Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates

In this paper we investigate the approximation properties of kernel interpolants on manifolds. The kernels we consider will be obtained by the restriction of positive definite kernels on $\R^d$, such as radial basis functions (RBFs), to a smooth, compact embedded submanifold \M\subset \R^d. For restricted kernels having finite smoothness, we provide a complete characterization of the native space on \M. After this and some preliminary setup, we present Sobolev-type error estimates for the interpolation problem. Numerical results verifying the theory are also presented for a one-dimensional curve embedded in $\R^3$ and a two-dimensional torus

A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces

In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded in $\mathbb{R}^3$, these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions (RBFs) and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at "scattered" locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented

A study of ion composition and dynamics at Comet Halley

This report details the participation by Lockheed co-investigators in the reduction, analysis, and interpretation of data obtained by the Ion Mass Spectrometer onboard the Giotto mission to Comet Halley. The data analysis activities and much of the scientific collaboration was shared by this team. One objective of the effort under this contract was to use data obtained by the Giotto Ion Mass Spectrometer (IMS) during the encounter with comet Halley for the purpose of advancing our understanding of the chemistry and physics of the interaction of the solar wind with comets and obtaining new information on the comet's composition. An additional objective was to make this unique data set available in a format which can be easily used by the reset of the cometary science community for other analysis in the future. The IMS has two sensors: the High Intensity Spectrometer (HIS) and the High Energy Range Spectrometer (HERS)

A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations

A new projection method based on radial basis functions (RBFs) is presented for discretizing the incompressible unsteady Stokes equations in irregular geometries. The novelty of the method comes from the application of a new technique for computing the Leray-Helmholtz projection of a vector field using generalized interpolation with divergence-free and curl-free RBFs. Unlike traditional projection methods, this new method enables matching both tangential and normal components of divergence-free vector fields on the domain boundary. This allows incompressibility of the velocity field to be enforced without any time-splitting or pressure boundary conditions. Spatial derivatives are approximated using collocation with global RBFs so that the method only requires samples of the field at (possibly scattered) nodes over the domain. Numerical results are presented demonstrating high-order convergence in both space (between 5th and 6th order) and time (up to 4th order) for some model problems in two dimensional irregular geometries.Comment: 34 pages, 8 figure

Evidence of component merging equatorward of the cusp

The Polar spacecraft passed through a region near the dayside magnetopause on May 29, 1996, at a geocentric distance of similar to 8 R-E and high, northern magnetic latitudes. The interplanetary magnetic field (IMF) was northward during the pass. Data from the Thermal Ion Dynamics Experiment revealed the existence of low-speed (similar to 50 km s(-1)) ion D-shaped distributions mixed with cold ions (similar to 2 eV) over a period of 2.5 hours. These ions were traveling parallel to the magnetic field toward the Northern Hemisphere ionosphere and were convecting primarily eastward. The D-shaped distributions are distinct from a convecting Maxwellian and, along with the magnetic field direction, are taken as evidence that the spacecraft was inside the magnetosphere and not in the magnetosheath. Furthermore, the absence of ions in the antiparallel direction is taken as evidence that low-shear merging was occurring at a location southward of the spacecraft and equatorward of the Southern Hemisphere cusp. The cold ions were of ionospheric origin, with initially slow field-aligned speeds, which were accelerated upon reflection from the magnetopause. These observations provide significant new evidence consistent with component magnetic merging sites equatorward of the cusp for northward IMF

Initiating a New Perspective: The Prevalence, Perceptions, and Attitudes Towards Hazing Across Student Organizations on a College Campus

Hazing is a widespread problem occurring on college campuses across the United States. This current study explored the prevalence, perceptions, and attitudes towards hazing at a medium-sized mid-west university. Specifically, the researchers were interested in undergraduate students’ involvement, attitudes toward hazing, and identification within student groups at the medium-sized university, Notably, though hazing is defined, it has been seen that students’ often do not identify the behaviors as such. Using a web-based questionnaire, a random sample of 1,300 undergraduate students at the participating university were asked questions regarding their participation and victimization of hazing behaviors. Results revealed that 53.8% of college students have a clear understanding of the hazing policy at their university. Sleep deprivation and associating with specific people and not others are hazing practices commonly reported. To date, hazing research on college campuses has focused on Greek life and sports while examining a generalized portion of demographic factors. Findings from this current study will target and identify hazing behaviors within a larger scale of student groups and identify demographical differences. Further, these quantitative results can be used for hazing prevention measures across college/universities