Wave asymptotics for waveguides and manifolds with infinite cylindrical ends

We describe wave decay rates associated to embedded resonances and spectral thresholds for waveguides and manifolds with infinite cylindrical ends. We show that if the cut-off resolvent is polynomially bounded at high energies, as is the case in certain favorable geometries, then there is an associated asymptotic expansion, up to a $O(t^{-k_0})$ remainder, of solutions of the wave equation on compact sets as $t \to \infty$. In the most general such case we have $k_0=1$, and under an additional assumption on the infinite ends we have $k_0 = \infty$. If we localize the solutions to the wave equation in frequency as well as in space, then our results hold for quite general waveguides and manifolds with infinite cylindrical ends. To treat problems with and without boundary in a unified way, we introduce a black box framework analogous to the Euclidean one of Sj\"ostrand and Zworski. We study the resolvent, generalized eigenfunctions, spectral measure, and spectral thresholds in this framework, providing a new approach to some mostly well-known results in the scattering theory of manifolds with cylindrical ends.Comment: In this revision we work in a more general black box setting than in the first version of the paper. In particular, we allow a boundary extending to infinity. The changes to the proofs of the main theorems are minor, but the presentation of the needed basic material from scattering theory is substantially expanded. New examples are included, both for the main results and for the black box settin

Discovery of TUG-770: a highly potent free fatty acid receptor 1 (FFA1/GPR40) agonist for treatment of type 2 diabetes

Free fatty acid receptor 1 (FFA1 or GPR40) enhances glucose-stimulated insulin secretion from pancreatic β-cells and currently attracts high interest as a new target for the treatment of type 2 diabetes. We here report the discovery of a highly potent FFA1 agonist with favorable physicochemical and pharmacokinetic properties. The compound efficiently normalizes glucose tolerance in diet-induced obese mice, an effect that is fully sustained after 29 days of chronic dosing

Resolvent estimates, wave decay, and resonance-free regions for star-shaped waveguides

Using coordinates $(x,y)\in \mathbb R\times \mathbb R^{d-1}$, we introduce the notion that an unbounded domain in $\mathbb R^d$ is star shaped with respect to $x=\pm \infty$. For such domains, we prove estimates on the resolvent of the Dirichlet Laplacian near the continuous spectrum. When the domain has infinite cylindrical ends, this has consequences for wave decay and resonance-free regions. Our results also cover examples beyond the star-shaped case, including scattering by a strictly convex obstacle inside a straight planar waveguide.Comment: 21 pages, 5 figure

Low energy scattering asymptotics for planar obstacles

We compute low energy asymptotics for the resolvent of a planar obstacle, and deduce asymptotics for the corresponding scattering matrix, scattering phase, and exterior Dirichlet-to-Neumann operator. We use an identity of Vodev to relate the obstacle resolvent to the free resolvent and an identity of Petkov and Zworski to relate the scattering matrix to the resolvent. The leading singularities are given in terms of the obstacle's logarithmic capacity or Robin constant. We expect these results to hold for more general compactly supported perturbations of the Laplacian on $\mathbb R^2$, with the definition of the Robin constant suitably modified, under a generic assumption that the spectrum is regular at zero.Comment: 26 pages, 1 figur

Preliminary evaluation of radar imagery of Yellowstone Park, Wyoming

Evaluation of radar imagery of Yellowstone Park, Wyomin

Complex Line Bundles over Simplicial Complexes and their Applications

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular, we obtain a discrete analogue of a theorem of Andr\'e Weil on the classification of hermitian line bundles. Moreover, we associate to each discrete hermitian line bundle with curvature a unique piecewise-smooth hermitian line bundle of piecewise constant curvature. This is then used to define a discrete Dirichlet energy which generalizes the well-known cotangent Laplace operator to discrete hermitian line bundles over Euclidean simplicial manifolds of arbitrary dimension

The first high-amplitude delta Scuti star in an eclipsing binary system

We report the discovery of the first high-amplitude delta Scuti star in an eclipsing binary, which we have designated UNSW-V-500. The system is an Algol-type semi-detached eclipsing binary of maximum brightness V = 12.52 mag. A best-fitting solution to the binary light curve and two radial velocity curves is derived using the Wilson-Devinney code. We identify a late A spectral type primary component of mass 1.49+/-0.02 M_sun and a late K spectral type secondary of mass 0.33+/-0.02 M_sun, with an inclination of 86.5+/-1.0 degrees, and a period of 5.3504751+/-0.0000006 d. A Fourier analysis of the residuals from this solution is performed using PERIOD04 to investigate the delta Scuti pulsations. We detect a single pulsation frequency of f_1 = 13.621+/-0.015 c/d, and it appears this is the first overtone radial mode frequency. This system provides the first opportunity to measure the dynamical mass for a star of this variable type; previously, masses have been derived from stellar evolution and pulsation models.Comment: 7 pages, 6 figures, 2 tables, for submission to MNRAS, v2: paper size change, small typographical changes to abstrac