Series: Universitext
2nd ed. 2012, 2012, XVI, 453 p. 11 illus.
Softcover Information
Softcover
ISBN 978-1-4471-2983-7
Provides a complete and thorough introduction into the theory of linear and nonlinear partial differential equations
Presents interesting applications to physics and differential geometry
Includes the basic methods from linear and nonlinear functional analysis
This two-volume textbook provides comprehensive coverage of partial differential equations, spanning elliptic, parabolic, and hyperbolic types in two and several variables.
In this second volume, special emphasis is placed on functional analytic methods and applications to differential geometry. The following topics are treated:
*solvability of operator equations in Banach spaces
*Linear operators in Hilbert spaces and spectral theory
*Schauder's theory of linear elliptic differential equations
*weak solutions of differential equations
*nonlinear partial differential equations and characteristics
*nonlinear elliptic systems
*boundary value problems from differential geometry
This new second edition of this volume has been thoroughly revised and a new chapter on boundary value problems from differential geometry has been added.
2012, 2012, XII, 432 p. 4 illus.
Hardcover Information
HardcoverHardcover version
ISBN 978-1-4614-3809-0
Due: May 31, 2012
.In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated, "Ramanujan's lost notebook." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony.
This volume is the third of five volumes that the authors plan to write on Ramanujan’s lost notebook and other manuscripts and fragments found in The Lost Notebook and Other Unpublished Papers, published by Narosa in 1988. The ordinary partition function p(n) is the focus of this third volume. In particular, ranks, cranks, and congruences for p(n) are in the spotlight. Other topics include the Ramanujan tau-function, the Rogers?Ramanujan functions, highly composite numbers, and sums of powers of theta functions.
Preface.- Introduction.- 1. Ranks and Cranks, Part I.- 2. Ranks and Cranks, Part II.- 3. Ranks and Cranks, Part III.- 4. Ramanujan's Unpublished Manuscript on the Partition and Tau Functions.- 5. Theorems about the Partition Function on Pages 189 and 182.- 6. Congruences for Generalized Tau Functions on Page 178.- 7. Ramanujan's Forty Identities for the Rogers-Ramanujan Functions.- 8. Circular Summation.- 9. Highly Composite Numbers.- Scratch Work.- Location Guide.- Provenance.- References.
Series: Sources and Studies in the History of Mathematics and Physical Sciences
2012, 2012, XXVI, 615 p. 207 illus., 68 in color.
Hardcover
ISBN 978-1-4614-3781-9
Due: May 31, 2012
Fully revised and expanded translation and analysis of the procedure texts of Babylonian mathematical astronomy
Babylonian mathematical astronomy is explained through simple concepts to increasingly complex concepts and algorithms
Incorporates a typological analysis of all astronomical procedures
Includes a glossary of Babylonian technical astronomical terms, many of which are not adequately explained in the available dictionaries
Includes over 100 photos of cuneiform tablets dating from 350-50 BCE
Incorporates recent insights from Assyriology and translation science
Contains updated and expanded astronomical interpretations and investigations that have previously ignored in linguistic, mathematical and other aspects
Babylonian Mathematical Astronomy: Procedure Texts contains a new analysis of the procedure texts of Babylonian mathematical astronomy. These cuneiform tablets, excavated in Babylon and Uruk and dating from 350?50 BCE, contain computational instructions that represent the earliest known form of mathematical astronomy of the ancient world. The targeted readership includes assyriologists, historians of science, astronomers and others with an interest in Babylonian astronomy.
The book includes new translations of all 108 available tablets that are based on a modern approach incorporating recent insights from assyriology and translation science. All translations are accompanied by commentaries and photographs of the tablets. The preceding chapters are devoted to documentary, lexical, semantic, mathematical and astronomical aspects of the procedure texts. Special attention is given to issues of mathematical representation, a topic that had previously been largely ignored. Mathematical concepts are presented in a didactic fashion, setting out from the most elementary ones (numbers and elementary operations) to more complex ones (algorithms and computational systems). Chapters devoted to the planets and the Moon contain updated and expanded reconstructions and astronomical interpretations of the algorithms.
The author intends to continue his study of Babylonian mathematical astronomy with a new publication devoted to the Tabular Texts?the end products of Babylonian mathematical astronomy, computed with algorithms that are formulated in the present volume. The upcoming volume will contain new editions and reconstructions of over 250 tabular texts and a new philological, astronomical, and mathematical analysis of these texts.
Preface.- Acknowledgements.- Abbreviations and symbols.- 1. Procedure texts.- 2. Mathematical concepts ? from numbers to computational systems.- 3. Planets.- 4. Moon.- 5. Critical editions.- Appendices.- Glossary.- Bibliography.- Indices.
2012, 2012, XII, 343 p. 36 illus.
Hardcover
ISBN 978-1-4614-3630-0
Due: July 31, 2012
Identifies the important topics in logic that mathematicians use in their proofs
Methodically presents the key strategies used in mathematical proofs
Each proof strategy is illustrated by a variety of theorems concerning the natural, rational and real numbers
An introduction to group theory and real analysis that presents proof strategies for dealing with the core concepts introduced in these subjects
A Logical Introduction to Proof is a unique textbook that uses a logic-first approach to train and guide undergraduates through a transition or “bridge” course between calculus and advanced mathematics courses. The author’s approach prepares the student for the rigors required in future mathematics courses and is appropriate for majors in mathematics, computer science, engineering, as well as other applied mathematical sciences. It may also be beneficial as a supplement for students at the graduate level who need guidance or reference for writing proofs. Core topics covered are logic, sets, relations, functions, and induction, where logic is the instrument for analyzing the structure of mathematical assertions and is a tool for composing mathematical proofs. Exercises are given at the end of each section within a chapter.
Chapter 1 focuses on propositional logic while Chapter 2 is devoted to the logic of quantifiers. Chapter 3 methodically presents the key strategies that are used in mathematical proofs; each presented as a proof diagram. Every proof strategy is carefully illustrated by a variety of mathematical theorems concerning the natural, rational, and real numbers. Chapter 4 focuses on mathematical induction and concludes with a proof of the fundamental theorem of arithmetic. Chapters 5 through 7 introduce students to the essential concepts that appear in all branches of mathematics. Chapter 8 introduces the basic structures of abstract algebra: groups, rings, quotient groups, and quotient rings. Finally, Chapter 9 presents proof strategies that explicitly show students how to deal with the fundamental definitions that they will encounter in real analysis, followed by numerous examples of proofs that use these strategies. The appendix provides a useful summary of strategies for dealing with proofs.
Preface.- The Greek Alphabet.- 1. Propositional Logic.- 2. Predicate Logic.- 3. Proof Strategies and Diagrams.- 4. Mathematical Induction.- 5. Set Theory.- 6. Functions.- 7. Relations.- 8. Core Concepts in Abstract Algebra.- 9. Core Concepts in Real Analysis.- A Summary of Strategies.- References.- List of Symbols. Index.
Series: Grundlehren der mathematischen Wissenschaften, Vol. 345
2012, 2012, XVIII, 440 p.
Hardcover
ISBN 978-3-642-29879-0
Due: August 31, 2012
.Unique in mathematical literature covers the most advanced theories about martingale approach to central limit theorems
Develops techniques that allow to deal with applications in statistical mechanics and engineering
Is of interest to probabilists, mathematical physicists and analysts
Diffusive phenomena in statistical mechanics and in other fields arise from markovian modeling and their study requires sophisticated mathematical tools. In infinite dimensional situations, time symmetry properties can be exploited in order to make martingale approximations, along the lines of the seminal work of Kipnis and Varadhan. The present volume contains the most advanced theories on the martingale approach to central limit theorems. Using the time symmetry properties of the Markov processes, the book develops the techniques that allow us to deal with infinite dimensional models that appear in statistical mechanics and engineering (interacting particle systems, homogenization in random environments, and diffusion in turbulent flows, to mention just a few applications). The first part contains a detailed exposition of the method, and can be used as a text for graduate courses. The second concerns application to exclusion processes, in which the duality methods are fully exploited. The third part is about the homogenization of diffusions in random fields, including passive tracers in turbulent flows (including the superdiffusive behavior).
There are no other books in the mathematical literature that deal with this kind of approach to the problem of the central limit theorem. Hence, this volume meets the demand for a monograph on this powerful approach, now widely used in many areas of probability and mathematical physics. The book also covers the connections with and application to hydrodynamic limits and homogenization theory, so besides probability researchers it will also be of interest to mathematical physicists and analysts.
s Preface.- Part I: General Theory.- 1.A Warming-up Example.- 2.Central Limit Theorems.- 3.RandomWalks in Random Environment.- 4.Bounds and Variational Principles for the Asymptotic Variance.- Part II: Simple Exclusion Processes.- 5.The Simple Exclusion Process.- 6.Self Diffusion.- 7.Equilibrium Fluctuations of the Density Field.- 8.Regularity of the Asymptotic Variance.- Part III: Diffusions in Random Environments.- 10.Variational Principles for the Limiting Variance.- 11.Diffusions with Divergence Free Drifts.- 12.Diffusions with Gaussian Drifts.- 13.Ornstein-Uhlenbeck Process with a Random Potential.- 14.Analytic Methods in Homogenization Theory.- References.- Notation.- Subject Index.