368 pages | 234x153mm
978-0-19-870249-8 | Hardback | September 2015 (estimated)
The first comprehensive monograph on an abstract theory of Feynman's operational calculus.
The presentation is detailed and mostly self-contained. Proofs and computations are presented with a great deal of detail
This book is a significant step in making Feynman's heuristic ideas mathematically rigorous.
Written for mathematicians, mathematically oriented physicists, graduate students in both mathematics and physics and other scientists of a mathematical bent who are interested in quantum theory.
This book is aimed at providing a coherent, essentially self-contained, rigorous and comprehensive abstract theory of Feynman's operational calculus for noncommuting operators. Although it is inspired by Feynman's original heuristic suggestions and time-ordering rules in his seminal 1951 paper An operator calculus having applications in quantum electrodynamics, as will be made abundantly clear in the introduction (Chapter 1) and elsewhere in the text, the theory developed in this book also goes well beyond them in a number of directions which were not anticipated in Feynman's work. Hence, the second part of the main title of this book.
The basic properties of the operational calculus are developed and certain algebraic and analytic properties of the operational calculus are explored. Also, the operational calculus will be seen to possess some pleasant stability properties. Furthermore, an evolution equation and a generalized integral equation obeyed by the operational calculus are discussed and connections with certain analytic Feynman integrals are noted.
This volume is essentially self-contained and we only assume that the reader has a reasonable, graduate level, background in analysis, measure theory and functional analysis or operator theory. Much of the necessary remaining background is supplied in the text itself.
Readership: Mathematicians, mathematically oriented physicists, graduate students in both mathematics
1: Introduction
2: Disentangling: Definitions, Properties and Elementary Examples
3: Disentangling via Tensor Products and Ordered Supports
4: Extraction of Multilinear Factors and Iterative Disentangling
5: Auxiliary Operations and Disentangling Algebras
6: Time-Dependent Feynman's Operational Calculus and Evolution Equations
7: Stability Properties of Feynman's Operational Calculi
8: Disentangling via Continuous and Discrete Measures
9: Derivational Derivatives and Feynman's Operational Calculi
10: Spectral Theory for Noncommuting Operators
11: Epilogue: Miscellaneous Topics and Possible Extensions
Hardcover | June 2015 | ISBN: 9780691161730
248 pp. | 9 x 10 | 103 color illus.
This lavishly illustrated book provides a hands-on, step-by-step introduction to the intriguing mathematics of symmetry. Instead of breaking up patterns into blocks?a sort of potato-stamp method?Frank Farris offers a completely new waveform approach that enables you to create an endless variety of rosettes, friezes, and wallpaper patterns: dazzling art images where the beauty of nature meets the precision of mathematics.
Featuring more than 100 stunning color illustrations and requiring only a modest background in math, Creating Symmetry begins by addressing the enigma of a simple curve, whose curious symmetry seems unexplained by its formula. Farris describes how complex numbers unlock the mystery, and how they lead to the next steps on an engaging path to constructing waveforms. He explains how to devise waveforms for each of the 17 possible wallpaper types, and then guides you through a host of other fascinating topics in symmetry, such as color-reversing patterns, three-color patterns, polyhedral symmetry, and hyperbolic symmetry. Along the way, Farris demonstrates how to marry waveforms with photographic images to construct beautiful symmetry patterns as he gradually familiarizes you with more advanced mathematics, including group theory, functional analysis, and partial differential equations. As you progress through the book, you’ll learn how to create breathtaking art images of your own.
Fun, accessible, and challenging, Creating Symmetry features numerous examples and exercises throughout, as well as engaging discussions of the history behind the mathematics presented in the book.
Frank A. Farris teaches mathematics at Santa Clara University. He is a former editor of Mathematics Magazine, a publication of the Mathematical Association of America. He lives in San Jose, Californi
New edition extensively revised and updated
Covers important topics such as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory
Fourth edition includes updates on the computer algebra and independent projects appendices
Features new central theoretical results such as the elimination theorem, the extension theorem, the closure theorem, and the nullstellensatz
Presents some of the newer approaches to computing Groebner bases
This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry?the elimination theorem, the extension theorem, the closure theorem and the Nullstellensatz?this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new Chapter (ten), which presents some of the essentials of progress made over the last decades in computing Grobner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D).
The book may serve as a first or second course in undergraduate abstract algebra and with some supplementation perhaps, for beginning graduate level courses in algebraic geometry or computational algebra. Prerequisites for the reader include linear algebra and a proof-oriented course. It is assumed that the reader has access to a computer algebra system. Appendix C describes features of Maple?, MathematicaR and Sage, as well as other systems that are most relevant to the text. Pseudocode is used in the text; Appendix B carefully describes the pseudocode used.
“…The book gives an introduction to Buchberger’s algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations and elimination theory. …The book is well-written. …The reviewer is sure that it will be an excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry.”
Peter Schenzel, zbMATH, 2007
“I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry.”
The American Mathematical Monthly
Prefers ideas to calculations
Explains the ideas without parsimony of words
Based on 35 years of teaching at Paris University
Blends mathematics skillfully with didactical and historical considerations
Analysis Volume IV introduces the reader to functional analysis (integration, Hilbert spaces, harmonic analysis in group theory) and to the methods of the theory of modular functions (theta and L series, elliptic functions, use of the Lie algebra of SL2). As in volumes I to III, the inimitable style of the author is recognizable here too, not only because of his refusal to write in the compact style used nowadays in many textbooks. The first part (Integration), a wise combination of mathematics said to be `modern' and `classical', is universally useful whereas the second part leads the reader towards a very active and specialized field of research, with possibly broad generalizations.
Proves the significance of mirror symmetry and chirality in topological insulators
Provides a detailed description of the various phenomena when two surface states of topological insulators hybridize, depending on the chirality
Examines those signs of the Dirac velocities that determine whether the interface states between two topological insulators are metallic or insulating.
Nominated as an outstanding contribution by the Tokyo Institute of Technology’s Physics Department in 2013
In this book, the author theoretically studies two aspects of topological states.
First, novel states arising from hybridizing surface states of topological insulators are theoretically introduced. As a remarkable example, the author shows the existence of gapless interface states at the interface between two different topological insulators, which belong to the same topological phase. While such interface states are usually gapped due to hybridization, the author proves that the interface states are in fact gapless when the two topological insulators have opposite chiralities. This is the first time that gapless topological novel interface states protected by mirror symmetry have been proposed.
Second, the author studies the Weyl semimetal phase in thin topological insulators subjected to a magnetic field. This Weyl semimetal phase possesses edge states showing abnormal dispersion, which is not observed without mirror symmetry. The author explains that the edge states gain a finite velocity by a particular form of inversion symmetry breaking, which makes it possible to observe the phenomenon by means of electric conductivity.