Author: Simon L. Altmann Publisher: Courier Corporation ISBN: 0486317730 Category : Mathematics Languages : en Pages : 315
Book Description
This self-contained text presents a consistent description of the geometric and quaternionic treatment of rotation operators, employing methods that lead to a rigorous formulation and offering complete solutions to many illustrative problems. Geared toward upper-level undergraduates and graduate students, the book begins with chapters covering the fundamentals of symmetries, matrices, and groups, and it presents a primer on rotations and rotation matrices. Subsequent chapters explore rotations and angular momentum, tensor bases, the bilinear transformation, projective representations, and the geometry, topology, and algebra of rotations. Some familiarity with the basics of group theory is assumed, but the text assists students in developing the requisite mathematical tools as necessary.
Author: Ernst Binz Publisher: American Mathematical Soc. ISBN: 0821844954 Category : Mathematics Languages : en Pages : 321
Book Description
"The three-dimensional Heisenberg group, being a quite simple non-commutative Lie group, appears prominently in various applications of mathematics. The goal of this book is to present basic geometric and algebraic properties of the Heisenberg group and its relation to other important mathematical structures (the skew field of quaternions, symplectic structures, and representations) and to describe some of its applications. In particular, the authors address such subjects as signal analysis and processing, geometric optics, and quantization. In each case, the authors present necessary details of the applied topic being considered." "This book manages to encompass a large variety of topics being easily accessible in its fundamentals. It can be useful to students and researchers working in mathematics and in applied mathematics."--BOOK JACKET.
Author: Tevian Dray Publisher: World Scientific ISBN: 981440182X Category : Mathematics Languages : en Pages : 229
Book Description
There are precisely two further generalizations of the real and complex numbers, namely, the quaternions and the octonions. The quaternions naturally describe rotations in three dimensions. In fact, all (continuous) symmetry groups are based on one of these four number systems. This book provides an elementary introduction to the properties of the octonions, with emphasis on their geometric structure. Elementary applications covered include the rotation groups and their spacetime generalization, the Lorentz group, as well as the eigenvalue problem for Hermitian matrices. In addition, more sophisticated applications include the exceptional Lie groups, octonionic projective spaces, and applications to particle physics including the remarkable fact that classical supersymmetry only exists in particular spacetime dimensions.Contents: Introduction"Number Systems: "The Geometry of the Complex NumbersThe Geometry of the QuaternionsThe Geometry of the OctonionsOther Number Systems"Symmetry Groups: "Some Orthogonal GroupsSome Unitary GroupsSome Symplectic GroupsSymmetry Groups over Other Division AlgebrasLie Groups and Lie AlgebrasThe Exceptional Groups"Applications: "Division Algebras in MathematicsOctonionic Eigenvalue ProblemsThe Physics of the OctonionsMagic Squares Readership: Advanced ubdergraduate and graduate students and faculty in mathematics and physics; non-experts with moderately sophisticated mathematics background. Key Features: This book is easily digestible by a large audience wanting to know the elementary introduction to octanionsSuitable for any reader with a grasp of the complex numbers, although familiarity with non-octonionic versions of some of the other topics would be helpfulMany open problems are very accessibleAdvanced topics covered are quite sophisticated, leading up to a clear discussion of (one representation of) the exceptional Lie algebras and their associated root diagrams, and of the octonionic projective spaces on which they act
Author: Stephen Bruce Sontz Publisher: Springer Nature ISBN: 3030407675 Category : Science Languages : en Pages : 299
Book Description
Since the 17th century, physical theories have been expressed in the language of mathematical equations. This introduction to quantum theory uses that language to enable the reader to comprehend the notoriously non-intuitive ideas of quantum physics. The mathematical knowledge needed for using this book comes from standard undergraduate mathematics courses and is described in detail in the section Prerequisites. This text is especially aimed at advanced undergraduate and graduate students of mathematics, computer science, engineering and chemistry among other disciplines, provided they have the math background even though lacking preparation in physics. In fact, no previous formal study of physics is assumed.
Author: Kenichi Kanatani Publisher: CRC Press ISBN: 1000070662 Category : Computers Languages : en Pages : 171
Book Description
3D rotation analysis is widely encountered in everyday problems thanks to the development of computers. Sensing 3D using cameras and sensors, analyzing and modeling 3D for computer vision and computer graphics, and controlling and simulating robot motion all require 3D rotation computation. This book focuses on the computational analysis of 3D rotation, rather than classical motion analysis. It regards noise as random variables and models their probability distributions. It also pursues statistically optimal computation for maximizing the expected accuracy, as is typical of nonlinear optimization. All concepts are illustrated using computer vision applications as examples. Mathematically, the set of all 3D rotations forms a group denoted by SO(3). Exploiting this group property, we obtain an optimal solution analytical or numerically, depending on the problem. Our numerical scheme, which we call the "Lie algebra method," is based on the Lie group structure of SO(3). This book also proposes computing projects for readers who want to code the theories presented in this book, describing necessary 3D simulation setting as well as providing real GPS 3D measurement data. To help readers not very familiar with abstract mathematics, a brief overview of quaternion algebra, matrix analysis, Lie groups, and Lie algebras is provided as Appendix at the end of the volume.
Author: Pertti Lounesto Publisher: Cambridge University Press ISBN: 0521005515 Category : Mathematics Languages : en Pages : 352
Book Description
This is the second edition of a popular work offering a unique introduction to Clifford algebras and spinors. The beginning chapters could be read by undergraduates; vectors, complex numbers and quaternions are introduced with an eye on Clifford algebras. The next chapters will also interest physicists, and include treatments of the quantum mechanics of the electron, electromagnetism and special relativity with a flavour of Clifford algebras. This edition has three new chapters, including material on conformal invariance and a history of Clifford algebras.
Author: George E. Martin Publisher: Springer Science & Business Media ISBN: 1461256801 Category : Mathematics Languages : en Pages : 251
Book Description
Transformation Geometry: An Introduction to Symmetry offers a modern approach to Euclidean Geometry. This study of the automorphism groups of the plane and space gives the classical concrete examples that serve as a meaningful preparation for the standard undergraduate course in abstract algebra. The detailed development of the isometries of the plane is based on only the most elementary geometry and is appropriate for graduate courses for secondary teachers.
Author: Robin Hartshorne Publisher: Springer Science & Business Media ISBN: 0387226761 Category : Mathematics Languages : en Pages : 535
Book Description
This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's Elements leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants, with complete proofs. Topics include the introduction of coordinates, the theory of area, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra.
Author: Simon L. Altmann
Author: Ernst Binz
Author: Tevian Dray
Author: Henri Bacry
Author: Stephen Bruce Sontz
Author: Kenichi Kanatani
Author: Pertti Lounesto
Author: George E. Martin
Author: Michael Davis
Author: Robin Hartshorne