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"Clifford algebras."
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A Note on Centralizers and Twisted Centralizers in Clifford Algebras
This paper investigates centralizers and twisted centralizers in degenerate and non-degenerate Clifford (geometric) algebras. We provide an explicit form of the centralizers and twisted centralizers of the subspaces of fixed grades, subspaces determined by the grade involution and the reversion, and their direct sums. The results can be useful for applications of Clifford algebras in computer science, physics, and engineering.
Journal Article
The power of geometric algebra computing
\"The Power of Geometric Algebra Computing for Engineering and Quantum Computing is based on GAALOPWeb, a new user-friendly, web-based tool for the generation of optimized code for different programming languages as well as for the visualization of Geometric Algebra algorithms for a wide range of engineering applications. This book includes applications from the fields of computer graphics, robotics and quantum computing and will help students, engineers and researchers interested in really computing with Geometric Algebra\"-- Provided by publisher.
Book
Clifford Odd and Even Objects in Even and Odd Dimensional Spaces Describing Internal Spaces of Fermion and Boson Fields
In a long series of works, it has been demonstrated that the spin-charge-family theory, assuming a simple starting action in even dimensional spaces with d≥(13+1), with massless fermions interacting with gravity only, offers the explanation for all assumed properties of the second quantized fermion and boson fields in the standard model, as well as offering predictions and explanations for several of the observed phenomena. The description of the internal spaces of the fermion and boson fields by the Clifford odd and even objects, respectively, justifies the choice of the simple starting action of the spin-charge-family theory. The main topic of the present article is the analysis of the properties of the internal spaces of the fermion and boson fields in odd dimensional spaces, d=(2n+1), which can again be described by the Clifford odd and even objects, respectively. It turns out that the properties of fermion and boson fields differ essentially from their properties in even dimensional spaces, resembling the ghosts needed when looking for final solutions with Feynman diagrams.
Journal Article
Representation Theory and Harmonic Analysis on Symmetric Spaces
This volume contains the proceedings of the AMS Special Session on Harmonic Analysis, in honor of Gestur Ólafsson's 65th birthday, held on January 4, 2017, in Atlanta, Georgia. The articles in this volume provide fresh perspectives on many different directions within harmonic analysis, highlighting the connections between harmonic analysis and the areas of integral geometry, complex analysis, operator algebras, Lie algebras, special functions, and differential operators. The breadth of contributions highlights the diversity of current research in harmonic analysis and shows that it continues to be a vibrant and fruitful field of inquiry.
eBook
Introduction to Clifford Algebra
This book pursues to exhibit how we can construct a Clifford type algebra from the classical one. The basic idea of these lecture notes is to show how to calculate fundamental solutions to either firstâorder differential operators of the form D=â_(i=0)^nâ'ãe_i Î'_iãor secondâorder elliptic differential operators¯D D, both with constant coefficients or combinations of this kind of operators. After considering in detail how to find the fundamental solution we study the problem of integral representations in a classical Clifford algebra and in a dependentâparameter Clifford algebra which generalizes the classical one. We also propose a basic method to extend the order of the operator, for instance D^n,nâN and how to produce integral representations for higher order operators and mixtures of them. Although the Clifford algebras have produced many applications concerning boundary value problems, initial value problems, mathematical physics, quantum chemistry, among others; in this book we do not discuss these topics as they are better discussed in other courses. Researchers and practitioners will find this book very useful as a source book. The reader is expected to have basic knowledge of partial differential equations and complex analysis. When planning and writing these lecture notes, we had in mind that they would be used as a resource by mathematics students interested in understanding how we can combine partial differential equations and Clifford analysis to find integral representations. This in turn would allow them to solve boundary value problems and initial value problems. To this end, proofs have been described in rigorous detail and we have included numerous worked examples. On the other hand, exercises have not been included.
eBook
Quadratic mappings and Clifford algebras
After a classical presentation of quadratic mappings and Clifford algebras over arbitrary rings (commutative, associative, with unit), other topics involve more original methods: interior multiplications allow an effective treatment of deformations of Clifford algebras; the relations between automorphisms of quadratic forms and Clifford algebras.
eBook
Invariant algebras and geometric reasoning
The demand for more reliable geometric computing in robotics, computer vision and graphics has revitalized many venerable algebraic subjects in mathematics — among them, Grassmann–Cayley algebra and Geometric Algebra. Nowadays, they are used as powerful languages for projective, Euclidean and other classical geometries.
eBook