Degenerate elliptic operators: capacity, flux and separation

Let $S=\{S_t\}_{t\geq0}$ be the semigroup generated on L_2(\Ri^d) by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients. Further let $\Omega$ be an open subset of \Ri^d with Lipschitz continuous boundary $\partial\Omega$. We prove that $S$ leaves $L_2(\Omega)$ invariant if, and only if, the capacity of the boundary with respect to $H$ is zero or if, and only if, the energy flux across the boundary is zero. The global result is based on an analogous local result.Comment: 18 page

Degenerate elliptic operators in one dimension

Let $H$ be the symmetric second-order differential operator on L_2(\Ri) with domain C_c^\infty(\Ri) and action $H\varphi=-(c \varphi')'$ where c\in W^{1,2}_{\rm loc}(\Ri) is a real function which is strictly positive on \Ri\backslash\{0\} but with $c(0)=0$. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of $H$. In particular if $\nu=\nu_+\vee\nu_-$ where $\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}$ then $H$ has a unique self-adjoint extension if and only if $\nu\not\in L_2(0,1)$ and a unique submarkovian extension if and only if $\nu\not\in L_\infty(0,1)$. In both cases the corresponding semigroup leaves $L_2(0,\infty)$ and $L_2(-\infty,0)$ invariant. In addition we prove that for a general non-negative c\in W^{1,\infty}_{\rm loc}(\Ri) the corresponding operator $H$ has a unique submarkovian extension.Comment: 28 page

Markov uniqueness of degenerate elliptic operators

Let $\Omega$ be an open subset of \Ri^d and $H_\Omega=-\sum^d_{i,j=1}\partial_i c_{ij} \partial_j$ a second-order partial differential operator on $L_2(\Omega)$ with domain $C_c^\infty(\Omega)$ where the coefficients $c_{ij}\in W^{1,\infty}(\Omega)$ are real symmetric and $C=(c_{ij})$ is a strictly positive-definite matrix over $\Omega$. In particular, $H_\Omega$ is locally strongly elliptic. We analyze the submarkovian extensions of $H_\Omega$, i.e. the self-adjoint extensions which generate submarkovian semigroups. Our main result establishes that $H_\Omega$ is Markov unique, i.e. it has a unique submarkovian extension, if and only if \capp_\Omega(\partial\Omega)=0 where \capp_\Omega(\partial\Omega) is the capacity of the boundary of $\Omega$ measured with respect to $H_\Omega$. The second main result establishes that Markov uniqueness of $H_\Omega$ is equivalent to the semigroup generated by the Friedrichs extension of $H_\Omega$ being conservative.Comment: 24 page

Hot Stars With Cool Companions

Young intermediate-mass stars have become high-priority targets for direct-imaging planet searches following the recent discoveries of planets orbiting e.g. HR 8799 and Beta Pictoris. Close stellar companions to these stars can affect the formation and orbital evolution of any planets, and so a census of the multiplicity properties of nearby intermediate mass stars is needed. Additionally, the multiplicity can help constrain the important binary star formation physics. We report initial results from a spectroscopic survey of 400 nearby A- and B-type stars. We search for companions by cross-correlating high resolution and high signal-to-noise ratio echelle spectra of the targets stars against model spectra for F- to M-type stars. We have so far found 18 new candidate companions, and have detected the spectral lines of the secondary in 4 known spectroscopic binary systems. We present the distribution of mass-ratios for close companions, and find that it differs from the distribution for wide ($a < 100$ AU) intermediate-mass binaries, which may indicate a different formation mechanism for the two populations.Comment: Submitted as part of the 18th Cambridge Workshop on Cool Stars, Stellar Systems, and the Sun Proceedings of Lowell Observatory (9-13 June 2014

Correcting For Telluric Absorption: Methods, Case Studies, And Release Of The TelFit Code

Ground-based astronomical spectra are contaminated by the Earth's atmosphere to varying degrees in all spectral regions. We present a Python code that can accurately fit a model to the telluric absorption spectrum present in astronomical data, with residuals of similar to 3%-5% of the continuum for moderately strong lines. We demonstrate the quality of the correction by fitting the telluric spectrum in a nearly featureless A0V star, HIP 20264, as well as to a series of dwarf M star spectra near the 819 nm sodium doublet. We directly compare the results to an empirical telluric correction of HIP 20264 and find that our model-fitting procedure is at least as good and sometimes more accurate. The telluric correction code, which we make freely available to the astronomical community, can be used as a replacement for telluric standard star observations for many purposes.UT Austin Hutchinson fellowshipUniversity of TexasAstronom

Epoxidation of Strained Alkenes Catalysed by (1,2-dimethyl-4(1H)pyridinone-3-olate)2MnIIICl

The mild epoxidation of strained alkenes using (DMPO)2MnCl catalyst (DMPO = 1,2-dimethyl-4(1H)-pyridinone-3-olate) in the presence of various oxidants was studied. Hydrogen peroxide and monopersulfate were found to be the best oxidants when used with imidazole in acetonitrile at 4 °C, with up to 94% conversion. Dismutation of hydrogen peroxide was also observed when used as an oxidant. The epoxidation using hydrogen peroxide or monoperoxysulfate appears to be mild and very selective for strained alkenes. A mechanism is proposed where imidazole is required for activation of the oxidant and where a detected MnV = O species is proposed as the active species. Competitive reaction between H2O2 and the substrate for the active species is proposed and homolytic vs heterolytic scissions of the Osingle bondO bond of the oxidant are discussed