Hypercomplex Analysis and Applications

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