Sabadini / Sommen Hypercomplex Analysis and Applications

1. Auflage 2010
ISBN: 978-3-0346-0246-4
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 284 Seiten, eBook

Reihe: Trends in Mathematics

The purpose of the volume is to bring forward recent trends of research in hypercomplex analysis. The list of contributors includes first rate mathematicians and young researchers working on several different aspects in quaternionic and Clifford analysis. Besides original research papers, there are papers providing the state-of-the-art of a specific topic, sometimes containing interdisciplinary fields.

The intended audience includes researchers, PhD students, postgraduate students who are interested in the field and in possible connection between hypercomplex analysis and other disciplines, including mathematical analysis, mathematical physics, algebra.

Zielgruppe

Research

Autoren/Hrsg.

Sommen, Frank

Sabadini, Irene

Sommen, Franciscus

Inhaltsverzeichnis

1;Hypercomplex Analysis and Applications;3
1.1;Contents;5
1.2;Preface;7
1.3;On the Geometry of the Quaternionic Unit Disc;9
1.3.1;1. Introduction;9
1.3.2;2. Basics of quaternionic invariant geometry;10
1.3.3;3. Poincaré and Kobayashi distances on the quaternionic unit disc;14
1.3.4;References;17
1.4;Bounded Perturbations of the Resolvent Operators Associated to the F-Spectrum;20
1.4.1;1. Introduction;20
1.4.2;2. Preliminary material;22
1.4.3;3. Examples of equations for the F-spectrum;25
1.4.4;4. Bounded perturbations of the SC-resolvent;27
1.4.5;5. Bounded perturbations of the F-resolvent;30
1.4.6;References;34
1.5;Harmonic and Monogenic Functions in Superspace;36
1.5.1;1. Introduction;36
1.5.2;2. Preliminaries;37
1.5.3;3. Monogenic functions theory in superspace;40
1.5.4;4. Basis for the space of symplectic harmonics;43
1.5.5;References;48
1.6;A Hyperbolic Interpretation of Cauchy-Type Kernels in Hyperbolic Function Theory;49
1.6.1;1. Introduction;49
1.6.2;2. Clifford Numbers;52
1.6.3;3. On the Poincaré Upper-Half Space;53
1.6.4;4. On Hyperbolic Function Theory;55
1.6.5;5. Hyperbolic Interpretations of the P- and Q-kernels;58
1.6.6;6. The Mean-Value Theorem for the P-Part of a Hypermonogenic Function;62
1.6.7;References;63
1.7;Gyrogroups in Projective Hyperbolic Clifford Analysis;66
1.7.1;1. Introduction;66
1.7.2;2. Gyrogroups;69
1.7.3;3. The projective hyperbolic space model;70
1.7.4;4. The Möbius gyrogroup (Bn1 ,.M);74
1.7.5;5. The Einstein gyrogroup (Bn1 ,.E);76
1.7.6;6. The proper velocity gyrogroup (Rn,.U);77
1.7.7;7. Relation between different velocities;77
1.7.8;8. Gyrovector spaces;79
1.7.9;9. Gyrovector space isomorphims;79
1.7.10;10. Möbius, Einstein and proper gyrations as spin representation of the group Spin(n);83
1.7.11;References;84
1.8;Invariant Operators of First Order Generalizing the Dirac Operator in 2 Variables;86
1.8.1;1. Introduction;86
1.8.1.1;1.1. Invariant differential operators;86
1.8.1.2;1.2. Dirac operator in k variables;87
1.8.1.3;1.3. Verma modules and invariant operators in parabolic geometry;88
1.8.2;2. Invariant operators acting between higher spin modules;92
1.8.2.1;2.1. Classification of first order operator on G/P in terms of weights;92
1.8.2.2;2.2. Explicit realizations in simple cases;93
1.8.3;References;97
1.9;The Zero Sets of Slice Regular Functions and the Open Mapping Theorem;99
1.9.1;1. Introduction;99
1.9.2;2. Preliminary results;103
1.9.3;3. Algebraic properties of the zero set;105
1.9.4;4. Topological properties of the zero set;106
1.9.5;5. The Maximum and Minimum Modulus Principles;106
1.9.6;6. The Open Mapping Theorem;108
1.9.7;References;110
1.10;A New Approach to Slice Regularity on Real Algebras;112
1.10.1;1. Introduction;112
1.10.2;2. The quadratic cone of a real alternative algebra;114
1.10.3;3. Slice functions;117
1.10.4;4. Slice regular functions;119
1.10.5;5. Product of slice functions;120
1.10.6;6. Zeros of slice functions;121
1.10.7;7. Examples;123
1.10.8;References;124
1.11;On the Incompressible Viscous Stationary MHD Equations and Explicit Solution Formulas for Some Three-dimensional Radially Symmetric Domains;127
1.11.1;1. Introduction;127
1.11.2;2. Preliminaries;129
1.11.2.1;2.1. The quaternionic operator calculus;129
1.11.3;3. The incompressible stationary MHD equations revisited in the quaternionic calculus;132
1.11.4;4. The highly viscous case;133
1.11.5;5. Outlook for the non-linear case;136
1.11.6;Acknowledgements;137
1.11.7;References;137
1.12;The Fischer Decomposition for the H-action and Its Applications;140
1.12.1;1. Introduction;140
1.12.2;2. The Fischer Decomposition for the H-action;141
1.12.3;3. Special Monogenic Polynomials;144
1.12.4;4. Inframonogenic Polynomials;146
1.12.5;Acknowledgment;148
1.12.6;References;148
1.13;Bochner’s Formulae for Dunkl-Harmonics and Dunkl-Monogenics;150
1.13.1;1. Introduction;150
1.13.2;2. Clifford Analysis and Dunkl Analysis;151
1.13.3;3. Bochner’s Formula for Dunkl-Harmonics;153
1.13.4;4. Bochner’s Formula for Dunkl-Monogenics;157
1.13.5;References;159
1.14;An Invitation to Split Quaternionic Analysis;161
1.14.1;1. Introduction;161
1.14.2;2. The Quaternionic Spaces HC, HR and M;164
1.14.3;3. Regular Functions on H and HC;168
1.14.4;4. Regular Functions on HR;169
1.14.5;5. Fueter Formula for Holomorphic Regular Functions on HR;170
1.14.6;6. Fueter Formula for Regular Functions on HR;173
1.14.7;7. Separation of the Series for SL(2,R);177
1.14.8;References;179
1.15;On the Hyperderivatives of Moisil–Théodoresco Hyperholomorphic Functions;181
1.15.1;1. Introduction;181
1.15.2;2. The left-i-hyperderivative;185
1.15.3;3. The directional left-i-hyperderivative;187
1.15.4;4. The left-i-hyperderivative and the Cauchy-type integral;188
1.15.5;5. The left j- and k-hyperderivatives;191
1.15.6;6. Comparison with one complex variable case;192
1.15.7;References;192
1.16;Deconstructing Dirac Operators. II: Integral Representation Formulas;194
1.16.1;1. Introduction;194
1.16.2;2. Integral Representation Formulas;198
1.16.2.1;2.1. The Setting;199
1.16.2.2;2.2. Related Integral Operators;200
1.16.2.3;2.3. Main Results;202
1.16.3;3. Auxiliary Results and Proofs;203
1.16.3.1;3.1. An Integral Formula;203
1.16.3.2;3.2. Integral Representation Formulas with Remainders;204
1.16.3.3;3.3. Proofs of Theorems A and B;206
1.16.3.4;3.4. Concluding Remarks;207
1.16.4;References;208
1.17;A Differential Form Approach to Dirac Operators on Surfaces;211
1.17.1;1. Introduction;211
1.17.2;2. Basic Language;212
1.17.2.1;2.1. Clifford Algebra;212
1.17.2.2;2.2. Differential Forms;213
1.17.2.3;2.3. Clifford Algebra-valued Differential Forms;214
1.17.2.4;2.4. Monogenic Differential Calculus;214
1.17.3;3. Clifford Algebraic Tools for Surfaces;215
1.17.4;4. Surface Monogenics;217
1.17.4.1;4.1. Restricted Dirac Operator;218
1.17.4.2;4.2. Connection with Lie Derivatives;220
1.17.4.3;4.3. Tangential Dirac Operator;224
1.17.5;5. Clifford Analysis on the Paraboloid;224
1.17.5.1;5.1. The Tangential Dirac Operator on the Paraboloid;225
1.17.5.2;5.2. On Surface Monogenics on the Paraboloid;227
1.17.5.2.1;Conclusions and Acknowledgments;229
1.17.6;References;229
1.18;Killing Tensor Spinor Forms and Their Application in Riemannian Geometry;231
1.18.1;1. Introduction;231
1.18.2;2. Killing spinor forms;233
1.18.2.1;2.1. Algebraic preliminaries;233
1.18.2.2;2.2. Geometric applications;241
1.18.3;3. Generalized Killing tensor spinors;243
1.18.4;References;245
1.19;Construction of Conformally Invariant Differential Operators;246
1.19.1;1. Introduction;246
1.19.2;2. Conformal geometry and the ambient construction;248
1.19.3;3. Construction of conformally invariant differential operators;252
1.19.4;4. Symmetry operators of the Laplace equation;254
1.19.5;References;257
1.20;Remarks on Holomorphicity in Three Settings: Complex, Quaternionic, and Bicomplex;258
1.20.1;1. Introduction;258
1.20.2;2. Algebraic Definitions;259
1.20.3;3. Differentiability and Regularity;262
1.20.4;4. Bicomplex Hyperfunctions in One and Several Variables;265
1.20.5;References;269
1.21;The Gauss-Lucas Theorem for Regular Quaternionic Polynomials;272
1.21.1;1. Introduction;272
1.21.2;2. Basic preliminary results for complex polynomials;273
1.21.3;3. The Gauss–Lucas Theorem for regular polynomials in H;274
1.21.4;References;278