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This volume presents the proceedings of the SeventhInternational Workshopon Computational Geometry, CG'91,held at the University of Berne,Switzerland, March 21/22,1991.Computational geometry is not a precisely defined field.Often, it is understood as a nearly mathematical discipline,dealing mainly with complexity questionsconcerninggeometrical problems and algorithms. But often too,andperhaps increasingly, questions of more practical relevanceare central, such as applicability, numerical behavior andperformance for allkinds of input size. Topics consideredin CG'91 include:-Generalizations and applications of the Voronoi diagram- Problems with rectangular objects- Path determination- Moving objects- Visibility questions- Layout problems- Representation of spatialobjects and spatial queries- Problems in higher dimensions- Implementation questions- Relations to artificial intelligence. ... Read more

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This monograph develops projective geometries and provides asystematic treatment of morphisms. It is unique in that it does notconfine itself to isomorphisms. This work introduces a new fundamentaltheorem and its applications describing homogeneous coordinates asmorphisms of projective geometries by semilinear maps. Other topicstreated include three equivalent definitions of projective geometriesand their correspondence with certain lattices; quotients ofprojective geometries and isomorphism theorems; recent results indimension theory; morphisms and homomorphisms of projectivegeometries; special morphisms; duality theory; morphisms of affinegeometries; polarities; orthogonalities; Hilbertian geometries andpropositional systems. The book concludes with a large section ofexercises.

Audience: This volume will be of interest to mathematicians andresearchers whose work involves projective geometries and theirmorphisms, semilinear maps and sesquilinear forms, lattices, categorytheory, and quantum mechanics. This book can also be recommended as atext in axiomatic geometry. ... Read more

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In algebraic topology some classical invariants - such as Betti numbers and Reidemeister torsion - are defined for compact spaces and finite group actions. They can be generalized using von Neumann algebras and their traces, and applied also to non-compact spaces and infinite groups. These new L2-invariants contain very interesting and novel information and can be applied to problems arising in topology, K-Theory, differential geometry, non-commutative geometry and spectral theory. It is particularly these interactions with different fields, that make L2-invariants very powerful and exciting. The book presents a comprehensive introduction to this area of research, as well as its most recent results and developments. It is written in a way which enables the reader to pick out his favourite topic and to find the result she or he is interested in quickly and without being forced to go through other material. ... Read more

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Contains 1 study guide master for every lesson in geometry. Full-size answers to make it easy for teacher or student to check their work. ... Read more

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This book introduces the basic ideas of mathematical proof to students embarking on university mathematics. The emphasis is on helping the reader to understand and construct proofs and write clear mathematics. The authors achieve this by exploring set theory, combinatorics and number theory, which include many fundamental mathematical ideas. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all time great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas. ... Read more

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The second edition of the Handbook of Discrete and Computational Geometry is a thoroughly revised version of the bestselling first edition. With the addition of 500 pages and 14 new chapters covering topics such as geometric graphs, collision detection, clustering, applications of computational geometry, and statistical applications, this is a significant update. This edition includes expanded coverage on the topics of mesh generation in two and three dimensions, aspect graphs, center points, and probabilistic roadmap algorithms. It also features new results on solutions of the Kepler conjecture, and honeycomb conjecture, new bounds on K sets, and new results on face numbers of polytopes. ... Read more

Customer Reviews (1)

Very comprehensive overview of computational geometry
This book, written by many well-known experts in the field, is a fine compendium of articles on the most active areas of computational geometry. Each article is supplemented with a glossary of terms needed for understanding the relevant concepts and frequently contains a list of open problems.An overview of the convex hull of a collection of random points in Euclidean n-space is given in one of the articles on discrete aspects of stochastic geometry, where also a very interesting discussion of generalizations of the Buffon needle problem is given.

There are a few articles overviewing Voronoi diagrams, such as the one on Voronoi diagrams and triangulations. The applications of Voronoi diagrams are many, and include tumour cell diagnosis, biometry, galaxy distributions, and pattern recognition. This article is a little short considering the importance of the subject.

The article on shortest paths and networks is somewhat disappointing since there is no in-depth discussion on network routing algorithms.

The article on computational topology highlights some of the results in this very important area. Many problems in topology have been tackled recently using computers, particularly the work of the mathematician A.T. Fomenko. Computational topology is a relatively young field, having been in existence only since the early 1990's. The applications are enormous, ranging from meshing, morphing, feature extraction, data compression, and in many scientific areas such as computational medicine, chemistry, and astrophysics. It can also be used in computer security via graphical passwords. It is an immense help in visualizing complicated topological objects, such as Lens spaces, horned spheres, and thickened knots. The article does not touch on the use of Mayer-Vietoris sequences to design efficient divide-and-conquer schemes for computing the homology of higher-dimensional complexes. The interplay between topology and finding better algorithms in computational geometry is one that will flourish no doubt in years to come.

The last section of the book covers applications with the most interesting article being the one on sphere packing and coding theory. The algorithms in sphere packing have direct applicability to error correctiong codes over the field GF(q). The author of this article does touch briefly on general algebraic-geometric codes, which is good considering their importance in applications.

The last article appropriately discusses available software for computational geometry. Although the list of Web sites is quite extensive, there are many more available since this book was first printed.

A very fine addition to the literature on computational geometry and should be on everyone's shelf who is interested in this important area. ... Read more

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Many of the earliest books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive. We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork. ... Read more

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Introduction to Differential Geometry with applications to Navier-Stokes Dynamics is an invaluable manuscript for anyone who wants to understand and use exterior calculus and differential geometry, the modern approach to calculus and geometry.

Author Troy Story makes use of over thirty years of research experience to provide a smooth transition from conventional calculus to exterior calculus and differential geometry, assuming only a knowledge of conventional calculus.Introduction to Differential Geometry with applications to Navier-Stokes Dynamics includes the topics:

• Geometry,
• Exterior calculus,
• Homology and co-homology,
• Applications of differential geometry and exterior calculus to: Hamiltonian mechanics, geometric optics, irreversible thermodynamics, black hole dynamics, electromagnetism, classical string fields, and Navier-Stokes dynamics. ... Read more

  Customer Reviews (3)

  Excellent Investment
  Beginning with a definiton of Euclidean geometry, this book presents a logical and rigorous introduction to differential geometry and exterior calculus. Rather than using the style of of writing theorems followed by proofs or sketches of proofs, the writer displays fundamental definitions followed by examples and a chapter on applications to dynamics. In contrast to one of the reviewers, my purchase of this book is an excellent investment.

  Insane
  Refering to this book as an introduction to anything is insane. Definitions occur rarely and invariably use terms which are themselves undefined.While it is reasonable for the author of a book on differentiable geometry to not define the derivative of real valued function of a single real variable, or a jacobian matrix, a notion such as a metric tensor should be fully defined.
  This book lacks any pretense of rigour (I do not recall seeing the word proof once!). This alone is not a damning feature but it is also lacking in tools to develop intuition as well. Do not expect to be able to prove or compute anything after reading this book. I got the e-book version and it is the worst 6 dollars I have ever spent.

  Highly recomended
  This book has the clearest and most direct introduction to differential geometry that I've seen, thus, I highly recommend it.
  
  One of the book's highlights is chapter 2, which presents a discussion of exterior calculus assuming only a knowledge of conventional calculus. Unlike most books on this topic(exterior calculus), this book includes a definition of the exterior derivative rather than a few examples.
  
  Another highlight is the chapter on dynamics, where it is shown that many areas of dynamics can be described by differential one-forms, including Navier-Stokes dynamics for incompressible fluids.
  
  
  
  ... Read more

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This work presents some classical as well as some very recentresults and techniques concerning the spectral geometry correspondingto the Laplace--Beltrami operator and the Hodge--de Rhamoperators. It treats many topics that are not usually dealt with in this field,such as the continuous dependence of the eigenvalues with respect tothe Riemannian metric in the C-topology, and some of theirconsequences, such as Uhlenbeck's genericity theorem; examples ofnon-isometric flat tori in all dimensions greater than or equal tofour; Gordon's classical technique for constructing isospectral closedRiemannian manifolds; a detailed presentation of Sunada's techniqueand Pesce's approach to isospectrality; Gordon and Webb's example ofnon-isometric convex domains in Rn (n>=4) thatare isospectral for both Dirichlet and Neumann boundary conditions;the Chanillo--Treves estimate for the first positiveeigenvalue of the Hodge--de Rham operator, etc. Significantapplications are developed, and many open problems, references andsuggestions for further reading are given. Several themes foradditional research are pointed out. Audience: This volume is designed as an introductory text formathematicians and physicists interested in global analysis, analysison manifolds, differential geometry, linear and multilinear algebra,and matrix theory. It is accessible to readers whose backgroundincludes basic Riemannian geometry and functional analysis. Thesemathematical prerequisites are covered in the first two chapters, thusmaking the book largely self-contained. ... Read more

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The theory of rigid (analytic) spaces, originally invented to describe degenerations, reductions, and moduli of algebraic curves and abelian varieties, has undergone significant growth in the last two decades; today the theory has applications to arithmetic algebraic geometry, number theory, the arithmetic of function fields, and p-adic differential equations. This work, a revised and greatly expanded new English edition of the earlier French text by the same authors, is an accessible introduction to the theory of rigid spaces and now includes a large number of exercises.

Key topics:

* Chapters on the applications of this theory to curves and abelian varieties: the Tate curve, stable reduction for curves, Mumford curves, Néron models, uniformization of abelian varieties

* Unified treatment of the concepts: points of a rigid space, overconvergent sheaves, Monsky--Washnitzer cohomology and rigid cohomology; detailed examination of Kedlaya's application of the Monsky--Washnitzer cohomology to counting points on a hyperelliptic curve over a finite field

* The work of Drinfeld on "elliptic modules" and the Langlands conjectures for function fields use a background of rigid étale cohomology; detailed treatment of this topic

* Presentation of the rigid analytic part of Raynaud's proof of the Abhyankar conjecture for the affine line, with only the rudiments of that theory

Assuming a basic knowledge of algebraic geometry, advanced graduate students and researchers in algebraic geometry, number theory, representation theory, and other areas of mathematics will benefit from the book's breadth and clarity. ... Read more

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This volume is the proceedings of the Conference on Algebra and Algebraic Geometry with Applications which was held July 19 - 26, 2000, at Purdue University to honor Professor Shreeram S. Abhyankar on the occasion of his seventieth birthday. Eighty-five of Professor Abhyankar's students, collaborators, and colleagues were invited participants. Sixty participants presented papers related to Professor Abhyankar's broad areas of mathematical interest. There were sessions on algebraic geometry, singularities, group theory, Galois theory, combinatorics, Drinfield modules, affine geometry, and the Jacobian problem. This volume offers an outstanding collection of papers by authors who are among the experts in their areas. ... Read more

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This book contains an introduction to the recently developedspectral theory of associative rings and Abelian categories, and itsapplications to the study of irreducible representations of classes ofalgebras which play an important part in modern mathematical physics.
Audience: A self-contained volume for researchers and graduatestudents interested in new geometric ideas in algebra, and in thespectral theory of noncommutative rings, currently invadingmathematical physics. Valuable reading for mathematicians working onrepresentation theory, quantum groups and related topics,noncommutative algebra, algebraic geometry, and algebraic K-theory.
... Read more

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"An introduction to the ideas of algebraic geometry in the motivated context of system theory." Thus the author describes his textbook that has been specifically written to serve the needs of students of systems and control. Without sacrificing mathematical care, the author makes the basic ideas of algebraic geometry accessible to engineers and applied scientists. The emphasis is on constructive methods and clarity rather than on abstraction.

The student will find here a clear presentation, with an applied flavor, of the core ideas in the algebro-geometric treatment of scalar linear system theory. The author introduces the four representations of a scalar linear system and establishes the major results of the underlying theory. He concludes with an examination of abstract affine varieties and an interlude which indicates the development of a similar theory for multivariable systems appearing in a succeeding volume ( Part II: Multivariable Linear Systems and Projective Algebraic Geometry ).

Prerequisites are the basics of linear algebra, some simple notions from topology and the elementary properties of groups, rings, and fields, and a basic course in linear systems. Exercises are an integral part of the treatment and are used where relevant in the main body of the text. ... Read more