Real analysis, real variables Books
Birkhauser Verlag AG A.P. Morse’s Set Theory and Analysis
Book SynopsisThis volume explores A.P. Morse’s (1911-1984) development of a formal language for writing mathematics, his application of that language in set theory and mathematical analysis, and his unique perspective on mathematics. The editor brings together a variety of Morse’s works in this compilation, including Morse's book A Theory of Sets, Second Edition (1986), in addition to material from another of Morse’s publications, Web Derivatives, and notes for a course on analysis from the early 1950's. Because Morse provided very little in the way of explanation in his written works, the editor’s commentary serves to outline Morse’s goals, give informal explanations of Morse’s formal language, and compare Morse’s often unique approaches to more traditional approaches. Minor corrections to Morse’s previously published works have also been incorporated into the text, including some updated axioms, theorems, and definitions. The editor’s introduction thoroughly details the corrections and changes made and provides readers with valuable insight on Morse’s methods.A.P. Morse’s Set Theory and Analysis will appeal to graduate students and researchers interested in set theory and analysis who also have an interest in logic. Readers with a particular interest in Morse’s unique perspective and in the history of mathematics will also find this book to be of interest.Table of ContentsPreface.- Editor's Introduction.- Language and Inference.- Logic.- Set Theory.- Elementary Analysis.- Metrics.- Measure.- Linear Measure and Total Variation.- Integration.- Product Measures.- Web Derivatives.- Classical Differentiation.- The Construction of Definition.- The Consistency of the Axiom of Size.- Suggested Reading.- Publications of A.P. Morse.- Errata to A Theory of Sets, Second Edition.- Integration with Respect to Addor Functions.- The Henstock-Kurzweil Integral.
£93.49
Springer International Publishing AG Excursions in Number Theory, Algebra, and
Book SynopsisThis textbook originates from a course taught by the late Ken Ireland in 1972. Designed to explore the theoretical underpinnings of undergraduate mathematics, the course focused on interrelationships and hands-on experience. Readers of this textbook will be taken on a modern rendering of Ireland’s path of discovery, consisting of excursions into number theory, algebra, and analysis. Replete with surprising connections, deep insights, and brilliantly curated invitations to try problems at just the right moment, this journey weaves a rich body of knowledge that is ideal for those going on to study or teach mathematics. A pool of 200 ‘Dialing In’ problems opens the book, providing fuel for active enquiry throughout a course. The following chapters develop theory to illuminate the observations and roadblocks encountered in the problems, situating them in the broader mathematical landscape. Topics cover polygons and modular arithmetic; the fundamental theorems of arithmetic and algebra; irrational, algebraic and transcendental numbers; and Fourier series and Gauss sums. A lively accompaniment of examples, exercises, historical anecdotes, and asides adds motivation and context to the theory. Return trips to the Dialing In problems are encouraged, offering opportunities to put theory into practice and make lasting connections along the way. Excursions in Number Theory, Algebra, and Analysis invites readers on a journey as important as the destination. Suitable for a senior capstone, professional development for practicing teachers, or independent reading, this textbook offers insights and skills valuable to math majors and high school teachers alike. A background in real analysis and abstract algebra is assumed, though the most important prerequisite is a willingness to put pen to paper and do some mathematics.Trade Review“Rather than being a book that one reads from cover to cover, Excursions is a curated collection problems followed by expository material aimed at providing background material useful for solving these problems. I imagine it would be a great experience to have a course taught out of this book. The second author clearly enjoyed the experience of studying this material under the guidance of the first author and wanted to make that experience available to others.” (John D. Cook, MAA Reviews, June 17, 2023)Table of ContentsPreface.- 1. Dialing In Problems.- 2. Polygons and Modular Arithmetic.- 3. The Fundamental Theorem of Arithmetic.- 4. The Fundamental Theorem of Algebra.- 5. Irrational, Algebraic and Transcendental Numbers.- 6. Fourier Series and Gauss Sums.- Epilogue.- Notation.- Bibliography.- Index.
£42.49
Springer Convex Functions and Their Applications
Book SynopsisConvex Functions at a First Glance.- More on Convex Functions on Intervals.- Convex Sets in Real Linear Spaces.- Convex Functions on a Normed Linear Space.- Differentiable Convex Functions. 6. Convexity and Majorization.- Convexity in Spaces of Matrices.- Convexity in Spaces of Matrices.- Duality and Convex Optimization.- Special Topics in Majorization Theory. Appendices.- Generalized Convexity on Intervals.- Background on Convex Sets.- Elementary Symmetric Functions.- Second Order Differentiability of Convex Functions.- The Variational Approach of PDE.
£66.49
Springer International Publishing AG An Excursion through Elementary Mathematics,
Book SynopsisThis book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. It expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This first volume covers Real Numbers, Functions, Real Analysis, Systems of Equations, Limits and Derivatives, and much more. As part of a collection, the book differs from other publications in this field by not being a mere selection of questions or a set of tips and tricks that applies to specific problems. It starts from the most basic theoretical principles, without being either too general or too axiomatic. Examples and problems are discussed only if they are helpful as applications of the theory. Propositions are proved in detail and subsequently applied to Olympic problems or to other problems at the Olympic level. The book also explores some of the hardest problems presented at National and International Mathematics Olympiads, as well as many essential theorems related to the content. An extensive Appendix offering hints on or full solutions for all difficult problems rounds out the book.Trade Review“This book develops several basic concepts in elementary mathematics and which are needed in the preparation of mathematical Olympiads around the world. … Throughout the book, the author puts great emphasis on explaining the issues discussed. … The presentation is pedagogical and coherent throughout with emphasis on concepts and underlying ideas … . The author has succeeded in crafting a comprehensive book to give a well-rounded picture of the role of real numbers and functions in elementary mathematics.” (Teodora Liliana Rădulescu, zbMATH 1371.97002, 2017)Table of ContentsChapter 1 The Set of Real Numbers.- Chapter 2 Algebraic Identities, Equations and Systems.- Chapter 3 Elementary Sequences.- Chapter 4 Induction and the Binomial Formula.- Chapter 5 Elementary Inequalities.- Chapter 6 The Concept of Function.- Chapter 7 More on Real Numbers.- Chapter 8 Continuous Functions.- Chapter 9 Limits and Derivatives.- Chapter 10 Riemann’s Integral.- Chapter 11 Series of Functions.- Bibliography.- Appendix A Glossary.- Appendix B Hints and Solutions.
£999.99
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG One-Dimensional Dynamics
Book SynopsisOne-dimensional dynamics has developed in the last decades into a subject in its own right. Yet, many recent results are inaccessible and have never been brought together. For this reason, we have tried to give a unified ac count of the subject and complete proofs of many results. To show what results one might expect, the first chapter deals with the theory of circle diffeomorphisms. The remainder of the book is an attempt to develop the analogous theory in the non-invertible case, despite the intrinsic additional difficulties. In this way, we have tried to show that there is a unified theory in one-dimensional dynamics. By reading one or more of the chapters, the reader can quickly reach the frontier of research. Let us quickly summarize the book. The first chapter deals with circle diffeomorphisms and contains a complete proof of the theorem on the smooth linearizability of circle diffeomorphisms due to M. Herman, J.-C. Yoccoz and others. Chapter II treats the kneading theory of Milnor and Thurstonj also included are an exposition on Hofbauer's tower construction and a result on fuB multimodal families (this last result solves a question posed by J. Milnor).Table of Contents0. Introduction.- I. Circle Diffeomorphisms.- 1. The Combinatorial Theory of Poincaré.- 2. The Topological Theory of Denjoy.- 2.a The Denjoy Inequality.- 2.b Ergodicity.- 3. Smooth Conjugacy Results.- 4. Families of Circle Diffeomorphisms; Arnol’d tongues.- 5. Counter-Examples to Smooth Linearizability.- 6. Frequency of Smooth Linearizability in Families.- 7. Some Historical Comments and Further Remarks.- II. The Combinatorics of One-Dimensional Endomorphisms.- 1. The Theorem of Sarkovskii.- 2. Covering Maps of the Circle as Dynamical Systems.- 3. The Kneading Theory and Combinatorial Equivalence.- 3.a Examples.- 3.b Hofbauer’s Tower Construction.- 4. Full Families and Realization of Maps.- 5. Families of Maps and Renormalization.- 6. Piecewise Monotone Maps can be Modelled by Polynomial Maps.- 7. The Topological Entropy.- 8. The Piecewise Linear Model.- 9. Continuity of the Topological Entropy.- 10. Monotonicity of the Kneading Invariant for the Quadratic Family.- 11. Some Historical Comments and Further Remarks.- III. Structural Stability and Hyperbolicity.- 1. The Dynamics of Rational Mappings.- 2. Structural Stability and Hyperbolicity.- 3. Hyperbolicity in Maps with Negative Schwarzian Derivative.- 4. The Structure of the Non-Wandering Set.- 5. Hyperbolicity in Smooth Maps.- 6. Misiurewicz Maps are Almost Hyperbolic.- 7. Some Further Remarks and Open Questions.- IV. The Structure of Smooth Maps.- 1. The Cross-Ratio: the Minimum and Koebe Principle.- l.a Some Facts about the Schwarzian Derivative.- 2. Distortion of Cross-Ratios.- 2.a The Zygmund Conditions.- 3. Koebe Principles on Iterates.- 4. Some Simplifications and the Induction Assumption.- 5. The Pullback of Space: the Koebe/Contraction Principle.- 6. Disjointness of Orbits of Intervals.- 7. Wandering Intervals Accumulate on Turning Points.- 8. Topological Properties of a Unimodal Pullback.- 9. The Non-Existence of Wandering Intervals.- 10. Finiteness of Attractors.- 11. Some Further Remarks and Open Questions.- V. Ergodic Properties and Invariant Measures.- 1. Ergodicity, Attractors and Bowen-Ruelle-Sinai Measures.- 2. Invariant Measures for Markov Maps.- 3. Constructing Invariant Measures by Inducing.- 4. Constructing Invariant Measures by Pulling Back.- 5. Transitive Maps Without Finite Continuous Measures.- 6. Frequency of Maps with Positive Liapounov Exponents in Families and Jakobson’s Theorem.- 7. Some Further Remarks and Open Questions.- VI. Renormalization.- 1. The Renormalization Operator.- 2. The Real Bounds.- 3. Bounded Geometry.- 4. The PullBack Argument.- 5. The Complex Bounds.- 6. Riemann Surface Laminations.- 7. The Almost Geodesic Principle.- 8. Renormalization is Contracting.- 9. Universality of the Attracting Cantor Set.- 10. Some Further Remarks and Open Questions.- VII. Appendix.- 1. Some Terminology in Dynamical Systems.- 2. Some Background in Topology.- 3. Some Results from Analysis and Measure Theory.- 4. Some Results from Ergodic Theory.- 5. Some Background in Complex Analysis.- 6. Some Results from Functional Analysis.
£87.99
Springer Verlag, Singapore Fundamentals of Analysis with Applications
Book SynopsisThis book serves as a textbook in real analysis. It focuses on the fundamentals of the structural properties of metric spaces and analytical properties of functions defined between such spaces. Topics include sets, functions and cardinality, real numbers, analysis on R, topology of the real line, metric spaces, continuity and differentiability, sequences and series, Lebesgue integration, and Fourier series. It is primarily focused on the applications of analytical methods to solving partial differential equations rooted in many important problems in mathematics, physics, engineering, and related fields. Both the presentation and treatment of topics are fashioned to meet the expectations of interested readers working in any branch of science and technology. Senior undergraduates in mathematics and engineering are the targeted student readership, and the topical focus with applications to real-world examples will promote higher-level mathematical understanding for undergraduates in sciences and engineering.Table of Contents1. Sets, Functions and Cardinality.- 2. The Real Numbers.- 3. Sequence and Series of Numbers.- 4. Analysis on R.- 5. Topology of the Real Line.- 6. Metric Spaces.- 7. Continuity and Differentiability.- 8. Sequences and Series of Functions.- 9. Lebesgue Integration.- 10. Fourier Series.
£38.24
Elsevier Science Vertex Operator Algebras and the Monster
Trade Review"This book is a detailed research monograph which contains the complete proofs of the authors' previously announced results. It is mostly self-contained and the exposition, given the technicalities involved, could not have been any better. It will be an asset to research mathematicians as well as physicists, and certainly deserves a place in every mathematics and physics library." --MATHEMATICAL REVIEWS "The present book shows how this group arises as the symmetry group of a certain vertex-operator algebra....'One fact, however, is undeniable. As the automorphism group of a distinguished conformal field theory, the Monster is fundamentally related to one of the most spectacular chapters of modern theoretical physics--string theroy." --N.J.A. SLoane, AT&T Laboratories quoted in AMERICAN SCIENTISTTable of ContentsLie Algebras. Formal Calculus: Introduction. Realizations of sl(2) by Twisted Vertex Operators. Realizations of sl(2) by Untwisted Vertex Operators. Central Extensions. The Simple Lie Algebras An, Dn, En. Vertex Operator Realizations of An, Dn, En. General Theory of Untwisted Vertex Operators. General Theory of Twisted Vertex Operators. The Moonshine Module. Triality. The Main Theorem. Completion of the Proof. Appendix: Complex Realization of Vertex Operator Algebras. Bibliography. Index of frequently used symbols. Index.
£44.99
Cambridge University Press Real Analysis and Probability 74 Cambridge Studies in Advanced Mathematics Series Number 74
Book SynopsisThis classic textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The first half of the book gives an exposition of real analysis: basic set theory, general topology, measure theory, integration, an introduction to functional analysis in Banach and Hilbert spaces, convex sets and functions and measure on topological spaces. The second half introduces probability based on measure theory, including laws of large numbers, ergodic theorems, the central limit theorem, conditional expectations and martingale's convergence. A chapter on stochastic processes introduces Brownian motion and the Brownian bridge. The edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical nTrade Review'A marvellous work which will soon become a standard text in the field for both teaching and reference … a complete and pedagogically perfect presentation of both the necessary preparatory material of real analysis and the proofs throughout the text. Some of the topics and proofs are rarely found in other textbooks.' Proceedings of the Edinburgh Mathematical Society'Careful, scholarly, and stimulating. It would be a pleasure to teach a mathematically-oriented graduate-level course from this text.' Short Book Reviews of the ISI'[It] will serve for a long time as a standard reference.' Zentralblatt fur und ihre Grenzgebiete'What makes the book special … is the care and scholarship with which the material is treated, and the choice of additional topics … not usually covered in first year graduate courses.' Mathematical Reviews'The book serves as a clear, rigorous, and complete introduction to modern probability theory using methods of mathematical analysis, and a description of relations between the two fields … it could be very useful for students interested in learning both topics, it can also serve as complementary reading to standard lectures. Teachers preparing their graduate level courses can use the book as an excellent, rigorously written and complete source.' EMS NewsletterTable of Contents1. Foundations: set theory; 2. General topology; 3. Measures; 4. Integration; 5. Lp spaces: introduction to functional analysis; 6. Convex sets and duality of normed spaces; 7. Measure, topology, and differentiation; 8. Introduction to probability theory; 9. Convergence of laws and central limit theorems; 10. Conditional expectations and martingales; 11. Convergence of laws on separable metric spaces; 12. Stochastic processes; 13. Measurability: Borel isomorphism and analytic sets; Appendixes: A. Axiomatic set theory; B. Complex numbers, vector spaces, and Taylor's theorem with remainder; C. The problem of measure; D. Rearranging sums of nonnegative terms; E. Pathologies of compact nonmetric spaces; Indices.
£54.14
Cambridge University Press Classical and Quantum Orthogonal Polynomials in One Variable 98 Encyclopedia of Mathematics and its Applications Series Number 98
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£73.30
Cambridge University Press Special Functions A Graduate Text 126 Cambridge Studies in Advanced Mathematics Series Number 126
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£44.64
Cambridge University Press Multidimensional Real Analysis I
Book SynopsisPart one of the authors' comprehensive and innovative work on multidimensional real analysis. Numerous illustrative exercises combined with an exhaustive and transparent treatment of subject matter make the book ideal as either the text for a course, a source of problems for a seminar or for self study.Trade Review'Throughout the notation is carefully organized and all proofs are complete and rigorous. The text is completed by carefully worked examples, many of them are illustrated by drawings. A special feature of the book is the extensive collection of exercises … The book is a good preparation for readers who wish to go on to more advanced studies in analysis. It can be also highly recommended as a text for a course or for self study.' Zentralblatt fur MathematikTable of ContentsPreface; Acknowledgements; Introduction; 1. Continuity; 2. Differentiation; 3. Inverse function and implicit function theorems; 4. Manifolds; 5. Tangent spaces; Exercises; Notation; Index.
£91.19
Cambridge University Press An Introduction to Nonlinear Analysis
Book SynopsisThe techniques used to solve non-linear problems differ greatly from those dealing with linear features. Deriving all the necessary theorems from first principles, this 2005 textbook should give upper undergraduates and graduate students a thorough understanding using as little background material as possible.Trade ReviewReview of the hardback: '… presents an introduction to critical point theory addressed to students with a modest background in Lebesgue integration and linear functional analysis. Many important methods from nonlinear analysis are introduced in a problem oriented way … well written … should be present in the library of any researcher interested in Lévy processes and Lie groups.' Zentralblatt MATHTable of Contents1. Extrema; 2. Critical points; 3. Boundary value problems; 4. Saddle points; 5. Calculus of variations; 6. Degree theory; 7. Conditional extrema; 8. Minimax methods; 9. Jumping nonlinearities; 10. Higher dimensions.
£42.74
Cambridge University Press Hausdorff Measures Cambridge Mathematical Library
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£53.19
Cambridge University Press Real Analysis and Probability 74 Cambridge Studies in Advanced Mathematics Series Number 74
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£164.35
Cambridge University Press Multidimensional Real Analysis II Integration Cambridge Studies in Advanced Mathematics Series Number 87
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£94.04
Cambridge University Press Singular Points of Plane Curves 63 London Mathematical Society Student Texts Series Number 63
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£122.55
Cambridge University Press An Introduction to Nonlinear Analysis
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£122.55
Cambridge University Press Higher Special Functions
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£90.25
Cambridge University Press Troilus and Cressida The New Cambridge Shakespeare
Book SynopsisThis two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and useful to graduates and researchers in pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. The first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transform. Then the higher-dimensional CalderÃnâZygmund and LittlewoodâPaley theories are developed. Probabilistic methods and their applications are discussed, as are applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to the Weyl calculus. The second volume goes beyond the classical to the highly contemporary and focuses on multilinear aspects of harmonic analysis: the bilinear Hilbert transform; CoifmanâMeyer theory; Carleson's resolution of the Lusin conjecture; CalderÃn's commutators and the Cauchy integral on Lipschitz curves. TTrade ReviewReview of the set: 'The two-volume set under review is a worthy addition to this tradition from two of the younger generation of researchers. It is remarkable that the authors have managed to fit all of this into [this number of] smaller-than-average pages without omitting to provide motivation and helpful intuitive remarks. Altogether, these books are a most welcome addition to the literature of harmonic analysis.' Gerald B. Folland, Mathematical ReviewsTable of ContentsPreface; Acknowledgements; 1. Leibniz rules and gKdV equations; 2. Classical paraproducts; 3. Paraproducts on polydiscs; 4. Calderón commutators and the Cauchy integral; 5. Iterated Fourier series and physical reality; 6. The bilinear Hilbert transform; 7. Almost everywhere convergence of Fourier series; 8. Flag paraproducts; 9. Appendix: multilinear interpolation; Bibliography; Index.
£59.84
