On the connection between the Nekhoroshev theorem and Arnold Diffusion

The analytical techniques of the Nekhoroshev theorem are used to provide estimates on the coefficient of Arnold diffusion along a particular resonance in the Hamiltonian model of Froeschl\'{e} et al. (2000). A resonant normal form is constructed by a computer program and the size of its remainder $||R_{opt}||$ at the optimal order of normalization is calculated as a function of the small parameter $\epsilon$. We find that the diffusion coefficient scales as $D\propto||R_{opt}||^3$, while the size of the optimal remainder scales as $||R_{opt}|| \propto\exp(1/\epsilon^{0.21})$ in the range $10^{-4}\leq\epsilon \leq 10^{-2}$. A comparison is made with the numerical results of Lega et al. (2003) in the same model.Comment: Accepted in Celestial Mechanics and Dynamical Astronom

An asynchronous leapfrog method II

A second order explicit one-step numerical method for the initial value problem of the general ordinary differential equation is proposed. It is obtained by natural modifications of the well-known leapfrog method, which is a second order, two-step, explicit method. According to the latter method, the input data for an integration step are two system states, which refer to different times. The usage of two states instead of a single one can be seen as the reason for the robustness of the method. Since the time step size thus is part of the step input data, it is complicated to change this size during the computation of a discrete trajectory. This is a serious drawback when one needs to implement automatic time step control. The proposed modification transforms one of the two input states into a velocity and thus gets rid of the time step dependency in the step input data. For these new step input data, the leapfrog method gives a unique prescription how to evolve them stepwise. The stability properties of this modified method are the same as for the original one: the set of absolute stability is the interval [-i,+i] on the imaginary axis. This implies exponential growth of trajectories in situations where the exact trajectory has an asymptote. By considering new evolution steps that are composed of two consecutive old evolution steps we can average over the velocities of the sub-steps and get an integrator with a much larger set of absolute stability, which is immune to the asymptote problem. The method is exemplified with the equation of motion of a one-dimensional non-linear oscillator describing the radial motion in the Kepler problem.Comment: 41 pages, 25 figure

Generic Tracking of Multiple Apparent Horizons with Level Flow

We report the development of the first apparent horizon locator capable of finding multiple apparent horizons in a ``generic'' numerical black hole spacetime. We use a level-flow method which, starting from a single arbitrary initial trial surface, can undergo topology changes as it flows towards disjoint apparent horizons if they are present. The level flow method has two advantages: 1) The solution is independent of changes in the initial guess and 2) The solution can have multiple components. We illustrate our method of locating apparent horizons by tracking horizon components in a short Kerr-Schild binary black hole grazing collision.Comment: 13 pages including figures, submitted to Phys Rev

Digitization of sunspot drawings by Sp\"orer made in 1861-1894

Most of our knowledge about the Sun's activity cycle arises from sunspot observations over the last centuries since telescopes have been used for astronomy. The German astronomer Gustav Sp\"orer observed almost daily the Sun from 1861 until the beginning of 1894 and assembled a 33-year collection of sunspot data covering a total of 445 solar rotation periods. These sunspot drawings were carefully placed on an equidistant grid of heliographic longitude and latitude for each rotation period, which were then copied to copper plates for a lithographic reproduction of the drawings in astronomical journals. In this article, we describe in detail the process of capturing these data as digital images, correcting for various effects of the aging print materials, and preparing the data for contemporary scientific analysis based on advanced image processing techniques. With the processed data we create a butterfly diagram aggregating sunspot areas, and we present methods to measure the size of sunspots (umbra and penumbra) and to determine tilt angles of active regions. A probability density function of the sunspot area is computed, which conforms to contemporary data after rescaling.Comment: 10 pages, 8 figures, accepted for publication in Astronomische Nachrichten/Astronomical Note

Tangent-ball techniques for shape processing

Shape processing defines a set of theoretical and algorithmic tools for creating, measuring and modifying digital representations of shapes.  Such tools are of paramount importance to many disciplines of computer graphics, including modeling, animation, visualization, and image processing.  Many applications of shape processing can be found in the entertainment and medical industries. In an attempt to improve upon many previous shape processing techniques, the present thesis explores the theoretical and algorithmic aspects of a difference measure, which involves fitting a ball (disk in 2D and sphere in 3D) so that it has at least one tangential contact with each shape and the ball interior is disjoint from both shapes. We propose a set of ball-based operators and discuss their properties, implementations, and applications.  We divide the group of ball-based operations into unary and binary as follows: Unary operators include: * Identifying details (sharp, salient features, constrictions) * Smoothing shapes by removing such details, replacing them by fillets and roundings * Segmentation (recognition, abstract modelization via centerline and radius variation) of tubular structures Binary operators include: * Measuring the local discrepancy between two shapes * Computing the average of two shapes * Computing point-to-point correspondence between two shapes * Computing circular trajectories between corresponding points that meet both shapes at right angles * Using these trajectories to support smooth morphing (inbetweening) * Using a curve morph to construct surfaces that interpolate between contours on consecutive slices The technical contributions of this thesis focus on the implementation of these tangent-ball operators and their usefulness in applications of shape processing. We show specific applications in the areas of animation and computer-aided medical diagnosis.  These algorithms are simple to implement, mathematically elegant, and fast to execute.Ph.D.Committee Chair: Jarek Rossignac; Committee Member: Greg Slabaugh; Committee Member: Greg Turk; Committee Member: Karen Liu; Committee Member: Maryann Simmon