Clifford Analysis and Related Topics: In Honor of Paul A. M. Dirac, CART 2014, Tallahassee, Florida, December 15–17 (Springer Proceedings in Mathematics & Statistics, 260)

This book, intended to commemorate the work of Paul Dirac, highlights new developments in the main directions of Clifford analysis. Just as complex analysis is based on the algebra of the complex numbers, Clifford analysis is based on the geometric Clifford algebras. Many methods and theorems from complex analysis generalize to higher dimensions in various ways. However, many new features emerge in the process, and much of this work is still in its infancy. Some of the leading mathematicians working in this field have contributed to this book in conjunction with "Clifford Analysis and Related Topics: a conference in honor of Paul A.M. Dirac," which was held at Florida State University, Tallahassee, on December 15-17, 2014. The content reflects talks given at the conference, as well as contributions from mathematicians who were invited but were unable to attend. Hence much of the mathematics presented here is not only highly topical, but also cannot be found elsewhere in print. Given its scope, the book will be of interest to mathematicians and physicists working in these areas, as well as students seeking to catch up on the latest developments.-- Provided by publisher Preface 6 Contents 8 Lambda-Harmonic Functions: An Expository Account 9 1 Introduction 9 2 λ-Harmonic Functions 10 3 The Invariant Laplacian (λ= n-22) 11 4 λ-Harmonic Functions Continued 15 4.1 The Hypergeometric Function 15 4.2 Homogeneous Harmonic Polynomials and Spherical Harmonics 16 4.3 Linearity, Möbius Invariance, and Polyharmonicity of Δλ 19 5 The λ-Poisson Kernel on mathbbBn 20 6 Conclusion 24 References 25 Some Applications of Parabolic Dirac Operators to the Instationary Navier-Stokes Problem on Conformally Flat Cylinders and Tori in mathbbR3 26 1 Introduction 26 2 Preliminaries 29 2.1 Quaternionic Function Theory 29 2.2 The Instationary Case 30 3 The Navier-Stokes Equations Shortly Revisited in Quaternions 33 4 The Linear Case 34 5 The Case of a Non Negligenciable Convective Term 36 6 The Navier-Stokes Equations in the More General Context of Some Conformally Flat Spin 3-Manifolds 39 References 43 From Hermitean Clifford Analysis to Subelliptic Dirac Operators on Odd Dimensional Spheres and Other CR Manifolds 45 1 Introduction 45 2 Preliminaries on Hermitean Clifford Analysis 46 3 Hermitean Clifford Analysis and Matrix Differential Operators 48 4 Conformal Transformations in the Hermitean Case 50 5 Kohn Dirac and Laplacian Operators in S2n-1 51 6 Realization on mathbbS3 53 7 Subelliptic Dirac Operator 56 8 Irreducible Representations in mathcalU'(n) 56 References 57 On Some Conformally Invariant Operators in Euclidean Space 58 1 Introduction 58 2 Preliminaries 60 2.1 Clifford Algebra 60 2.2 Irreducible Representations of the Spin Group 63 3 Properties of the Rarita-Schwinger Operator 68 3.1 A Counterexample 68 3.2 Conformal Invariance 68 3.3 Intertwining Operators of Rk 70 4 Rarita-Schwinger Type Operators 72 4.1 Conformal Invariance 73 References 76 Notions of Regularity for Functions of a Split-Quaternionic Variable 78 1 Introduction 78 1.1 The Split-Quaternions 80 2 Notions of Holomorphy 83 2.1 Analogues of the Cauchy-Riemann Operator 83 2.2 Difference Quotients 87 2.3 Regularity and John's Equation 91 2.4 A Brief Discussion of Split-Quaternionic Power Series 91 3 A Theory of Left-Regular Functions 92 3.1 A Class of Left Regular Functions 93 3.2 Generating Left Regular Functions 95 References 100 Decomposition of the Twisted Dirac Operator 102 1 Introduction 102 2 Definitions and Notations 103 3 The Twisted Dirac Operator 104 4 An Extremal Projector for the Symplectic Lie Algebra 106 5 Embedding Factors 109 6 An Explicit Decomposition of the Twisted Dirac Operator 110 6.1 The Case k=1 111 6.2 The Case k=2 111 6.3 General Case 113 7 Conclusion 115 References 115 Norms and Moduli on Multicomplex Spaces 117 1 Introduction 117 2 Bicomplex and Multicomplex Spaces 118 2.1 Bicomplex Space 119 2.2 The Tricomplex Space mathbbBmathbbC3 125 3 Representations of Bicomplex Holomorphic Functions 128 4 Norms and Moduli on the Bicomplex Space 130 4.1 The Euclidean Norm 131 4.2 The Set of Hyperbolic Numbers and the Hyperbolic Valued Norm 131 4.3 Finsler-Type Modulus 133 4.4 Lie and Dual Lie Norms 134 5 Norms and Moduli on the Multicomplex Space 135 5.1 The Set of Multi-hyperbolic Numbers and the Multi-hyperbolic-Valued Norm 136 5.2 Finsler-Type Modulus 137 6 Bicomplex Manifolds 138 6.1 Examples of Bicomplex Manifolds 139 6.2 Remarks on Bicomplex Manifolds 139 7 Multicomplex Manifolds 140 8 Conclusions and Future Endeavors 142 References 143 Associated Operators to the Space of Meta-q-Monogenic Functions 145 1 Introduction 145 2 The Meta-q-Monogenic Operator 147 3 Associated Spaces 149 3.1 Necessary and Sufficient Conditions on the Coefficients of mathcalF 150 4 Application of Associated Spaces to Initial Value Problems 153 4.1 Solvability of Initial Value Problems 153 5 Concluding Remarks 155 References 155 Front Matter ....Pages i-vii Lambda-Harmonic Functions: An Expository Account (K. Ballenger-Fazzone, C. A. Nolder)....Pages 1-17 Some Applications of Parabolic Dirac Operators to the Instationary Navier-Stokes Problem on Conformally Flat Cylinders and Tori in \(\mathbb {R}^3\) (P. Cerejeiras, U. Kähler, R. S. Kraußhar)....Pages 19-37 From Hermitean Clifford Analysis to Subelliptic Dirac Operators on Odd Dimensional Spheres and Other CR Manifolds (P. Cerejeiras, U. Kähler, J. Ryan)....Pages 39-51 On Some Conformally Invariant Operators in Euclidean Space (C. Ding, J. Ryan)....Pages 53-72 Notions of Regularity for Functions of a Split-Quaternionic Variable (J. A. Emanuello, C. A. Nolder)....Pages 73-96 Decomposition of the Twisted Dirac Operator (T. Raeymaekers)....Pages 97-111 Norms and Moduli on Multicomplex Spaces (M. B. Vajiac)....Pages 113-140 Associated Operators to the Space of Meta-q-Monogenic Functions (C. J. Vanegas, F. A. Vargas)....Pages 141-152 Correction to: Clifford Analysis and Related Topics (Paula Cerejeiras, Craig A. Nolder, John Ryan, Carmen Judith Vanegas Espinoza)....Pages E1-E1