Series: Cambridge Studies in Probability, Induction and Decision Theory
Paperback (ISBN-13: 9780521037549)
Page extent: 274 pages
Size: 228 x 152 mm
Weight: 0.422 kg
The term probability can be used in two main senses. In the frequency interpretation it is a limiting ratio in a sequence of repeatable events. In the Bayesian view, probability is a mental construct representing uncertainty. This book is about these two types of probability and investigates how, despite being adopted by scientists and statisticians in the eighteenth and nineteenth centuries, Bayesianism was discredited as a theory of scientific inference during the 1920s and 1930s. Through the examination of a dispute between two British scientists, the author argues that a choice between the two interpretations is not forced by pure logic or the mathematics of the situation, but depends on the experiences and aims of the individuals involved. The book should be of interest to students and scientists interested in statistics and probability theories and to general readers with an interest in the history, sociology and philosophy of science.
? Makes an important contribution to the on-going debate on the foundations of statistical inference ? Convincingly shows how probability theories are products of the social and cultural backgrounds of their authors ? Continues the tradition of Cambridge books in the history of probability
Contents
Acknowledgements; 1. Introduction; 2. Probability up to the twentieth century; 3. R. A. Fisher and statistical probability; 4. Harold Jeffreys and inverse probability; 5. The Fisher-Jeffreys exchange, 1932?1934; 6. Probability during the 1930s; 7. Epilogue and conclusions; Appendices; Bibliography; Index.
Paperback (ISBN-13: 9780521891400)
100 line diagrams 15 half-tones 18 tables 84 exercises
Page extent: 466 pages
Size: 247 x 174 mm
Quantum information theory is at the frontiers of physics, mathematics and information science, offering a variety of solutions that are impossible using classical theory. This book provides an introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. After a gentle introduction to the necessary mathematics the authors describe the geometry of quantum state spaces. Focusing on finite dimensional Hilbert spaces, they discuss the statistical distance measures and entropies used in quantum theory. The final part of the book is devoted to quantum entanglement - a non-intuitive phenomenon discovered by Schrodinger, which has become a key resource for quantum computation. This richly-illustrated book is useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.
? The first book to focus on the geometry of quantum states ? Stresses the similarities and differences between classical and quantum theory ? Uses a non-technical style and numerous figures to make the book accessible to non-specialists
Contents
Preface; 1. Convexity, colours and statistics; 2. Geometry of probability distributions; 3. Much ado about spheres; 4. Complex projective spaces; 5. Outline of quantum mechanics; 6. Coherent states and group actions; 7. The stellar representation; 8. The space of density matrices; 9. Purification of mixed quantum states; 10. Quantum operations; 11. Duality: maps versus states; 12. Density matrices and entropies; 13. Distinguishability measures; 14. Monotone metrics and measures; 15. Quantum entanglement; Epilogue; Appendices; References; Index.
Series: Cambridge Studies in Advanced Mathematics (No. 110)
Hardback (ISBN-13: 9780521719308)
6 line diagrams 95 exercises 20 worked examples
Page extent: 318 pages
Size: 228 x 152 mm
The subject of analysis and Lie groups comprises an eclectic group of topics which can be treated from many different perspectives. This self-contained text concentrates on the perspective of analysis, to the topics and methods of non-commutative harmonic analysis, assuming only elementary knowledge of linear algebra and basic differential calculus. The author avoids unessential technical discussions and instead describes in detail many interesting examples, including formulae which have not previously appeared in book form. Topics covered include the Haar measure and invariant integration, spherical harmonics, Fourier analysis and the heat equation, Poisson kernel, the Laplace equation and harmonic functions. Perfect for advanced undergraduates and graduates in geometric analysis, harmonic analysis and representation theory, the tools developed will also be useful for specialists in stochastic calculation and the statisticians. With numerous exercises and worked examples, the text is ideal for a graduate course on analysis on Lie groups.
* Self-contained and elementary presentation of Lie group theory, concentrating
on analysis on Lie groups * Numerous applications to classical harmonic
analysis, useful for the study of the theory of random matrices * Many
exercises and worked examples mean this is ideal for a graduate course
on analysis on Lie groups
Contents
Preface; 1. The linear group; 2. The exponential map; 3. Linear Lie groups; 4. Lie algebras; 5. Haar measure; 6. Representations of compact groups; 7. The groups SU(2) and SO(3), Haar measure; 8. Analysis on the group SU(2); 9. Analysis on the sphere; 10. Analysis on the spaces of symmetric and Hermitian matrices; 11. Irreducible representations of the unitary group; 12. Analysis on the unitary group; Bibliography; Index.
Series: Encyclopedia of Mathematics and its Applications (No. 120)
Hardback (ISBN-13: 9780521875561)
85 exercises 25 worked examples
Page extent: 350 pages
Size: 234 x 156 mm
Absolute measurable space and absolute null space are very old topological notions, developed from well-known facts of descriptive set theory, topology, Borel measure theory and analysis. This monograph systematically develops and returns to the topological and geometrical origins of these notions. Motivating the development of the exposition are the action of the group of homeomorphisms of a space on Borel measures, the Oxtoby-Ulam theorem on Lebesgue-like measures on the unit cube, and the extensions of this theorem to many other topological spaces. Existence of uncountable absolute null space, extension of the Purves theorem and recent advances on homeomorphic Borel probability measures on the Cantor space, are among the many topics discussed. A brief discussion of set-theoretic results on absolute null space is given, and a four-part appendix aids the reader with topological dimension theory, Hausdorff measure and Hausdorff dimension, and geometric measure theory.
? First book on absolute measurable space to emphasize topological, geometrical and analytical properties of such spaces ? Four-part appendix aids the reader with topics such as topological dimension theory, Hausdorff measure and Hausdorff dimension, and geometric measure theory ? Presents complete proofs of the Oxtaby-Ulam theorem, and of several theorems on the existence of uncountable absolute null space
Contents
Preface; 1. The absolute property; 2. The universally measurable property; 3. The Homeomorphism Group of X; 4. Real-valued functions; 5. Hausdorff measure and dimension; 6. Martin axiom; Appendix A. Preliminary material; Appendix B. Probability theoretic approach; Appendix C. Cantor spaces; Appendix D. Dimensions and measures; Bibliography.
Series: Encyclopedia of Mathematics and its Applications
Hardback (ISBN-13: 9780521887625)
131 line diagrams
Page extent: 800 pages
Size: 234 x 156 mm
The Hilbert transform has many uses, including solving problems in aerodynamics, condensed matter physics, optics, fluids, and engineering. Written in a style that will suit a wide audience (including the physical sciences), this book will become the reference of choice on the topic, whatever the subject background of the reader. It explains all the common Hilbert transforms, mathematical techniques for evaluating them, and has detailed discussions of their application. Especially useful for researchers are the tabulation of analytically evaluated Hilbert transforms, and an atlas that immediately illustrates how the Hilbert transform alters a function. A collection of exercises helps the reader to test their understanding of the material in each chapter. The bibliography is a wide-ranging collection of references both to the classical mathematical papers, and to a diverse array of applications.
? Informal style opens up the material to anyone working in the physical sciences ? The only book to contain an extensive table of Hilbert transforms, and it has a mini atlas to show reader immediately how the Hilbert transform alters a function ? Exercises are included to help test understanding, and a large bibliography points to classical papers and a wide range of applications
Contents
Preface; 1. Introduction; 2. Review of some background mathematics; 3. Derivation of the Hilbert transform relations; 4. Some basic properties of the Hilbert transform; 5. Relationship of the Hilbert transform to some common transforms; 6. The Hilbert transform of periodic functions; 7. Inequalities for the Hilbert transforms; 8. Asymptotic behavior of the Hilbert transform; 9. Hilbert transforms of some special functions; 10. Hilbert transforms involving distributions; 11. The finite Hilbert transform; 12. Some singular integral equations; 13. Discrete Hilbert transforms; 14. Numerical evaluation of Hilbert transforms; 15. Hilbert transforms involving distributions; 16. Some further extensions of the classical Hilbert transforms; 17. Linear systems and causality; 18. The Hilbert transform of waveforms and signal processing; 19. Kramers-Kronig relations; 20. Dispersion relations for some linear optical properties; 21. Dispersion relations for magneto-optical and natural optical activity; 22. Dispersion relations for nonlinear optical properties; 23. Some further applications of Hilbert transforms; Appendix 1. Table of selected Hilbert transforms; Appendix 2. Atlas of selected Hilbert transform pairs; References; Author index; Notational index; Subject index.