Codomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation

We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold. If the Riemannian manifold is modelled as an elastic medium, this operator represents the data available to an observer on the boundary of the manifold when the manifold is set into motion through boundary vibrations. We study the Dirichlet to Neumann operator when vibrations are imposed and data recorded on disjoint sets, a useful setting for applications. We prove that this operator determines the Dirichlet to Neumann operator where sources and observations are on the same set, provided a spectral condition on the Laplace-Beltrami operator for the manifold is satisfied. We prove this by providing an implementable procedure for determining a portion of the Riemannian manifold near the area where sources are applied. Drawing on established results, an immediate corollary is that a compact Riemannian manifold can be reconstructed from the Dirichlet to Neumann operator where sources and observations are on disjoint sets.

Codomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation

We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold. If the Riemannian manifold is modelled as an elastic medium, this operator represents the data available to an observer on the boundary of the manifold when the manifold is set into motion through boundary vibrations. We study the Dirichlet to Neumann operator when vibrations are imposed and data recorded on disjoint sets, a useful setting for applications. We prove that this operator determines the Dirichlet to Neumann operator where sources and observations are on the same set, provided a spectral condition on the Laplace-Beltrami operator for the manifold is satisfied. We prove this by providing an implementable procedure for determining a portion of the Riemannian manifold near the area where sources are applied. Drawing on established results, an immediate corollary is that a compact Riemannian manifold can be reconstructed from the Dirichlet to Neumann operator where sources and observations are on disjoint sets.

Design and Implementation of Tools for STM Studies in UHV and at the Solid-Liquid Interface

This thesis presents details of the design and development of novel tools and instruments for scanning tunneling microscopy (STM), and may be considered as a repository for several years' worth of development work. The author presents design goals and implementations for two microscopes. First, a novel Pan-type STM was built that could be operated in an ambient environment as a liquid-phase STM. Unique features of this microscope include a unibody frame, for increased microscope rigidity, a novel slider component with large Z-range, a unique wiring scheme and damping mechanism, and a removable liquid cell. The microscope exhibits a high level of mechanical isolation at the tunnel junction, and operates excellently as an ambient tool. Experiments in liquid are on-going. Simultaneously, the author worked on designs for a novel low temperature, ultra-high vacuum (LT-UHV) instrument, and these are presented as well. A novel stick-slip vertical coarse approach motor was designed and built. To gauge the performance of the motor, an in situ motion sensing apparatus was implemented, which could measure the step size of the motor to high precision. A new driving circuit for stick-slip inertial motors is also presented, that o ffers improved performance over our previous driving circuit, at a fraction of the cost. The circuit was shown to increase step size performance by 25%. Finally, a horizontal sample stage was implemented in this microscope. The build of this UHV instrument is currently being fi nalized. In conjunction with the above design projects, the author was involved in a collaborative project characterizing N-heterocyclic carbene (NHC) self-assembled monolayers (SAMs) on Au(111) films. STM was used to characterize Au substrate quality, for both commercial substrates and those manufactured via a unique atomic layer deposition (ALD) process by collaborators. Ambient and UHV STM was then also used to characterize the NHC/Au(111) films themselves, and several key properties of these films are discussed. During this study, the author discovered an unexpected surface contaminant, and details of this are also presented. Finally, two models are presented for the nature of the NHC-Au(111) surface interaction based on the observed film properties, and some preliminary theoretical work by collaborators is presented.