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Discrete Differential Geometry
An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. For a given ...
Computational Approach to Riemann Surfaces
This volume offers a well-structured overview of existent computational approaches to Riemann surfaces and those currently in development. The authors of the contributions represent the groups providing publically available numerical codes in this field. Thus this volume illustrates which software tools are available and how they can be used in practice. In addition examples for solutions to partial differential equations and in surface theory are presented. The intended audience of this book is twofold. It can be used as a textbook for a graduate course in numerics of Riemann surfaces, in which case the standard undergraduate background, i.e., calculus and linear algebra, is required. In pa...
Non-Euclidean Laguerre Geometry and Incircular Nets
This textbook is a comprehensive and yet accessible introduction to non-Euclidean Laguerre geometry, for which there exists no previous systematic presentation in the literature. Moreover, we present new results by demonstrating all essential features of Laguerre geometry on the example of checkerboard incircular nets. Classical (Euclidean) Laguerre geometry studies oriented hyperplanes, oriented hyperspheres, and their oriented contact in Euclidean space. We describe how this can be generalized to arbitrary Cayley-Klein spaces, in particular hyperbolic and elliptic space, and study the corresponding groups of Laguerre transformations. We give an introduction to Lie geometry and describe how these Laguerre geometries can be obtained as subgeometries. As an application of two-dimensional Lie and Laguerre geometry we study the properties of checkerboard incircular nets.
Integrable Systems
Designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors, this book has its origins in a lecture series given by the internationally renowned authors. Written in an accessible, informal style, it fills a gap in the existing literature.
Electromagnetic Theory and Computation
This book explores the connection between algebraic structures in topology and computational methods for 3-dimensional electric and magnetic field computation. The connection between topology and electromagnetism has been known since the 19th century, but there has been little exposition of its relevance to computational methods in modern topological language. This book is an effort to close that gap. It will be of interest to people working in finite element methods for electromagnetic computation and those who have an interest in numerical and industrial applications of algebraic topology.
Geometry II
A very clear account of the subject from the viewpoints of elementary geometry, Riemannian geometry and group theory – a book with no rival in the literature. Mostly accessible to first-year students in mathematics, the book also includes very recent results which will be of interest to researchers in this field.
Bäcklund and Darboux Transformations
This book explores the deep and fascinating connections that exist between a ubiquitous class of physically important waves known as solitons and the theory of transformations of a privileged class of surfaces as they were studied by eminent geometers of the nineteenth century. Thus, nonlinear equations governing soliton propagation and also mathematical descriptions of their remarkable interaction properties are shown to arise naturally out of the classical differential geometry of surfaces and what are termed Bäcklund-Darboux transformations.This text, the first of its kind, is written in a straightforward manner and is punctuated by exercises to test the understanding of the reader. It is suitable for use in higher undergraduate or graduate level courses directed at applied mathematicians or mathematical physics.
Algebraic and Geometric Aspects of Integrable Systems and Random Matrices
This volume contains the proceedings of the AMS Special Session on Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, held from January 6-7, 2012, in Boston, MA. The very wide range of topics represented in this volume illustrates
Concrete Mathematics
This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline. Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas,...
An Excursion Through Discrete Differential Geometry
Discrete Differential Geometry (DDG) is an emerging discipline at the boundary between mathematics and computer science. It aims to translate concepts from classical differential geometry into a language that is purely finite and discrete, and can hence be used by algorithms to reason about geometric data. In contrast to standard numerical approximation, the central philosophy of DDG is to faithfully and exactly preserve key invariants of geometric objects at the discrete level. This process of translation from smooth to discrete helps to both illuminate the fundamental meaning behind geometric ideas and provide useful algorithmic guarantees. This volume is based on lectures delivered at the 2018 AMS Short Course ``Discrete Differential Geometry,'' held January 8-9, 2018, in San Diego, California. The papers in this volume illustrate the principles of DDG via several recent topics: discrete nets, discrete differential operators, discrete mappings, discrete conformal geometry, and discrete optimal transport.
