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The scheduling and Control Group at the University of South Australia has been studying the optimal control of trains for many years, and has developed in-cab devices that help drivers stay on time and minimise energy use. In this thesis, we re-examine the optimal control theory for the train control problem. In particular, we study the optimal control around steep sections of track. To calculate an optimal driving strategy we need a realistic model of train performance. In particular, we need to know a coefficient of rolling resistance and a coefficient of aerodynamic drag. In practice, these coefficients are different for every train and difficult to predict. In the thesis, we study the use of mathematical filters to estimate model parameters from observations of actual train performance.

Qualified difference sets are a class of combinatorial configuration. The sets are related to the residue difference sets that were first discussed in detail in 1953 by Emma Lehmer. Qualified difference sets consist of a set of residues modulo an integer v and they possess attractive properties that suggest potential applications in areas such as image formation, signal processing and aperture synthesis. This thesis outlines the theory behind qualified difference sets and gives conditions for the existence and nonexistence of these sets in various cases. A special case of the qualified difference sets is the qualified residue difference sets. These consist of the set of nth power residues of certain types of prime. Necessary and sufficient conditions for the existence of qualified residue difference sets are derived and the precise conditions for the existence of these sets are given for n = 2, 4 and 6. Qualified residue difference sets are proved nonexistent for n = 8, 10, 12, 14 and 18. A generalisation of the qualified residue difference sets is introduced. These are the qualified difference sets composed of unions of cyclotomic classes. A cyclotomic class is defined for an integer power n and the results of an exhaustive computer search are presented for n = 4, 6, 8, 10 and 12. Two new families of qualified difference set were discovered in the case n = 8 and some isolated systems were discovered for n = 6, 10 and 12. An explanation of how qualified difference sets may be implemented in physical applications is given and potential applications are discussed.

Qualified difference sets are a class of combinatorial configuration. The sets are related to the residue difference sets that were first discussed in detail in 1953 by Emma Lehmer. Qualified difference sets consist of a set of residues modulo an integer v and they possess attractive properties that suggest potential applications in areas such as image formation, signal processing and aperture synthesis. This thesis outlines the theory behind qualified difference sets and gives conditions for the existence and nonexistence of these sets in various cases. A special case of the qualified difference sets is the qualified residue difference sets. These consist of the set of nth power residues of certain types of prime. Necessary and sufficient conditions for the existence of qualified residue difference sets are derived and the precise conditions for the existence of these sets are given for n = 2, 4 and 6. Qualified residue difference sets are proved nonexistent for n = 8, 10, 12, 14 and 18. A generalisation of the qualified residue difference sets is introduced. These are the qualified difference sets composed of unions of cyclotomic classes. A cyclotomic class is defined for an integer power n and the results of an exhaustive computer search are presented for n = 4, 6, 8, 10 and 12. Two new families of qualified difference set were discovered in the case n = 8 and some isolated systems were discovered for n = 6, 10 and 12. An explanation of how qualified difference sets may be implemented in physical applications is given and potential applications are discussed.

Srinivasa Ramanujan (1887-1920) was one of the world's greatest mathematical geniuses. He work extensively in a branch of mathematics called "q-series". Around 1913, he found an important formula which now is known as Ramanujan's 1ψ1summation formula. The aim of this thesis is to investigate Ramanujan's 1ψ1summation formula and explore its applications to number theory and combinatorics. First, we consider several classical important results on elliptic functions and then give new proofs of these results using Ramanujan's 1ψ1 summation formula. For example, we will present a number of classical and new solutions for the problem of representing an integer as sums of squares (one of the most celebrated in number theory and combinatorics) in this thesis. This will be done by using q-series and Ramanujan's 1ψ1 summation formula. This in turn will give an insight into how Ramanujan may have proven many of his results, since his own proofs are often unknown, thereby increasing and deepening our understanding of Ramanujan's work.

Qualified difference sets are a class of combinatorial configuration. The sets are related to the residue difference sets that were first discussed in detail in 1953 by Emma Lehmer. Qualified difference sets consist of a set of residues modulo an integer v and they possess attractive properties that suggest potential applications in areas such as image formation, signal processing and aperture synthesis. This thesis outlines the theory behind qualified difference sets and gives conditions for the existence and nonexistence of these sets in various cases. A special case of the qualified difference sets is the qualified residue difference sets. These consist of the set of nth power residues of certain types of prime. Necessary and sufficient conditions for the existence of qualified residue difference sets are derived and the precise conditions for the existence of these sets are given for n = 2, 4 and 6. Qualified residue difference sets are proved nonexistent for n = 8, 10, 12, 14 and 18. A generalisation of the qualified residue difference sets is introduced. These are the qualified difference sets composed of unions of cyclotomic classes. A cyclotomic class is defined for an integer power n and the results of an exhaustive computer search are presented for n = 4, 6, 8, 10 and 12. Two new families of qualified difference set were discovered in the case n = 8 and some isolated systems were discovered for n = 6, 10 and 12. An explanation of how qualified difference sets may be implemented in physical applications is given and potential applications are discussed.