A new proof of scattering below the ground state for the non-radial focusing NLS

We revisit the scattering result of Duyckaerts, Holmer, and Roudenko for the non-radial $\dot H^{1/2}$-critical focusing NLS. By proving an interaction Morawetz inequality, we give a simple proof of scattering below the ground state in dimensions $d\geq 3$ that avoids the use of concentration compactness.Comment: 14 page

Semiclassical Approach to Heterogeneous Vacuum Decay

We derive the decay rate of an unstable phase of a quantum field theory in the presence of an impurity in the thin-wall approximation. This derivation is based on the how the impurity changes the (flat spacetime) geometry relative to case of pure false vacuum. Two examples are given that show how to estimate some of the additional parameters that enter into this heterogeneous decay rate. This formalism is then applied to the Higgs vacuum of the Standard Model (SM), where baryonic matter acts as an impurity in the electroweak Higgs vacuum. We find that the probability for heterogeneous vacuum decay to occur is suppressed with respect to the homogeneous case. That is to say, the conclusions drawn from the homogeneous case are not modified by the inclusion of baryonic matter in the calculation. On the other hand, we show that Beyond the Standard Model physics with a characteristic scale comparable to the scale that governs the homogeneous decay rate in the SM, can in principle lead to an enhanced decay rate.Comment: v3: version published in JHEP, very minor changes from v

Who\u27s That Girl? The Many Faces of Mexican Women

Undergraduate Winner: 2nd Place, 2011. 24th Annual Carl Neureuther Student Book Collection Competitio

Doctor of Philosophy

dissertationTransport in disordered composite media is a problem that arises throughout the sciences and engineering and has attracted significant theoretical, computational, and experimental interest. One of the key features of these types of problems is the critical dependence of the effective transport properties on system parameters, such as volume fraction, component contrast ratio, applied field strength, etc. In recent years a broad range of mathematical techniques have been developed to study phase transitions exhibited by such composites, revealing features which are virtually ubiquitous in disordered systems. Here we construct a multifaceted mathematical framework describing phase transitions exhibited by two phase random media, using techniques from: statistical mechanics, percolation theory, random matrix theory, and a critical theory for Stieltjes functions of a complex variable involving the spectral measure of a self-adjoint random operator (or matrix). In particular, we present a general theory for critical behavior of transport in two phase random media. The theory holds for lattice and continuum percolation models in both the static case with real parameters and the frequency dependent quasi-static case with complex parameters. Through a direct, analytic correspondence between the magnetization of the Ising model and the effective parameter problem of two phase random media, we show that the critical exponents of the transport coefficients satisfy the standard scaling relations for phase transitions in statistical mechanics. Our work also shows that delta components form in the underlying spectral measures at the spectral endpoints precisely at the percolation threshold pc and 1 − pc. This is analogous to the Lee-Yang-Ruelle characterization of the Ising model phase transition, and identifies these transport transitions with the collapse of spectral gaps in these measures. Using random matrix theory, we also characterize these transport transitions via transitions in the eigenvalue statistics of the underlying random matrix. Finally, we construct a canonical ensemble statistical mechanics framework for general transport models of two phase random dielectric media, which parallels the Ising model. Our physically consistent model is formulated from first principles in physics, and is both physically transparent and mathematically tractable

The Merry Death Collector

Documentary filmmaking puts life on lens and who knows what the outcome will be. The creative team chose Arnie Meredith to be the subject of the short documentary The Merry Death Collector. The oddball LaFollette, Tennessee resident collects and sells antique items from estate sales and recently purchased two buildings to house and run his business. The story follows Arnie’s struggle to open and maintain a small-town business. He attends a convention in Knoxville, Tennessee to raise funds, and even attempts to hang a biplane in one of his buildings. With his projected opening date around the corner, an exhausted Arnie finds himself two weeks behind schedule. The journey to his opening was a late one but one of true community support. The Merry Death Collector will make its official World Premiere at the Nashville Film Festival in late April. This Academy-Award qualifying film festival is the oldest running festival in the South and accepts only 6% of its over 3500 entrees