracking capability are determined by the spatial window for narrowband signals the spatial window m

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

signal and traveltime parameter estimation using singular value decomposition abstractsignal detection and traveltime parameter estimation can be performed by computing a coherence function in a data window centered around a traveltime function defined by its parameters we used singular value decomposition of the data matrix not eigendecomposition of a covariance matrix to review the most commonly used coherence measures this resulted in a new reduced semblance coefficient defined from the first eigenimage assuming that the signal amplitude was the same on all data channels as in classical semblance in a second signal model the time signal was constant on each channel but the amplitude changed then the semblance coefficient is the square of the first singular value divided by the energy of the data two normalized crosscorrelation coefficients derived from the first eigenimage can also be used as a coherence measure the normalized crosscorrelation of the spatial singular vector with a vector with all elements equal to one and the normalized crosscorrelation of the tem

metastable bubble solutions for the allen cahn equation with mass conservation in a multidimensional domain the slow motion behavior of internal layer solutions with spherical interfaces referred to as bubble solutions is analyzed for the nonlocal allen cahn equation with mass conservation this problem represents the simplest model for the phase separation of a binary mixture in the presence of a mass constraint the bubble is shown to drift exponentially slowly across the domain without change of shape toward the closest point on the boundary of the domain an explicit ordinary differential equation for the motion of the center of the bubble is derived by extending to a multidimensional setting the asymptotic projection method developed previously by the author to treat metastable problems in one spatial dimension an asymptotic formula for the time of collapse of the bubble against the boundary of the domain is derived in terms of the principal radii of curvature of the boundary at the initial contact point an analogy between slow bubble motion and the classical exit prob

dispersive and friction induced stabilization of the cahn hilliard inverse cascade we discuss the stabilization of the inverse cascade in the large scale instability of the kolmogorov flow described by the complete cahn hilliard equation with inclusion of effect large scale friction and deformation radius the friction and the values halting the inverse cascade at the various possible intermediate states are calculated by means of singular perturbation techniques and compared to the values resulting from numerical simulation of the complete cahn hilliard equation the excellent agreement validates the theory our main result is that the critical values of friction or halting the inverse cascade scale exponentially as a function of the jet separation in the final flow contrary to previous theories and phenomenological approach 2002 elsevier science b v all rights reserved pacs 47 10 g 92 60 ek 47 27 i 47 27 ak 47 35 i

metastable patterns for the cahn hilliard equation part i abstract in this paper we study the dynamics of the one dimensional cahn hilliard equation in a neighborhood of an equilibrium having n 1 transition layers an approximation for an n dimensional invariant manifold and slow channel are defined and it is proved that ff a solution of the cahn hilliard equation starts outside but close to the slow channel around the approximate manifold then it will approach the channel at an exponentially large speed after it enters the slow channel it will follow the approximate manifold with speed o e c and stay in the channel for exponentially large time

stft with adaptive window width based on the chirp rate an adaptive time frequency representation tfr with higher energy concentration usually requires higher complexity recently a low complexity adaptive short time fourier transform astft based on the chirp rate has been proposed to enhance the performance this method is substantially modified in this paper i because the wavelet transform used for instantaneous frequency if estimation is not signal dependent a low complexity astft based on a novel concentration measure is addressed ii in order to increase robustness to if estimation error the principal component analysis pca replaces the difference operator for calculating the chirp rate and iii a more robust gaussian kernel with time frequency varying window width is proposed simulation results show that our method has higher energy concentration than the other astfts especially for multicomponent signals and nonlinear fm signals also for if estimation our method is superior to many other adaptive tfrs in low signal to noise ratio snr environments

traveltime approximations for a layered transversely isotropic medium we consider multiple transmitted reflected and converted qp qsv waves or multiple transmitted and reflected sh waves in a horizontally layered medium that is transversely isotropic with a vertical symmetry axis vti traveltime and offset horizontal distance between a source and receiver not necessarily in the same layer are expressed as functions of horizontal slowness these functions are given in terms of a taylor series in slowness in exactly the same form as for a layered isotropic medium the coefficients depend on the parameters of the anisotropic layers through which the wave has passed and there is no weak anisotropy assumption using classical formulas the traveltime or traveltime squared can then be expressed as a taylor series in even powers of offset these taylor series give rise to a shifted hyperbola traveltime approximation and a new continued fraction approximation described by four parameters that match the taylor series up to the sixth power in offset further approximations give several simplified continued fraction approximations all of which depend on three parameters zero offset traveltime nmo velocity and a heterogeneity coefficient the approximations break down when there is a cusp in the group velocity for the qsv wave numerical studies indicate that approximations of traveltime squared are generally better than those for traveltime a new continued fraction approximation that depends on three parameters is more accurate than the commonly used continued fraction approximation and the shifted hyperbola

high resolution velocity gathers and offset space reconstruction we present a high resolution procedure to reconstruct common midpoint cmp gathers first we describe the forward and inverse transformations between offset and velocity space then we formulate an underdetermined linear inverse problem in which the target is the artifacts free aperture compensated velocity gather we show that a sparse inversion leads to a solution that resembles the infinite aperture velocity gather the latter is the velocity gather that should have been estimated with a simple conjugate operator designed from an infinite aperture seismic array this high resolution velocity gather is then used to reconstruct the offset space the algorithm is formally derived using two basic principles first we use the principle of maximum entropy to translate prior information about the unknown parameters into a probabilistic framework in other words to assign a probability density function to our model second we apply bayes s rule to relate the a priori probability density function pdf with the pdf corresponding to the experimental uncertainties likelihood function to construct the a posteriori distribution of the unknown parameters finally the model is evaluated by maximizing the a posteriori distribution when the problem is correctly regularized the algorithm converges to a solution characterized by different degrees of sparseness depending on the required resolution the solutions exhibit minimum entropy when the entropy is measured in terms of burg s definition we emphasize two crucial differences in our approach with the familiar burg method of maximum entropy spectral analysis first burg s entropy is minimized rather than maximized which is equivalent to inferring as much as possible about the model from the data second our approach uses the data as constraints in contrast with the classic maximum entropy spectral analysis approach where the autocorrelation function is the constraint this implies that we recover not only amplitude information but also phase information which serves to extrapolate the data outside the original aperture of the array the tradeoff is controlled by a single parameter that under asymptotic conditions reduces the method to a damped least squares solution finally the high resolution or aperture compensated velocity gather is used to extrapolate near and far offset traces

random noise attenuation using local signal and noise orthogonalization abstractwe have developed a novel approach to attenuate random noise based on local signal and noise orthogonalization in this approach we first removed from a seismic section using one of the conventional denoising operators and then applied a weighting operator to the initially denoised section to predict the signal leakage energy as well as retrieve it from the initial noise section the weighting operator was obtained by solving a least squares minimization problem via shaping regularization with a smoothness constraint next the initially denoised section and the retrieved signal were combined to form the final denoised section the proposed denoising approach corresponded to orthogonalizing the initially denoised signal and noise in a local manner we evaluated the denoising performance using local similarity to test the orthogonalization property of the estimated signal and noise we calculated the local similarity map between the denoised signal section and removed noise section low values o

the statistical performance of the music and the minimum norm algorithms in resolving plane waves in noise this paper presents an asymptotic statistical analysis of the null spectra of two eigen assisted methods music 1 and minimum norm 2 for resolving independent closely spaced plane waves in noise particular attention is paid to the average deviation of the null spectra from zero at the true angles of arrival for the plane waves these deviations are expressed as functions of signal to noise ratios number of array elements angular separation of emitters and the number of snapshots in the case of music an approximate expression is derived for the resolution threshold of two plane waves with equal power in noise this result is validated by monte carlo simulations

wigner based formulation of the chirplet transform using the wigner distribution we derive and analyze a matrix formulation for the chirplet transform a signal analysis tool that generalizes the wavelet and short time fourier transforms the formulation expresses the translations scalings and shears of the chirplet transform in terms of affine matrix transformations on the time frequency plane our approach leads naturally to several new signal transforms which we derive analyze and extend

a resolution comparison of several time frequency representations two signal components are considered resolved in a time frequency representation when two distinct peaks can be observed the time frequency resolution limit of two gaussian components alike except for their time and frequency centers is determined for the wigner distribution the pseudo wigner distribution the smoother wigner distribution the squared magnitude of the short time fourier transform and the choi williams distribution the relative performance of the various distributions depends on the signal the pseudo wigner distribution is best for signals of this class with only one frequency component at any one time the choi williams distribution is most attractive for signals in which all components have constant frequency content and the matched filter short time fourier transform is best for signal components with significant frequency modulation a relationship between the short time fourier transform and the cross wigner distribution is used to argue that with a properly chosen window the short time fourier transform of the cross wigner distribution must provide better signal component separation that the wigner distribution

the relationship between semblance and eigenstructure velocity estimators this short presentation gives for the first time a formulation of the semblance coefficient in terms of data covariance matrix eigenstructure because the high resolution wavefront or spectral eigenstructure methods have received so much interest over the past decade it is necessary to analytically tie the conventionally used semblance produce to eigenstructure thereby allowing the seismic signal analyst an opportunity to relate the various displays of velocity spectra using more than visual appearance the eigenstructure form of semblance is compared to a number of the now well known eigenstructure based spectral estimators that separate signal and noise vector subspaces because of the inclusion of noise space energy in the coherence measure conventional semblance does not have the resolving power of the newer methods we suggest an enhanced semblance based on the signal and noise subspace separation concept a brief simulation verifies the visual improvement in the velocity spectrum obtained from

constrained full waveform inversion by model reparameterization the waveform inversion problem is inherently ill posed traditionally regularization schemes are used to address this issue for waveform inversion where the model is expected to have many details reflecting the physical properties of the earth regularization and data fitting can work in opposite directions the former smoothing and the latter adding details to the model we propose constraining estimated velocity fields by reparameterizing the model this technique also called model space preconditioning is based on directional laplacian filters it preserves most of the details of the velocity model while smoothing the solution along known geological dips preconditioning also yields faster convergence at early iterations the laplacian filters have the property to smooth or kill local planar events according to a local dip field by construction these filters can be inverted and used in a preconditioned waveform inversion strategy to yield geologically meaningful models we illustrate with 2d syn

spinodal decomposition for the cahn hilliard equation in higher dimensions part i probability and wavelength estimate this paper is the first in a series of two papers addressing the phenomenon of spinodal decomposition for the cahn hilliard equation where is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u3 we will present the main ideas of our approach and explain in what way our method differs from known results in one space dimension due to grant 26 furthermore we derive certain probability and wavelength estimates the probability estimate is needed to understand why in a neighborhood of a homogeneous equilibrium u0 of the cahn hilliard equation with mass in the spinodal region a strongly unstable manifold has dominating effects this is demonstrated for the linearized equation but will be essential for the nonlinear setting in the second paper 37 as well moreover we introduce the notion of a characteristic wavelength for the strongly unstable directions

fast acoustic velocity tomography of focusing operators abstracta 2d velocity model was estimated by tomographic imaging of overlapping focusing operators that contain one way traveltimes from common focus points to receivers in an aperture along the earth s surface the stability and efficiency of convergence and the quality of the resulting models were improved by a sequence of ideas we used a hybrid parameterization that has an underlying grid upon which is superimposed a flexible pseudolayer model we first solved for the low wavenumber parts of the model approximating it as constant velocity pseudo layers then we allowed intermediate wavenumbers allowing the layers to have linear velocity gradients and finally did unconstrained iterations to add the highest wavenumber details layer boundaries were implicitly defined by focus points that align along virtual marker reflector horizons each focus point sampled an area bounded by the first and last rays in the data aperture at the surface this reduced the amount of computation and the size of th

on computing equally spaced samples of a complex gaussian function this correspondence extends the simple nonlinear recursion presented by kaiser in 3 to the fast generation of equally spaced samples of a complex gaussian chirp function exp c 0 c 1 t c 2 t 2 using only two complex multiplications per point

canonical metrics on the moduli space of riemann surfaces i in this paper we continue our study of the canonical metrics on the moduli space of curves we first prove the bounded geometry of the ricci and perturbed ricci metrics by carefully choosing the pertubation constant and by studying the asymptotics we show that the ricci and holomorphic sectional curvatures of the perturbed ricci metric are bounded from above and below by negative constants based on our understanding of the kahler einstein metric we show that the logarithmic cotangent bundle over the deligne mumford moduli space is stable with respect to the canonical polarization finally in the last section we prove the strongly bounded geometry of the kahler einstein metric by using the kahler ricci flow and a priori estimates of the complex monge ampere equation

coarsening dynamics of the one dimensional cahn hilliard model the dynamics of one dimensional cahn hilliard model is studied the stationary and particle type solutions the bubbles are perused as a function of initial conditions boundary conditions and system size we characterize the bubble solutions which are involved in the coarsening dynamics and establish the bifurcation scenarios of the system a set of ordinary differential equation permits us to describe the coarsening dynamics in very good agreement with numerical simulations we also compare these dynamics with the bubble dynamics deduced from the classical kink interaction computation where our model seems to be more appropriated in the case of two bubbles we deduce analytical expressions for the bubble s position and the bubble s width besides a simple description of the ulterior dynamics is presented

new approach to seismic reflection event detection and velocity determination recently a number of high resolution techniques for event detection and parameter estimation velocity have emerged in the field of multichannel sonar array processing we find that a practical and computationally efficient implementation of these methods is possible for seismic reflection data this implementation results in development of the covariance measure for event detection and velocity estimation in seismic reflection data the measure is based on the eigenstructure of the sampled data covariance matrix this decomposition is carried out within hyperbolic windows that are moved through common depth point cdp data eigenvalues of the data covariance matrix allow simultaneous estimation of the noise and signal energy present within each window resulting in the development of a coherency measure computer implementation of the covariance measure provides a high resolution method for determination of velocity spectra relative to currently used techniques the method applied to synthetic data res

dispersion based short time fourier transform applied to dispersive wave analysis although time frequency analysis is effective for characterizing dispersive wave signals the time frequency tilings of most conventional analysis methods do not take into account dispersion phenomena an adaptive time frequency analysis method is introduced whose time frequency tiling is determined with respect to the wave dispersion characteristics in the dispersion based time frequency tiling each time frequency atom is adaptively rotated in the time frequency plane depending on the local wave dispersion although this idea can be useful in various problems its application to the analysis of dispersive wave signals has not been made in this work the adaptive time frequency method was applied to the analysis of dispersive elastic waves measured in waveguide experiments and a theoretical investigation on its time frequency resolution was presented the time frequency resolution of the proposed transform was then compared with that of the standard short time fourier transform to show its effectiveness in dealing with dispersive wave signals in addition to facilitate the adaptive time frequency analysis of experimentally measured signals whose dispersion relations are not known an iterative scheme for determining the relationships was developed the validity of the present approach in dealing with dispersive waves was verified experimentally

dimension and rank for mapping class groups we study the large scale geometry of the mapping class group mcg s our main result is that for any asymptotic cone of mcg s the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of mcg s an application is a proof of brock farb s rank conjecture which asserts that mcg s has quasi flats of dimension n if and only if it has a rank n free abelian subgroup hamenstadt has also given a proof of this conjecture using different methods we also compute the maximum dimension of quasi flats in teichmuller space with the weil petersson metric

time varying filtering and signal estimation using wigner distribution synthesis techniques the short time fourier transform stft the ambiguity function af and the wigner distribution wd are mixed time frequency signal representations that use fourier transform techniques to map a one dimensional function of time into a two dimensional function of time and frequency these mixed time frequency mappings have been used to analyze the local frequency characteristics of a variety of signals and systems although much work has also been done to develop stft and af synthesis algorithms that can be used to implement a variety of time varying signal processing operations no such synthesis techniques have thus far been developed for the wd in this paper a signal synthesis algorithm that works directly with the real valued high resolution wd will be derived examples of how this wd synthesis procedure can be used to perform time varying filtering operations or signal separation will be given

shadowing matching errors for wave front like solutions abstract consider a singularly perturbed system formula assume that the system has a sequence of regular and internal layers occurring alternatively along the x direction these multiple wave solutions can formally be constructed by matched asymptotic expansions to obtain a genuine solution we derive a spatial shadowing lemma which assures the existence of an exact solution that is near the formal asymptotic series provided 1 the residual errors are small in all the layers and 2 the matching errors are small along the lateral boundaries of the adjacent layers the method should work on some other systems like eu t e 2 d xx m u

wave propagation and sampling theory part ii sampling theory and complex waves morlet et al 1982 this issue showed the advantages of using complex values for both waves and characteristics of the media we simulated the theoretical tools we present here using the goupillaud kunetz algorithm now we present sampling methods for complex signals or traces corresponding to received waves and sampling methods for complex characterization of multilayered or heterogeneous media regarding the complex signals we present a twodimecsional 2 d method of sampling in the time frequency domain using a special or extended gabor expansion on a set of basic wavelets adapted to phase preservation such a 2 d expansion permits us to handle in a proper manner instantaneous frequency spectra we show the differences between wavelet resolution and sampling grid resolution we also show the importance of phase preservation in high resolution seismic regarding the media we show how analytical studies of wave propagation in periodic structured layers could help when trying to characterize the physical properties of the layers and their large scale granularity as a result of complex deconvolution analytical studies of wave propagation in periodic structures are well known in solid state physics and lead to the so called bloch waves the introduction of complex waves leads to replacing the classical wave equation by a schriidinger equation finally we show that complex wave equations gabor expansion and bloch waves are three different ways of introducing the tools of quantum mechanics in highresolution seismic gabor 1946 kittel 1976 morlet 1975 and conversely the goupillaud kunetz algorithm and an extended gabor expansion may be of some use in solid state physics

high resolution bootstrapped differential semblance abstractwe formulated two coherency measures based on the bootstrapped differential semblance bds estimator that offered higher resolution in parameter tracking than did standard normalized differential semblance bootstrapping is a statistical resampling procedure used to infer estimates of standard errors and confidence intervals from data samples for which the statistical properties are unattainable via simple means or when the probability density function is unkown or difficult to estimate the first proposed estimator was based on a deterministic sorting of original offset traces by alternating near and far offsets to achieve maximized time shifts between adjacent traces the near offsets were indexed with odd integers while the even integers were used to index far offsets that were located at a constant index increment from the previous trace the second was the product of several bds terms with the first term being the deterministic bds defined above the other terms were generated by random

dynamics as a mechanism preventing the formation of finer and finer microstructure we study the dynamics of pattern formation in the one dimensional partial differential equation u u w u x x u xxt u 0 text u u x t text x in 0 1 text t 0 proposed recently by ball holmes james pego swart bhjps as a mathematical cartoon for the dynamic formation of microstructures observed in various crystalline solids here w is a double well potential like 1 4 ux 2 1 2 what makes this equation interesting and unusual is that it possesses as a lyapunov function a free energy consisting of kinetic energy plus a nonconvex elastic energy but no interfacial energy contribution which does not attain a minimum but favours the formation of finer and finer phase mixtures e u u t int limits 0 1 frac u t 2 2 w u x frac u 2 2 dx our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics conjectured in bhjps on the basis of numerical simulations of swart holmes sh while minimizing the above energy predicts infinitely fine patterns mathematically weak but not strong convergence of all minimizing sequences unvn of e u v in the sobolev space w1p 0 1 l2 0 1 solutions to the evolution equation of ball et al typically develop patterns of small but finite length scale mathematically strong convergence in w1p 0 1 l2 0 1 of all solutions u t ut t with low initial energy as time t

velocity analysis using similarity weighted semblance weighted semblance can be used for improving the performance of traditional semblance for specific data sets we have developed a novel approach for prestack velocity analysis using weighted semblance the novelty came from a different weighting criteria in which the local similarity between each trace and a reference trace is used on the one hand low similarity corresponded to a noise point or a point indicating incorrect moveout which should be given a small weight on the other hand high similarity corresponded to a point indicating correct moveout that should be given a high weight our approach could also be effectively used for analyzing avo anomalies with increased resolution compared with ab semblance synthetic and field common midpoint gathers demonstrated higher resolution using the approach we developed applications of the proposed method on a prestack data set further confirmed that the stacked data using the similarity weighted semblance could obtain better energy focused events which indicated a more precise velocity picking

high resolution spectral analysis of images using the pseudo wigner distribution several methods for the computation of the discrete wigner distribution dwd through the use of two dimensional 2 d analytic signals have been proposed depending of the direction of the phase shift most of the methods cope with the problem of aliasing of the dwd by low pass prefiltering the spectrum but reducing spatial frequency support a new method for 2 d dwd free of aliasing and simultaneously increasing the spatial frequency support is proposed through the use of a new analytic signal in this way local spectral analysis for small window sizes can be accomplished improving the resolution but reducing cross terms and other interferences inherent in the dwd computation for the three analytic images used in the comparison the method proposed here also performs better than the other two for such tasks

the keller segel model with logistic sensitivity function and small diffusivity the keller segel model is the classical model for chemotaxis of cell populations it consists of a drift diffusion equation for the cell density coupled to an equation for the chemoattractant here a variant of this model is studied in one dimensional position space where the chemotactic drift is turned off for a limiting cell density by a logistic term and where the chemoattractant density solves an elliptic equation modeling a quasi stationary balance of reaction and diffusion with production of the chemoattractant by the cells the case of small cell diffusivity is studied by asymptotic and numerical methods on a time scale characteristic for the convective effects convergence of solutions to weak entropy solutions of the limiting nonlinear hyperbolic conservation law is proven numerical and analytic evidence indicates that solutions of this problem converge to irregular patterns of cell aggregates separated by entropic shocks from vacuum regions as time tends to infinity close to each of these p

an exhaustion of locally symmetric spaces by compact submanifolds with corners letx be a riemannian symmetric space of noncompact type and rank 2 and let be a non uniform irreducible lattice on the locally symmetric quotientv x we construct an exhaustion functionh v 0 whose sublevel sets h s are compact submanifolds ofv with corners the top dimensional boundary faces of h s are parts of certain horospheres that join together at the corners it can be shown that actually h s is a submanifold with corners isomorphic to the borel serre compactification ofv

optimum smoothing of the wigner ville distribution a compromise is found between the different requirements that we would like to be fulfilled by a time frequency distribution namely positivity and obtention of a distribution close to the dirac one for the unimodular signal s t exp i phi t the fulfillment of the marginal conditions being of less interest in signal theory starting from the usual wigner ville distribution we define an optimum smoothing by minimizing the width of the different functions approximating the desired dirac distribution the smoothing is obtained by a convolution through a double gaussian of width t and such that t 1 2 two possibilities appear in the first one we do not introduce any correlation between t and in the convolution kernel and obtain a simple result in the second one extrapolating the frequency variation and still using a gaussian we obtain a better result although the smoothing process becomes more complex these results to be physically meaningful impose inequalities on the successive derivatives of which are equivalent to those used for the obtention of the classical limit for the corresponding quantum problem

optimality of high resolution array processing using the eigensystem approach in the classical approach to underwater passive listening the medium is sampled in a convenient number of look directions from which the signals are estimated in order to build an image of the noise field in contrast a modern trend is to consider the noise field as a global entity depending on few parameters to be estimated simultaneously in a gaussian context it is worthwhile to consider the application of likelihood methods in order to derive a detection test for the number of sources and estimators for their locations and spectral levels this paper aims to compute such estimators when the wavefront shapes are not assumed known a priori this justifies results previously found using the asymptotical properties of the eigenvalue eigenvector decomposition of the estimated spectral density matrix of the sensor signals they have led to a variety of high resolution array processing methods more specifically a covariance matrix test for equality of the smallest eigenvalues is presented for source detection for source localization a best fit method and a test of orthogonality between the smallest eigenvectors and the source vectors are discussed

positive time frequency distributions based on joint marginal constraints the authors study the formulation of members of the cohen posch 1985 class of positive time frequency energy distributions minimization of cross entropy measures with respect to different priors and the case of no prior or maximum entropy were considered they conclude that in general the information provided by the classical marginal constraints is very limited and thus the final distribution heavily depends on the prior distribution to overcome this limitation joint time and frequency marginals are derived based on a direction invariance criterion on the time frequency plane that are directly related to the fractional fourier transform

high resolution velocity spectra using eigenstructure methods stacking spectra provide maximum likelihood estimates for the stacking velocity or for the ray parameter of well separated reflections in additive white noise however the resolution of stacking spectra is limited by the aperture of the array and the frequency of the data despite these limitations parametric spectral estimation methods achieve better resolution than does stacking to improve resolution the parametric methods introduce a parsimonious model for the spectrum of the data in particular when the data are modeled as the superposition of wavefronts the properties of the eigenstructure of the data covariance matrix can be used to obtain high resolution spectra the traditional stacking spectra can also be expressed as a function of the data covariance matrix and directly compared to the eigenstructure spectra the superiority of the latter in separating closely interfering reflections is then apparent from a simple geometric interpretation eigenstructure methods were originally developed