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Progress in Mathematical Physics

Analysis of Dirac Systems and Computational Algebra

Authors: Colombo, F., Sabadini, I., Sommen, F., Struppa, D.C.

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About this Textbook

The subject of Clifford algebras has become an increasingly rich area of research with a significant number of important applications not only to mathematical physics but to numerical analysis, harmonic analysis, and computer science.

The main treatment is devoted to the analysis of systems of linear partial differential equations with constant coefficients, focusing attention on null solutions of Dirac systems. In addition to their usual significance in physics, such solutions are important mathematically as an extension of the function theory of several complex variables. The term "computational" in the title emphasizes two main features of the book, namely, the heuristic use of computers to discover results in some particular cases, and the application of Gröbner bases as a primary theoretical tool.

Knowledge from different fields of mathematics such as commutative algebra, Gröbner bases, sheaf theory, cohomology, topological vector spaces, and generalized functions (distributions and hyperfunctions) is required of the reader. However, all the necessary classical material is initially presented.

The book may be used by graduate students and researchers interested in (hyper)complex analysis, Clifford analysis, systems of partial differential equations with constant coefficients, and mathematical physics.

Reviews

From the reviews:

"The book presents a uniform treatment of some fundamental differential equations for physics. Maxwell and Dirac equations are particular examples that fall into this study. The authors concentrate on systems of linear partial differential equations with constatn coefficients n the Clifford algebra setting...The material is presented in a very accessible format...The book ends with a list of open problems that pertain to the topic." ---Internationale Mathematische Nachrichtén, Nr. 201

"The first 138 pages of this book are a good introduction to algebraic analysis (in the sense of Sato), and some computational aspects, in the setting of quaternionic analysis. But the core of the book is the study of different important systems of partial differential equations in the setting of Clifford analysis...The last chapter states some open problems and avenues of further research. A rich list of references, an alphabetic index and a list of notation close the volume. Well-written and with many explicit results, the book is interesting and is addressed to Ph.D. students and researchers interested in this field." ---Revue Roumaine de Mathématiques Pures et Appliquées

“Altogether the book is a pioneering, and quite successful, attempt to apply computational and algebraic techniques to several branches of hypercomplex analysis … The book provides a very different way to look at some important questions which arise when one tries to develop multi-dimensional theories.”(MATHEMATICAL REVIEWS)


Table of contents (6 chapters)

  • Background Material

    Colombo, Fabrizio (et al.)

    Pages 1-91

  • Computational Algebraic Analysis for Systems of Linear Constant Coefficients Differential Equations

    Colombo, Fabrizio (et al.)

    Pages 93-138

  • The Cauchy-Fueter System and Its Variations

    Colombo, Fabrizio (et al.)

    Pages 139-207

  • Special First Order Systems in Clifford Analysis

    Colombo, Fabrizio (et al.)

    Pages 209-266

  • Some First Order Linear Operators in Physics

    Colombo, Fabrizio (et al.)

    Pages 267-306

Buy this book

eBook $79.99
price for USA (gross)
  • ISBN 978-0-8176-8166-1
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $104.00
price for USA
  • ISBN 978-0-8176-4255-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $104.00
price for USA
  • ISBN 978-1-4612-6469-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Analysis of Dirac Systems and Computational Algebra
Authors
Series Title
Progress in Mathematical Physics
Series Volume
39
Copyright
2004
Publisher
Birkhäuser Basel
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-0-8176-8166-1
DOI
10.1007/978-0-8176-8166-1
Hardcover ISBN
978-0-8176-4255-6
Softcover ISBN
978-1-4612-6469-9
Series ISSN
1544-9998
Edition Number
1
Number of Pages
XV, 332
Topics