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This collection of essays on the legacy of mathematican Donald Coxeter is a mixture of surveys, updates, history, storytelling and personal memories covering both applied and abstract maths. Subjects include: polytopes, Coxeter groups, equivelar polyhedra, Ceva's theorum, and Coxeter and the artists.
Combinatorics '81
A total foliation is an example of a geometric structure on a manifold. A total foliation of an [script]n dimensional manifold consists of a [script]n codimension one foliations that are transverse at every point. If a manifold admits a total foliation where all [script]n foliations are transverse oriented, it is said to be totally parallelizable. A necessary condition for total parallelizability is that the manifold be parallelizable. Whether or not this is also a sufficient condition is not known. In this memoir, the author proves a theorem: All compact orientable three dimensional manifolds admit total foliations. This theorem is proven by explicitly constructing total foliations for all compact orientable three manifolds
The articles in this collection are based on lectures given at the 20th Annual Symposium on Some Mathematical Questions in Biology, held in May 1986, and sponsored jointly by the AMS, the Society for Industrial and Applied Mathematics, and Section A of the American Association for the Advancement of Science. For the past thirty years, due particularly to the fundamental work of Pittendrigh, Aschoff, and Wever, theoretical analysis of circadian rhythms and sleep have gone hand in hand with experimental and clinical studies. Circadian rhythms have been investigated at levels ranging from cell fragments to humans, from biochemistry to behavior. This experimental diversity is reflected in a diversity of modeling approaches, several of which are represented in this collection. One class of models focuses on the circadian sleep and activity cycles of humans, for which some investigators postulate pacemaker systems with two coupled oscillators, while others propose single oscillator models. Other analyses focus upon the activity patterns of small vertebrates or upon anatomical data and physiological recordings. The mathematical formulations and analyses utilize nonlinear dynamical systems, stochastic models, and computer simulations. The articles in this volume discuss, analyze, and compare these various experimental, theoretical, and mathematical approaches.
Contains lectures from the CBMS Regional Conference held at Harvey Mudd College, June 1977. This monograph consists of applications to nonlinear differential equations of the author's coincidental degree. It includes an bibliography covering many aspects of the modern theory of nonlinear differential equations and the theory of nonlinear analysis.
Consists of three chapters covering the following topics: foundations, bilinear forms and presentations of certain 2-step nilpotent Lie groups, discrete subgroups of the Heisenberg group, the automorphism group of the Heisenberg group, fundamental unitary representations of the Heisenberg group, and the Fourier transform and the Weil-Brezin map.
An illuminating biography of one of the greatest geometers of the twentieth century Driven by a profound love of shapes and symmetries, Donald Coxeter (1907–2003) preserved the tradition of classical geometry when it was under attack by influential mathematicians who promoted a more algebraic and austere approach. His essential contributions include the famed Coxeter groups and Coxeter diagrams, tools developed through his deep understanding of mathematical symmetry. The Man Who Saved Geometry tells the story of Coxeter’s life and work, placing him alongside history’s greatest geometers, from Pythagoras and Plato to Archimedes and Euclid—and it reveals how Coxeter’s boundless creativity reflects the adventurous, ever-evolving nature of geometry itself. With an incisive, touching foreword by Douglas R. Hofstadter, The Man Who Saved Geometry is an unforgettable portrait of a visionary mathematician.