This paper investigates the pressure field generated at the bottom of a high-contrast randomly layered slab by an acoustical pulse emitted at the surface of the slab. This analysis takes place in the framework introduced by Asch, Kohler, Papanicolaou, Postel and White [1] where the incident pulse wave length is long compared to the correlation length of the random inhomogeneities, but short compared to the size of the slab. This problem has been studied in the one-dimensional case simultaneously by Clouet and Fouque [4] and Lewicki, Burridge and Papanicolaou [6] or for multimode plane wave pulses in Lewicki, Burridge and De Hoop [7]. These situations require only the use of classical diffusion-approximation results whereas the point-source problem studied in this paper requires a non-trivial combination of diffusion-approximation results with stationary phase methods. The stationary phase method has been used by De Hoop, Chang and Burridge [5] for weakly fluctuating media and in [1] fo...

We consider acoustic pulse propagation in inhomogeneous media over relatively long propagation distances. Our main objective is to characterize the spreading of the travelling pulse due to microscale variations in the medium parameters. The pulse is generated by a point source and the medium is modeled by a smooth three dimensional background that is modulated by stratified random fluctuations. We refer to such media as locally layered. We show that, when the pulse is observed relative to its random arrival time, it stabilizes to a shape determined by the slowly varying background convoluted with a Gaussian. The width of the Gaussian and the random travel time are determined by the medium parameters along the ray connecting the source and the point of observation. The ray is determined by high frequency asymptotics (geometrical optics). If we observe the pulse in a deterministic frame moving with the effective slowness, it does not stabilize and its mean is broader because of the random component of the travel time. The analysis of this phenomenon involves the asymptotic solution of partial differential equations with randomly varying coefficients and is based on a new representation of the field in terms of generalized plane waves that travel in opposite directions relative to the layering.

Choudhury and Lucantoni recently developed an algorithm for calculating moments of a probability distribution by numerically inverting its moment generating function. They also showed that high-order moments can be used to calculate asymptotic parameters of the complementary cumulative distribution function when an asymptotic form is assumed, such as F c (x) ~ ax b e -hx as x . Moment-based algorithms for computing asymptotic parameters are especially useful when the transforms are not available explicitly, as in models of busy periods or polling systems. Here we provide additional theoretical support for this moment-based algorithm for computing asymptotic parameters and new refined estimators for the case b 0. The new refined estimators converge much faster (as a function of moment order) than the previous estimators, which means that fewer moments are needed, thereby speeding up the algorithm. We also show how to compute all the parameters in a multi-term asymptote of the form F c (x) ~ k = 1 S m a k x b - k + 1 e -hx . We identify conditions under which the estimators converge to the asymptotic parameters and we determine rates of convergence, focusing especially on the case b 0. Even when b = 0, we show that it is necessary to assume the asymptotic form for the complementary distribution function; the asymptotic form is not implied by convergence of the moment-based estimators alone. In order to get good estimators of the asymptotic decay rate h and the asymptotic power b when b 0, a multiple-term asymptotic expansion is required. Such asymptotic expansions typically hold when b 0, corresponding to the dominant singularity of the transform being a multiple pole (b a positive integer) or an algebraic singularity (branch point, b non-integer)...

A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type (e x 2 ), that is, Gaussian. We exhibit here a new class of \universal" phenomena that are of the exponential-cubic type (e ix 3 ), corresponding to nonstandard distributions that involve the Airy function. Such Airy phenomena are expected to be found in a number of applications, when conuences of critical points and singularities occur. About a dozen classes of planar maps are treated in this way, leading to the occurrence of a common Airy distribution that describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and ne optimization of random generation algorithms for multiply connected planar graphs.

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Abstract — In this paper we present a multiple-scattering solver for non-convex geometries such as those obtained as the union of a finite number of convex surfaces. For a prescribed error tolerance this algorithm exhibits a fixed computational cost for arbitrarily high frequencies. At the core of the method is an extension of the method of stationary phase, together with the use of an ansatz for the unknown density in a combined-field boundary integral formulation. Index Terms — Wave scattering, boundary integral equations, spectral methods, high frequency methods I.

The Green’s function of acoustic or elastic wave propagation can, for loss-less media, be retrieved by correlating the wave field that is excited by random sources and is recorded at two locations. Here the generalization of this idea to attenuating acoustic waves in an inhomogeneous medium is addressed, and it is shown that the Green’s function can be retrieved from waves that are excited throughout the volume by spatially uncorrelated injection sources with a power spectrum that is proportional to the local dissipation rate. For a finite volume, one needs both volume sources and sources at the bounding surface for the extraction of the Green’s functions. For the special case of a homogeneous attenuating medium defined over a finite volume, the phase and geometrical spreading of the Green’s function is correctly retrieved when the volume sources are ignored, but the attenuation is not.

Asymptotic formulae for two-dimensional arrays (fr,s)r,s≥0 where the associated generating function F (z, w): = � fr,szrw s is meromorphic are provided. Our ap-r,s≥0 proach is geometrical. To a big extent it generalizes and completes the asymptotic description of the coefficients fr,s along a compact set of directions specified by smooth points of the singular variety of the denominator of F (z, w). The scheme we develop can lead to a high level of complexity. However, it provides the leading asymptotic order of fr,s if some unusual and pathological behavior is ruled out. It relies on the asymptotic analysis of a certain type of stationary phase integral of the form � e −s·P (d,θ) A(d, θ)dθ, which describes up to an exponential factor the asymptotic be-havior of the coefficients fr,s along the direction d = r s in the (r, s)-lattice. The cases of interest are when either the phase term P (d, θ) or the amplitude term A(d, θ) exhibits a change of degree as d approaches a degenerate direction. These are han-dled by a generalized version of the stationary phase and the coalescing saddle point method which we propose as part of this dissertation. The occurrence of two spe-cial functions related to the Airy function is established when two simple saddles of the phase term coalesce. A scheme to study the asymptotic behavior of big powers of generating functions is proposed as an additional application of these generalized methods. ii Dedicated to my mother, father and sister. iii ACKNOWLEDGMENTS I would like to thank to my advisor, Robin Pemantle, for his support and guidance throughout my graduate years at Ohio State. I am deeply indebted for he having supported me as his research assistant for an extended period of time. I also offer my gratitude for his unconditional commitment to connect and keep me in touch with

Posterior distributions in limited information analysis of the simultaneous equations model using the

A fine analysis is given of the transitional behavior of the average cost of quicksort with median-of-three. Asymptotic formulae are derived for the stepwise improvement of the average cost of quicksort when iterating median-of-three k rounds for all possible values of k. The methods used are general enough to apply to quicksort with median-of-(2t + 1) and to explain in a precise manner the transitional behaviors of the average cost from insertion sort to quicksort proper. Our results also imply nontrivial bounds on the expected height, "saturation level", and width in a random locally balanced binary search tree.