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Advances in Analysis and Geometry

New Developments Using Clifford Algebras

  • Tao Qian
  • Thomas Hempfling
  • Alan McIntosh
  • Frank Sommen

Part of the Trends in Mathematics book series (TM)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Differential Equations and Operator Theory

    1. Front Matter
      Pages 1-1
    2. Andreas Axelsson, Alan McIntosh
      Pages 3-29
    3. S. Bernstein, K. Gürlebeck, L. F. Reséndis, Luis M. Tovar S.
      Pages 51-63
    4. Fred Brackx, Richard Delanghe, Frank Sommen
      Pages 65-96
    5. Sirkka-Liisa Eriksson, Heinz Leutwiler
      Pages 97-112
    6. K. Gürlebeck, A. El-Sayed Ahmed
      Pages 113-129
  3. Global Analysis and Differential Geometry

  4. Applications

About this book

Introduction

On the 16th of October 1843, Sir William R. Hamilton made the discovery of the quaternion algebra H = qo + qli + q2j + q3k whereby the product is determined by the defining relations ·2 ·2 1 Z =] = - , ij = -ji = k. In fact he was inspired by the beautiful geometric model of the complex numbers in which rotations are represented by simple multiplications z ----t az. His goal was to obtain an algebra structure for three dimensional visual space with in particular the possibility of representing all spatial rotations by algebra multiplications and since 1835 he started looking for generalized complex numbers (hypercomplex numbers) of the form a + bi + cj. It hence took him a long time to accept that a fourth dimension was necessary and that commutativity couldn't be kept and he wondered about a possible real life meaning of this fourth dimension which he identified with the scalar part qo as opposed to the vector part ql i + q2j + q3k which represents a point in space.

Keywords

Algebra Clifford algebras Clifford analysis Cohomology Distribution Elliptic functions Hypermonogenic functions Mathematical physics Operator theory Spinor geometry calculus

Editors and affiliations

  • Tao Qian
    • 1
  • Thomas Hempfling
    • 2
  • Alan McIntosh
    • 3
  • Frank Sommen
    • 4
  1. 1.Faculty of Science and TechnologyUniversity of MacauMacauChina
  2. 2.BaselSwitzerland
  3. 3.Centre for Mathematics and its Applications Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  4. 4.Department of Mathematical AnalysisUniversity of GhentGhentBelgium

Bibliographic information