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

Introduction to Graph Polynomials

With graph polynomials being a fairly new but intricate realm of graph theory, I will begin with a brief historical background and progress to elucidate each polynomial’s unique characteristics and mathematical underpinnings. Through illustrative examples, the paper elucidates the practical applications of these graph polynomials, showcasing their efficacy in real-world scenarios. My research contributes to the broader understanding of graph polynomials and inspires further research in the intersection of mathematics and technology

Using character varieties: Presentations, invariants, divisibility and determinants

If G is a finitely generated group, then the set of all characters from G into a linear algebraic group is a useful (but not complete) invariant of G . In this thesis, we present some new methods for computing with the variety of SL2C -characters of a finitely presented group. We review the theory of Fricke characters, and introduce a notion of presentation simplicity which uses these results. With this definition, we give a set of GAP routines which facilitate the simplification of group presentations. We provide an explicit canonical basis for an invariant ring associated with a symmetrically presented group\u27s character variety. Then, turning to the divisibility properties of trace polynomials, we examine a sequence of polynomials rn(a) governing the weak divisibility of a family of shifted linear recurrence sequences. We prove a discriminant/determinant identity about certain factors of rn( a) in an intriguing manner. Finally, we indicate how ordinary generating functions may be used to discover linear factors of sequences of discriminants. Other novelties include an unusual binomial identity, which we use to prove a well-known formula for traces; the use of a generating function to find the inverse of a map xn ∣→ fn(x); and a brief exploration of the relationship between finding the determinants of a parametrized family of matrices and the Smith Normal Forms of the sequence

Graph theory in America 1876-1950

This narrative is a history of the contributions made to graph theory in the United States of America by American mathematicians and others who supported the growth of scholarship in that country, between the years 1876 and 1950. The beginning of this period coincided with the opening of the first research university in the United States of America, The Johns Hopkins University (although undergraduates were also taught), providing the facilities and impetus for the development of new ideas. The hiring, from England, of one of the foremost mathematicians of the time provided the necessary motivation for research and development for a new generation of American scholars. In addition, it was at this time that home-grown research mathematicians were first coming to prominence. At the beginning of the twentieth century European interest in graph theory, and to some extent the four-colour problem, began to wane. Over three decades, American mathematicians took up this field of study - notably, Oswald Veblen, George Birkhoff, Philip Franklin, and Hassler Whitney. It is necessary to stress that these four mathematicians and all the other scholars mentioned in this history were not just graph theorists but worked in many other disciplines. Indeed, they not only made significant contributions to diverse fields but, in some cases, they created those fields themselves and set the standards for others to follow. Moreover, whilst they made considerable contributions to graph theory in general, two of them developed important ideas in connection with the four-colour problem. Grounded in a paper by Alfred Bray Kempe that was notorious for its fallacious 'proof' of the four-colour theorem, these ideas were the concepts of an unavoidable set and a reducible configuration. To place the story of these scholars within the history of mathematics, America, and graph theory, brief accounts are presented of the early years of graph theory, the early years of mathematics and graph theory in the USA, and the effects of the founding of the first institute for postgraduate study in America. Additionally, information has been included on other influences by such global events as the two world wars, the depression, the influx of European scholars into the United States of America, mainly during the 1930s, and the parallel development of graph theory in Europe. Until the end of the nineteenth century, graph theory had been almost entirely the prerogative of European mathematicians. Perhaps the first work in graph theory carried out in America was by Charles Sanders Peirce, arguably America's greatest logician and philosopher at the time. In the 1860s, he studied the four-colour conjecture and claimed to have written at least two papers on the subject during that decade, but unfortunately neither of these has survived. William Edward Story entered the field in 1879, with unfortunate consequences, but it was not until 1897 that an American mathematician presented a lecture on the subject, albeit only to have the paper disappear. Paul Wernicke presented a lecture on the four-colour problem to the American Mathematician Society, but again the paper has not survived. However, his 1904 paper has survived and added to the story of graph theory, and particularly the four-colour conjecture. The year 1912 saw the real beginning of American graph theory with Veblen and Birkhoff publishing major contributions to the subject. It was around this time that European mathematicians appeared to lose interest in graph theory. In the period 1912 to 1950 much of the progress made in the subject was from America and by 1950 not only had the United States of America become the foremost country for mathematics, it was the leading centre for graph theory

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website

Whitworth College Bulletin 1989-1991

https://digitalcommons.whitworth.edu/whitworthcatalogs/1078/thumbnail.jp

Theoretical screening of organic conjugated materials

As the urgency to address the effects of climate change increases, so does the need to discover new materials for the generation of renewable energy, to replace the environmentally dam- aging combustion of fossil fuels. Two main areas of research are focused on the design and discovery of photoactive materials for photovoltaics and for photocatalysts for water splitting. Although there are currently high-efficiency photovoltaics commercially available, there is mo- tivation to replace them with materials with non-toxic and earth-abundant compositions. As they generally meet these criteria, organic conjugated materials are very desirable candidates for these applications. Computational chemistry methods can accelerate materials discovery, eliminating the need to synthesize large libraries of molecules in the preliminary screening stages. Both high-accuracy, expensive methods and fast, cheap, lower-accuracy methods have their merits and in conjunction with one another can provide a detailed and informative description of chemical systems. The high-throughput virtual screening methodology used throughout this thesis provides the opportunity to efficiently explore property space and high- light potential candidates for given applications, such as polymeric photocatalysts, organic photovoltaics and dye sensitizers in solar cells. In this thesis this methodology is explored for a small aromatic molecules, diketopyrrolopyrrole-based dyes and both ordered and disordered polymers. Through the high-throughput virtual screening, large datasets of chemical com- pounds were investigated and analysed, highlighting the patterns in the optical and electronic properties influenced by building block sequence, conformerism and composition. The use of high-accuracy, expensive methods is also explored in this thesis, demonstrating the difficulties in pushing such methods to larger chemical structures

Whitworth College Bulletin 1987-1989

https://digitalcommons.whitworth.edu/whitworthcatalogs/1077/thumbnail.jp

The development of a weighted directed graph model for dynamic systems and application of Dijkstra’s algorithm to solve optimal control problems.

Master of Science (Chemical Engineering). University of KwaZulu-Natal. Durban, 2017.Optimal control problems are frequently encountered in chemical engineering process control applications as a result of the drive for more regulatory compliant, efficient and economical operation of chemical processes. Despite the significant advancements that have been made in Optimal Control Theory and the development of methods to solve this class of optimization problems, limitations in their applicability to non-linear systems inherent in chemical process unit operations still remains a challenge, particularly in determining a globally optimal solution and solutions to systems that contain state constraints. The objective of this thesis was to develop a method for modelling a chemical process based dynamic system as a graph so that an optimal control problem based on the system can be solved as a shortest path graph search problem by applying Dijkstra’s Algorithm. Dijkstra’s algorithm was selected as it is proven to be a robust and global optimal solution based algorithm for solving the shortest path graph search problem in various applications. In the developed approach, the chemical process dynamic system was modelled as a weighted directed graph and the continuous optimal control problem was reformulated as graph search problem by applying appropriate finite discretization and graph theoretic modelling techniques. The objective functional and constraints of an optimal control problem were successfully incorporated into the developed weighted directed graph model and the graph was optimized to represent the optimal transitions between the states of the dynamic system, resulting in an Optimal State Transition Graph (OST Graph). The optimal control solution for shifting the system from an initial state to every other achievable state for the dynamic system was determined by applying Dijkstra’s Algorithm to the OST Graph. The developed OST Graph-Dijkstra’s Algorithm optimal control solution approach successfully solved optimal control problems for a linear nuclear reactor system, a non-linear jacketed continuous stirred tank reactor system and a non-linear non-adiabatic batch reactor system. The optimal control solutions obtained by the developed approach were compared with solutions obtained by the variational calculus, Iterative Dynamic Programming and the globally optimal value-iteration based Dynamic Programming optimal control solution approaches. Results revealed that the developed OST Graph-Dijkstra’s Algorithm approach provided a 14.74% improvement in the optimality of the optimal control solution compared to the variational calculus solution approach, a 0.39% improvement compared to the Iterative Dynamic Programming approach and the exact same solution as the value–iteration Dynamic Programming approach. The computational runtimes for optimal control solutions determined by the OST Graph-Dijkstra’s Algorithm approach were 1 hr 58 min 33.19 s for the nuclear reactor system, 2 min 25.81s for the jacketed reactor system and 8.91s for the batch reactor system. It was concluded from this work that the proposed method is a promising approach for solving optimal control problems for chemical process-based dynamic systems

Whitworth College Bulletin 1993-1995

https://digitalcommons.whitworth.edu/whitworthcatalogs/1080/thumbnail.jp