Weighted skeletons and fixed-share decomposition

AbstractWe introduce the concept of weighted skeleton of a polygon and present various decomposition and optimality results for this skeletal structure when the underlying polygon is convex

The Polytope Formalism: isomerism and associated unimolecular isomerisation

This thesis concerns the ontology of isomerism, this encompassing the conceptual frameworks and relationships that comprise the subject matter; the necessary formal definitions, nomenclature, and representations that have impacts reaching into unexpected areas such as drug registration and patent specifications; the requisite controlled and precise vocabulary that facilitates nuanced communication; and the digital/computational formalisms that underpin the chemistry software and database tools that empower chemists to perform much of their work. Using conceptual tools taken from Combinatorics, and Graph Theory, means are presented to provide a unified description of isomerism and associated unimolecular isomerisation spanning both constitutional isomerism and stereoisomerism called the Polytope Formalism. This includes unification of the varying approaches historically taken to describe and understand stereoisomerism in organic and inorganic compounds. Work for this Thesis began with the synthesis, isolation, and characterisation of compounds not adequately describable using existing IUPAC recommendations. Generalisation of the polytopal-rearrangements model of stereoisomerisation used for inorganic chemistry led to the prescriptions that could deal with the synthesised compounds, revealing an unrecognised fundamental form of isomerism called akamptisomerism. Following on, this Thesis describes how in attempting to place akamptisomerism within the context of existing stereoisomerism reveals significant systematic deficiencies in the IUPAC recommendations. These shortcomings have limited the conceptualisation of broad classes of compounds and hindered development of molecules for medicinal and technological applications. It is shown how the Polytope Formalism can be applied to the description of constitutional isomerism in a practical manner. Finally, a radically different medicinal chemistry design strategy with broad application, based upon the principles, is describe