Hypercomplex Analysis and Applications (eBook)

Hypercomplex Analysis and Applications 3
Contents 5
Preface 7
On the Geometry of the Quaternionic Unit Disc 9
1. Introduction 9
2. Basics of quaternionic invariant geometry 10
3. Poincaré and Kobayashi distances on the quaternionic unit disc 14
References 17
Bounded Perturbations of the Resolvent Operators Associated to the F-Spectrum 20
1. Introduction 20
2. Preliminary material 22
3. Examples of equations for the F-spectrum 25
4. Bounded perturbations of the SC-resolvent 27
5. Bounded perturbations of the F-resolvent 30
References 34
Harmonic and Monogenic Functions in Superspace 36
1. Introduction 36
2. Preliminaries 37
3. Monogenic functions theory in superspace 40
4. Basis for the space of symplectic harmonics 43
References 48
A Hyperbolic Interpretation of Cauchy-Type Kernels in Hyperbolic Function Theory 49
1. Introduction 49
2. Clifford Numbers 52
3. On the Poincaré Upper-Half Space 53
4. On Hyperbolic Function Theory 55
5. Hyperbolic Interpretations of the P- and Q-kernels 58
6. The Mean-Value Theorem for the P-Part of a Hypermonogenic Function 62
References 63
Gyrogroups in Projective Hyperbolic Clifford Analysis 66
1. Introduction 66
2. Gyrogroups 69
3. The projective hyperbolic space model 70
4. The Möbius gyrogroup (Bn1 ,.M) 74
5. The Einstein gyrogroup (Bn1 ,.E) 76
6. The proper velocity gyrogroup (Rn,.U) 77
7. Relation between different velocities 77
8. Gyrovector spaces 79
9. Gyrovector space isomorphims 79
10. Möbius, Einstein and proper gyrations as spin representation of the group Spin(n) 83
References 84
Invariant Operators of First Order Generalizing the Dirac Operator in 2 Variables 86
1. Introduction 86
1.1. Invariant differential operators 86
1.2. Dirac operator in k variables 87
1.3. Verma modules and invariant operators in parabolic geometry 88
2. Invariant operators acting between higher spin modules 92
2.1. Classification of first order operator on G/P in terms of weights 92
2.2. Explicit realizations in simple cases 93
References 97
The Zero Sets of Slice Regular Functions and the Open Mapping Theorem 99
1. Introduction 99
2. Preliminary results 103
3. Algebraic properties of the zero set 105
4. Topological properties of the zero set 106
5. The Maximum and Minimum Modulus Principles 106
6. The Open Mapping Theorem 108
References 110
A New Approach to Slice Regularity on Real Algebras 112
1. Introduction 112
2. The quadratic cone of a real alternative algebra 114
3. Slice functions 117
4. Slice regular functions 119
5. Product of slice functions 120
6. Zeros of slice functions 121
7. Examples 123
References 124
On the Incompressible Viscous Stationary MHD Equations and Explicit Solution Formulas for Some Three-dimensional Radially Symmetric Domains 127
1. Introduction 127
2. Preliminaries 129
2.1. The quaternionic operator calculus 129
3. The incompressible stationary MHD equations revisited in the quaternionic calculus 132
4. The highly viscous case 133
5. Outlook for the non-linear case 136
Acknowledgements 137
References 137
The Fischer Decomposition for the H-action and Its Applications 140
1. Introduction 140
2. The Fischer Decomposition for the H-action 141
3. Special Monogenic Polynomials 144
4. Inframonogenic Polynomials 146
Acknowledgment 148
References 148
Bochner’s Formulae for Dunkl-Harmonics and Dunkl-Monogenics 150
1. Introduction 150
2. Clifford Analysis and Dunkl Analysis 151
3. Bochner’s Formula for Dunkl-Harmonics 153
4. Bochner’s Formula for Dunkl-Monogenics 157
References 159
An Invitation to Split Quaternionic Analysis 161
1. Introduction 161
2. The Quaternionic Spaces HC, HR and M 164
3. Regular Functions on H and HC 168
4. Regular Functions on HR 169
5. Fueter Formula for Holomorphic Regular Functions on HR 170
6. Fueter Formula for Regular Functions on HR 173
7. Separation of the Series for SL(2,R) 177
References 179
On the Hyperderivatives of Moisil–Théodoresco Hyperholomorphic Functions 181
1. Introduction 181
2. The left-i-hyperderivative 185
3. The directional left-i-hyperderivative 187
4. The left-i-hyperderivative and the Cauchy-type integral 188
5. The left j- and k-hyperderivatives 191
6. Comparison with one complex variable case 192
References 192
Deconstructing Dirac Operators. II: Integral Representation Formulas 194
1. Introduction 194
2. Integral Representation Formulas 198
2.1. The Setting 199
2.2. Related Integral Operators 200
2.3. Main Results 202
3. Auxiliary Results and Proofs 203
3.1. An Integral Formula 203
3.2. Integral Representation Formulas with Remainders 204
3.3. Proofs of Theorems A and B 206
3.4. Concluding Remarks 207
References 208
A Differential Form Approach to Dirac Operators on Surfaces 211
1. Introduction 211
2. Basic Language 212
2.1. Clifford Algebra 212
2.2. Differential Forms 213
2.3. Clifford Algebra-valued Differential Forms 214
2.4. Monogenic Differential Calculus 214
3. Clifford Algebraic Tools for Surfaces 215
4. Surface Monogenics 217
4.1. Restricted Dirac Operator 218
4.2. Connection with Lie Derivatives 220
4.3. Tangential Dirac Operator 224
5. Clifford Analysis on the Paraboloid 224
5.1. The Tangential Dirac Operator on the Paraboloid 225
5.2. On Surface Monogenics on the Paraboloid 227
Conclusions and Acknowledgments 229
References 229
Killing Tensor Spinor Forms and Their Application in Riemannian Geometry 231
1. Introduction 231
2. Killing spinor forms 233
2.1. Algebraic preliminaries 233
2.2. Geometric applications 241
3. Generalized Killing tensor spinors 243
References 245
Construction of Conformally Invariant Differential Operators 246
1. Introduction 246
2. Conformal geometry and the ambient construction 248
3. Construction of conformally invariant differential operators 252
4. Symmetry operators of the Laplace equation 254
References 257
Remarks on Holomorphicity in Three Settings: Complex, Quaternionic, and Bicomplex 258
1. Introduction 258
2. Algebraic Definitions 259
3. Differentiability and Regularity 262
4. Bicomplex Hyperfunctions in One and Several Variables 265
References 269
The Gauss-Lucas Theorem for Regular Quaternionic Polynomials 272
1. Introduction 272
2. Basic preliminary results for complex polynomials 273
3. The Gauss–Lucas Theorem for regular polynomials in H 274
References 278