2011, XVII, 416 p. 2 illus.
Hardcover, ISBN 978-0-85729-807-2
As computer software becomes more complex, the question of how its correctness can be assured grows ever more critical. Formal logic embodied in computer programs is an important part of the answer to this problem.
This must-read text presents the pioneering work of the late Professor Jacob (Jack) T. Schwartz on computational logic and set theory and its application to proof verification techniques, culminating in the AtnaNova system, a prototype computer program designed to verify the correctness of mathematical proofs presented in the language of set theory. Taking a systematic approach, the book begins with a survey of traditional branches of logic before describing in detail the underlying design of the AtnaNova system. Major classical results on undecidability and unsolvability are then recast for this system. Readers do not require great knowledge of formal logic in order to follow the text, but a good understanding of standard programming techniques, and a familiarity with mathematical definitions and proofs reflecting the usual levels of rigor is assumed.
1st Edition., 2011, XXIV, 446 p. 123 illus.
Hardcover, ISBN 978-0-85729-810-2
Due: August 31, 2011
Geometric algebra (GA), also known as Clifford algebra, is a powerful unifying framework for geometric computations that extends the classical techniques of linear algebra and vector calculus in a structural manner. Its benefits include cleaner computer-program solutions for known geometric computation tasks, and the ability to address increasingly more involved applications.
This highly practical Guide to Geometric Algebra in Practice reviews algebraic techniques for geometrical problems in computer science and engineering, and the relationships between them. The topics covered range from powerful new theoretical developments, to successful applications, and the development of new software tools. Contributions are included from an international community of experts spanning a broad range of disciplines.
How to Read this Guide to Geometric Algebra in Practice
Leo Dorst and Joan Lasenby
Part I: Rigid Body Motion
Part II: Interpolation and Tracking
Part III: Image Processing
Part IV: Theorem Proving and Combinatorics
Part V: Applications of Line Geometry
Part VI: Alternatives to Conformal Geometric Algebra
Part VII: Towards Coordinate-Free Differential Geometry
Tutorial: Structure Preserving Representation of Euclidean Motions through Conformal Geometric Algebra
Leo Dorst
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1st Edition., 2011, XXII, 358 p.
Broche, ISBN 978-2-8178-0193-3
Les sciences non lineaires ont pour objet l’ensemble des phenomenes dont l’analyse resiste au principe de superposition. Elles concernent en grande partie les systemes dits ≪ complexes ≫ dont l’interaction et l’interdependance entre les parties empechent de predire precisement l’evolution du systeme.
Pour expliquer ces phenomenes, deux approches complementaires ont ete proposees : la theorie des bifurcations et la theorie des catastrophes. Mais la pleine comprehension et la modelisation de la non-linearite restent chacune un defi pour les scientifiques du XXIe siecle. C’est dans la perspective d’accompagner tous ceux qui voudront le relever que ce livre a ete concu.
Son objectif est d’exposer au lecteur le langage et le formalisme necessaires a l’etude de la non-linearite. Partant d’exemples simples, pour ensuite atteindre un niveau d’abstraction visant l’universalite, l’auteur explore les divers scenarios possibles de bifurcations et les catastrophes elementaires caracteristiques du changement qualitatif de comportement d’un systeme ; l’etude de l’evolution temporelle est abordee a travers la mise en equation de phenomenes aux solutions stationnaires ou oscillantes ; l’analyse de l’evolution spatiale des systemes non lineaires nous introduit quant a elle au probleme fascinant de la morphogenese.
Accessible des le premier cycle universitaire aux etudiants de toutes les disciplines concernees par les phenomenes non lineaires (physique, mathematiques, chimie, geologie, economie, etc.), cet ouvrage constituera aussi une synthese riche et utile pour les enseignants et chercheurs de ces differents domaines.
1 Presentation.- 2 Introduction elementaire aux bifurcations a une dimension.- 3 Les autres bifurcations generiques.- 4 Classification des sept catastrophes elementaires.- 5 Bifurcation de Hopf.- 6 Equation d’amplitude pour une bifurcation de Hopf.- 7 Instabilite parametrique et autres instabilites.- 8 Introduction au chaos.- 9 Naissance de l’ordre spatial unidimensionnel.- 10 Universalite au voisinage du seuil.- 11 Fronts entre domaines.- 12 ordres et desordres spatial et temporel.- 13 Structures bidimensionnelles.- 14 Conclusion.
Series: Lecture Notes in Mathematics, Vol. 2028
Subseries: C.I.M.E. Foundation Subseries
1st Edition., 2011, XIII, 224 p. 8 illus., 7 in color.
Softcover, ISBN 978-3-642-21718-0
his volume collects the notes of the CIME course "Nonlinear PDE’s and applications" held in Cetraro (Italy) on June 23?28, 2008. It consists of four series of lectures, delivered by Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), and Cedric Villani (Ecole Normale Superieure de Lyon).
They presented a broad overview of far-reaching findings and exciting new developments concerning, in particular, optimal transport theory, nonlinear evolution equations, functional inequalities, and differential geometry. A sampling of the main topics considered here includes optimal transport, Hamilton-Jacobi equations, Riemannian geometry, and their links with sharp geometric/functional inequalities, variational methods for studying nonlinear evolution equations and their scaling properties, and the metric/energetic theory of gradient flows and of rate-independent evolution problems.
The book explores the fundamental connections between all of these topics and points to new research directions in contributions by leading experts in these fields.
Transport Rays and Applications to Hamilton-Jacobi Equations.- Functional Inequalities and Dynamics.- Differential, Energetic, and Metric Formulations for Rate-independent Processes.- Optimal Transport and Curvature.
Series: Universitext
2011, 2011, XII, 388 p. 28 illus.
Softcover, ISBN 978-1-4471-2169-5
Due: September 30, 2011
It is commonly believed that chaos is linked to non-linearity, however many (even quite natural) linear dynamical systems exhibit chaotic behavior. The study of these systems is a young and remarkably active field of research, which has seen many landmark results over the past two decades. Linear dynamics lies at the crossroads of several areas of mathematics including operator theory, complex analysis, ergodic theory and partial differential equations. At the same time its basic ideas can be easily understood by a wide audience.
Written by two renowned specialists, Linear Chaos provides a welcome introduction to this theory. Split into two parts, part I presents a self-contained introduction to the dynamics of linear operators, while part II covers selected, largely independent topics from linear dynamics. More than 350 exercises and many illustrations are included, and each chapter contains a further ‘Sources and Comments’ section.
The only prerequisites are a familiarity with metric spaces, the basic theory of Hilbert and Banach spaces and fundamentals of complex analysis. More advanced tools, only needed occasionally, are provided in two appendices.
A self-contained exposition, this book will be suitable for self-study and will appeal to advanced undergraduate or beginning graduate students. It will also be of use to researchers in other areas of mathematics such as partial differential equations, dynamical systems and ergodic theory.
Topological dynamics.- Hypercyclic and chaotic operators.- The Hypercyclicity Criterion.- Classes of hypercyclic and chaotic operators.- Necessary conditions for hypercyclicity and chaos.- Connectedness arguments in linear dynamics.- Dynamics of semigroups, with applications to differential equations.- Existence of hypercyclic operators.- Frequently hypercyclic operators.- Hypercyclic subspaces.- Common hypercyclic vectors.- Linear dynamics in topological vector spaces