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This paper investigates whether it is reasonable to bestow credit to one person or group for the famed theorem that relates to the side lengths of any right-angled triangle, a theorem routinely referred to as the “Pythagorean Theorem”. The author investigates the first-documented occurrences of the theorem, along with its first proofs. In addition, proofs that stem from different branches of mathematics and science are analyzed in an effort to display that credit for the development of the theorem should be shared amongst its many contributors rather than crediting the whole of the theorem to one man and his supporters.

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The usual virial theorem, relating kinetic and potential energy, is extended to a molecule by the use of the true wave function. The virial theorem is also obtained for a molecule from a trial wave function which is scaled separately for electronic and nuclear coordinates.

The motion of an incompressible fluid caused by the oscillation of a plane at plate of in nite length is termed Stokes' second problem. We assume zero velocity normal to the plate and thus simpli ed Navier-Stokes equations. For the unsteady Stokes' second problem, solutions may be obtained by using Laplace transforms, perturbation techniques, homotopy, di erential transform method or Adomian decomposition method. Stokes' second problem is discussed for second-grade and Oldroyd-B non-Newtonian fluids. This dissertation summarizes previously published work.

As functions which further the state of a proof in automated theorem proving, tactics are an important development in automated deduction. This thesis describes a method to tackle the problem of tactic formation. Tactics must currently be developed by hand, which can be a complicated and time-consuming process. A method is presented for the automatic production of useful tactics. The method presented works on the principle that commonly occurring patterns within proof corpora may have some significance and could therefore be exploited to provide novel tactics. These tactics are discovered using a three step process. Firstly a suitable corpus is chosen and processed. One example of a suitable corpus is that of the Isabelle theorem prover. A number of possible abstractions are presented for this corpus. Secondly, machine learning techniques are used to data-mine each corpus and find sequences of commonly occurring proof steps. The specifics of a proof step are defined by the specified abstraction. The formation of these tactics is completed using evolutionary techniques to combine these patterns into compound tactics. These new tactics are applied using a naive prover as well as undergoing manual evalutation. The tactics show favourable results across a selection of tests, justifying the claim that this project provides a novel method of automatically producing tactics which are both viable and useful.