Computational Noncommutative Algebra And Applications

by Byrnes, Jim; NATO Advanced Study Institute on Computational Noncommutative Algebra and Applications (2003 : Il Ciocco, Italy)

• ISBN13: 9781402019838
• ISBN10: 1402019831
• Format: Paperback
• Copyright: 2004-09-01
• Publisher: Kluwer Academic Pub
• More Book Details
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Summary

The fusion of algebra, analysis and geometry, and their application to real world problems, have been dominant themes underlying mathematics for over a century. Geometric algebras, introduced and classified by Clifford in the late 19th century, have played a prominent role in this effort, as seen in the mathematical work of Cartan, Brauer, Weyl, Chevelley, Atiyah, and Bott, and in applications to physics in the work of Pauli, Dirac and others. One of the most important applications of geometric algebras to geometry is to the representation of groups of Euclidean and Minkowski rotations. This aspect and its direct relation to robotics and vision will be discussed in several chapters of this multi-authored textbook, which resulted from the ASI meeting. Moreover, group theory, beginning with the work of Burnside, Frobenius and Schur, has been influenced by even more general problems. As a result, general group actions have provided the setting for powerful methods within group theory and for the use of groups in applications to physics, chemistry, molecular biology, and signal processing. These aspects, too, will be covered in detail. With the rapidly growing importance of, and ever expanding conceptual and computational demands on signal and image processing in remote sensing, computer vision, medical image processing, and biological signal processing, and on neural and quantum computing, geometric algebras, and computational group harmonic analysis, the topics of the book have emerged as key tools. The list of authors includes many of the world's leading experts in the development of new algebraic modeling and signal representation methodologies, novel Fourier-based and geometric transforms, and computational algorithms required for realizing the potential of these new application fields.

Table of Contents

Dedication

Preface

Acknowledgments

Clifford Geometric Algebras in Multilinear Algebra and Non-Euclidean Geometries

Geometric algebra Projective Geometries

Affine and other geometries

Affine Geometry of pseudo-euclidean space

Conformal Geometry and the Horosphere

References

Content-Based Information Retrieval by Group Theoretical Methods

Introduction

Motivating Examples

General Concept; Fault Tolerance

Applications, Prototypes, and Test Results

Related Work and Future Research

References

Four Problems in Radar

Introduction

Radar Fundamentals

Radar Waveforms

Signal Processing

Space-Time Adaptive Processing

Four Problems in Radar

Conclusions

Introduction to Generalized Classical and Quantum Signal and System Theories on Groups and Hypergroups

Valeriy Labunets

Generalized classical signal/system theory on hypergroups

Generalized quantum signal/system theory on hypergroups

Conclusion

References

Lie Groups and Lie Algebras in Robotics

Introduction

''Rigid Body Motions

Lie Groups

Finite Screw Motions

Mechanical Joints

Invisible Motion and Gripping

Forward Kinematics

Lie Algebra

The Adjoint Representation

The Exponential Map Derivatives of Exponentials

Jacobians

Concluding Remarks

References

Quantum/Classical Interface: a Geometric Approach from the Classical Side

Introduction

Paravector Space as Spacetime

Eigenspinors

Spin

Dirac Equation

Bell''s Theorem

Qubits and Entanglement

Conclusions

References

PONS, Reed-Muller Codes, and Group Algebras

Introduction

Analytic Theory of One-Dimensional PONS (Welti)

Shapiro Sequences, Reed-Muller Codes, and Functional Equations

Group Algebras

Reformulation of Classical PONS

Group Algebra of Classical PONS

Group Algebra Convolution

Splitting Sequences

Historical Appendix on PONS

References

Clifford Algebras as a Unified Language

Valeriy Labunets

Introduction

Clifford algebras as models of physical spaces

Clifford Algebras as Models of Perceptual Multicolor Spaces

Hypercomplex-Valued invariants of nD multicolor images

Conclusions

Acknowledgments

References

Recent Progress and Applications in Group FFTs

Introduction

Finite group FFTs

FFTs for compact groups

Noncompact groups

References

Group Filters and Image Processing

Introduction: Classical Digital Signal Processing

Abelian Group DSP

Nonabelian Groups

Examples; Group Transforms

Group Filters

Line-like Images

Acknowledgments

References

A Geometric Algebra Approach to Some Problems of Robot Vision

Introduction

Local Analysis of Multi-dimensional Signals

Knowledge Based Neural Computing

Acknowledgments

References

Group Theory in Radar and Signal Processing

Introduction

How a Radar Works

Representations

Representations and Radar

Ambiguity Functions

The Wide Band Case

References

Geometry of Paravector Space with Applications to Relativistic Physics

Clifford Algebras in Physics

Paravector Space as Spacetime

Interpretation

Eigenspinors

Maxwell''s Equation

Conclusions

References

A Unified Approach to Fourier-Clifford-Prometheus Transforms

Introduction

New construction of classical and multiparametric Prometheus transforms

PONS associated with Abelian groups

Fast Fourier-Prometheus Transforms

Conclusions

Acknowledgments

References

Fast Color Wavelet Transforms

Introduction

Color images

Color Wavelet-Haar-Prometheus transforms

Edge detection and compression of color images

Conclusion

Acknowledgments

References

Selected Problems

Various Authors

Transformations of Euclidean Space and Clifford Geometric

Algebra

References

On the Distribution of Kloosterman Sums on Polynomials over Quaternions

References

Harmonic Sliding

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