The maximum number of systoles for genus two Riemann surfaces with abelian differentials

In this article, we provide bounds on systoles associated to a holomorphic $1$-form $\omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,\omega)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $\omega$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.Comment: 41 page

The maximum number of systoles for genus two Riemann surfaces with abelian differentials

This article explores the length and number of systoles associated to holomorphic $1$-forms on surfaces. In particular, we show that up to homotopy, there are at most $10$ systolic loops on such a genus two surface and that the bound is realized by a unique translation surface up to homothety. We also provide sharp upper bounds on the the number of homotopy classes of systoles for a holomorphic $1$-form with a single zero in terms of the genus

Space Shuttle Solid Rocket Motor Plume Pressure and Heat Rate Measurements

The Solid Rocket Booster (SRB) Main Flame Deflector (MFD) at Launch Complex 39A was instrumented with sensors to measure heat rates, pressures, and temperatures on the last three Space Shuttle launches. Because the SRB plume is hot and erosive, a robust Tungsten Piston Calorimeter was developed to compliment the measurements made by off-the-shelf sensors. Witness materials were installed and their melting and erosion response to the Mach 2 / 4500 F / 4-second duration plume was observed. The data show that the specification document used for the design of the MFD thermal protection system over-predicted heat rates by a factor of 3 and under-predicted pressures by a factor of 2. These findings will be used to baseline NASA Computational Fluid Dynamics models and develop innovative MFD designs for the Space Launch System (SLS) before this vehicle becomes operational in 2017