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Computational Noncommutative Algebra and Applications: Proceedings of the NATO Advanced Study Institute, on Computatoinal Noncommutative Algebra and Applications, ... II: Mathematics, Physics and Chemistry) PDF

435 Pages·2004·5.78 MB·English
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Computational Noncommutative Algebra and Applications NATO Science Series A Series presenting the results of scientific meetings supported under the NATO Science Programme. The Series is published by IOS Press, Amsterdam, and Kluwer Academic Publishers in conjunction with the NATO Scientific Affairs Division Sub-Series I. Life and Behavioural Sciences IOS Press II. Mathematics,Physics and Chemistry Kluwer Academic Publishers III.Computer and Systems Science IOS Press IV.Earth and Environmental Sciences Kluwer Academic Publishers V. Science and Technology Policy IOS Press The NATO Science Series continues the series of books published formerly as the NATO ASI Series. The NATO Science Programme offers support for collaboration in civil science between scientists of countries of the Euro-Atlantic Partnership Council.The types of scientific meeting generally supported are “Advanced Study Institutes” and “Advanced Research Workshops”, although other types of meeting are supported from time to time.The NATO Science Series collects together the results of these meetings.The meetings are co-organized bij scientists from NATO countries and scientists from NATO’s Partner countries – countries of the CIS and Central and Eastern Europe. Advanced Study Institutes are high-level tutorial courses offering in-depth study of latest advances in a field. Advanced Research Workshops are expert meetings aimed at critical assessment of a field, and identification of directions for future action. As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO Science Series has been re-organised and there are currently Five Sub-series as noted above.Please consult the following web sites for information on previous volumes published in the Series, as well as details of earlier Sub-series. http://www.nato.int/science http://www.wkap.nl http://www.iospress.nl http://www.wtv-books.de/nato-pco.htm Series II:Mathematics,Physics and Chemistry – Vol.136 Computational Noncommutative Algebra and Applications edited by Jim Byrnes Prometheus Inc., Newport, Rhode Island, U.S.A. KLUWER ACADEMIC PUBLISHERS NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW eBookISBN: 1-4020-2307-3 Print ISBN: 1-4020-1982-3 ©2005 Springer Science + Business Media, Inc. Print ©2004 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook maybe reproducedor transmitted inanyform or byanymeans,electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: http://ebooks.springerlink.com and the Springer Global Website Online at: http://www.springeronline.com Dedication This book is dedicated to the cherished memory of Edward Joseph Sanchas, a most decent and trusted human being whom we always counted on to take charge of any and all problems. He was a great man. We miss him deeply. Contents Preface xi Acknowledgments xiv CliffordGeometricAlgebrasinMultilinearAlgebraandNon-EuclideanGe- ometries 1 Garret Sobczyk 1 Geometric algebra 2 2 Projective Geometries 11 3 Affine and other geometries 17 4 Affine Geometry of pseudo-euclidean space 19 5 Conformal Geometry and the Horosphere 21 References 27 Content-Based Information Retrieval by Group Theoretical Methods 29 Michael Clausen, Frank Kurth 1 Introduction 29 2 Motivating Examples 32 3 General Concept 39 4 Fault Tolerance 45 5 Applications, Prototypes, and Test Results 48 6 Related Work and Future Research 51 References 54 Four Problems in Radar 57 Michael C. Wicks and Braham Himed 1 Introduction 57 2 Radar Fundamentals 58 3 Radar Waveforms 60 4 Signal Processing 63 5 Space-Time Adaptive Processing 67 6 Four Problems in Radar 71 7 Conclusions 72 Introduction to Generalized Classical and Quantum Signal and System Theories on Groups and Hypergroups 75 Valeriy Labunets 1 Generalized classical signal/system theory on hypergroups 77 2 Generalized quantum signal/system theory on hypergroups 91 3 Conclusion 99 References 99 vi COMPUTATIONAL NONCOMMUTATIVE ALGEBRA Lie Groups and Lie Algebras in Robotics 101 J.M. Selig 1 Introduction—Rigid Body Motions 102 2 Lie Groups 104 3 Finite Screw Motions 106 4 Mechanical Joints 108 5 Invisible Motion and Gripping 110 6 Forward Kinematics 111 7 Lie Algebra 112 8 The Adjoint Representation 113 9 The Exponential Map 115 10 Derivatives of Exponentials 119 11 Jacobians 122 12 Concluding Remarks 124 References 125 Quantum/Classical Interface: a Geometric Approach from the Classical Side 127 William E. Baylis 1 Introduction 127 2 ParavectorSpace as Spacetime 128 3 Eigenspinors 129 4 Spin 132 5 Dirac Equation 137 6 Bell’s Theorem 145 7 Qubits and Entanglement 146 8 Conclusions 153 References 154 PONS, Reed-Muller Codes, and Group Algebras 155 Myoung An, Jim Byrnes, William Moran, B. Saffari, Harold S. Shapiro, and Richard Tolimieri 1 Introduction 156 2 Analytic theory of one-dimensional PONS 156 3 Shapiro sequences, Reed-Muller codes, and functional equations165 4 Group Algebras 174 5 Reformulation of classical PONS 178 6 Group Algebra of Classical PONS 179 7 Group Algebra Convolution 182 8 Splitting Sequences 188 References 195 Contents vii Clifford Algebras as a Unified Language for Image Processing and Pattern Recognition 197 Valeriy Labunets 1 Introduction 197 2 Clifford algebras as models of physical spaces 200 3 Clifford Algebras as Models of Perceptual Multicolor Spaces 205 4 Hypercomplex-valued invariants of nD multicolor images 217 5 Conclusions 223 Acknowledgments 224 References 224 Recent Progress and Applications in Group FFTs 227 Daniel N. Rockmore 1 Introduction 227 2 Finite group FFTs 233 3 FFTs for compact groups 244 4 Noncompact groups 249 References 251 Group Filters and Image Processing 255 Richard Tolimieri and Myoung An 1 Introduction: Classical Digital Signal Processing 256 2 Abelian Group DSP 258 3 Nonabelian Groups 265 4 Examples 285 5 Group Transforms 291 6 Group Filters 295 7 Line-like Images 304 References 308 A Geometric Algebra Approach to Some Problems of Robot Vision 309 Gerald Sommer 1 Introduction 309 2 Local Analysis of Multi-dimensional Signals 312 3 Knowledge Based Neural Computing 324 Acknowledgments 335 References 335 viii COMPUTATIONAL NONCOMMUTATIVE ALGEBRA Group Theory in Radar and Signal Processing 339 William Moran, Jonathan H. Manton 1 Introduction 339 2 How a Radar Works 341 3 Representations 343 4 Representations and Radar 347 5 Ambiguity Functions 357 6 The Wide Band Case 360 References 362 Geometry of ParavectorSpace with Applications to Relativistic Physics 363 William E. Baylis 1 Clifford Algebras in Physics 364 2 ParavectorSpace as Spacetime 368 3 Interpretation 375 4 Eigenspinors 381 5 Maxwell’s Equation 382 6 Conclusions 386 References 387 A Unified Approach to Fourier-Clifford-PrometheusSequences,Transforms and Filter Banks 389 Ekaterina L.-Rundblad, Valeriy Labunets, and Ilya Nikitin 1 Introduction 389 2 NewconstructionofclassicalandmultiparametricPrometheus transforms 391 3 PONS associated with Abelian groups 393 4 Fast Fourier-Prometheus Transforms 397 5 Conclusions 399 Acknowledgments 399 References 400 FastColorWavelet-Haar-Hartley-PrometheusTransformsforImageProcessing 401 Ekaterina L.-Rundblad, Alexei Maidan, Peter Novak, Valeriy Labunets 1 Introduction 2 Color images 402 3 Color Wavelet-Haar-Prometheustransforms 404 4 Edge detection and compression of color images 406 5 Conclusion 408 Acknowledgments 410 References 410

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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
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