Planar graphs : a historical perspective.

The field of graph theory has been indubitably influenced by the study of planar graphs. This thesis, consisting of five chapters, is a historical account of the origins and development of concepts pertaining to planar graphs and their applications. The first chapter serves as an introduction to the history of graph theory, including early studies of graph theory tools such as paths, circuits, and trees. The second chapter pertains to the relationship between polyhedra and planar graphs, specifically the result of Euler concerning the number of vertices, edges, and faces of a polyhedron. Counterexamples and generalizations of Euler\u27s formula are also discussed. Chapter III describes the background in recreational mathematics of the graphs of K5 and K3,3 and their importance to the first characterization of planar graphs by Kuratowski. Further characterizations of planar graphs by Whitney, Wagner, and MacLane are also addressed. The focus of Chapter IV is the history and eventual proof of the four-color theorem, although it also includes a discussion of generalizations involving coloring maps on surfaces of higher genus. The final chapter gives a number of measurements of a graph\u27s closeness to planarity, including the concepts of crossing number, thickness, splitting number, and coarseness. The chapter conclused with a discussion of two other coloring problems - Heawood\u27s empire problem and Ringel\u27s earth-moon problem

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 exploration of a category theory-based virtual Geometrical product specification system for design and manufacturing

In order to ensure quality of products and to facilitate global outsourcing, almost all the so-called “world-class” manufacturing companies nowadays are applying various tools and methods to maintain the consistency of a product’s characteristics throughout its manufacturing life cycle. Among these, for ensuring the consistency of the geometric characteristics, a tolerancing language − the Geometrical Product Specification (GPS) has been widely adopted to precisely transform the functional requirements from customers into manufactured workpieces expressed as tolerance notes in technical drawings. Although commonly acknowledged by industrial users as one of the most successful efforts in integrating existing manufacturing life-cycle standards, current GPS implementations and software packages suffer from several drawbacks in their practical use, possibly the most significant, the difficulties in inferring the data for the “best” solutions. The problem stemmed from the foundation of data structures and knowledge-based system design. This indicates that there need to be a “new” software system to facilitate GPS applications. The presented thesis introduced an innovative knowledge-based system − the VirtualGPS − that provides an integrated GPS knowledge platform based on a stable and efficient database structure with knowledge generation and accessing facilities. The system focuses on solving the intrinsic product design and production problems by acting as a virtual domain expert through translating GPS standards and rules into the forms of computerized expert advices and warnings. Furthermore, this system can be used as a training tool for young and new engineers to understand the huge amount of GPS standards in a relative “quicker” manner. The thesis started with a detailed discussion of the proposed categorical modelling mechanism, which has been devised based on the Category Theory. It provided a unified mechanism for knowledge acquisition and representation, knowledge-based system design, and database schema modelling. As a core part for assessing this knowledge-based system, the implementation of the categorical Database Management System (DBMS) is also presented in this thesis. The focus then moved on to demonstrate the design and implementation of the proposed VirtualGPS system. The tests and evaluations of this system were illustrated in Chapter 6. Finally, the thesis summarized the contributions to knowledge in Chapter 7. After thoroughly reviewing the project, the conclusions reached construe that the III entire VirtualGPS system was designed and implemented to conform to Category Theory and object-oriented programming rules. The initial tests and performance analyses show that the system facilitates the geometric product manufacturing operations and benefits the manufacturers and engineers alike from function designs, to a manufacturing and verification

American pure and applied mathematics, 1940-1975

Thesis (Ph. D. in History, Anthropology, and Science, Technology and Society (HASTS))--Massachusetts Institute of Technology, Program in Science, Technology and Society, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (pages 317-336).This study investigates the status of mathematical knowledge in mid-century America. It is motivated by questions such as: when did mathematical theories become applicable to a wide range of fields from medicine to the social science? How did this change occur? I ask after the implications of this transformation for the development of mathematics as an academic discipline and how it affected what it meant to be a mathematician. How did mathematicians understand the relation between abstractions and generalizations on the one hand and their manifestation in concrete problems on the other? Mathematics in Cold War America was caught between the sciences and the humanities. This dissertation tracks the ways this tension between the two shaped the development of professional identities, pedagogical regimes, and the epistemological commitments of the American mathematical community in the postwar period. Focusing on the constructed division between pure and applied mathematics, it therefore investigates the relationship of scientific ideas to academic and governmental institutions, showing how the two are mutually inclusive. Examining the disciplinary formation of postwar mathematics, I show how ideas about what mathematics is and what it should be crystallized in institutional contexts, and how in turn these institutions reshaped those ideas. Tuning in to the ways different groups of mathematicians strove to make sense of the transformations in their fields and the way they struggled to implement their ideological convictions into specific research agendas and training programs sheds light on the co-construction of mathematics, the discipline, and mathematics as a body of knowledge. The relation between pure and applied mathematics and between mathematics and the rest of the sciences were disciplinary concerns as much as they were philosophical musings. As the reconfiguration of the mathematical field during the second half of the twentieth century shows, the dynamic relation between the natural and the human sciences reveals as much about institutions, practices, and nations as it does about epistemological commitments.by Alma Steingart.Ph.D.in History, Anthropology, and Science, Technology and Society (HAST

The exploration of a category theory-based virtual geometrical product specification system for design and manufacturing

In order to ensure quality of products and to facilitate global outsourcing, almost all the so-called “world-class” manufacturing companies nowadays are applying various tools and methods to maintain the consistency of a product’s characteristics throughout its manufacturing life cycle. Among these, for ensuring the consistency of the geometric characteristics, a tolerancing language − the Geometrical Product Specification (GPS) has been widely adopted to precisely transform the functional requirements from customers into manufactured workpieces expressed as tolerance notes in technical drawings. Although commonly acknowledged by industrial users as one of the most successful efforts in integrating existing manufacturing life-cycle standards, current GPS implementations and software packages suffer from several drawbacks in their practical use, possibly the most significant, the difficulties in inferring the data for the “best” solutions. The problem stemmed from the foundation of data structures and knowledge-based system design. This indicates that there need to be a “new” software system to facilitate GPS applications. The presented thesis introduced an innovative knowledge-based system − the VirtualGPS − that provides an integrated GPS knowledge platform based on a stable and efficient database structure with knowledge generation and accessing facilities. The system focuses on solving the intrinsic product design and production problems by acting as a virtual domain expert through translating GPS standards and rules into the forms of computerized expert advices and warnings. Furthermore, this system can be used as a training tool for young and new engineers to understand the huge amount of GPS standards in a relative “quicker” manner. The thesis started with a detailed discussion of the proposed categorical modelling mechanism, which has been devised based on the Category Theory. It provided a unified mechanism for knowledge acquisition and representation, knowledge-based system design, and database schema modelling. As a core part for assessing this knowledge-based system, the implementation of the categorical Database Management System (DBMS) is also presented in this thesis. The focus then moved on to demonstrate the design and implementation of the proposed VirtualGPS system. The tests and evaluations of this system were illustrated in Chapter 6. Finally, the thesis summarized the contributions to knowledge in Chapter 7. After thoroughly reviewing the project, the conclusions reached construe that the III entire VirtualGPS system was designed and implemented to conform to Category Theory and object-oriented programming rules. The initial tests and performance analyses show that the system facilitates the geometric product manufacturing operations and benefits the manufacturers and engineers alike from function designs, to a manufacturing and verification.EThOS - Electronic Theses Online ServiceGBUnited Kingdo