Geodesics on Lie groups: Euler equations and totally geodesic subgroup

The geodesic motion on a Lie group equipped with a left or right invariant Riemannian metric is governed by the Euler-Arnold equation. This paper investigates conditions on the metric in order for a given subgroup to be totally geodesic. Results on the construction and characterisation of such metrics are given. The setting works both in the classical nite dimensional case, and in the category of in nite dimensional Fr echet Lie groups, in which di eomorphism groups are included. Using the framework we give new examples of both nite and in nite dimensional totally geodesic subgroups. In particular, based on the cross helicity, we construct right invariant metrics such that a given subgroup of exact volume preserving di eomorphisms is totally geodesic. The paper also gives a general framework for the representation of Euler-Arnold equations in arbitrary choice of dual pairing

Spontaneous radiative decay of translational levels of an atom near a dielectric surface

We study spontaneous radiative decay of translational levels of an atom in the vicinity of a semi-infinite dielectric. We systematically derive the microscopic dynamical equations for the spontaneous decay process. We calculate analytically and numerically the radiative linewidths and the spontaneous transition rates for the translational levels. The roles of the interference between the emitted and reflected fields and of the transmission into the evanescent modes are clearly identified. Our numerical calculations for the silica--cesium interaction show that the radiative linewidths of the bound excited levels with large enough but not too large vibrational quantum numbers are moderately enhanced by the emission into the evanescent modes and those for the deep bound levels are substantially reduced by the surface-induced red shift of the transition frequency

Pseudo-High-Order Symplectic Integrators

Symplectic N-body integrators are widely used to study problems in celestial mechanics. The most popular algorithms are of 2nd and 4th order, requiring 2 and 6 substeps per timestep, respectively. The number of substeps increases rapidly with order in timestep, rendering higher-order methods impractical. However, symplectic integrators are often applied to systems in which perturbations between bodies are a small factor of the force due to a dominant central mass. In this case, it is possible to create optimized symplectic algorithms that require fewer substeps per timestep. This is achieved by only considering error terms of order epsilon, and neglecting those of order epsilon^2, epsilon^3 etc. Here we devise symplectic algorithms with 4 and 6 substeps per step which effectively behave as 4th and 6th-order integrators when epsilon is small. These algorithms are more efficient than the usual 2nd and 4th-order methods when applied to planetary systems.Comment: 14 pages, 5 figures. Accepted for publication in the Astronomical Journa

Calibration of the Ames Anechoic Facility. Phase 1: Short range plan

A calibration was made of the acoustic and aerodynamic characteristics of a small, open-jet wind tunnel in an anechoic room. The jet nozzle was 102 mm diameter and was operated subsonically. The anechoic-room dimensions were 7.6 m by 5.5 m by 3.4 m high (wedge tip to wedge tip). Noise contours in the chamber were determined by various jet speeds and exhaust collector positions. The optimum nozzle/collector separation from an acoustic standpoint was 2.1 m. Jet velocity profiles and turbulence levels were measured using pressure probes and hot wires. The jet was found to be symmetric, with no unusual characteristics. The turbulence measurements were hampered by oil mist contamination of the airflow

Application of Gene Shaving and Mixture Models to Cluster Microarray Gene Expression Data

Researchers are frequently faced with the analysis of microarray data of a relatively large number of genes using a small number of tissue samples. We examine the application of two statistical methods for clustering such microarray expression data: EMMIX-GENE and GeneClust. EMMIX-GENE is a mixture-model based clustering approach, designed primarily to cluster tissue samples on the basis of the genes. GeneClust is an implementation of the gene shaving methodology, motivated by research to identify distinct sets of genes for which variation in expression could be related to a biological property of the tissue samples. We illustrate the use of these two methods in the analysis of Affymetrix oligonucleotide arrays of well-known data sets from colon tissue samples with and without tumors, and of tumor tissue samples from patients with leukemia. Although the two approaches have been developed from different perspectives, the results demonstrate a clear correspondence between gene clusters produced by GeneClust and EMMIX-GENE for the colon tissue data. It is demonstrated, for the case of ribosomal proteins and smooth muscle genes in the colon data set, that both methods can classify genes into co-regulated families. It is further demonstrated that tissue types (tumor and normal) can be separated on the basis of subtle distributed patterns of genes. Application to the leukemia tissue data produces a division of tissues corresponding closely to the external classification, acute myeloid meukemia (AML) and acute lymphoblastic leukemia (ALL), for both methods. In addition, we also identify genes specific for the subgroup of ALL-Tcell samples. Overall, we find that the gene shaving method produces gene clusters at great speed; allows variable cluster sizes and can incorporate partial or full supervision; and finds clusters of genes in which the gene expression varies greatly over the tissue samples while maintaining a high level of coherence between the gene expression profiles. The intent of the EMMIX-GENE method is to cluster the tissue samples. It performs a filtering step that results in a subset of relevant genes, followed by gene clustering, and then tissue clustering, and is favorable in its accuracy of ranking the clusters produced

Geometric Integration of Hamiltonian Systems Perturbed by Rayleigh Damping

Explicit and semi-explicit geometric integration schemes for dissipative perturbations of Hamiltonian systems are analyzed. The dissipation is characterized by a small parameter $\epsilon$, and the schemes under study preserve the symplectic structure in the case $\epsilon=0$. In the case $0<\epsilon\ll 1$ the energy dissipation rate is shown to be asymptotically correct by backward error analysis. Theoretical results on monotone decrease of the modified Hamiltonian function for small enough step sizes are given. Further, an analysis proving near conservation of relative equilibria for small enough step sizes is conducted. Numerical examples, verifying the analyses, are given for a planar pendulum and an elastic 3--D pendulum. The results are superior in comparison with a conventional explicit Runge-Kutta method of the same order

Smart automotive technology adherence to the law: (de)constructing road rules for autonomous system development, verification and safety

Driving is an intuitive task that requires skill, constant alertness and vigilance for unexpected events. The driving task also requires long concentration spans, focusing on the entire task for prolonged periods, and sophisticated negotiation skills with other road users including wild animals. Modern motor vehicles include an array of smart assistive and autonomous driving systems capable of subsuming some, most, or in limited cases, all of the driving task. Building these smart automotive systems requires software developers with highly technical software engineering skills, and now a lawyer’s in-depth knowledge of traffic legislation as well. This article presents an approach for deconstructing the complicated legalese of traffic law and representing its requirements and flow. Our approach (de)constructs road rules in legal terminology and specifies them in ‘structured English logic’ that is expressed as ‘Boolean logic’ for automation and ‘Lawmaps’ for visualization. We demonstrate an example using these tools leading to the construction and validation of a ‘Bayesian Network model’. We strongly believe these tools to be approachable by programmers and the general public, useful in development of Artificial Intelligence to underpin motor vehicle smart systems, and in validation to ensure these systems are considerate of the law when making decisions.fals

Interaction of Nonlinear Schr\"odinger Solitons with an External Potential

Employing a particularly suitable higher order symplectic integration algorithm, we integrate the 1-$d$ nonlinear Schr\"odinger equation numerically for solitons moving in external potentials. In particular, we study the scattering off an interface separating two regions of constant potential. We find that the soliton can break up into two solitons, eventually accompanied by radiation of non-solitary waves. Reflection coefficients and inelasticities are computed as functions of the height of the potential step and of its steepness.Comment: 14 pages, uuencoded PS-file including 10 figure