Computing Jacobi's θ\theta in quasi-linear time

Jacobi's $\theta$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $\theta(z,\tau)$, for $z, \tau$ verifying certain conditions, with precision $P$ in $O(\mathcal{M}(P) \sqrt{P})$ bit operations, where $\mathcal{M}(P)$ denotes the number of operations needed to multiply two complex $P$-bit numbers. We generalize an algorithm which computes specific values of the $\theta$ function (the \textit{theta-constants}) in asymptotically faster time; this gives us an algorithm to compute $\theta(z, \tau)$ with precision $P$ in $O(\mathcal{M}(P) \log P)$ bit operations, for any $\tau \in \mathcal{F}$ and $z$ reduced using the quasi-periodicity of $\theta$

Analytic Construction of Periodic Orbits in the Restricted Three-Body Problem

This dissertation explores the analytical solution properties surrounding a nominal periodic orbit in two different planes, the plane of motion of the two primaries and a plane perpendicular to the line joining the two primaries, in the circular restricted three-body problem. Assuming motion can be maintained in the plane and motion of the third body is circular, Jacobi\u27s integral equation can be analytically integrated, yielding a closed-form expression for the period and path expressed with elliptic integral and elliptic function theory. In this case, the third body traverses a circular path with nonuniform speed. In a strict sense, the in-plane assumption cannot be maintained naturally. However, there may be cases where the assumption is approximately maintained over a finite time period. More importantly, the nominal solution can be used as the basis for an iterative analytical solution procedure for the three dimensional periodic trajectory where corrections are computable in closed-form. In addition, the in-plane assumption can be strictly enforced with the application of modulated thrust acceleration. In this case, the required thrust control inputs are found to be nonlinear functions in time. Total velocity increment, required to maintain the nominal orbit, for one complete period of motion of the third body is expressed as a function of the orbit characteristics

Orbital Tori Construction Using Trajectory Following Spectral Methods

By assuming the motion of a satellite about the earth’s geopotential mimics the known Kolmogorov-Arnold-Moser (KAM) solution of a lightly perturbed integrable Hamiltonian system, this research focused on applying trajectory following spectral methods to estimate orbital tori from sampled orbital data. From an estimated basis frequency set, orbital data was decomposed into multi-periodic Fourier series, essentially compressing ephemerides for long-term use. Real-world Global Positioning System (GPS) orbital tracks were decomposed and reconstructed with error from as low as few kilometers per coordinate axis over a 10-week span to tens of kilometers per coordinate axis over the same time period, depending on the method chosen. These less-than-precision-level results were due primarily to the resonant orbits of the GPS constellation. Additionally, the trajectory following spectral methods chosen experienced difficulties converging on a complete basis set when using data time spans much smaller than the period of the slowest system frequency. However, the lessons learned from GPS led to a new orbital tori construction method. This approach focused on fitting local spectral structures, denoted as frequency clusters, within the sampled orbital data to the analytical form of the windowed, truncated, continuous Fourier transform. Methods employing direct use of the observed spectrum as well as least squares fitting techniques were developed with considerable success. For portions of the low-earth-orbit regime, maximum errors per coordinate axis in orbital tori fits were kept below 5 meters over a time period of 1 year. Simulations using the Hubble Space Telescope yielded 1-dimensional root mean square errors of less than 2 meters in each coordinate axis in the initial and predicted ephemeris fits, both of which used 1-year-long tracks of numerically integrated data

Elastic and inelastic X-ray scattering studies of the low dimensional spin-1/2 quantum magnet TiOCl

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2007.MIT Science Library Copy: printed in pages.Also issued printed in pages.Includes bibliographical references (leaves 177-183).The ground state for a one dimensional spin 1/2 Heisenberg chain coupled to phonons is a dimerized singlet state known as a "spin-Peierls" state. Currently, the spin-Peierls state is realized in only a handful of known compounds. Even after decades of scientific scrutiny, there is an absence of direct measurements of the lattice dynamics associated with the transition. In this work we present an extensive study of a new one dimensional spin-Peierls compound, TiOC1. The magnetic susceptibility strongly indicates a singlet ground state, with two apparent anomalies observed at T, ,=65 K and Tc2=92 K. Specific heat measurements have been performed and the associated entropy changes quantified. The 65 K transition exhibits a thermal hysteresis, indicative of a first order phase transition. A detailed synchrotron x-ray study of the structure reveals the appearance of superlattice peaks at ( ... ) below 65 K. The intensity of the peaks drop very sharply above T, and a thermal hysteresis is observed which is consistent with a first order phase transition at 65 K. We find that the temperature region between 65 K and 92 K is characterized by a novel incommensurate state. The incommensurate reflections appear at ( ... ). The temperature dependence of the intensity of the incommensurate peaks shows a more gradual onset, with no thermal hysteresis. The incommensurate wavevectors change continuously as a function of temperature and can be analyzed in terms of a mean field theory of phase shifted discommensurations. The observation of the third harmonics enabled a careful characterization of the underlying real space superstructure. We find that all of the observed scattering can be reproduced by a one dimensional long-wavelength modulation of a locally dimerized structure.(cont.) The lattice dynamics above T 2were characterized by inelastic x-ray scattering measurements. By analyzing the data in terms of a damped harmonic oscillator response function, we are able to extract the phonon frequency and damping for all observed modes. We find a longitudinal acoustic phonon branch whose damping increases for q-vectors close to the zone boundary, which is also associated with an apparent softening of the frequency. Both of these anharmonic effects increase as T2 is approached, and are consistent with a soft phonon description of the dimerization. The anomalous phonon damping and softening are then analyzed using the Cross & Fisher theory of spin-phonon interaction leading to a spin-Peierls transition. We find that the theory succeeds in describing the data for a narrow temperature range about Tc2, for q near the zone boundary. It does not account for the anharmonic effects observed at high temperatures. Our experimental analysis represents one of the most in-depth quantitative tests of the Cross & Fisher theory to date. In addition our results suggest that TiOCl is a particularly ideal realization of a spin-Peierls system.by Eric T. Abel.Ph.D

Aeronautical engineering: A continuing bibliography with indexes (supplement 270)

This bibliography lists 600 reports, articles, and other documents introduced into the NASA scientific and technical information system in September, 1991. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics

Second International Workshop on Harmonic Oscillators

The Second International Workshop on Harmonic Oscillators was held at the Hotel Hacienda Cocoyoc from March 23 to 25, 1994. The Workshop gathered 67 participants; there were 10 invited lecturers, 30 plenary oral presentations, 15 posters, and plenty of discussion divided into the five sessions of this volume. The Organizing Committee was asked by the chairman of several Mexican funding agencies what exactly was meant by harmonic oscillators, and for what purpose the new research could be useful. Harmonic oscillators - as we explained - is a code name for a family of mathematical models based on the theory of Lie algebras and groups, with applications in a growing range of physical theories and technologies: molecular, atomic, nuclear and particle physics; quantum optics and communication theory