Computational Noncommutative Algebra and Applications (eBook)

Dedication. Preface. Acknowledgments. Clifford Geometric Algebras in Multilinear Algebra and Non-Euclidean Geometries; Garret Sobczyk.- 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 Michael Clausen, Frank Kurth.- Introduction; Motivating Examples; General Concept;Fault Tolerance.- Applications, Prototypes, and Test Results; Related Work and Future Research; References.- Four Problems in Radar; Michael C. Wicks and Braham Himed.-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; J.M. Selig.- 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; William E. Baylis.- Introduction Paravector Space as Spacetime; Eigenspinors; Spin; Dirac Equation; Bell’s Theorem; Qubits and Entanglement; Conclusions; References. PONS, Reed-Muller Codes, and Group Algebras; Myoung An et al.- 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; Daniel N. Rockmore.-Introduction; Finite group FFTs; FFTs for compact groups; Noncompact groups; References. Group Filters and Image Processing; Richard Tolimieri and Myoung An.- 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; Gerald Sommer.- Introduction; Local Analysis of Multi-dimensional Signals; Knowledge Based Neural Computing; Acknowledgments; References. Group Theory in Radar and Signal Processing; William Moran, Jonathan H. Manton.- 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; William E. Baylis.- Clifford Algebras in Physics; Paravector Space as Spacetime; Interpretation; Eigenspinors; Maxwell’s Equation; Conclusions; References. A Unified Approach to Fourier-Clifford-Prometheus Transforms Ekaterima L.-Rundbla et al.- 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; Ekaterina L.-Rundblad et al.- 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 Analysis Problems; References; Spectral Analysis under Conditions of Uncertainty; A Canonical Basis for Maximal Tori of the Reductive Centrizer of a Nilpotent Element; References;6 The Quantum Chaos Conjecture References; Four Problems in Radar; Topic Index; Author Index