Stochastic geometry and topology of non-Gaussian fields

Gaussian random fields pervade all areas of science. However, it is often the departures from Gaussianity that carry the crucial signature of the nonlinear mechanisms at the heart of diverse phenomena, ranging from structure formation in condensed matter and cosmology to biomedical imaging. The standard test of non-Gaussianity is to measure higher order correlation functions. In the present work, we take a different route. We show how geometric and topological properties of Gaussian fields, such as the statistics of extrema, are modified by the presence of a non-Gaussian perturbation. The resulting discrepancies give an independent way to detect and quantify non-Gaussianities. In our treatment, we consider both local and nonlocal mechanisms that generate non-Gaussian fields, both statically and dynamically through nonlinear diffusion.Comment: 8 pages, 4 figure

Nonlinear r-modes in a spherical shell: issues of principle

We use a simple physical model to study the nonlinear behaviour of the r-mode instability. We assume that r-modes (Rossby waves) are excited in a thin spherical shell of rotating incompressible fluid. For this case, exact Rossby wave solutions of arbitrary amplitude are known. We find that: (a) These nonlinear Rossby waves carry ZERO physical angular momentum and positive physical energy, which is contrary to the folklore belief that the r-mode angular momentum and energy are negative. (b) Within our model, we confirm the differential drift reported by Rezzolla, Lamb and Shapiro (1999). Radiation reaction is introduced into the model by assuming that the fluid is electrically charged; r-modes are coupled to electromagnetic radiation through current (magnetic) multipole moments. We find that: (c) To linear order in the mode amplitude, r-modes are subject to the CFS instability, as expected. (d) Radiation reaction decreases the angular velocity of the shell and causes differential rotation (which is distinct from but similar in magnitude to the differential drift reported by Rezzolla et al.) prior to saturation of the r-mode growth. This is contrary to the phenomenological treatments to date, which assume that the loss of stellar angular momentum is accounted for by the r-mode growth. We demonstrate, for the first time, that r-mode radiation reaction leads to differential rotation. (e) We show that for l=2 r-mode electromagnetic radiation reaction is equivalent to gravitational radiation reaction in the lowest post-Newtonian order.Comment: 8 pages, no figures, uses MNRAS style, abstract abridged to fit into 24 line

Analysis of false waves in numerical sea simulations

[EN] It is common practice to consider the random sea waves as a succession of discrete waves characterized by individual amplitudes and periods. The zero-up-crossing criterion for discretizing waves, as well as other criteria proposed by different authors, has been found to isolate some discrete waves that do not correspond to physical waves. These false waves alter the wave statistics of random sea waves. A new orbital criterion is proposed to avoid this problem. The orbital criterion has been shown to be consistent and robust with respect to the zero-up-crossing criterion. Furthermore, the new criterion produces a distribution of wave heights in better agreement with the Rayleigh distribution. The mean period of the discrete waves corresponding to the orbital criterion is proved to be T01, while the mean period of the zero-up-crossing waves is T02. A formula relating the Longuet-Higgins spectral bandwidth nu with the relative number of false waves is given.Gimenez Valentin, MH.; Sánchez Carratalá, CR.; Medina, JR. (1994). Analysis of false waves in numerical sea simulations. Ocean Engineering. 21(8):751-764. doi:10.1016/0029-8018(94)90050-7S75176421

Geometrical phase driven predissociation: Lifetimes of 2^2 A' levels of H_3

We discuss the role of the geometrical phase in predissociation dynamics of vibrational states near a conical intersection of two electronic potential surfaces of a D_{3h} molecule. For quantitative description of the predissociation driven by the coupling near a conical intersection, we developed a method for calculating lifetimes and positions of vibrational predissociated states (Feshbach resonances) for X_3 molecule. The method takes into account the two coupled three-body potential energy surfaces, which are degenerate at the intersection. As an example, we apply the method to obtain lifetimes and positions of resonances of predissociated vibrational levels of the 2^2 A' electronic state of the H_3 molecule. The three-body recombination rate coefficient for the H+H+H -> H_2+H process is estimated.Comment: 4 pages, 4 figure

On the wave-induced difference in mean sea level between the two sides of a submerged breakwater

Very simple formulae are derived for the difference in mean level between the two sides of a submerged breakwater when waves are incident on it at an arbitrary angle. The formulae apply also to waves undergoing refraction due to changes in depth and to waves in open channel transitions

On the statistical distribution of the heights of sea waves

The statistical distribution of wave-heights is derived theoretically on the assumptions (a) that the wave spectrum contains a single narrow band of frequencies, and (b) that the wave energy is being received from a large number of different sources whose phases are random. Theoretical relations are found between the root-meansquare wave-height, the mean height of the highest one-third (or highest one-tenth) waves and the most probable height of the largest wave in a given interval of time. There is close agreement with observation

Quasi-exact-solution of the Generalized Exe Jahn-Teller Hamiltonian

We consider the solution of a generalized Exe Jahn-Teller Hamiltonian in the context of quasi-exactly solvable spectral problems. This Hamiltonian is expressed in terms of the generators of the osp(2,2) Lie algebra. Analytical expressions are obtained for eigenstates and eigenvalues. The solutions lead to a number of earlier results discussed in the literature. However, our approach renders a new understanding of ``exact isolated'' solutions

Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions

Several theories for weakly damped free-surface flows have been formulated. In this paper we use the linear approximation to the Navier-Stokes equations to derive a new set of equations for potential flow which include dissipation due to viscosity. A viscous correction is added not only to the irrotational pressure (Bernoulli's equation), but also to the kinematic boundary condition. The nonlinear Schr\"odinger (NLS) equation that one can derive from the new set of equations to describe the modulations of weakly nonlinear, weakly damped deep-water gravity waves turns out to be the classical damped version of the NLS equation that has been used by many authors without rigorous justification

Empirically testing <i>Tonnetz</i>, voice-leading, and spectral models of perceived triadic distance

We compare three contrasting models of the perceived distance between root-position major and minor chords and test them against new empirical data. The models include a recent psychoacoustic model called spectral pitch class distance, and two well-established music theoretical models – Tonnetz distance and voice-leading distance. To allow a principled challenge, in the context of these data, of the assumptions behind each of the models, we compare them with a simple “benchmark” model that simply counts the number of common tones between chords. Spectral pitch class and Tonnetz have the highest correlations with the experimental data and each other, and perform significantly better than the benchmark. The voice-leading model performs worse than the benchmark. We suggest that spectral pitch class distance provides a psychoacoustic explanation for perceived harmonic distance and its music theory representation, the Tonnetz. Scores and MIDI files of the stimuli, the experimental data, and the computational models are available in the online supplement