Cross-Spectral Analysis of Midfrequency Acoustic Waves Reflected by the Seafloor

Direct path measurements of a single-bottom interacting path on a vertical array are used to probe the seabed structure. The phase of the cross-spectrum, commonly used in engineering acoustics, permits examination of the importance of subbottom paths. When the cross-spectral phase is linear with frequency it implies that source to receiver propagation is dominated by a single path. A linear cross-spectral phase would also satisfy the linear seabed reflection coefficient phase approximation sometimes employed in forward modeling and geoacoustic inversion approaches. Shallow water measurements of the cross-spectrum, however, evidence a strongly nonlinear phase, below about 1500 Hz at one site, and 600 Hz at another site, implying that: 1) the subbottom structure plays an important role (i.e., a seabed half-space approximation would be inappropriate); and 2) the linear reflection phase approximation would be violated at those frequencie

Multivariate Davenport series

We consider series of the form $\sum a_n \{n\cdot x\}$, where $n\in\Z^{d}$ and $\{x\}$ is the sawtooth function. They are the natural multivariate extension of Davenport series. Their global (Sobolev) and pointwise regularity are studied and their multifractal properties are derived. Finally, we list some open problems which concern the study of these series.Comment: 43 page

Fast computation of the matrix exponential for a Toeplitz matrix

The computation of the matrix exponential is a ubiquitous operation in numerical mathematics, and for a general, unstructured $n\times n$ matrix it can be computed in $\mathcal{O}(n^3)$ operations. An interesting problem arises if the input matrix is a Toeplitz matrix, for example as the result of discretizing integral equations with a time invariant kernel. In this case it is not obvious how to take advantage of the Toeplitz structure, as the exponential of a Toeplitz matrix is, in general, not a Toeplitz matrix itself. The main contribution of this work are fast algorithms for the computation of the Toeplitz matrix exponential. The algorithms have provable quadratic complexity if the spectrum is real, or sectorial, or more generally, if the imaginary parts of the rightmost eigenvalues do not vary too much. They may be efficient even outside these spectral constraints. They are based on the scaling and squaring framework, and their analysis connects classical results from rational approximation theory to matrices of low displacement rank. As an example, the developed methods are applied to Merton's jump-diffusion model for option pricing