The Green’s function of the mild-slope equation:: The case of a monotonic bed profile

In the present work the Green’s function of the mild-slope and the modified mild-slope equations is studied. An effective numerical Fourier inversion scheme has been developed and applied to the construction and study of the source-generated water-wave potential over an uneven bottom profile with different depths at infinity. In this sense, the present work is a prerequisite to the study of the diffraction of water waves by localized bed irregularities superimposed over an uneven bottom. In the case of a monotonic bed profile, the main characteristics of the far-field are: (i) the formation of a shadow zone with an ever expanding width, which is located along the bottom irregularity, and (ii) in each of the two sectors not including the bottom irregularity the asymptotic behavior of the wave field approaches the form of an outgoing cylindrical wave, propagating with an amplitude of order O(R^−1/2), where R is the distance from the source, and wavelength corresponding to the sector-depth at infinity. Moreover, the weak wave system propagating in the shadow zone is of order O(R^−3/2), and along the bottom irregularity consists of the superposition of two outgoing waves with wavelengths corresponding to the two depths at infinity.     

Exact Results and Bounds for the Joint Tail and Moments of the Recurrence Times in a Renewal Process

Methodology and Computing in Applied Probability - The best known result about the joint distribution of the backward and forward recurrence times in a renewal process concerns the asymptotic...     

Systematic classical continuum limits of integrable spin chains and emerging novel dualities

We examine certain classical continuum long wave-length limits of prototype integrable quantum spin chains. We define the corresponding construction of classical continuum Lax operators. Our discussion starts with the XXX chain, the anisotropic Heisenberg model and their generalizations and extends to the generic isotropic and anisotropic gln$g{l}_n$ magnets. Certain classical and quantum integrable models emerging from special “dualities” of quantum spin chains, parametrized by c-number matrices, are also presented.