Geometric discretization through primal-dual meshes

This thesis introduces new tools for geometric discretization in computer graphics and computational physics. Our work builds upon the duality between weighted triangulations and power diagrams to provide concise, yet expressive discretization of manifolds and differential operators. Our exposition begins with a review of the construction of power diagrams, followed by novel optimization procedures to fully control the local volume and spatial distribution of power cells. Based on this power diagram framework, we develop a new family of discrete differential operators, an effective stippling algorithm, as well as a new fluid solver for Lagrangian particles. We then turn our attention to applications in geometry processing. We show that orthogonal primal-dual meshes augment the notion of local metric in non-flat discrete surfaces. In particular, we introduce a reduced set of coordinates for the construction of orthogonal primal-dual structures of arbitrary topology, and provide alternative metric characterizations through convex optimizations. We finally leverage these novel theoretical contributions to generate well-centered primal-dual meshes, sphere packing on surfaces, and self-supporting triangulations.

Part I - The radiation of elastic waves from a spherical cavity in a half space. Part II - Precision determination of focal depths and epicenters of earthquakes.

Part I: The dynamic response of an elastic half space to an explosion in a buried spherical cavity is investigated by two methods. The first is implicit, and the final expressions for the displacements at the free surface are given as a series of spherical wave functions whose coefficients are solutions of an infinite set of linear equations. The second method is based on Schwarz's technique to solve boundary value problems, and leads to an iterative solution, starting with the known expression for the point source in a half space as first term. The iterative series is transformed into a system of two integral equations, and into an equivalent set of linear equations. In this way, a dual interpretation of the physical phenomena is achieved. The systems are treated numerically and the Rayleigh wave part of the displacements is given in the frequency domain. Several comparisons with simpler cases are analyzed to show the effect of the cavity radius-depth ratio on the spectra of the displacements.

Part II: A high speed, large capacity, hypocenter location program has been written for an IBM 7094 computer. Important modifications to the standard method of least squares have been incorporated in it. Among them are a new way to obtain the depth of shocks from the normal equations, and the computation of variable travel times for the local shocks in order to account automatically for crustal variations. The multiregional travel times, largely based upon the investigations of the United States Geological Survey, are confronted with actual traverses to test their validity.

It is shown that several crustal phases provide control enough to obtain good solutions in depth for nuclear explosions, though not all the recording stations are in the region where crustal corrections are considered. The use of the European travel times, to locate the French nuclear explosion of May 1962 in the Sahara, proved to be more adequate than previous work.

A simpler program, with manual crustal corrections, is used to process the Kern County series of aftershocks, and a clearer picture of tectonic mechanism of the White Wolf fault is obtained.

Shocks in the California region are processed automatically and statistical frequency-depth and energy depth curves are discussed in relation to the tectonics of the area.

A perfect day for zebrafish : neuromodulation of sleep in a diurnal vertebrate

Every day, we shift among various states of sleep and arousal to meet the many demands of our bodies and environment. A central puzzle in neurobiology is how the brain controls these behavioral states, which are essential to an animal's well-being and survival. Mammalian models have predominated sleep and arousal research, although in the past decade, invertebrate models have made significant contributions to our understanding of the genetic underpinnings of behavioral states. More recently, the zebrafish (Danio rerio), a diurnal vertebrate, has emerged as a promising model system for sleep and arousal research.

In this thesis, I describe two studies on sleep/arousal pathways that I conducted using zebrafish, and I discuss how the findings can be combined in future projects to advance our understanding of vertebrate sleep/arousal pathways. In the first study, I discovered a neuropeptide that regulates zebrafish sleep and arousal as a result of a large-scale effort to identify molecules that regulate behavioral states. Taking advantage of facile zebrafish genetics, I constructed mutants for the three known receptors of this peptide and identified the one receptor that exclusively mediates the observed behavioral effects. I further show that the peptide exerts its behavioral effects independently of signaling at a key module of a neuroendocrine signaling pathway. This finding contradicts the hypothesis put forth in mammalian systems that the peptide acts through the classical neuroendocrine pathway; our data further generate new testable hypotheses for determining the central nervous system or alternative neuroendocrine pathways involved.

Second, I will present the development of a chemigenetic method to non-invasively manipulate neurons in the behaving zebrafish. I validated this technique by expressing and inducing the chemigenetic tool in a restricted population of sleep-regulating neurons in the zebrafish. As predicted by established models of this vertebrate sleep regulator, chemigenetic activation of these neurons induced hyperactivity, whereas chemigenetic ablation of these neurons induced increased sleep behavior. Given that light is a potent modulator of behavior in zebrafish, our proof-of-principle data provide a springboard for future studies of sleep/arousal and other light-dependent behaviors to interrogate genetically-defined populations of neurons independently of optogenetic tools.