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.

Preparation of Polymer Nanoparticles by Ring Opening Metathesis Polymerization in Aqueous Dispersed Systems with Water-Soluble Ruthenium-based Catalyst

Ring opening metathesis polymerization (ROMP) is a variant of olefin metathesis used to polymerize strained cyclic olefins. Ruthenium-based Grubbs’ catalysts are widely used in ROMP to produce industrially important products. While highly efficient in organic solvents such as dichloromethane and toluene, these hydrophobic catalysts are not typically applied in aqueous systems. With the advancements in emulsion and miniemulsion polymerization, it is promising to conduct ROMP in an aqueous dispersed phase to generate well-defined latex nanoparticles while improving heat transfer and reducing the use of volatile organic solvents (VOCs). Herein I report the efforts made using a PEGylated ruthenium alkylidene as the catalyst to initiate ROMP in an oil-in-water miniemulsion. 1H NMR revealed that the synthesized PEGylated catalyst was stable and reactive in water. Using 1,5-cyclooctadiene (COD) as monomer, we showed the highly efficient catalyst yielded colloidally stable polymer latexes with ~ 100% conversion at room temperature. Kinetic studies demonstrated first-order kinetics with good livingness as confirmed by the shift of gel permeation chromatography (GPC) traces. Depending on the surfactants used, the particle sizes ranged from 100 to 300 nm with monomodal distributions. The more strained cyclic olefin norbornene (NB) could also be efficiently polymerized with a PEGylated ruthenium alkylidene in miniemulsion to full conversion and with minimal coagulum formation.