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.
Inhaltsverzeichnis
A. Differential Equations and Operator Theory. - Hodge Decompositions on Weakly Lipschitz Domains. - Monogenic Functions of Bounded Mean Oscillation in the Unit Ball. - Bp, q-Functions and their Harmonic Majorants. - Spherical Means and Distributions in Clifford Analysis. - Hypermonogenic Functions and their Cauchy-Type Theorems. - On Series Expansions of Hyperholomorphic BqFunctions. - Pointwise Convergence of Fourier Series on the Unit Sphere of R4with the Quaternionic Setting. - Cauchy Kernels for some Conformally Flat Manifolds. - Clifford Analysis on the Space of Vectors, Bivectors and ? -vectors. - B. Global Analysis and Differential Geometry. - Universal Bochner-Weitzenböck Formulas for Hyper-Kählerian Gradients. - Cohomology Groups of Harmonic Spinors on Conformally Flat Manifolds. - Spin Geometry, Clifford Analysis, and Joint Seminormality. - A Mean Value Laplacian for Strongly Kähler Finsler Manifolds. - C. Applications. - Non-commutative Determinants and Quaternionic Monge-Ampère Equations. - Galpern Sobolev Type Equations with Non-constant Coefficients. - A Theory of Modular Forms in Clifford Analysis, their Applications and Perspectives. - Automated Geometric Theorem Proving, Clifford Bracket Algebra and Clifford Expansions. - Quaternion-valued Smooth Orthogonal Wavelets with Short Support and Symmetry.
