Numerical Calculations in Clifford Algebra

Numerical Calculations in Clifford Algebra
Author: Andrew Seagar
Publisher: John Wiley & Sons
Total Pages: 532
Release: 2023-07-31
Genre: Mathematics
ISBN: 1394173245

NUMERICAL CALCULATIONS IN CLIFFORD ALGEBRA An intuitive combination of the theory of Clifford algebra with numerous worked and computed examples and calculations Numerical Calculations in Clifford Algebra: A Practical Guide for Engineers and Scientists is an accessible and practical introduction to Clifford algebra, with comprehensive coverage of the theory and calculations. The book offers many worked and computed examples at a variety of levels of complexity and over a range of different applications making extensive use of diagrams to maintain clarity. The author introduces and documents the Clifford Numerical Suite, developed to overcome the limitations of existing computational packages and to enable the rapid creation and deployment of sophisticated and efficient code. Applications of the suite include Fourier transforms for arrays of any types of Clifford numbers and the solution of linear systems in which the coefficients are Clifford numbers of particular types, including scalars, bicomplex numbers, quaternions, Pauli matrices, and extended electromagnetic fields. Readers will find: A thorough introduction to Clifford algebra, with a combination of theory and practical implementation in a range of engineering problems Comprehensive explorations of a variety of worked and computed examples at various levels of complexity Practical discussions of the conceptual and computational tools for solving common engineering problems Detailed documentation on the deployment and application of the Clifford Numerical Suite Perfect for engineers, researchers, and academics with an interest in Clifford algebra, Numerical Calculations in Clifford Algebra: A Practical Guide for Engineers and Scientists will particularly benefit professionals in the areas of antenna design, digital image processing, theoretical physics, and geometry.

Clifford Analysis and Its Applications

Clifford Analysis and Its Applications
Author: F. Brackx
Publisher: Springer Science & Business Media
Total Pages: 440
Release: 2001-07-31
Genre: Mathematics
ISBN: 9780792370444

In its traditional form, Clifford analysis provides the function theory for solutions of the Dirac equation. From the beginning, however, the theory was used and applied to problems in other fields of mathematics, numerical analysis, and mathematical physics. recently, the theory has enlarged its scope considerably by incorporating geometrical methods from global analysis on manifolds and methods from representation theory. New, interesting branches of the theory are based on conformally invariant, first-order systems other than the Dirac equation, or systems that are invariant with respect to a group other than the conformal group. This book represents an up-to-date review of Clifford analysis in its present form, its applications, and directions for future research. Readership: Mathematicians and theoretical physicists interested in Clifford analysis itself, or in its applications to other fields.

Quaternion and Clifford Fourier Transforms and Wavelets

Quaternion and Clifford Fourier Transforms and Wavelets
Author: Eckhard Hitzer
Publisher: Springer Science & Business Media
Total Pages: 358
Release: 2013-06-24
Genre: Mathematics
ISBN: 3034806035

Quaternion and Clifford Fourier and wavelet transformations generalize the classical theory to higher dimensions and are becoming increasingly important in diverse areas of mathematics, physics, computer science and engineering. This edited volume presents the state of the art in these hypercomplex transformations. The Clifford algebras unify Hamilton’s quaternions with Grassmann algebra. A Clifford algebra is a complete algebra of a vector space and all its subspaces including the measurement of volumes and dihedral angles between any pair of subspaces. Quaternion and Clifford algebras permit the systematic generalization of many known concepts. This book provides comprehensive insights into current developments and applications including their performance and evaluation. Mathematically, it indicates where further investigation is required. For instance, attention is drawn to the matrix isomorphisms for hypercomplex algebras, which will help readers to see that software implementations are within our grasp. It also contributes to a growing unification of ideas and notation across the expanding field of hypercomplex transforms and wavelets. The first chapter provides a historical background and an overview of the relevant literature, and shows how the contributions that follow relate to each other and to prior work. The book will be a valuable resource for graduate students as well as for scientists and engineers.

Clifford Algebras with Numeric and Symbolic Computations

Clifford Algebras with Numeric and Symbolic Computations
Author: Rafal Ablamowicz
Publisher: Springer Science & Business Media
Total Pages: 328
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461581575

This edited survey book consists of 20 chapters showing application of Clifford algebra in quantum mechanics, field theory, spinor calculations, projective geometry, Hypercomplex algebra, function theory and crystallography. Many examples of computations performed with a variety of readily available software programs are presented in detail.

Analysis of Dirac Systems and Computational Algebra

Analysis of Dirac Systems and Computational Algebra
Author: Fabrizio Colombo
Publisher: Springer Science & Business Media
Total Pages: 344
Release: 2012-12-06
Genre: Mathematics
ISBN: 0817681663

* The main treatment is devoted to the analysis of systems of linear partial differential equations (PDEs) with constant coefficients, focusing attention on null solutions of Dirac systems * All the necessary classical material is initially presented * Geared toward graduate students and researchers in (hyper)complex analysis, Clifford analysis, systems of PDEs with constant coefficients, and mathematical physics

Clifford Algebras: An Introduction

Clifford Algebras: An Introduction
Author: D. J. H. Garling
Publisher: Cambridge University Press
Total Pages: 209
Release: 2011-06-23
Genre: Mathematics
ISBN: 1107096383

A straightforward introduction to Clifford algebras, providing the necessary background material and many applications in mathematics and physics.

New Foundations in Mathematics

New Foundations in Mathematics
Author: Garret Sobczyk
Publisher: Springer Science & Business Media
Total Pages: 373
Release: 2012-10-26
Genre: Mathematics
ISBN: 0817683852

The first book of its kind, New Foundations in Mathematics: The Geometric Concept of Number uses geometric algebra to present an innovative approach to elementary and advanced mathematics. Geometric algebra offers a simple and robust means of expressing a wide range of ideas in mathematics, physics, and engineering. In particular, geometric algebra extends the real number system to include the concept of direction, which underpins much of modern mathematics and physics. Much of the material presented has been developed from undergraduate courses taught by the author over the years in linear algebra, theory of numbers, advanced calculus and vector calculus, numerical analysis, modern abstract algebra, and differential geometry. The principal aim of this book is to present these ideas in a freshly coherent and accessible manner. New Foundations in Mathematics will be of interest to undergraduate and graduate students of mathematics and physics who are looking for a unified treatment of many important geometric ideas arising in these subjects at all levels. The material can also serve as a supplemental textbook in some or all of the areas mentioned above and as a reference book for professionals who apply mathematics to engineering and computational areas of mathematics and physics.

Clifford Algebras

Clifford Algebras
Author: Rafal Ablamowicz
Publisher: Springer Science & Business Media
Total Pages: 635
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461220440

The invited papers in this volume provide a detailed examination of Clifford algebras and their significance to analysis, geometry, mathematical structures, physics, and applications in engineering. While the papers collected in this volume require that the reader possess a solid knowledge of appropriate background material, they lead to the most current research topics. With its wide range of topics, well-established contributors, and excellent references and index, this book will appeal to graduate students and researchers.

Application of Geometric Algebra to Electromagnetic Scattering

Application of Geometric Algebra to Electromagnetic Scattering
Author: Andrew Seagar
Publisher: Springer
Total Pages: 187
Release: 2015-11-12
Genre: Technology & Engineering
ISBN: 9811000891

This work presents the Clifford-Cauchy-Dirac (CCD) technique for solving problems involving the scattering of electromagnetic radiation from materials of all kinds. It allows anyone who is interested to master techniques that lead to simpler and more efficient solutions to problems of electromagnetic scattering than are currently in use. The technique is formulated in terms of the Cauchy kernel, single integrals, Clifford algebra and a whole-field approach. This is in contrast to many conventional techniques that are formulated in terms of Green's functions, double integrals, vector calculus and the combined field integral equation (CFIE). Whereas these conventional techniques lead to an implementation using the method of moments (MoM), the CCD technique is implemented as alternating projections onto convex sets in a Banach space. The ultimate outcome is an integral formulation that lends itself to a more direct and efficient solution than conventionally is the case, and applies without exception to all types of materials. On any particular machine, it results in either a faster solution for a given problem or the ability to solve problems of greater complexity. The Clifford-Cauchy-Dirac technique offers very real and significant advantages in uniformity, complexity, speed, storage, stability, consistency and accuracy.

An Introduction to Clifford Algebras and Spinors

An Introduction to Clifford Algebras and Spinors
Author: Jayme Vaz Jr.
Publisher: Oxford University Press
Total Pages: 257
Release: 2016
Genre: Mathematics
ISBN: 0198782926

This work is unique compared to the existing literature. It is very didactical and accessible to both students and researchers, without neglecting the formal character and the deep algebraic completeness of the topic along with its physical applications.