Analytic topology Books
John Wiley & Sons Inc Fractal Geometry
Book SynopsisThis comprehensive and popular textbook makes fractal geometry accessible to final-year undergraduate math or physics majors, while also serving as a reference for research mathematicians or scientists. This up-to-date edition covers introductory multifractal theory, random fractals, and modern applications in finance and science.Trade Review“Falconer’s book is excellent in many respects and the reviewer strongly recommends it. May every university library own a copy, or three! And if you’re a student reading this, go check it out today!.” (Mathematical Association of America, 11 June 2014)Table of ContentsPreface to the first edition ix Preface to the second edition xiii Preface to the third edition xv Course suggestions xvii Introduction xix PART I FOUNDATIONS 1 1 Mathematical background 3 1.1 Basic set theory 3 1.2 Functions and limits 7 1.3 Measures and mass distributions 11 1.4 Notes on probability theory 17 1.5 Notes and references 24 Exercises 24 2 Box-counting dimension 27 2.1 Box-counting dimensions 27 2.2 Properties and problems of box-counting dimension 34 *2.3 Modified box-counting dimensions 38 2.4 Some other definitions of dimension 40 2.5 Notes and references 41 Exercises 42 3 Hausdorff and packing measures and dimensions 44 3.1 Hausdorff measure 44 3.2 Hausdorff dimension 47 3.3 Calculation of Hausdorff dimension – simple examples 51 3.4 Equivalent definitions of Hausdorff dimension 53 *3.5 Packing measure and dimensions 54 *3.6 Finer definitions of dimension 57 *3.7 Dimension prints 58 *3.8 Porosity 60 3.9 Notes and references 63 Exercises 64 4 Techniques for calculating dimensions 66 4.1 Basic methods 66 4.2 Subsets of finite measure 75 4.3 Potential theoretic methods 77 *4.4 Fourier transform methods 80 4.5 Notes and references 81 Exercises 81 5 Local structure of fractals 83 5.1 Densities 84 5.2 Structure of 1-sets 87 5.3 Tangents to s-sets 92 5.4 Notes and references 96 Exercises 96 6 Projections of fractals 98 6.1 Projections of arbitrary sets 98 6.2 Projections of s-sets of integral dimension 101 6.3 Projections of arbitrary sets of integral dimension 103 6.4 Notes and references 105 Exercises 106 7 Products of fractals 108 7.1 Product formulae 108 7.2 Notes and references 116 Exercises 116 8 Intersections of fractals 118 8.1 Intersection formulae for fractals 119 *8.2 Sets with large intersection 122 8.3 Notes and references 128 Exercises 128 PART II APPLICATIONS AND EXAMPLES 131 9 Iterated function systems – self-similar and self-affine sets 133 9.1 Iterated function systems 133 9.2 Dimensions of self-similar sets 139 CONTENTS vii 9.3 Some variations 143 9.4 Self-affine sets 149 9.5 Applications to encoding images 155 *9.6 Zeta functions and complex dimensions 158 9.7 Notes and references 167 Exercises 167 10 Examples from number theory 169 10.1 Distribution of digits of numbers 169 10.2 Continued fractions 171 10.3 Diophantine approximation 172 10.4 Notes and references 176 Exercises 176 11 Graphs of functions 178 11.1 Dimensions of graphs 178 *11.2 Autocorrelation of fractal functions 188 11.3 Notes and references 192 Exercises 192 12 Examples from pure mathematics 195 12.1 Duality and the Kakeya problem 195 12.2 Vitushkin’s conjecture 198 12.3 Convex functions 200 12.4 Fractal groups and rings 201 12.5 Notes and references 204 Exercises 204 13 Dynamical systems 206 13.1 Repellers and iterated function systems 208 13.2 The logistic map 209 13.3 Stretching and folding transformations 213 13.4 The solenoid 217 13.5 Continuous dynamical systems 220 *13.6 Small divisor theory 225 *13.7 Lyapunov exponents and entropies 228 13.8 Notes and references 231 Exercises 232 14 Iteration of complex functions – Julia sets and the Mandelbrot set 235 14.1 General theory of Julia sets 235 14.2 Quadratic functions – the Mandelbrot set 243 14.3 Julia sets of quadratic functions 248 14.4 Characterisation of quasi-circles by dimension 256 14.5 Newton’s method for solving polynomial equations 258 14.6 Notes and references 262 Exercises 262 15 Random fractals 265 15.1 A random Cantor set 266 15.2 Fractal percolation 272 15.3 Notes and references 277 Exercises 277 16 Brownian motion and Brownian surfaces 279 16.1 Brownian motion in ℝ 279 16.2 Brownian motion in ℝn 285 16.3 Fractional Brownian motion 289 16.4 Fractional Brownian surfaces 294 16.5 Lévy stable processes 296 16.6 Notes and references 299 Exercises 299 17 Multifractal measures 301 17.1 Coarse multifractal analysis 302 17.2 Fine multifractal analysis 307 17.3 Self-similar multifractals 310 17.4 Notes and references 320 Exercises 320 18 Physical applications 323 18.1 Fractal fingering 325 18.2 Singularities of electrostatic and gravitational potentials 330 18.3 Fluid dynamics and turbulence 332 18.4 Fractal antennas 334 18.5 Fractals in finance 336 18.6 Notes and references 340 Exercises 341 References 342 Index 357
£40.80
Oxford University Press Visual Complex Analysis
Book SynopsisComplex Analysis is the powerful fusion of the complex numbers (involving the ''imaginary'' square root of -1) with ordinary calculus, resulting in a tool that has been of central importance to science for more than 200 years.This book brings this majestic and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. The 501 diagrams of the original edition embodied geometrical arguments that (for the first time) replaced the long and often opaque computations of the standard approach, in force for the previous 200 years, providing direct, intuitive, visual access to the underlying mathematical reality.This new 25th Anniversary Edition introduces brand-new captions that fully explain the geometrical reasoning, making it possible to read the work in an entirely new wayas a highbrow comic book!Trade ReviewVisual Complex Analysis is a delight, and a book after my own heart. By his innovative and exclusive use of the geometrical perspective, Tristan Needham uncovers many surprising and largely unappreciated aspects of the beauty of complex analysis. * Sir Roger Penrose *...it is comparable with Feynman's Lectures on Physics. At every point it asks "why" and finds a beautiful visual answer. * Newsletter of the European Mathematical Society *Newton would have approved... a fascinating and refreshing look at a familiar subject... essential reading for anybody with any interest at all in this absorbing area of mathematics. * Times Higher Education Supplement *One of the saddest developments in school mathematics has been the downgrading of the visual for the formal. I'm not lamenting the loss of traditional Euclidean geometry, despite its virtues, because it too emphasised stilted formalities. But to replace our rich visual tradition by silly games with 2 x 2 matrices has always seemed to me to be the height of folly. It is therefore a special pleasure to see Tristan Needham's Visual Complex Analysis with its elegantly illustrated visual approach. Yes, he has 2 x 2 matrices--but his are interesting. * Ian Stewart, New Scientist *an engaging, broad, thorough, and often deep, development of undergraduate complex analysis and related areas from a geometric point of view. The style is lucid, informal, reader-friendly, and rich with helpful images (e.g. the complex derivative as an "amplitwist"). A truly unusual and notably creative look at a classical subject. * Paul Zorn, American Mathematical Monthly *If your budget limits you to only buying one mathematics book in a year then make sure that this is the one that you buy. * Mathematical Gazette *I was delighted when I came across Visual Complex Analysis. As soon as I thumbed through it, I realized that this was the book I was looking for ten years ago. * Ed Catmull, former president of Pixar and Disney Animation Studios *The new ideas and exercises bring together a body of information potentially invaluable to researchers in fields from topology to number theory... this is only the beginning of a long list of famous facts for which Needham offers attractive visual proofs: Cauchy's theorem is a satisfying example: you can see the contribution to the integral from each infinitesimal square vanish before your eyes. * Frank Farris, American Mathematical Monthly *This informal style is excellently judged and works extremely well. Many of the arguments presented will be new even to experts, and the book will be of great interest to professionals working in either complex analysis or in any field where complex analysis is used. * David Armitage, Mathematical Reviews *The arguments constructed are highly innovative; even veterans of the field will find new ideas here. This is a special book. Tristan Needham has not only completely rethought a classical field of mathematics, but has presented it in a clear and compelling way. Visual Complex Analysis is worthy of the accolades it has received * MAA Reviews *This new edition of Visual Complex Analysis applies Newton's geometrical methods from the Principia and his concept of ultimate equality to Complex Analysis. * MathSciNet *Table of Contents1: Geometry and Complex Arithmetic 2: Complex Functions as Transformations 3: Möbius Transformations and Inversion 4: Differentiation: The Amplitwist Concept 5: Further Geometry of Differentiation 6: Non-Euclidean Geometry 7: Winding Numbers and Topology 8: Complex Integration: Cauchy's Theorem 9: Cauchy's Formula and Its Applications 10: Vector Fields: Physics and Topology 11: Vector Fields and Complex Integration 12: Flows and Harmonic Functions
£40.84
Oxford University Press Fractals
Book SynopsisMany are familiar with the beauty and ubiquity of fractal forms within nature. Unlike the study of smooth forms such as spheres, fractal geometry describes more familiar shapes and patterns, such as the complex contours of coastlines, the outlines of clouds, and the branching of trees. In this Very Short Introduction, Kenneth Falconer looks at the roots of the ''fractal revolution'' that occurred in mathematics in the 20th century, presents the ''new geometry'' of fractals, explains the basic concepts, and explores the wide range of applications in science, and in aspects of economics.This is essential introductory reading for students of mathematics and science, and those interested in popular science and mathematics.ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.Trade ReviewFractals: A Very Short Introduction is an obvious starting point for lay readers interested in fractals. It presents the key ideas and explains their context and significance, while introducing and using some very basic mathematics. * Danny Yee's Book Reviews *a most enjoyable, 'short read' * Institute of Mathematics *[A] very well-written introduction to fractals for non-specialists ... Highly recommended. * CHOICE *Table of ContentsPreface ; 1. The fractal concept ; 2. Self-similarity ; 3. Fractal dimension ; 4. Julia sets and the Mandelbrot set ; 5. Random walks and Brownian motion ; 6. Fractals in the real world ; 7. A little history ; Further reading
£9.49
The University of Chicago Press Lectures on Lie Groups Midway Reprints Series
Book Synopsis"["Lectures in Lie Groups"] fulfills its aim admirably and should be a useful reference for any mathematician who would like to learn the basic results for compact Lie groups. . . . The book is a well written basic text [and Adams] has done a service to the mathematical community." Irving Kaplansky"
£23.00
The University of Chicago Press Mathematics and the Unexpected
Book SynopsisIn this brief treatise, Ekelund explains some philosophical implications of recent mathematics. He examines randomness, the geometry involved in making predictions, and why general trends are easy to project, but particulars are practically impossible.
£21.00
De Gruyter Proximity Approach to Problems in Topology and
Book SynopsisDieses Buch konzentriert das aktuelle Gesamtwissen zum Proximity-Konzept und stellt es dem Leser in gut strukturierter Form dar. Hauptaugenmerk liegt auf den vielfältigen Möglichkeiten, die sich aus dem Proximity-Konzept der räumlichen Nähe und seiner Verallgemeinerung im Nearness-Konzept ergeben.
£76.00
Oxford University Press (UK) Introduction to Banach Spaces and Algebras
Book SynopsisBanach spaces and algebras are a key topic of pure mathematics. Graham Allan''s careful and detailed introductory account will prove essential reading for anyone wishing to specialise in functional analysis and is aimed at final year undergraduates or masters level students. Based on the author''s lectures to fourth year students at Cambridge University, the book assumes knowledge typical of first degrees in mathematics, including metric spaces, analytic topology, and complex analysis. However, readers are not expected to be familiar with the Lebesgue theory of measure and integration.The text begins by giving the basic theory of Banach spaces, including dual spaces and bounded linear operators. It establishes forms of the theorems that are the pillars of functional analysis, including the Banach-Alaoglu, Hahn-Banach, uniform boundedness, open mapping, and closed graph theorems. There are applications to Fourier series and operators on Hilbert spaces.The main body of the text is an intTrade ReviewThis well-crafted and scholarly book ...leaves nothing to be desired: this is a fine way to get into this beautiful in a subject and will serve to reel in a huge number of futureews devotees. * Michael Berg, MAA Reviews *Table of ContentsPART I INTRODUCTION TO BANACH SPACES ; 1. Preliminaries ; 2. Elements of normed spaces ; 3. Banach spaces ; PART II BANACH ALGEBRAS ; 4. Banach algebras ; 5. Representation theory ; 6. Algebras with an involution ; 7. The Borel functional calculus ; PART III SCV AND BANACH ALGEBRAS ; 8. Introduction to several complex variables ; 9. The holomorphic functional calculus in several variables ; Bibliography ; Index
£92.00
Cambridge University Press Duality and Perturbation Methods in Critical Point Theory Cambridge Tracts in Mathematics Series Number 107
Book SynopsisThe calculus of variations has been an active area of mathematics for over 300 years. Its main use is to find stable critical points of functions for the solution of problems. To find unstable values, new approaches (Morse theory and min-max methods) were developed, and these are still being refined to overcome difficulties when applied to the theory of partial differential equations. Here, Professor Ghoussoub describes a point of view that may help when dealing with such problems. Building upon min-max methods, he systematically develops a general theory that can be applied in a variety of situations. In so doing he also presents a whole array of duality and perturbation methods. The prerequisites for following this book are relatively few; an appendix sketching certain methods in analysis makes the book reasonably self-contained. Consequently, it should be accessible to all mathematicians, pure or applied, economists and engineers working in nonlinear analysis or optimization.Trade Review"The excellent material presentation of this small book follows the tradition of the series." Jean Mawhin, SIAM Review"...I recommend Ghoussoub's book to anyone working on variational problems, in particular on LS-theory. The reader will find a systematic presentation and detailed exposition of useful techniques." T. Bartsch, Bulletin of the American Mathematical Society"...Each chapter ends with some historical notes and comments, which offer a very useful guide through the literature on the subjects mentioned." Maria Letizia Bertotti, Mathematical ReviewsTable of Contents1. Lipschitz and smooth perturbed minimization principles; 2. Linear and plurisubharmonic perturbed minimization principles; 3. The classical min-max theorem; 4. A strong form of the min-max principle; 5. Relaxed boundary conditions in the presence of a dual set; 6. The critical set in the mountain pass theorem; 7. Group actions and multiplicity of critical points; 8. The Palais–Smale condition around a dual set - examples; 9. Morse indices of min-max critical points - the non-degenerate case; 10. Morse indices of min-max critical points - the degenerate case; 11. Morse-type information on Palais–Smale sequences; Appendix; References; Index.
£53.99
Cambridge University Press Chain Conditions in Topology 79 Cambridge Tracts in Mathematics Series Number 79
Book SynopsisA chain condition is a property, typically involving considerations of cardinality, of the family of open subsets of a topological space. (Sample questions: (a) How large a fmily of pairwise disjoint open sets does the space admit? (b) From an uncountable family of open sets, can one always extract an uncountable subfamily with the finite intersection property. This monograph, which is partly fresh research and partly expository (in the sense that the authors co-ordinate and unify disparate results obtained in several different countries over a period of several decades) is devoted to the systematic use of infinitary combinatorial methods in topology to obtain results concerning chain conditions. The combinatorial tools developed by P. ErdÃs and the Hungarian school, by ErdÃs and Rado in the 1960s and by the Soviet mathematician Shanin in the 1940s, are adequate to handle many natural questions concerning chain conditions in product spaces.Table of Contents1. Some infinitary combinatorics; 2. Introducing the chain conditions; 3. Chain conditions in products; 4. Classes of calibres, using Σ –products; 5. Calibres of compact spaces; 6. Strictly positive measures; 7. Between property (K) and the countable chain condition; 8. Classes of compact-calibres, using spaces of ultralilters; 9. Pseudo-compactness numbers: examples; 10. Continuous functions on product spaces.
£36.99
Cambridge University Press Characteristic Classes and the Cohomology of Finite Groups 9 Cambridge Studies in Advanced Mathematics Series Number 9
Book SynopsisThe purpose of this book is to study the relation between the representation ring of a finite group and its integral cohomology by means of characteristic classes. In this way it is possible to extend the known calculations and prove some general results for the integral cohomology ring of a group G of prime power order. Among the groups considered are those of p-rank less than 3, extra-special p-groups, symmetric groups and linear groups over finite fields. An important tool is the Riemann - Roch formula which provides a relation between the characteristic classes of an induced representation, the classes of the underlying representation and those of the permutation representation of the infinite symmetric group. Dr Thomas also discusses the implications of his work for some arithmetic groups which will interest algebraic number theorists. Dr Thomas assumes the reader has taken basic courses in algebraic topology, group theory and homological algebra, but has included an appendix in wTable of Contents1. Group cohomology; 2. Products and change of group; 3. Relations with subgroups and duality; 4. Spectral sequences; 5. Representations and vector bundles; 6. Bundles over the classifying space for a discrete group; 7. The symmetric group; 8. Finite groups with p-rank less than or equal to 2; 9. Linear groups over finite fields.
£42.49
Cambridge University Press LMS 50 Commutator Calculus London Mathematical Society Lecture Note Series Series Number 50
Book SynopsisA fundamental problem of algebraic topology is the classification of homotopy types and homotopy classes of maps. In this work the author extends results of rational homotopy theory to a subring of the rationale. The methods of proof employ classical commutator calculus of nilpotent group and Lie algebra theory and rely on an extensive and systematic study of the algebraic properties of the classical homotopy operations (composition and addition of maps, smash products, Whitehead products and higher order James-Hopi invariants). The account is essentially self-contained and should be accessible to non-specialists and graduate students with some background in algebraic topology and homotopy theory.Table of Contents1. Commuter calculus; 2. Distributivity laws in homotopy theory; 3. Homotopy operations on spheres; 4. Higher order Hopf invariants on spheres; 5. The homotopy Lie algebra and the spherical cohomotopy algebra; 6. Groups of homotopy classes; 7. The Hilton-Milnor theorum and its dual.
£41.99
Cambridge University Press LMS 77 Isolated Singular Points London Mathematical Society Lecture Note Series Series Number 77
Book SynopsisSingularity theory is not a field in itself, but rather an application of algebraic geometry, analytic geometry and differential analysis. The adjective 'singular' in the title refers here to singular points of complex-analytic or algebraic varieties or mappings.Table of Contents1. Examples of isolated singular points; 2. The milnor fibration; 3. Picard-Lefschetz formulas; 4. Critical space and discriminant space; 5. Relative monodromy; 6. Deformations; 7. Vanishing lattices, monodromy groups and adjacency; 8. The local Guass-Manin connection; 9. Applications of the local Gauss-Manin connection.
£59.31
Cambridge University Press LMS 104 Elliptic Struct 3 Manifold London Mathematical Society Lecture Note Series Series Number 104
Book SynopsisThis volume will give a systematic exposition of known results for free actions by finite groups on S. The text begins with preliminary material on Seifert manifolds and group classification. This is followed by sections dealing with related topics including free bZe/2 and bZe/3 actions on lens/prism manifolds, the reduction theorem and tangential structure.Table of Contents1. Introduction; 2. Seifert manifolds; 3. Groups with periodic cohomology; 4. Free C2 and C3 actions on certain Seifert manifolds; 5. The reduction theorem; 6. Tangenital structure; 7. SL(2, F5); 7. Finite Poincare complexes and homology spheres; 8. Workpoints.
£37.99
Cambridge University Press The Geometry of Fractal Sets 85 Cambridge Tracts in Mathematics Series Number 85
Book SynopsisThis book contains a rigorous mathematical treatment of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Questions of local density and the existence of tangents of such sets are studied, as well as the dimensional properties of their projections in various directions. In the case of sets of integral dimension the dramatic differences between regular 'curve-like' sets and irregular 'dust like' sets are exhibited. The theory is related by duality to Kayeka sets (sets of zero area containing lines in every direction). The final chapter includes diverse examples of sets to which the general theory is applicable: discussions of curves of fractional dimension, self-similar sets, strange attractors, and examples from number theory, convexity and so on. There is an emphasis on the basic tools of the subject such as the Vitali covering lemma, net measures and Fourier transform methods.Trade Review'This book is filled with the geometric jewels of fractional and integral Hausdorff dimension. It contains a much-needed unified notation and includes many recent results with simplified proofs. The theory is classically and rigorously presented with applications only alluded to in the introduction. Each chapter contains a short and important problem set. This is a lovely introduction to the mathematics of fractal sets for the pure mathematician.' American Mathematical MonthlyTable of ContentsPreface; Introduction; Notation; 1. Measure and dimension; 2. Basic density properties; 3. Structure of sets of integral dimension; 4. Structure of sets of non-integral dimension; 5. Comparable net measures; 6. Projection properties; 7. Besicovitch and Kakeya sets; 8. Miscellaneous examples of fractal sets; References; Index.
£58.12
Cambridge University Press Hamiltonian Systems Chaos and Quantization Cambridge Monographs on Mathematical Physics
Book SynopsisThis introduction to the theory of Hamiltonian chaos outlines the main results in the field, and goes on to consider implications for quantum mechanics. The study of nonlinear dynamics, and in particular of chaotic systems, is one of the fastest growing and most productive areas in physics and applied mathematics. In its first six chapters, this timely book introduces the theory of classical Hamiltonian systems. The aim is not to be comprehensive but, rather, to provide a mathematical trunk from which the reader will be able to branch out. The main focus is on periodic orbits and their neighbourhood, as this approach is especially suitable as an introduction to the implications of the theory of chaos in quantum mechanics, which are discussed in the last three chapters.Trade Review' … it successfully gives a concise treatment of well-chosen key elements of the field that are suitable for an upper-level graduate physics course.' ScienceTable of ContentsPreface; 1. Linear dynamical systems; 2. Nonlinear systems; 3. Chaotic systems; 4. Normal forms; 5. Maps of the circle; 6. Integrable and quasi-integrable systems; 7. Torus quantization; 8. Quantization of ergodic systems; 9. Periodic orbits in quantum field theory; References; Index.
£38.99
Cambridge University Press Dynamical Systems and Ergodic Theory 40 London Mathematical Society Student Texts Series Number 40
Book SynopsisThis book is essentially a self-contained introduction to topological dynamics and ergodic theory. It is divided into a number of relatively short chapters with the intention that each may be used as a component of a lecture course tailored to the particular audience. Parts of the book are suitable for a final year undergraduate course or for a master's level course. A number of applications are given, principally to number theory and arithmetic progressions (through van der Waerden's theorem and Szemerdi's theorem).Trade Review' … the volume achieves its goals well. It covers a broad range of topics clearly and succinctly … There is much material here to interest and stimulate the reader … I thoroughly recommend it to anyone of has some knowledge of the subject matter and wants a concise and well presented reference for more advanced concepts.' UK Non-Linear NewsTable of ContentsIntroduction and preliminaries; Part I. Topological Dynamics: 1. Examples and basic properties; 2. An application of recurrence to arithmetic progressions; 3. Topological entropy; 4. Interval maps; 5. Hyperbolic toral automorphisms; 6. Rotation numbers; Part II. Measurable Dynamics: 7. Invariant measures; 8. Measure theoretic entropy; 9. Ergodic measures; 10. Ergodic theorems; 11. Mixing; 12. Statistical properties; Part III. Supplementary Chapters: 13. Fixed points for the annulus; 14. Variational principle; 15. Invariant measures for commuting transformations; 16. An application of ergodic theory to arithmetic progressions.
£106.00
Cambridge University Press LMSST 40 Dynamical Sys Ergodic Thy London Mathematical Society Student Texts Series Number 40
Book SynopsisThis book is essentially a self-contained introduction to topological dynamics and ergodic theory. It is divided into a number of relatively short chapters with the intention that each may be used as a component of a lecture course tailored to the particular audience. Parts of the book are suitable for a final year undergraduate course or for a master's level course. A number of applications are given, principally to number theory and arithmetic progressions (through van der Waerden's theorem and Szemerdi's theorem).Trade Review' … the volume achieves its goals well. It covers a broad range of topics clearly and succinctly … There is much material here to interest and stimulate the reader … I thoroughly recommend it to anyone of has some knowledge of the subject matter and wants a concise and well presented reference for more advanced concepts.' UK Non-Linear NewsTable of ContentsIntroduction and preliminaries; Part I. Topological Dynamics: 1. Examples and basic properties; 2. An application of recurrence to arithmetic progressions; 3. Topological entropy; 4. Interval maps; 5. Hyperbolic toral automorphisms; 6. Rotation numbers; Part II. Measurable Dynamics: 7. Invariant measures; 8. Measure theoretic entropy; 9. Ergodic measures; 10. Ergodic theorems; 11. Mixing; 12. Statistical properties; Part III. Supplementary Chapters: 13. Fixed points for the annulus; 14. Variational principle; 15. Invariant measures for commuting transformations; 16. An application of ergodic theory to arithmetic progressions.
£52.20
Cambridge University Press Harmonic Maps Loop Groups and Integrable Systems 38 London Mathematical Society Student Texts Series Number 38
Book SynopsisThis book is an introduction to an important area of mathematics and mathematical physics. It is accessible to university students, but leads to topics of current research in the theory of harmonic maps. It is the first book on this subject at this level.Trade Review'… an accessible introduction to some of the fundamental connections beween differental geometry, Lie groups and integrable Hamiltonian systems.' L'Enseignement Mathématique'It is very rare to find a book that can take a student from the very basics of a subject to the frontiers of active research. The author is to be congratulated on having produced just such a rarity!' Bulletin of the London Mathematics Society'The book will certainly be appreciated by mathematicians as well as theoretical physics interested in the subject.' European Mathematical SocietyTable of ContentsPreface; Acknowledgements; Part I. One-Dimensional Integrable Systems: 1. Lie groups; 2. Lie algebras; 3. Factorizations and homogeneous spaces; 4. Hamilton's equations and Hamiltonian systems; 5. Lax equations; 6. Adler-Kostant-Symes; 7. Adler-Kostant-Symes (continued); 8. Concluding remarks on one-dimensional Lax equations; Part II. Two-Dimensional Integrable Systems: 9. Zero-curvature equations; 10. Some solutions of zero-curvature equations; 11. Loop groups and loop algebras; 12. Factorizations and homogeneous spaces; 13. The two-dimensional Toda lattice; 14. T-functions and the Bruhat decomposition; 15. Solutions of the two-dimensional Toda lattice; 16. Harmonic maps from C to a Lie group G; 17. Harmonic maps from C to a Lie group (continued); 18. Harmonic maps from C to a symmetric space; 19. Harmonic maps from C to a symmetric space (continued); 20. Application: harmonic maps from S2 to CPn; 21. Primitive maps; 22. Weierstrass formulae for harmonic maps; Part III. One-Dimensional and Two-Dimensional Integrable Systems: 23. From 2 Lax equations to 1 zero-curvature equation; 24. Harmonic maps of finite type; 25. Application: harmonic maps from T2 to S2; 26. Epilogue; References; Index.
£106.00
Cambridge University Press LMSST 38 Harmonic Maps Loop Group London Mathematical Society Student Texts Series Number 38
Book SynopsisHarmonic maps are generalisations of the concept of geodesics. They encompass many fundamental examples in differential geometry and have recently become of widespread use in many areas of mathematics and mathematical physics. This is an accessible introduction to some of the fundamental connections between differential geometry, Lie groups, and integrable Hamiltonian systems. The specific goal of the book is to show how the theory of loop groups can be used to study harmonic maps. By concentrating on the main ideas and examples, the author leads up to topics of current research. The book is suitable for students who are beginning to study manifolds and Lie groups, and should be of interest both to mathematicians and to theoretical physicists.Trade Review'… an accessible introduction to some of the fundamental connections beween differental geometry, Lie groups and integrable Hamiltonian systems.' L'Enseignement Mathématique'It is very rare to find a book that can take a student from the very basics of a subject to the frontiers of active research. The author is to be congratulated on having produced just such a rarity!' Bulletin of the London Mathematics Society'The book will certainly be appreciated by mathematicians as well as theoretical physics interested in the subject.' European Mathematical SocietyTable of ContentsPreface; Acknowledgements; Part I. One-Dimensional Integrable Systems: 1. Lie groups; 2. Lie algebras; 3. Factorizations and homogeneous spaces; 4. Hamilton's equations and Hamiltonian systems; 5. Lax equations; 6. Adler-Kostant-Symes; 7. Adler-Kostant-Symes (continued); 8. Concluding remarks on one-dimensional Lax equations; Part II. Two-Dimensional Integrable Systems: 9. Zero-curvature equations; 10. Some solutions of zero-curvature equations; 11. Loop groups and loop algebras; 12. Factorizations and homogeneous spaces; 13. The two-dimensional Toda lattice; 14. T-functions and the Bruhat decomposition; 15. Solutions of the two-dimensional Toda lattice; 16. Harmonic maps from C to a Lie group G; 17. Harmonic maps from C to a Lie group (continued); 18. Harmonic maps from C to a symmetric space; 19. Harmonic maps from C to a symmetric space (continued); 20. Application: harmonic maps from S2 to CPn; 21. Primitive maps; 22. Weierstrass formulae for harmonic maps; Part III. One-Dimensional and Two-Dimensional Integrable Systems: 23. From 2 Lax equations to 1 zero-curvature equation; 24. Harmonic maps of finite type; 25. Application: harmonic maps from T2 to S2; 26. Epilogue; References; Index.
£53.19
Cambridge University Press Axiomatic Domain Theory 14 Distinguished Dissertations in Computer Science Series Number 14
Book SynopsisAxiomatic categorical domain theory is crucial for understanding the meaning of programs and reasoning about them. This book is the first systematic account of the subject and studies mathematical structures suitable for modelling functional programming languages in an axiomatic (i.e. abstract) setting. In particular, the author develops theories of partiality and recursive types and applies them to the study of the metalanguage FPC; for example, enriched categorical models of the FPC are defined. Furthermore, FPC is considered as a programming language with a call-by-value operational semantics and a denotational semantics defined on top of a categorical model. To conclude, for an axiomatisation of absolute non-trivial domain-theoretic models of FPC, operational and denotational semantics are related by means of computational soundness and adequacy results. To make the book reasonably self-contained, the author includes an introduction to enriched category theory.Trade Review' … the author succeeds in the difficult task of finding the right level of abstraction. Moreover, the exposition is very precise and technically outstanding.' Daniele Turi, Science of Computer Programming (1998)Table of Contents1. Introduction; 2. Categorical preliminaries; 3. Partiality; 4. Order-enriched categories of partial maps; 5. Data types; 6. Recursive types; 7. Recursive types in Cpo-categories; 8. FPC; 9. Computational soundness and adequacy; 10. Summary and future research; Appendices; References; Indices.
£38.99
Cambridge University Press Soliton Equations and Their AlgebroGeometric Solutions Volume 2 11Dimensional Discrete Models Cambridge Studies in Advanced Mathematics Series Number 114
Book SynopsisAs a partner to Volume 1: Dimensional Continuous Models, this monograph provides a self-contained introduction to algebro-geometric solutions of completely integrable, nonlinear, partial differential-difference equations, also known as soliton equations. The systems studied in this volume include the Toda lattice hierarchy, the Kac-van Moerbeke hierarchy, and the Ablowitz-Ladik hierarchy. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The theory presented includes trace formulas, algebro-geometric initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses basic techniques from the theory of difference equations and spectral analysis, some elements of algebraic geometry and especially, the theory of compact Riemann surfaces. The presentation is constructive and rigorous, with ample background material provided in various appenTrade ReviewReviews of Volume 1: '… this is a book that I would recommend to any student of mine, for clarity and completeness of exposition … Any expert as well would enjoy the book and learn something stimulating from the sidenotes that point to alternative developments. We look forward to volumes two and three!' Mathematical Reviews'The book is very well organized and carefully written. It could be particularly useful for analysts wanting to learn new methods coming from algebraic geometry.' EMS NewsletterReview of Volume 2: 'As with the first part, the book is very well written and carefully organised and it is a pleasure to read it.' EMS NewsletterTable of ContentsAcknowledgments; Introduction; 1. The Toda hierarchy; 2. The Kac–van Moerbeke hierarchy; 3. The Ablowitz–Ladik hierarchy; A. Algebraic Curves and Their Theta Functions in a Nutshell; B. Hyperelliptic Curves of the Toda-Type; C. Asymptotic Spectral Parameter Expansions; D. Lagrange Interpolation; List of Symbols; Bibliography; Index; Errata and Addenda for Volume I.
£122.55
Cambridge University Press Harmonic Maps between Riemannian Polyhedra 142 Cambridge Tracts in Mathematics Series Number 142
Book SynopsisHarmonic maps between smooth Riemannian manifolds play a ubiquitous role in differential geometry. Examples include geodesics viewed as maps, minimal surfaces, holomorphic maps and Abelian integrals viewed as maps to a circle. The theory of such maps has been extensively developed over the last 40 years, and has significant applications throughout mathematics. This 2001 book extends that theory in full detail to harmonic maps between broad classes of singular Riemannian polyhedra, with many examples being given. The analytical foundation is based on existence and regularity results which use the potential theory of Riemannian polyhedral domains viewed as Brelot harmonic spaces and geodesic space targets in the sense of Alexandrov and Busemann. The work sets out much material on harmonic maps between singular spaces and will hence serve as a concise source for all researchers working in related fields.Trade Review'This book can be highly recommended, both to specialists in the field, who will find a direct interest, and to geometers and analysts, who will find a source containing a large amount of material, with precise references. The organization of the chapters is excellent.' Luc Lemaire, Bulletin of the London Mathematical SocietyTable of ContentsGromov's preface; Preface; 1. Introduction; Part I. Domains, Targets, Examples: 2. Harmonic spaces, Dirichlet spaces and geodesic spaces; 3. Examples of domains and targets; 4. Riemannian polyhedra; Part II. Potential Theory on Polyhedra: 5. The Sobolev space W1,2(X). Weakly harmonic functions; 6. Harnack inequality and Hölder continuity for weakly harmonic functions; 7. Potential theory on Riemannian polyhedra; 8. Examples of Riemannian polyhedra and related spaces; Part III. Maps between Polyhedra: 9. Energy of maps; 10. Hölder continuity of energy minimizers; 11. Existence of energy minimizers; 12. Harmonic maps - totally geodesic maps; 13. Harmonic morphisms; 14. Appendix A. Energy according to Korevaar-Schoen; 15. Appendix B. Minimizers with small energy decay; Bibliography; Special symbols; Index.
£111.00
Cambridge University Press The Mandelbrot Set Theme and Variations 274 London Mathematical Society Lecture Note Series Series Number 274
Book SynopsisThis volume provides a systematic exposition of current knowledge about the Mandelbrot set and presents important results in complex dynamics hitherto unpublished or difficult to find in the literature. For graduates and researchers in dynamical systems, Kleinian groups or related subjects.Trade Review'… this collection presents important results hitherto unpublished or difficult to find in the literature.' European Maths Society Journal'The analytic techniques employed cover an exceptionally broad range and students of mainstream science in search of an appropriate mathematical model to fit their dynamical scheme will find a very solid theoretical base going way beyond the basic concepts.' Contemporary Physics… should be studied in depth by any potential worker in this field. This book should remain popular for many years to come.' The Mathematical GazetteTable of ContentsIntroduction L.Tan; Preface J. Hubbard; 1. The Mandelbrot set is universal C. McMullen; 2. Baby Mandelbrot sets are born in cauliflowers A. Douady, X. Buff, R. Devaney and P. Sentenac; 3. Modulation dans l'ensemble de Mandelbrot P. Haïssinsky; 4. Local connectivity of Julia sets: expository lectures J. Milnor; 5. Holomorphic motions and puzzles (following M. Shishikura) P. Roesch; 6. Local properties of the Mandelbrot set at parabolic points L.Tan; 7. Convergence of rational rays in parameter spaces C. Petersen and G. Ryd; 8. Bounded recurrence of critical points and Jakobson's Theorem S. Luzzatto; 9. The Herman–Swiatek theorems with applications C. Petersen; 10. Perturbations d'une fonction linéarisable H. Jellouli; 11. Indice holomorphe et multiplicateur H. Jellouli; 12. An alternative proof of Mañé's theorem on non-expanding Julia sets M. Shishikura and L.Tan; 13. Geometry and dimension of Julia sets Y. -C. Yin; 14. On a theorem of Mary Rees for the matings of polynomials M. Shishikura; 15. Le théorème d'intégrabilité des structures presque complexes A. Douady and X. Buff; 16. Bifurcation of parabolic fixed points M. Shishikura.
£79.90
Cambridge University Press Analysis on Fractals 143 Cambridge Tracts in Mathematics Series Number 143
Book SynopsisA self-contained introduction to analysis on fractals, a developing area of mathematics which focuses on the dynamical aspects of fractals. Suitable for graduate students and researchers in analysis and probability theory. Also useful as a supplementary text for graduate courses covering fractals, for both mathematicians and physicists.Trade Review'… the most recent introduction to the analysis of 'Laplacians' on what physicists call finitely ramified self-similar fractals.' Volker Metz, Zentralblatt MATH'Anyone with a background in the analysis of linear field equations, with an interest in heterogeneous media, or who is looking to breathe new life into their research, should read this book.' A. J. Mulholland, Proceedings of the Edinburgh Mathematical Society'This book is an introduction to the subject written by one of the active researchers in the area. It is recommended to those who would like to go from the basics to current research topics.' Acta. Sci. Math.Table of ContentsIntroduction; 1. Geometry of self-similar sets; 2. Analysis on limits of networks; 3. Construction of Laplacians on P. C. F. self-similar structures; 4. Eigenvalues and eigenfunctions of Laplacians; 5. Heat kernels; Appendix A: Additional fact; Appendix B: Mathematical backgrounds; Bibliography; List of notations; Index.
£111.00
Cambridge University Press Introduction to Foliations and Lie Groupoids 91
Book SynopsisThis book gives a quick introduction to the theory of foliations, Lie groupoids and Lie algebroids. An important feature is the emphasis on the interplay between these concepts: Lie groupoids form an indispensable tool to study the transverse structure of foliations as well as their noncommutative geometry, while the theory of foliations has immediate applications to the Lie theory of groupoids and their infinitesimal algebroids. The book starts with a detailed presentation of the main classical theorems in the theory of foliations then proceeds to Molino's theory, Lie groupoids, constructing the holonomy groupoid of a foliation and finally Lie algebroids. Among other things, the authors discuss to what extent Lie's theory for Lie groups and Lie algebras holds in the more general context of groupoids and algebroids. Based on the authors' extensive teaching experience, this book contains numerous examples and exercises making it ideal for graduate students and their instructors.Trade Review'The book is based on course lecture notes and it still keeps its qualities and nice presentation.' EMS Newsletter'The book by Moerdijk and Mrcun serves as a concise and user-friendly introduction to the foundations of foliation theory and Lie groupoids, delving into just about all of the above-mentioned facets of the subject, and more. … the book reflects a great deal of care and thought on the part of the authors. … there is no apparent neglect of essential details, and the clarity of exposition is maintained right to the end. Those interested in the subject should find the book a delight to work through, and should thereby find a pathway into the more specialized topics mentioned earlier. … the treatment of each topic is as admirably self-contained as one might hope for. The instructional value of the book seems quite high, and it could certainly be recommended as text on which to base a graduate course.' Bulletin London Math SocietyTable of ContentsPreface; Prerequisites; 1. Foliations; 2. Holonomy and stability; 3. Two classical theorems; 4. Molino's theory; 5. Lie groupoids; 6. Lie algebroids; References and further reading; Index.
£65.99
Princeton University Press Hodge Theory
Book SynopsisProvides an introduction to Hodge theory - one of the central and most vibrant areas of contemporary mathematics - from leading specialists on the subject. This book includes topics that range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps.Trade Review"Charles and Schnell's chapter beautifully surveys the theory of absolute Hodge classes, giving in particular a complete proof of Deligne's theorem on absolute Hodge classes on abelian varieties... A welcome addition to the literature and should be useful to both graduate students and researchers working in Hodge theory."--Dan Petersen, MathSciNetTable of Contents*FrontMatter, pg. i*Contributors, pg. v*Contents, pg. vii*Preface, pg. xv*Chapter One. Introduction to Kahler Manifolds, pg. 1*Chapter Two. From Sheaf Cohomology to the Algebraic de Rham Theorem, pg. 70*Chapter Three. Mixed Hodge Structures, pg. 123*Chapter Four. Period Domains and Period Mappings, pg. 217*Chapter Five. The Hodge Theory of Maps, pg. 257*Chapter Six The Hodge Theory of Maps, pg. 273*Chapter Seven. Introduction to Variations of Hodge Structure, pg. 297*Chapter Eight. Variations of Mixed Hodge Structure, pg. 333*Chapter Nine. Lectures on Algebraic Cycles and Chow Groups, pg. 410*Chapter Ten. The Spread Philosophy in the Study of Algebraic Cycles, pg. 449*Chapter Eleven. Notes on Absolute Hodge Classes, pg. 469*Chapter Twelve. Shimura Varieties: A Hodge-Theoretic Perspective, pg. 531*Bibliography, pg. 574*Index, pg. 577
£78.20
American Mathematical Society Foliations Volume 2
Book SynopsisA guide to the qualitative theory of foliations. It features topics including: analysis on foliated spaces, characteristic classes of foliations and foliated manifolds. It is suitable as a supplementary text for a topics course at the advanced graduate level.Table of ContentsPart 1: Analysis and geometry on foliated spaces: Foreword to part 1 The $C^*$-algebra of a foliated space Harmonic measures for foliated spaces Generic leaves Part 2: Characteristic classes and foliations: Foreword to part 2 The Euler class of circle bundles The Chern-Weil construction Characteristic classes and integrability The Godbillon-Vey classes Part 3: Foliated 3-manifolds: Foreword to part 3 Constructing foliations Reebless foliations Foliations and the Thurston norm Disk decomposition and foliations of link complements $C^*$-Algebras Riemannian geometry and heat diffusion Brownian motion Planar foliations Bibliography Index.
£100.80
World Scientific Publishing Co Pte Ltd Foliations 2005 - Proceedings Of The
Book SynopsisThis volume takes a look at the current state of the theory of foliations, with surveys and research articles concerning different aspects. The focused aspects cover geometry of foliated Riemannian manifolds, Riemannian foliations and dynamical properties of foliations and some aspects of classical dynamics related to the field. Among the articles readers may find a study of foliations which admit a transverse contractive flow, an extensive survey on non-commutative geometry of Riemannian foliations, an article on contact structures converging to foliations, as well as a few articles on conformal geometry of foliations. This volume also contains a list of open problems in foliation theory which were collected from the participants of the Foliations 2005 conference.Table of ContentsMorphisms of Pseudogroups and Foliated Maps (J Alvarez Lopez & X Masa); On Infinitesimal Derivative of the Bott Class (T Asuke); Hirsch Foliations in Codimension Greater Than One (A Bi , S Hurder & J Shive); Extrinsic Geometry of Foliations on 3-Manifolds (D Bolotov); Extrinsic Geometry of Foliations (M Czarnecki & P Walczak); Transversal Twistor Spinors on a Riemannian Foliation (S Jung); A Survey on Simplicial Volume and Invariants of Foliations and Laminations (T Kuessner); Harmonic Foliations of the Plane: A Conformal Approach (R Langevin); Consecutive Shifts Along Orbits of Vector Fields (S Maksymenko); Generalized Equivariant Index Theory (K Richardson); Vanishing Results for Spectral Terms of a Riemannian Foliation (V Slesar); On the Group of Foliation Preserving Diffeomorphisms (T Tsuboi); and other papers.
£154.85
Oxford University Press Visual Complex Analysis
Book SynopsisThis new 25th anniversary edition of Visual Complex Analysis introduces this powerful method combining complex numbers with ordinary calculus, and includes new introductory content and brand-new captions that fully explain the geometrical reasoning.Trade ReviewVisual Complex Analysis is a delight, and a book after my own heart. By his innovative and exclusive use of the geometrical perspective, Tristan Needham uncovers many surprising and largely unappreciated aspects of the beauty of complex analysis. * Sir Roger Penrose *...it is comparable with Feynman's Lectures on Physics. At every point it asks "why" and finds a beautiful visual answer. * Newsletter of the European Mathematical Society *Newton would have approved... a fascinating and refreshing look at a familiar subject... essential reading for anybody with any interest at all in this absorbing area of mathematics. * Times Higher Education Supplement *One of the saddest developments in school mathematics has been the downgrading of the visual for the formal. I'm not lamenting the loss of traditional Euclidean geometry, despite its virtues, because it too emphasised stilted formalities. But to replace our rich visual tradition by silly games with 2 x 2 matrices has always seemed to me to be the height of folly. It is therefore a special pleasure to see Tristan Needham's Visual Complex Analysis with its elegantly illustrated visual approach. Yes, he has 2 x 2 matrices--but his are interesting. * Ian Stewart, New Scientist *an engaging, broad, thorough, and often deep, development of undergraduate complex analysis and related areas from a geometric point of view. The style is lucid, informal, reader-friendly, and rich with helpful images (e.g. the complex derivative as an "amplitwist"). A truly unusual and notably creative look at a classical subject. * Paul Zorn, American Mathematical Monthly *If your budget limits you to only buying one mathematics book in a year then make sure that this is the one that you buy. * Mathematical Gazette *I was delighted when I came across Visual Complex Analysis. As soon as I thumbed through it, I realized that this was the book I was looking for ten years ago. * Ed Catmull, former president of Pixar and Disney Animation Studios *The new ideas and exercises bring together a body of information potentially invaluable to researchers in fields from topology to number theory... this is only the beginning of a long list of famous facts for which Needham offers attractive visual proofs: Cauchy's theorem is a satisfying example: you can see the contribution to the integral from each infinitesimal square vanish before your eyes. * Frank Farris, American Mathematical Monthly *This informal style is excellently judged and works extremely well. Many of the arguments presented will be new even to experts, and the book will be of great interest to professionals working in either complex analysis or in any field where complex analysis is used. * David Armitage, Mathematical Reviews *The arguments constructed are highly innovative; even veterans of the field will find new ideas here. This is a special book. Tristan Needham has not only completely rethought a classical field of mathematics, but has presented it in a clear and compelling way. Visual Complex Analysis is worthy of the accolades it has received * MAA Reviews *This new edition of Visual Complex Analysis applies Newton's geometrical methods from the Principia and his concept of ultimate equality to Complex Analysis. * MathSciNet *Table of Contents1: Geometry and Complex Arithmetic 2: Complex Functions as Transformations 3: Möbius Transformations and Inversion 4: Differentiation: The Amplitwist Concept 5: Further Geometry of Differentiation 6: Non-Euclidean Geometry 7: Winding Numbers and Topology 8: Complex Integration: Cauchy's Theorem 9: Cauchy's Formula and Its Applications 10: Vector Fields: Physics and Topology 11: Vector Fields and Complex Integration 12: Flows and Harmonic Functions
£999.99
CRC Press Integrable Hamiltonian Systems Geometry Topology
Book SynopsisIntegrable Hamiltonian systems have been of growing interest over the past 30 years and represent one of the most intriguing and mysterious classes of dynamical systems. This book explores the topology of integrable systems and the general theory underlying their qualitative properties, singularites, and topological invariants.The authors, both of whom have contributed significantly to the field, develop the classification theory for integrable systems with two degrees of freedom. This theory allows one to distinguish such systems up to two natural equivalence relations: the equivalence of the associated foliation into Liouville tori and the usual orbital equaivalence. The authors show that in both cases, one can find complete sets of invariants that give the solution of the classification problem.The first part of the book systematically presents the general construction of these invariants, including many examples and applications. In the second part, the authors apply the general methods of the classification theory to the classical integrable problems in rigid body dynamics and describe their topological portraits, bifurcations of Liouville tori, and local and global topological invariants. They show how the classification theory helps find hidden isomorphisms between integrable systems and present as an example their proof that two famous systems--the Euler case in rigid body dynamics and the Jacobi problem of geodesics on the ellipsoid--are orbitally equivalent.Integrable Hamiltonian Systems: Geometry, Topology, Classification offers a unique opportunity to explore important, previously unpublished results and acquire generally applicable techniques and tools that enable you to work with a broad class of integrable systems.Table of ContentsBasic Notions. The Topology of Foliations on Two-Dimensional Surfaces Generated by Morse Functions. Rough Liouville Equivalence of Integrable Systems with Two Degrees of Freedom. Liouville Equivbalence of Integrable Systems with Two Degrees of Freedom. Orbital Classification of Integrable Systems with Two Degrees of Freedom. Classification of Hamiltonian Flows on Two-Dimensional Surfaces up to Topological Conjugacy. Smooth Conjugacy of Hamiltonian Flows on Two-Dimensional Surfaces. Orbital Classification of Integrable Hamiltonian Systems with Two Degrees of Freedom. The Second Step. Liouville Classification of Integrable Systems with Two Degrees of Freedom in Four-Dimensional Neighborhoods of Singular Points. Methods of Calculation of Topological Invariants of Integrable Hamiltonian Systems. Integrable Geodesic Flows on Two-Dimensional Surfaces. Liouville Classification of Integrable Geodesic Flows on Two-Dimensional Surfaces. Orbital Classification of Integrable Geodesic Flows on Two-Dimensional Surfaces. The Topology of Liouville Foliations in Classical Integrable Cases in Rigid Body Dynamics. Maupertuis Principle and Geodesic Equivalence.
£61.99
Krieger Publishing Company Introduction to Topology
Book Synopsis
£34.95
World Scientific Publishing Co Pte Ltd Quantum Topology
Book SynopsisThis book constitutes a review volume on the relatively new subject of Quantum Topology. Quantum Topology has its inception in the 1984/1985 discoveries of new invariants of knots and links (Jones, Homfly and Kauffman polynomials). These invariants were rapidly connected with quantum groups and methods in statistical mechanics. This was followed by Edward Witten's introduction of methods of quantum field theory into the subject and the formulation by Witten and Michael Atiyah of the concept of topological quantum field theories.This book is a review volume of on-going research activity. The papers derive from talks given at the Special Session on Knot and Topological Quantum Field Theory of the American Mathematical Society held at Dayton, Ohio in the fall of 1992. The book consists of a self-contained article by Kauffman, entitled Introduction to Quantum Topology and eighteen research articles by participants in the special session.This book should provide a useful source of ideas and results for anyone interested in the interface between topology and quantum field theory.
£108.00
World Scientific Publishing Co Pte Ltd Hamiltonian Dynamics
Book SynopsisThis is both a textbook and a monograph. It is partially based on a two-semester course, held by the author for third-year students in physics and mathematics at the University of Salerno, on analytical mechanics, differential geometry, symplectic manifolds and integrable systems.As a textbook, it provides a systematic and self-consistent formulation of Hamiltonian dynamics both in a rigorous coordinate language and in the modern language of differential geometry. It also presents powerful mathematical methods of theoretical physics, especially in gauge theories and general relativity.As a monograph, the book deals with the advanced research topic of completely integrable dynamics, with both finitely and infinitely many degrees of freedom, including geometrical structures of solitonic wave equations.Table of ContentsPart 1 Analytical mechanics: the Lagrangian coordinates; Hamiltonian systems; transformation theory; the integration methods. Part 2 Basic ideas of differential geometry: manifolds and tangent spaces; differential forms; integration theory; Lie groups and Lie algebras. Part 3 Geometry and physics: symplectic manifolds and Hamiltonian systems; the orbits method; classical electrodynamics. Part 4 Integrable field theories: "KdV" equation; general structures; meaning and existence of recursion operators; miscellanea; integrability of fermionic dynamics.
£95.00
World Scientific Publishing Co Pte Ltd Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory
Book SynopsisThis book contains an in-depth overview of the current state of the recently emerged and rapidly growing theory of Gnk groups, picture-valued invariants, and braids for arbitrary manifolds. Equivalence relations arising in low-dimensional topology and combinatorial group theory inevitably lead to the study of invariants, and good invariants should be strong and apparent. An interesting case of such invariants is picture-valued invariants, whose values are not algebraic objects, but geometrical constructions, like graphs or polyhedra.In 2015, V O Manturov defined a two-parametric family of groups Gnk and formulated the following principle: if dynamical systems describing a motion of n particles possess a nice codimension 1 property governed by exactly k particles then these dynamical systems possess topological invariants valued in Gnk.The book is devoted to various realisations and generalisations of this principle in the broad sense. The groups Gnk have many epimorphisms onto free products of cyclic groups; hence, invariants constructed from them are powerful enough and easy to compare. However, this construction does not work when we try to deal with points on a 2-surface, since there may be infinitely many geodesics passing through two points. That leads to the notion of another family of groups — Γnk, which give rise to braids on arbitrary manifolds yielding invariants of arbitrary manifolds.
£99.00
