Fan, Sin Tsun Edward. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 85-87). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 1.1 --- Motivation --- p.5 / Chapter 1.2 --- Main results --- p.10 / Chapter 2 --- Preliminaries --- p.14 / Chapter 2.1 --- Z-gradations of Semisimple Lie Algebras --- p.14 / Chapter 2.2 --- Basic Facts about Algebraic Groups --- p.15 / Chapter 3 --- Weight Multiplicity Free Representations and Pre- homogeneous Vector Spaces --- p.18 / Chapter 3.1 --- Weight Multiplicity Free Representations --- p.18 / Chapter 3.2 --- Prehomogeneous Vector Spaces --- p.22 / Chapter 4 --- Augmentations of Dynkin Diagrams --- p.25 / Chapter 5 --- Orbit Finiteness and Prehomogeneity --- p.32 / Chapter 6 --- Termination of Z-grading --- p.36 / Chapter 7 --- Explicit Construction of Generic Elements in Simply- laced Cases --- p.42 / Chapter 8 --- The Ambient Lie Algebras of Parabolic PVS's --- p.47 / Chapter 9 --- PVS's of Twisted Affine Type --- p.52 / Chapter 10 --- "Orbit Structure of (GL2 x SL2m+1,C2 x A2C2m+1)" --- p.55 / Chapter 11 --- Nilvarieties and Generalisation of Open Orbits --- p.59 / Chapter 11.1 --- Nilvarieties and Visible Representations --- p.59 / Chapter 11.2 --- Augmeantations of Affine Dynkin Diagrams --- p.62 / Chapter 11.3 --- Classification of Irreducible Visible Representations --- p.67 / Chapter 12 --- Real Forms of PVS of Parabolic Type --- p.70 / Chapter 12.1 --- Representations of Real Reductive Lie Algebras and Satake Diagrams --- p.70 / Chapter 12.2 --- Real Forms of PVS of Parabolic Type --- p.77 / Chapter 13 --- Tables --- p.81 / Bibliography --- p.85

A notion of entropy is defined for the non-singular action of finite co-ordinate changes on X - the infinite product of two- point spaces. This quantity - average co-ordinate or AC entropy - is calculated for product measures and G-measures on X, and an equivalence relation is established for which AC entropy is an invariant. The Inverse Vitali Lemma is discussed in a measure preserving context, and it is shown that for a certain class of measures on X known as odometer bounded, the result will still hold for odometer actions. The foundations for a non-singular version of Rudolph's restricted orbit equivalence are established, and a size for non-singular orbit equivalence is introduced. It is shown that provided the Inverse Vitali Lemma still holds, the non-singular orbit equivalence classes can be described using this new size.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 154-157). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 154-157). Also available on the Internet.

The historical method of precise orbit determination is a dynamic approach. However, with the improvement of GPS tracking data and associated tracking networks, two newer methods have been developed: reduced-dynamic and kinematic. In addition to orbit determination, alternative methods of gravity field recovery have been developed using kinematic orbits which do not rely on any force modeling. However, one significant drawback of kinematic orbits is that they lack any velocity or acceleration information. These have to be derived numerically. Based on the results of this thesis, the Savitzky-Golay filter, without using a remove-restore procedure, is recommended for deriving kinematic velocities of the GRACE mission. In addition, the numerical differentiation methods are tested to see how well accurately they represent the satellite's acceleration for all three orbit types. Finally, with the kinematic orbits properly reconstructed, the results can also be compared to dynamic and reduced-dynamic orbits through K-Band Ranging residuals. / text

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This research effort is concerned with the methods of simplified orbit determination and orbit representation and their applications for Low Earth Orbit (LEO) satellite missions, particularly addressing the operational needs of the FedSat mission. FedSat is the first Australian-built satellite in over thirty years. The microsatellite is approximately 50cm cubed with a mass of 58 kg. The satellite was successfully placed into a low-earth near-polar orbit at an altitude of 780km by the Japanese National Space Development Agency (NASDA) H-IIA launch vehicle on 14, December 2002. Since then, it has been streaming scientific data to its ground station in Adelaide almost daily. This information is used by Australian and international researchers to study space weather, to help improve the design of space computers, communication systems and other satellite technology, and for research into navigation and satellite tracking. This research effort addresses four practical issues regarding the FedSat mission and operations. First, unlike most satellite missions, the GPS receiver onboard FedSat operates in a duty-cycle mode due to the limitations of the FedSat power supply. This causes significant difficulties for orbit tracking, Precise Orbit Determination and scientific applications. A covariance analysis was performed before the mission launch to assess the orbit performance under different operational modes. The thesis presents the analysis methods and results. Second, FedSat supports Ka-band tracking experiments that require a pointing accuracy of 0.03 degree. The QUT GPS group is obligated to provide the GPS precise orbit solution to meet this requirement. Ka-band tracking requests satellite orbital position at any instant time with respect to any of the observation stations. Because orbit determination and prediction software only provide satellite orbital data at a discrete time point, it is necessary to find a way to represent the satellite orbit as a continuous trajectory with discrete observation data, able to obtain the position of the satellite at the time of interest. For this purpose, an orbit interpolation algorithm using the Chebyshev polynomial was developed and applied to Ka-band tracking applications. The thesis will describe the software and results. Third, since the launch of FedSat, investigators have received much flight GPS data. Some research was invested in the analysis of FedSat orbit performance, GPS data quality and the quality of the onboard navigation solutions. Studies have revealed that there are many gross errors in the FedSat onboard navigation solution (ONS). Although the 1-sigma accuracy of each component is about 20 m, there are more than 11 %positioning errors that fall outside +/-50m, and 5% of the errors are outside the 100mbound. The 3D RMS values would be 35m, 87m, and 173m for the above three cases respectively. The FedSat ONS uncertainties are believed to be approximately three times greater than those from other satellite missions. Due to the high percentage of outlier solutions, it would be dangerous to use these without first applying data detection and exclusion procedures. Therefore, this thesis presents two simplified orbit determination methods that can improve the ONS. One is the &quotgeometric method", which makes use of delta-position solutions derived from carrier phase differences between two epochs to smooth the code-based navigation solutions. The algorithms were tested using SAC-C GPS data and showing some improvement. The second method is the &quotdynamic method", which uses orbit dynamics information for orbit improvements. Fourth, the FedSat ground tracking team at Adelaide use the NORAD TLE orbit for daily FedSat tracking. Research was undertaken to convert an orbit trajectory into these Two Line Elements (TLE). Algorithms for the estimation of TLE solutions from the FedSat onboard GPS navigation solutions are outlined. Numerical results have shown the effects of the unmodelled forces/perturbations in the SPG4 models for the FedSat orbit determination would reach a level of ±1000m. This only includes the orbit representation errors with TLE data sets. The total FedSat orbit propagation should include both the orbit propagation and orbit representation terms. The analysis also demonstrates that the orbit presentation error can be reduced to ±200m and ±100mlevels with the EGM4x4 and EGM10x10 gravity field models respectively. This can meet the requirements for Ka-band tracking. However, a simplified tracking program based on numerical integration has to be developed to replace the SPG4 models.

Thesis(M.S.)--Auburn University, 2007. / Abstract. Vita. Includes bibliographic references (p.35-36).