Generalized Morse Wavelets as a Superfamily of Analytic Wavelets

The generalized Morse wavelets are shown to constitute a superfamily that essentially encompasses all other commonly used analytic wavelets, subsuming eight apparently distinct types of analysis filters into a single common form. This superfamily of analytic wavelets provides a framework for systematically investigating wavelet suitability for various applications. In addition to a parameter controlling the time-domain duration or Fourier-domain bandwidth, the wavelet {\em shape} with fixed bandwidth may be modified by varying a second parameter, called $\gamma$. For integer values of $\gamma$, the most symmetric, most nearly Gaussian, and generally most time-frequency concentrated member of the superfamily is found to occur for $\gamma=3$. These wavelets, known as "Airy wavelets," capture the essential idea of popular Morlet wavelet, while avoiding its deficiencies. They may be recommended as an ideal starting point for general purpose use

The Distribution of the Area under a Bessel Excursion and its Moments

A Bessel excursion is a Bessel process that begins at the origin and first returns there at some given time $T$. We study the distribution of the area under such an excursion, which recently found application in the context of laser cooling. The area $A$ scales with the time as $A \sim T^{3/2}$, independent of the dimension, $d$, but the functional form of the distribution does depend on $d$. We demonstrate that for $d=1$, the distribution reduces as expected to the distribution for the area under a Brownian excursion, known as the Airy distribution, deriving a new expression for the Airy distribution in the process. We show that the distribution is symmetric in $d-2$, with nonanalytic behavior at $d=2$. We calculate the first and second moments of the distribution, as well as a particular fractional moment. We also analyze the analytic continuation from $d2$. In the limit where $d\to 4$ from below, this analytically continued distribution is described by a one-sided L\'evy $\alpha$-stable distribution with index $2/3$ and a scale factor proportional to $[(4-d)T]^{3/2}$

Area distribution of the planar random loop boundary

We numerically investigate the area statistics of the outer boundary of planar random loops, on the square and triangular lattices. Our Monte Carlo simulations suggest that the underlying limit distribution is the Airy distribution, which was recently found to appear also as area distribution in the model of self-avoiding loops.Comment: 10 pages, 2 figures. v2: minor changes, version as publishe

Quasi-Resonant Theory of Tidal Interactions

When a spinning system experiences a transient gravitational encounter with an external perturber, a quasi-resonance occurs if the spin frequency of the victim matches the peak orbital frequency of the perturber. Such encounters are responsible for the formation of long tails and bridges of stars during galaxy collisions. For high-speed encounters, the resulting velocity perturbations can be described within the impulse approximation. The traditional impulse approximation, however, does not distinguish between prograde and retrograde encounters, and therefore completely misses the resonant response. Here, using perturbation theory, we compute the effects of quasi-resonant phenomena on stars orbiting within a disk. Explicit expressions are derived for the velocity and energy change to the stars induced by tidal forces from an external gravitational perturber passing either on a straight line or parabolic orbit. Comparisons with numerical restricted three-body calculations illustrate the applicability of our analysis.Comment: 22 pages, 13 figures, ApJ submitted, numerical routines for evaluation of special functions and analytical results are provided upon reques

The band spectrum of the periodic airy-schrodinger operator on the real line

We introduce the periodic Airy-Schr\"odinger operator and we study its band spectrum. This is an example of an explicitly solvable model with a periodic potential which is not differentiable at its minima and maxima. We define a semiclassical regime in which the results are stated for a fixed value of the semiclassical parameter and are thus estimates instead of asymptotic results. We prove that there exists a sequence of explicit constants, which are zeroes of classical functions, giving upper bounds of the semiclassical parameter for which the spectral bands are in the semiclassical regime. We completely determine the behaviour of the edges of the first spectral band with respect to the semiclassical parameter. Then, we investigate the spectral bands and gaps situated in the range of the potential. We prove precise estimates on the widths of these spectral bands and these spectral gaps and we determine an upper bound on the integrated spectral density in this range. Finally, in the semiclassical regime, we get estimates of the edges of every spectral bands and thus of the widths of every spectral bands and spectral gaps

Tail estimates for the Brownian excursion area and other Brownian areas

Several Brownian areas are considered in this paper: the Brownian excursion area, the Brownian bridge area, the Brownian motion area, the Brownian meander area, the Brownian double meander area, the positive part of Brownian bridge area, the positive part of Brownian motion area. We are interested in the asymptotics of the right tail of their density function. Inverting a double Laplace transform, we can derive, in a mechanical way, all terms of an asymptotic expansion. We illustrate our technique with the computation of the first four terms. We also obtain asymptotics for the right tail of the distribution function and for the moments. Our main tool is the two-dimensional saddle point method.Comment: 34 page

Uniform asymptotics of area-weighted Dyck paths

Using the generalized method of steepest descents for the case of two coalescing saddle points, we derive an asymptotic expression for the bivariate generating function of Dyck paths, weighted according to their length and their area in the limit of the area generating variable tending towards 1. The result is valid uniformly for a range of the length generating variable, including the tricritical point of the model.Comment: 14 pages, 5 figure