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. Wellwritten 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 multidimensional theories.”(MATHEMATICAL REVIEWS)
 Table of contents (6 chapters)


Background Material
Pages 191

Computational Algebraic Analysis for Systems of Linear Constant Coefficients Differential Equations
Pages 93138

The CauchyFueter System and Its Variations
Pages 139207

Special First Order Systems in Clifford Analysis
Pages 209266

Some First Order Linear Operators in Physics
Pages 267306

Table of contents (6 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Analysis of Dirac Systems and Computational Algebra
 Authors

 Fabrizio Colombo
 Irene Sabadini
 Franciscus Sommen
 Daniele C. Struppa
 Series Title
 Progress in Mathematical Physics
 Series Volume
 39
 Copyright
 2004
 Publisher
 Birkhäuser Basel
 Copyright Holder
 Springer Science+Business Media New York
 eBook ISBN
 9780817681661
 DOI
 10.1007/9780817681661
 Hardcover ISBN
 9780817642556
 Softcover ISBN
 9781461264699
 Series ISSN
 15449998
 Edition Number
 1
 Number of Pages
 XV, 332
 Topics