Ebook: Advances in Analysis and Geometry: New Developments Using Clifford Algebras

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.

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The study of systems of special partial differential operators that arise naturally from the use of Clifford algebra as a calculus tool lies in the heart of Clifford analysis. The focus is on the study of Dirac operators and related ones, together with applications in mathematics, physics and engineering. At the present time, the study of Clifford algebra and Clifford analysis has grown into a major research field.

There are two sources of papers in this collection. One is from a satellite conference to the ICM 2002 in Beijing, held August 15-18 at the University of Macau; and the other stems from invited contributions by top-notch experts in the field. All articles were strictly refereed and contain unpublished new results. Some of them are incorporated with comprehensive surveys in the particular areas that the authors work in.

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The study of systems of special partial differential operators that arise naturally from the use of Clifford algebra as a calculus tool lies in the heart of Clifford analysis. The focus is on the study of Dirac operators and related ones, together with applications in mathematics, physics and engineering. At the present time, the study of Clifford algebra and Clifford analysis has grown into a major research field.

There are two sources of papers in this collection. One is from a satellite conference to the ICM 2002 in Beijing, held August 15-18 at the University of Macau; and the other stems from invited contributions by top-notch experts in the field. All articles were strictly refereed and contain unpublished new results. Some of them are incorporated with comprehensive surveys in the particular areas that the authors work in.

Content:
Front Matter....Pages i-xv
Front Matter....Pages 1-1
Hodge Decompositions on Weakly Lipschitz Domains....Pages 3-29
Monogenic Functions of Bounded Mean Oscillation in the Unit Ball....Pages 31-50
Bp,q-Functions and their Harmonic Majorants....Pages 51-63
Spherical Means and Distributions in Clifford Analysis....Pages 65-96
Hypermonogenic Functions and their Cauchy-Type Theorems....Pages 97-112
On Series Expansions of Hyperholomorphic Bq Functions....Pages 113-129
Pointwise Convergence of Fourier Series on the Unit Sphere of R4 with the Quaternionic Setting....Pages 131-147
Cauchy Kernels for some Conformally Flat Manifolds....Pages 149-160
Clifford Analysis on the Space of Vectors, Bivectors and ?-vectors....Pages 161-185
Front Matter....Pages 187-187
Universal Bochner-Weitzenb?ck Formulas for Hyper-K?hlerian Gradients....Pages 189-208
Cohomology Groups of Harmonic Spinors on Conformally Flat Manifolds....Pages 209-225
Spin Geometry, Clifford Analysis, and Joint Seminormality....Pages 227-255
A Mean Value Laplacian for Strongly K?hler-Finsler Manifolds....Pages 257-286
Front Matter....Pages 287-287
Non-commutative Determinants and Quaternionic Monge-Amp?re Equations....Pages 289-300
Galpern—Sobolev Type Equations with Non-constant Coefficients....Pages 301-310
A Theory of Modular Forms in Clifford Analysis, their Applications and Perspectives....Pages 311-343
Automated Geometric Theorem Proving, Clifford Bracket Algebra and Clifford Expansions....Pages 345-363
Quaternion-valued Smooth Orthogonal Wavelets with Short Support and Symmetry....Pages 365-376

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The study of systems of special partial differential operators that arise naturally from the use of Clifford algebra as a calculus tool lies in the heart of Clifford analysis. The focus is on the study of Dirac operators and related ones, together with applications in mathematics, physics and engineering. At the present time, the study of Clifford algebra and Clifford analysis has grown into a major research field.

There are two sources of papers in this collection. One is from a satellite conference to the ICM 2002 in Beijing, held August 15-18 at the University of Macau; and the other stems from invited contributions by top-notch experts in the field. All articles were strictly refereed and contain unpublished new results. Some of them are incorporated with comprehensive surveys in the particular areas that the authors work in.

Content:
Front Matter....Pages i-xv
Front Matter....Pages 1-1
Hodge Decompositions on Weakly Lipschitz Domains....Pages 3-29
Monogenic Functions of Bounded Mean Oscillation in the Unit Ball....Pages 31-50
Bp,q-Functions and their Harmonic Majorants....Pages 51-63
Spherical Means and Distributions in Clifford Analysis....Pages 65-96
Hypermonogenic Functions and their Cauchy-Type Theorems....Pages 97-112
On Series Expansions of Hyperholomorphic Bq Functions....Pages 113-129
Pointwise Convergence of Fourier Series on the Unit Sphere of R4 with the Quaternionic Setting....Pages 131-147
Cauchy Kernels for some Conformally Flat Manifolds....Pages 149-160
Clifford Analysis on the Space of Vectors, Bivectors and ?-vectors....Pages 161-185
Front Matter....Pages 187-187
Universal Bochner-Weitzenb?ck Formulas for Hyper-K?hlerian Gradients....Pages 189-208
Cohomology Groups of Harmonic Spinors on Conformally Flat Manifolds....Pages 209-225
Spin Geometry, Clifford Analysis, and Joint Seminormality....Pages 227-255
A Mean Value Laplacian for Strongly K?hler-Finsler Manifolds....Pages 257-286
Front Matter....Pages 287-287
Non-commutative Determinants and Quaternionic Monge-Amp?re Equations....Pages 289-300
Galpern—Sobolev Type Equations with Non-constant Coefficients....Pages 301-310
A Theory of Modular Forms in Clifford Analysis, their Applications and Perspectives....Pages 311-343
Automated Geometric Theorem Proving, Clifford Bracket Algebra and Clifford Expansions....Pages 345-363
Quaternion-valued Smooth Orthogonal Wavelets with Short Support and Symmetry....Pages 365-376
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