Quaternion Electromagnetism and the Relation with 2-Spinor Formalism

By using complex quaternion, which is the system of quaternion representation extended to complex numbers, we show that the laws of electromagnetism can be expressed much more simply and concisely. We also derive the quaternion representation of rotations and boosts from the spinor representation of Lorentz group. It is suggested that the imaginary 'i' should be attached to the spatial coordinates, and observe that the complex conjugate of quaternion representation is exactly equal to parity inversion of all physical quantities in the quaternion. We also show that using quaternion is directly linked to the two-spinor formalism. Finally, we discuss meanings of quaternion, octonion and sedenion in physics as n-fold rotationComment: Version published in journal Universe (2019

Automated Conjecturing in QuickSpec

A key component of mathematical reasoning is the ability to formulate interesting conjectures about a problem domain at hand. This task has not yet been widely studied by the automated reasoning and AI communities, but we believe interest is growing. In this paper, we give a brief overview of a theory exploration system called QuickSpec, able to automatically discover interesting conjectures about a given set of functions. QuickSpec works by interleaving term generation with random testing to form candidate equational conjectures. This is made tractable by starting from small sizes and ensuring that only terms that are irreducible with respect to already discovered equalities are considered. QuickSpec has been successfully applied to generate lemmas for automated inductive theorem proving as well as to generate specifications of functional programs. We also give a small survey of different approaches to conjecture discovery, and speculate about future directions combining symbolic methods and machine learning

Probabilistic Arguments in Mathematics

This thesis addresses a question that emerges naturally from some observations about contemporary mathematical practice. Firstly, mathematicians always demand proof for the acceptance of new results. Secondly, the ability of mathematicians to tell if a discourse gives expression to a proof is less than perfect, and the computers they use are subject to a variety of hardware and software failures. So false results are sometimes accepted, despite insistence on proof. Thirdly, over the past few decades, researchers have also developed a variety of methods that are probabilistic in nature. Even if carried out perfectly, these procedures only yield a conclusion that is very likely to be true. In some cases, these chances of error are precisely specifiable and can be made as small as desired. The likelihood of an error arising from the inherently uncertain nature of these probabilistic algorithms can therefore be made vanishingly small in comparison to the chances of an error arising when implementing an equivalent deductive algorithm. Moreover, the structure of probabilistic algorithms tends to minimise these Implementation Errors too. So overall, probabilistic methods are sometimes more reliable than deductive ones. This invites the question: ‘Are mathematicians rational in continuing to reject these probabilistic methods as a means of establishing mathematical claims?