On the realization of Riemannian symmetric spaces in Lie groups

In this paper we give a realization of some symmetric space G/K as a closed submanifold P of G. We also give several equivalent representations of the submanifold P. Some properties of the set gK\cap P are also discussed, where gK is a coset space in G.Comment: 8 page

On the realization of Riemannian symmetric spaces in Lie groups II

In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of fixed points of involutions are also proved.Comment: 6 page

Holomorphic harmonic analysis on complex reductive groups

We define the holomorphic Fourier transform of holomorphic functions on complex reductive groups, prove some properties like the Fourier inversion formula, and give some applications. The definition of the holomorphic Fourier transform makes use of the notion of $K$-admissible measures. We prove that $K$-admissible measures are abundant, and the definition of holomorphic Fourier transform is independent of the choice of $K$-admissible measures.Comment: 15 pages, revision of a preprint by the first author in 200

A Generalization of Random Matrix Ensemble I: General Theory

We give a generalization of the random matrix ensembles, including all lassical ensembles. Then we derive the joint density function of the generalized ensemble by one simple formula, which give a direct and unified way to compute the density functions for all classical ensembles and various kinds of new ensembles. An integration formula associated with the generalized ensemble is also given. We also give a classification scheme of the generalized ensembles, which will include all classical ensembles and some new ensembles which were not considered before.Comment: 15 page

Command and Control Software Development

As NASAs role in spaceflight operations change, so too does its design priorities. NASAs flagship venture, the Space Launch System (SLS), is being created primarily to explore deep space. While commercial partners conduct low Earth orbit resupplies, tests, and even transportation of astronauts, NASA can divert more resources to more exciting projects - going back to the moon, or onwards to Mars. In the short term, this goal manifests in the upcoming Exploration Mission-1 (EM-1) test of the SLS, slated for as early as 2019. SLS, with the Orion capsule, will travel farther than any human-rated spacecraft has gone before. As the most powerful launch vehicle ever created, SLS requires many new innovations to ensure mission success. One such technology is the launch control software, which is the focus of this internship. The SLS launch control software is composed of many functions, all of which require rigorous testing to meet the standard of life-critical code. To facilitate easier testing, the first project I undertook was to customize the open-source tool Wireshark to the software teams needs. Wireshark is a network protocol analyzer that can take in information about custom information packets. The launch control software will have several protocols that are custom to NASA. I ensured that all necessary component files were the correct file type and in the correct structure for Wireshark to compile. Having allowed Wireshark access to understand custom NASA data packets, testers of the launch control software will be able to find anomalies easier