Provides a rigorous introduction to complex analysis

Arranges the material effectively in 50 class-tested lectures

Uses ample illustrations and examples to explain the subject

Provides problems for practice

StudentsStandard (0)978-1-4899-9716-6XIV, 331 p. 94 illus.Planned978-1-4020-2695-9AJAYIDeborah AJAYI, Pennsylavania State University Dept. Mathematics, University Park, PA, USA; David Hurtubise, Penn State University Department of Mathematics and Statistics, Altoona, PA, USALectures on Morse Homology+Texts in the Mathematical Sciences (closed) IX, 326 p.SCM12082)Global Analysis and Analysis on ManifoldsPBKSSCM280272Manifolds and Cell Complexes (incl. Diff.Topology)PBMSSpringer Netherlands\1. Introduction.- 2. The CW-Homology Theorem.- 3. Basic Morse Theory.- 4. The Stable/Unstable Manifold Theorem.- 5. Basic Differential Topology.- 6. Morse-Smale Functions.- 7. The Morse Homology Theorem.- 8. Morse Theory On Grassmann Manifolds.- 9. An Overview of Floer Homology Theories.- Hints and References for Selected Problems.- Symbol Index.GThis book presents in great detail all the results one needs to prove the Morse Homology Theorem using classical techniques from algebraic topology and homotopy theory. Most of these results can be found scattered throughout the literature dating from the mid to late 1900's in some form or other, but often the results are proved in different contexts with a multitude of different notations and different goals. This book collects all these results together into a single reference with complete and detailed proofs. The core material in this book includes CW-complexes, Morse theory, hyperbolic dynamical systems (the Lamba-Lemma, the Stable/Unstable Manifold Theorem), transversality theory, the Morse-Smale-Witten boundary operator, and Conley index theory. More advanced topics include Morse theory on Grassmann manifolds and Lie groups, and an overview of Floer homology theories. With the stress on completeness and by its elementary approach to Morse homology, this book is suitable as a textbook for a graduate level course, or as a reference for working mathematicians and physicists.978-90-481-6705-0978-3-642-27892-1Alabau-BoussouirawFatiha Alabau-Boussouira, Universit Paul Verlaine-Metz LMAM, Metz, France; Roger Brockett, Harvard University Engineering and Applied Sciences, Cambridge, MA, USA; Olivier Glass, Universit Paris-Dauphine CEREMADE, Paris, France; Jrme Le Rousseau, Universit d'Orlans Laboratoire MAPMO, Orlans, France; Enrique Zuazua, Basque Center for Applied Mathematics, Derio, Spain)Control of Partial Differential EquationsCCetraro, Italy 2010, Editors: Piermarco Cannarsa, Jean-Michel CoronLecture Notes in Mathematics+XIII, 344 p. 66 illus., 49 illus. in color.SCM12155Partial Differential EquationsPBKJSCM13070Systems Theory, ControlGPFCContributed volume1 On some recent advances on stabilization for hyperbolic equations.- 2 Notes on the Control of the Liouville Equation.- 3 Some questions of control in uid mechanics.- 4 Carleman estimates and some applications to control theory.- 5 The Wave Equation: Control and Numerics`The term control theory refers to the body of results - theoretical, numerical and algorithmic - which have been developed to influence the evolution of the state of a given system in order to meet a prescribed performance criterion. Systems of interest to control theory may be of very different natures. This monograph is concerned with models that can be described by partial differential equations of evolution. It contains five major contributions and is connected to the CIME Course on Control of Partial Differential Equations that took place in Cetraro (CS, Italy), July 19 - 23, 2010. Specifically, it covers the stabilization of evolution equations, control of the Liouville equation, control in fluid mechanics, control and numerics for the wave equation, and Carleman estimates for elliptic and parabolic equations with application to control. We are confident this work will provide an authoritative reference work for all scientists who are interested in this field, representing at the same time a friendly introduction to, and an updated account of, some of the most active trends in current research.LSelf-contained: the book can be used by researchers in partial differential equations who want to learn about the control of such equations without requiring a special background in control theory.

Covers the main recent progress in control of partial differential equations.

Includes numerous challenging open problems

The approach taken is the unifying viewpoint of basic sequences

978-1-4419-2099-7978-1-4020-2186-2Alekseev V.B. Alekseev(Abel s Theorem in Problems and Solutions.Based on the lectures of Professor V.I. ArnoldXIV, 269 p.SCM11078 Group Theory and GeneralizationsPBGFrom the contents: Preface for the English edition; V.I. Arnold.- Preface.- Introduction.- 1: Groups.- 2: The complex numbers.- 3: Hints, Solutions and Answers.- Appendix. Solvability of equations by explicit formulae; A. Khovanskii.- Bibliography.- Appendix; V.I. Arnold.- Index. Do formulas exist for the solution to algebraical equations in one variable of any degree like the formulas for quadratic equations? The main aim of this book is to give new geometrical proof of Abel's theorem, as proposed by Professor V.I. Arnold. The theorem states that for general algebraical equations of a degree higher than 4, there are no formulas representing roots of these equations in terms of coefficients with only arithmetic operations and radicals. A secondary, and more important aim of this book, is to acquaint the reader with two very important branches of modern mathematics: group theory and theory of functions of a complex variable. This book also has the added bonus of an extensive appendix devoted to the differential Galois theory, written by Professor A.G. Khovanskii. As this text has been written assuming no specialist prior knowledge and is composed of definitions, examples, problems and solutions, it is suitable for self-study or teaching students of mathematics, from high school to graduate. 978-90-481-6609-1978-0-387-87822-5AlinhacLSerge Alinhac, Universit Paris-Sud XI Dpt. Mathmatiques, Orsay CX, France)Hyperbolic Partial Differential EquationsUniversitextVector Fields and Integral Curves.- Operators and Systems in the Plane.- Nonlinear First Order Equations.- Conservation Laws in One-Space Dimension.- The Wave Equation.- Energy Inequalities for the Wave Equation.- Variable Coefficient Wave Equations and Systems.This excellent introduction to hyperbolic differential equations is devoted to linear equations and symmetric systems, as well as conservation laws. The book is divided into two parts. The first, which is intuitive and easy to visualize, includes all aspects of the theory involving vector fields and integral curves; the second describes the wave equation and its perturbations for two- or three-space dimensions. Over 100 exercises are included, as well as do it yourself instructions for the proofs of many theorems. Only an understanding of differential calculus is required. Notes at the end of the self-contained chapters, as well as references at the end of the book, enable ease-of-use for both the student and the independent researcher.<P>Contains over 100 exercises</P> <P> Do it yourself instructions included for theorems</P> <P>An elementary approach to partial differential equations presented, with minimal prerequisites </P> <P>Self-contained chapters, split into two main parts</P>978-0-387-95298-7AllaireGregoire Allaire/Shape Optimization by the Homogenization MethodApplied Mathematical SciencesXV, 456 p. 54 illus.SCP21018 MechanicsPHD1 Homogenization.- 1.1 Introduction to Periodic Homogenization.- 1.2 Definition of H-convergence.- 1.3 Proofs and Further Results.- 1.4 Generalization to the Elasticity System.- 2 The Mathematical Modeling of Composite Materials.- 2.1 Homogenized Properties of Composite Materials.- 2.2 Conductivity.- 2.3 Elasticity.- 3 Optimal Design in Conductivity.- 3.1 Setting of Optimal Shape Design.- 3.2 Relaxation by the Homogenization Method.- 4 Optimal Design in Elasticity.- 4.1 Two-phase Optimal Design.- 4.2 Shape Optimization.- 5 Numerical Algorithms.- 5.1 Algorithms for Optimal Design in Conductivity.- 5.2 Algorithms for Structural Optimization.=This book provides an introduction to the theory and numerical developments of the homogenization method. Its main features are: a comprehensive presentation of homogenization theory; an introduction to the theory of two-phase composite materials;a detailed treatment of structural optimization by using homogenization; a complete discussion of the resulting numerical algorithms with many documented test problems. It will be of interest to researchers, engineers, and advanced graduate students in applied mathematics, mechanical engineering, and structural optimization.978-1-4419-2942-6978-3-0348-0077-8AlpayiDaniel Alpay, Ben-Gurion University of the Negev Dept. Mathematics & Computer Science, Beer Sheva, IsraelA Complex Analysis Problem BookX, 526p.Springer BaselPrologue.- I Complex numbers.- 1 Complex numbers: algebra.- 2 Complex numbers: geometry.- 3 Complex numbers and analysis.- 4. Remarks and generalizations: quaternions, etc.- II Functions of a complex variable.- 5 C-differentiable functions.- 6 Cauchy's theorem.- 7 First applications.- 8 Laurent expansions and applications.- 9 Computations of definite integrals.- 10 Harmonic functions.- 11 Conformal mappings.-III Complements.- 12 Some useful theorems.- 13 Some topology.- References.- Index.VThis is a collection of exercises in the theory of analytic functions, with completed and detailed solutions. We wish to introduce the student to applications andaspects of the theory of analytic functions not always touched upon in a first course. Using appropriate exercises we wish to show to the students some aspects of what lies beyond a first course in complex variables. We also discuss topics of interest for electrical engineering students (for instance, the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). Examples of important Hilbert spaces of analytic functions (in particular the Hardy space and the Fock space) are given. The book also includes apart where relevant facts from topology, functional analysis and Lebesgue integration are reviewed.Connections with electrical engineering and the theory of linear systems

A variety of non trivial and interesting examples are given as exercises (for instance the Bohr phenomenon,integral representations of certain analytic functions, Blaschke products, the Schur algorithm)

Examples using positive definite functions and reproducing kernel spaces are given in the exercises

978-3-7643-7472-3AmannHerbert Amann, Zrich, Switzerland; Joachim Escher, Leibniz Universitt Hannover Inst. Angewandte Mathematik, Hannover, GermanyAnalysis II< XII, 400p.Undergraduate textbookBirkhuser BaselPreface.- VI. Integral Calculus in One Variable - 1. Step Continuous Functions - 2. Continuous Extensions - 3. The Cauchy-Riemann Integral - 4. Properties of the Integral - 5. The Technology of Integration - 6. Sums and Integrals - 7. Fourier Series - 8. Improper Integrals - 9. The Gamma Function.- VII. Differential Calculus in Several Variables - 1. Continuous Linear Mappings - 2. Differentiability - 3. Calculation Rules - 4. Multilinear Mappings - 5. Higher Derivatives - 6. Nemytski Operators and Calculus of Variations - 7. Inverse Mappings - 8. Implicit Functions - 9. Manifolds - 10. Tangents and Normals.- VIII. Line Integrals - 1. Curves and Their Length - 2. Curves in Rn - 3. Pfaff Forms - 4. Line Integrals - 5. Holomorphic Functions - 6. Meromorphic Functions.- Bibliography.- Index.The second volume of this introduction into analysis deals with the integration theory of functions of one variable, the multidimensional differential calculus and the theory of curves and line integrals. The modern and clear development that started in Volume I (3-7643-7153-6) continues. In this way a sustainable basis will be created which allows to deal with interesting applications that sometimes go considerably beyond the material that is represented in traditional textbooks. This applies, for instance, to the exploration of Nemytskii operators which enable a transparent introduction into the calculus of variations and the derivation of the Euler-Lagrange equations. Another example is the presentation of the local theory of submanifolds of Rn.<P>Cauchy s integral theorems and the theory of holomorphic functions including the homological version of the residue theorem are derived as an application of the theory of line integrals</P> <P>In addition to the calculation of important definite integrals which appear in Mathematics and in Physics, theoretic properties of the Gamma function and Riemann s Zeta function are explored</P> <P>Numerous examples with varying degrees of difficulty and many informative figures</P>978-0-8176-8113-5 AmbrosettiAntonio Ambrosetti, SISSA Department of Mathematics, Trieste, Italy; David Arcoya lvarez, Universidad Granada Facultad de Ciencias, Granada, SpainFAn Introduction to Nonlinear Functional Analysis and Elliptic ProblemsCProgress in Nonlinear Differential Equations and Their ApplicationsXII, 199p. 12 illus..Birkhuser BostonNotation.- Preliminaries.- Some Fixed Point Theorems.- Local and Global Inversion Theorems.- Leray-Schauder Topological Degree.- An Outline of Critical Points.- Bifurcation Theory.- Elliptic Problems and Functional Analysis.- Problems with A Priori Bounds.- Asymptotically Linear Problems.- Asymmetric Nonlinearities.- Superlinear Problems.- Quasilinear Problems.- Stationary States of Evolution Equations.- Appendix A Sobolev Spaces.- Exercises.- Index.- Bibliography.PThis self-contained textbook provides the basic, abstracttoolsused innonlinear analysisand their applications to semilinear elliptic boundary value problems.By firstoutlining the advantages and disadvantages of each method, this comprehensive textdisplays how variousapproachescan easily beappliedto a range of model cases.An Introduction to Nonlinear Functional Analysis and Elliptic Problemsis divided into two parts: the first discusses keyresults such as the Banach contraction principle, a fixed point theorem for increasing operators, local and global inversion theory, Leray Schauder degree, critical point theory, and bifurcation theory; the second part shows how these abstract results apply to Dirichlet elliptic boundary value problems. The exposition is driven by numerous prototype problems and exposes a variety of approaches tosolving them.Complete with a preliminary chapter, an appendix that includes further results on weak derivatives, and chapter-by-chapter exercises, this book is apractical text for an introductory course or seminar on nonlinear functional analysis.Provides the basic, abstracttoolsused innonlinear analysis

Keyresults such as the Banach contraction principle, a fixed point theorem for increasing operators, local and global inversion theory, Leray--Schauder degree, critical point theory, and bifurcation theory

Outlines a variety of approaches and displays how they can easily beappliedto a range of model cases

Clear exposition driven by numerous prototype problems

An extensive appendix that includes further results on weak derivatives

978-3-642-32159-7AmbrosioLuigi Ambrosio, Scuola Normale Superiore Department of Mathematics, Pisa, Italy; Alberto Bressan, Penn State University State College, University Park, PA, USA; Dirk Helbing, ETH Zurich, Swiss Federal Institute of T Chair of Sociology, in particular of Mod, Zurich, Switzerland; Axel Klar, Technische Universitt Kaiserslautern, Kaiserslautern, Germany; Enrique Zuazua, BCAM Basque Center for Applied Mathema, Bilbao, Spain/Modelling and Optimisation of Flows on Networks>Cetraro, Italy 2009, Editors: Benedetto Piccoli, Michel Rascle+XIV, 497 p. 141 illus., 32 illus. in color.SCM140680Mathematical Modeling and Industrial MathematicsPBWH|A User s Guide to Optimal Transport.- Hyperbolic Conservation Laws: an Illustrated Tutorial.- Derivation of Non-Local Macroscopic Traffic Equations and Consistent Traffic Pressures from Microscopic Car-Following Models.- On the Controversy around Daganzo s Requiem for and Aw-Rascle s Resurrection of Second-Order Traffic Flow Models.- Theoretical vs. Empirical Classification and Prediction of Congested Traffic States.- Self-Organized Network Flows.- Operation Regim< es and Slower-is-Faster-Effect in the Control of Traffic Intersections.- Modeling and Optimization of Scalar Flows on Networks.- The Wave Equation: Control and NumericspIn recent years flows in networks have attracted the interest of many researchers from different areas, e.g. applied mathematicians, engineers, physicists, economists. The main reason for this ubiquity is the wide and diverse range of applications, such as vehicular traffic, supply chains, blood flow, irrigation channels, data networks and others. This book presents an extensive set of notes by world leaders on the main mathematical techniques used to address such problems, together with investigations into specific applications. The main focus is on partial differential equations in networks, but ordinary differential equations and optimal transport are also included. Moreover, the modeling is completed by analysis, numerics, control and optimization of flows in networks. The book will be a valuable resource for every researcher or student interested in the subject.Rich notes on most recent advances on traffic flow on networks, including control and optimization. Tutorials on conservation laws,,wave equations and optimal transport theory Diverse applications such as vehicular traffic, supply chains and others978-0-8176-3517-6 AndreescuTitu Andreescu, University of Texas at Dallas Natural Sciences and Mathematics, Richardson, TX, USA; Oleg Mushkarov, Bulgarian Academy of Sciences Inst. Mathematics, Sofia, Bulgaria; Luchezar Stoyanov, University of Western Australia Perth Dept. Mathematics, Nedlands, WA, Australia'Geometric Problems on Maxima and MinimaX, 264 p., 262 illus.SCM21006GeometryPBMSCM26008OptimizationPBUMethods for Finding Geometric Extrema.- Selected Types of Geometric Extremum Problems.- Miscellaneous.- Hints and Solutions to the Exercises.jQuestions of maxima and minima have great practical significance, with applications to physics, engineering, and economics; they have also given rise to theoretical advances, notably in calculus and optimization. Indeed, while most texts view the study of extrema within the context of calculus, this carefully constructed problem book takes a uniquely intuitive approach to the subject: it presents hundreds of extreme value problems, examples, and solutions primarily through Euclidean geometry. Written by a team of established mathematicians and professors, this work draws on the authors experience in the classroom and as Olympiad coaches. By exposing readers to a wealth of creative problem-solving approaches, the text communicates not only geometry but also algebra, calculus, and topology. Ideal for use at the junior and senior undergraduate level, as well as in enrichment programs and Olympiad training for advanced high school students, this book s breadth and depth will appeal to a wide audience, from secondary school teachers and pupils to graduate students, professional mathematicians, and puzzle enthusiasts.Presents hundreds of extreme value problems, examples, and solutions primarily through Euclidean geometry

Unified approach to the subject, with emphasis on geometric, algebraic, analytic, and combinatorial reasoning

Applications to physics, engineering, and economics

Ideal for use at the junior and senior undergraduate level, with wide appeal to students, teachers, professional mathematicians, and puzzle enthusiasts

978-0-8176-3245-8Titu Andreescu, University of Texas at Dallas Natural Sciences and Mathematics, Richardson, TX, USA; Dorin Andrica, Babe_-Bolyai University Faculty of Mathematics and Computer Scie, Cluj-Napoca, Romania Number Theory"Structures, Examples, and ProblemsSCM25001PBHSCM00009Mathematics, generalPBFundamentals.- Divisibility.- Powers of Integers.- Floor Function and Fractional Part.- Digits of Numbers.- Basic Principles in Number Theory.- Arithmetic Functions.- More on Divisibility.- Diophantine Equations.- Some Special Problems in Number Theory.- Problems Involving Binomial Coefficients.- Miscellaneous Problems.- Solutions to Additional Problems.- Divisibility.- Powers of Integers.- Floor Function and Fractional Part.- Digits of Numbers.- Basic Principles in Number Theory.- Arithmetic Functions.- More on Divisibility.- Diophantine Equations.- Some Special Problems in Number Theory.- Problems Involving Binomial Coefficients.- Miscellaneous Problems.lNumber theory, an ongoing rich area of mathematical exploration, is noted for its theoretical depth, with connections and applications to other fields from representation theory, to physics, cryptography, and more. While the forefront of number theory is replete with sophisticated and famous open problems, at its foundation are basic, elementary ideas that can stimulate and challenge beginning students. This lively introductory text focuses on a problem-solving approach to the subject. Key features of Number Theory: Structures, Examples, and Problems: * A rigorous exposition starts with the natural numbers and the basics. * Important concepts are presented with an example, which may also emphasize an application. The exposition moves systematically and intuitively to uncover deeper properties. * Topics include divisibility, unique factorization, modular arithmetic and the Chinese Remainder Theorem, Diophantine equations, quadratic residues, binomial coefficients, Fermat and Mersenne primes and other special numbers, and special sequences. Sections on mathematical induction and the pigeonhole principle, as well as a discussion of other number systems are covered. * Unique exercises reinforce and motivate the reader, with selected solutions to some of the problems. * Glossary, bibliography, and comprehensive index round out the text. Written by distinguished research mathematicians and renowned teachers, this text is a clear, accessible introduction to the subject and a source of fascinating problems and puzzles, from advanced high school students to undergraduates, their instructors, and general readers at all levels.Approaches number theory from a problem-solving standpoint, presenting each concept in the framework of an example or problem

The text progresses incrementally from simpler to more complex principles

Covers modular arithmetic and the Chinese Remainder Theorem, Diophantine equations, binomial coefficients, Fermat and Mersenne primes and other special numbers, special sequences, and problems of density

Includes sections on mathematical induction and the pigeonhole principle, and discusses other number systems

Unconventional, essay-type, non-routine examples, exercises and problems, presented in an original fasion

978-0-387-25529-3AndrewsGeorge E. Andrews, The Pennsylvania State University, University Park, PA, < USA; Bruce C. Berndt, University of Illinois, Urbana, IL, USARamanujan's Lost NotebookPart IXIV, 437 p.SCM11019Algebraic GeometryPBMWSCM1218XSequences, Series, SummabilityJInroduction.- Rogers-Ramanujan Continued Fraction and Its Modular Properties.- Explicit Evaluations of the Rogers-Ramanujan Continued Fraction.- A Fragment on the Rogers-Ramanujan and Cubic Continued Fractions.- The Rogers-Ramanujan Continued Fraction and Its Connections with Partitions and Lambert Series.- Finite Rogers-Ramanujan Continued Fractions.- Other q-continued Fractions.- Asymptotic Formulas for Continued Fractions.- Ramanujan s Continued Fraction for (q2;q3)?/(q;q3)?.- The Rogers-Fine Identity.- An Empirical Study of the Rogers-Ramanujan Identities.- Rogers-Ramanujan-Slater Type Identities.- Partial Fractions.- Hadamard Products for Two q-Series.- Integrals of Theta Functions.- Incomplete Elliptic Integrals.- Infinite Integrals of q-Products.- Modular Equations in Ramanujan s Lost Notebook.- Fragments on Lambert Series.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. The 'lost notebook' contains considerable material on mock theta functions and so undoubtedly emanates from the last year of Ramanujan's life. It should be emphasized that the material on mock theta functions is perhaps Ramanujan's deepest work. Mathematicians are probably several decades away from a complete understanding of those functions. More than half of the material in the book is on q-series, including mock theta functions; the remaining part deals with theta function identities, modular equations, incomplete elliptic integrals of the first kind and other integrals of theta functions, Eisenstein series, particular values of theta functions, the Rogers-Ramanujan continued fraction, other q-continued fractions, other integrals, and parts of Hecke's theory of modular forms.Most of this material has never before been published in book form

Includes letters to G.H. Hardy

Authors have organized, and provided commentary on, Ramanujan's results

978-1-4419-2062-1978-3-642-30897-0AnnabyMahmoud H. Annaby, Cairo University Faculty of Science, Giza, Egypt; Zeinab S. Mansour, King Saud University Faculty of Science, Riyadh, Saudi Arabia#q-Fractional Calculus and EquationsXIX, 318 p. 6 illus.SCM12031#Difference and Functional Equations21 Preliminaries.- 2 q-Difference Equations.- 3 q-Sturm Liouville Problems.- 4 Riemann Liouville q-Fractional Calculi.- 5 Other q-Fractional Calculi.- 6 Fractional q-Leibniz Rule and Applications.- 7 q-Mittag Leffler Functions.- 8 Fractional q-Difference Equations.- 9 Applications of q-Integral Transforms.This nine-chapter monograph introduces a rigorous investigationof q-difference operators in standard and fractional settings. It starts with elementary calculus of q-differences and integration of Jackson s type before turning to q-difference equations. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with retarded arguments. Regular q-Sturm Liouville theory is also introduced; Green s function is constructed and the eigenfunction expansion theorem is given. The monograph also discusses some integral equations of Volterra and Abel type, as introductory material for the study of fractional q-calculi. Hence fractional q-calculi of the types Riemann Liouville; Grnwald Letnikov; Caputo; Erdlyi Kober and Weyl aredefined analytically. Fractional q-Leibniz rules with applications in q-series are also obtained with rigorous proofs of the formal results of Al-Salam-Verma, which remained unproved for decades. In working towards the investigation of q-fractional difference equations; families of q-Mittag-Leffler functions are defined and their properties are investigated, especially the q-Mellin Barnes integral and Hankel contour integral representation of the q-Mittag-Leffler functions under consideration, the distribution, asymptotic and reality of their zeros, establishing q-counterparts of Wiman s results. Fractional q-difference equations are studied; existence and uniqueness theorems are given and classes of Cauchy-type problems are completely solved in terms of families of q-Mittag-Leffler functions. Among many q-analogs of classical results and concepts, q-Laplace, q-Mellin and q2-Fourier transforms are studied and their applications are investigated.3First detailed rigorous study of q-calculiFirst detailed rigorous study of q-difference equations

First detailed rigorous study of q-fractional calculi and equations

Proofs of many classical unproved results are given

Illustrative examples and figures helps readers to digest the new approaches978-1-4614-1808-5Arapura8Donu Arapura, Purdue University, West Lafayette, IN, USA+Algebraic Geometry over the Complex Numbers(XII, 329p. 17 illus< ., 1 illus. in color.SCM12198-Several Complex Variables and Analytic SpacesPreface.- 1. Plane Curves.- 2. Manifolds and Varieties via Sheaves.- 3. More Sheaf Theory.- 4. Sheaf Cohomology.- 5. de Rham Cohomoloy of Manifolds.- 6. Riemann Surfaces.- 7. Simplicial Methods.- 8. The Hodge Theorem for Riemann Manifolds.- 9. Toward Hodge Theory for Complex Manifolds.- 10. Kahler Manifolds.- 11. A Little Algebraic Surface Theory.- 12. Hodge Structures and Homological Methods.- 13. Topology of Families.- 14. The Hard Lefschez Theorem.- 15. Coherent Sheaves.- 16. Computation of Coherent Sheaves.- 17. Computation of some Hodge numbers.- 18. Deformation Invariance of Hodge Numbers.- 19. Analogies and Conjectures.-References.- Index.This is a relatively fast paced graduate level introduction to complex algebraic geometry, from the basics to the frontier of the subject. It covers sheaf theory, cohomology, some Hodge theory, as well as some of the more algebraic aspects of algebraic geometry. The author frequently refers the reader if the treatment of a certain topic is readily available elsewhere but goes into considerable detail on topics for which his treatment puts a twist or a more transparent viewpoint. His cases of exploration and are chosen very carefully and deliberately. The textbook achieves its purpose of taking new students of complex algebraic geometry through this a deep yet broad introduction to a vast subject, eventually bringing them to the forefront of the topic via a non-intimidating style.$

Contains a rapid introduction to complex algebraic geometry Includes background material on topology, manifold theory and sheaf theory

Analytic and algebraic approaches are developed somewhat in parallel

Easy-going style will not intimidate newcomers to algebraic geometry

978-0-8176-8339-9ArnoldV.I. Arnold, Russian Academy of Sciences, Moscow, Russia; S.M. Gusein-Zade, Moscow State University, Moscow, Russia; Alexander N. Varchenko, University of North Carolina Department of Mathematics, Chapel Hill, NC, USA.Singularities of Differentiable Maps, Volume 1;Classification of Critical Points, Caustics and Wave FrontsModern Birkhuser ClassicsXII, 282 p. 67 illus.( Part I. Basic concepts.- The simplest examples.- The classes Sigma^ I .- The quadratic differential of a map.- The local algebra of a map and the Weierstrass preparation theorem.- The local multiplicity of a holomorphic map.- Stability and infinitesimal stability.- The proof of the stability theorem.- Versal deformations.- The classification of stable germs by genotype.- Review of further results.- Part II. Critical points of smooth functions.- A start to the classification of critical points.- Quasihomogeneous and semiquasihomogeneous singularities.- The classification of quasihomogeneous functions.- Spectral sequences for the reduction to normal forms.- Lists of singularities.- The determinator of singularities.- Real, symmetric and boundary singularities.- Part III. Singularities of caustics and wave fronts.- Lagrangian singularities.- Generating families.- Legendrian singularities.- The classification of Lagrangian and Legendrian singularities.- The bifurcation of caustics and wave fronts.- References.- Further references.- Subject Index. Singularity theory is a far-reaching extension of maxima and minima investigations of differentiable functions, with implications for many different areas of mathematics, engineering (catastrophe theory and the theory of bifurcations), and science. The three parts of this first volume of a two-volume set deal with the stability problem for smooth mappings, critical points of smooth functions, and caustics and wave front singularities. The second volume describes the topological and algebro-geometrical aspects of the theory: monodromy, intersection forms, oscillatory integrals, asymptotics, and mixed Hodge structures of singularities. The first volume has been adapted for the needs of non-mathematicians, presupposing a limited mathematical background and beginning at an elementary level. With this foundation, the book's sophisticated development permits readers to explore more applications than previous books on singularities.9Affordable reprint of a classic monograph written by experts in the field

Provides a uniquely sophisticated investigation of the topics discussed

Useful for a wide range of applications across disciplines in fields such as differential equations, dynamical systems, optimal control, and optics

978-0-387-94947-5oVladimir I. Arnold, Russian Academy of Sciences Steklov Mathematical Institute, Moscow, Russia; Boris A. Khesin$Topological Methods in Hydrodynamics XV, 376 p.SCP21026Fluid- and AerodynamicsPHDFGroup and Hamiltonian Structures of Fluid Dynamics.- Topology of Steady Fluid Flows.- Topological Properties of Magnetic and Vorticity Fields.- Differential Geometry of Diffeomorphism Groups.- Kinematic Fast Dynamo Problems.- Dynamical Systems with Hydrodynamical Background.Topological hydrodynamics is a young branch of mathematics studying topological features of flows with complicated trajectories, as well as their applications to fluid motions. It is situated at the crossroad of hyrdodynamical stability theory, Riemannian and symplectic geometry, magnetohydrodynamics, theory of Lie algebras and Lie groups, knot theory, and dynamical systems. Applications of this approach include topological classification of steady fluid flows, descriptions of the Korteweg-de Vries equation as a geodesic flow, and results on Riemannian geometry of diffeomorphism groups, explaining, in particular, why longterm dynamical weather forecasts are not reliable. Topological Methods in Hydrodynamics is the first monograph to treat topological, group-theoretic, and geometric problems of ideal hydrodynamics and magnetohydrodynamics for a unified point of view. The necessary preliminary notions both in hydrodynamics and pure mathematics are described with plenty of examples and figures. The book is accessible to graduate students as well as to both pure and applied mathematicians working in the fields of hydrodynam< ics, Lie groups, dynamical systems and differential geometry.978-0-387-00211-8AsmussenSoeren AsmussenApplied Probability and Queues,Stochastic Modelling and Applied ProbabilityXII, 439 p.SCM27004+Probability Theory and Stochastic ProcessesPBTSCM26024'Operations Research, Management ScienceKJT Simple Markovian Models.- Markov Chains.- Markov Jump Processes.- Queueing Theory at the Markovian Level.- Queueing Networks and Insensitivity.- Some General Tools and Methods.- Renewal Theory.- Regenerative Processes.- Further Topics in Renewal Theory and Regenerative Processes.- Random Walks.- Lvy Processes, Reflection and Duality.- Special Models and Methods.- Steady-State Properties of of GI/G/1.- Markov Additive Models.- Many-Server Queues.- Exponential Change of Measure.- Dams, Inventories and Insurance Risk.uThis book serves as an introduction to queuing theory and provides a thorough treatment of tools like Markov processes, renewal theory, random walks, Levy processes, matrix-analytic methods and change of measure. It also treats in detail basic structures like GI/G/1 and GI/G/s queues, Markov-modulated models and queuing networks, and gives an introduction to areas such as storage, inventory, and insurance risk. Exercises are included and a survey of mathematical prerequisites is given in an appendix This much updated and expanded second edition of the 1987 original contains an extended treatment of queuing networks and matrix-analytic methods as well as additional topics like Poisson's equation, the fundamental matrix, insensitivity, rare events and extreme values for regenerative processes, Palm theory, rate conservation, Levy processes, reflection, Skorokhod problems, Loynes' lemma, Siegmund duality, light traffic, heavy tails, the Ross conjecture and ordering, and finite buffer problems. Students and researchers in statistics, probability theory, operations research, and industrial engineering will find this book useful.978-1-4419-1809-3978-3-540-64983-0YJean-Pierre Aubin, Universit Paris IX Rseau de Recherche Viabilit, Paris CX 16, FranceOptima and Equilibria%An Introduction to Nonlinear AnalysisXVII, 437 p.SCW12065Economic TheoryKCA1 Minimisation Problems: General Theorems.- 2 Convex Functions and Proximation, Projection and Separation Theorems.- 3 Conjugate Functions and Convex Minimisation Problems.- 4 Subdifferentials of Convex Functions.- 5 Marginal Properties of Solutions of Convex Minimisation Problems.- 6 Generalised Gradients of Locally Lipschitz Functions.- 7 Two-person Games. Fundamental Concepts and Examples.- 8 Two-person Zero-sum Games: Theorems of Von Neumann and Ky Fan.- 9 Solution of Nonlinear Equations and Inclusions.- 10 Introduction to the Theory of Economic Equilibrium.- 11 The Von Neumann Growth Model.- 12 n-person Games.- 13 Cooperative Games and Fuzzy Games.- 14 Exercises.- 15 Statements of Problems.- 16 Solutions to Problems.- 17 Compendium of Results.- References.kProgress in the theory of economic equilibria and in game theory has proceeded hand in hand with that of the mathematical tools used in the field, namely nonlinear analysis and, in particular, convex analysis. Jean-Pierre Aubin, one of the leading specialists in nonlinear analysis and its application to economics, has written a rigorous and concise - yet still elementary and self-contained - textbook providing the mathematical tools needed to study optima and equilibria, as solutions to problems, arising in economics, management sciences, operations research, cooperative and non-cooperative games, fuzzy games etc. It begins with the foundations of optimization theory, and mathematical programming, and in particular convex and nonsmooth analysis. Nonlinear analysis is then presented, first game-theoretically, then in the framework of set valued analysis. These results are then applied to the main classes of economic equilibria. The book contains numerous exercises and problems: the latter allow the reader to venture into areas of nonlinear analysis that lie beyond the scope of the book and of most graduate courses.978-3-642-08446-1978-3-540-60752-6Aubin Thierry Aubin.Some Nonlinear Problems in Riemannian Geometry"Springer Monographs in MathematicsXVII, 397 p.SCM21022Differential GeometryPBMP1 Riemannian Geometry.- 2 Sobolev Spaces.- 3 Background Material.- 4 Complementary Material.- 5 The Yamabe Problem.- 6 Prescribed Scalar Curvature.- 7 Einstein Khler Metrics.- 8 Monge Ampre Equations.- 9 The Ricci Curvature.- 10 Harmonic Maps.- Bibliography*.- Notation.During the last few years, the field of nonlinear problems has undergone great development. This book consisting of the updated Grundlehren volume 252 by the author and of a newly written part, deals with some important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved. Each problem is explained, up-to-date results are given and proofs are presented. Thus, the reader is given access, for each specific problem, to its present status of solution as well as to the most up-to-date methods for approaching it. The main objective of the book is to explain some methods and new techniques, and to apply them. It deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber.978-3-642-08236-8978-3-642-17853-5AudinlMichle Audin, Universit de Strasbourg et CNRS Inst. Recherche Mathmatiques Avance, Strasbourg CX, FranceFatou, Julia, Montel<The Great Prize of Mathematical Sciences of 1918, and BeyondVIII, 332 p.SCM23009 History of Mathematical SciencesPBXSCM1204X$Dynamical Systems and Ergodic TheoryPBWRI The Great Prize, the framework.- I.1 The iteration problem in 1915.- I.2 The protagonists around 1917 1918.- I.3 The war.-I.4 Iteration, a few definitions and notation.- I.5 Normal families.- I.6 Relation to functional equations.- II The Great Prize of Mathematical.- II.1 Year 1917.- II.2 Year 1918.- III The memoirs.- III.1 Julia s memoir.- III.2 The (three) memoir(s) of Fatou.- III.3 Comments (in the first person).- III.4 To summarise.- IV After Fatou and Julia.- IV.1 Stop.- IV.2 Hausdorff distance (1914) and dimension (1919).- IV.3 Irregular points, J-points, O-points (1925 1927).- IV.4 The centre problem (1927 1942).-IV.5 Holomorphic dynamics.-V On Pierre Fatou.-V.1 Childhood and youth of Fatou.- V.2 What do we know of Pierre Fatou?.- V.3 Continuation of Fatou s career.- V.4 Fatou s thesis.- V.5 Fatou as a mathematician.-V.6 Fatou as an astronomer.- V.7 Teaching and candidatures of Fatou.- V.8 Fatou and other mathematicians.- V.9 Death of Fatou.- VI A controversy in 1965.- VI.1 The protagonists, from 1918 to 1965.- VI.2 Relations between Julia and Montel< , in the 1930 s.- VI.3 The third centenary of the Institut de France VI.4.- As a conclusion: O for a biography of Gaston Julia.- References.-Index.How did Pierre Fatou and Gaston Julia create what we now call Complex Dynamics, in the context of the early twentieth century and especially of the First World War? The book is based partly on new, unpublished sources. Who were Pierre Fatou, Gaston Julia, Paul Montel? New biographical information is given on the little known mathematician that was Pierre Fatou. How did the WW1 injury of Julia influence mathematical life in France? From the reviews of the French version: 'Audin s book is & filled with marvelous biographical information and analysis, dealing not just with the men mentioned in the book s title but a large number of other players, too & [It] addresses itself to scholars for whom the history of mathematics has a particular resonance and especially to mathematicians active, or even with merely an interest, in complex dynamics. & presents it all to the reader in a very appealing form.' (Michael Berg, The Mathematical Association of America, October 2009)..978-3-642-32156-6Bachar Mostafa Bachar, King Saud University College of Sciences, Riyadh, Saudi Arabia; Jerry J. Batzel, University of Graz Mathematics and Scientific Computing, Graz, Austria; Susanne Ditlevsen, University of Copenhagen Department of Mathematical Sciences, Copenhagen, Denmark!Stochastic Biomathematical Models&with Applications to Neuronal Modeling*XVI, 206 p. 34 illus., 13 illus. in color.1 Introduction to stochastic models in biology.- 2 One-dimensional homogeneous diffusions.- 3 A brief introduction to large deviations theory.- 4 Some numerical methods for rare events simulation and analysis.- 5 Stochastic Integrate and Fire models: a review on mathematical methods and their applications.- 6 Stochastic partial differential equations in Neurobiology: linear and nonlinear models for spiking neurons.- 7 Deterministic and stochastic FitzHugh-Nagumo systems.- 8 Stochastic modeling of spreading cortical depressionStochastic biomathematical models are becoming increasingly important as new light is shed on the role of noise in living systems. In certain biological systems, stochastic effects may even enhance a signal, thus providing a biological motivation for the noise observed in living systems. Recent advances in stochastic analysis and increasing computing power facilitate the analysis of more biophysically realistic models, and this book provides researchers in computational neuroscience and stochastic systems with an overview of recent developments. Key concepts are developed in chapters written by experts in their respective fields. Topics include: one-dimensional homogeneous diffusions and their boundary behavior, large deviation theory and its application in stochastic neurobiological models, a review of mathematical methods for stochastic neuronal integrate-and-fire models, stochastic partial differential equation models in neurobiology, and stochastic modeling of spreading cortical depression.Written by current leading experts in the field Focus on interdisciplinary (physiological and biological) applications of stochastic methods Representation of key theoretical ideas but also clear and motivated examples of application and implementation issues978-0-387-98668-5BadescuLucian Silvestru BadescuAlgebraic Surfaces XI, 259 p.l1 Cohomological Intersection Theory and the Nakai-Moishezon Criterion of Ampleness.- 2 The Hodge Index Theorem and the Structure of the Intersection Matrix of a Fiber.- 3 Criteria of Contractability and Rational Singularities.- 4 Properties of Rational Singularities.- 5 Noether s Formula, the Picard Scheme, the Albanese Variety, and Plurigenera.- 6 Existence of Minimal Models.- 7 Morphisms from a Surface to a Curve. Elliptic and Quasielliptic Fibrations.- 8 Canonical Dimension of an Elliptic or Quasielliptic Fibration.- 9 The Classification Theorem According to Canonical Dimension.- 10 Surfaces with Canonical Dimension Zero (char(k) ? 2, 3).- 11 Ruled Surfaces. The Noether-Tsen Criterion.- 12 Minimal Models of Ruled Surfaces.- 13 Characterization of Ruled and Rational Surfaces.- 14 Zariski Decomposition and Applications.- 15 Appendix: Further Reading.- References.This book presents fundamentals from the theory of algebraic surfaces, including areas such as rational singularities of surfaces and their relation with Grothendieck duality theory, numerical criteria for contractibility of curves on an algebraic surface, and the problem of minimal models of surfaces. In fact, the classification of surfaces is the main scope of this book and the author presents the approach developed by Mumford and Bombieri. Chapters also cover the Zariski decomposition of effective divisors and graded algebras.978-1-4419-3149-8978-0-387-76895-3BainmAlan Bain, BNP Paribas, London, UK; Dan Crisan, Imperial College London Department of Mathematics, London, UK$Fundamentals of Stochastic FilteringXIII, 390 p.SCT19000Control, Robotics, MechatronicsTJFMProfessional bookFiltering Theory.- The Stochastic Process ?.- The Filtering Equations.- Uniqueness of the Solution to the Zakai and the Kushner Stratonovich Equations.- The Robust Representation Formula.- Finite-Dimensional Filters.- The Density of the Conditional Distribution of the Signal.- Numerical Algorithms.- Numerical Methods for Solving the Filtering Problem.- A Continuous Time Particle Filter.- Particle Filters in Discrete Time.The objective of stochastic filtering is to determine the best estimate for the state of a stochastic dynamical system from partial observations. The solution of this problem in the linear case is the well k< nown Kalman-Bucy filter which has found widespread practical application. The purpose of this book is to provide a rigorous mathematical treatment of the non-linear stochastic filtering problem using modern methods. Particular emphasis is placed on the theoretical analysis of numerical methods for the solution of the filtering problem via particle methods. The book should provide sufficient background to enable study of the recent literature. While no prior knowledge of stochastic filtering is required, readers are assumed to be familiar with measure theory, probability theory and the basics of stochastic processes. Most of the technical results that are required are stated and proved in the appendices. The book is intended as a reference for graduate students and researchers interested in the field. It is also suitable for use as a text for a graduate level course on stochastic filtering (suitable exercises and solutions are included).The authors are an authority in the stochastic filtering field

An assortment of Measure Theory, Probability Theory and Stochastic Analysis results are included in order to make this book as self contained as possible

Exercises and solutions included throughout

Professionals978-1-4419-2642-5978-0-387-97586-3UThomas Banchoff, Brown University Dept. Mathematics, Providence, RI, USA; John WermerLinear Algebra Through Geometry"Undergraduate Texts in MathematicsXII, 305 p. 92 illus.SCM11094.Linear and Multilinear Algebras, Matrix TheoryPBF1.0 Vectors in the Line.- 2.0 The Geometry of Vectors in the Plane.- 2.1 Transformations of the Plane.- 2.2 Linear Transformations and Matrices.- 2.3 Sums and Products of Linear Transformations.- 2.4 Inverses and Systems of Equations.- 2.5 Determinants.- 2.6 Eigenvalues.- 2.7 Classification of Conic Sections.- 3.0 Vector Geometry in 3-Space.- 3.1 Transformations of 3-Space.- 3.2 Linear Transformations and Matrices.- 3.3 Sums and Products of Linear Transformations.- 3.4 Inverses and Systems of Equations.- 3.5 Determinants.- 3.6 Eigenvalues.- 3.7 Symmetric Matrices.- 3.8 Classification of Quadric Surfaces.- 4.0 Vector Geometry in n-Space, n ? 4.- 4.1 Transformations of n-Space, n ? 4.- 4.2 Linear Transformations and Matrices.- 4.3 Homogeneous Systems of Equations in n-Space.- 4.4 Inhomogeneous Systems of Equations in n-Space.- 5.0 Vector Spaces.- 5.1 Bases and Dimensions.- 5.2 Existence and Uniqueness of Solutions.- 5.3 The Matrix Relative to a Given Basis.- 6.0 Vector Spaces with an Inner Product.- 6.1 Orthonormal Bases.- 6.2 Orthogonal Decomposition of a Vector Space.- 7.0 Symmetric Matrices in n Dimensions.- 7.1 Quadratic Forms in n Variables.- 8.0 Differential Systems.- 8.1 Least Squares Approximation.- 8.2 Curvature of Function Graphs.Linear Algebra Through Geometry introduces the concepts of linear algebra through the careful study of two and three-dimensional Euclidean geometry. This approach makes it possible to start with vectors, linear transformations, and matrices in the context of familiar plane geometry and to move directly to topics such as dot products, determinants, eigenvalues, and quadratic forms. The later chapters deal with n-dimensional Euclidean space and other finite-dimensional vector space. Topics include systems of linear equations in n variable, inner products, symmetric matrices, and quadratic forms. The final chapter treats application of linear algebra to differential systems, least square approximations and curvature of surfaces in three spaces. The only prerequisite for reading this book (with the exception of one section on systems of differential equations) are high school geometry, algebra, and introductory trigonometry.978-3-642-24973-0Bangert@Patrick Bangert, algorithmica technologies GmbH, Bremen, Germany$Optimization for Industrial Problems+XXII, 246 p. 64 illus., 30 illus. in color.Overview of Heuristic Optimization.- Statistical Analysis in Solution Space.- Project Management.- Pre-processing: Cleaning up Data.- Data Mining: Knowledge from Data.- Modeling: Neural Networks.- Optimization: Simulated Annealing.- The Human Aspect in Sustainable Change and Innovation.Industrial optimization lies on the crossroads between mathematics, computer science, engineering and management. This book presents these fields in interdependence as a conversation between theoretical aspects of mathematics and computer science and the mathematical field of optimization theory at a practical level. The 19 case studies that were conducted by the author in real enterprises in cooperation and co-authorship with some of the leading industrial enterprises, including RWE, Vattenfall, EDF, PetroChina, Vestolit, Sasol, and Hella, illustrate the results that may be reasonably expected from an optimization project in a commercial enterprise. The book is aimed at persons working in industrial facilities as managers or engineers; it is also suitable for university students and their professors as an illustration of how the academic material may be used in real life. It will not make its reader a mathematician but it will help its reader in improving his plant.6Presentation of modern methods to solve real and pressing industrial optimization problems in a practical way together with real-life examples

With examples usually hard to come by or not published at all

Presents new methods or new results about existing methods to information science

978-0-387-98948-8BaoD. Bao; S.-S. Chern; Z. Shen+An Introduction to Riemann-Finsler Geometry XX, 435 p.hOne Finsler Manifolds and Their Curvature.- 1 Finsler Manifolds and the Fundamentals of Minkowski Norms.- 2 The Chern Connection.- 3 Curvature and Schur s Lemma.- 4 Finsler Surfaces and a Generalized Gauss Bonnet Theorem.- Two Calculus of Variations and Comparison Theorems.- 5 Variations of Arc Length, Jacobi Fields, the Effect of Curvature.- 6 The Gauss Lemma and the Hopf-Rinow Theorem.- 7 The Index Form and the Bonnet-Myers Theorem.- 8 The Cut and Conjugate Loci, and Synge s Theorem.- 9 The Cartan-Hadamard Theorem and Rauch s First Theorem.- Three Special Finsler Spaces over the Reals.- 10 Berwald Spaces and Szab s Theorem for Berwald Surfaces.- 11 Randers Spaces and an Elegant Theorem.- 12 Constant Flag Curvature Spaces and Akbar-Zadeh s Theorem.- 13 Riemannian Manifolds and Two of Hopf s Theorems.- 14 Minkowski Spaces, the Theorems of Deicke and Brickell.In Riemannian geometry, measurements are made with both yardsticks and protractors. These tools are represented by a family of inner-products. In Riemann-Finsler geometry (or Finsler geometry for short), one is in principle equipped with only a family of Minkowski norms. So ardsticks are assigned but protractors are not. With such a limited tool kit, it is natural to wonder just how much geometry one can uncover and describe? It now appears that there is a reasonable answer. Finsler geometry encompasses a solid repertoire of rigidity and comparison theorems, most of them founded upon a fruitful analogue of the sectional curvature. There is also a bewildering array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. This book focuses on the elementary but essential items among these results. Much t< hought has gone into making the account a teachable one.978-0-387-95529-2BarbeauOEdward J. Barbeau, University of Toronto Dept. Mathematics, Toronto, ON, CanadaPell s EquationProblem Books in MathematicsIX, 212 p. 9 illus.Reference workQThe Square Root of 2.- Problems Leading to Pel?s Equation and Preliminary Investigations.- Quadratic Surds.- The Fundamental Solution.- Tracking Down the Fundamental Solution.- Pel?s Equation and Pythagorean Triples.- The Cubic Analogue of Pel?s Equation.- Analogues of the Fourth and Higher Degrees.- A Finite Version of Pel?s Equation.~Pell's equation is part of a central area of algebraic number theory that treats quadratic forms and the structure of the rings of integers in algebraic number fields. It is an ideal topic to lead college students, as well as some talented and motivated high school students, to a better appreciation of the power of mathematical technique. Even at the specific level of quadratic diophantine equations, there are unsolved problems, and the higher degree analogues of Pell's equation, particularly beyond the third, do not appear to have been well studied. In this focused exercise book, the topic is motivated and developed through sections of exercises which will allow the readers to recreate known theory and provide a focus for their algebraic practice. There are several explorations that encourage the reader to embark on their own research. A high school background in mathematics is all that is needed to get into this book, and teachers and others interested in mathematics who do not have (or have forgotten) a background in advanced mathematics may find that it is a suitable vehicle for keeping up an independent interest in the subject.978-1-4419-3040-8978-0-387-40627-5E.J. BarbeauPolynomialsXXII, 455 p. 36 illus.SCM11000Algebraz1 Fundamentals.- 1.1 The Anatomy of a Polynomial of a Single Variable.- 1.2 Quadratic Polynomials.- 1.3 Complex Numbers.- 1.4 Equations of Low Degree.- 1.5 Polynomials of Several Variables.- 1.6 Basic Number Theory and Modular Arithmetic.- 1.7 Rings and Fields.- 1.8 Problems on Quadratics.- 1.9 Other Problems.- Hints.- 2 Evaluation, Division, and Expansion.- 2.1 Horner s Method.- 2.2 Division of Polynomials.- 2.3 The Derivative.- 2.4 Graphing Polynomials.- 2.5 Problems.- Hints.- 3 Factors and Zeros.- 3.1 Irreducible Polynomials.- 3.2 Strategies for Factoring Polynomials over Z.- 3.3 Finding Integer and Rational Roots: Newton s Method of Divisors.- 3.4 Locating Integer Roots: Modular Arithmetic.- 3.5 Roots of Unity.- 3.6 Rational Functions.- 3.7 Problems on Factorization.- 3.8 Other Problems.- Hints.- 4 Equations.- 4.1 Simultaneous Equations in Two or Three Unknowns.- 4.2 Surd Equations.- 4.3 Solving Special Polynomial Equations.- 4.4 The Fundamental Theorem of Algebra: Intersecting Curves.- 4.5 The Fundamental Theorem: Functions of a Complex Variable.- 4.6 Consequences of the Fundamental Theorem.- 4.7 Problems on Equations in One Variable.- 4.8 Problems on Systems of Equations.- 4.9 Other Problems.- Hints.- 5 Approximation and Location of Zeros.- 5.1 Approximation of Roots.- 5.2 Tests for Real Zeros.- 5.3 Location of Complex Roots.- 5.4 Problems.- Hints.- 6 Symmetric Functions of the Zeros.- 6.1 Interpreting the Coefficients of a Polynomial.- 6.2 The Discriminant.- 6.3 Sums of the Powers of the Roots.- 6.4 Problems.- Hints.- 7 Approximations and Inequalities.- 7.1 Interpolation and Extrapolation.- 7.2 Approximation on an Interval.- 7.3 Inequalities.- 7.4 Problems on Inequalities.- 7.5 Other Problems.- Hints.- 8 Miscellaneous Problems.- E.62 Zeros of z?1[(1 + z)n ? 1 ? zn].- E.63 Two trigonometric products.- E.64 Polynomials all of whose derivatives have integer zeros.- E.65 Polynomials with equally spaced zeros.- E.66 Composition of polynomials of several variables.- E.67 The Mandelbrot set.- E.68 Sums of two squares.- E.69 Quaternions.- Hint.- Answers to Exercises and Solutions to Problems.- Notes on Explorations.- Further Reading.The book extends the high school curriculum and provides a backdrop for later study in calculus, modern algebra, numerical analysis, and complex variable theory. Exercises introduce many techniques and topics in the theory of equations, such as evolution and factorization of polynomials, solution of equations, interpolation, approximation, and congruences. The theory is not treated formally, but rather illustrated through examples. Over 300 problems drawn from journals, contests, and examinations test understanding, ingenuity, and skill. Each chapter ends with a list of hints; there are answers to many of the exercises and solutions to all of the problems. In addition, 69 'explorations' invite the reader to investigate research problems and related topics.978-3-642-32881-7BatzelJerry J. Batzel, University of Graz Mathematics and Scientific Computing, Graz, Austria; Mostafa Bachar, King Saud University College of Sciences, Riyadh, Saudi Arabia; Franz Kappel, University of Graz Institute for Mathematics, Graz, Austria2Mathematical Modeling and Validation in Physiology:Applications to the Cardiovascular and Respiratory Systems)XX, 254 p. 83 illus., 34 illus. in color.SCM31000&Mathematical and Computational BiologyPDESCB13004Human PhysiologyMFG1 Merging Mathematical and Physiological Knowledge: Dimensions and Challenges.- 2 Mathematical Modeling of Physiological Systems.- 3Parameter Selection Methods in Inverse Problem Formulation.-4 Application of the Unscented Kalman Filtering to Parameter Estimation.- 5 Integrative and Reductionist Approaches to Modeling of Control of Breathing.- 6 Parameter Identification in a Respiratory Control System Model with Delay.- 7 Experimental Studies of Respiration and Apnea.- 8 Model Validation and Control Issues in the Respiratory System.- 9 Experimental Studies of the Baroreflex.- 10 Development of Patient Specific Cardiovascular Models Predicting Dynamics in Response to Orthostatic Stress Challen< ges.- 11 Parameter Estimation of a Model for Baroreflex Control of Unstressed Volume.This volume synthesizes theoretical and practical aspects of both the mathematical and life science viewpoints needed for modeling of the cardiovascular-respiratory system specifically and physiological systems generally. Theoretical points include model design, model complexity and validation in the light of available data, as well as control theory approaches to feedback delay and Kalman filter applications to parameter identification. State of the art approaches using parameter sensitivity are discussed for enhancing model identifiability through joint analysis of model structure and data. Practical examples illustrate model development at various levels of complexity based on given physiological information. The sensitivity-based approaches for examining model identifiability are illustrated by means of specific modeling examples. The themes presented address the current problem of patient-specific model adaptation in the clinical setting, where data is typically limited.Focused study of modeling from model design to model identifiability and validation

Written by current leading experts in the field and including topics of current research interest in state of the art questions and methods

Focus on interdisciplinary (physiological and mathematical) collaboration and applications of modeling with clinical relevance

Presentation of key theoretical ideas and current areas of research interest through clear and motivated examples of application and implementation

978-3-642-18323-2BuerleuNicole Buerle, Karlsruhe Institute of Technology, Karlsruhe, Germany; Ulrich Rieder, University of Ulm, Ulm, Germany6Markov Decision Processes with Applications to FinanceXVI, 388p. 24 illus..SCM13062Quantitative FinanceKFFPreface.- 1.Introduction and First Examples.- Part I Finite Horizon Optimization Problems and Financial Markets.- 2.Theory of Finite Horizon Markov Decision Processes.- 3.The Financial Markets.- 4.Financial Optimization Problems.- Part II Partially Observable Markov Decision Problems.- 5.Partially Observable Markov Decision Processes.- 6.Partially Observable Markov Decision Problems in Finance.- Part III Infinite Horizon Optimization Problems.- 7.Theory of Infinite Horizon Markov Decision Processes.- 8.Piecewise Deterministic Markov Decision Processes.- 9.Optimization Problems in Finance and Insurance.- Part IV Stopping Problems.- 10.Theory of Optimal Stopping Problems.- 11.Stopping Problems in Finance.- Part V Appendix.- A.Tools from Analysis.- B.Tools from Probability.- C.Tools from Mathematical Finance.- References.- Index.zThe theory of Markov decision processes focuses on controlled Markov chains in discrete time. The authors establish the theory for general state and action spaces and at the same time show its application by means of numerous examples, mostly taken from the fields of finance and operations research. By using a structural approach many technicalities (concerning measure theory) are avoided. They cover problems with finite and infinite horizons, as well as partially observable Markov decision processes, piecewise deterministic Markov decision processes and stopping problems. The book presents Markov decision processes in action and includes various state-of-the-art applications with a particular view towards finance. It is useful for upper-level undergraduates, Master's students and researchers in both applied probability and finance, and provides exercises (without solutions).Contains various applications with a particular viewtowards finance/insurance

Avoids many technical (e.g. measure theoretic) problems

The collection of topics is unique

Approach is problem-oriented and illustrated by many examples

978-0-387-90925-7BergIC. van den Berg, Ruinerwold, Netherlands; J. P. R. Christensen; P. ResselHarmonic Analysis on Semigroups1Theory of Positive Definite and Related Functions X, 292 p.SCM11132Topological Groups, Lie Groups1 Introduction to Locally Convex Topological Vector Spaces and Dual Pairs.- 1. Locally Convex Vector Spaces.- 2. Hahn-Banach Theorems.- 3. Dual Pairs.- Notes and Remarks.- 2 Radon Measures and Integral Representations.- 1. Introduction to Radon Measures on Hausdorff Spaces.- 2. The Riesz Representation Theorem.- 3. Weak Convergence of Finite Radon Measures.- 4. Vague Convergence of Radon Measures on Locally Compact Spaces.- 5. Introduction to the Theory of Integral Representations.- Notes and Remarks.- 3 General Results on Positive and Negative Definite Matrices and Kernels.- 1. Definitions and Some Simple Properties of Positive and Negative Definite Kernels.- 2. Relations Between Positive and Negative Definite Kernels.- 3. Hubert Space Representation of Positive and Negative Definite Kernels.- Notes and Remarks.- 4 Main Results on Positive and Negative Definite Functions on Semigroups.- 1. Definitions and Simple Properties 86 2. Exponentially Bounded Positive Definite Functions on Abelian Semigroups.- 3. Negative Definite Functions on Abelian Semigroups.- 4. Examples of Positive and Negative Definite Functions.- 5. T-Positive Functions.- 6. Completely Monotone and Alternating Functions.- Notes and Remarks.- 5 Schoenberg-Type Results for Positive and Negative Definite Functions.- 1. Schoenberg Triples.- 2. Norm Dependent Positive Definite Functions on Banach Spaces.- 3. Functions Operating on Positive Definite Matrices.- 4. Schoenberg s Theorem for the Complex Hilbert Sphere.- 5. The Real Infinite Dimensional Hyperbolic Space.- Notes and Remarks.- 6 Positive Definite Functions and Moment Functions.- 1. Moment Functions.- 2. The One-Dimensional Moment Problem.- 3. The Multi-Dimensional Moment Problem.- 4. The Two-Sided Moment Problem.- 5. Perfect Semigroups.- Notes and Remarks.- 7 Hoeffding s Inequality and Multivariate Majorization.- 1. The Discrete Case.- 2. Extension to Nondiscrete Semigroups.- 3. Completely Negative Definite Functions and Schur-Monotonicity.- Notes and Remarks.- 8 P< ositive and Negative Definite Functions on Abelian Semigroups Without Zero.- 1. Quasibounded Positive and Negative Definite Functions.- 2. Completely Monotone and Completely Alternating Functions.- Notes and Remarks.- References.- List of Symbols.978-3-540-70996-1BergerXMarcel Berger, Institut des Hautes tudes Scientifiques (IHES), Bures-sur-Yvette, FranceGeometry Revealed*A Jacob's Ladder to Modern Higher GeometryXII, 860p. 666 illus..Points and lines in the plane.- Circles and spheres.- The sphere by itself: can we distribute points on it evenly?.- Conics and quadrics.- Plane curves.- Smooth surfaces.- Convexity and convex sets.- Polygons, polyhedra, polytopes.- Lattices, packings and tilings in the plane.- Lattices and packings in higher dimensions.- Geometry and dynamics I: billiards.- Geometry and dynamics II: geodesic flow on a surface.kBoth classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built 'above' the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended... In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called 'elementary' geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations. Marcel Berger is the author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.u

Newest work of best known mathematician Marcel Berger

Offers readers elements of a modern geometric culture

Demonstrate the unceasingly renewed spirit of geometry

Visually rich and inviting

Includes series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction

978-3-642-66453-3BerghJ. Bergh; Jorgen LfstrmInterpolation SpacesAn Introduction5 figs. X,207 pages. 1. Some Classical Theorems.- 1.1. The Riesz-Thorin Theorem.- 1.2. Applications of the Riesz-Thorin Theorem.- 1.3. The Marcinkiewicz Theorem.- 1.4. An Application of the Marcinkiewicz Theorem.- 1.5. Two Classical Approximation Results.- 1.6. Exercises.- 1.7. Notes and Comment.- 2. General Properties of Interpolation Spaces.- 2.1. Categories and Functors.- 2.2. Normed Vector Spaces.- 2.3. Couples of Spaces.- 2.4. Definition of Interpolation Spaces.- 2.5. The Aronszajn-Gagliardo Theorem.- 2.6. A Necessary Condition for Interpolation.- 2.7. A Duality Theorem.- 2.8. Exercises.- 2.9. Notes and Comment.- 3. The Real Interpolation Method.- 3.1. The K-Method.- 3.2. The J-Method.- 3.3. The Equivalence Theorem.- 3.4. Simple Properties of ??, q.- 3.5. The Reiteration Theorem.- 3.6. A Formula for the K-Functional.- 3.7. The Duality Theorem.- 3.8. A Compactness Theorem.- 3.9. An Extremal Property of the Real Method.- 3.10. Quasi-Normed Abelian Groups.- 3.11. The Real Interpolation Method for Quasi-Normed Abelian Groups.- 3.12. Some Other Equivalent Real Interpolation Methods.- 3.13. Exercises.- 3.14. Notes and Comment.- 4. The Complex Interpolation Method.- 4.1. Definition of the Complex Method.- 4.2. Simple Properties of ?[?].- 4.3. The Equivalence Theorem.- 4.4. Multilinear Interpolation.- 4.5. The Duality Theorem.- 4.6. The Reiteration Theorem.- 4.7. On the Connection with the Real Method.- 4.8. Exercises.- 4.9. Notes and Comment.- 5. Interpolation of Lp-Spaces.- 5.1. Interpolation of Lp-Spaces: the Complex Method.- 5.2. Interpolation of Lp-Spaces: the Real Method.- 5.3. Interpolation of Lorentz Spaces.- 5.4. Interpolation of Lp-Spaces with Change of Measure: p0 = p1.- 5.5. Interpolation of Lp-Spaces with Change of Measure: p0 ? p1.- 5.6. Interpolation of Lp-Spaces of Vector-Valued Sequences.- 5.7. Exercises.- 5.8. Notes and Comment.- 6. Interpolation of Sobolev and Besov Spaces.- 6.1. Fourier Multipliers.- 6.2. Definition of the Sobolev and Besov Spaces.- 6.3. The Homogeneous Sobolev and Besov Spaces.- 6.4. Interpolation of Sobolev and Besov Spaces.- 6.5. An Embedding Theorem.- 6.6. A Trace Theorem.- 6.7. Interpolation of Semi-Groups of Operators.- 6.8. Exercises.- 6.9. Notes and Comment.- 7. Applications to Approximation Theory.- 7.1. Approximation Spaces.- 7.2. Approximation of Functions.- 7.3. Approximation of Operators.- 7.4. Approximation by Difference Operators.- 7.5. Exercises.- 7.6. Notes and Comment.- References.- List of Symbols.978-0-8176-8387-0BernhardPierre Bernhard, INRIA Sophia Antipolis-Mediterranee, Sophia Antipolis cedex, France; Jacob C. Engwerda, Tilburg University School of Economics and Management, Tilburg, Netherlands; Berend Roorda, University of Twente, Enschede, Netherlands; J.M. Schumacher, Tilburg University, Tilburg, Netherlands; Vassili Kolokoltsov, University of Warwick, Warwick, UK; Patrick Saint-Pierre, Universit Paris Dauphine, Paris Cedex 16, France; Jean-Pierre Aubin, VIMADES, Paris, France1The Interval Market Model in Mathematical FinanceGame-Theoretic Methods8Static & Dynamic Game Theory: Foundations & ApplicationsXVI, 346 p. 64 illus.SCM130112Game Theory, Economics, Social and Behav. SciencesPBUDSCW12248 Game Theory/Mathematical MethodsPreface.- Part I Revisiting Two Classic Results in Dynamic Portfolio Management.- Merton s Optimal Dynamic Portfolio Revisited.- Option Pricing: Classic Results.- Introduction.- Part II Hedging in Interval Models.- Fair Price Intervals.- Optimal Hedging Under Robust-Cost Constraints.- Appendix: Proofs.- Continuous and Discrete-Time Option Pricing and Interval Market Model.- Part III Robust-Control Approach to Option Pricing.- Vanilla Options.- Digital Options.- Validation.- Introduction.- Part IV Game-Theoretic Analysis of Rainbow Options in Incomplete Markets.- Emergence of Risk-Neutral Probabilities.- Rainbow Options in Discrete Time, I.- Rainbow Options in Discrete Time, II.- Continuous-Time Limits.- Credit Derivatives.- Computational Methods Based on the Guaranteed Capture Basin Algorithm.- Viability Approach to Complex Option Pricing and Portfolio Insurance.- Asset and Liability Insurance Management (ALIM) for Risk Eradication.- References.- Index. < Toward the late 1990s, several research groups independently began developing new, related theories in mathematical finance.These theories didaway with the standard stochastic geometric diffusion Samuelson market model (also known as the Black-Scholes model because it is used in that most famous theory), instead opting for models that allowed minimax approachesto complement or replace stochastic methods.Among the most fruitful models were those utilizing game-theoretic tools and the so-called interval market model. Over time, these models have slowly but steadily gained influence in the financial community, providing a useful alternative to classical methods.A self-contained monograph, The Interval Market Model in Mathematical Finance: Game-Theoretic Methodsassembles some of the most important results, old and new, in this area of research. Written by seven of the most prominent pioneers of the interval market model and game-theoretic finance, the work provides a detailed account of several closely relatedmodeling techniquesfor an array of problems in mathematical economics. The book isdivided into five parts, which successively address topics including: probability-free Black-Scholes theory; fair-price interval of an option; representation formulas and fast algorithms for option pricing; rainbow options; tychastic approach of mathematical finance based upon viability theory.This book providesa welcome addition to the literature, complementing myriad titles on the market that take a classical approach to mathematical finance. Itis a worthwhile resource for researchers in applied mathematics and quantitative finance,and has also beenwritten in a manneraccessible to financially-inclined readers with a limited technical background.First book on the market to highlight the interval market model in mathematical finance

Combines several related paths of researchinto a single source, while providing numerous unpublished results

Presented in a manner accessible to readers specializing in both mathematics and finance

Includes many features toclarify concepts and facilitate referencing, such as figures, tables, biographicaldata, and subject, author, and notation indices

978-1-4899-8580-4978-3-540-88701-0 BernsteinDaniel J. Bernstein, University of Illinois, Chicago Dept. Computer Science, Chicago, IL, USA; Johannes Buchmann, TU Darmstadt FB Informatik, Darmstadt, Germany; Erik Dahmen, TU Darmstadt FB Informatik, Darmstadt, GermanyPost-Quantum CryptographySCI15033Data EncryptionURYto post-quantum cryptography.- Quantum computing.- Hash-based Digital Signature Schemes.- Code-based cryptography.- Lattice-based Cryptography.- Multivariate Public Key Cryptography.Quantum computers will break today's most popular public-key cryptographic systems, including RSA, DSA, and ECDSA. This book introduces the reader to the next generation of cryptographic algorithms, the systems that resist quantum-computer attacks: in particular, post-quantum public-key encryption systems and post-quantum public-key signature systems. Leading experts have joined forces for the first time to explain the state of the art in quantum computing, hash-based cryptography, code-based cryptography, lattice-based cryptography, and multivariate cryptography. Mathematical foundations and implementation issues are included. This book is an essential resource for students and researchers who want to contribute to the field of post-quantum cryptography.The contributors to the book take on the big challenge in cryptography, namely: what to do when someone will break the crypto-systems of today978-3-642-10019-2978-3-642-24078-2 BertoluzzaSilvia Bertoluzza, CNR Istituto di Matematica Applicata, Pavia, Italy; Ricardo H. Nochetto, University of Maryland Department of Mathematics, College Park, MD, USA; Alfio Quarteroni, cole Polytechnique Fdrale de Lausanne Chaire de Modelisation, Lausanne, Switzerland; Kunibert G. Siebert, Universitt Stuttgart Fakultt fr Mathematik und Physik, Stuttgart, Germany; Andreas Veeser, Universit degli Studi di Milano Dipartimento di Matematica, Milano, Italy>Multiscale and Adaptivity: Modeling, Numerics and Applications+C.I.M.E. Summer School, Cetraro, Italy 2009*XII, 314 p. 72 illus., 24 illus. in color.SCM14050Numerical AnalysisSCM14026%Computational Science and EngineeringAdaptiveWavelet Methods.- Heterogeneous Mathematical Models in Fluid Dynamics and Associated Solution Algorithms.- Primer of Adaptive Finite Element Methods.- Mathematically Founded Design of Adaptive Finite Element Software.This book is a collection of lecture notes for the CIME course on 'Multiscale and Adaptivity: Modeling, Numerics and Applications,' held in Cetraro (Italy), in July 2009. Complex systems arise in several physical, chemical, and biological processes, in which length and time scales may span several orders of magnitude. Traditionally, scientists have focused on methods that are particularly applicable in only one regime, and knowledge of the system on one scale has been transferred to another scale only indirectly. Even with modern computer power, the complexity of such systems precludes their being treated directly with traditional tools, and new mathematical and computational instruments have had to be developed to tackle such problems. The outstanding and internationally renowned lecturers, coming from different areas of Applied Mathematics, have themselves contributed in an essential way to the development of the theory and techniques that constituted the subjects of the courses.ZGives an overview of state-of-the-art mathematical tools in multiscale modeling

Contains a chapter on multiscale modeling and related numerical methods in fluid dynamics

Provides a detailed description of adaptive finite element methods, ranging from the mathematical theory to implementation details and software development

978-0-387-95331-1BetounesiDavid Betounes, University of Southern Mississippi Dept. Mathematics, Hattiesburg, MS, USA; Mylan RedfernMathematical Computing+An Introduction to Programming Using MapleBook w. online files/update%XII, 412 p. With online files/update.SCI14010Programming TechniquesUMz1 Preliminaries.- 1.< 1 Maple as a Programming Language.- 1.2 Analyzing Programming Tasks.- 1.3 Documentation and Coding.- 1.4 Maple/Calculus Notes.- 2 Basic Aspects of Maple.- 2.1 Variables and Constants.- 2.2 Expressions and Assignments.- 2.3 Notation in Mathematics and in Maple.- 2.4 Sequences, Lists, Sets, and Arrays.- 2.5 The Do Loop.- 2.6 Procedures: A First Glance.- 2.7 Evaluation Rules.- 2.8 Maple/Calculus Notes.- 3 Looping and Repetition.- 3.1 The Basic Loop.- 3.2 The Do Loop with All Its Features.- 3.3 Case Study: Iterated Maps.- 3.4 Maple/Calculus Notes.- 4 Conditionals Flow of Control.- 4.1 Logic in Mathematics.- 4.2 Relational and Logical Operators.- 4.3 Boolean Expressions.- 4.4 The if-then-else Statement.- 4.5 The if-then-elif-then Statement.- 4.6 Case Study: Riemann Sums for a Double Integral.- 4.7 Maple/Calculus Notes.- 5 Procedures.- 5.1 Maple s Procedure Statement.- 5.2 Procedures Some Details.- 5.3 Groups of Related Procedures.- 5.4 Case Study: Trig Integrals.- 5.5 Maple/Calculus Notes.- 6 Data Structures.- 6.1 Expressions and Operands.- 6.2 Quotes and Strings.- 6.3 Numbers.- 6.4 Lists: Vector Methods in Geometry.- 6.5 Arrays and Tables.- 6.6 Sets: The Cantor Set and Limiting Covers.- 6.7 Polynomials.- 6.8 Case Study: Partial Fractions.- 7 Graphics Programming.- 7.1 Preliminary Examples.- 7.2 Maple s Plot Structures.- 7.3 Approximating Curves and Surfaces.- 7.4 The GRID and MESH Objects.- 7.5 Animations.- 7.6 Maple/Calculus Notes.- 8 Recursion.- 8.1 Recurrence Relations Series Solutions.- 8.2 Reduction Formulas for Integration.- 8.3 Sorting.- 8.4 Numbers.- 8.5 Maple/Calculus Notes.- 9 Programming Projects.- 9.1 Projects on Crystal Growth.- 9.2 Projects on Inscribed Polygons.- 9.3 Projects on Random Walks.- 9.4 Projects on Newton s Second Law.- A Maple Reference.- A.1 Expressions and Functions.- A.2 Plotting and Visualization.- A.3 Programming.- A.4 Packages.This book is designed to teach introductory computer programming using Maple. It aims to infuse more mathematically oriented programming exercises and problems than those found in traditional programming courses while reinforcing and applying concepts and techniques of calculus. All the important, basic elements of computer programming can be easily learned within the interactive and user friendly environment of a Computer Algebra System (CAS) such as Maple. Most chapters feature case studies that provide greater depth on some topics and also serve to illustrate the methodology of analysis and design of code for more complex problems. This book is directed at undergraduates in the fields of math, science, or secondary education.978-3-540-44238-7BjornerAnders Bjorner, Royal Institute of Technology (KTH) Dept. Mathematics, Stockholm, Sweden; Francesco Brenti, Universit di Roma - Tor Vergata Dipto. Matematica, Roma, ItalyCombinatorics of Coxeter GroupsXIV, 370 p. 81 illus.I.- The basics.- Bruhat order.- Weak order and reduced words.- Roots, games, and automata.- II.- Kazhdan-Lusztig and R-polynomials.- Kazhdan-Lusztig representations.- Enumeration.- Combinatorial Descriptions.This book is a carefully written exposition of Coxeter groups, an area of mathematics which appears in algebra, geometry, and combinatorics. In this book, the combinatorics of Coxeter groups has mainly to do with reduced expressions, partial order of group elements, enumeration, associated graphs and combinatorial cell complexes, and connections with combinatorial representation theory. While Coxeter groups have already been exposited from algebraic and geometric perspectives, this book will be presenting the combinatorial aspects of Coxeter groups. The authors have included an exposition of Coxeter groups along with a rich variety of exercises, ranging from easy to very difficult, giving the book the unique character of serving as both a textbook and a monograph.Coxeter groups have already been exposited from algebraic and geometric perspectives, but this book will be presenting the combinatorial aspects of Coxeter groups

978-3-642-07922-1978-0-387-72176-7BlochBEthan D. Bloch, Bard College Dept. Mathematics, Annandale, NY, USA"The Real Numbers and Real AnalysisXXVIII, 553p. 42 illus..SCM12171Real FunctionsPBKBAPreface.-To the Student.-To the Instructor.-1. Construction of the Real Numbers.- 2. Properties of the Real Numbers.- 3. Limits and Continuity.- 4. Differentiation.- 5. Integration.- 6. Limits to Infinity.-7. Transcental Functions.-8.Sequences.- 9. Series.- 10. Sequences and Series of Functions.- Bibliography.- Index.This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs.It is organized in a distinctive, flexible way that would make it equally appropriate to undergraduate mathematics majors who want to continue in mathematics, and to future mathematics teachers who want to understand the theory behind calculus.The Real Numbers and Real Analysis will serve as an excellent one-semester text for undergraduates majoring in mathematics, and for students in mathematics education who want a thorough understanding of the theory behind the real number system and calculus.Provides an unusually thorough treatment of the real numbers, emphasizing their importance as the basis of real analysis

Presents material in an order resembling that of standard calculus courses, for the sake of student familiarity, and for helping future teachers use real analysis to better understand calculus

Emphasizes the direct role of the Least Upper Bound Property in the study of limits, derivatives and integrals, rather than making use of sequences for proofs

Presents the equivalence of various important theorems of real analysis with the Least Upper Bound Property

Relates real analysis to previously learned materal, including detailed discussion of such topics as the transcendental functions, area and the number pi

Offers three different entryways into the study of real numbers, depending on the student audience

Contains historical context, biographical anecdotes, and reflections on the material in each chapter

Includes over 350 exercises, reinforcing concepts

978-1-4899-983< 4-7XXVIII, 553 p. 42 illus.978-0-387-98281-6Blum~Lenore Blum; Felipe Cucker; Michael Shub; Steve Smale, University of California, Berkeley Dept. Mathematics, Berkeley, CA, USAComplexity and Real Computation%XVI, 453 p. With online files/update.SCM24005"Mathematical Logic and FoundationsPBCSCI16005Theory of ComputationUY51 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem P ? NP? .- 8 Newton s Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bzout s Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.The classical theory of computation has its origins in the work of Goedel, Turing, Church, and Kleene and has been an extraordinarily successful framework for theoretical computer science. The thesis of this book, however, is that it provides an inadequate foundation for modern scientific computation where most of the algorithms are real number algorithms. The goal of this book is to develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing. Along the way, the authors consider such fundamental problems as: * Is the Mandelbrot set decidable? * For simple quadratic maps, is the Julia set a halting set? * What is the real complexity of Newton's method? * Is there an algorithm for deciding the knapsack problem in a ploynomial number of steps? * Is the Hilbert Nullstellensatz intractable? * Is the problem of locating a real zero of a degree four polynomial intractable? * Is linear programming tractable over the reals? The book is divided into three parts: The first part provides an extensive introduction and then proves the fundamental NP-completeness theorems of Cook-Karp and their extensions to more general number fields as the real and complex numbers. The later parts of the book develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing.* Unique work on this core topic * Written by internationally recognised specialists in mathematics and computing * Provides the basics for numerous practical industrial applications, e.g. AI, robotics, digital cash978-0-387-98654-8George Bluman; Stephen Anco;Symmetry and Integration Methods for Differential Equations X, 422 p.SCM13003Applications of MathematicsPBWSCP190053Theoretical, Mathematical and Computational PhysicsPHUDimensional Analysis, Modeling, and Invariance.- Lie Groups of Transformations and Infinitesimal Transformations.- Ordinary Differential Equations (ODEs).- Partial Differential Equations (PDEs).This book provides a comprehensive treatment of symmetry methods and dimensional analysis. The authors discuss aspects of Lie groups of point transformations, contact symmetries, and higher order symmetries that are essential for solving differential equations. Emphasis is given to an algorithmic, computational approach to finding integrating factors and first integrals. Numerous examples including ordinary differential equations arising in applied mathematics are used for illustration and exercise sets are included throughout the text. This book is designed for advanced undergraduate or beginning graduate students of mathematics and physics, as well as researchers in mathematics, physics, and engineering.Bluman was the first modern mathematician in the West to seriously investigate the applications of Lie Groups to differential equations and remains a well-recognized authority on the subjectIncludes lots of nice exercise sets978-1-4419-3147-4978-3-642-25633-2 BogatyrevbAndrei Bogatyrev, Institute of Numerical Mathematics of the Russian Acad. Sciences, Moscow, Russia)Extremal Polynomials and Riemann SurfacesXXV, 150 p. 47 illus.SCM12023Approximations and Expansions 1 Least deviation problems.- 2 Chebyshev representation of polynomials.- 3 Representations for the moduli space.- 4 Cell decomposition of the moduli space.- 5 Abel s equations.- 6 Computations in moduli spaces.- 7 The problem of the optimal stability polynomial.- Conclusion.- References. The problems of conditional optimization of the uniform (or C-) norm for polynomials and rational functions arise in various branches of science and technology. Their numerical solution is notoriously difficult in case of high degree functions. The book develops the classical Chebyshev's approach which gives analytical representation for the solution in terms of Riemann surfaces. The techniques born in the remote (at the first glance) branches of mathematics such as complex analysis, Riemann surfaces and Teichmller theory, foliations, braids, topology are applied to approximation problems. The key feature of this book is the usage of beautiful ideas of contemporary mathematics for the solution of applied problems and their effective numerical realization. This is one of the few books where the computational aspects of the higher genus Riemann surfaces are illuminated. Effective work with the moduli spaces of algebraic curves provides wide opportunities for numerical experiments in mathematics and theoretical physics.

Includes numerous problems and exercises which provide a deep insight in the subject and allow to conduct independent research in this top< ic

Contains many pictures which visualize involved theory

Description of effective computational algorithms for higher genus algebraic curves provides wide opportunities for numerical experiments in mathematics and theoretical physics

978-3-642-44332-9978-0-8176-3247-2BorelArmand Borel, Princeton University Inst. Advanced Study, Princeton, NJ, USA; Lizhen Ji, University of Michigan Dept. Mathematics, Ann Arbor, MI, USA;Compactifications of Symmetric and Locally Symmetric Spaces"Mathematics: Theory & ApplicationsXIII, 479 p.SCM28019Algebraic TopologyPBPDCompactifications of Riemannian Symmetric Spaces.- Review of Classical Compactifications of Symmetric Spaces.- Uniform Construction of Compactifications of Symmetric Spaces.- Properties of Compactifications of Symmetric Spaces.- Smooth Compactifications of Semisimple Symmetric Spaces.- Smooth Compactifications of Riemannian Symmetric Spaces G/K.- Semisimple Symmetric Spaces G/H.- The Real Points of Complex Symmetric Spaces Defined over ?.- The DeConcini-Procesi Compactification of a Complex Symmetric Space and Its Real Points.- The Oshima-Sekiguchi Compactification of G/K and Comparison with (?).- Compactifications of Locally Symmetric Spaces.- Classical Compactifications of Locally Symmetric Spaces.- Uniform Construction of Compactifications of Locally Symmetric Spaces.- Properties of Compactifications of Locally Symmetric Spaces.- Subgroup Compactifications of ??G.- Metric Properties of Compactifications of Locally Symmetric Spaces ??X.Noncompact symmetric and locally symmetric spaces naturally appear in many mathematical theories, including analysis (representation theory, nonabelian harmonic analysis), number theory (automorphic forms), algebraic geometry (modulae) and algebraic topology (cohomology of discrete groups). In most applications it is necessary to form an appropriate compactification of the space. The literature dealing with such compactifications is vast. The main purpose of this book is to introduce uniform constructions of most of the known compactifications with emphasis on their geometric and topological structures. The book is divided into three parts. Part I studies compactifications of Riemannian symmetric spaces and their arithmetic quotients. Part II is a study of compact smooth manifolds. Part III studies the compactification of locally symmetric spaces. Familiarity with the theory of semisimple Lie groups is assumed, as is familiarity with algebraic groups defined over the rational numbers in later parts of the book, although most of the pertinent material is recalled as presented. Otherwise, the book is a self-contained reference aimed at graduate students and research mathematicians interested in the applications of Lie theory and representation theory to diverse fields of mathematics.Introduces uniform constructions of most of the known compactifications of symmetric and locally symmetric spaces, with emphasis on their geometric and topological structures

Relatively self-contained reference aimed at graduate students and research mathematicians interested in the applications of Lie theory and representation theory to analysis, number theory, algebraic geometry and algebraic topology

"Wissensch. Buch /Science Book (WI)978-1-4614-0121-6BorweinJonathan M. Borwein, The University of Newcastle, Callaghan, NSW, Australia; Matthew P. Skerritt, The University of Newcastle, Callaghan, NSW, Australia0An Introduction to Modern Mathematical ComputingWith Maple"!:Springer Undergraduate Texts in Mathematics and Technology)XVI, 216p. 86 illus., 81 illus. in color.SCM1400X0Computational Mathematics and Numerical AnalysisSCM14042Mathematical SoftwareUFMk-Preface. -Conventions and Notation.-1. Number Theory (Introduction to Maple, Putting it together, Enough code, already. Show me some maths!, Problems and Exercises, Further Explorations). -2. Calculus(Revision and Introduction, Univariate Calculus, Multivariate Calculus, Exercises, Further Explorations). -3. Linear Algebra (Introduction and Review, Vector Spaces, Linear Transformations, Exercises, Further Explorations). -4. Visualisation and Geometry: a postscript (Useful Visualisation Tools, Geometry and Geometric Constructions). A. Sample Quizzes (Number Theory, Calculus, Linear Algebra). Index. ReferencesThirty years ago mathematical, as opposed to applied numerical, computation was difficult to perform and so relatively little used. Three threads changed that: the emergence of the personal computer; the discovery of fiber-optics and the consequent development of the modern internet; and the building of the Three M s Maple, Mathematica and Matlab.We intend to persuade that Maple and other like tools are worth knowing assuming only that one wishes to be a mathematician, a mathematics educator, a computer scientist, an engineer or scientist, or anyone else who wishes/needs to use mathematics better. We also hope to explain how to become an `experimental mathematician' while learning to be better at proving things. To accomplish this our material is divided into three main chapters followed by a postscript. These cover elementary number theory, calculus of one and several variables, introductory linear algebra, and visualization and interactive geometric computation.gPlaces primary importance on the mathematics, rather than being a 'how to'manual for making computations

Integrates numerous worked examples and introduces all key programming constructions

Includes exercises, sample tests, and a careful selection of 'explorations' suitable for either independent studies or for term projects

978-0-387-94509-5pPeter Borwein, Simon Fraser University Ctr Experimental & Constructive Math., Burnaby, BC, Canada; Tamas Erdelyi'Polynomials and Polynomial Inequalities X, 480 p.Chaptern 1 Introduction and Basic Properties.- 2 Some Special Polynomials.- 3 Chebyshev and Descartes Systems.- 4 Denseness Questions.- 5 Basic Inequalities.- 6 Inequalities in Mntz Spaces.- Inequalities for Rational Function Spaces.- Appendix A1 Algorithms and Computational Concerns.- Appendix A2 Orthogonality and Irrationality.- Appendix A3 An Interpolation Theorem.- Appendix A5 Inequalities for Polynomials with Constraints.- Notation.!Polynomials pervade mathematics, virtually every branch of mathematics from algebraic number theory and algebraic geometry to applied analysis and computer science, has a corpus of theory arising from polynomials. < The material explored in this book primarily concerns polynomials as they arise in analysis; it focuses on polynomials and rational functions of a single variable. The book is self-contained and assumes at most a senior-undergraduate familiarity with real and complex analysis. After an introduction to the geometry of polynomials and a discussion of refinements of the Fundamental Theorem of Algebra, the book turns to a consideration of various special polynomials. Chebyshev and Descartes systems are then introduced, and Mntz systems and rational systems are examined in detail. Subsequent chapters discuss denseness questions and the inequalities satisfied by polynomials and rational functions. Appendices on algorithms and computational concerns, on the interpolation theorem, and on orthogonality and irrationality conclude the book.978-1-4471-4828-9Bosch\Siegfried Bosch, Westflische Wilhelms-Universitt Mathematisches Institut, Mnster, Germany*Algebraic Geometry and Commutative Algebra X, 504 p.SCM11043Commutative Rings and AlgebrasSpringer London#Rings and Modules.- The Theory of Noetherian Rings.- Integral Extensions.- Extension of Coefficients and Descent.- Homological Methods: Ext and Tor.- Affine Schemes and Basic Constructions.- Techniques of Global Schemes.- Etale and Smooth Morphisms.- Projective Schemes and Proper Morphisms.#Algebraic geometry is a fascinating branch of mathematics that combines methods from both, algebra and geometry. It transcends the limited scope of pure algebra by means of geometric construction principles. Moreover, Grothendieck s schemes invented in the late 1950s allowed the application of algebraic-geometric methods in fields that formerly seemed to be far away from geometry, like algebraic number theory. The new techniques paved the way to spectacular progress such as the proof of Fermat s Last Theorem by Wiles and Taylor.The scheme-theoretic approach to algebraic geometry is explained for non-experts. More advanced readers can use the book to broaden their view on the subject. A separate part deals with the necessary prerequisites from commutative algebra. On a whole, the book provides a very accessible and self-contained introduction to algebraic geometry, up to a quite advanced level.Every chapter of the book is preceded by a motivating introduction with an informal discussion of the contents. Typical examples and an abundance of exercises illustrate each section. This way the book is an excellent solution for learning by yourself or for complementing knowledge that is already present. It can equally be used as a convenient source for courses and seminars or as supplemental literature.Explains schemes in algebraic geometry from a beginner's level up to advanced topics such as smoothness and ample invertible sheaves

Is self-contained and well adapted for self-study

Includes prerequisites from commutative algebra in a separate part

Gives motivating introductions to the different themes, illustrated by typical examples

Offers an abundance of exercises, specially adapted to the different sections

978-1-4614-1685-2BrauerFred Brauer, University of British Columbia Dept. Mathematics, Vancouver, BC, Canada; Carlos Castillo-Chavez, Arizona State University Dept. Mathematics & Statistics, Tempe, AZ, USA:Mathematical Models in Population Biology and EpidemiologyTexts in Applied MathematicsXXIV, 508 p.SCL19120Community & Population EcologyPSVSPart I: Simple Single-Species Models. 1. Continuous population models. 2. Discrete population models. 3. Continuous single-species population models with delay.- Part II: Models for Interacting Species. 4. Introduction and mathematical preliminaries. 5. Continuous models for two interacting populations. 6. Harvesting in two-species population models. 7. Multi-species population models.- Part III: Structured Population Models. 8. Models for population with age structure. 9. Models for populations with spatial distribution.This textbook provides an introduction to the field of mathematical biology through the integration of classical applications in ecology with more recent applications to epidemiology, particularly in the context of spread of infectious diseases. It integrates modeling, mathematics, and applications in a semi-rigorous way, stating theoretical results and giving references but not necessarily giving detailed proofs, providing a solid introduction to the field to undergraduates (junior and senior level), graduate students in applied mathematics, ecology, epidemiology or evolutionary biology, sustainability scientists, and to researchers who must routinely read the practical and theoretical results that come from modeling in ecology and epidemiology.This new edition has been updated throughout. In particular the chapters on epidemiology have been updated and extended considerably, and there is a new chapter on spatially structured populations that incorporates dispersal. The number of problems has been increased and the number of projects has more than doubled, in particular those stressing connections to data. In addition some examples, exercises, and projects include use of Maple and Matlab. Review of first edition:'A strength of the book is the large number of biologically-motivated problem sets. These and the references to the original biological papers would be valuable resources for an instructor.' (UK Nonlinear News, 2001)Free supplementary material available on the author's website involving problems using both Mathematica and Maple

Text offers nice balance of theory and application

Concentration is on applications in population biology, epidemiology, and resource management

A rigorous and thorough mathematical introduction to the foundations of the subject

A clear and concise treatment of modern fast solution techniques

978-1-4419-2611-1978-0-387-97606-8BressoudDavid BressoudSecond Year Calculus.From Celestial Mechanics to Special RelativityXI, 400 pp. 98 illus.1 F=ma.- 1.1 Prelude to Newton s Principia.- 1.2 Equal Area in Equal Time.- 1.3 The Law of Gravity.- 1.4 Exercises.- 1.5 Reprise with Calculus.- 1.6 Exercises.- 2 Vector Algebra.- 2.1 Basic Notions.- 2.2 The Dot Product.- 2.3 The Cross Product.- 2.4 Using Vector Algebra.- 2.5 Exercises.- 3 Celestial Mechanics.- 3.1 The Calculus of Curves.- 3.2 Exercises.- 3.3 Orbital Mechanics.- 3.4 Exercises.- 4 Differential Forms.- 4.1 Some History.- 4.2 Differential 1-Forms.- 4.3 Exercises.- 4.4 Constant Differential 2-Forms.- 4.5 Exercises.- 4.6 Constant Differential k-Forms.- 4.7 Prospects.- 4.8 Exercises.- 5 Line Integrals, Multiple Integrals.- 5.1 The Riemann Integral.- 5.2 Line Integrals.- 5.3 Exercises.- 5.4 Multiple Integrals.- 5.5 Using Multiple Integrals.- 5.6 Exercises.- 6 Linear Transformations.- 6.1 Basic Notions.- 6.2 Determinants.- 6.3 History and Comments.- 6.4 Exercises.- 6.5 Invertibility.- 6.6 Exercises.- 7 Differential Calculus.- 7.1 Limits.- 7.2 Exercises.- 7.3 Directional Derivatives.- 7.4 The Derivative.- 7.5 Exercises.- 7.6 The Chain Rule.- 7.7 Using the Gradient.- 7.8 Exercises.- 8 Integration by Pullback.- 8.1 Change of Variables.- 8.2 Interlude with Lagrange.- 8.3 Exercises.- 8.4 The Surface Integral.- 8.5 Heat Flow.- 8.6 Exercises.- 9 Techniques of Differential Calculus.- 9.1 Implicit Differentiation.- 9.2 Invertibility.- 9.3 Exercises.- 9.4 Locating Extrema.- 9.5 Taylor s Formula in Several Variables.- 9.6 Exercises.- 9.7 Lagrange Multipliers.- 9.8 Exercises.- 10 The Fundamental Theorem of Calculus.- 10.1 Overview.- 10.2 Independence of Path.- 10.3 Exercises.- 10.4 The Divergence Theorems.- 10.5 Exercises.- 10.6 Stokes Theorem.- 10.7 Summary for R3.- 10.8 Exercises.- 10.9 Potential Theory.- 11 E = mc2.- 11.1 Prelude to Maxwell s Dynamical Theory.- 11.2 Flow in Space-Time.- 11.3 Electromagnetic Potential.- 11.4 Exercises.- 11.5 Special Relativity.- 11.6 Exercises.- Appendices.- A An Opportunity Missed 361.- B Bibliography 365.- C Clues and Solutions 367.- Index 382.Second Year Calculus: From Celestial Mechanics to Special Relativity covers multi-variable and vector calculus, emphasizing the historical physical problems which gave rise to the concepts of calculus. The book carries us from the birth of the mechanized view of the world in Isaac Newton's Mathematical Principles of Natural Philosophy in which mathematics becomes the ultimate tool for modelling physical reality, to the dawn of a radically new and often counter-intuitive age in Albert Einstein's Special Theory of Relativity in which it is the mathematical model which suggests new aspects of that reality. The development of this process is discussed from the modern viewpoint of differential forms. Using this concept, the student learns to compute orbits and rocket trajectories, model flows and force fields, and derive the laws of electricity and magnetism. These exercises and observations of mathematical symmetry enable the student to better understand the interaction of physics and mathematics.978-1-4612-7824-5BrezziFranco Brezzi, CNR Pavia Ist. Matematica Applicata e, Pavia, Italy; Michel Fortin, Universit Laval Dpt. Mathematique et Statistique, Quebec, QC, Canada'Mixed and Hybrid Finite Element Methods,Springer Series in Computational MathematicsIX, 350 pp. 65 figs.oI: Variational Formulations and Fini< te Element Methods.- 1. Classical Methods.- 2. Model Problems and Elementary Properties of Some Functional Spaces.- 3. Duality Methods.- 4. Domain Decomposition Methods, Hybrid Methods.- 5. Augmented Variational Formulations.- 6. Transposition Methods.- 7. Bibliographical remarks.- II: Approximation of Saddle Point Problems.- 1. Existence and Uniqueness of Solutions.- 2. Approximation of the Problem.- 3. Numerical Properties of the Discrete Problem.- 4. Solution by Penalty Methods, Convergence of Regularized Problems.- 5. Iterative Solution Methods. Uzawa s Algorithm.- 6. Concluding Remarks.- III: Function Spaces and Finite Element Approximations.- 1. Properties of the spaces Hs(?) and H(div; ?).- 2. Finite Element Approximations of H1(?) and H2(?).- 3. Approximations of H (div; ?).- 4. Concluding Remarks.- IV: Various Examples.- 1. Nonstandard Methods for Dirichlet s Problem.- 2. Stokes Problem.- 3. Elasticity Problems.- 4. A Mixed Fourth-Order Problem.- 5. Dual Hybrid Methods for Plate Bending Problems.- V: Complements on Mixed Methods for Elliptic Problems.- 1. Numerical Solutions.- 2. A Brief Analysis of the Computational Effort.- 3. Error Analysis for the Multiplier.- 4. Error Estimates in Other Norms.- 5. Application to an Equation Arising from Semiconductor Theory.- 6. How Things Can Go Wrong.- 7. Augmented Formulations.- VI: Incompressible Materials and Flow Problems.- 1. Introduction.- 2. The Stokes Problem as a Mixed Problem.- 3. Examples of Elements for Incompressible Materials.- 4. Standard Techniques of Proof for the inf-sup Condition.- 5. Macroelement Techniques and Spurious Pressure Modes.- 6. An Alternative Technique of Proof and Generalized Taylor-Hood Element.- 7. Nearly Incompressible Elasticity, Reduced Integration Methods and Relation with Penalty Methods.- 8. Divergence-Free Basis, Discrete Stream Functions.- 9. Other Mixed and Hybrid Methods for Incompressible Flows.- VII: Other Applications.- 1. Mixed Methods for Linear Thin Plates.- 2. Mixed Methods for Linear Elasticity Problems.- 3. Moderately Thick Plates.- References.Research on non-standard finite element methods is evolving rapidly and in this text Brezzi and Fortin give a general framework in which the development is taking place. The presentation is built around a few classic examples: Dirichlet's problem, Stokes problem, Linear elasticity. The authors provide with this publication an analysis of the methods in order to understand their properties as thoroughly as possible.978-1-4419-6869-2BroerHenk Broer, University of Groningen Dept. Mathematics, Groningen, Netherlands; Floris Takens, Rijksuniversiteit Groningen Mathematisch Instituut, Bedum, NetherlandsDynamical Systems and ChaosXVI, 313 p.Examples and definitions of dynamical phenomena.- Qualitative properties and predictability of evolutions.- Persistence of dynamical properties.- Global structure of dynamical systems.- On KAM Theory.- Reconstruction and time series analysis.Over the last four decades there has been extensive development in the theory of dynamical systems. This book aims at a wide audience where the first four chapters have been used for an undergraduate course in Dynamical Systems. Material from the last two chapters and from the appendices has been used quite a lot for master and PhD courses. All chapters are concluded by an exercise section. The book is also directed towards researchers, where one of the challenges is to help applied researchers acquire background for a better understanding of the data that computer simulation or experiment may provide them with the development of the theory.Authors are pioneers in dynamical systems Offers a fresh, modern perspective Highly illustrated with many exercises accessible.978-1-4614-2712-4978-3-0348-0527-8BrownB. Malcolm Brown, Cardiff University Cardiff School of Computer Science, Cardiff, UK; Michael S.P. Eastham, Cardiff University Cardiff School of, Cardiff, UK; Karl Michael Schmidt, Cardiff University, Cardiff, UKPeriodic Differential Operators*Operator Theory: Advances and ApplicationsVIII, 216 p. 1 illus. in color.SCM12147Ordinary Differential EquationsSCM12139Operator TheoryZPreface.- 1 Floquet Theory.- 2 Oscillations.- 3 Asymptotics.- 4 Spectra.- 5 Perturbations.2Periodic differential operators have a rich mathematical theory as well as important physical applications. They have been the subject of intensive development for over a century and remain a fertile research area. This book lays out the theoretical foundations and then moves on to give a coherent account of more recent results, relating in particular to the eigenvalue and spectral theory of the Hill and Dirac equations. The book will be valuable to advanced students and academics both for general reference and as an introduction to active research topics.;Comprehensive treatment of one-dimensional periodic systems from the differential equation, boundary-value problem and operator-theoretic points of view

Unified analysis of periodic Sturm-Liouville- and Dirac-type operators by the use of oscillation methods

Study of perturbed periodic problems

978-3-0348-0754-8978-3-540-31486-8BushnellColin J. Bushnell, King's College London Dept. Mathematics, London, UK; Guy Henniart, Universit Paris-Sud XI Dpt. Mathmatiques, Orsay CX, France(The Local Langlands Conjecture for GL(2) XI, 351 p.Smooth Representations.- Finite Fields.- Induced Representations of Linear Groups.- Cuspidal Representations.- Parametrization of Tame Cuspidals.- Functional Equation.- Representations of Weil Groups.- The Langlands Correspondence.- The Weil Representation.- Arithmetic of Dyadic Fields.- Ordinary Representations.- The Dyadic Langlands Correspondence.- The Jacquet-Langlands Correspondence.YIf F is a non-Archimedean local field, local class field theory can be viewed as giving a canonical bijection between the characters of the multiplicative group GL(1,F) of F and the characters of the Weil grou< p of F. If n is a positive integer, the n-dimensional analogue of a character of the multiplicative group of F is an irreducible smooth representation of the general linear group GL(n,F). The local Langlands Conjecture for GL(n) postulates the existence of a canonical bijection between such objects and n-dimensional representations of the Weil group, generalizing class field theory. This conjecture has now been proved for all F and n, but the arguments are long and rely on many deep ideas and techniques. This book gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groups and the structure theory of local fields. It uses only local methods, with no appeal to harmonic analysis on adele groups.Contributes an unprededented text to the so-called "Langlands theory"

An ambitious research program of already 40 years

Masterly exposition by authors who have contributed significantly to the Langlands program

978-3-642-06853-9978-1-4419-5526-5CampbellStephen L. Campbell, North Carolina State University Dept. Mathematics, Raleigh, NC, USA; Jean-Philippe Chancelier, Ecole Nationale des Ponts et Chausses Centre d'Enseignement et de Recherche en, Marne-la-Vallee, France; Ramine Nikoukhah, INRIA Rocquencourt, Le Chesnay CX, France;Modeling and Simulation in Scilab/Scicos with ScicosLab 4.4 XI, 330p.A1 Scilab.- General Information.- to Scilab.- Modeling and Simulation in Scilab.- Optimization.- Examples.- 2 Scicos.- Getting Started.- Scicos Formalism.- Scicos Blocks.- Examples and Applications.- Batch Processing in Scilab.- Code Generation.- Debugging.- Implicit Scicos and Modelica.- Inside Scicos.- Coding Examples."Scilab is a free open-source software package for scientific computation. It includes hundreds of general purpose and specialized functions for numerical computation, organized in libraries called toolboxes, which cover such areas as simulation, optimization, systems and control, and signal processing. One important Scilab toolbox is Scicos. Scicos provides a block diagram graphical editor for the construction and simulation of dynamical systems. The objective of this book is to provide a tutorial for the use of Scilab/Scicos with a special emphasis on modeling and simulation tools. While it will provide useful information to experienced users it is designed to be accessible to beginning users from a variety of disciplines. Students and academic and industrial scientists and engineers should find it useful. The book is divided into two parts. The first part concerns Scilab and includes a tutorial covering the language features, the data structures and specialized functions for doing graphics, importing, exporting data and interfacing external routines. It also covers in detail Scilab numerical solvers for ordinary differential equations and differential-algebraic equations. Even though the emphasis is placed on modeling and simulation applications, this part provides a global view of Scilab. The second part is dedicated to modeling and simulation of dynamical systems in Scicos. This type of modeling tool is widely used in industry because it provides a means for constructing modular and reusable models. This part contains a detailed description of the editor and its usage, which is illustrated through numerous examples. All codes used in the book is made available to the reader. This new edition includes expanded chapters, new exercises and major rewrites for examples to work with the new Maple.First book to focus on simulation and modeling

First book to put a major emphasis on Scicos and discuss it in depth

Over 100 illustrations

Large number of carefully worked out examples and illustrations

Does not presuppose familiarity with Scilab/Scicos

Discusses some related aspects of simulation and modeling

Provides guidance in the use of Scilab

All code used in the book is available on the web

978-3-0348-0480-6CanoAngel Cano, UNAM, Unidad Cuernavaca, Cuernavaca, Mexico; Juan Pablo Navarrete, Universidad Autnoma de Yucatn, Mrida, Mexico; Jos Seade, UNAM, Unidad Cuernavaca, Cuernavaca, MexicoComplex Kleinian GroupsProgress in Mathematics'XX, 271 p. 7 illus., 3 illus. in color. Preface.- Introduction.- Acknowledgments.- 1 A glance of the classical theory.- 2 Complex hyperbolic geometry.- 3 Complex Kleinian groups.- 4 Geometry and dynamics of automorphisms of P2C.- 5 Kleinian groups with a control group.- 6 The limit set in dimension two.- 7 On the dynamics of discrete subgroups of PU(n,1).- 8 Projective orbifolds and dynamics in dimension two.- 9 Complex Schottky groups.- 10 Kleinian groups and twistor theory.- Bibliography.- Index. This monograph lays down the foundations of the theory of complex Kleinian groups, a newly born area of mathematics whose origin traces back to the work of Riemann, Poincar, Picard and many others. Kleinian groups are, classically, discrete groups of conformal automorphisms of the Riemann sphere, and these can be regarded too as being groups of holomorphic automorphisms of the complex projective line CP1. When going into higher dimensions, there is a dichotomy: Should we look at conformal automorphisms of the n-sphere?, or should we look at holomorphic automorphisms of higher dimensional complex projective spaces? These two theories are different in higher dimensions. In the first case we are talking about groups of isometries of real hyperbolic spaces, an area of mathematics with a long-standing tradition. In the second case we are talking about an area of mathematics that still is in its childhood, and this is the focus of study in this monograph. This brings together several important areas of mathematics, as for instance classical Kleinian group actions, complex hyperbolic geometry, chrystallographic groups and the uniformization problem for complex manifolds. }Lays down the foundations of a new field of mathematics including areas as important as real and complex hyperbolic geometry, discrete group actions in complex geometry and the uniformization problem

First book of its kind in the literature

Accessible to a wide audience

Serves also as an introduction to the study of real and complex hyperbolic geometry

978-3-0348-0805-7978-3-642-32905-0Capietto< Anna Capietto, Universit di Torino Dipartimento di Matematica, Torino, Italy; Peter Kloeden, Universitt Frankfurt Institut fr Mathematik, Frankfurt, Germany; Jean Mawhin, Universit Catholique de Louvain Institut de Recherche, Louvain-la-Neuve, Belgium; Sylvia Novo, E.T.S. de Ingenieros Industriales de Valladolid, Valladolid, Spain; Miguel Ortega, Universidad de Granada Departamento de Mtematica Aplicada, Granada, SpainJStability and Bifurcation Theory for Non-Autonomous Differential EquationsBCetraro, Italy 2011, Editors: Russell Johnson, Maria Patrizia Pera(IX, 303 p. 26 illus., 9 illus. in color.9The Maslov index and global bifurcation for nonlinear boundary value problems.- Discrete-time nonautonomous dynamical systems.- Resonance problems for some non-autonomous ordinary differential equations.- Non-autonomous functional differential equations and applications.- Twist mappings with non-periodic angles.This volume contains the notes from five lecture courses devoted to nonautonomous differential systems, in which appropriate topological and dynamical techniques were described and applied to a variety of problems. The courses took place during the C.I.M.E. Session 'Stability and Bifurcation Problems for Non-Autonomous Differential Equations,' held in Cetraro, Italy, June 19-25 2011. Anna Capietto and Jean Mawhin lectured on nonlinear boundary value problems; they applied the Maslov index and degree-theoretic methods in this context. Rafael Ortega discussed the theory of twist maps with nonperiodic phase and presented applications. Peter Kloeden and Sylvia Novo showed how dynamical methods can be used to study the stability/bifurcation properties of bounded solutions and of attracting sets for nonautonomous differential and functional-differential equations. The volume will be of interest to all researchers working in these and related fields.Up-to-date study of ordinary and functional differential equations

Use of topological and dynamical methods and comparison of such methods

Speakers are all leading experts in the field

978-0-8176-8354-2Cardaliaguet~Pierre Cardaliaguet, Universit Paris-Dauphine, Paris, France; Ross Cressman, Wilfrid Laurier University, Waterloo, ON, CanadaAdvances in Dynamic GamesQTheory, Applications, and Numerical Methods for Differential and Stochastic Games4Annals of the International Society of Dynamic Games+XVII, 421 p. 83 illus., 50 illus. in color.Part I Evolutionary Games.- 1 Some Generalizations of a Mutual Mate Choice Problem with Age Preferences.- 2 Signalling Victory to Ensure Dominance: A Continuous Model.- 3 Evolutionary Games for Multiple Access Control.- 4 Join Forces or Cheat: Evolutionary Analysis of a Consumer-Resource System.- Part II Dynamic and Differential Games: Theoretical Developments.- 5 Strong Strategic Support of Cooperative Solutions in Differential Games.- 6 Characterization of Feedback Nash Equilibrium for Differential Games.- 7 Nash Equilibrium Payoffs in Mixed Strategies.- 8 A Penalty Method Approach for Open-Loop Variational Games with Equality Constraints.- 9 Nash Equilibrium Seeking for Dynamic Systems with Non-Quadratic Payoffs.- 10 A Uniform Tauberian Theorem in Optimal Control.- 11 E-equilibria for Multicriteria Games.- 12 Mean Field Games with Quadratic Hamiltonian: A Constructive Scheme.- Part III Pursuit-Evasion Games and Search Games.- 13 Differential Game-Theoretic Approach to a Spatial Jamming Problem.- 14 Study of Linear Game with Two Pursuers and One Evader: Different Strength of Pursuers.- 15 Salvo Enhanced No Escape Zone.- 16 One Method of Solving Differential Games Under Integrally Constrained Controls.- 17 Anglers' Fishing Problem.- 18 A Nonzero-Sum Search Game with Two Competitive Searchers and a Target.- Part IV Applications of Dynamic Games.- 19 Advertising and Price to Sustain the Brand Value in a Licensing Contract.- 20 Cost-Revenue Sharing in a Closed-Loop Supply Chain.This book focuses on various aspects of dynamic game theory, presenting state-of-the-art research and serving as a testament to the vitality and growth of the field of dynamic games and their applications. Its contributions, written by experts in their respective disciplines, are outgrowths of presentations originally given at the 14th International Symposium of Dynamic Games and Applications held in Banff. Advances in Dynamic Games covers a variety of topics, ranging from evolutionary games, theoretical developments in game theory and algorithmic methods to applications, examples, and analysis in fields as varied as mathematical biology, environmental management, finance and economics, engineering, guidance and control, and social interaction. Featured throughout are valuable tools and resources for researchers, practitioners, and graduate students interested in dynamic games and their applications to mathematics, engineering, economics, and management science. <p>A unified, self-contained collection of selected contributions giving a state-of-the-art account of recent developments in dynamic game theory and its applications</p><p>Real-world applications to dynamic games in communication and transportation networks, economics, finance, management science, environmental science, biology, and social science</p><p>Interdisciplinary audience of researchers, practitioners, and advanced graduate students </p>978-0-387-97942-7Carleson"Lennart Carleson; Theodore GamelinComplex DynamicsIX, 175 p. 28 illus.I. Conformal and Quasiconformal Mappings.- 1. Some Estimates on Conformal Mappings.- 2. The Riemann Mapping.- 3. Montel s Theorem.- 4. The Hyperbolic Metric.- 5. Quasiconformal Mappings.- 6. Singular Integral Operators.- 7. The Beltrami Equation.- II. Fixed Points and Conjugations.- 1. Classification of Fixed Points.- 2. Attracting Fixed Points.- 3. Repelling Fixed Points.- 4. Superattracting Fixed Points.- 5. Rationally Neutral Fixed Points.- 6. Irrationally Neutral Fixed Points.- 7. Homeomorphisms of the Circle.- III. Basic Rational Iteration.- 1. The Julia Set.- 2. Counting Cycles.- 3. Density of Repelling Periodic Points.-< 4. Polynomials.- IV. Classification of Periodic Components.- 1. Sullivan s Theorem.- 2. The Classification Theorem.- 3. The Wolff-Denjoy Theorem.- V. Critical Points and Expanding Maps.- 1. Siegel Disks.- 2. Hyperbolicity.- 3. Subhyperbolicity.- 4. Locally Connected Julia Sets.- VI. Applications of Quasiconformal Mappings.- 1. Polynomial-like Mappings.- 2. Quasicircles.- 3. Herman Rings.- 4. Counting Herman Rings.- 5. A Quasiconformal Surgical Procedure.- VII. Local Geometry of the Fatou Set.- 1. Invariant Spirals.- 2. Repelling Arms.- 3. John Domains.- VIII. Quadratic Polynomials.- 1. The Mandelbrot Set.- 2. The Hyperbolic Components of ?.- 3. Green s Function of ?c.- 4. Green s Function of ?.- 5. External Rays with Rational Angles.- 6. Misiurewicz Points.- 7. Parabolic Points.- Epilogue.- References.- Symbol Index.~Complex Dynamics discusses the properties of conformal mappings in the complex plane, a subject that is closely connected to the study of fractals and chaos. Indeed the culmination of the book is a detailed study of the famous Mandelbrot set, which describes very general properties of such mappings. The book focuses on the analytic side of this contemporary subject. The text was developed out of a course taught over several semesters; its focus is to help students and instructors to familiarize themselves with complex dynamics. Topics covered include: conformal and quasi-conformal mappings, fixed points and conjugations, basic rational iteration (the Julia set), classification of periodic components, critical points and expanding maps, some applications of conformal mappings (e.g. Hermann rings), the local geometry of the Fatou set, and quadratic polynomials and the Mandelbrot set.978-0-387-95012-9CarterM. Carter; B. van BruntThe Lebesgue-Stieltjes IntegralA Practical Introduction IX, 230 p.1 Real Numbers.- 1.1 Rational and Irrational Numbers.- 1.2 The Extended Real Number System.- 1.3 Bounds.- 2 Some Analytic Preliminaries.- 2.1 Monotone Sequences.- 2.2 Double Series.- 2.3 One-Sided Limits.- 2.4 Monotone Functions.- 2.5 Step Functions.- 2.6 Positive and Negative Parts of a Function.- 2.7 Bounded Variation and Absolute Continuity.- 3 The Riemann Integral.- 3.1 Definition of the Integral.- 3.2 Improper Integrals.- 3.3 A Nonintegrable Function.- 4 The Lebesgue-Stieltjes Integral.- 4.1 The Measure of an Interval.- 4.2 Probability Measures.- 4.3 Simple Sets.- 4.5 Definition of the Integral.- 4.6 The Lebesgue Integral.- 5 Properties of the Integral.- 5.1 Basic Properties.- 5.2 Null Functions and Null Sets.- 5.3 Convergence Theorems.- 5.4 Extensions of the Theory.- 6 Integral Calculus.- 6.1 Evaluation of Integrals.- 6.2 IWo Theorems of Integral Calculus.- 6.3 Integration and Differentiation.- 7 Double and Repeated Integrals.- 7.1 Measure of a Rectangle.- 7.2 Simple Sets and Simple Functions in Two Dimensions.- 7.3 The Lebesgue-Stieltjes Double Integral.- 7.4 Repeated Integrals and Fubini s Theorem.- 8 The Lebesgue SpacesLp.- 8.1 Normed Spaces.- 8.2 Banach Spaces.- 8.3 Completion of Spaces.- 8.4 The SpaceL1.- 8.5 The LebesgueLp.- 8.6 Separable Spaces.- 8.7 ComplexLpSpaces.- 8.8 The Hardy SpacesHp.- 8.9 Sobolev SpacesWk,p.- 9 Hilbert Spaces andL2.- 9.1 Hilbert Spaces.- 9.2 Orthogonal Sets.- 9.3 Classical Fourier Series.- 9.4 The Sturm-Liouville Problem.- 9.5 Other Bases forL2.- 10 Epilogue.- 10.1 Generalizations of the Lebesgue Integral.- 10.2 Riemann Strikes Back.- 10.3 Further Reading.- Appendix: Hints and Answers to Selected Exercises.- References.AMathematics students generally meet the Riemann integral early in their undergraduate studies, then at advanced undergraduate or graduate level they receive a course on measure and integration dealing with the Lebesgue theory. However, those whose interests lie more in the direction of applied mathematics will in all probability find themselves needing to use the Lebesgue or Lebesgue-Stieltjes Integral without having the necessary theoretical background. It is to such readers that this book is addressed. The authors aim to introduce the Lebesgue-Stieltjes integral on the real line in a natural way as an extension of the Riemann integral. They have tried to make the treatment as practical as possible. The evaluation of Lebesgue-Stieltjes integrals is discussed in detail, as are the key theorems of integral calculus as well as the standard convergence theorems. The book then concludes with a brief discussion of multivariate integrals and surveys ok L^p spaces and some applications. Exercises, which extend and illustrate the theory, and provide practice in techniques, are included. Michael Carter and Bruce van Brunt are senior lecturers in mathematics at Massey University, Palmerston North, New Zealand. Michael Carter obtained his Ph.D. at Massey University in 1976. He has research interests in control theory and differential equations, and has many years of experience in teaching analysis. Bruce van Brunt obtained his D.Phil. at the University of Oxford in 1989. His research interests include differential geometry, differential equations, and analysis. His publications include978-1-4614-4580-7CarvalhoAlexandre Carvalho, Instituto de Cincias Matemticas e de C Universidade de So Paulo, So Carlos SP, Brazil; Jos A. Langa, Universidad de Sevilla, Seville, Spain; James Robinson, University of Warwick Mathematics Institute, Coventry, UKDAttractors for infinite-dimensional non-autonomous dynamical systemsXXXVI, 409 p. 12 illus.The pullback attractor.- Existence results for pullback attractors.- Continuity of attractors.- Finite-dimensional attractors.- Gradient semigroups and their dynamical properties.- Semiline< ar Differential Equations.- Exponential dichotomies.- Hyperbolic solutions and their stable and unstable manifolds.- A non-autonomous competitive Lotka-Volterra system.- Delay differential equations.-The Navier Stokes equations with non-autonomous forcing.- Applications to parabolic problems.- A non-autonomous Chafee Infante equation.- Perturbation of diffusion and continuity of attractors with rate.- A non-autonomous damped wave equation.- References.- Index.- The book treats the theory of attractors for non-autonomous dynamical systems. The aim of the book is to give a coherent account of the current state of the theory, using the framework of processes to impose the minimum of restrictions on the nature of the non-autonomous dependence. The book is intended as an up-to-date summary of the field, but much of it will be accessible to beginning graduate students. Clear indications will be given as to which material is fundamental and which is more advanced, so that those new to the area can quickly obtain an overview, while those already involved can pursue the topics we cover more deeply.Key Researcher

Obtains new results on the characterization of global attractors for processes and their perturbations

An up-to-date summary of the field

978-1-4899-9176-8978-3-642-31207-6CazalsFrdric Cazals, INRIA Sophia Antipolis - Mditerrane, Sophia Antipolis, France; Pierre Kornprobst, INRIA Sophia Antipolis - Mditerrane, Sophia Antipolis, France1Modeling in Computational Biology and BiomedicineA Multidisciplinary Endeavor+XXVI, 315 p. 85 illus., 65 illus. in color.AForeword by Olivier Faugeras.- Foreword by Jol Janin.- Preface.- Part I Bioinformatics.- 1.Modeling Macro-molecular Complexes: a Journey Across Scales. F.Cazals, T.Dreyfus, and C.H. Robert.- 1.1.Introduction.- 1.2.Modeling Atomic Resolution.- 1.3.Modeling Large Assemblies.- 1.4.Outlook.- 1.5.Online Resources.- References.- 2.Modeling and Analysis of Gene Regulatory Networks. G.Bernot, J-P.Comet, A.Richard, M.Chaves, J-L.Gouz, and F.Dayan.- 2.1.Introduction.- 2.2.Continuous and Hybrid Models of Genetic Regulatory Networks.- 2.3.Discrete Models of GRN.- 2.4.Outlook.- 2.5.Online Resources.- 2.6.Acknowledgments.- References.- Part II Biomedical Signal and Image Analysis.- 3.Noninvasive Cardiac Signal Analysis Using Data Decomposition Techniques. V.Zarzoso, O.Meste, P.Comon, D.G.Latcu, and N.Saoudi.- 3.1.Preliminaries and Motivation.- 3.2.T-Wave Alternans Detection via Principal Component Analysis.- 3.3.Atrial Activity Extraction via Independent Component Analysis.- 3.4.Conclusion and Outlook.- 3.5.Online Resources.- References.- 4.Deconvolution and Denoising for Confocal Microscopy. P.Pankajakshan, G.Engler, L.Blanc-Fraud, and J.Zerubia.- 4.1.Introduction.- 4.2.Development of the Auxiliary Computational Lens.- 4.3.Outlook.- 4.4.Selected Online Resources.- References.- 5.Statistical Shape Analysis of Surfaces in Medical Images Applied to the Tetralogy of Fallot Heart. K.McLeod, T.Mansi, M.Sermesant, G.Pongiglione, and X.Pennec.- 5.1.Introduction.- 5.2.Statistical Shape Analysis.- 5.3.Shape Analysis of ToF Data.- 5.4.Conclusion.- 5.5.Online Resources.- References.- 6.From Diffusion MRI to Brain Connectomics. A.Ghosh and R.Deriche.- 6.1.Introduction.- 6.2.A Brief History of NMR and MRI.- 6.3.Nuclear Magnetic Resonance and Diffusion.- 6.4.From Diffusion MRI to Tissue Microstructure.- 6.5.Computational Framework for Processing Diffusion MR Images.- 6.6.Tractography: Inferring the Connectivity.- 6.7.Clinical Applications 6.8.Conclusion.- 6.9.Online Resources.- References.- Part III Modeling in neuroscience.- 7.Single-Trial Analysis of Bioelectromagnetic Signals: The Quest for Hidden Information. M.Clerc, T.Papadopoulo, and C.Bnar.- 7.1.Introduction.- 7.2.Data-driven Approaches: Non-linear Dimensionality Reduction.- 7.3.Model-Driven Approaches: Matching Pursuit and its Extensions.- 7.4.Success Stories.- 7.5.Conclusion.- 7.6.Selected Online Resources.- References.- 8 Spike Train Statistics from Empirical Facts to Theory: The Case of the Retina. B.Cessac and A.Palacios.- 8.1.Introduction.- 8.2.Unraveling the Neural Code in the Retina via Spike Train Statistics Analysis.- 8.3.Spike Train Statistics from a Theoretical Perspective.- 8.4.Using Gibbs Distributions to Analysing Spike Trains Statistics.- 8.5.Conclusion.- 8.6.Outlook.- 8.7.Online Resources.- References.- Biology, Medicine and Biophysics.- Mathematics and Computer Science.- Index.Computational biology, mathematical biology, biology and biomedicine are currently undergoing spectacular progresses due to a synergy between technological advances and inputs from physics, chemistry, mathematics, statistics and computer science. The goal ofthis book is to evidence this synergy by describing selected developments in the following fields: bioinformatics, biomedicine and neuroscience. This work is unique in two respects - first, by the variety and scales of systems studied and second, by its presentation: Each chapter provides the biological or medical context, follows up with mathematical or algorithmic developments triggered by a specific problem and concludes with one or two success stories, namely new insights gained thanks to these methodological developments. It also highlights some unsolved and outstanding theoretical questions, with a potentially high impact on these disciplines. Two communities will be particularly interested in this book. The first one is the vast community of applied mathematicians and computer scientists, whose interests should be captured by the added value generated by the application of advanced concepts and algorithms to challenging biological or medical problems. The second is the equally vast community of biologists. Whether scientists or engineers, they will find in this book a clear and self-contained account of concepts and techniques from mathematics and computer science, together with success stories on their favorite systems. The variety of systems described represents a panoply of complementary conceptual tools. On a practical level, the resources listed at the end of each chapter (databases, software) offer invaluable support for getting started on a specific topic in the fields of biomedicine, bioinformatics and neuroscience.O<p> First book that strikes a balance between biology and biomedicine on the one hand, and applied mathematics and computer science on the other hand </p><p>Presents a panoply of systems ranging from atoms and molecules to organs and biomedicine </p><p>Three communities are addressed: Bioinformatics, Biomedicine, and Neuroscience</p>978-3-642-44669-6978-3-642-30900-7 CegielskiZAndrzej Cegielski, University of Zielona Gra Faculty of Mathematics, Zielona Gra, Poland<Iterative Methods for Fixed Point Problems in Hilbert Spaces)XVI, 298 p. 61 illus., 3 illus. in color.1 Introduction.- 2 Algorithmic Operators.- 3 Convergence of Iterative Methods.- 4 Algorithmic Projection Operators.- 5 Projection methods.*Iterative methods for finding fixed points of non-expansive operators in Hilbert spaces have been described in many publications. In this monograph we try to present the methods in a consolidated way. We introduce several classes of operators, examine their properties, define iterative methods generated by operators from these classes and present gene< ral convergence theorems. On this basis we discuss the conditions under which particular methods converge. A large part of the results presented in this monograph can be found in various forms in the literature (although several results presented here are new). We have tried, however, to show that the convergence of a large class of iteration methods follows from general properties of some classes of operators and from some general convergence theorems.AThe projection methods for fixed point problems are presented in a consolidated way

Over 60 figures help to understand the properties of important classes of algorithmic operators

The convergence properties of projection methods follow from a few general convergence theorems presented in the monograph

978-0-387-98503-9ChambersJohn M. ChambersProgramming with DataA Guide to the S Language XV, 469 p. StatisticsManualHighlights.- Concepts.- Quick Reference.- Computations in S.- Objects, Databases, and Chapters.- Creating Functions.- Creating Classes.- Creating Methods.- Documentation.- Connections.- Interfaces to C and Fortran.- Programming in C with S Objects.- Compatibility with Older Versions.]Here is a thorough and authoritative guide to the latest version of the S language and to its programming environment, the premier software platform for computing with data. Programming with Data describes a new and greatly extended version of S, and is written by the chief designer of the language. The book is a guide to the complete programming process, starting from simple, interactive use and continuing through ambitious software projects. S is designed for computing with data - for any project in which organizing, visualizing, summarizing, or modeling data is a central concern. Its focus is on the needs of the programmer/user, and its goal is 'to turn ideas into software, quickly and faithfully.' S is a functional, object-based language with a huge library of functions for all aspects of computing with data. Its long and enthusiastic use in statistics and applied fields has also led to many valuable libraries of user-written functions. The new version of S provides a powerful class/method structure, new techniques to deal with large objects, extended interfaces to other languages and files, object-based documentation compatible with HTML, and powerful new interactive programming techniques. This version of S underlies the S-Plus system, versions 5.0 and higher. John Chambers has been a member of the technical staff in research at Bell Laboratories since 1966. In 1977, he became the first statistician to be named a Bell Labs Fellow, cited for 'pioneering contributions to the field of statistical computing.' His research has touched on nearly all aspects of computing with data, but he is best known for the design of the S language. He is the author or co-author of seven books on S, on computational methods, and on graphical methods; and he is a Fellow of the American Statistical Association and the American Association for the Advancement of Science.978-3-642-15006-7ChenLouis H.Y. Chen, National University of Singapore, Singapore, Singapore; Larry Goldstein, University of Southern California, Los Angeles, CA, USA; Qi-Man Shao, Hong Kong University of Science, Hong Kong, Hong Kong SAR&Normal Approximation by Stein s Method Probability and Its ApplicationsXII, 408 p.2Preface.- 1.Introduction.- 2.Fundamentals of Stein's Method.- 3.Berry-Esseen Bounds for Independent Random Variables.- 4.L^1 Bounds.- 5.L^1 by Bounded Couplings.- 6 L^1: Applications.- 7.Non-uniform Bounds for Independent Random Variables.- 8.Uniform and Non-uniform Bounds under Local Dependence.- 9.Uniform and Non-Uniform Bounds for Non-linear Statistics.- 10.Moderate Deviations.- 11.Multivariate Normal Approximation.- 12.Discretized normal approximation.- 13.Non-normal Approximation.- 14.Extensions.- References.- Author Index .- Subject Index.- Notation.Since its introduction in 1972, Stein s method has offered a completely novel way of evaluating the quality of normal approximations. Through its characterizing equation approach, it is able to provide approximation error bounds in a wide variety of situations, even in the presence of complicated dependence. Use of the method thus opens the door to the analysis of random phenomena arising in areas including statistics, physics, and molecular biology. Though Stein's method for normal approximation is now mature, the literature has so far lacked a complete self contained treatment. This volume contains thorough coverage of the method s fundamentals, includes a large number of recent developments in both theory and applications, and will help accelerate the appreciation, understanding, and use of Stein's method by providing the reader with the tools needed to apply it in new situations. It addresses researchers as well as graduate students in Probability, Statistics and Combinatorics.First book that presents a complete self-contained treatment of Stein's method Includes many recent applications Important reference for researchers and suitable for graduate student seminars978-3-642-26565-5978-1-4614-5130-3Chinchuluun3Altannar Chinchuluun, National University of Mongolia, Ulaanbaatar, Mongolia; Panos Pardalos, University of Florida, Gainesville, FL, USA; Rentsen Enkhbat, National University of Mongolia The School of Economic Studies, Ulaanbaatar, Mongolia; Efstratios N. Pistikopoulos, Imperial College London, London, UK%Optimization, Simulation, and Control*Springer Optimization and Its Applications*XII, 345 p. 75 illus., 25 illus. in color.Preface.-On the Composition of Convex Envelopes for Quadrilinear Terms (P. Belotti, S. Cafieri, J. Lee, L. Liberti, A. Miller).-An Oriented Distance Function Application to Gap Functions for Vector Variational Inequalities (L. Altangerel, G. Wanka).-Optimal Inscribing of Two Balls into Polyhedral Set (R. Enkhbat, B. Barsbold).-Mathematical Programs with Equilibrium Constraints: A Brief Survey of Methods and optimality conditions (I. Tseveendorj).-Linear programming with interval data: a Two-Level Programming Approach (C. Kao, S. Liu).-Quantifying Retardation in Simulation Based Optimization (A. Griewank, A. Hamdi, E. Ozkaya).-Evolutionary Algorithm for Generalized Nash Equilibrium Problems (M. Majig, R. Enkhbat, M. Fukushima).-8. Scalar and Vector Optimization with Composed Objective Functions and Constraints (N. Lorenze, G. Wanka).-9. A PTAS for Weak Minimum Routing Cost Connected Dominating Set of Unit Disk Graph(Q. Liu, Z. Zhang, Y. Hong, W. Wu, D. Du).-10. Power Control in Wireless Ad-Hoc Networks: Stability and Convergence Under Uncertainties (T. Charalambous).-The Changing Role of Optimization in Urban Planning (J. Keirstead, N. Shah).-Parametric Optimization Approach to the Solow Growth Theory (R. Enkhbat, D. Bayanjargal ).-Cyclical Fluctuations in Continuous Time Dynamic Optimization Models: Survey of General Theory and an Application to Dynamic Limit Pricing (T. Asada).-Controlling Of Processes By Optimized Expert Systems (W. Lippe).-Using Homotopy Method to Solve Ban< g-bang Optimal Control Problems (Z. Gao, H. Baoyin).-A Collection of Test Multiextremal Optimal Control Problems (A. Gornov, T. Zarodnyuk, T. Madzhara, A. Daneeva, I. Veyalko).-A Multimethod Technique for Solving Optimal Control Problem (A. Tyatyushkin).-Algorithm for Solving Nonconvex Optimal Control Problems (A. Gornov, T. Zarodnyuk).-Solving Linear Systems with Polynomial Parameter Dependency with Application to the Verified Solution of Problems in Structural Mechanics (J. Garloff, E. Popova, A. Smith).-A fast block Krylov implicit Runge-Kutta method for solving large-scale ordinary differential equations (A. Bouhamidi, K. Jbilou).-Semilocal convergence with R-order Three Theorems for the Chebyshev Method and its Modifications (Z. Tugal, K. Dashdondog)(Optimization, simulation and control play an increasingly important role in science and industry. Because of their numerous applications in various disciplines, research in these areas is accelerating at a rapid pace.This volume brings together the latest developments in these areas of research as well as presents applications of these results to a wide range of real-world problems. The book is composed of invited contributions by experts from around the world who work to develop and apply new optimization, simulation and control techniques either at a theoretical level or in practice. Some key topics presented include: equilibrium problems, multi-objective optimization, variational inequalities, stochastic processes, numerical analysis, optimization in signal processing, and various other interdisciplinary applications.This volume can serve as a useful resource for researchers, practitioners, and advanced graduate students of mathematics and engineering working in research areas where results in optimization, simulation and control can be applied.Collection of selected contributions giving a state-of-the-art account of recent developments in the field

Covers a broad range of topics in optimization and optimal control, including unique applications

Written by an international group of experts in their respective disciplines

Broad audience of researchers, practitioners, and advanced graduate students in applied mathematics and engineering

978-1-4899-8781-5978-3-7643-9981-8ChipotQMichel Chipot, University of Zrich Institute of Mathematics, Zrich, Switzerland*Elliptic Equations: An Introductory Course-Birkhuser Advanced Texts Basler Lehrbcher7Basic Techniques.- Hilbert Space Techniques.- A Survey of Essential Analysis.- Weak Formulation of Elliptic Problems.- Elliptic Problems in Divergence Form.- Singular Perturbation Problems.- Asymptotic Analysis for Problems in Large Cylinders.- Periodic Problems.- Homogenization.- Eigenvalues.- Numerical Computations.- More Advanced Theory.- Nonlinear Problems.- L?-estimates.- Linear Elliptic Systems.- The Stationary Navier Stokes System.- Some More Spaces.- Regularity Theory.- The p-Laplace Equation.- The Strong Maximum Principle.- Problems in the Whole Space.The aim of this book is to introduce the reader to different topics of the theory of elliptic partial differential equations by avoiding technicalities and refinements. Apart from the basic theory of equations in divergence form it includes subjects such as singular perturbation problems, homogenization, computations, asymptotic behaviour of problems in cylinders, elliptic systems, nonlinear problems, regularity theory, Navier-Stokes system, p-Laplace equation. Just a minimum on Sobolev spaces has been introduced, and work or integration on the boundary has been carefully avoided to keep the reader's attention on the beauty and variety of these issues. The chapters are relatively independent of each other and can be read or taught separately. Numerous results presented here are original and have not been published elsewhere. The book will be of interest to graduate students and faculty members specializing in partial differential equations.Simple presentation

Large spectrum of issues on elliptic equations

Many original results

Independent chapters

978-0-8176-4802-2 ChirikjianlGregory S. Chirikjian, The Johns Hopkins University Department of Mechanical Engineering, Baltimore, MD, USA?Stochastic Models, Information Theory, and Lie Groups, Volume 1'Classical Results and Geometric Methods'Applied and Numerical Harmonic AnalysisXXII, 383p. 13 illus..SCT110065Appl.Mathematics/Computational Methods of EngineeringTBJGaussian Distributions and the Heat Equation.- Probability and Information Theory.- Stochastic Differential Equations.- Geometry of Curves and Surfaces.- Differential Forms.- Polytopes and Manifolds.- Stochastic Processes on Manifolds.- Summary.nThe subjects of stochastic processes, information theory, and Lie groups are usually treated separately from each other. This unique two-volumeset presents these topics in a unified setting, thereby building bridges between fields that are rarely studied by the same people. Unlike the many excellent formal treatments available for each of these subjects individually, the emphasis in both of these volumes is on the use of stochastic, geometric, and group-theoretic concepts in the modeling of physical phenomena.Volume 1 establishes the geometric and statistical foundations required to understand the fundamentals of continuous-time stochastic processes, differential geometry, and the probabilistic foundations of information theory. Extensive exercises and motivating examples make the work suitable as a textbook for use in courses that emphasize applied stochastic processes or differential geometry.Stochastic Models, Information Theory, and Lie Groups will be of interest to advanced undergraduate and graduate students, researchers, and practitionersworking in applied mathematics, the physical sciences, and engineering.<p>Unique work:the only book to usetools and concepts from several mathematical areas usually treated in separate books stochastic processes, information theory, andLie theory thereby building bridges between topics rarely studied by the same individuals.</p><p>Extensive exercises</p><p>Concrete presentation makes it easy for readers to obtain numerical solutions for their own problems</p><p>Numerous examples used to motivate concepts with an emphasis on modeling physical phenomena</p><p>Suitable as a textbook for advanced undergraduate and graduate courses in applied stochastic processes or differential geometry</p><p>For a broad audience of advanced undergraduate and graduate students, researchers, and practitioners in applied mathematics, the physical sciences, and engineering</p>978-0-8176-4943-2?Stochastic Models, Information Theory, and Lie Groups, Volume 2(Analytic Methods and Modern Application< sXXVII, 435p.aLie Groups I: Introduction and Examples.- Lie Groups II: Differential Geometric Properties.- Lie Groups III: Integration, Convolution, and Fourier Analysis.- Variational Calculus on Lie Groups.- Statistical Mechanics and Ergodic Theory.- Parts Entropy and the Principal Kinematic Formula.- Estimation and Multivariate Analysis in R^n.- Information, Communication, and Group Therapy.- Algebraic and Geometric Coding Theory.- Information Theory on Lie Groups.- Stochastic Processes on Lie Groups.- Locomotion and Perception as Communication over Principal Fiber Bundles; and A Survey of Additional Applications.The subjects of stochastic processes, information theory, and Lie groups are usually treated separately from each other. This unique two-volume set presents these topics in a unified setting, thereby building bridges between fields that are rarely studied by the same people. Unlike the many excellent formal treatments available for each of these subjects individually, the emphasis in both of these volumes is on the use of stochastic, geometric, and group-theoretic concepts in the modeling of physical phenomena.Volume 2 builds on the fundamentals presented in Volume 1, delving deeper into relationshipsamong stochastic geometry, geometric aspects of the theory of communications and coding, multivariate statistical analysis, and error propagation on Lie groups. Extensive exercises, motivating examples, and real-world applicationsmake the work suitable as a textbook for use in courses that emphasize applied stochastic processes or differential geometry.Stochastic Models, Information Theory, and Lie Groups will be of interest to advanced undergraduate and graduate students, researchers, and practitioners working in applied mathematics, the physical sciences, and engineering.<p>Unique work: the only book to use tools and concepts from several mathematical areas usually treated in separate books stochastic processes, information theory, and Lie theory thereby building bridges between topics rarely studied by the same individuals</p><p>Extensive exercises and numerous examples used to motivate concepts with an emphasis on modeling physical phenomena</p><p>Concrete presentation makes it easy for readers to obtain numerical solutions for their own problems</p><p>Applications to a variety of areas, includingconformational fluctuations of DNA, infotaxis, statistical mechanics, and biomolecular information theory</p><p>Suitable as a textbook for advanced undergraduate and graduate courses in applied stochastic processes or differential geometry</p><p>For a broad audience of advanced undergraduate and graduate students, researchers, and practitioners in applied mathematics, the physical sciences, and engineering</p>978-0-8176-4295-2Christensen[Ole Christensen, Technical University of Denmark Department of Mathematics, Lyngby, Denmark)An Introduction to Frames and Riesz BasesXXI, 440 p.Preface Frames in Finite-dimensional Inner Product Spaces Infinite-dimensional Vector Spaces and Sequences Bases Bases and their Limitations Frames in Hilbert Spaces Frames versus Riesz Bases Frames of Translates Gabor Frames in L2(R) Selected Topics on Gabor Frames Gabor Frames in l2(Z) General Wavelet Frames Dyadic Wavelet Frames Frame Multiresolution Analysis Wavelet Frames via Extension Principles Perturbation of Frames Approximation of the Inverse Frame Operator Expansions in Banach Spaces Appendix List of Symbols References IndexThe theory for frames and bases has developed rapidly in recent years because of its role as a mathematical tool in signal and image processing. In this self-contained work, frames and Riesz bases are presented from a functional analytic point of view, emphasizing their mathematical properties. This is the first comprehensive book to focus on the general properties and interplay of frames and Riesz bases, and thus fills a gap in the literature.Basic results presented in an accessible way for both pure and applied mathematicians

Extensive exercises make the work suitable as a textbook for use in graduate courses

Full proofs included in introductory chapters; only basic knowledge of functional analysis required

Explicit constructions of frames with applications and connections to time-frequency analysis, wavelets, and nonharmonic Fourier series

978-0-387-98336-3ClarkeFFrancis H. Clarke; Yuri S. Ledyaev; Ronald J. Stern; Peter R. Wolenski%Nonsmooth Analysis and Control TheoryXIII, 278 p.SCM260168Calculus of Variations and Optimal Control; OptimizationPBKQProximal Calculus in Hilbert Space.- Generalized Gradients in Banach Space.- Special Topics.- A Short Course in Control Theory.In the last decades the subject of nonsmooth analysis has grown rapidly due to the recognition that nondifferentiable phenomena are more widespread, and play a more important role, than had been thought. In recent years, it has come to play a role in functional analysis, optimization, optimal design, mechanics and plasticity, differential equations, control theory, and, increasingly, in analysis. This volume presents the essentials of the subject clearly and succinctly, together with some of its applications and a generous supply of interesting exercises. The book begins with an introductory chapter which gives the reader a sampling of what is to come while indicating at an early stage why the subject is of interest. The next three chapters constitute a course in nonsmooth analysis and identify a coherent and comprehensive approach to the subject leading to an efficient, natural, yet powerful body of theory. The last chapter, as its name implies, is a self-contained introduction to the theory of control of ordinary differential equations. End-of-chapter problems also offer scope for deeper understanding. The authors have incorporated in the text a number of new results which clarify the relationships between the different schools of thought in the subject. Their goal is to make nonsmooth analysis accessible to a wider audience. In this spirit, the book is written so as to be used by anyone who has taken a course in functional analysis.978-1-4419-9836-1CohenLHarold Cohen, California State University, Los Angeles, Los Angeles, CA, USANumerical Approximation Methods H" 355/113XIII, 485p. 21 illus..SCP19021#Numerical and Computational PhysicsPreface. -1 INTERPOLATION and CURVE FITTING. -2 ZEROS of a FUNCTION. -3 SERIES. -4 INTEGRATION. -5 DETERMINANTS and MATRICES. -6 ORDINARY FIRST ORDER DIFFERENTIAL EQUATIONS. -7 ORDINARY SECOND ORDER DIFFERENTIAL EQUATIONS. -8 PARTIAL DIFFERENTIAL EQUATIONS. -9 LINEAR INTEGRAL EQUATIONS IN ONE VARIABLE. -APPENDIX 1 PADE APPROXIMANTS. -APPENDIX 2 INFINITE SERIES CONVERGENCE TESTS. -APPENDIX 3 GAMMA an< d BETA FUNCTIONS. -APPENDIX 4 PROPERTIES OF DETERMINANTS. -APPENDIX 5 PROOF OF THE SINGULARITY OF A MATRIX. References. -INDEX. This book presents numerical and other approximation techniques for solving various types of mathematical problems that cannot be solved analytically. In addition to well known methods, it contains some non-standard approximation techniques that are now formally collected as well as original methods developed by the author that do not appear in the literature.This book contains an extensive treatment of approximate solutions to various types of integral equations, a topic that is not often discussed in detail. There are detailed analyses of ordinary and partial differential equations and descriptions of methods for estimating the values of integrals that are presented in a level of detail that will suggest techniques that will be useful for developing methods for approximating solutions to problems outside of this text.The book is intended for researchers who must approximate solutions to problems that cannot be solved analytically. It is also appropriate for students taking courses in numerical approximation techniques.Focuses on the methods and algorithms of computation

Addresses concrete applications in science and engineering

Includes plenty of illustrated examples, with separate index table for quick access

978-1-4899-9159-1XIII, 485 p. 21 illus.978-3-0348-0293-2[Leon Cohen, City University of New York Hunter College & Graduate Center, New York, NY, USA(The Weyl Operator and its GeneralizationPseudo-Differential OperatorsXII, 159 p.Introduction.- The Fundamental Idea, Terminology, and Operator Algebra.- The Weyl Operator.- The Algebra of the Weyl Operator.- Product of Operators, Commutators, and the Moyal Sin Bracket.- Some Other Ordering Rules.- Generalized Operator Association.- The Fourier, Monomial, and Delta Function Associations.- Transformation Between Associations.- Path Integral Approach.- The Distribution of a Symbol and Operator.- The Uncertainty Principle.- Phase-Space Distributions.- Amplitude, Phase, Instantaneous Frequency, and the Hilbert Transform.- Time - Frequency Analysis.- The Transformation of Differential Equations into Phase Space.- The Representation of Functions.- The N Operator Case.`The discovery of quantum mechanics in the years 1925-1930 necessitated the consideration of associating ordinary functions with non-commuting operators. Methods were proposed by Born/Jordan, Kirkwood, and Weyl. Sometime later, Moyal saw the connection between the Weyl rule and the Wigner distribution, which had been proposed by Wigner in 1932 as a way of doing quantum statistical mechanics. The basic idea of associating functions with operators has since been generalized and developed to a high degree. It has found several application fields, including quantum mechanics, pseudo-differential operators, time-frequency analysis, quantum optics, wave propagation, differential equations, image processing, radar, and sonar. This book aims at bringing together the results from the above mentioned fields in a unified manner and showing the reader how the methods have been applied. A wide audience is addressed, particularly students and researchers who want to obtain an up-to-date working knowledge of the field. The mathematics is accessible to the uninitiated reader and is presented in a straightforward manner.Numerous examples are worked, out illustrating each result

The applications to various fields are discussed in elementary fashion

The methods are extended to operators other than the standard ones

978-1-85233-587-8CohnPaul M. Cohn Basic AlgebraGroups, Rings and FieldsXII, 465 p.1. Sets.- 2. Groups.- 3. Lattices and Categories.- 4. Rings and Modules.- 5. Algebras.- 6. Multilinear Algebra.- 7. Field Theory.- 8. Quadratic Forms and Ordered Fields.- 9. Valuation Theory.- 10. Commutative Rings.- 11. Infinite Field Extensions.- List of Notations.- Author Index. Basic Algebra is the first volume of a new and revised edition of P.M. Cohn's classic three-volume text Algebra which is widely regarded as one of the most outstanding introductory algebra textbooks. For this edition, the text has been reworked and updated into two self-contained, companion volumes, covering advanced topics in algebra for second- and third-year undergraduate and postgraduate research students. In this first volume, the author covers the important results of algebra; the companion volume, Further Algebra and Applications, brings more advanced topics and focuses on the applications. Readers should have some knowledge of linear algebra and have met groups and fields before, although all the essential facts and definitions are recalled. The coverage is comprehensive and includes topics such as: - Groups - lattices and categories - rings, modules and algebras - fields The author gives a clear account, supported by worked examples, with full proofs. There are numerous exercises with occasional hints, and some historical remarks.A revised and updated version of a classic text

Advanced topics in algebra are presented by one of the foremost algebraists in the UK

978-3-642-33130-5ColletPierre Collet, CNRS et Centre de Physique Thorique, Palaiseau, France; Servet Martnez, University of Chile Faculty of Mathematical, Santiago, Chile; Jaime San Martn, University of Chile Faculty of Mathematical, Santiago, ChileQuasi-Stationary Distributions/Markov Chains, Diffusions and Dynamical Systems)XV, 280 p. 15 illus., 12 illus. in color.<1.Introduction.- 2.Quasi-stationary Distributions: General Results.- 3.Markov Chains on Finite Spaces.- 4.Markov Chains on Countable Spaces.- 5.Birth and Death Chains.- 6.Regular Diffusions on [0,").- 7.Infinity as Entrance Boundary.- 8.Dynamical Systems.- References.- Index.- Table of Notations.- Citations Index.Main concepts of quasi-stationary distributions (QSDs) for killed processes are the focus of the present volume. For diffusions, the killing is at the boundary and for dynamical systems there is a trap. The authors present the QSDs as the ones that allow describing the long-term behavior conditioned to not being killed. Studies in this research area started with Kolmogorov and Yaglom and in the last few decades have received a great deal of attention. The authors provide the exponential distribution property of the killing time for QSDs, present the more general result on their existence and study the process of trajectories that survive forever. For birth-and-death chains and diffusions, the existence of a single or a continuum of QSDs is described. They study the convergence to the extremal QSD and give the classification of the survival process. In this monograph, the authors discuss Gibbs QSDs for symbolic systems and absolutely continuous QSDs for repellers. The findings described are relevant to researchers in the fields of Markov chains, diffusions, potential theory, dynamical systems, and in areas where extinction is a central concept. The theory is illustrated with numerous examples. The volume uniquely presents the distribution behavior of individuals who survive in a decaying population for a very long time. It also provides the background for applications in mathematical ecology, statistical physics, computer sciences, and economics.<p>Deals withan area that has received a lot of attention in last decades </p><p>Provides numerous examples </p><p>Focuses on selected topics </p>978-3-642-42888-3< 978-1-4614-4941-6ColtonDavid Colton, University of Delaware Dept. Mathematical Science, Newark, DE, USA; Rainer Kress, Georg-August-Universitt Gttingen Institut fuer Numerische und Angewandte, Gttingen, Germany6Inverse Acoustic and Electromagnetic Scattering TheoryXIV, 405 p. 8 illus.SCM12090Integral EquationsPBKL]Introduction.- The Helmholtz Equation.- Direct Acoustic Obstacle Scattering.- III-Posed Problems.- Inverse Acoustic Obstacle Scattering.- The Maxwell Equations.- Inverse Electromagnetic Obstacle Scattering.- Acoustic Waves in an Inhomogeneous Medium.- Electromagnetic Waves in an Inhomogeneous Medium.- The Inverse Medium Problem.-References.- IndexrThe inverse scattering problem is central to many areas of science and technology such as radar and sonar, medical imaging, geophysical exploration and nondestructive testing. This book is devoted to the mathematical and numerical analysis of the inverse scattering problem for acoustic and electromagnetic waves. In this third edition, new sections have been added on the linear sampling and factorization methods for solving the inverse scattering problem as well as expanded treatments of iteration methods and uniqueness theorems for the inverse obstacle problem. These additions have in turn required an expanded presentation of both transmission eigenvalues and boundary integral equations in Sobolev spaces. As in the previous editions, emphasis has been given to simplicity over generality thus providing the reader with an accessible introduction to the field of inverse scattering theory.Review of earlier editions: Colton and Kress have written a scholarly, state of the art account of their view of direct and inverse scattering. The book is a pleasure to read as a graduate text or to dip into at leisure. It suggests a number of open problems and will be a source of inspiration for many years to come. SIAM Review, September 1994 This book should be on the desk of any researcher, any student, any teacher interested in scattering theory. Mathematical Intelligencer, June 1994New sections included

Chapters updated throughout

This book has become the standard reference book in the field of inverse scattering theory

978-1-4899-9983-2978-3-7643-7002-2Corry Leo Corry6Modern Algebra and the Rise of Mathematical Structures451 p.Introduction: Structures in Mathematics.- One: Structures in the Images of Mathematics.- 1 Structures in Algebra: Changing Images.- 2 Richard Dedekind: Numbers and Ideals.- 3 David Hilbert: Algebra and Axiomatics.- 4 Concrete and Abstract: Numbers, Polynomials, Rings.- 5 Emmy Noether: Ideals and Structures.- Two: Structures in the Body of Mathematics.- 6 Oystein Ore: Algebraic Structures.- 7 Nicolas Bourbaki: Theory ofStructures.- 8 Category Theory: Early Stages.- 9 Categories and Images of Mathematics.- Author Index.o The book describes two stages in the historical development of the notion of mathematical structures: first, it traces its rise in the context of algebra from the mid-nineteenth century to its consolidation by 1930, and then it considers several attempts to formulate elaborate theories after 1930 aimed at elucidating, from a purely mathematical perspective, the precise meaning of this idea. First published in the series Science Networks Historical Studies, Vol. 17 (1996). In the second rev. edition the author has eliminated misprints, revised the chapter on Richard Dedekind, and updated the bibliographical index..Updated, improved, and revised

978-3-642-34099-4Costa9Oswaldo Luiz do Valle Costa, University of So Paulo Polytechnic School, So Paulo, Brazil; Marcelo D. Fragoso, Nat. Lab. for Scientific Computing Department of Systems and Control, Petrpolis, Brazil; Marcos G. Todorov, Nat. Lab. for Scientific Computing Department of Systems and Control, Petrpolis, RJ, Brazil*Continuous-Time Markov Jump Linear Systems)XII, 286 p. 17 illus., 9 illus. in color. 1.Introduction.- 2.A Few Tools and Notations.- 3.Mean Square Stability.- 4.Quadratic Optimal Control with Complete Observations.- 5.H2 Optimal Control With Complete Observations.- 6.Quadratic and H2 Optimal Control with Partial Observations.- 7.Best Linear Filter with Unknown (x(t), (t)).- 8.H_$infty$ Control.- 9.Design Techniques.- 10.Some Numerical Examples.- A.Coupled Differential and Algebraic Riccati Equations.- B.The Adjoint Operator and Some Auxiliary Results.- References. - Notation and Conventions.- Index."It has been widely recognized nowadays the importance of introducing mathematical models that take into account possible sudden changes in the dynamical behavior of a high-integrity systems or a safety-critical system. Such systems can be found in aircraft control, nuclear power stations, robotic manipulator systems, integrated communication networks and large-scale flexible structures for space stations, and are inherently vulnerable to abrupt changes in their structures caused by component or interconnection failures. In this regard, a particularly interesting class of models is the so-called Markov jump linear systems (MJLS), which have been used in numerous applications including robotics, economics and wireless communication. Combining probability and operator theory, the present volume provides a unified and rigorous treatment of recent results in control theory of < continuous-time MJLS. This unique approach is of great interest to experts working in the field of linear systems with Markovian jump parameters or in stochastic control. The volume focuses on one of the few cases of stochastic control problems with an actual explicit solution and offers material well-suited to coursework, introducing students to an interesting and active research area. The book is addressed to researchers working in control and signal processing engineering. Prerequisites include a solid background in classical linear control theory, basic familiarity with continuous-time Markov chains and probability theory, and some elementary knowledge of operator theory. <p>Introduces an active and interesting research area </p><p>Presents a unified and rigorous treatment of recent results in control theory </p><p>Numerous applications in safety critical and high-integrity systems including robotics, economics and wireless communication </p>978-3-642-43112-8978-0-8176-8393-1de MouraCarlos A. de Moura, Rio de Janeiro State University, Rio de Janeiro, Brazil; Carlos S. Kubrusly, Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil+The Courant Friedrichs Lewy (CFL) Condition80 Years After Its Discovery+XII, 237 p. 118 illus., 40 illus. in color.Foreword.- Stability of Different Schemes.- Mathematical Intuition: Poincar, Plya, Dewey.-Three-dimensional Plasma Arc Simulation using Resistive MHD.- A Numerical Algorithm for Ambrosetti-Prodi Type Operators.- On the Quadratic Finite Element Approximation of 1-D Waves: Propagation, Observation, Control, and Numerical Implementation.- Space-Time Adaptive Mutilresolution Techniques for Compressible Euler Equations.- A Framework for Late-time/stiff Relaxation Asymptotics.- Is the CFL Condition Sufficient? Some Remarks.- Fast Chaotic Artificial Time Integration.- Appendix A.- Hans Lewy's Recovered String Trio.- Appendix B.- Appendix C.- Appendix D.Thisvolume comprises a carefully selectedcollection ofarticlesemerging from and pertinent to the 2010 CFL-80 conference in Rio de Janeiro,celebrating the80th anniversary of the Courant-Friedrichs-Lewy (CFL) condition. A major result in the field of numerical analysis, the CFL condition has influenced the research of many important mathematicians over the past eight decades, and this work is meant to take stock of its most important and current applications.The Courant Friedrichs Lewy (CFL) Condition: 80 Years After its Discovery will be of interest to practicing mathematicians, engineers, physicists, and graduate students who work with numerical methods.All articles carefully selected and written by well-known experts

Provides a survey of the current state of the field

Includes original research results

978-0-387-95227-7DebarreOlivier Debarre%Higher-Dimensional Algebraic GeometryXIII, 234 p.1 Curves and Divisors on Algebraic Varieties.- 2 Parametrizing Morphisms.- 3 Bend-and-Break Lemmas.- 4 Uniruled and Rationally Connected Varieties.- 5 The Rational Quotient.- 6 The Cone of Curves in the Smooth Case.- 7 Cohomological Methods.- References.hHigher-Dimensional Algebraic Geometry studies the classification theory of algebraic varieties. This very active area of research is still developing, but an amazing quantity of knowledge has accumulated over the past twenty years. The author's goal is to provide an easily accessible introduction to the subject. The book covers in the beginning preparatory and standard definitions and results, moves on to discuss various aspects of the geometry of smooth projective varieties with many rational curves, and finishes in taking the first steps towards Mori's minimal model program of classification of algebraic varieties by proving the cone and contraction theorems. The book is well-organized and the author has kept the number of concepts that are used but not proved to a minimum to provide a mostly self-contained introduction to graduate students and researchers. 978-1-4419-2917-4978-0-8176-8264-4DebnathFLokenath Debnath, University of Texas, Pan American, Edinburg, TX, USAENonlinear Partial Differential Equations for Scientists and EngineersXXIII, 860p. 104 illus..Preface to the Third Edition.- Preface.- Linear Partial Differential Equations.- Nonlinear Model Equations and Variational Principles.- First-Order, Quasi-Linear Equations and Method of Characteristics.- First-Order Nonlinear Equations and Their Applications.- Conservation Laws and Shock Waves.- Kinematic Waves and Real-World Nonlinear Problems.- Nonlinear Dispersive Waves and Whitham's Equations.- Nonlinear Diffusion-Reaction Phenomena.- Solitons and the Inverse Scattering Transform.- The Nonlinear Schroedinger Equation and Solitary Waves.- Nonlinear Klein--Gordon and Sine-Gordon Equations.- Asymptotic Methods and Nonlinear Evolution Equations.- Tables of Integral Transforms.- Answers and Hints to Selected Exercises.- Bibliography.- Index.-The revised and enlarged third edition of this successful book presents a comprehensive and systematic treatment of linear and nonlinear partial differential equations and their varied and updated applications. In an effort to make the book more useful for a diverse readership, updated modern examples of applications are chosen from areas of fluid dynamics, gas dynamics, plasma physics, nonlinear dynamics, quantum mechanics, nonlinear optics, acoustics, and wave propagation.Nonlinear Partial Differential Equations for Scientists and Engineers, Third Edition,improves onanalready highlycomplete and accessible resource for graduate students and professionals in mathematics, physics, science, and engineering. It may be used to great effect as a course textbook, research reference, or self-study guide.MExtraordinary breadth of coverage

Highly systematic presentation

Hundreds of examples and exercises, with hints, selected solutions, and a complete solutions manual for instructors

Useful for students and researchers alike in not only mathematics, but also engineering, physics, and other natural sciences

978-3-642-23898-7 Deuflhard<Peter Deuflhard, Zuse-Institut Berlin (ZIB), Berlin, Germany%Newton Methods for Non< linear Problems)Affine Invariance and Adaptive AlgorithmsXII, 424p. 49 illus..This book deals with the efficient numerical solution of challenging nonlinear problems in science and engineering, both in finite dimension (algebraic systems) and in infinite dimension (ordinary and partial differential equations). Its focus is on local and global Newton methods for direct problems or Gauss-Newton methods for inverse problems. The term 'affine invariance' means that the presented algorithms and their convergence analysis are invariant under one out of four subclasses of affine transformations of the problem to be solved. Compared to traditional textbooks, the distinguishing affine invariance approach leads to shorter theorems and proofs and permits the construction of fully adaptive algorithms. Lots of numerical illustrations, comparison tables, and exercises make the text useful in computational mathematics classes. At the same time, the book opens many directions for possible future research.978-0-387-90317-0Dhrymes P. J. DhrymesIntroductory Econometrics480 p.SCS0000XStatistics, general1 The General Linear Model I.- 1.1 Introduction.- 1.2 Model Specification and Estimation.- 1.3 Goodness of Fit.- Questions and Problems.- 2 The General Linear Model II.- 2.1 Generalities.- 2.2 Distribution of the Estimator of ?.- 2.3 General Linear Restriction: Estimation and Tests.- 2.4 Mixed Estimators and the Bayesian Approach.- Questions and Problems.- 3 The General Linear Model III.- 3.1 Generalities.- 3.2 Violation of Standard Error Process Assumptions.- Questions and Problems.- 4 The General Linear Model IV.- 4.1 Multicollinearity: Failure of the Rank Condition.- 4.2 Analysis of Variance: Categorical Explanatory Variables.- 4.3 Analysis of Covariance: Some Categorical and Some Continuous Explanatory Variables.- 5 Misspecification Analysis and Errors in Variables.- 5.1 Introduction.- 5.2 Misspecification Analysis.- 5.3 Errors in Variables (EIV): Bivariate Model.- 5.4 Errors in Variables (EIV): General Model.- 5.5 Misspecification Error Analysis for EIV Models.- Questions and Problems.- 6 Systems of Simultaneous Equations.- 6.1 Introduction.- 6.2 The Simultaneous Equations Model (SEM): Definitions, Conventions, and Notation.- 6.3 The Identification Problem.- 6.4 Estimation of the GLSEM.- 6.5 Prediction from the GLSEM.- 6.6 The GLSEM and Undersized Samples.- 6.7 Maximum Likelihood (ML) Estimators.- Questions and Problems.- 7 Discrete Choice Models: Logit and Probit Analysis.- 7.1 Introduction.- 7.2 The Nature of Discrete Choice Models.- 7.3 Formulation of Dichotomous Choice Models.- 7.4 A Behavioral Justification for the Dichotomous Choice Model.- 7.5 Inapplicability of OLS Procedures 3.- 7.6 Maximum Likelihood Estimation.- 7.7 Inference for Discrete Choice Models.- 7.8 Polytomous Choice Models.- 8 Statistical and Probabilistic Background.- 8.1 Multivariate Density and Distribution Functions.- 8.2 The Multivariate Normal Distribution.- 8.3 Point Estimation.- 8.4 Elements of Bayesian Inference.- Questions and Problems.- Tables for Testing Hypotheses on the Autoregressive Structure of the Errors in a GLM.- References.This book represents a first course in econometrics, assuming only some knowledge of elementary probability theory and statistics on the part of the student. Its rigorous and comprehensive discussion concentrates on the general linear model, treating the standard case as well as the consequences resulting from violation of the underlying assumptions. Extensively documented chapters also cover the misspecification problem and errors in the variable model, simultaneous equations models and, uniquely, Multiple Comparison Test, Durbin- Watson Theory, Power Functions and Bayesian Analysis. Each chapter concludes with carefully selected exercises.978-0-8176-4526-7DiBenedettoZEmmanuele DiBenedetto, Vanderbilt University Department of Mathematics, Nashville, TN, USAClassical Mechanics Theory and Mathematical ModelingCornerstonesXX, 351p. 63 illus..Preface.- Geometry of Motion.- Constraints and Lagrangian Coordinates.- Dynamics of a Point Mass.- Geometry of Masses.- Systems Dynamics.- The Lagrange Equations.- Precessions.- Variational Principles.- Bibliography.- Index.eClassical mechanics is a chief example of the scientific method organizing a 'complex' collection of information into theoretically rigorous, unifying principles; in this sense, mechanics represents one of the highest forms of mathematical modeling. This textbook covers standard topics of a mechanics course, namely, the mechanics of rigid bodies, Lagrangian and Hamiltonian formalism, stability and small oscillations, an introduction to celestial mechanics, and Hamilton Jacobi theory, but at the same time features unique examples such as the spinning top including friction and gyroscopic compass seldom appearing in this context. In addition, variational principles like Lagrangian and Hamiltonian dynamics are treated in great detail. Using a pedagogical approach, the author covers many topics that are gradually developed and motivated by classical examples. Through `Problems and Complements' sections at the end of each chapter, the work presents various questions in an extended presentation that is extremely useful for an interdisciplinary audience trying to master the subject. Beautiful illustrations, unique examples, and useful remarks are key features throughout the text. Classical Mechanics: Theory and Mathematical Modeling may serve as a textbook for advanced graduate students in mathematics, physics, engineering, and the natural sciences, as well as an excellent reference or self-study guide for applied mathematicians and mathematical physicists. Prerequisites include a working knowledge of linear algebra, multivariate calculus, the basic theory of ordinary differential equations, and elementary physics.* Offers a rigorous mathematical treatment of mechanics as a text or reference

* Revisits beautiful classical material, including gyroscopes, precessions, spinning tops, effects of rotation of the Earth on gravity motions, and variational principles

* Employs mathematics not only as a "unifying" language, but also to exemplify its role as a catalyst behind new concepts and discoveries

978-0-387-94020-5Degenerate Parabolic EquationsXV, 387 pp. 12 figs.<I. Notation and function spaces.- 1. Some notation.- 2. Basic facts aboutW1,p(?) andWo1,p(?).- 3. Parabolic spaces and embeddings.- 4. Auxiliary lemmas< .- 5. Bibliographical notes.- II. Weak solutions and local energy estimates.- 1. Quasilinear degenerate or singular equations.- 2. Boundary value problems.- 3. Local integral inequalities.- 4. Energy estimates near the boundary.- 5. Restricted structures: the levelskand the constant ?.- 6. Bibliographical notes.- III. Hlder continuity of solutions of degenerate parabolic equations.- 1. The regularity theorem.- 2. Preliminaries.- 3. The main proposition.- 4. The first alternative.- 5. The first alternative continued.- 6. The first alternative concluded.- 7. The second alternative.- 8. The second alternative continued.- 9. The second alternative concluded.- 10. Proof of Proposition 3.1.- 11. Regularity up tot= 0.- 12. Regularity up toST. Dirichlet data.- 13. Regularity atST. Variational data.- 14. Remarks on stability.- 15. Bibliographical notes.- IV. Hlder continuity of solutions of singular parabolic equations.- 1. Singular equations and the regularity theorems.- 2. The main proposition.- 3. Preliminaries.- 4. Rescaled iterations.- 5. The first alternative.- 6. Proof of Lemma 5.1. Integral inequalities.- 7. An auxiliary proposition.- 8. Proof of Proposition 7.1 when (7.6) holds.- 9. Removing the assumption (6.1).- 10. The second alternative.- 11. The second alternative concluded.- 12. Proof of the main proposition.- 13. Boundary regularity.- 14. Miscellaneous remarks.- 15. Bibliographical notes.- V. Boundedness of weak solutions.- 1. Introduction.- 2. Quasilinear parabolic equations.- 3. Sup-bounds.- 4. Homogeneous structures. 2.- 5. Homogeneous structures. The singular case 1max\left\{ {1;\frac{{2N}} {{N + 2}}} \right\}} \right) $$.- 9. Global iterative inequalities.- 10. Homogeneous structures and $$ 1 < p \leqslant max\left\{ {1;\frac{{2N}} {{N + 2}}} \right\} $$.- 11. Proof of Theorems 3.1 and 3.2.- 12. Proof of Theorem 4.1.- 13. Proof of Theorem 4.2.- 14. Proof of Theorem 4.3.- 15. Proof of Theorem 4.5.- 16. Proof of Theorems 5.1 and 5.2.- 17. Natural growth conditions.- 18. Bibliographical notes.- VI. Harnack estimates: the casep>2.- 1. Introduction.- 2. The intrinsic Harnack inequality.- 3. Local comparison functions.- 4. Proof of Theorem 2.1.- 5. Proof of Theorem 2.2.- 6. Global versus local estimates.- 7. Global Harnack estimates.- 8. Compactly supported initial data.- 9. Proof of Proposition 8.1.- 10. Proof of Proposition 8.1 continued.- 11. Proof of Proposition 8.1 concluded.- 12. The Cauchy problem with compactly supported initial data.- 13. Bibliographical notes.- VII. Harnack estimates and extinction profile for singular equations.- 1. The Harnack inequality.- 2. Extinction in finite time (bounded domains).- 3. Extinction in finite time (in RN).- 4. An integral Harnack inequality for all 1 2).- 4. Hlder continuity ofDu (the case 1

2).- 5. Estimating the local average of |Dw| (the casep> 2).- 6. Estimating the local averages of w (the casep> 2).- 7. Comparing w and y (the case max $$ \left\{ {1;\tfrac{{2N}} {{N + 2}}} \right\Mathematicians have only recently begun to understand the local structure of solutions of degenerate and singular parabolic partial differential equations. The problem originated in the mid '60s with the work of DeGiorgi, Moser, Ladyzenskajia and Uraltzeva. This book will be an account of the developments in this field over the past five years. It evolved out of the 1990-Lipschitz Lectures given by Professor DiBenedetto at the Institut fr angewandte Mathematik of the University, Bonn.978-0-387-76634-8Diwekar(Urmila Diwekar, Clarendon Hills, IL, USA$Introduction to Applied OptimizationXXVI, 292p. 100 illus..SCC27000)Industrial Chemistry/Chemical EngineeringTDCLinear Programming.- Nonlinear Programming.- Discrete Optimization.- Optimization Under Uncertainty.- Multiobjective Optimization.- Optimal Control and Dynamic Optimization.The wide scope of optimization mandates extensive interaction between various disciplines in the development of the methods and algorithms, and in their fruitful application to real-world problems. This book presents a discipline-independent view of optimization, providing opportunities for students to identify and apply algorithms, methods, and tools from the diverse areas of optimization to their own fields without getting into too much detail about the underlying theories. The second edition of this book includes two new chapters: a chapter on global optimization and a real-world case study that uses principles from each chapter. Key Features: (1) Provides self-contained chapters, including problem sets and exercises; (2) Introduces applied optimization with several unique applications, i.e., hazardous waste blending problem; (3) Explores a number or important methods, i.e., the simplex method, weighting method, constraint method, and goal programming method; (4) Explores several different types of optimization, i.e., discrete, global, multi-objective, and dynamic optimization; (5) Includes an extensive bibliography at the end of each chapter. This book is intended for a variety of scientists, engineers, researchers, and advanced students interested in applied optimization.

Implemented in a classroom setting or used for independent study

Self-contained chapters that include problem sets and exercises

Provides a thorough introduction to applied optimization with unique applications

Introduces a number or important results in the field

Includes an extensive bibliography at the end of each chapter

Solutions manual available upon adoptions

978-1-4419-4570-9978-0-387-94599-6DixonJohn D. Dixon; Brian MortimerPermutation GroupsXII, 348 p.SCM11086K-Theory1. The Basic Ideas.- 1.1. Symmetry.- 1.2. Symmetric Groups.- 1.3. Group Actions.- 1.4. Orbits and Stabilizers.- 1.5. Blocks and Primitivity.- 1.6. Permutation Representations and Normal Subgroups.- 1.7. Orbits and Fixed Points.- 1.8. Some Examples from the Early History of Permutation Groups.- 1.9. Notes.- 2. Examples and Constructions.- 2.1. Actions on k-tuples and Subsets.- 2.2. Automorphism Groups of Algebraic Structures.- 2.3. Graphs.- 2.4. Relations.- 2.5. Semidirect Products.- 2.6. Wreath Products and Imprimitive Groups.- 2.7. Primitive Wreath Products.- 2.8. Affine and Projective Groups.- 2.9. The Transitive Groups of Degree at Most 7.- 2.10. Notes.- 3. The Action of a Permutation Group.- 3.1. Introduction.- 3.2. Orbits of the Stabilizer.- 3.3. Minimal Degree and Bases.- 3.4. Frobenius Groups.- 3.5. Permutation Groups Which Contain a Regular Subgroup.- 3.6.< Computing in Permutation Groups.- 3.7. Notes.- 4. The Structure of a Primitive Group.- 4.1. Introduction.- 4.2. Centralizers and Normalizers in the Symmetric Group.- 4.3. The Socle.- 4.4. Subnormal Subgroups and Primitive Groups.- 4.5. Constructions of Primitive Groups with Nonregular Socles.- 4.6. Finite Primitive Groups with Nonregular Socles.- 4.7. Primitive Groups with Regular Socles.- 4.8. Applications of the O Nan-Scott Theorem.- 4.9. Notes.- 5. Bounds on Orders of Permutation Groups.- 5.1. Orders of Elements.- 5.2. Subgroups of Small Index in Finite Alternating and Symmetric Groups.- 5.3. The Order of a Simply Primitive Group.- 5.4. The Minimal Degree of a 2-transitive Group.- 5.5. The Alternating Group as a Section of a Permutation Group.- 5.6. Bases and Orders of 2-transitive Groups.- 5.7. The Alternating Group as a Section of a Linear Group.- 5.8. Small Subgroups of Sn.- 5.9. Notes.- 6. The Mathieu Groups and Steiner Systems.- 6.1. The Mathieu Groups.- 6.2. Steiner Systems.- 6.3. The Extension of AG2 (3).- 6.4. The Mathieu Groups M 11 and M12.- 6.5. The Geometry of PG 2 (4).- 6.6. The Extension of PG 2 (4) and the Group M 22.- 6.7. The Mathieu Groups M 23 and M 24.- 6.8. The Geometry of W24.- 6.9. Notes.- 7. Multiply Transitive Groups.- 7.1. Introduction.- 7.2. Normal Subgroups.- 7.3. Limits to Multiple Transitivity.- 7.4. Jordan Groups.- 7.5. Transitive Extensions.- 7.6. Sharply k-transitive Groups.- 7.7. The Finite 2-transitive Groups.- 7.8. Notes.- 8. The Structure of the Symmetric Groups.- 8.1. The Normal Structure of Sym(?).- 8.2. The Automorphisms of Sym(?).- 8.3. Subgroups of F Sym(?).- 8.4. Subgroups of Small Index in Sym(?).- 8.5. Maximal Subgroups of the Symmetric Groups.- 8.6. Notes.- 9. Examples and Applications of Infinite Permutation Groups.- 9.1. The Construction of a Finitely Generated Infinite p-group.- 9.2. Groups Acting on Trees.- 9.3. Highly Transitive Free Subgroups of the Symmetric Group.- 9.4. Homogeneous Groups.- 9.5. Automorphisms of Relational Structures.- 9.6. The Universal Graph.- 9.7. Notes.- Appendix A. Classification of Finite Simple Groups.- Appendix B. The Primitive Permutation Groups of Degree Less than 1000.- References.Permutation Groups form one of the oldest parts of group theory. Through the ubiquity of group actions and the concrete representations which they afford, both finite and infinite permutation groups arise in many parts of mathematics and continue to be a lively topic of research in their own right. The book begins with the basic ideas, standard constructions and important examples in the theory of permutation groups.It then develops the combinatorial and group theoretic structure of primitive groups leading to the proof of the pivotal O'Nan-Scott Theorem which links finite primitive groups with finite simple groups. Special topics covered include the Mathieu groups, multiply transitive groups, and recent work on the subgroups of the infinite symmetric groups. This text can serve as an introduction to permutation groups in a course at the graduate or advanced undergraduate level, or for self- study. It includes many exercises and detailed references to the current literature.978-3-540-52018-4Dodson`C. T. J. Dodson, University of Manchester Mathematical Institute, Manchester, UK; Timothy PostonTensor Geometry$The Geometric Viewpoint and its UsesXIV, 434 p.u0. Fundamental Not(at)ions.- I. Real Vector Spaces.- II. Affine Spaces.- III. Dual Spaces.- IV. Metric Vector Spaces.- V. Tensors and Multilinear Forms.- VI Topological Vector Spaces.- VII. Differentiation and Manifolds.- VIII. Connections and Covariant Differentiation.- IX. Geodesics.- X. Curvature.- XI. Special Relativity.- XII. General Relativity.- Index of Notations.This treatment of differential geometry and the mathematics required for general relativity makes the subject of this book accessible for the first time to anyone familiar with elementary calculus in one variable and with a knowledge of some vector algebra. The emphasis throughout is on the geometry of the mathematics, which is greatly enhanced by the many illustrations presenting figures of three and more dimensions as closely as book form will allow. The imaginative text is a major contribution to expounding the subject of differential geometry as applied to studies in relativity, and will prove of interest to a large number of mathematicians and physicists. Review from L'Enseignement Mathmatique978-3-642-27460-2 Donati-MartinCatherine Donati-Martin, Universit de Versailles, Versailles, France; Antoine Lejay, Nancy-Universit, INRIA IECN, Campus scientifique, BP 239, Vandoeuvre-ls-Nancy, France; Alain Rouault, Universit de Versailles-St-Quentin Laboratoire de Mathmatiques, Versailles, FranceSminaire de Probabilits XLIV+VIII, 469 p. 17 illus., 10 illus. in color.Context trees, variable length Markov chains and dynamical sources.- Martingale property of generalized stochastic exponentials.- Some classes of proper integrals and generalized Ornstein-Uhlenbeck processes.- Martingale representations for diffusion processes and backward stochastic differential equations.- Quadratic Semimartingale BSDEs Under an Exponential Moments Condition.- The derivative of the intersection local time of Brownian motion through Wiener chaos.- On the occupation times of Brownian excursions and Brownian loops.- Discrete approximation to solutio< n flows of Tanaka s SDE related to Walsh Brownian motion.- Spectral Distribution of the Free unitary Brownian motion: another approach.- Another failure in the analogy between Gaussian and semicircle laws.- Global solutions to rough differential equations with unbounded vector fields.- Asymptotic behavior of oscillatory fractional processes.- Time inversion property for rotation invariant self-similar diffusion processes.- On Peacocks: a general introduction to two articles.- Some examples of peacocks in a Markovian set-up.- Peacocks obtained by normalisation; strong and very strong peacocks.- Branching Brownian motion: Almost sure growth along scaled paths.- On the delocalized phase of the random pinning model.- Large deviations for Gaussian stationary processes and semi-classical analysis.- Girsanov theory under a finite entropy condition.LAs usual, some of the contributions to this 44th Sminaire de Probabilits were presented during the Journes de Probabilits held in Dijon in June 2010. The remainder were spontaneous submissions or were solicited by the editors. The traditional and historical themes of the Sminaire are covered, such as stochastic calculus, local times and excursions, and martingales. Some subjects already touched on in the previous volumes are still here: free probability, rough paths, limit theorems for general processes (here fractional Brownian motion and polymers), and large deviations. Lastly, this volume explores new topics, including variable length Markov chains and peacocks. We hope that the whole volume is a good sample of the main streams of current research on probability and stochastic processes, in particular those active in France.]This volume provides a broad insights on current, high level researches in probability theory978-0-387-87820-1DontchevAssen L. Dontchev, University of Michigan Dept. Mathematics, Ann Arbor, MI, USA; R. Tyrrell Rockafellar, University of Washington Dept. Mathematics, Seattle, WA, USA(Implicit Functions and Solution Mappings A View from Variational AnalysisXII, 376p. 12 illus..%Functions Defined Implicitly by Equations.- Implicit Function Theorems for Variational Problems.- Regularity properties of set-valued solution mappings.- Regularity Properties Through Generalized Derivatives.- Regularity in infinite dimensions.- Applications in Numerical Variational Analysis.The implicit function theorem is one of the most important theorems in analysis and its many variants are basic tools in partial differential equations and numerical analysis. This book treats the implicit function paradigm in the classical framework and beyond, focusing largely on properties of solution mappings of variational problems. The purpose of this self-contained work is to provide a reference on the topic and to provide a unified collection of a number of results which are currently scattered throughout the literature. The first chapter of the book treats the classical implicit function theorem in a way that will be useful for students and teachers of undergraduate calculus. The remaining part becomes gradually more advanced, and considers implicit mappings defined by relations other than equations, e.g., variational problems. Applications to numerical analysis and optimization are also provided. This valuable book is a major achievement and is sure to become a standard reference on the topic.An attractive blend of modern topics and classical results

The authors have included a nice selection of exercises for classroom use

Ideal reference work which contains a large amount of historical material and references

978-1-4419-2771-2978-0-387-94055-7Doob J.L. DoobMeasure TheoryXII, 212 p.v0. Conventions and Notation.- I. Operations on Sets.- II. Classes of Subsets of a Space.- III. Set Functions.- IV. Measure Spaces.- V. Measurable Functions.- VI. Integration.- VII. Hilbert Space.- VIII. Convergence of Measure Sequences.- IX. Signed Measures.- X. Measures and Functions of Bounded Variation on R.- XI. Conditional Expectations ; Martingale Theory.- Notation.:This book is different from other books on measure theory in that it accepts probability theory as an essential part of measure theory. This means that many examples are taken from probability; that probabilistic concepts such as independence, Markov processes, and conditional expectations are integrated into the text rather than being relegate to an appendix; that more attention is paid to the role of algebras than is customary; and that the metric defining the distance between sets as the measure of their symmetric difference is exploited more than is customary.978-1-4614-1700-2DuDing-Zhu Du, University of Texas, Dallas, Richardson, TX, USA; Ker-I Ko, Stony Brook University, Stony Brook, NY, USA; Xiaodong Hu, Beijing, China, People's Republic/Design and Analysis of Approximation AlgorithmsXII, 440 p.SCI16021)Algorithm Analysis and Problem ComplexityUMBPreface.- 1. Introduction.- 2. Greedy Strategy.- 3. Restriction.- 4. Partition.- 5. Guillotine Cut.- 6. Relaxation.- 7. Linear Programming.- 8. Primal-Dual Scheme and Local Ratio.- 9. Semidefinite Programming.- 10. Inapproximability.- Bibliography.- Index.This book is intended to be used as a textbook for graduate students studying theoretical computer science. It can also be used as a reference book for researchers in the area of design and analysis of approximation algorithms. Design and Analysis of Approximation Algorithms is a graduate course in theoretical computer science taught widely in the universities, both in the United States and abroad. There are, however, very few textbooks available for this course. Among those available in the market, most books follow a problem-oriented format; that is, they collected many important combinatorial optimization problems and their approximation algorithms, and organized them based on the types, or applications, of problems, such as geometric-type problems, algebraic-type problems, etc. Such arrangement of materials is perhaps convenient for a researcher to look for the problems and algorithms related to his/her work, but is difficult for a student to capture the ideas underlying the various algorithms. In the new book proposed here, we follow a more structured, technique-oriented presentation. We organize approximation algorithms into different chapters, based on the design techniques for the algorithms, so that the reader can study approximation algorithms of the same nature together. It helps the reader to better understand the design and analysis techniques for approximation algorithms, and also helps the teacher to present the ideas and techniques of approximation algorithms in a more unified way.The technique-oriented approach provides a unified view of the design techniques for approximation algorithms

Detailed algorithms, as well as complete proofs and analyses, are presented for each technique

Numerous examples help the reader to better understand the design and analysis techniques

Collects a great number of applications, many from recent research papers

Includes a large collection of approximation algorithms of geometric problems

978-1-4899-9844-6978-0-387-96162-0DubrovinB.A. Dubrovin; A.T. Fomenko, Lo< monosov Moscow State University Dept. Mathematics & Mechanics, Moskva, Russia; S.P. Novikov, MIAN, Moscow GSP 1, Russia)Modern Geometry Methods and Applications/Part II: The Geometry and Topology of Manifolds XV, 432 p.1 Examples of Manifolds.- 1. The concept of a manifold.- 2. The simplest examples of manifolds.- 3. Essential facts from the theory of Lie groups.- 4. Complex manifolds.- 5. The simplest homogeneous spaces.- 6. Spaces of constant curvature (symmetric spaces).- 7. Vector bundles on a manifold.- 2 Foundational Questions. Essential Facts Concerning Functions on a Manifold. Typical Smooth Mappings.- 8. Partitions of unity and their applications.- 9. The realization of compact manifolds as surfaces in ?N.- 10. Various properties of smooth maps of manifolds.- 11. Applications of Sard s theorem.- 3 The Degree of a Mapping. The Intersection Index of Submanifolds. Applications.- 12. The concept of homotopy.- 13. The degree of a map.- 14. Applications of the degree of a mapping.- 15. The intersection index and applications.- 4 Orientability of Manifolds. The Fundamental Group. Covering Spaces (Fibre Bundles with Discrete Fibre).- 16. Orientability and homotopies of closed paths.- 17. The fundamental group.- 18. Covering maps and covering homotopies.- 19. Covering maps and the fundamental group. Computation of the fundamental group of certain manifolds.- 20. The discrete groups of motions of the Lobachevskian plane.- 5 Homotopy Groups.- 21. Definition of the absolute and relative homotopy groups. Examples.- 22. Covering homotopies. The homotopy groups of covering spaces and loop spaces.- 23. Facts concerning the homotopy groups of spheres. Framed normal bundles. The Hopf invariant.- 6 Smooth Fibre Bundles.- 24. The homotopy theory of fibre bundles.- 25. The differential geometry of fibre bundles.- 26. Knots and links. Braids.- 7 Some Examples of Dynamical Systems and Foliations on Manifolds.- 27. The simplest concepts of the qualitative theory of dynamical systems. Two-dimensional manifolds.- 28. Hamiltonian systems on manifolds. Liouville s theorem. Examples.- 29. Foliations.- 30. Variational problems involving higher derivatives.- 8 The Global Structure of Solutions of Higher-Dimensional Variational Problems.- 31. Some manifolds arising in the general theory of relativity (GTR).- 32. Some examples of global solutions of the Yang-Mills equations. Chiral fields.- 33. The minimality of complex submanifolds.978-0-8176-8246-0DuistermaatSJ.J. Duistermaat, Universiteit Utrecht Mathematisch Instituut, Utrecht, NetherlandsKThe Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac OperatorVIII, 247p.1 Introduction.- 2 The Dolbeault-Dirac Operator.- 3 Clifford Modules.- 4 The Spin Group and the Spin-c Group.- 5 The Spin-c Dirac Operator.- 6 Its Square.- 7 The Heat Kernel Method.- 8 The Heat Kernel Expansion.- 9 The Heat Kernel on a Principal Bundle.- 10 The Automorphism.- 11 The Hirzebruch-Riemann-Roch Integrand.- 12 The Local Lefschetz Fixed Point Formula.- 13 Characteristic Case.- 14 The Orbifold Version.- 15 Application to Symplectic Geometry.- 16 Appendix: Equivariant Forms.Reprinted as it originally appeared in the 1990s,this work is as an affordable textthat will be of interest to a range of researchers in geometric analysis and mathematical physics. Thebook covers avarietyof concepts fundamental tothe study and applications of the spin-c Dirac operator, making use of the heat kernels theory of Berline, Getzlet, and Vergne. True to the precision and clarity for which J.J. Duistermaat was so well known, the exposition is elegant and concise.Affordable softcover edition ofthe only book everpublished on the subject

Written by one of the leading geometric analysts of the late 20th century

Presents interesting applications of theory

Givesan accessible approach to the field

978-0-387-98945-7Dullerud#Geir E. Dullerud; Fernando Paganini!A Course in Robust Control TheoryA Convex Approach XX, 419 p.0 Introduction.- 1 Preliminaries in Finite Dimensional Space.- 2 State Space System Theory.- 3 Linear Analysis.- 4 Model Realizations and Reduction.- 5 Stabilizing Controllers.- 6 H2 Optimal Control.- 7 H? Synthesis.- 8 Uncertain Systems.- 9 Feedback Control of Uncertain Systems.- 10 Further Topics: Analysis.- 11 Further Topics: Synthesis.- A Some Basic Measure Theory.- A.1 Sets of zero measure.- A.2 Terminology.- Notes and references.- B Proofs of Strict Separation.- Notes and references.- Notes and references.- Notation.- References.Research in robust control theory has been one of the most active areas of mainstream systems theory since the late 70s. This research activity has been at the confluence of dynamical systems theory, functional analysis, matrix analysis, numerical methods, complexity theory, and engineering applications. The discipline has involved interactions between diverse research groups including pure mathematicians, applied mathematicians, computer scientists and engineers. This research effort has produced a rather extensive set of approaches using a wide variety of mathematical techniques, and applications of robust control theory are spreading to areas as diverse as control of fluids, power networks, and the investigation of feddback mechanisms in biology. During the 90's the theory has seen major advances and achieved a new maturity, centered around the notion of convexity. The goal of this book is to give a graduate-level course on robust control theory that emphasizes these new developments, but at the same time conveys the main principles and ubiquitous tools at the heart of the subject. Its pedagogical objectives are to introduce a coherent and unified framework for studying robust control theory, to provide students with the control-theoretic background required to < read and contribute to the research literature, and to present the main ideas and demonstrations of the major results of robust control theory. The book will be of value to mathematical researchers and computer scientists wishing to learn about robust control theory, graduate students planning to do research in the area, and engineering practitioners requiring advanced control techniques.978-1-4419-3189-4978-1-4419-0235-1DzemydaGintautas Dzemyda, Vilnius University, Vilnius, Lithuania; Olga Kurasova, Vilnius University, Vilnius, Lithuania; Julius }ilinskas, Vilnius University, Vilnius, Lithuania#Multidimensional Data VisualizationMethods and Applications+XII, 250 p. 122 illus., 38 illus. in color.SCI21025Simulation and ModelingUGK7Preface.- 1.Multidimensional Data and the Concept of Visualization.- 2. Strategies for Multidimensional Data Visualization.- 3. Optimization-Based Visualization.- 4. Combining Multidimensional Scaling with Artificial Neural Networks.- 5. Applications of Visualizations.- A. Test Data Sets.- References.- Index.This book highlights recent developments in multidimensional data visualization, presentingboth new methodsandmodifications onclassic techniques.Throughout thebook, various applications ofmultidimensional data visualizationare presented including itsuses insocial sciences (economy, education, politics, psychology), environmetrics, andmedicine (ophthalmology, sport medicine, pharmacology, sleep medicine).The book provides recent research results in optimization-based visualization. Evolutionary algorithmsand atwo-level optimization method, based on combinatorial optimization and quadratic programming, are analyzed in detail. The performance of thesealgorithms and thedevelopment of parallelversionsare discussed.Theutilization of newvisualization techniquesto improve thecapabilies ofartificial neural networks (self-organizing maps, feed-forward networks)is alsodiscussed.The bookincludesover 100 detailedimages presenting examples of the manydifferent visualization techniques thatthe book presents.This book is intended for scientists and researchers inany field of studywhere complex andmultidimensionaldata must be represented visually.<P>" Presents an overview of multidimensional data visualization</P> <P>" Provides backgroud to construction, analysis, and implementation of optimization algorithms for visualization of multidimensional data</P> <P><EM>" </EM> Shows benefits of artificial neural networks and their integrated use with other methods for visualization of multidimensional data</P> <P><EM>" </EM> Presents various applications of multidimensional data visualization: from social sciences to medicine</P>978-1-4899-9000-6978-0-387-94258-2cH.-D. Ebbinghaus; J. Flum; Wolfgang Thomas, Aachen University LS fr Informatik 7, Thomas, GermanyMathematical Logic X, 291 p.SCO25000Mathematics EducationJNUA.- I Introduction.- II Syntax of First-Order Languages.- III Semantics of First-Order Languages.- IV A Sequent Calculus.- V The Completeness Theorem.- VI The Lwenheim-Skolem and the Compactness Theorem.- VII The Scope of First-Order Logic.- VIII Syntactic Interpretations and Normal Forms.- B.- IX Extensions of First-Order Logic.- X Limitations of the Formal Method.- XI Free Models and Logic Programming.- XII An Algebraic Characterization of Elementary Equivalence.- XIII Lindstrm s Theorems.- References.- Symbol Index.qThis junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete - it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. Fraiss's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic.978-0-85729-020-5 EinsiedlerManfred Einsiedler, ETH Zrich Departement Mathematik, Zrich, Switzerland; Thomas Ward, University of East Anglia School of Mathematics, Norwich, UKErgodic Theory!with a view towards Number Theory XVIII, 482 p.Motivation.- Ergodicity, Recurrence and Mixing.- Continued Fractions.- Invariant Measures for Continuous Maps.- Conditional Measures and Algebras.- Factors and Joinings.- Furstenberg s Proof of Szemeredi s Theorem.- Actions of Locally Compact Groups.- Geodesic Flow on Quotients of the Hyperbolic Plane.- Nilrotation.- More Dynamics on Quotients of the Hyperbolic Plane.- Appendix A: Measure Theory.- Appendix B: Functional Analysis.- Appendix C: Topological GroupsThis text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence. Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits. Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.With a rigorous development of basic ergodic theory and homogeneous dynamics, no background in Ergodic theory or Lie theory is assumed Offers both complete and motivated treatments of Weyl and Szemeredi theorems P< rovides a number of exercises and hints to problems978-1-4471-2591-4978-3-540-60931-5 EmbrechtsPaul Embrechts, ETH Zrich, Zrich, Switzerland; Claudia Klppelberg, Munich University of Technology Center for Mathematical Sciences, Garching by Munich, Germany; Thomas Mikosch, University Copenhagen Inst. Mathematical Sciences, Copenhagen, DenmarkModelling Extremal Eventsfor Insurance and Finance XV, 648 p.Reader Guidelines.- Risk Theory.- Fluctuations of Sums.- Fluctuations of Maxima.- Fluctuations of Upper Order Statistics.- An Approach to Extremes via Point Processes.- Statistical Methods for Extremal Events.- Time Series Analysis for Heavy-Tailed Processes.- Special Topics.Both in insurance and in finance applications, questions involving extremal events (such as large insurance claims, large fluctuations in financial data, stock market shocks, risk management, ...) play an increasingly important role. This book sets out to bridge the gap between the existing theory and practical applications both from a probabilistic as well as from a statistical point of view. Whatever new theory is presented is always motivated by relevant real-life examples. The numerous illustrations and examples, and the extensive bibliography make this book an ideal reference text for students, teachers and users in the industry of extremal event methodology.978-3-642-08242-9978-88-470-2426-7Emmer@Michele Emmer, Universit di Roma Dipto. Matematica, Roma, ItalyImagine MathBetween Culture and MathematicsVII, 289 p.SCM320001Mathematics in the Humanities and Social SciencesSpringer MilanThe Many Faces of Lorenz Knots by M. Abate.- Lost in a Good Book: Jorge Borges Inescapable Labyrinth by W.G. Bloch.- The Mobius Strip by E. Blondeau.- The Unreasonable Effectiveness of Mathematics in Human Sciences: The Attribution of Texts to Antonio Gramsci by D. Benedetto, E. Caglioti, M. Degli Esposti.- Exactitude and Extravagance: Andrea Pozzo s Viewpoint by F. Camerota.- Numeracy, Metrology and Mathematics in Mesopotamia: Social and Cultural Practices by G. Chambon.- The FantasticWorld of Tor Bled-Nam by M. Emmer.- From Brigitte Bardot to Angelina Jolie by M. Emmer.- Visual Harmonies: an Exhibition on Art and Math by M. Emmer.- The Reconstruction of the Teatro La Fenice: splendidezze and dorature (Gleam and Gilding) by E. Fabbri.- Hypatia s Dream by M. Vincenzi.- E Pluribus Unum by M. LiCalzi.- Origami and Partial Differential Equations by P. Marcellini, E. Paolini.- Modern geometry versus modern architecture by I. Birindelli, R. Cedrone.- The Apse Scenes in the Prospective Inventions of Andrea Pozzo by S. Carandini.- Emilio Prini, Alison Knowles, and Art s Logic by C. Lauf.- Connecting Ventricular Assist Devices to the Aorta: a Numerical Model by J. Bonnemain, S. Deparis, A. Quarteroni.- Aperiodic Tiling by G.M. Todesco.- Andrea Pozzo: Art, Culture and Mathematics by M. Costamagna.- Women s Contributions to the Progress of Mathematics: Lights and Shadows by E. Strickland.- The Mathematical Ideas of Luca Pacioli Depicted by Iacopo de Barbari in the Doppio ritratto by E. Gamba.- All the numbers end in numbers. On a work by Alighiero Boetti by A. Valle.- De divina proportione: From a Renaissance Treatise to a Multimedia Work for Theatre by S. Sorini.- Waiting for ABRACADABRA. Occurrences of words and leading numbers by E. De Santis, F. Spizzichino.-0Imagine mathematics, imagine with the help of mathematics, imagine new worlds, new geometries, new forms. This bookis intended to contribute to grasping how much that is interesting and new is happening in the relationships between mathematics, imagination and culture. With a look at the past, at figures and events, that help to understand the phenomena of today.It is no coincidence that this volume contains an homage to the great Italian artist of the 1700s, Andrea Pozzo, and his perspective views. Theatre, art and architecture are the topics of choice, along with music, literature and cinema. No less important are applications of mathematics to medicine and economics.The treatment is rigorous but captivating, detailed but full of evocations, an all-embracing look at the world of mathematics and cultureA very unique book with many papers on the various aspects of mathematics and culture

Papers by experts in different topics, with a relevant numbers of images

An interesting story, that continues the series of math and culture

978-3-540-20100-7Michele Emmer, University of Rome, Italy, Roma, Italy; Doris Schattschneider, Moravian College Dept. Mathematics, Bethlehem, PA, USAM.C. Escher s LegacyA Centennial CelebrationXVI, 458 p.SCA11007Science, generalYQSEscher s Fondness for Animals.- Selection is Distortion.- Ravello: An Escherian Place.- Mystery, Classicism, Elegance: an Endless Chase After Magic.- M.C. Escher and C.v.S. Roosevelt.- Escher s Sense of Wonder.- In Search of M.C. Escher s Metaphysical Unconscious.- Parallel Worlds: Escher and Mathematics, Revisited.- M.C. Escher in Italy: The Trail Back.- Islamic Patterns: The Spark in Escher s Genius.- Space Time with M.C. Escher and R. Buckminster Fuller.- Between Illusion and Reality.- Painting After M.C. Escher.- M.C. Escher: Art, Math, and Cinema.- Organic Structures Related to M.C. Escher s Work.- Extending Escher s Recognizable-Motif Tilings to Multiple-Solution Tilings and Fractal Tilings.- A Circle Limit in Stone.- Portrait of Escher: Behind the Mirror.- Life After Escher: A (Young) Artist s Journey.- Shar< ing some Common Interests of M.C. Escher.- New Expressions in Tessellating Art: Layered Three-Dimensional Tessellations.- The Mirrors of the Master.- Tilings and Other Unusual Escher-Related Prints.- Escher-Like Patterns from Pentagonal Tilings.- Not the Tiles, but the Joints: A little Bridge Between M.C. Escher and Leonardo da Vinci.- Architecture, Perspective and Scenography in the Graphic Work of M.C. Escher: From Vredeman de Vries to Luca Ronconi.- Hand with Reflective Sphere to Six-Point Perspective Sphere.- Families of Escher Patterns.- The Trigonometry of Escher s Woodcut Circle Limit III.- Escher in the Classroom.- Chaotic Geodesic Motion: An Extension of M.C. Escher s Circle Limit Designs.- Rotations and Notations.- Folding Rings of Eight Cubes.- Dethronement of the Symmetry Plane.- Computer Games Based on Escher s Spatial Illusions.- Escher s World: Structure, Symmetry, Sense.- Adapting Escher s Rules for Regular Division of the Plane to Create TesselMania!.- M.C. Escher at the Museum: An Educator s Perspective.- Escher, Napoleon, Fermat and the Nine-point Centre.- The Symmetry Mystique.- Escher-Like Tessellations on Spherical Models.!From the reviews of the hardcover edition: ... This conference [... to celebrate the centennial of the birth of Escher] resulted in an immensely interesting collection of articles ... Although Escher himself is no longer among us, M.C.Escher's Legacy, like a garden of continually blooming flowers, allows us to appreciate his heritage anew. Notices of the AMS April 2003 ... It is a handsome volume, and contains articles from 41 people, which cover a wide range of artistic and analytical endeavour. ... A quick dip into each section produces small gems. ... there is enough here to provide rich pickings for any interested party, no matter what their particular discipline is. Embedded in the various articles are even snippets which illuminate Escher's intentions, and his relationships with his mathematician friends ... Even though short, these are rewarding to read. ... the CD-ROM ... is an excellent addition to the book, and contains much more material, including 'video' excerpts from some of the lectures.' Australian Math. Soc. GAZETTE May 2003Rich with illustrations, both of Escher's work and of work by contemporary artists whose art is inspired by that of Escher

New insights into Escher's work, including articles by the well-known commentators on M.C. Escher, D. Hofstadter and B. Ernst

CD-ROM with color illustrations, movies, animations, and other demonstrations accompanies the text and provides essential add-on value to readers

Trade978-0-387-31341-2EngelKlaus-Jochen Engel, Universita L'Aquila Facolta di Ingegneria, Roio Poggio, Italy; Rainer Nagel, Universitt Tbingen Mathematisches Institut, Tbingen, Germany%A Short Course on Operator Semigroups X, 247 p.Semigroups, Generators, and Resolvents.- Perturbation of Semigroups.- Approximation of Semigroups.- Spectral Theory and Asymptotics for Semigroups.- Positive Semigroups.The book offers a direct and up-to-date introduction to the theory of one-parameter semigroups of linear operators on Banach spaces. It contains the fundamental results of the theory such as the Hille-Yoshida generation theorem, the bounded perturbation theorem, and the Trotter-Kato approximation theorem, but also treats the spectral theory of semigroups and its consequences for the qualitative behavior. The book is intended for students and researchers who want to become acquainted with the concept of semigroups in order to work with it in fields like partial and functional differential equations, stochastic processes, infinite dimensional control theory, or dynamical systems coming from physics or biology.#Special feature -- treatment of spectral theory leading to a detailed qualitative theory for semigroups

Concise introduction providing modern tools for the study of linear evolution equations

Exercises at the end of chapters

High quality and careful exposition

978-1-4419-2174-1978-0-387-75338-6Enns=Richard H. Enns, Simon Fraser University, Burnaby, BC, CanadaIt's a Nonlinear World)XII, 384p. 145 illus., 8 illus. in color.}Preface.- Part I. World of Mathematics.- 1. World of Nonlinear Systems.- 2. World of Nonlinear ODEs.- 3. World of Nonlinear Maps.- 4. World of Solitons.- Part II. Our Nonlinear World.- 5. World of Motion.- 6. World of Sports.- 7. World of Electromagnetism.- 8. World of Weather Prediction.- 9. World of Chemistry.- 10. World of Disease.- 11. World of War.- Bibliography.- Index.~Drawing examplesfrom mathematics, physics, chemistry, biology, engineering, economics, medicine, politics, and sports, this book illustrates how nonlinear dynamics plays a vital role in our world. Examples cover a wide range from the spread and possible control of communicable diseases, to the lack of predictability in long-range weather forecasting, to competition between political groups and nations.After an introductorychapter that explores what it means to be nonlinear, the book covers the mathematical conceptssuch as limit cycles, fractals, chaos, bifurcations, and solitons, that will be applied throughout the book. Numerous computer simulations and exercises allow students to explore topics in greater depth using the Maple computer algebra system. The mathematical level of the text assumes prior exposure to ordinary differential equations and familiarity with the wave and diffusion equations.No prior knowledge of Maple is assumed.The book may be used at the undergraduate or graduate level to prepare science and engineering students for problems in the 'real world', or for self-study by practicing scientists and engineers.Explores in each chapter nonlinear phenomena drawn from a unique discipline within the physical and social sciences, engineering, and medicine978-0-387-96824-7Euler&Leonhard Euler, St. Petersburg, Russia(Introduction to Analysis of the InfiniteBook I XV, 327 p.KI. On Functions in General.- II. On the Transformation of Functions.- III. On the Transformation of Functions by Substitution.- IV. On the Development of Functions in Infinite Series.- V. Concerning Functions of Two or More Variables.- VI. On Exponentials and Logarithms.- VII. Exponentials and Logarithms Expressed through Series.- VIII. On Transcendental Quantities Which Arise from the Circle.- IX. On Trinomial Factors.- X. On the Use of the Discovered Factors to Sum Infinite Series.- XI. On Other Infinite Expressions for Arcs and Sines.- XII. On the Development of Real Rational Functions.- XIII. On Recurrent Series.- XIV. On the Multiplication and Division of Angles.- XV. On Series Which Arise from Products.- XVI. On the Partition of Numbers.- XVII. Using Recurrent Series to Find Roots of Equations.- XVIII. On Continued Fractions.From the preface of the author: '...I have divided this work into two books; in the first of these I have confined myself to those mat< ters concerning pure analysis. In the second book I have explained those thing which must be known from geometry, since analysis is ordinarily developed in such a way that its application to geometry is shown. In the first book, since all of analysis is concerned with variable quantities and functions of such variables, I have given full treatment to functions. I have also treated the transformation of functions and functions as the sum of infinite series. In addition I have developed functions in infinite series...'978-0-387-20158-0Evertsz; MandelbrotiBenoit Mandelbrot, Yale University Dept. Mathematics, New Haven, CT, USA; C.J.G. Evertsz; M.C. GutzwillerFractals and ChaosThe Mandelbrot Set and BeyondXII, 308 p.Collected worksList of Chapters.- C1 Introduction to papers on quadratic dynamics: a progression from seeing to discovering (2003).- C2 Acknowledgments related to quadratic dynamics (2003).- C3 Fractal aspects of the iteration of z ? ? z (1-z) for complex A and z (M1980n).- C4 Cantor and Fatou dusts; self-squared dragons (M 1982F).- C5 The complex quadratic map and its M-set (M1983p).- C6 Bifurcation points and the n squared approximation and conjecture (M1985g), illustrated by M.L Frame and K Mitchell.- C7 The normalized radical of the M-set (M1985g).- C8 The boundary of the M-set is of dimension 2 (M1985g).- C9 Certain Julia sets include smooth components (M1985g).- C10 Domain-filling sequences of Julia sets, and intuitive rationale for the Siegel discs (M1985g).- C11 Continuous interpolation of the quadratic map and intrinsic tiling of the interiors of Julia sets (M1985n).- C12 Introduction to chaos in nonquadratic dynamics: rational functions devised from doubling formulas (2003).- C13 The map z ? ? (z + 1/z) and roughening of chaos from linear to planar (computer-assisted homage to K Hokusai) (M1984k).- C14 Two nonquadratic rational maps, devised from Weierstrass doubling formulas (1979 2003).- C15 Introduction to papers on Kleinian groups, their fractal limit sets, and IFS: history, recollections, and acknowledgments (2003).- C16 Self-inverse fractals, Apollonian nets, and soap (M 1982F).- C17 Symmetry by dilation or reduction, fractals, roughness (M2002w).- C18 Self-inverse fractals osculated by sigma-discs and limit sets of inversion ( Kleinian ) groups (M1983m).- C19 Introduction to measures that vanish exponentially almost everywhere: DLA and Minkowski (2003).- C20 Invariant multifractal measures in chaotic Hamiltonian systems and related structures (Gutzwiller & M 1988).- C21 The Minkowski measure and multifractal anomalies in invariant measures of parabolic dynamic systems (M1993s).- C22 Harmonic measure on DLA and extended self-similarity (M & Evertsz 1991).- C23 The inexhaustible function z squared plus c (1982 2003).- C24 The Fatou and Julia stories (2003).- C25 Mathematical analysis while in the wilderness (2003).- Cumulative Bibliography.aIt has only been a couple of decades since Benoit Mandelbrot published his famous picture of what is now called the Mandelbrot set. That picture, now seeming graphically primitive, has changed our view of the mathematical and physical universe. The properties and circumstances of the discovery of the Mandelbrot Set continue to generate much interest in the research community and beyond. This book contains the hard-to-obtain original papers, many unpublished illustrations dating back to 1979 and extensive documented historical context showing how Mandelbrot helped change our way of looking at the world.RDocumented history of discovery of the Mandelbrot Set written by Mandelbrot

978-1-4419-1897-0978-0-387-95219-2FabianMarian Fabian, ASCR Praha Mathematical Institute, Praha 1, Czech Republic; Petr Habala, Czech Technical University Praha Faculty of Electrical Engineering, Praha, Czech Republic; Petr Hajek, ASCR Praha Mathematical Institute, Praha 1, Czech Republic; Vicente Montesinos Santalucia, Univ. Politecnica de Valencia Dept. Matematica Aplicada, Valencia, Spain; Jan Pelant, ASCR Praha Inst. Mathematics, Praha 1, Czech Republic; Vaclav Zizler, ASCR Praha Mathematical Institute, Praha 1, Czech Republic5Functional Analysis and Infinite-Dimensional GeometryCMS Books in Mathematics IX, 451 p.~1 Basic Concepts in Banach Spaces.- 2 Hahn Banach and Banach Open Mapping Theorems.- 3 Weak Topologies.- 4 Locally Convex Spaces.- 5 Structure of Banach Spaces.- 6 Schauder Bases.- 7 Compact Operators on Banach Spaces.- 8 Differentiability of Norms.- 9 Uniform Convexity.- 10 Smoothness and Structure.- 11 Weakly Compactly Generated Spaces.- 12 Topics in Weak Topology.- References.This book introduces the reader to the basic principles of functional analysis and to areas of Banach space theory that are close to nonlinear analysis and topology. In the first part, the book develops the classical theory, including weak topologies, locally convex spaces, Schauder bases, and compact operator theory. The presentation is self-contained, including many folklore results, and the proofs are accessible to students with the usual background in real analysis and topology. The second part covers topics in convexity and smoothness, finite representability, variational principles, homeomorphisms, weak compactness and more. Several results are published here for the first time in a monograph. The text can be used in graduate courses or for independent study. It includes a large number of exercises of different levels of difficulty, accompanied by hints. The book is also directed to young researchers in functional analysis and can serve as a reference book.< 978-1-4419-2912-9978-0-387-95369-4FallCChristopher P. Fall; Eric S. Marland; John M. Wagner; John J. TysonComputational Cell Biology%Interdisciplinary Applied Mathematics XX, 469 p.Introductory Course.- Dynamic Phenomena in Cells.- Voltage Gated Ionic Currents.- Transporters and Pumps.- Fast and Slow Time Scales.- Whole Cell Models.- Intercellular Communication.- Advanced Material.- Spatial Modeling.- Modeling Intracellular Calcium Waves and Sparks.- Biochemical Oscillations.- Cell Cycle Controls.- Modeling the Stochastic Gating of Ion Channels.- Molecular Motors: Theory.- Molecular Motors: Examples.This textbook provides an introduction to dynamic modeling in molecular cell biology, taking a computational and intuitive approach. Selected biological examples are used to motivate concepts and techniques used in computational cell biology through a progression of increasingly more complex cellular functions modeled with increasingly complex mathematical and computational techniques. Detailed illustrations, examples, and exercises are included throughout the text. Appendices containing mathematical and computational techniques are provided as a reference tool. Advanced undergraduate and graduate theoretical biologists, and mathematics students and researchers who wish to learn about modeling in cell biology will find this book useful. 'What better tribute to the late Joel Keizer than to expand his unfinished accounts of teaching and research to a splendid book. COMPUTATIONAL CELL BIOLOGY performs much more than it promises, for it also deals with considerable analytical material and with aspects of molecular biology. There's something for everybody interested in how modeling leads to greater understanding in the core of the biological sciences.' -Lee Segal (Weizmann Institute)978-1-4419-2975-4978-1-4614-4468-8Fasano[Giorgio Fasano, Thales Alenia Space SpA, Turin, Italy; Jnos D. Pintr, Halifax, NS, Canada.Modeling and Optimization in Space Engineering+XII, 404 p. 147 illus., 90 illus. in color.SCT17050%Aerospace Technology and AstronauticsTRPPModel Development and Optimization for Space Engineering: Concepts, Tools, Applications and Perspectives. Jnos D. Pintr and Giorgio Fasano. -Practical Direct Collocation Methods for Computational Optimal Control. Victor M. Becerra. - Formation Flying Control for Satellites: Anti-windup Based Approach. Josep Boada, Christophe Prieur, Sophie Tarbouriech, Christelle Pittet and Catherine Charbonnel. - The ESA NLP Solver WORHP. Christof Bskens and Dennis Wassel. -Global Optimization Approaches for Optimal Trajectory Planning. Andrea Cassioli, David Di Lorenzo, Marco Locatelli and Fabio Schoen. -Indirect Methods for the Optimization of Spacecraft Trajectories. Guido Colasurdo and Lorenzo Casalino. Launch and Re-entry Vehicles. Francesco Cremaschi. -Global Optimization of Interplanetary Transfers with Deep Space Manoeuvres using Differential Algebra. Pierluigi Di Lizia, Roberto Armellin, Francesco Topputo, Franco Bernelli-Zazzera, and Martin Berz. -A Traffic Model for the International Space Station: An MIP Approach. Giorgio Fasano. -Global Optimization Approaches to Sensor Placement: Model Versions and Illustrative Results. Giorgio Fasano and Jnos D. Pintr. -Space Module On-Board Stowage Optimization Exploiting Container Empty Volumes. Giorgio Fasano and Maria Chiara Vola. -Optimization Models for theThree-dimensional Container Loading Problem with Practical Constraints. Leonardo Junqueira, Reinaldo Morabito, Denise SatoYamashita and Horacio Hideki Yanasse. -Optimal Magnetic Cleanliness Modelling of Spacecraft. Klaus Mehlem. -Integrated Design-Trajectory Optimization for Hybrid Rocket Motors. Dario Pastrone and Lorenzo Casalino. -Mathematical Models of Placement Optimisation: Two-and-three-dimensional Problems and Applications. Yuri Stoyan and Tatiana Romanova. -Optimization of Low Energy Transfers. Francesco Topputo and Edward Belbruno. ,This volume presents a selection of advanced case studies that address a substantial range of issues and challenges arising in space engineering. The contributing authors are well-recognized researchers and practitioners in space engineering and in applied optimization. The key mathematical modeling and numerical solution aspects of each application case study are presented in sufficient detail. Classic and more recent space engineering problems including cargo accommodation and object placement, flight control of satellites, integrated design and trajectory optimization, interplanetary transfers with deep space manoeuvres, low energy transfers, magnetic cleanliness modeling, propulsion system design, sensor system placement, systems engineering, space traffic logistics, and trajectory optimization are discussed. Novel points of view related to computational global optimization and optimal control, and to multidisciplinary design optimization a< re also given proper emphasis. A particular attention is paid also to scenarios expected in the context of future interplanetary explorations. Modeling and Optimization in Space Engineering will benefit researchers and practitioners working on space engineering applications. Academics, graduate and post-graduate students in the fields of aerospace and other engineering, applied mathematics, operations research and optimal control will also find the book useful, since it discusses a range of advanced model development and solution techniques and tools in the context of real-world applications and new challenges.Offers practical, real-world applications in space engineering

Answersand raises questions on thefuture space exploration

Includes detailed numerical case studies

978-1-4899-9737-1978-1-4419-7235-4FerrarioDavide L. Ferrario, Universit Milano-Bicocca Dipto. Matematica e Applicazioni, Milano, Italy; Renzo A. Piccinini, Dalhousie University Dept. Mathematics & Statistics, Halifax, NS, Canada!Simplicial Structures in TopologyXVI, 243 p.Preface.- Fundamental Concepts.- Simplicial Complexes.- Homology of Polyhedra.- Cohonology.- Triangulable Manifolds.- Homotopy Groups.- Bibliography.- IndexSimplicial Structures in Topology provides a clear and comprehensive introduction to the subject. Ideas are developed in the first four chapters. The fifth chapter studies closed surfaces and gives their classification. The last chapter of the book is devoted to homotopy groups, which are used in short introduction on obstruction theory. The text is more in tune with the original development of algebraic topology as given by Henry Poincar (singular homology is discussed). Illustrative examples throughout and extensive exercises at the end of each chapter for practice enhance the text. Advanced undergraduate and beginning graduate students will benefit from this book. Researchers and professionals interested in topology and applications of mathematics will also find this book useful.Contains extensive exercises for student practice

Creates a strong foundation in general topology before moving on to more specialized topics

Clarifies the text with many illustrative examples

978-1-4614-2698-1978-3-7643-8349-7 FerreirsPJos Ferreirs, Universidad de Sevilla Depto. Filosofia y Logica, Sevilla, SpainLabyrinth of Thought:A History of Set Theory and Its Role in Modern MathematicsXXV, 466 p. 7 illus.SCM35000Mathematical PhysicsThe Emergence of Sets within Mathematics.- Institutional and Intellectual Contexts in German Mathematics, 1800 1870.- A New Fundamental Notion: Riemann s Manifolds.- Dedekind and the Set-theoretical Approach to Algebra.- The Real Number System.- Origins of the Theory of Point-Sets.- Entering the Labyrinth-Toward Abstract Set Theory.- The Notion of Cardinality and the Continuum Hypothesis.- Sets and Maps as a Foundation for Mathematics.- The Transfinite Ordinals and Cantor s Mature Theory.- In Search of an Axiom System.- Diffusion, Crisis, and Bifurcation: 1890 to 1914.- Logic and Type Theory in the Interwar Period.- Consolidation of Axiomatic Set Theory.Labyrinth of Thought discusses the emergence and development of set theory and the set-theoretic approach to mathematics during the period 1850-1940. Rather than focusing on the pivotal figure of Georg Cantor, it analyzes his work and the emergence of transfinite set theory within the broader context of the rise of modern mathematics. The text has a tripartite structure. Part 1, The Emergence of Sets within Mathematics, surveys the initial motivations for a mathematical notion of a set within several branches of the discipline (geometry, algebra, algebraic number theory, real and complex analysis), emphasizing the role played by Riemann in fostering acceptance of the set-theoretic approach. In Part 2, Entering the Labyrinth, attention turns to the earliest theories of sets, their evolution, and their reception by the mathematical community; prominent are the epoch-making contributions of Cantor and Dedekind, and the complex interactions between them. Part 3, In Search of an Axiom System, studies the four-decade period from the discovery of set-theoretic paradoxes to Gdel s independence results, an era during which set theory gradually became assimilated into mainstream mathematics; particular attention is given to the interactions between axiomatic set theory and modern systems of formal logic, especially the interplay between set theory and type theory. A new Epilogue for this second edition offers further reflections on the foundations of set theory, including the 'dichotomy conception' and the well-known iterative conception.978-0-387-94657-3FineBenjamin Fine, Fairfield University Dept. Mathematics, Fairfield, CT, USA; Gerhard Rosenberger, Universitt Dortmund FB Mathematik, Dortmund, Germany"The Fundamental Theorem of Algebra XI, 210 p. 1 Introduction and Historical Remarks.- 2 Complex Numbers.- 2.1 Fields and the Real Field.- 2.2 The Complex Number Field.- 2.3 Geometrical Representation of Complex Numbers.- 2.4 Polar Form and Euler s Identity.- 2.5 DeMoivre s Theorem for Powers and Roots.- Exercises.- 3 Polynomials and Complex Polynomials.- 3.1 The Ring of Polynomials over a Field.- 3.2 Divisibility and Unique Factorization of Polynomials.- 3.3 Roots of Polynomials and Factorization.- 3.4 Real and Complex Polynomials.- 3.5 The Fu< ndamental Theorem of Algebra: Proof One.- 3.6 Some Consequences of the Fundamental Theorem.- Exercises.- 4 Complex Analysis and Analytic Functions.- 4.1 Complex Functions and Analyticity.- 4.2 The Cauchy-Riemann Equations.- 4.3 Conformal Mappings and Analyticity.- Exercises.- 5 Complex Integration and Cauchy s Theorem.- 5.1 Line Integrals and Green s Theorem.- 5.2 Complex Integration and Cauchy s Theorem.- 5.3 The Cauchy Integral Formula and Cauchy s Estimate.- 5.4 Liouville s Theorem and the Fundamental Theorem of Algebra: Proof Ttvo.- 5.5 Some Additional Results.- 5.6 Concluding Remarks on Complex Analysis.- Exercises.- 6 Fields and Field Extensions.- 6.1 Algebraic Field Extensions.- 6.2 Adjoining Roots to Fields.- 6.3 Splitting Fields.- 6.4 Permutations and Symmetric Polynomials.- 6.5 The Fundamental Theorem of Algebra: Proof Three.- 6.6 An Application The Transcendence of e and ?.- 6.7 The Fundamental Theorem of Symmetric Polynomials.- Exercises.- 7 Galois Theory.- 7.1 Galois Theory Overview.- 7.2 Some Results From Finite Group Theory.- 7.3 Galois Extensions.- 7.4 Automorphisms and the Galois Group.- 7.5 The Fundamental Theorem of Galois Theory.- 7.6 The Fundamental Theorem of Algebra: Proof Four.- 7.7 Some Additional Applications of Galois Theory.- 7.8 Algebraic Extensions of ? and Concluding Remarks.- Exercises.- 8 7bpology and Topological Spaces.- 8.1 Winding Number and Proof Five.- 8.2 Tbpology An Overview.- 8.3 Continuity and Metric Spaces.- 8.4 Topological Spaces and Homeomorphisms.- 8.5 Some Further Properties of Topological Spaces.- Exercises.- 9 Algebraic Zbpology and the Final Proof.- 9.1 Algebraic lbpology.- 9.2 Some Further Group Theory Abelian Groups.- 9.3 Homotopy and the Fundamental Group.- 9.4 Homology Theory and Triangulations.- 9.5 Some Homology Computations.- 9.6 Homology of Spheres and Brouwer Degree.- 9.7 The Fundamental Theorem of Algebra: Proof Six.- 9.8 Concluding Remarks.- Exercises.- Appendix A: A Version of Gauss s Original Proof.- Appendix B: Cauchy s Theorem Revisited.- Appendix C: Three Additional Complex Analytic Proofs of the Fundamental Theorem of Algebra.- Appendix D: Two More Ibpological Proofs of the Fundamental Theorem of Algebra.- Bibliography and References.RThe Fundamental Theorem of Algebra states that any complex polynomial must have a complex root. This basic result, whose first accepted proof was given by Gauss, lies really at the intersection of the theory of numbers and the theory of equations, and arises also in many other areas of mathematics. The purpose of this book is to examine three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. The first proof in each pair is fairly straightforward and depends only on what could be considered elementary mathematics. However, each of these first proofs lends itself to generalizations, which in turn, lead to more general results from which the fundamental theorem can be deduced as a direct consequence. These general results constitute the second prooof in each pair. To arrive at each of the proofs, enough of the general theory of each relevant area is developed to understand the proof. In addition to the proofs and techniques themselves, many applications such as the insolvability of the quintic and the trascendence of e and pi are presented. Finally, a series of appendices give six additional proofs including a version of Gauss' original first proof. The book is intended for junior/senior level undergraduate mathematics students or first year graduate students. It is ideal for a 'capstone' course in mathematics. It could also be used as an alternative approach to an undergraduate abstract algebra course. Finally, because of the breadth of topics it covers it would also be ideal for a graduate course for mathmatics teachers.978-0-387-26045-7FlemingWendell H. Fleming, Brown University Div. Applied Mathematics, Providence, RI, USA; Halil Mete Soner, ETH Zrich, Zrich, Switzerland3Controlled Markov Processes and Viscosity SolutionsXVII, 429 p.Deterministic Optimal Control.- Viscosity Solutions.- Optimal Control of Markov Processes: Classical Solutions.- Controlled Markov Diffusions in ?n.- Viscosity Solutions: Second-Order Case.- Logarithmic Transformations and Risk Sensitivity.- Singular Perturbations.- Singular Stochastic Control.- Finite Difference Numerical Approximations.- Applications to Finance.- Differential Games.bThis book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. A new Chapter X gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter VI of the First Edition has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter XI gives a concise introduction to two-controller, zero-sum differential games. Also covered are controlled Markov diffusions and viscosity solutions of Hamilton-Jacobi-Bellman equations. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g. large deviations theory) and with applications to engineering, physics, management, and finance. In this Second Edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.Provides a luckd introduction to optimal stochastic control for continuous time Markov processes and to the theo< ry of viscosity solutions

Also offers a concise introduction to risk-sensitive control theory, nonlinear H-infinity control and differential games

Several all-new chapters have been added, and others completely rewritten

For the Second Edition, new material has been added on application to mathematical finance

978-1-4419-2078-2978-3-642-20553-8FreitagNEberhard Freitag, Universitt Heidelberg Inst. Mathematik, Heidelberg, GermanyComplex Analysis 2XRiemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular FunctionsXIII, 506p. 51 illus..Chapter I. Riemann Surfaces.- Chapter II. Harmonic Functions on Riemann Surfaces.- Chapter III. Uniformization.- Chapter IV. Compact Riemann Surfaces.- Appendices to Chapter IV.- Chapter V. Analytic Functions of Several Complex Variables.- Chapter V. Analytic Functions of Several Complex Variable.- Chapter VI. Abelian Functions.- Chapter VII. Modular Forms of Several Variables.- Chapter VIII. Appendix: Algebraic Tools.- References.- Index.GThe book contains a complete self-contained introduction to highlights of classical complex analysis. New proofs and some new results are included. All needed notions are developed within the book: with the exception of some basic facts which can be found in the rst volume. There is no comparable treatment in the literature.IAll needed notions are developed within the book with the exception of fundamentals, which are presented in introductory lectures; no other knowledge is assumed

Provides a more in-depth introduction to the subject than other existing books in this area

Manyexercises including hints for solutions are included

978-3-642-24608-1Frmond_Michel Frmond, Universit di Roma "Tor Vergata" Dipartimento di Ingegneria Civile, Roma, ItalyPhase Change in Mechanics/Lecture Notes of the Unione Matematica Italiana*XIII, 303p. 66 illus., 36 illus. in color.SCP25099(Phase Transitions and Multiphase SystemsPHFCPhysics_1 Introduction.- 2 The State Quantities and the Quantities Describing the Evolution.- 3 The Basic Laws of Mechanics.- 4 Solid-liquid Phase Change.- 5 Shape Memory Alloys.- 6 Damage.- 7 Contact with Adhesion.- 8 Damage of Solids Glued on One Another. Coupling of Volume and Surface Damages.- 9 Phase Change with Discontinuity of Temperature: Warm Water in Contact with Cold Ice.- 10 Phase Change and Collisions.- 11 Collisions of Deformable Bodies and Phase Change.- 12 Phase Change Depending on a State Quantity: Liquid-vapor Phase Change.- 13 Clouds: Mixture of Air, Vapor and Liquid Water.- 14 Conclusion.Predictive theories of phenomena involving phase change with applications in engineering are investigated in this volume, e.g. solid-liquid phase change, volume and surface damage, and phase change involving temperature discontinuities. Many other phase change phenomena such as solid-solid phase change in shape memory alloys and vapor-liquid phase change are also explored. Modeling is based on continuum thermo-mechanics. This involves a renewed principle of virtual power introducing the power of the microscopic motions responsible for phase change. This improvement yields a new equation of motion related to microscopic motions, beyond the classical equation of motion for macroscopic motions. The new theory sensibly improves the phase change modeling. For example, when warm rain falls on frozen soil, the dangerous black ice phenomenon can be comprehensively predicted. In addition, novel equations predict the evolution of clouds, which are themselves a mixture of air, liquid water and vapor.Thermomechanical predictive theories of phase change are improved by a renewed principle of virtual power

Coupling of volume damage and of surface adhesion are highly innovative for civil and mechanical engineering

A novel theory of macroscopic clouds evolution innovates in fluid mechanics

Mechanical and thermal effects of collisions, involving phase change are accurately and comprehensively predicted

978-0-8176-4206-8Fresnel Jean Fresnel; Marius van der Put,Rigid Analytic Geometry and Its Applications XI, 299 p.F 1 Valued Fields and Normed Spaces.- 1.1 Valued fields.- 1.2 Banach spaces and Banach algebras.- 2 The Projective Line.- 2.1 Some definitions.- 2.2 Holomorphic functions on an affinoid subset.- 2.3 The residue theorem.- 2.4 The Grothendieck topology on P.- 2.5 Some sheaves on P.- 2.6 Analytic subspaces of P.- 2.7 Cohomology on an analytic subspace of P.- 3 Affinoid Algebras.- 3.1 Definition of an affinoid algebra.- 3.2 Consequences of the Weierstrass theorem.- 3.3 Affinoid spaces, Examples.- 3.4 Properties of the spectral (semi-)norm.- 3.5 Integral extensions of affinoid algebras.- 3.6 The differential module ?A/kf.- 3.7 Products of affinoid spaces, Picard groups.- 4 Rigid Spaces.- 4.1 Rational subsets.- 4.2 The weak G-topology and Tate s theorem.- 4.3 General rigid spaces.- 4.4 Sheaves on a rigid space.- 4.5 Coherent analytic sheaves.- 4.6 The sheaf of meromorphic functions.- 4.7 Rigid vector bundles.- 4.8 Analytic reductions and formal schemes.- 4.9 Analytic reductions of a subspace of Pk1, an.- 4.10 Separated and proper rigid spaces.- 5 Curves and Their Reductions.- 5.1 The Tate curve.- 5.2 Nron models for abelian varieties.- 5.3 The Nron model of an elliptic curve.- 5.4 Mumford curves and Schottky groups.- 5.5 Stable reduction of curves.- 5.6 A rigid proof of stable reduction for curves.- 5.7 The universal analytic covering of a curve.- 6 Abelian Varieties.- 6.1 The complex case.- 6.2 The non-archimedean case.- 6.3 The analytification of an algebraic torus.- 6.4 Lattices and analytic tori.- 6.5 Meromorphic functions on an analytic torus.- 6.6 Analytic tori and abelian varieties.- 6.7 Nron models and uniformization.- 7 Points of Rigid Spaces, Rigid Cohomology.- 7.1 Points and sheaves on an affinoid space.- 7.2 Explicit examples in dimension 1.- 7.3 $$ \mathcal{P} $$(X) and the reductions of X.- 7.4 Base change for overconvergent sheaves.- 7.5 Overconvergent affinoid spaces.- 7.6 Monsky-Washnitzer cohomology.- 7.7 Rigid cohomology.- 8 Etale Cohomology o< f Rigid Spaces.- 8.1 Etale morphisms.- 8.2 The tale site.- 8.3 Etale points, overconvergent tale sheaves.- 8.4 Etale cohomology in dimension 1.- 8.5 Higher dimensional rigid spaces.- 9 Covers of Algebraic Curves.- 9.1 Introducing the problem.- 9.2 I. Serre s result.- 9.3 II. Rigid construction of coverings.- 9.4 III. Reductions of curves modulo p.- References.- List of Notation.yThe theory of rigid (analytic) spaces, originally invented to describe degenerations, reductions, and moduli of algebraic curves and abelian varieties, has undergone significant growth in the last two decades; today the theory has applications to arithmetic algebraic geometry, number theory, the arithmetic of function fields, and p-adic differential equations. This work, a revised and greatly expanded new English edition of the earlier French text by the same authors, is an accessible introduction to the theory of rigid spaces and now includes a large number of exercises. Key topics: - Chapters on the applications of this theory to curves and abelian varieties: the Tate curve, stable reduction for curves, Mumford curves, Nron models, uniformization of abelian varieties - Unified treatment of the concepts: points of a rigid space, overconvergent sheaves, Monsky--Washnitzer cohomology and rigid cohomology; detailed examination of Kedlaya s application of the Monsky--Washnitzer cohomology to counting points on a hyperelliptic curve over a finite field - The work of Drinfeld on 'elliptic modules' and the Langlands conjectures for function fields use a background of rigid tale cohomology; detailed treatment of this topic - Presentation of the rigid analytic part of Raynaud s proof of the Abhyankar conjecture for the affine line, with only the rudiments of that theory A basic knowledge of algebraic geometry is a sufficient prerequisite for this text. Advanced graduate students and researchers in algebraic geometry, number theory, representation theory, and other areas of mathematics will benefit from the book s breadth and clarity.<P>Chapters on the applications of this theory to curves and abelian varieties</P> <P>The work of Drinfeld on "elliptic modules" and the Langlands conjectures for function fields use a background of rigid tale cohomology; detailed treatment of this topic</P> <P>Presentation of the rigid analytic part of Raynaud s proof of the Abhyankar conjecture for the affine line, with only the rudiments of that theory</P>978-0-387-98361-5FriedmanRobert Friedman1Algebraic Surfaces and Holomorphic Vector Bundles IX, 329 p. 1 Curves on a Surface.- Invariants of a surface.- Divisors on a surface.- Adjunction and arithmetic genus.- The Riemann-Roch formula.- Algebraic proof of the Hodge index theorem.- Ample and nef divisors.- Exercises.- 2 Coherent Sheaves.- What is a coherent sheaf?.- A rapid review of Chern classes for projective varieties.- Rank 2 bundles and sub-line bundles.- Elementary modifications.- Singularities of coherent sheaves.- Torsion free and reflexive sheaves.- Double covers.- Appendix: some commutative algebra.- Exercises.- 3 Birational Geometry.- Blowing up.- The Castelnuovo criterion and factorization of birational morphisms.- Minimal models.- More general contractions.- Exercises.- 4 Stability.- Definition of Mumford-Takemoto stability.- Examples for curves.- Some examples of stable bundles on ?2.- Gieseker stability.- Unstable and semistable sheaves.- Change of polarization.- The differential geometry of stable vector bundles.- Exercises.- 5 Some Examples of Surfaces.- Rational ruled surfaces.- General ruled surfaces.- Linear systems of cubics.- An introduction toK3 surfaces.- Exercises.- 6 Vector Bundles over Ruled Surfaces.- Suitable ample divisors.- Ruled surfaces.- A brief introduction to local and global moduli.- A Zariski open subset of the moduli space.- Exercises.- 7 An Introduction to Elliptic Surfaces.- Singular fibers.- Singular fibers of elliptic fibrations.- Invariants and the canonical bundle formula.- Elliptic surfaces with a section and Weierstrass models.- More general elliptic surfaces.- The fundamental group.- Exercises.- 8 Vector Bundles over Elli< ptic Surfaces.- Stable bundles on singular curves.- Stable bundles of odd fiber degree over elliptic surfaces.- A Zariski open subset of the moduli space.- An overview of Donaldson invariants.- The 2-dimensional invariant.- Moduli spaces via extensions.- Vector bundles with trivial determinant.- Even fiber degree and multiple fibers.- Exercises.- 9 Bogomolov s Inequality and Applications.- Statement of the theorem.- The theorems of Bombieri and Reider.- The proof of Bogomolov s theorem.- Symmetric powers of vector bundles on curves.- Restriction theorems.- Appendix: Galois descent theory.- Exercises.- 10 Classification of Algebraic Surfaces and of Stable.- Bundles.- Outline of the classification of surfaces.- Proof of Castelnuovo s theorem.- The Albanese map.- Proofs of the classification theorems for surfaces.- The Castelnuovo-deFranchis theorem.- Classification of threefolds.- Classification of vector bundles.- Exercises.- References.This book covers the theory of algebraic surfaces and holomorphic vector bundles in an integrated manner. It is aimed at graduate students who have had a thorough first year course in algebraic geometry (at the level of Hartshorne's ALGEBRAIC GEOMETRY), as well as more advanced graduate students and researchers in the areas of algebraic geometry, gauge thoery, or 4-manifold topolgogy. Many of the results on vector bundles should also be of interest to physicists studying string theory. A novel feature of the book is its integrated approach to algebraic surface theory and the study of vector bundle theory on both curves and surfaces. While the two subjects remain separate through the first few chapters, and are studied in alternate chapters, they become much more tightly interconnected as the book progresses. Thus vector bundles over curves are studied to understand ruled surfaces, and then reappear in the proof of Bogomolov's inequality for stable bundles, which is itself applied to study canonical embeddings of surfaces via Reider's method. Similarly, ruled and elliptic surfaces are discussed in detail, and then the geometry of vector bundles over such surfaces is analyzed. Many of the results on vector bundles appear for the first time in book form, suitable for graduate students. The book also has a strong emphasis on examples, both of surfaces and vector bundles. There are over 100 exercises which form an integral part of the text.JOne of the books primary assets is its method of presentation which makes the subject rather accessibleThe only prerequisite is a good working knowledge of elementary algebraic geometry

Unified introduction to the study of algebraic surfaces and vector bundles

Algebraic geometry is an active area of current research978-1-4614-0337-1FuhrmannmPaul A. Fuhrmann, Ben-Gurion University of the Negev Dept. Mathematics & Computer Science, Beer Sheva, Israel'A Polynomial Approach to Linear Algebra XVI, 411p.Preliminaries.- Linear Spaces.- Determinants.- Linear Transformations.- The Shift Operator.- Structure Theory of Linear Transformations.- Inner Product Spaces.- Quadratic Forms.- Stability.- Elements of System Theory.- Hankel Norm Approximation.A Polynomial Approach to Linear Algebra is a text which is heavily biased towards functional methods. In using the shift operator as a central object, it makes linear algebra a perfect introduction to other areas of mathematics, operator theory in particular. This technique is very powerful as becomes clear from the analysis of canonical forms (Frobenius, Jordan). It should be emphasized that these functional methods are not only of great theoretical interest, but lead to computational algorithms. Quadratic forms are treated from the same perspective, with emphasis on the important examples of Bezoutian and Hankel forms. These topics are of great importance in applied areas such as signal processing, numerical linear algebra, and control theory. Stability theory and system theoretic concepts, up to realization theory, are treated as an integral part of linear algebra. This new edition has been updated throughout, in particular new sections have been added on rational interpolation, interpolation using H^{\nfty} functions, and tensor products of models.Review from first edition: & the approach pursed by the author is of unconventional beauty and the material covered by the book is unique. (Mathematical Reviews)

Many additions including Rational interpolation, interpolation using H^{\nfty} functions and tensor products and models (scalar case)

Almost 100 new pages added

Allows the reader to access ideas and results on the frontier of current research

978-0-387-98549-7FultonWilliam FultonIntersection TheoryXIII, 470 p.1. Rational Equivalence.- 2. Divisors.- 3. Vector Bundles and Chern Classes.- 4. Cones and Segre Classes.- 5. Deformation to the Normal Cone.- 6. Intersection Products.- 7. Intersection Multiplicities.- 8. Intersections on Non-singular Varieties.- 9. Excess and Residual Intersections.- 10. Families of Algebraic Cycles.- 11. Dynamic Intersections.- 12. Positivity.- 13. Rationality.- 14. Degeneracy Loci and Grassmannians.- 15. Riemann-Roch for Non-singular Varieties.- 16. Correspondences.- 17. Bivariant Intersection Theory.- 18. Riemann-Roch for Singular Varieties.- 19. Algebraic, Homological and Numerical Equivalence.- 20. Generalizations.- Appendix A. Algebra.- Appendix B. Algebraic Geometry (Glossary).- Notation.=From the ancient origins of algebraic geometry in the solutions of polynomial equations, through the triumphs of algebraic geometry during the last two centuries, intersection theory has played a central role. The aim of this book is to develop the foundations of this theory, and to indicate the range of classical and modern applications. Although a comprehensive history of this vast subject is not attempted, the author points out some of the striking early appearances of the ideas of intersection theory. A suggested prerequisite for t< he reading of this book is a first course in algebraic geometry. Fulton's introduction to intersection theory has been well used for more than 10 years. It is still the only existing complete modern treatise of the subject and received the Steele Prize for best exposition in August 1996.978-1-4614-2199-3GalbisAntonio Galbis, Universidad de Valencia, Burjasot (Valencia), Spain; Manuel Maestre, Universidad de Valencia, Burjasot (Valencia), Spain&Vector Analysis Versus Vector Calculus*XIII, 375p. 79 illus., 59 illus. in color.Preface.- 1 Vectors and Vector Fields.- 2 Line Integrals.- 3 Regular k-surfaces.- 4 Flux of a Vector Field.- 5 Orientation ofa Surface.- 6 Differential Forms.- Integration on Surfaces.- 8 Surfaces with Boundary.- 9 The General Stokes' Theorem.- Solved Exercises .- References.- Index.6The aim of this book is to facilitate the use of Stokes' Theorem in applications. The text takes a differential geometric point of view and provides for the student a bridge between pure and applied mathematics by carefully building a formal rigorous development of the topic and following this through to concrete applications in two and three variables. Several practical methods and many solved exercises are provided. This book tries to show that vector analysis and vector calculus are not always at odds with one another. Key topics include: -vectors and vector fields; -line integrals; -regular k-surfaces; -flux of a vector field; -orientation of a surface; -differential forms; -Stokes' theorem; -divergence theorem. This book is intended for upper undergraduate students who have completed a standard introduction to differential and integral calculus for functions of several variables. The book can also be useful to engineering and physics students who know how to handle the theorems of Green, Stokes and Gauss, but would like to explore the topic further.<p>Presents a precise and rigorous exposition of Stokes' theorem</p><p>Takes a differential geometric point of view on vector calculus and analysis</p><p>Designed as a textbook for upper-undergraduate students, and can also be useful for engineering and physics students </p>978-1-4419-8046-5Gallier?Jean Gallier, University of Pennsylvania, Philadelphia, PA, USADiscrete Mathematics+XIII, 465p. 220 illus., 20 illus. in color.SCI22005:Computer Imaging, Vision, Pattern Recognition and GraphicsUYQV Mathematical Reasoning, Proof Principles and Logic.- Relations, Functions, Partial Functions.- Graphs, Part I: Basic Notions.- Some Counting Problems; Multinomial Coefficients.- Partial Orders, GCD's, RSA, Lattices.- Graphs, Part II: More Advanced Notions.- Answers to Selected Problems.This books gives an introduction to discrete mathematics for beginning undergraduates. One of original features of this book is that it begins with a presentation of the rules of logic as used in mathematics. Many examples of formal and informal proofs are given. With this logical framework firmly in place, the book describes the major axioms of set theory and introduces the natural numbers. The rest of the book is more standard. It deals with functions and relations, directed and undirected graphs, and an introduction to combinatorics. There is a section on public key cryptography and RSA, with complete proofs of Fermat's little theorem and the correctness of the RSA scheme, as well as explicit algorithms to perform modular arithmetic. The last chapter provides more graph theory. Eulerian and Hamiltonian cycles are discussed. Then, we study flows and tensions and state and prove the max flow min-cut theorem. We also discuss matchings, covering, bipartite graphs.Summarizing the rules of mathematical reasoning and how to construct proofs

Presents examples of formal and informal proofs

Includes examples of proofs by induction

Discusses public key cryptography, with a complete proof of the correctness of RSA Explicit, detailed algorithms for modular arithmetic

Explores graph flows and the max-flow min-cut theorem

Covers planar graphs

978-3-540-21127-3GanderWalter Gander, ETH Zrich Institut fr Computational Science, Zrich, Switzerland; Jiri Hrebicek, Masaryk University Fac. Informatics, Brno, Czech Republic@Solving Problems in Scientific Computing Using Maple and MATLABXXII, 476 p.1. The Tractrix and Similar Curves.- 2. Trajectory of a Spinning Tennis Ball.- 3. The Illumination Problem.- 4. Orbits in the Planar Three-Body Problem.- 5. The Internal Field in Semiconductors.- 6. Some Least Squares Problems.- 7. The Generalized Billiard Problem.- 8. Mirror Curves.- 9. Smoothing Filters.- 10. The Radar Problem.- 11. Conformal Mapping of a Circle.- 12. The Spinning Top.- 13. The Calibration Problem.- 14. Heat Flow Problems.- 15. Modeling Penetration Phenomena.- 16. Heat Capacity of System of Bose Particles.- 17. Free Metal Compression.- 18. Gauss Quadrature.- 19. Symbolic Computation of Explicit Runge-Kutta Formulas.- 20. Transient Response of a Two-Phase Half-Wave Rectifier.- 21. Circuits in Power Electronics.- 22. Newton s and Kepler s laws.- 23. Least Squares Fit of Point Clouds.- 24. Modeling Social Processes.- 25. Contour Plots of Analytic Functions.- 26. Non Linear Least Squares: Finding the most accurate location of an aircraft.- 27. Computing Plane Sundials.- 28. Agriculture Kinematics.- 29. The Catenary Curve.- 30. Least Squares Fit with Piecewise Functions.- 31. Portfolio Problems Solved Online.- Appendix A. Shared knowledge of Maple and Matlab.- A.1 Introduction.- A.2 Application Centers.- A.3 Conclusions.From the reviews: '... An excellent reference on undergraduate mathematical computing.' American Mathematical Monthly '... the book is worth buying if you want guidance in applying Maple and MATLAB to problems in the workplace...' Computing Reviews '... The presentation is unique, and extremely interesting. I was thrilled to read this text, and to learn the powerful problem-solving skills presented by these authors. I recommend the text highly, as a learning experience, not only to engineering students, but also to anyone interested in computation.' Mathematics of Computation For this edition four chapters have been added. Some of the chapters of the previous editions were revised using new possibilities offered by Maple and MATLAB. Some i< nteresting web pages related to Maple and MATLAB have been added in an appendix. Moreover, the editors have created a web page (www.SolvingProblems.inf.ethz.ch), where all Maple and MATLAB programs are available.VOne of the most popular applied mathematics textbooks for interdisciplinary use

978-0-8176-8258-3Gautschi;Walter Gautschi, Purdue University, West Lafayette, IN, USAXXVI, 588p. 59 illus..Preface to the Second Edition.- Preface.- Prologue.- Chapter 1. Machine Arithmetic and Related Matters.- Chapter 2. Approximation and Interpolation.- Chapter 3. Numerical Differentiation and Integration.- Chapter 4. Nonlinear Equations.- Chapter 5. Initial Value Problems for ODEs --- One-Step Methods.- Chapter 6. Initial Value Problems for ODEs --- Multi-Step Methods.- Chapter 7. Two-Point Boundary Value Problems for ODEs.- References.- Subject Index.Revised andupdated, this second edition of Walter Gautschi's successful Numerical Analysisexplorescomputational methodsfor problems arising in the areas of classical analysis, approximation theory, and ordinary differential equations, among others. Topics included in the book are presented with a view toward stressing basic principles and maintaining simplicity and teachability as far as possible, while subjects requiring a higher level of technicality are referenced in detailed bibliographic notes at the end of each chapter. Readers are thus given the guidance and opportunity to pursue advanced modern topics in more depth. Along with updated references, new biographical notes, and enhanced notational clarity,this second editionincludes the expansion of an alreadylarge collection of exercises and assignments, both the kind that deal with theoretical and practical aspects of the subject and those requiring machine computation and the use of mathematical software. Perhaps most notably,the edition also comes with a complete solutions manual, carefully developed and polished by the author, which will serve as an exceptionally valuable resource for instructors.LHighly acclaimedstyle is simple and teachable

Detailed bibliographic notes for further study

Integrated with mathematical software for advanced computations

Abundant exercisescover both theoretical and practicalmethods

Complete, 400-pagesolutions manual provides exceptional instructional value

978-3-642-16193-3 Gawarecki}Leszek Gawarecki, Kettering University, Flint, MI, USA; Vidyadhar Mandrekar, Michigan State University, East Lansing, MI, USA8Stochastic Differential Equations in Infinite Dimensions>with Applications to Stochastic Partial Differential EquationsXVI, 292 p.Preface.- Part I: Stochastic Differential Equations in Infinite Dimensions.- 1.Partial Differential Equations as Equations in Infinite.- 2.Stochastic Calculus.- 3.Stochastic Differential Equations.- 4.Solutions by Variational Method.- 5.Stochastic Differential Equations with Discontinuous Drift.- Part II: Stability, Boundedness, and Invariant Measures.- 6.Stability Theory for Strong and Mild Solutions.- 7.Ultimate Boundedness and Invariant Measure.- References.- Index.The systematic study of existence, uniqueness, and properties of solutions to stochastic differential equations in infinite dimensions arising from practical problems characterizes this volume that is intended for graduate students and for pure and applied mathematicians, physicists, engineers, professionals working with mathematical models of finance. Major methods include compactness, coercivity, monotonicity, in a variety of set-ups. The authors emphasize the fundamental work of Gikhman and Skorokhod on the existence and uniqueness of solutions to stochastic differential equations and present its extension to infinite dimension. They also generalize the work of Khasminskii on stability and stationary distributions of solutions. New results, applications, and examples of stochastic partial differential equations are included. This clear and detailed presentation gives the basics of the infinite dimensional version of the classic books of Gikhman and Skorokhod and of Khasminskii in one concise volume that covers the main topics in infinite dimensional stochastic PDE s. By appropriate selection of material, the volume can be adapted for a 1- or 2-semester course, and can prepare the reader for research in this rapidly expanding area.A- Most comprehensive coverage of the modern techniques used for solving problems in infinite dimensional stochastic differential equations
- Presents major methods, including compactness, coercivity, monotonicity, in different set-ups
- Provides a broad range of new results and applications

Introduces the principles of real analysis, as a formidable counterpart to calculus

Offers self-contained introduction to the calculus of functions of one variable

The text is sequenced to emphasize the structural development of calculus

Places appropriate emphasis on computational techniques and applications

978-1-4419-2145-1978-0-8176-8309-2 GiaquintaMariano Giaquinta, Scuola Normale Superiore Dipartimento di Matematica, Pisa, Italy; Giuseppe Modica, Universit Firenze, Firenze, ItalyMathematical AnalysisFFoundations and Advanced Techniques for Functions of Several VariablesXIII, 405p. 66 illus..1Preface.- Spaces of Summable Functions and Partial Differential Equations.- Convex Sets and Convex Functions.- The Formalism of the Calculus of Variations.- Differential Forms.- Measures and Integrations.- Hausdorff and Radon Measures.- Mathematicians and Other Scientists.- Bibliographical Notes.- Index.PMathematical Analysis: Foundations and Advanced Techniques for Functions of Several Variables builds upon the basic ideas and techniques of differential and integral calculus for functions of several variables, as outlined in an earlier introductory volume. The presentation is largely focused on the foundations of measure and integration theory.The book begins with a discussion of the geometry of Hilbert spaces, convex functions and domains, and differential forms, particularly k-forms. The exposition continues with an introduction to the calculus of variations with applications to geometric optics and mechanics.The authorsconclude with the study of measure and integration theory Borel, Radon, and Hausdorff measures and the derivation of measures. An appendix highlights important mathematicians and other scientists whose contributions have made a great impact on the development of theories in analysis.This work may be used as a supplementary text in the classroom or for self-study by advanced undergraduate and graduate students and as a valuable reference for researchers in mathematics, physics, and engineering. One of the key strengths of this presentation, along with the other four books on analysis published by the authors, is the motivation for understanding the subject through examples, observations, exercises, and illustrations.<Provides theoretical foundation for analysis of functions of several variables

Motivates the topics with examples, observations, exercises, and illustrations

Includes appendix of mathematicians who made important contributions to analysis

Exciting historical background motivates the subject

978-0-387-95241-3GodsilChris Godsil; Gordon F. RoyleAlgebraic Graph TheoryXIX, 443 p.SCM29010 CombinatoricsPBVkGraphs.- Groups.- Transitive Graphs.- Arc-Transitive Graphs.- Generalized Polygons and Moore Graphs.- Homomorphisms.- Kneser Graphs.- Matrix Theory.- Interlacing.- Strongly Regular Graphs.- Two-Graphs.- Line Graphs and Eigenvalues.- The Laplacian of a Graph.- Cuts and Flows.- The Rank Polynomial.- Knots.- Knots and Eulerian Cycles.- Glossary of Symbols.- Index.Algebraic graph theory is a combination of two strands. The first is the study of algebraic objects associated with graphs. The second is the use of tools from algebra to derive properties of graphs. The authors' goal has been to present and illustrate the main tools and ideas of algebraic graph theory, with an emphasis on current rather than classical topics. While placing a strong emphasis on concrete examples, the authors tried to keep the treatment self-contained.978-0-387-98464-3 GoldblattRobert GoldblattLectures on the Hyperreals'An Introduction to Nonstandard AnalysisXIV, 293 p.I Foundations.- 1 What Are the Hyperreals?.- 2 Large Sets.- 3 Ultrapower Construction of the Hyperreals.- 4 The Transfer Principle.- 5 Hyperreals Great and Small.- II Basic Analysis.- 6 Convergence of Sequences and Series.- 7 Continuous Functions.- 8 Differentiation.- 9 The Riemann Integral.- 10 Topology of the Reals.- III Internal and External Entities.- 11 Internal and External Sets.- 12 Internal Functions and Hyperfinite Sets.- IV Nonstandard Frameworks.- 13 Universes and Frameworks.- 14 The Existence of Nonstandard Entities.- 15 Permanence, Comprehensiveness, Saturation.- V Applications.- 16 Loeb Measure.- 17 Ramsey Theory.- 18 Completion by Enlargement.- 19 Hyperfinite Approximation.- 20 Books on Nonstandard Analysis.]This is an introduction to nonstandard analysis based on a course of lectures given several times by the author. It is suitable for use as a text at the beginning graduate or upper undergraduate level, or for self-study by anyone familiar with elementary real analysis. It presents nonstandard analysis not just as a theory about infinitely small and large numbers, but as a radically different way of viewing many standard mathematical concepts and constructions; a source of new ideas, objects and proofs; and a wellspring of powerful new principles of reasoning (transfer, overflow, saturation, enlargement, hyperfinite approximation etc.). The book begins with the ultrapower construction of hyperreal number systems, and proceeds to develop one-variable calculus, analysis and topology from the nonstandard perspective, emphasizing the role of the transfer principle as a working tool of mathematical practice. It then sets out the theory of enlargements of fragments of the mathematical universe, providing a foundation for the full-scale development of the nonstandard methodology. The final chapters apply this to a number of topics, including Loeb measure theory and its relation to Lebesgue measure on the real line, Ramsey's Theorem, nonstandard constructions of p-adic numbers and power series, and nonstandard proofs of the Stone representation theorem for Boolean algebras and the Hahn-Banach theorem. Features of the text include an early introduction of the ideas of internal, external and hyperfinite sets, and a more axiomatic set- theoretic approach to enlargements than the usual one based on superstructures.978-0-387-92153-2Gonzlez-VelascoSEnrique A. Gonzlez-Velasco, University of Massachusetts at Lowell, Lowell, MA, USAJourney through Mathematics Creative Episodes in Its HistoryXI, 466p. 146 illus..Preface.- 1 Trigonometry.- 2 Logarithms.- 3 Complex Numbers.- 4 Infinite Series.- 5 The Calculus.- 6 Convergence.- Bibliography.- IndexThis book offers an accessible and in-depth look at some of the most important episodes of two thousand years of mathematical history. Beginning with trigonometry and< moving on through logarithms, complex numbers, infinite series, and calculus, this book profiles some of the lesser known but crucial contributors to modern day mathematics. It is unique in its use of primary sources as well as its accessibility; a knowledge of first-year calculus is theonly prerequisite. But undergraduate and graduate students alike will appreciate this glimpse into the fascinating process of mathematical creation.The history of math is an intercontinental journey, and this book showcases brilliant mathematicians from Greece, Egypt, and India, as well as Europe and the Islamic world. Several of the primary sources have never before beentranslated into English. Their interpretation is thorough and readable, andoffers an excellent background for teachers of high school mathematics as wellas anyone interested in the history of math.An accessible and in-depth look at some of the most important episodes ofabout two thousand years of mathematical history

Interpretation and analysis of dozens of primary sources, and several never-before-seen documents translated into English

Requires only a knowledge of first-year calculus--an excellent resource for undergraduates, high school teachers, and anyone interested in the history of math

978-1-4899-8842-3XI, 466 p. 146 illus.978-0-387-79851-6GoodmanRoe Goodman, Rutgers University Dept. Mathematics, Piscataway, NJ, USA; Nolan R. Wallach, University of California, San Diego Dept. Mathematics, La Jolla, CA, USA)Symmetry, Representations, and InvariantsSCP19013Mathematical Methods in PhysicslLie Groups and Algebraic Groups.- Structure of Classical Groups.- Highest-Weight Theory.- Algebras and Representations.- Classical Invariant Theory.- Spinors.- Character Formulas.- Branching Laws.- Tensor Representations of GL(V).- Tensor Representations of O(V) and Sp(V).- Algebraic Groups and Homogeneous Spaces.- Representations on Spaces of Regular Functions.cSymmetry is a key ingredient in many mathematical, physical, and biological theories. Using representation theory and invariant theory to analyze the symmetries that arise from group actions, and with strong emphasis on the geometry and basic theory of Lie groups and Lie algebras, Symmetry, Representations, and Invariants is a significant reworking of an earlier highly-acclaimed work by the authors. The result is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups, in a new presentation that is more accessible to students and includes a broader range of applications. The philosophy of the earlier book is retained, i.e., presenting the principal theorems of representation theory for the classical matrix groups as motivation for the general theory of reductive groups. The wealth of examples and discussion prepares the reader for the complete arguments now given in the general case. Key Features of Symmetry, Representations, and Invariants: (1) Early chapters suitable for honors undergraduate or beginning graduate courses, requiring only linear algebra, basic abstract algebra, and advanced calculus; (2) Applications to geometry (curvature tensors), topology (Jones polynomial via symmetry), and combinatorics (symmetric group and Young tableaux); (3) Self-contained chapters, appendices, comprehensive bibliography; (4) More than 350 exercises (most with detailed hints for solutions) further explore main concepts; (5) Serves as an excellent main text for a one-year course in Lie group theory; (6) Benefits physicists as well as mathematicians as a reference work.Early chapters suitable for honors undergraduate or beginning graduate courses, requiring only linear algebra, basic abstract algebra, and advanced calculus

Applications to geometry (curvature tensors), topology (Jones polynomial via symmetry), and combinatorics (symmetric group and Young tableaux)

Self-contained chapters, appendices, comprehensive bibliography

More than 350 exercises (most with detailed hints for solutions) further explore main concepts

Serves as an excellent main text for a one-year course in Lie group theory

Benefits physicists as well as mathematicians as a reference work

978-1-4419-2729-3978-1-4419-7918-6GoodwineeBill Goodwine, University of Notre Dame Department of Aerospace and Mechanical E, Notre Dame, IN, USA"Engineering Differential EquationsTheory and ApplicationsXVI, 748 p. 313 illus.Preface.-Introduction and Preliminaries.- First-Order Ordinary Differential Equations.- Second-Order Linear Constant-Coefficient Ordinary Differential Equations.- Single Degreeof Freedom Vibrations.- Variable-Coefficient Linear Ordinary Differential Equations.- Systems of First-Order Linear Constant-Coefficient Ordinary Differential Equations.- Applications of Systems of First-Order Equations.- The Laplace Transform.- Classical Control Theory: Analysis.-Classical Control Theory: Design.- Partial Differential Equations.- Numerical Methods.- Introduction to Nonlinear Systems.-A: Some Complex Variable Theory.- B: Linear Algebra Review.- C: Detailed Computation.- D: Example Programs.- References.- Index.JThis book is a comprehensive treatment of engineering undergraduate differential equations as well as linear vibrations and feedback control. While this material has traditionally been separated into different courses in undergraduate engineering curricula. This text provides a streamlined and efficient treatment of material normally covered in three courses. Ultimately, engineering students study mathematics in order to be able to solve problems within the engineering realm. Engineering Differential Equations: Theory and Applications guides students to approach the mathematical theory with much greater interest and enthusiasm by teaching the theory together with applications. Additionally, it includes an abundance of detailed examples. Appendices include numerous C and FORTRAN example programs. This book is intended for engineering undergraduate students, particularly aerospace and mechanical engineers and students in other disciplines concerned with mechanical systems analysis and control. Prerequisites include basic and advanced calculus with an introduction to linear algebra.Utilizes an abundance of examples

Streamlines three courses into one two semester course

Guides students to approach the mathematical subjects with much greater interest

978-1-4899-8167-7978-3-7643-9991-7GossonEMaurice A. de Gosson, Universitt Wien Fak. Mathematik, Wien, AustriaCSymplectic Methods in Harmonic Analysis and in Mathematical PhysicsXXIV, 338p.The aim of this book is to give a rigorous and complete treatment of various topics from harmonic analysis with a strong emphasis on symplectic invariance properties, which are often ignored or underestimated in the time-frequency literature. The topics that are addressed include (but are not limited to) the theory of the Wigner transform, the uncertainty principle (from the point of view of symplectic topology), Weyl calculus and its symplectic covariance, Shubin s global theory of pseudo-differential operators, and Feichtinger s theory of modulation spaces. Several applications to time-frequency analysis and quantum mechanics are given, many of them concurrent with ongoing research. For instance, a non-standard pseudo-differential ca< lculus on phase space where the main role is played by Bopp operators (also called Landau operators in the literature) is introduced and studied. This calculus is closely related to both the Landau problem and to the deformation quantization theory of Flato and Sternheimer, of which it gives a simple pseudo-differential formulation where Feichtinger s modulation spaces are key actors.This book is primarily directed towards students or researchers in harmonic analysis (in the broad sense) and towards mathematical physicists working in quantum mechanics. It can also be read with profit by researchers in time-frequency analysis, providing a valuable complement to the existing literature on the topic.A certain familiarity with Fourier analysis (in the broad sense) and introductory functional analysis (e.g. the elementary theory of distributions) is assumed. Otherwise, the book is largely self-contained and includes an extensive list of references.Deformation quantization is a "hot" topic in pure mathematics

Absolutely new approach making use of well-established tools of time-frequency analysis

Probably the first text in mathematical physics using Feichtinger's modulation spaces

978-0-387-90110-7GreubWerner H. GreubLinear AlgebraXVII, 451 pp. 5 figs.: 0. Prerequisites.- I. Vector spaces.- 1. Vector spaces.- 2. Linear mappings.- 3. Subspaces and factor spaces.- 4. Dimension.- 5. The topology of a real finite dimensional vector space.- II. Linear mappings.- 1. Basic properties.- 2. Operations with linear mappings.- 3. Linear isomorphisms.- 4. Direct sum of vector spaces.- 5. Dual vector spaces.- 6. Finite dimensional vector spaces.- III. Matrices.- 1. Matrices and systems of linear equations.- 2. Multiplication of matrices.- 3. Basis transformation.- 4. Elementary transformations.- IV. Determinants.- 1. Determinant functions.- 2. The determinant of a linear transformation.- 3. The determinant of a matrix.- 4. Dual determinant functions.- 5. The adjoint matrix.- 6. The characteristic polynomial.- 7. The trace.- 8. Oriented vector spaces.- V. Algebras.- 1. Basic properties.- 2. Ideals.- 3. Change of coefficient field of a vector space.- VI. Gradations and homology.- 1. G-graded vector spaces.- 2. G-graded algebras.- 3. Differential spaces and differential algebras.- VII. Inner product spaces.- 1. The inner product.- 2. Orthonormal bases.- 3. Normed determinant functions.- 4. Duality in an inner product space.- 5. Normed vector spaces.- 6. The algebra of quaternions.- VIII. Linear mappings of inner product spaces.- 1. The adjoint mapping.- 2. Selfadjoint mappings.- 3. Orthogonal projections.- 4. Skew mappings.- 5. Isometric mappings.- 6. Rotations of Euclidean spaces of dimension 2, 3 and 4.- 7. Differentiate families of linear automorphisms.- IX. Symmetric bilinear functions.- 1. Bilinear and quadratic functions.- 2. The decomposition of E.- 3. Pairs of symmetric bilinear functions.- 4. Pseudo-Euclidean spaces.- 5. Linear mappings of Pseudo-Euclidean spaces.- X. Quadrics.- 1. Affine spaces.- 2. Quadrics in the affine space.- 3. Affine equivalence of quadrics.- 4. Quadrics in the Euclidean space.- XI. Unitary spaces.- 1. Hermitian functions.- 2. Unitary spaces.- 3. Linear mappings of unitary spaces.- 4. Unitary mappings of the complex plane.- 5. Application to Lorentz-transformations.- XII. Polynomial algebra.- 1. Basic properties.- 2. Ideals and divisibility.- 3. Factor algebras.- 4. The structure of factor algebras.- XIII. Theory of a linear transformation.- 1. Polynomials in a linear transformation.- 2. Generalized eigenspaces.- 3. Cyclic spaces.- 4. Irreducible spaces.- 5. Application of cyclic spaces.- 6. Nilpotent and semisimple transformations.- 7. Applications to inner product spaces.This textbook gives a detailed and comprehensive presentation of the linear algebra based on axiomatic treatment of linear spaces. The author maintains a good balance between modern algebraic interests and traditional linear algebra. Several chapters have been substantially rewritten for clarity of exposition, although their basic content is unchanged. A considerable number of exer- cises covering new material has also been added.978-1-4471-2169-5Grosse-ErdmannKarl-G. Grosse-Erdmann, Universit de Mons Institut de Mathmatique, Mons, Belgium; Alfred Peris Manguillot, Universitat Politcnica de Valncia Institut de Matemtica Pura i Aplicada, Valncia, SpainLinear ChaosXII, 388p. 28 illus..Topological dynamics.- Hypercyclic and chaotic operators.- The Hypercyclicity Criterion.- Classes of hypercyclic and chaotic operators.- Necessary conditions for hypercyclicity and chaos.- Connectedness arguments in linear dynamics.- Dynamics of semigroups, with applications to differential equations.- Existence of hypercyclic operators.- Frequently hypercyclic operators.- Hypercyclic subspaces.- Common hypercyclic vectors.- Linear dynamics in topological vector spacesIt is commonly believed that chaos is linked to non-linearity, however many (even quite natural) linear dynamical systems exhibit chaotic behavior. The study of these systems is a young and remarkably active field of research, which has seen many landmark results over the past two decades. Linear dynamics lies at the crossroads of several areas of mathematics including operator theory, complex analysis, ergodic theory and partial differential equations. At the same time its basic ideas can be easily understood by a wide audience. Written by two renowned specialists, Linear Chaos provides a welcome introduction to this theory. Split into two parts, part I presents a self-contained introduction to the dynamics of linear operators, while part II covers selected, largely independent topics from linear dynamics. More than 350 exercises and many illustrations are included, and each chapter contains a further Sources and Comments section. The only prerequisites are a familiarity with metric spaces, the basic theory of Hilbert and Banach spaces and fundame< ntals of complex analysis. More advanced tools, only needed occasionally, are provided in two appendices. A self-contained exposition, this book will be suitable for self-study and will appeal to advanced undergraduate or beginning graduate students. It will also be of use to researchers in other areas of mathematics such as partial differential equations, dynamical systems and ergodic theory.u<p>More than 350 exercises and a self-contained exposition makes the book suitable for both classroom use and self-study</p><p>Presents simplified proofs, as well as improvements, of known results</p><p>An exhaustive bibliography and the inclusion of a Sources and comments section at the end of each chapter provides the reader with ample links to further literature</p>978-3-642-23668-6GuedjgVincent Guedj, Universit Paul Sabatier Institut de Mathmatiques de Toulouse, Toulouse Cedex 9, FranceKComplex Monge Ampre Equations and Geodesics in the Space of Khler MetricsVIII, 310p. 4 illus..1.Introduction.- I. The Local Homogenious Dirichlet Problem.-2. Dirichlet Problem in Domains of Cn.- 3. Geometric Maximality.- II. Stochastic Analysis for the Monge-Ampre Equation.- 4. Probabilistic Approach to Regularity.- III. Monge-Ampre Equations on Compact Manifolds.- 5.The Calabi-Yau Theorem.- IV Geodesics in the Space of Khler Metrics.- 6. The Riemannian Space of Khler Metrics.- 7. MA Equations on Manifolds with Boundary.- 8. Bergman Geodesics.The purpose of these lecture notes is to provide an introduction to the theory of complex Monge Ampre operators (definition, regularity issues, geometric properties of solutions, approximation) on compact Khler manifolds (with or without boundary). These operators are of central use in several fundamental problems of complex differential geometry (Khler Einstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after Bedford Taylor), Monge Ampre foliations and laminated currents, polynomial hulls and Perron envelopes with no analytic structure, a self-contained presentation of Krylov regularity results, a modernized proof of the Calabi Yau theorem (after Yau and Kolodziej), an introduction to infinite dimensional riemannian geometry, geometric structures on spaces of Khler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity theory of Caffarelli Kohn Nirenberg Spruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after Phong Sturm and Berndtsson).Each chapter can be read independently and is based on a series of lectures byR. Berman, Z. Blocki, S. Boucksom, F. Delarue, R. Dujardin, B. Kolev and A. Zeriahi, delivered to non-experts. The book is thus addressed to any mathematician with some interest in one of the following fields, complex differential geometry, complex analysis, complex dynamics, fully non-linear PDE's and stochastic analysis.The first self contained presentation of Krylov's stochastic analysis for the complex Monge-Ampere equation

A comprehensive presentation of Yau's proof of the Calabi conjecture

A great part of the material (both classical results and more recent 4.

A pedagogical style, lectures accessible to non experts.developments) has not previously appeared in book form

Written in pedagogicalcal style, lectures accessible to non experts

978-3-0348-0100-3GupurAGeni Gupur, Xinjiang University, Urumqi, China, People's Republic2Functional Analysis Methods for Reliability ModelsVIII, 277p.iPreface.- C0-Semigroup of Linear Operators and Cauchy Problems.- Statement of the Problems.- The System Consisting of a Reliable Machine, an Unreliable Machine and a Storage Buffer with Finite Capacity.- Transfer Line Consisting of a Reliable Machine, an Unreliable Machine and a Storage Buffer with Infinite Capacity.- Further Research Problems.- Bibliography.The main goal of this book is to introduce readers to functional analysis methods, in particular, time dependent analysis, for reliability models. Understanding the concept of reliability is of key importance schedule delays, inconvenience, customer dissatisfaction, and loss of prestige and even weakening of national security are common examples of results that are caused by unreliability of systems and individuals. The book begins with an introduction to C0-semigroup theory. Then, after a brief history of reliability theory, methods that study the well-posedness, the asymptotic behaviors of solutions and reliability indices for varied reliability models are presented. Finally, further research problems are explored.Functional Analysis Methods for Reliability Models is an excellent reference for graduate students andresearchers in operations research, applied mathematics and systems engineering.Reliability models with importance to daily life are covered

The mathematics of reliability is clearly explained

Functional analysis methods are discussed

978-0-387-87861-4GusakDmytro Gusak, National Academy of Sciences of Ukraine Inst. Mathematics, Kyiv, Ukraine; Alexander Kukush, National Taras Shevchenko University of Kiev, Kiev, Ukraine; Alexey Kulik, National Academy of Sciences of Ukraine Inst. Mathematics, Kyiv, Ukrain< e; Yuliya Mishura, National Taras Shevchenko University of Kiev, Kiev, Ukraine; Andrey Pilipenko, National Academy of Sciences of Ukraine Inst. Mathematics, Kyiv, UkraineTheory of Stochastic Processes:With Applications to Financial Mathematics and Risk TheoryXII, 376p. 8 illus..SCS13004@Statistics for Business/Economics/Mathematical Finance/InsuranceDefinition of stochastic process. Cylinder σ-algebra, finite-dimensional distributions, the Kolmogorov theorem.- Characteristics of a stochastic process. Mean and covariance functions. Characteristic functions.- Trajectories. Modifications. Filtrations.- Continuity. Differentiability. Integrability.- Stochastic processes with independent increments. Wiener and Poisson processes. Poisson point measures.- Gaussian processes.- Martingales and related processes in discrete and continuous time. Stopping times.- Stationary discrete- and continuous-time processes. Stochastic integral over measure with orthogonal values.- Prediction and interpolation.- Markov chains: Discrete and continuous time.- Renewal theory. Queueing theory.- Markov and diffusion processes.- Itô stochastic integral. Itô formula. Tanaka formula.- Stochastic differential equations.- Optimal stopping of random sequences and processes.- Measures in a functional spaces. Weak convergence, probability metrics. Functional limit theorems.- Statistics of stochastic processes.- Stochastic processes in financial mathematics (discrete time).- Stochastic processes in financial mathematics (continuous time).- Basic functionals of the risk theory.This book is a collection of exercises covering all the main topics in the modern theory of stochastic processes and its applications, including finance, actuarial mathematics, queuing theory, and risk theory. The aim of this book is to provide the reader with the theoretical and practical material necessary for deeper understanding of the main topics in the theory of stochastic processes and its related fields. The book is divided into chapters according to the various topics. Each chapter contains problems, hints, solutions, as well as a self-contained theoretical part which gives all the necessary material for solving the problems. References to the literature are also given. The exercises have various levels of complexity and vary from simple ones, useful for students studying basic notions and technique, to very advanced ones that reveal some important theoretical facts and constructions. This book is one of the largest collections of problems in the theory of stochastic processes and its applications. The problems in this book can be useful for undergraduate and graduate students, as well as for specialists in the theory of stochastic processes.Contains over 1000 high quality exercises on stochastic processes

Presents a modern approach to topics such as sample paths and optimal stopping

Ideal for professors who need exercises for exams, and graduate students wishing to learn about stochastic processes

978-1-4614-2506-9978-3-642-21865-1 GustafsonStephen J. Gustafson, University of British Columbia Dept. Mathematics, Vancouver, BC, Canada; Israel Michael Sigal, University of Toronto, Toronto, ON, Canada*Mathematical Concepts of Quantum MechanicsXIII, 382 p. 37 illus.c1 Physical Background.- 2 Dynamics.- 3 Observables.- 4 Quantization.- 5 Uncertainty Principle and Stability of Atoms and Molecules.- 6 Spectrum and Dynamics.- 7 Special Cases.- 8 Bound States and Variational Principle.- 9 Scattering States.- Existence of Atoms and Molecules.- 11 Perturbation Theory: Feshbach-Schur Method.- 12 General Theory of Many-particle Systems.- 13 Self-consistent Approximations.- 14 The Feynman Path Integral.- 15 Quasi-classical Analysis.- 16 Resonances.- 17 Quantum Statistics.- 18 The Second Quantization.- 19 Quantum Electro-Magnetic Field Photons.- 20 Standard Model of Non-relativistic Matter and Radiation.- 21 Theory of Radiation.- 22 Renormalization Group.- 23 Mathematical Supplement: Spectral Analysis.- 24 Mathematical Supplement: The Calculus of Variations.- 25 Comments on Literature, and Further Reading.- References.- IndexThe book gives a streamlined introduction to quantum mechanics while describing the basic mathematical structures underpinning this discipline. Starting with an overview of key physical experiments illustrating the origin of the physical foundations, the book proceedswith a description of the basic notions of quantum mechanics and their mathematical content.It then makes its way to topics of current interest, specifically those in which mathematics plays an important role. The more advanced topics presented include many-body systems, modern perturbation theory, path integrals, the theory of resonances, quantum statistics, mean-field theory, second quantization, the theory of radiation (non-relativistic quantum electrodynamics), and the renormalization group.With different selections of chapters, the book can serve as a text for an introductory, intermediate, or advanced course in quantum mechanics. The last four chapters could also serve as an introductory course in quantum field theory.JA very readable introduction to modern mathematical topics in quantum mechanics

Solves the problem of how to teach quantum mechanics to mathematically oriented students in an optimal way

Shows how the mathematical treatment of quantum mechanics brings insights to physics

Useful guide to the literature

978-0-387-94076-2Hale%Jack K. Hale; Sjoerd M. Verduyn Lunel1Introduction to Functional Differential Equations X, 449 p.The present book builds upon the earlier work of J. Hale, 'Theory of Functional Differential Equations' published in 1977. The authors have attempted to maintain the spirit of that book and have retained approximately one-third of the material intact. One major change was a completely new presentation of linear systems (Chapter 6-9) for retarded and neutral functional differential equations. The theory of dissipative systems (Chapter 4) and global attractors was thoroughly revamped as well as the invariant manifold theory (Chapter 10) near equilibrium points and periodic orbits. A more complete theory of neutral equations is presented (Chapters 1,2,3,9,10). Chapter 12 is also entirely new and contains a guide to active topics of research. In the sections on supplementary remarks, the authors have included many references to recent literature, but, of course, not nearly all, because the subject is so extensive.978-0-387-90685-0P.R. HalmosA Hilbert Space Problem Book XVII, 369 pp.x1. Vectors.- 2. Spaces.- 3. Weak Topology.- 4. Analytic Functions.- 5. Infinite Matrices.- 6. Boundedness and Invertibility.- 7. Multiplication Operators.- 8. Operator Matrices.- 9. Properties of Spectra.- 10. Examples of Spectra.- 11. Spectral Radius.- 12. Norm Topology.- 13. Operator Topologies.- 14. Strong Operator Topology.- 15. Partial Isometries.- 16. Polar Decomposition.- 17. Unilateral Shift.- 18. Cyclic Vectors.- 19. Properties of Compactness.- 20. Examples of Compactness.- 21. Subnormal Operators.- 22. Numerical Range.- 23. Unitary Dilations.- 24. Commutators.- 25. Toeplitz Operators.- References.- List of Symbols.vFrom the Preface: 'This < book was written for the active reader. The first part consists of problems, frequently preceded by definitions and motivation, and sometimes followed by corollaries and historical remarks... The second part, a very short one, consists of hints... The third part, the longest, consists of solutions: proofs, answers, or contructions, depending on the nature of the problem.... This is not an introduction to Hilbert space theory. Some knowledge of that subject is a prerequisite: at the very least, a study of the elements of Hilbert space theory should proceed concurrently with the reading of this book.'978-1-4614-5939-2HanyWeimin Han, University of Iowa City, Iowa City, IA, USA; B. Daya Reddy, University of Cape Town, Rondebosch, South Africa Plasticity*Mathematical Theory and Numerical AnalysisXV, 421 p. 41 illus.SCT15001!Theoretical and Applied MechanicsTGMDRPreface to the Second Edition.- Preface to the First Edition.-Preliminaries.- Continuum Mechanics and Linearized Elasticity.- Elastoplastic Media.- The Plastic Flow Law in a Convex-Analytic Setting.- Basics of Functional Analysis and Function Spaces.- Variational Equations and Inequalities.- The Primal Variational Problem of Elastoplasticity.- The Dual Variational Problem of Classical Elastoplasticity.- Introduction to Finite Element Analysis.- Approximation of Variational Problems.- Approximations of the Abstract Problem.- Numerical Analysis of the Primal Problem.- References.- Index.- This book focuses on the theoretical aspects of small strain theory of elastoplasticity with hardening assumptions. It provides a comprehensive and unified treatment of the mathematical theory and numerical analysis. It is divided into three parts, with the first part providing a detailed introduction to plasticity, the second part covering the mathematical analysis of the elasticity problem, and the third part devoted to error analysis of various semi-discrete and fully discrete approximations for variational formulations of the elastoplasticity. This revised and expanded edition includes material on single-crystal and strain-gradient plasticity. In addition, the entire book has been revised to make it more accessible to readers who are actively involved in computations but less so in numerical analysis. Reviews of earlier edition: The authors have written an excellent book which can be recommended for specialists in plasticity who wish to know more about the mathematical theory, as well as those with a background in the mathematical sciences who seek a self-contained account of the mechanics and mathematics of plasticity theory. (ZAMM, 2002) In summary, the book represents an impressive comprehensive overview of the mathematical approach to the theory and numerics of plasticity. Scientists as well as lecturers and graduate students will find the book very useful as a reference for research or for preparing courses in this field. (Technische Mechanik) 'The book is professionally written and will be a useful reference to researchers and students interested in mathematical and numerical problems of plasticity. It represents a major contribution in the area of continuum mechanics and numerical analysis.' (Math Reviews)Book bridging mechanics and mathematics

Provides a comprehensive and unified treatment of the mathematical theory and numerical analysis

Focuses on theoretical aspects of the small-strain theory of hardening elastoplasticity

978-1-4899-9594-0978-1-4419-6054-2HelgasonLSigurdur Helgason, Massachusetts Institute of Technology, Cambridge, MA, USA&Integral Geometry and Radon TransformsXIII, 301p. 28 illus..SCM12112)Integral Transforms, Operational CalculusgThe Radon Transformon Rn .- A Duality in Integral Geometry.- The Radon Transform on Two-Point Homogeneous Spaces.- The X-Ray Transform on a Symmetric Space.- Orbital Integrals.- The Mean-Value Operator.- Fourier Transforms and Distribution: A Rapid Course.- Lie Transformation Groups and Differential Operators.- Bibliography.- Notational Conventions.- Index.In this text, integral geometry deals with Radon s problem of representing a function on a manifold in terms of its integrals over certain submanifolds hence the term the Radon transform. Examples and far-reaching generalizations lead to fundamental problems such as: (i) injectivity, (ii) inversion formulas, (iii) support questions, (iv) applications (e.g., to tomography, partial di erential equations and group representations). For the case of the plane, the inversion theorem and the support theorem have had major applications in medicine through tomography and CAT scanning. While containing some recent research, the book is aimed at beginning graduate students for classroom use or self-study. A number of exercises point to further results with documentation. From the reviews: Integral Geometry is a fascinating area, where numerous branches of mathematics meet together. the contents of the book is concentrated around the duality and double vibration, which is realized through the masterful treatment of a variety of examples. the book is written by an expert, who has made fundamental contributions to the area. Boris Rubin, Louisiana State University<Presents material accessible to advanced undergraduates

Contains the required expository material on Lie group theory

Features self-contained chapters with bibliographical notes, exercises, and further results (with documentation)

978-1-4899-9420-2XIII, 301 p. 28 illus.978-1-84882-318-1HelmsOLester Helms, University of Illinois, Urbana Dept. Mathematics, Urbana, IL, USAPreliminaries.- Laplace's Equation.- The Dirichlet Problem.- Green Functions.- Negligible Sets.- Dirichlet Problem for Unbounded Regions.- Energy.- Interpolation and Monotonicity.- Newtonian Potential.- Elliptic Operators.- Apriori Bounds.- Oblique Derivative Problem.This book presents a clear path from calculus to classical potential theory and beyond with the aim of moving the reader into a fertile area of mathematical research as quickly as possible. The first half of the book develops the subject matter from first principles using only calculus. The second half comprises more advanced material for those with a senior undergraduate or beginning graduate course in real analysis. For specialized regions, solutions of Laplace s equation are constructed having prescribed normal derivatives on the flat portion of the boundary and prescribed values on the remaining portion of the boundary. By means of transformations known as diffeomorphisms, these solutions are morphed into local solutions on regions with curved boundaries. The Perron-Weiner-Brelot method is then used to construct global solutions for elliptic PDEs involving a mixture of prescribed values of a boundary differential operator on part of the boundary and prescribed values on the remainder of the boundary._Written by the author of Introduction to Potential Theory, this is a new and modern textbook that introduces all the important concepts of classical potential theory

Equips readers for further study in elliptic partial differential equations, axiomatic potential theory, and the interplay between probability theory and potential theory

978-3-540-10557-2HenryDaniel Henry2Geometric Theory of Semilinear Parabolic Equations VI, 350 p.Preliminaries.- Examples of nonlinear parabolic equations in physical, biological and engineering problems.- Existence, uniqueness and continuous dependence.- Dynamical systems and liapunov stability.- Neighborhood of an equilibrium point.- Invariant manifolds near an equilibrium point.- Linear nonautonomous equations.- Neighborhood of a periodic solution.- The neighborhood of an invariant manifold.- Two examples.978-0-85729-105-9HerzogJrgen Herzog, Universitt Duisburg-Essen Fachbereich Mathematik, Essen, Germany; Takayuki Hibi, Osaka University Graduate School of Inf. Science & Techn., Toyonaka, JapanMonomial IdealsXVI, 308 p.Part I Grbner bases: Monomial Ideals.- A short introduction to Grbner bases.- Monomial orders and weights.- Generic initial ideals.- The exterior algebra.- Part II: Hilbert functions and resolutions.- Hilbert functions and the theorems of Macaulay and Kruskal-Katona.- Resolutions of monomial ideals and the Eliahou-Kervaire formula.- Alexander duality and resolutions.- Part III Combinatorics: Alexander duality and finite graphs.- Powers of monomial ideals.- Shifting theory.- Discrete Polymatroids.- Some homological algebra.- GeometryThis book demonstrates current trends in research on combinatorial and computational commutative algebra with a primary emphasis on topics related to monomial ideals. Providing a useful and quick introduction to areas of research spanning these fields, Monomial Ideals is split into three parts. Part I offers a quick introduction to the modern theory of Grbner bases as well as the detailed study of generic initial ideals. Part II supplies Hilbert functions and resolutions and some of the combinatorics related to monomial ideals including the Kruskal Katona theorem and algebraic aspects of Alexander duality. Part III discusses combinatorial applications of monomial ideals, providing a valuable overview of some of the central trends in algebraic combinatorics. Main subjects include edge ideals of finite graphs, powers of ideals, algebraic shifting theory and an introduction to discrete polymatroids. Theory is complemented by a number of examples and exercises throughout, bringing the reader to a deeper understanding of concepts explored within the text. Self-contained and concise, this book will appeal to a wide range of readers, including PhD students on advanced courses, experienced researchers, and combinatorialists and non-specialists with a basic knowledge of commutative algebra. Since their first meeting in 1985, Juergen Herzog (Universitt Duisburg-Essen, Germany) and Takayuki Hibi (Osaka University, Japan), have worked together on a number of research projects, of which recent results are presented in this monograph.-Theory is complemented by a number of examples and exercises throughout, bringing the reader to a deeper understanding of concepts explored within the text

- Provides a quick and usefulintroduction to research spanning the fields of combinatorial and computational commutative algebra, with a special focus on monomial ideals

- Only a basic knowledge of commutative algebra is required, making this accessible to specialists and non-specialists alike

978-1-4471-2594-5978-1-4419-9487-5HijabNOmar Hijab, Temple University Department of Mathematics, Philadelphia, PA, USA/Introduction to Calculus and Classical AnalysisXII, 364 p.Preface.- 1 The Set of Real Numbers.- 2 Continuity.- 3 Differ< entiation.- 4 Integration.- 5 Applications.- A Solutions.- References.- IndexThis text is intended for an honors calculus course or for an introduction to analysis. Involving rigorous analysis, computational dexterity, and a breadth of applications, it is ideal for undergraduate majors. This third edition includes corrections as well as some additional material.Some features of the text:* The text is completely self-contained and starts with the real number axioms;* The integral is defined as the area under the graph, while the area is defined for every subset of the plane;* There is a heavy emphasis on computational problems, from the high-school quadratic formula to the formula for the derivative of the zeta function at zero;* There are applications from many parts of analysis, e.g., convexity, the Cantor set, continued fractions, the AGM, the theta and zeta functions, transcendental numbers, the Bessel and gamma functions, and many more;* Traditionally transcendentally presented material, such as infinite products, the Bernoulli series, and the zeta functional equation, is developed over the reals;* There are 385 problems with all the solutions at the back of the text.Review from the first edition:'This is a very intriguing, decidedly unusual, and very satisfying treatment of calculus and introductory analysis. It's full of quirky little approaches to standard topics that make one wonder over and over again, 'Why is it never done like this?''-John Allen Paulos, author of Innumeracy and A Mathematician Reads the NewspaperApproaches calculus and introductory analysis in a nonstandard way

New edition extensively revised and updated

Completely self-contained text

978-1-4614-2842-8978-1-4471-2130-5HindryaMarc Hindry, Universit Paris 7 Denis Diderot Institut de Mathmatiques de Jussieu, Paris, FranceArithmeticsXVIII, 322p. 5 illus..Finite Structures.- Applications: Algorithms, Primality and Factorization, Codes.- Algebra and Diophantine Equations.- Analytic Number Theory.- Elliptic Curves.- Developments and Open Problems.- Factorization.- Elementary Projective Geometry.- Galois TheorydNumber theory is a branch of mathematics which draws its vitality from a rich historical background. It is also traditionally nourished through interactions with other areas of research, such as algebra, algebraic geometry, topology, complex analysis and harmonic analysis. More recently, it has made a spectacular appearance in the field of theoretical computer science and in questions of communication, cryptography and error-correcting codes. Providing an elementary introduction to the central topics in number theory, this book spans multiple areas of research. The first part corresponds to an advanced undergraduate course. All of the statements given in this part are of course accompanied by their proofs, with perhaps the exception of some results appearing at the end of the chapters. A copious list of exercises, of varying difficulty, are also included here. The second part is of a higher level and is relevant for the first year of graduate school. It contains an introduction to elliptic curves and a chapter entitled Developments and Open Problems , which introduces and brings together various themes oriented toward ongoing mathematical research. Given the multifaceted nature of number theory, the primary aims of this book are to: - provide an overview of the various forms of mathematics useful for studying numbers - demonstrate the necessity of deep and classical themes such as Gauss sums - highlight the role that arithmetic plays in modern applied mathematics - include recent proofs such as the polynomial primality algorithm - approach subjects of contemporary research such as elliptic curves - illustrate the beauty of arithmetic The prerequisites for this text are undergraduate level algebra and a little topology of Rn. It will be of use to undergraduates, graduates and phd students, and may also appeal to professional mathematicians as a reference text.Explores the multi-faceted nature of number theory, spanning several areas of research in one text

Begins at undergraduate level and takes the reader through to graduate level

Includes recent proofs, such as the polynomial primality algorithm

978-3-540-56850-6Hiriart-UrrutyJean-Baptiste Hiriart-Urruty, Universit Toulouse III Dpt. Mathmatiques, Toulouse CX 09, France; Claude Lemarechal, INRIA Rhne-Alpes, St. Ismier CX, France-Convex Analysis and Minimization Algorithms IFundamentalsXVII, 420 p.Table of Contents Part I.- I. Convex Functions of One Real Variable.- II. Introduction to Optimization Algorithms.- III. Convex Sets.- IV. Convex Functions of Several Variables.- V. Sublinearity and Support Functions.- VI. Subdifferentials of Finite Convex Functions.- VII. Constrained Convex Minimization Problems: Minimality Conditions, Elements of Duality Theory.- VIII. Descent Theory for Convex Minimization: The Case of Complete Information.- Appendix: Notations.- 1 Some Facts About Optimization.- 2 The Set of Extended Real Numbers.- 3 Linear and Bilinear Algebra.- 4 Differentiation in a Euclidean Space.- 5 Set-Valued Analysis.- 6 A Bird s Eye View of Measure Theory and Integration.- Bibliographical Comments.- References.1Convex Analysis may be considered as a refinement of standard calculus, with equalities and approximations replaced by inequalities. As such, it can easily be integrated into a graduate study curriculum. Minimization algorithms, more specifically those adapted < to non-differentiable functions, provide an immediate application of convex analysis to various fields related to optimization and operations research. These two topics making up the title of the book, reflect the two origins of the authors, who belong respectively to the academic world and to that of applications. Part I can be used as an introductory textbook (as a basis for courses, or for self-study); Part II continues this at a higher technical level and is addressed more to specialists, collecting results that so far have not appeared in books.978-3-642-08161-3978-0-387-90148-0HirschMorris W. HirschDifferential TopologyX, 222 p. 40 illus.v1 : Manifolds and Maps.- 0. Submanifolds of ?n+k.- 1. Differential Structures.- 2. Differentiable Maps and the Tangent Bundle.- 3. Embeddings and Immersions.- 4. Manifolds with Boundary.- 5. A Convention.- 2 : Function Spaces.- 1. The Weak and Strong Topologies on Cr(M, N).- 2. Approximations.- 3. Approximations on ?-Manifolds and Manifold Pairs.- 4. Jets and the Baire Property.- 5. Analytic Approximations.- 3 : Transversality.- 1. The Morse-Sard Theorem.- 2. Transversality.- 4 : Vector Bundles and Tubular Neighborhoods.- 1. Vector Bundles.- 2. Constructions with Vector Bundles.- 3. The Classification of Vector Bundles.- 4. Oriented Vector Bundles.- 5. Tubular Neighborhoods.- 6. Collars and Tubular Neighborhoods of Neat Submanifolds.- 7. Analytic Differential Structures.- 5 : Degrees, Intersection Numbers, and the Euler Characteristic.- 1. Degrees of Maps.- 2. Intersection Numbers and the Euler Characteristic.- 3. Historical Remarks.- 6 : Morse Theory.- 1. Morse Functions.- 2. Differential Equations and Regular Level Surfaces.- 3. Passing Critical Levels and Attaching Cells.- 4. CW-Complexes.- 7 : Cobordism.- 1. Cobordism and Transversality.- 2. The Thorn Homomorphism.- 8 : Isotopy.- 1. Extending Isotopies.- 2. Gluing Manifolds Together.- 3. Isotopies of Disks.- 9 : Surfaces.- 1. Models of Surfaces.- 2. Characterization of the Disk.- 3. The Classification of Compact Surfaces.This book gives the reader a thorough knowledge of the basic topological ideas necessary for studying differential manifolds. These topics include immersions and imbeddings, approach techniques, and the Morse classification of surfaces and their cobordism. The author keeps the mathematical prerequisites to a minimum; this and the emphasis on the geometric and intuitive aspects of the subject make the book an excellent and useful introduction for the student. There are numerous excercises on many different levels ranging from practical applications of the theorems to significant further development of the theory and including some open research problems.978-3-540-33260-2Hjelleyvind Hjelle, University of Oslo Simula Research Laboratory, Lysaker, Norway; Morten Dhlen, University of Oslo Simula Research Laboratory, Lysaker, NorwayTriangulations and ApplicationsMathematics and Visualization XI, 234 p.SCM14034 VisualizationSCM14018 AlgorithmssTriangles and Triangulations.- Graphs and Data Structures.- Delaunay Triangulations and Voronoi Diagrams.- Algorithms for Delaunay Triangulation.- Data Dependent Triangulations.- Constrained Delaunay Triangulation.- Delaunay Refinement Mesh Generation.- Least Squares Approximation of Scattered Data.- Programming Triangulations: The Triangulation Template Library (TTL).This book is entirely about triangulations. With emphasis on computational issues, the basic theory necessary to construct and manipulate triangulations is presented. In particular, a tour through the theory behind the Delaunay triangulation, including algorithms and software issues, is given. Various data structures used for the representation of triangulations are discussed. Throughout the book, the theory is related to selected applications, in particular surface construction, meshing and visualization. LApplications-oriented

Well-written textbook on triangulations

978-3-642-06988-8978-3-540-14887-6HochstttlerWinfried Hochstttler, Fernuniversitt Hagen FB Mathematik, Hagen, Germany; Alexander Schliep, Max-Planck-Gesellschaft Molekulare Genetik, Berlin, GermanyCATBox3An Interactive Course in Combinatorial OptimizationXII, 190p. 36 illus..Discrete Problems from Applications.- Basics, Notation and Data Structures.- Minimum Spanning Trees.- Linear Programming Duality.- Shortest Paths.- Maximal Flows.- Minimum-Cost Flows.- Matching.- Weighted Matching.Graph algorithms are easy to visualize and indeed there already exists a variety of packages to animate the dynamics when solving problems from graph theory. Still it can be difficult to understand the ideas behind the algorithm from the dynamic display alone. CATBox consists of a software system for animating graph algorithms and a course book which we developed simultaneously. The software system presents both the algorithm and the graph and puts the user always in control of the actual code that is executed. In the course book, intended for readers at advanced undergraduate or graduate level, computer exercises and examples replace the usual static pictures of algorithm dynamics. For this volume we have chosen solely algorithms for classical problems from combinatorial optimization, such as minimum spanning trees, shortest paths, maximum flows, minimum cost flows, weighted and unweighted matchings both for bipartite and non-bipartite graphs. Find more information at http://schliep.org/CATBox/.978-3-0348-0103-4HoferuHelmut Hofer, Institute for Advanced Study (IAS), Princeton, NJ, USA; Eduard Zehnder, ETH Zrich, Zrich, Switzerland.Symplectic Invariants and Hamiltonian DynamicsXIV, 341p. 49 illus..1 Introduction.- 2 Symplectic capacities.- 3 Existence of a capacity.- 4 Existence of closed characteristics.- 5 Compactly supported symplectic mappings in R2n.- 6 The Arnold conjecture, Floer homology and symplectic homology.- Appendix.- Index.- Bibliography.The discoveries of the last decades have opened new perspectives for the old field of Hamiltonian systems and led to the creation of a new field: symplectic topology. Surprising rigidity phenomena demonstrate that the nature of symplectic mappings is very different from that of volume preserving mappings. This raises new questions, many of them still unanswered. On the other hand, analysis of an old variational principle in classical mechanics has established global periodic phenomena in Hamiltonian systems. As it turns out, these seemingly different phenomena are mysteriously related. One of the links is a class of symplectic invariants, called symplectic capacities. These invariants are the main theme of this book, which includes such topics as basic symplectic geometry, symplectic capacities and rigidity, periodic orbits for Hamiltonian systems and the action principle, a bi-invariant metric on the symplectic diffeomorphism group and its geometry, symplectic fixed point theory, the Arnold conjectures and first order elliptic systems, and finally a survey on Floer homology and symplectic homology. The exposition is self-contained and addressed to researchers and students from the graduate level onwards.Opened new perspectives for the old field of Hamiltonian systems and let to the creation of a new field: symplectic topology

Invariants which grew out of lectures given by the authors

Selection by a single principle: the action principle of mechanics

978-0-387-95410-3HohmannMAndreas Hohmann; Peter Deuflhard, Zuse-Institut Berlin (ZIB), Berlin, Germany1Numerical Analysis in Modern Scientific ComputingXVIII, 337 p. 65 illus.1 Linear Systems.- 1.1 Solution of Triangular Systems.- 1.2 Gaussian Elimination.- 1.3 Pivoting Strategies and Iterative Refinement.- 1.4 Cholesky Decomposition for Symmetric Positive Definite Matrices.- Exercises.- 2 Error Analysis.- 2.1 Sources of Errors.- 2.2 Condition of Problems.- 2.3 Stability of Algorithms.- 2.4 Application to Linear Systems.- Exercises.- 3 Linear Least-Squares Problems.- 3.1 Least-Squares Method of Gauss.- 3.2 Orthogonalization Methods.- 3.3 Generalized Inverses.- Exe< rcises.- 4 Nonlinear Systems and Least-Squares Problems.- 4.1 Fixed-Point Iterations.- 4.2 Newton Methods for Nonlinear Systems.- 4.3 Gauss-Newton Method for Nonlinear Least-Squares Problems.- 4.4 Nonlinear Systems Depending on Parameters.- Exercises.- 5 Linear Eigenvalue Problems.- 5.1 Condition of General Eigenvalue Problems.- 5.2 Power Method.- 5.3 QR-Algorithm for Symmetric Eigenvalue Problems.- 5.4 Singular Value Decomposition.- 5.5 Stochastic Eigenvalue Problems.- Exercises.- 6 Three-Term Recurrence Relations.- 6.1 Theoretical Background.- 6.2 Numerical Aspects.- 6.3 Adjoint Summation.- Exercises.- 7 Interpolation and Approximation.- 7.1 Classical Polynomial Interpolation.- 7.2 Trigonometric Interpolation.- 7.3 Bzier Techniques.- 7.4 Splines.- Exercises.- 8 Large Symmetric Systems of Equations and Eigenvalue Problems.- 8.1 Classical Iteration Methods.- 8.2 Chebyshev Acceleration.- 8.3 Method of Conjugate Gradients.- 8.4 Preconditioning.- 8.5 Lanczos Methods.- Exercises.- 9 Definite Integrals.- 9.1 Quadrature Formulas.- 9.2 Newton-Cotes Formulas.- 9.3 Gauss-Christoffel Quadrature.- 9.4 Classical Romberg Quadrature.- 9.5 Adaptive Romberg Quadrature.- 9.6 Hard Integration Problems.- 9.7 Adaptive Multigrid Quadrature.- Exercises.- References.- Software.This book provides a well-written and clear introduction to the main topics of modern numerical analysis - sequence of linear equations, error analysis, least squares, nonlinear systems, symmetric eigenvalue problems, three-term recursions, interpolation and approximation, large systems and numerical integrations. It contains a large number of examples and exercises and many figures. This text is suitable for courses in numerical analysis and can also form the basis for a numerical linear algebra course.978-1-4419-2990-7978-0-387-89487-4Holden*Helge Holden, Norges Teknisk Naturvitenskap. Universitet NTNU, Trondheim, Norway; Bernt ksendal, University of Oslo CMA, Oslo, Norway; Jan Ube, Norwegian School of Economics and Business Administration, Bergen, Norway; Tusheng Zhang, University of Manchester School of Mathematics, Manchester, UK)Stochastic Partial Differential Equations+A Modeling, White Noise Functional ApproachmPreface to the Second Edition.- Preface to the First Edition.- Introduction.- Framework.- Applications to stochastic ordinary differential equations.- Stochastic partial differential equations driven by Brownian white noise.- Stochastic partial differential equations driven by Lvy white noise.- Appendix A. The Bochner-Minlos theorem.- Appendix B. Stochastic calculus based on Brownian motion.- Appendix C. Properties of Hermite polynomials.- Appendix D. Independence of bases in Wick products.- Appendix E. Stochastic calculus based on Lvy processes- References.- List of frequently used notation and symbols.- Index.The first edition of Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, gave a comprehensive introduction to SPDEs. In this, the second edition, the authors build on the theory of SPDEs driven by space-time Brownian motion, or more generally, space-time Lvy process noise, and introduce new applications of the field. Because the authors allow the noise to be in both space and time, the solutions to SPDEs are usually of the distribution type, rather than a classical random field. To make this study rigorous and as general as possible, the discussion of SPDEs is therefore placed in the context of Hida white noise theory. The first part of the book deals with the classical Brownian motion case; the second extends the Hida white noise theory to the Lvy white noise case. Applications of this theory are emphasized throughout. In particular, the stochastic pressure equation for fluid flow in porous media is treated, as are applications to finance. Graduate students in pure and applied mathematics as well as researchers in SPDEs, physics, and engineering will find this introduction indispensible. Useful exercises are collected at the end of each chapter.Focuses on the development of SPDEs and their application both to real-life problems and abstract mathematical topics

Includes new discussions of fractional Brownian motion, Lvy processes and Lvy random fields, and applications to finance

Provides an excellent introduction to the field and areas of current research

Exercises at the end of each chapter

978-1-4614-5476-2Holmes`Mark H. Holmes, Rensselaer Polytechnic Institute Academic Science of the Material, Troy, NY, USA$Introduction to Perturbation MethodsXVII, 436 p. 117 illus.Preface.- Preface to Second Edition.- Introduction to Asymptotic Approximations.- Matched Asymptotic Expansions.- Multiple Scales.- The WKB and Related Methods.- The Method of Homogenization- Introduction to Bifurcation and Stability.- References.- Index.7This introductory graduate text is based on a graduate course the author has taught repeatedly over the last ten years to students in applied mathematics, engineering sciences, and physics. Each chapter begins with an introductory development involving ordinary differential equations, and goes on to cover such traditional topics as boundary layers and multiple scales. However, it also contains material arising from current research interest, including homogenisation, slender body theory, symbolic computing, and discrete equations. Many of the excellent exercises are derived from problems of up-to-date research and are drawn from a wide range of application areas.One hundred new pages added includingnew material ontranscedentally small terms, Kummer's function, weakly coupled oscillatorsand wave interactions.iLots of examples and exercises

Class test for over 20 years

Includes delay equations

978-1-4899-9613-8978-0-8176-4363-8HottaRyoshi Hotta, Okayama University of Science Fac. Sciences, Okayama, Japan; Kiyoshi Takeuchi, University of Tsukuba, Tsukuba, Ibaraki, Japan; Toshiyuki Tanisaki, Osaka City University Graduate School of Science, Osaka, Japan6D-Modules, Perverse Sheaves, and Representation Theory XI, 412 p.D-Modules and Perverse Sheaves.- Preliminary Notions.- Coherent D-Modules.- Holonomic D-Modules.- Analytic D-Modules and the de Rham Functor.- Theory of Meromorphic Connections.- Regular Holonomic D-Modules.- Riemann Hilbert Correspondence.- Perverse Sheaves.- Representation Theory.- Algebraic Groups and Lie Algebras.- Conjugacy Classes of Semisimple Lie Algebras.- Representations of Lie Algebras and D-Modules.- Character Formula of HighestWeight Modules.- Hecke Algebras and Hodge Modules.D-modules continues to be an active area of stimulating research in such mathematical areas as algebra, analysis, differential equations, and representation theory. Key to D-modules, Perverse Sheaves, and Representation Theory is the authors' essential algebraic-analytic approach to the theory, which connects D-modules to representation theory and other areas of mathematics. Significant concepts and topics that have emerged over the last few decades are presented, including a treatment of the theory of holonomic D-modules, perverse sheaves, the all-important Riemann-Hilbert correspondence, Hodge modules, and the solution to the Kazhdan-Lusztig conjecture using D-module theory. To further aid the reader, and to make the work as self-contained as possible, appendices are provided as background for the theory of derived categories and algebraic varieties. The book is intended to serve graduate students in a classroom setting and as self-study for researchers in algebraic geometry, and representation theory.*D*-modules a sti< mulating and active area of research

The unique text treating an algebraic-analytic approach to *D*-module theory

Examines *D*-module theory, connecting algebraic geometry and representation theory

Clusters with many Springer books written by the authors, Kashiwara, Schapira and others

Uses *D*-module theory to prove the celebrated Kazhdan-Lusztig polynomials

Detailed examination with excellent proof of the Riemann-Hilbert correspondence

978-1-4419-7915-5HuangWeizhang Huang, University of Kansas, Lawrence, KS, USA; Robert D. Russell, Simon Fraser University Department of Mathematics, Burnaby, BC, CanadaAdaptive Moving Mesh Methods XVIII, 434 p.mPreface.- Introduction.- Adaptive Mesh Movement in 1D.- Discretization of PDEs on Time-Varying Meshes.- Basic Principles of Multidimensional Mesh Adaption.- Monitor Functions.- Variational Mesh Adaptive Methods.- Velocity-Based Adaptive Methods.- Appendix: Sobolev Spaces.- Appendix: Arithmetic Mean Geometric Mean Inequality and Jensen's Inequality.- Bibliography.Moving mesh methods are an effective, mesh-adaptation-based approach for the numerical solution of mathematical models of physical phenomena. Currently there exist three main strategies for mesh adaptation, namely, to use mesh subdivision, local high order approximation (sometimes combined with mesh subdivision), and mesh movement. The latter type of adaptive mesh method has been less well studied, both computationally and theoretically. This book is about adaptive mesh generation and moving mesh methods for the numerical solution of time-dependent partial differential equations. It presents a general framework and theory for adaptive mesh generation and gives a comprehensive treatment of moving mesh methods and their basic components, along with their application for a number of nontrivial physical problems. Many explicit examples with computed figures illustrate the various methods and the effects of parameter choices for those methods. The partial differential equations considered are mainly parabolic (diffusion-dominated, rather than convection-dominated). The extensive bibliography provides an invaluable guide to the literature in this field. Each chapter contains useful exercises. Graduate students, researchers and practitioners working in this area will benefit from this book.Weizhang Huang is a Professor in the Department of Mathematics at the University of Kansas. Robert D. Russell is a Professor in the Department of Mathematics at Simon Fraser University.,First ever comprehensive treatment of moving mesh methods for solving time-dependent Partial Differential Equations

General error analysis for adaptive mesh generation using equidistribution and alignment is covered

Numerous numerical examples and several Matlab codes are included

978-1-4614-2708-7978-3-540-21290-4 HuybrechtsQDaniel Huybrechts, Universit Paris VII UFR de Mathematiques, Paris CX 05, FranceComplex GeometryXII, 309 p.Local Theory.- Complex Manifolds.- Khler Manifolds.- Vector Bundles.- Applications of Cohomology.- Deformations of Complex Structures.Complex geometry studies (compact) complex manifolds. It discusses algebraic as well as metric aspects. The subject is on the crossroad of algebraic and differential geometry. Recent developments in string theory have made it an highly attractive area, both for mathematicians and theoretical physicists. The author s goal is to provide an easily accessible introduction to the subject. The book contains detailed accounts of the basic concepts and the many exercises illustrate the theory. Appendices to various chapters allow an outlook to recent research directions. Daniel Huybrechts is currently Professor of Mathematics at the University Denis Diderot in Paris.*Textbook on core topic in complex analysis978-1-4614-0334-0 InspergerTams Insperger, Budapest University of Technology and Ec, Budapest, Hungary; Gbor Stpn, Budapest University of Technology and Ec, Budapest, Hungary*Semi-Discretization for Time-Delay Systems&Stability and Engineering Applications X, 174 p.SCT19010Control~Introducing delay.- Basic delay differential equations.- Newtonian examples.- Engineering applications.- Summary.- References.This book presents the recently introduced and already widely referred semi-discretization method for the stability analysis of delayed dynamical systems. Delay differential equations often come up in different fields of engineering, like feedback control systems, machine tool vibrations, balancing/stabilization with reflex delay. The behavior of such systems is often counter-intuitive and closed form analytical formulas can rarely be given even for the linear stability conditions. If parametric excitation is coupled with the delay effect, then the governing equation is a delay differential equation with time periodic coefficients, and the stability properties are even more intriguing. The semi-discretization method is a simple but efficient method that is based on the discretization with respect to the delayed term and the periodic coefficients only. The method can effectively be used to construct stability diagrams in the space of system parameters.Stability diagrams for some basic delay systems appearing in engineering problems are summarized in the introduction

The semi-discretization method is presented in a simple and clear way with examples

Different approaches for the application of the semi-discretization method is presented considering the rate convergence and numerical efficiency Basic engineering models described by time-periodic and time-delayed systems are presented

978-1-4614-3013-1978-0-387-25364-0IsakovXVictor Isakov, Wichita State University Dept. Mathematics & Statistics, Wichita, KS, USA3Inverse Problems for Partial Differential EquationsXII, 344 p.SInverse Problems.- Ill-Posed Problems and Regularization.- Uniqueness and Stability in the Cauchy Problem.- Elliptic Equations: Single Boundary Measurements.- Elliptic Equations: Many Boundary Measurements.- Scattering Problems.- Integral Geometry and Tomography.- Hyperbolic Problems.- Inverse parabolic problems.- Some Numerical Methods.@The topic of the inverse problems is of substantial and rapidly growing interest for many scientists and engineers. The second edition covers most important recent developments in the field of inverse problems, describing theoretical and computational methods, and emphasizing new ideas and techniques. It also reflects new changes since the first edition, including some corrections. This edition is considerably expanded, with some concepts such as pseudo-convexity, and proofs simplified. New material is added to reflect recent progress in theory of inverse problems. This book is intended for mathematicians working with partial differential equations and their applications, and physicists, geophysicists and engineers involved with experiments in nondestructive evaluation, seismic exploration, remote sensing and tomography.mCovers most important recent developments in inverse problems

Presented in a readable and informative manner

Introduces both scientific and engineering researchers as well as graduate students to the significant work done in this area in recent years, relating it to broader themes in mathematical analysis

Contains numero< us exercises

978-1-4419-2054-6978-3-540-45895-1Jahren; DundasBjorn Ian Dundas, University of Bergen Dept. Mathematics, Bergen, Norway; Bjrn Jahren, Department of Mathematics University of Oslo, Oslo, Norway; Marc Levine, Northeastern University Dept. Mathematics, Boston, MA, USA; P.A. stvr; Oliver Rndigs, Universitt Osnabrck Institut fr Mathematik, Osnabrck, Germany; Vladimir Voevodsky, Princeton University Inst. Advanced Study, Princeton, NJ, USAMotivic Homotopy Theory@Lectures at a Summer School in Nordfjordeid, Norway, August 2002 X, 221 p.Prerequisites in Algebraic Topology the Nordfjordeid Summer School on Motivic Homotopy Theory.- Basic Properties and Examples.- Deeper Structure: Simplicial Sets.- Model Categories.- Motivic Spaces and Spectra.- Background from Algebraic Geometry.- Elementary Algebraic Geometry.- Sheaves for a Grothendieck Topology.- Voevodsky s Nordfjordeid Lectures: Motivic Homotopy Theory.- Voevodsky s Nordfjordeid Lectures: Motivic Homotopy Theory.This book is based on lectures given at a summer school on motivic homotopy theory at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. Aimed at graduate students in algebraic topology and algebraic geometry, it contains background material from both of these fields, as well as the foundations of motivic homotopy theory. It will serve as a good introduction as well as a convenient reference for a broad group of mathematicians to this important and fascinating new subject.978-3-540-41662-3JensenA. Jensen; Anders la Cour-HarboRipples in MathematicsThe Discrete Wavelet Transform IX, 246 p._ 1. Introduction.- 1.1 Prerequisites.- 1.2 Guide to the Book.- 1.3 Background Information.- 2. A First Example.- 2.1 The Example.- 2.2 Generalizations.- Exercises.- 3. The Discrete Wavelet Transform via Lifting.- 3.1 The First Example Again.- 3.2 Definition of Lifting.- 3.3 A Second Example.- 3.4 Lifting in General.- 3.5 DWT in General.- 3.6 Further Examples.- Exercises.- 4. Analysis of Synthetic Signals.- 4.1 The Haar Transform.- 4.2 The CDF(2,2) Transform.- Exercises.- 5. Interpretation.- 5.1 The First Example.- 5.2 Further Results on the Haar Transform.- 5.3 Interpretation of General DWT.- Exercises.- 6. Two Dimensional Transforms.- 6.1 One Scale DWT in Two Dimensions.- 6.2 Interpretation and Examples.- 6.3 A 2D Transform Based on Lifting.- Exercises.- 7. Lifting and Filters I.- 7.1 Fourier Series and the z-Transform.- 7.2 Lifting in the z-Transform Representation.- 7.3 Two Channel Filter Banks.- 7.4 Orthonormal and Biorthogonal Bases.- 7.5 Two Channel Filter Banks in the Time Domain.- 7.6 Summary of Results on Lifting and Filters.- 7.7 Properties of Orthogonal Filters.- 7.8 Some Examples.- Exercises.- 8. Wavelet Packets.- 8.1 From Wavelets to Wavelet Packets.- 8.2 Choice of Basis.- 8.3 Cost Functions.- Exercises.- 9. The Time-Frequency Plane.- 9.1 Sampling and Frequency Contents.- 9.2 Definition of the Time-Frequency Plane.- 9.3 Wavelet Packets and Frequency Contents.- 9.4 More about Time-Frequency Planes.- 9.5 More Fourier Analysis. The Spectrogram.- Exercises.- 10. Finite Signals.- 10.1 The Extent of the Boundary Problem.- 10.2 DWT in Matrix Form.- 10.3 Gram-Schmidt Boundary Filters.- 10.4 Periodization.- 10.5 Moment Preserving Boundary Filters.- Exercises.- 11. Implementation.- 11.1 Introduction to Software.- 11.2 Implementing the Haar Transform Through Lifting.- 11.3 Implementing the DWT Through Lifting.- 11.4 The Real Time Method.- 11.5 Filter Bank Implementation.- 11.6 Construction of Boundary Filters.- 11.7 Wavelet Packet Decomposition.- 11.8 Wavelet Packet Bases.- 11.9 Cost Functions.- Exercises.- 12. Lifting and Filters II.- 12.1 The Three Basic Representations.- 12.2 From Matrix to Equation Form.- 12.3 From Equation to Filter Form.- 12.4 From Filters to Lifting Steps.- 12.5 Factoring Daubechies 4 into Lifting Steps.- 12.6 Factorizing Coiflet 12 into Lifting Steps.- Exercises.- 13. Wavelets in Matlab.- 13.1 Multiresolution Analysis.- 13.2 Frequency Properties of the Wavelet Transform.- 13.3 Wavelet Packets Used for Denoising.- 13.4 Best Basis Algorithm.- 13.5 Some Commands in Uvi_Wave.- Exercises.- 14. Applications and Outlook.- 14.1 Applications.- 14.2 Outlook.- 14.3 Some Web Sites.- References.\This book gives an introduction to the discrete wavelet transform and some of its applications. It is based on a novel approach to discrete wavelets called lifting. The first part is a completely elementary introduction to the subject, and the prerequisites for this part are knowledge of basic calculus and linear algebra. The second part requires some knowledge of Fourier series and digital signal analysis. The connections between lifting and filter theory are presented and the wavelet packet transforms are defined. The time-frequency plane is used for interpretation of signals. The problems with finite length signals are treated in detail. MATLAB is used as the computational environment for examples and implementation of transforms. The book is well suited for undergraduate mathematics and electrical engineering students and engineers in industry.* Based on a novel approach to discrete wavelets called lifting* Elementary introduction

* Problems with finite length signals are treated in detail

* MATLAB is used as the computational environment for examples and implementations

978-3-642-00540-4Jost[Jrgen Jost, Max Planck Institut fr Mathematik in den Naturwissenschafte, Leipzig, GermanyGeometry and PhysicsXIV, 217p. 1 illus. in color.1.Geometry.- 1.1.Riemannian and Lorentzian manifolds.- 1.2.Bundles and connections.- 1.3.Tensors and spinors.- 1.4.Riemann surfaces and moduli spaces.- 1.5.Supermanifolds.- 2.Physics.- 2.1.Classical and quantum physics.- 2.2.Lagrangians.-2.3.Variational aspects.- 2.4.The sigma model.- 2.5.Functional integrals.- 2.6.Conformal field theory.- 2.7.String theory.- Bibliography.- Index.'Geometry and Physics' addresses mathematicians wanting to understand modern physics, and physicists wanting to learn geometry. It gives an introduction to modern quantum field theory and related areas of theoretical high-energy physics from the perspective of Riemannian geometry, and an introduction to modern geometry as needed and utilized in modern physics. Jrgen Jost, a well-known research mathematician and advanced textbook author, also develops important geometric concepts and methods that can be used for the structures of physics. In particular, he discusses the Lagrangians of the standard model and its supersymmetric extensions from a geometric perspective.IVery good introductory text on the interplay between geometry and physics978-3-642-42070-2XIV, 217 p. 1 illus. in color.978-1-4614-4808-2XIII, 410 p. 10 illus.Preface.- Introduction: What are Partial Differential Equations?.- 1 The Laplace equation as the Prototype of an Elliptic Partial Differential Equation of Second Order.- 2 The Maximum Principle.- 3 Existence Techniques I:Methods Based on the Maximum Principle.- 4 Existence Techniques II: Parabolic Methods. The Heat Equation.- 5 Reaction-Diffusion Equations and Systems.- 6 Hyperbolic Equations.- 7 The Heat Equation, Semigroups, and Brownian Motion.-8 Relationshipsbetween Different Partial Differential Equations.- 9 TheDirichlet Principle. Variational Methods for the Solutions of PDEs (Existence Techniques III).- 10Sobolev Spaces and L^2 Regularity theory.- 11 Strong solutions.- 12 The Regularity Theory of Schauder and the Continuity Method (Existence Techniques IV).- 13The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash.- Appendix: Banach and Hilbert spaces. The L^p-Spaces.- References.- Index of Notation.- Ind< ex.lThis book offers an ideal graduate-level introduction to the theory of partial differential equations. The first part of the book describes the basic mathematical problems and structures associated with elliptic, parabolic, and hyperbolic partial differential equations, and explores the connections between these fundamental types. Aspects of Brownian motion or pattern formation processes are also presented. The second part focuses on existence schemes and developsestimates for solutions of elliptic equations, such as Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. In particular, the reader will learn the basic techniques underlying current research in elliptic partial differential equations.This revised and expanded third edition is enhanced with many additional examples that will help motivate the reader. New features include a reorganized and extended chapter on hyperbolic equations, as well as a new chapter on the relations between different types of partial differential equations, including first-order hyperbolic systems, Langevin and Fokker-Planck equations, viscosity solutions for elliptic PDEs, and much more. Also, the new edition contains additional material on systems of elliptic partial differential equations, and it explains in more detail how the Harnack inequality can be used for the regularity of solutions.

New edition extensively revised and updated

Features a systematic discussion of the relations between different types of partial differential equations

Presents new Harnack type techniques

978-1-4939-0247-7978-0-8176-8255-2JoynerDavid Joyner, U.S. Naval Academy Chauvenet Hall, Annapolis, MD, USA; Jon-Lark Kim, University of Louisville Deptartment of Mathematics, Louisville, KY, USA+Selected Unsolved Problems in Coding TheoryXII, 248p. 17 illus..SCM13038'Information and Communication, CircuitsSCI15041Coding and Information TheoryGPJPreface.- Background.- Codes and Lattices.- Kittens and Blackjack.- RH and Coding Theory.- Hyperelliptic Curves and QR Codes.- Codes from Modular Curves.- Appendix.- Bibliography.- Index.This original monograph investigates several unsolved problems that currently exist in coding theory. A highly relevant branch of mathematical computer science, the theory of error-correcting codes is concerned with reliably transmitting data over a noisy channel.Despite its fairly long history and consistent prominence, the field still contains interesting problems that have resisted solution by some of the most prominent mathematicians of recent decades.Employing SAGE a free open-source mathematics software system to illustrate ideas, this book is intended for graduate students and researchers in algebraic coding theory, especially those who are interested in finding some current unsolved problems. Familiarity with concepts in algebra, number theory, and modular forms is assumed. The work may be used as supplementary reading material in a graduate course on coding theory or for self-study.SAGE, a free open-source mathematics software system, is used with concrete examples in order to emphasize the computational aspects of coding theory

Introduces and explores unsolved open problems in an effort to illuminate the fundamentals of coding theory and stimulate further research

Appeals to academia and industry with real-world applications in electrical engineering and digital communication

978-3-642-24507-7Jukna9Stasys Jukna, University of Frankfurt, Frankfurt, GermanyBoolean Function ComplexityAdvances and FrontiersAlgorithms and CombinatoricsXV, 617p. 70 illus..Part I Basics.- Part II Communication Complexity.- Part III Circuit Complexity.- Part IV Bounded Depth Circuits.- Part V Branching Programs.- Part VI Fragments of Proof Complexity.- A Epilog.- B Mathematical Background.- References.- Index.Boolean circuit complexity is the combinatorics of computer science and involves many intriguing problems that are easy to state and explain, even for the layman. This book is a comprehensive description of basic lower bound arguments, covering many of the gems of this complexity Waterloo that have been discovered over the past several decades, right up to results from the last year or two. Many open problems, marked as Research Problems, are mentioned along the way. The problems are mainly of combinatorial flavor but their solutions could have great consequences in circuit complexity and computer science. The book will be of interest to graduate students and researchers in the fields of computer science and discrete mathematics.<p>This is the first book covering the happening in circuit complexity during the last 20 years</p><p>Includes non-standard topics, like graph complexity or circuits with arbitrary gates </p><p>Includes about 40 open problems as potential research topics for students </p>978-3-642-43144-9XV, 617 p. 70 illus.978-1-4419-9636-7JungKSoon-Mo Jung, Hongik University, Jochiwon, Korea, Republic of (South Korea)JHyers-Ulam-Rassias Stability of Functional Equations in Nonlinear AnalysisXIV, 362 p. -1. Introduction.-2. Additive Cauchy Equation (Behavior of additive functions, Hyers-Ulam stability, Hyers-Ulam-Rassias stability, Stability on a restricted domain, Method of invariant means, Fixed point method, Composite functional congruences, Pexider equation, Remarks). -3. Generalized Additive Cauchy Equations (Functional equation f(ax+by)=af(x)+bf(y), Additive Cauchy equations of general form, Functional equation f(x+y)2=(f(x)+f(y))2). -4. Hossz s Functional Equation (Stability in the sense of Borelli, Hyers-Ulam stability, General< ized Hossz s equation is not stable on the unit interval, Hossz s functional equation of Pexider type). -5. Homogeneous Functional Equation(Homogeneous equation between Banach algebras, Superstability on a restricted domain, Homogeneous equation between vector spaces, Homogeneous equation of Pexide type). -6. Linear Functional Equations (A system for linear functions, Functional equation f(x+cy)=f(x)+cf(y), Stability for other equations).-7. Jensen s Functional Equation (Hyers-Ulam-Rassias stability, Stability on a restriced domain, Fixed point method, Loba evsk;s functional equation). -8. Quadratic Functional Equations (Hyers-Ulam-Rassias stability, Stability on a restricted domain, Fixedpoint method, Quadratic functional equation of other type, Quadratic functional equation of Pexider type). -9. Exponential Functional Equations (Superstability, Stability in the sense of Ger, Stability on a restricted domain, Exponential functional equation of other type). -10. Multiplicative Functional Equations (Superstability,-multiplicative functional, Theory of AMNM algebras, Functional equation f(xy)= f(x)y, Functional equation f(x+y)= f(x)f(y)f(1/x+1/y)). -11. Logarithmic Functional Equations (Functional equation f(xy)= yf(x), Superstability of equation f(xy)= yf(x), Functional equation of Heuvers). -12. Trigonometric Functional Equations (Cosine functional equation, Sine functional equation, Trigonometric equations with two unknowns, Butler-Rassias functional equation, Remarks). -13. Isometric Functional Equation (Hyers-Ulam stability, Stability on a restricted domain, Fixed point method, Wigner equation). -14. Miscellaneous (Associativity equation, Equation of multiplicative derivation, Gamma functional equation). -Bibliography. -Index.No books dealing with a comprehensive illustration of the fast developing field of nonlinear analysis had been published for the mathematicians interested in this field for more than a half century until D. H. Hyers, G. Isac and Th. M. Rassias published their book, 'Stability of Functional Equations in Several Variables'. This book will complement the books of Hyers, Isac and Rassias and of Czerwik (Functional Equations and Inequalities in Several Variables) by presenting mainly the results applying to the Hyers-Ulam-Rassias stability. Many mathematicians have extensively investigated the subjects on the Hyers-Ulam-Rassias stability. This book covers and offers almost all classical results on the Hyers-Ulam-Rassias stability in an integrated and self-contained fashion.<p>Organized such that highly advanced undergraduate as well as graduate level students will be able to follow the materials</p><p> Contains unified notation and refined formulae</p><p>All the necessary materials and information are included in this book .<i></p><p>Offers almost all classical results on the Hyers Ulam Rassias stability in an integrated and self-contained fashion</p>978-1-4614-2862-6978-0-8176-8110-4Kaiser<Gerald Kaiser, Center for Signals & Waves, Portland, OR, USAA Friendly Guide to WaveletsXVIII, 300p. 33 illus..SCT24051#Signal, Image and Speech ProcessingTTBM0Preface.- Suggestions to the Reader.- Symbols, Conventions, and Transforms.- Part I: Basic Wavelet Analysis. Preliminaries: Background and Notation.- Windowed Fourier Transforms.- Continuous Wavelet Transforms.- Generalized Frames: Key to Analysis and Synthesis.- Discrete Time-Frequency Analysis and Sampling.- Discrete Time-Scale Analysis.- Multiresolution Analysis.- Daubechies Orthonormal Wavelet Bases.- Part II: Physical Wavelets.- Introduction to Wavelet Electromagnetics.- Applications to Radar and Scattering.- Wavelet Acoustics.- References.- Index.This volume is designed as a textbook for an introductory course on wavelet analysis and time-frequency analysis aimed at graduate students or advanced undergraduates in science and engineering. It can also be used as a self-study or reference book by practicing researchers in signal analysis and related areas. Since the expected audience is not presumed to have a high level of mathematical background, much of the needed analytical machinery is developed from the beginning. The only prerequisites for the first eight chapters are matrix theory, Fourier series, and Fourier integral transforms. Each of these chapters ends with a set of straightforward exercises designed to drive home the concepts just covered, and the many graphics should further facilitate absorption. @Considered one of the best textbooks on this topic for mathematicians and physicists interested in wavelet analysis

Covers all the fundamental concepts of wavelets in an elegant, straightforward way

Has a number of unique features, which makes the book particularly valuable for newcomers to the field

978-3-540-88866-6KanamoriTAkihiro Kanamori, Boston University Dept. Manufacturing Engineering, Boston, MA, USAThe Higher Infinite3Large Cardinals in Set Theory from Their BeginningsSCM28000TopologyPBPPreliminaries.- Beginnings.- Partition Properties.- Forcing and Sets of Reals.- Aspects of Measurability.- Strong Hypotheses.- Determinacy.This is the softcover reprint of the very popular hardcover edition. The theory of large cardinals is currently a broad mainstream of modern set theory, the main area of investigation for the analysis of the relative consistency of mathematical propositions and possible new axioms for mathematics. The first of a proj< ected multi-volume series, this book provides a comprehensive account of the theory of large cardinals from its beginnings and some of the direct outgrowths leading to the frontiers of contemporary research. A genetic approach is taken, presenting the subject in the context of its historical development. With hindsight the consequential avenues are pursued and the most elegant or accessible expositions given. With open questions and speculations provided throughout the reader should not only come to appreciate the scope and coherence of the overall enterprise but also become prepared to pursue research in several specific areas by studying the relevant sections.EHas become a standard reference and guide in the set theory community978-0-8176-4912-8KapovichSMichael Kapovich, University of California, Davis Dept. Mathematics, Davis, CA, USA(Hyperbolic Manifolds and Discrete GroupsXXVII, 467p. 78 illus..vThree-Dimensional Topology.- Thurston Norm.- Geometry of Hyperbolic Space.- Kleinian Groups.- Teichmller Theory of Riemann Surfaces.- to Orbifold Theory.- Complex Projective Structures.- Sociology of Kleinian Groups.- Ultralimits of Metric Spaces.- to Group Actions on Trees.- Laminations, Foliations, and Trees.- Rips Theory.- Brooks Theorem and Circle Packings.- Pleated Surfaces and Ends of Hyperbolic Manifolds.- Outline of the Proof of the Hyperbolization Theorem.- Reduction to the Bounded Image Theorem.- The Bounded Image Theorem.- Hyperbolization of Fibrations.- The Orbifold Trick.- Beyond the Hyperbolization Theorem.This classic book is at the crossroads of several branches of mathematics: hyperbolic geometry, discrete groups, 3-dimensional topology, geometric group theory, and complex analysis. The main focus throughout the text is on Thurston s hyperbolization theorem, one of the central results of 3-dimensional topology that has completely changed the landscape of the field. The book contains a number of open problems and conjectures related to the hyperbolization theorem as well as rich discussions on related topics including geometric structures on 3-manifolds, higher dimensional negatively curved manifolds, and hyperbolic groups. Featuring beautiful illustrations, a rich set of examples, numerous exercises, and an extensive bibliography and index, Hyperbolic Manifolds and Discrete Groups continues to serve as an ideal graduate text and comprehensive reference. The book is very clearly written and fairly self-contained. It will be useful to researchers and advanced graduate students in the field and can serve as an ideal guide to Thurston's work and its recent developments. Mathematical Reviews. Beyond the hyperbolization theorem, this is an important book which had to be written; some parts are still technical and will certainly be streamlined and shortened in the next years, but together with Otal's work a complete published proof of the hyperbolization theorem is finally available. Apart from the proof itself, the book contains a lot of material which will be useful for various other directions of research. Zentralbatt MATH. This book can act as source material for a postgraduate course and as a reference text on the topic as the references are full and extensive. ... The text is self-contained and very well illustrated. ASLIB Book Guide. <P>Includes beautiful illustrations, a rich set of examples of key concepts, numerous exercises</P> <P>An extensive bibliography and index are complemented by a glossary of terms</P> <P>Presents the first complete proof of the generic case of Thurston s hyperbolization theorem</P>978-3-0348-0536-0 KarlovichYuri I. Karlovich, Universidad Autnoma del Estado de Morelos, Cuernavaca, Mexico; Luigi Rodino, Universit di Torino, Torino, Italy; Bernd Silbermann, Technische Universitt Chemnitz, Chemnitz, Germany; Leiba Rodman, College of William and Mary, Williamsburg, VA, USAHOperator Theory, Pseudo-Differential Equations, and Mathematical Physics*The Vladimir Rabinovich Anniversary Volume*XXV, 406 p. 25 illus., 16 illus. in color.Collection of essays<Preface.- Contributions by renowned scientists.- References.This volume is a collection of papers devoted to the 70th birthday of Professor Vladimir Rabinovich. The opening article (by Stefan Samko) includes a short biography of Vladimir Rabinovich, along with some personal recollections and bibliography of his work. It is followed by twenty research and survey papers in various branches of analysis (pseud< odifferential operators and partial differential equations, Toeplitz, Hankel, and convolution type operators, variable Lebesgue spaces, etc.) close to Professor Rabinovich's research interests. Many of them are written by participants of the International workshop Analysis, Operator Theory, and Mathematical Physics (Ixtapa, Mexico, January 23 27, 2012) having a long history of scientific collaboration with Vladimir Rabinovich, and are partially based on the talks presented there.The volume will be of great interest to researchers and graduate students in differential equations, operator theory, functional and harmonic analysis, and mathematical physics. <p>Wide spectrum of important problems in operator theory, PDEs, mathematical physics and numerical analysis </p><p>Modern methods and approaches </p><p>Dedicated to Vladimir Rabinovich </p>978-3-0348-0772-2978-3-7091-0444-6KauersManuel Kauers, Universitt Linz Research Institute for Symbolic, Linz, Austria; Peter Paule, Johannes Kepler University, Linz, AustriaThe Concrete TetrahedronOSymbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates*Texts & Monographs in Symbolic Computation IX, 203 p.SCM29000PBDSpringer Vienna1 Introduction.- 2 Formal Power Series.- 3 Polynomials.- 4 C-Finite Sequences.- 5 Hypergeometric Series.- 6 Algebraic Functions.- 7 Holonomic Sequences and Power Series.- Appendix.- References.- Index.>The book treats four mathematical concepts which play a fundamental role in many different areas of mathematics: symbolic sums, recurrence (difference) equations, generating functions, and asymptotic estimates.Their key features, in isolation or in combination, their mastery by paper and pencil or by computer programs, and their applications to problems in pure mathematics or to 'real world problems' (e.g. the analysis of algorithms) are studied. The book is intended as an algorithmic supplement to the bestselling 'Concrete Mathematics' by Graham, Knuth and Patashnik.Concrete mathematics from a computer algebra perspective

Informal style: as simple as possible, as complicated as necessary

Includes notivating applications from combinatorics, computer science, number theory, special functions

EBOP11645Computer Science978-0-387-94374-9KechrisZAlexander Kechris, California Institute of Technology Dept. Mathematics, Pasadena, CA, USA Classical Descriptive Set TheoryXVIII, 402 pp. 34 figs.=I Polish Spaces.- 1. Topological and Metric Spaces.- 2. Trees.- 3. Polish Spaces.- 4. Compact Metrizable Spaces.- 5. Locally Compact Spaces.- 6. Perfect Polish Spaces.- 7.Zero-dimensional Spaces.- 8. Baire Category.- 9. Polish Groups.- II Borel Sets.- 10. Measurable Spaces and Functions.- 11. Borel Sets and Functions.- 12. Standard Borel Spaces.- 13. Borel Sets as Clopen Sets.- 14. Analytic Sets and the Separation Theorem.- 15. Borel Injections and Isomorphisms.- 16. Borel Sets and Baire Category.- 17. Borel Sets and Measures.- 18. Uniformization Theorems.- 19. Partition Theorems.- 20. Borel Determinacy.- 21. Games People Play.- 22. The Borel Hierarchy.- 23. Some Examples.- 24. The Baire Hierarchy.- III Analytic Sets.- 25. Representations of Analytic Sets.- 26. Universal and Complete Sets.- 27. Examples.- 28. Separation Theorems.- 29. Regularity Properties.- 30. Capacities.- 31. Analytic Well-founded Relations.- IV Co-Analytic Sets.- 32. Review.- 33. Examples.- 34. Co-Analytic Ranks.- 35. Rank Theory.- 36. Scales and Uniformiiatiou.- V Projective Sets.- 37. The Projective Hierarchy.- 38. Projective Determinacy.- 39. The Periodicity Theorems.- 40. Epilogue.- Appendix A. Ordinals and Cardinals.- Appendix B. Well-founded Relations.- Appendix C. On Logical Notation.- Notes and Hints.- References.- Symbols and Abbreviations._Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. It includes a wide variety of examples, exercises (over 400), and applications, in order to illustrate the general concepts and results of the theory. This text provides a first basic course in classical descriptive set theory and covers material with which mathematicians interested in the subject for its own sake or those that wish to use it in their field should be familiar. Over the years, researchers in diverse areas of mathematics, such as logic and set theory, analysis, topology, probability theory, etc., have brought to the subject of descriptive set theory their own intuitions, concepts, terminology and notation.978-3-642-03544-9KemperQGregor Kemper, Technische Universitt Zentrum Mathematik - M11, Garching, GermanyA Course in Commutative AlgebraXII, 248 p.)Introduction.- Part I The Algebra Geometry Lexicon: 1 Hilbert's Nullstellensatz; 2 Noetherian and Artinian Rings; 3 The Zariski Topology; 4 A Summary of the Lexicon.- Part II Dimension: 5 Krull Dimension and Transcendence Degree; 6 Localization; 7 The Principal Ideal Theorem; 8 Integral Extensions.- Part III Computational Methods: 9 Grbner Bases; 10 Fibers and Images of Morphisms Revisited; 11 Hilbert Series and Dimension.- Part IV Local Rings: 12 Dimension Theory; 13 Regular Local Rings; 14 Rings of Dimension One.- References.- Notation.- Index.3This textbook offers a thorough, modern introduction into commutative algebra. It is intented mainly to serve as a guide for a course of one or two semesters, or for self-study. The carefully selected subject matter concentrates on the concepts and results at the center of the field. The book maintains a constant view on the natural geometric context, enabling the reader to gain a deeper understanding of the material. Although it emphasizes theory, three chapters are devoted to computational aspects. Many illustrative examples and exercises enrich the text.Excellently written textbook in commutative algebra Book makes getting into the subject easier for students than with existing works Many illustrative examples and exercises978-3-642-26632-4978-1-4614-0501-6 Kielhfer=Hansjrg Kielhfer, University of Augsburg, Augsburg, GermanyBifurcation TheoryCAn Introduction with Applications to Partial Differential EquationsVIII, 400 p.,Introduction.- Global Theory.- Applications.In the past three decades, bifurcation theory has matured into a well-established and vibrant branch of mathematics. This book gives a unified presentation in an abstract setting of the main theorems in bifurcation theory, as well as more recent and lesser known results. It covers both the local and global theory of one-parameter bifurcations for operators acting in infinite-dimensional Banach spaces, and shows how to apply the theory to problems involving partial differential equations. In addition to existence, qualitative properties such as stability and nodal structure of bifurcating solutions are treated in depth. This volume will serve as an important reference for mathematicians, physicists, and theoretically-inclined engineers working in bifurcation theory and its applications to partial differential equations.The second edition is substantially and formally revised and new material is added. Among this is bifurcati< on with a two-dimensional kernel with applications, the buckling of the Euler rod, the appearance of Taylor vortices, the singular limit process of the Cahn-Hilliard model, and an application of this method to more complicated nonconvex variational problems.Gives a unified presentation in an abstract setting

Two new sections along with many revisions

More references included

978-1-4939-0140-1978-0-387-94102-8KinseyL.Christine KinseyTopology of Surfaces X, 281 p.,1. Introduction to topology.- 1.1. An overview.- 2. Point-set topology in ?n.- 2.1. Open and closed sets in ?n.- 2.2. Relative neighborhoods.- 2.3. Continuity.- 2.4. Compact sets.- 2.5. Connected sets.- 2.6. Applications.- 3. Point-set topology.- 3.1. Open sets and neighborhoods.- 3.2. Continuity, connectedness, and compactness.- 3.3. Separation axioms.- 3.4. Product spaces.- 3.5. Quotient spaces.- 4. Surfaces.- 4.1. Examples of complexes.- 4.2. Cell complexes.- 4.3. Surfaces.- 4.4. Triangulations.- 4.5. Classification of surfaces.- 4.6. Surfaces with boundary.- 5. The euler characteristic.- 5.1. Topological invariants.- 5.2. Graphs and trees.- 5.3. The euler characteristic and the sphere.- 5.4. The euler characteristic and surfaces.- 5.5. Map-coloring problems.- 5.6. Graphs revisited.- 6. Homology.- 6.1. The algebra of chains.- 6.2. Simplicial complexes.- 6.3. Homology.- 6.4. More computations.- 6.5. Betti numbers and the euler characteristic.- 7. Cellular functions.- 7.1. Cellular functions.- 7.2. Homology and cellular functions.- 7.3. Examples.- 7.4. Covering spaces.- 8. Invariance of homology.- 8.1. Invariance of homology for surfaces.- 8.2. The Simplicial Approximation Theorem.- 9. Homotopy.- 9.1. Homotopy and homology.- 9.2. The fundamental group.- 10. Miscellany.- 10.1. Applications.- 10.2. The Jordan Curve Theorem.- 10.3. 3-manifolds.- 11. Topology and calculus.- 11.1. Vector fields and differential equations in ?n.- 11.2. Differentiable manifolds.- 11.3. Vector fields on manifolds.- 11.4. Integration on manifolds.- Appendix: Groups.- References.4This book aims to provide undergraduates with an understanding of geometric topology. Topics covered include a sampling from point-set, geometric, and algebraic topology. The presentation is pragmatic, avoiding the famous pedagogical method 'whereby one begins with the general and proceeds to the particular only after the student is too confused to understand it.' Exercises are an integral part of the text. Students taking the course should have some knowledge of linear algebra. An appendix provides a brief survey of the necessary background of group theory.978-0-8176-4790-2Klauder<John R. Klauder, University of Florida, Gainesville, FL, USA+A Modern Approach to Functional IntegrationXVI, 282p. 9 illus..SCM12120Measure and IntegrationSCP19080Quantum PhysicsPHQPreface.- Introduction.- Part I: Stochastic Theory.- Probability.- Infinite-Dimensional Integrals.- Stochastic Variable Theory.- Part II: Quantum Theory.- Background to an Analysis of Quantum Mechanics.- Quantum Mechanical Path Integrals.- Coherent State Path Integrals.- Continuous-Time Regularized Path Integrals.- Classical and Quantum Constraints.- Part III: Quantum Field Theory.- Application to Quantum Field Theory.- A Modern Approach to Nonrenormalizable Models.- References.- Index.This text takes advantage of recent developments in the theory of path integration to provide an improved treatment of quantization of systems that either have no constraints or instead involve constraints with demonstratively improved procedures. The modern treatment used by the author is an attempt to make a major paradigm shift in how the art of functional integration is practiced. The techniques developed in the work will prove valuable to graduate students and researchers in physics, chemistry, mathematical physics, and applied mathematics who find it necessary to deal with solutions to wave equations, both quantum and beyond. A Modern Approach to Functional Integration offers insight into a number of contemporary research topics, which may lead to improved methods and results that cannot be found elsewhere in the textbook literature. Exercises are included in most chapters, making the book suitable for a one-semester graduate course on functional integration; prerequisites consist mostly of some basic knowledge of quantum mechanics.A major paradigm shift in how the art of functional integration is practiced

Will prove valuable to a broad audience of graduate students and researchers in physics, chemistry, mathematical physics, and applied mathematics

Offers insight into contemporary research topics that cannot be found elsewhere in the textbook literature

Includes exercises in most chapters

978-0-8176-8267-5Kleiner]Israel Kleiner, York University Department of Mathematics and Statistics, Toronto, ON, Canada(Excursions in the History of MathematicsXXI, 347p. 36 illus..A. Number Theory.- 1. Highlights in the History of Number Theory: 1700 BC - 2008.- 2. Fermat: The Founder of Modern Number Theory.- 3. Fermat's Last Theorem: From Fermat to Wiles.- B. Calculus/Analysis.- 4. A History of the Infinitely Small and the Infinitely Large in Calculus, with Remarks for the Teacher.- 5. A Brief History of the Function Concept.- 6. More on the History of Functions, Including Remarks on Teaching.- C. Proof.- 7. Highlights in the Practice of Proof: 1600 BC - 2009.- 8. Paradoxes: What are they Good for?.- 9. Principle of Continuity: 16th - 19th centuries.- 10. Proof: A Many-Splendored Thing.- D. Courses Inspired by History.- 11. Numbers as a Source of Mathematical Ideas.- 12. History of Complex Numbers, with a Moral for Teachers.- 13. A History-of-Mathematics Course for Teachers, Based on Great Quotations.- 14. Famous Problems in Mathematics.- E. Brief Biographies of Selected Mathematicians.- 15. The Biographies.- Index.5This book comprises five parts. The first three contain ten historical essays on important topics: number theory, calculus/analysis, and proof, respectively. Part four deals with several historically oriented courses, and Part five provides biographies of five mathematicians Dedekind, Euler, Gauss, Hilbert, andWeierstrass who played major roles in the historical events described in the first four parts of the work.Excursions in the History of Mathematics was written with several goals in mind: to arouse mathematics teachers interest in the history of their subject; to encourage mathematics teachers with at least some knowledge of the history of mathematics to offer courses with a strong historical component; and to provide an historical perspective on a number of basic topics taught in mathematics courses. 978-3-540-54062-5KloedenvPeter E. Kloeden; Eckhard Platen, University of Technology, Sydney Dept. Mathematical Sciences, Sydney, NSW, Australia7Numerical Solution of Stochastic Differential Equations XXXVI, 636 p.1. Probability and Statistics.- 2. Probability and Stochastic Processes.- 3. Ito Stochastic Calculus.- 4. Stochastic Differential Equations.- 5. Stochastic Taylor Expansions.- 6. Modelling with Stochastic Differential Equations.- 7. Applications of Stochastic Differential Equations.- 8. Time Discrete Approximation of Deterministic Differential Equations.- 9. I< ntroduction to Stochastic Time Discrete Approximation.- 10. Strong Taylor Approximations.- 11. Explicit Strong Approximations.- 12. Implicit Strong Approximations.- 13. Selected Applications of Strong Approximations.- 14. Weak Taylor Approximations.- 15. Explicit and Implicit Weak Approximations.- 16. Variance Reduction Methods.- 17. Selected Applications of Weak Approximations.- Solutions of Exercises.- Bibliographical Notes.7The numerical analysis of stochastic differential equations differs significantly from that of ordinary differential equations, due to the peculiarities of stochastic calculus. The book proposes to the reader whose background knowledge is limited to undergraduate level methods for engineering and physics, and easily accessible introductions to SDE and then applications as well as the numerical methods for dealing with them. To help the reader develop an intuitive understanding and hand-on numerical skills, numerous exercises including PC-exercises are included.The book is interdisciplinary in its appoach and orientationIt places equal emphasis on both theory and applications

Besides serving as a basic text on stochastic differential equations it derives and discusses the numerical methods needed to solve such equations978-3-642-08107-1978-3-540-63446-1KoblitzJNEAL Koblitz, University of Washington Dept. Mathematics, Seattle, WA, USA!Algebraic Aspects of Cryptography)Algorithms and Computation in Mathematics IX, 206 p.SCI150092Data Structures, Cryptology and Information Theory1. Cryptography.- 1. Early History.- 2. The Idea of Public Key Cryptography.- 3. The RSA Cryptosystem.- 4. Diffie-Hellman and the Digital Signature Algorithm.- 5. Secret Sharing, Coin Flipping, and Time Spent on Homework.- 6. Passwords, Signatures, and Ciphers.- 7. Practical Cryptosystems and Useful Impractical Ones.- 2. Complexity of Computations.- 1. The Big-O Notation.- 2. Length of Numbers.- 3. Time Estimates.- 4. P, NP, and NP-Completeness.- 5. Promise Problems.- 6. Randomized Algorithms and Complexity Classes.- 7. Some Other Complexity Classes.- 3. Algebra.- 1. Fields.- 2. Finite Fields.- 3. The Euclidean Algorithm for Polynomials.- 4. Polynomial Rings.- 5. Grbner Bases.- 4. Hidden Monomial Cryptosystems.- 1. The Imai-Matsumoto System.- 2. Patarin s Little Dragon.- 3. Systems That Might Be More Secure.- 5. Combinatorial-Algebraic Cryptosystems.- 1. History.- 2. Irrelevance of Brassard s Theorem.- 3. Concrete Combinatorial-Algebraic Systems.- 4. The Basic Computational Algebra Problem.- 5. Cryptographic Version of Ideal Membership.- 6. Linear Algebra Attacks.- 7. Designing a Secure System.- 6. Elliptic and Hyperelliptic Cryptosystems.- 1. Elliptic Curves.- 2. Elliptic Curve Cryptosystems.- 3. Elliptic Curve Analogues of Classical Number Theory Problems.- 4. Cultural Background: Conjectures on Elliptic Curves and Surprising Relations with Other Problems.- 5. Hyperelliptic Curves.- 6. Hyperelliptic Cryptosystems.- 1. Basic Definitions and Properties.- 2. Polynomial and Rational Functions.- 3. Zeros and Poles.- 4. Divisors.- 5. Representing Semi-Reduced Divisors.- 6. Reduced Divisors.- 7. Adding Reduced Divisors.- Exercises.- Answers to Exercises.This is a textbook for a course (or self-instruction) in cryptography with emphasis on algebraic methods. The first half of the book is a self-contained informal introduction to areas of algebra, number theory, and computer science that are used in cryptography. Most of the material in the second half - 'hidden monomial' systems, combinatorial-algebraic systems, and hyperelliptic systems - has not previously appeared in monograph form. The appendix by Menezes, Wu, and Zuccherato gives an elementary treatment of hyperelliptic curves. This book is intended for graduate students, advanced undergraduates, and scientists working in various fields of data security.

Cryptography is one of the hot topics right now used for manifold applications, such as telecommunication, secrecy for internet etc.

Leads readers into advanced methods in number theory which are used for cryptography

978-3-642-08332-7978-0-387-96017-33p-adic Numbers, p-adic Analysis, and Zeta-FunctionsXII, 153 p.I p-adic numbers.- 1. Basic concepts.- 2. Metrics on the rational numbers.- Exercises.- 3. Review of building up the complex numbers.- 4. The field of p-adic numbers.- 5. Arithmetic in ?p.- Exercises.- II p-adic interpolation of the Riemann zeta-function.- 1. A formula for ?(2k).- 2. p-adic interpolation of the function f(s) = as.- Exercises.- 3. p-adic distributions.- Exercises.- 4. Bernoulli distributions.- 5. Measures and integration.- Exercises.- 6. The p-adic ?-function as a Mellin-Mazur transform.- 7. A brief survey (no proofs).- Exercises.- III Building up ?.- 1. Finite fields.- Exercises.- 2. Extension of norms.- Exercises.- 3. The algebraic closure of ?p.- 4. ?.- Exercises.- IV p-adic power series.- 1. Elementary functions.- Exercises.- 2. The logarithm, gamma and Artin-Hasse exponential functions.- Exercises.- 3. Newton polygons for polynomials.- 4. Newton polygons for power series.- Exercises.- V Rationality of the zeta-function of a set of equations over a finite field.- 1. Hypersurfaces and their zeta-functions.- Exercises.- 2. Characters and their lifting.- 3. A linear map on the vector space of power series.- 4. p-adic analytic expression for the zeta-function.- Exercises.- 5. The end of the proof.- Answers and Hints for the Exercises.Neal Koblitz was a student of Nicholas M. Katz, under whom he received his Ph.D. in mathematics at Princeton in 1974. He spent the year 1974 -75 and the spring semester 1978 in Moscow, where he did research in p -adic analysis and also translated Yu. I. Manin's 'Course in Mathematical Logic' (GTM 53). He taught at Harvard from 1975 to 1979, and since 1979 has been at the University of Washington in Seattle. He has published papers in number theory, algebraic geometry, and p-adic analysis, and he is the author of 'p-adic Analysis: A Short Course on Recent Work' (Cambridge University Press and GTM 97: 'Introduction to Elliptic Curves and Modular Forms (Springer-Verlag).978-0-8176-4456-7KockJoachim Kock, Universitat Autnoma de Barcelona Fac. Cincies, Bellaterra, Spain; Israel Vainsencher, Universidade Federal de Pernambuco Depto. Matematica, Recife, Brazil#An Invitation to Quantum Cohomology.Kontsevich's < Formula for Rational Plane CurvesPrologue: Warming Up with Cross Ratios, and the Definition of Moduli Space.- Stable n-pointed Curves.- Stable Maps.- Enumerative Geometry via Stable Maps.- Gromov Witten Invariants.- Quantum Cohomology.This book is an elementary introduction to stable maps and quantum cohomology, starting with an introduction to stable pointed curves, and culminating with a proof of the associativity of the quantum product. The viewpoint is mostly that of enumerative geometry, and the red thread of the exposition is the problem of counting rational plane curves. Emphasis is given throughout the exposition to examples, heuristic discussions, and simple applications of the basic tools to best convey the intuition behind the subject. The book demystifies these new quantum techniques by showing how they fit into classical algebraic geometry. Some familiarity with basic algebraic geometry and elementary intersection theory is assumed. Each chapter concludes with some historical comments and an outline of key topics and themes as a guide for further study, followed by a collection of exercises that complement the material covered and reinforce computational skills. As such, the book is ideal for self-study, as a text for a mini-course in quantum cohomology, or as a special topics text in a standard course in intersection theory. The book will prove equally useful to graduate students in the classroom setting as to researchers in geometry and physics who wish to learn about the subject.Elementary introduction to stable maps and quantum cohomology presents the problem of counting rational plane curves

Viewpoint is mostly that of enumerative geometry

Emphasis is on examples, heuristic discussions, and simple applications to best convey the intuition behind the subject

Ideal for self-study, for a mini-course in quantum cohomology, or as a special topics text in a standard course in intersection theory

978-3-642-24487-2KorteBernhard Korte, Universitt Bonn Forschungsinstitut fr, Bonn, Germany; Jens Vygen, Universitt Bonn Forschungsinstitut fr, Bonn, GermanyCombinatorial OptimizationTheory and AlgorithmsXIX, 659p. 77 illus..91 Introduction.- 2 Graphs.- 3 Linear Programming.- 4 Linear Programming Algorithms.- 5 Integer Programming.- 6 Spanning Trees and Arborescences.- 7 Shortest Paths.- 8 Network Flows.- 9 Minimum Cost Flows.- 10 Maximum Matchings.- 11 Weighted Matching.- 12 b-Matchings and T -Joins.- 13 Matroids.- 14 Generalizations of Matroids.- 15 NP-Completeness.- 16 Approximation Algorithms.- 17 The Knapsack Problem.- 18 Bin-Packing.- 19 Multicommodity Flows and Edge-Disjoint Paths.- 20 Network Design Problems.- 21 The Traveling Salesman Problem.- 22 Facility Location.- Indices. This comprehensive textbook on combinatorial optimization places specialemphasis on theoretical results and algorithms with provably goodperformance, in contrast to heuristics. It is based on numerous courses on combinatorial optimization and specialized topics, mostly at graduate level. This book reviews the fundamentals, covers the classical topics (paths, flows, matching, matroids, NP-completeness, approximation algorithms) in detail, and proceeds to advanced and recent topics, some of which have not appeared in a textbook before. Throughout,it contains complete but concise proofs, and also provides numerousexercises and references. This fifth edition has again been updated, revised, and significantlyextended, with more than 60 new exercises and new material on varioustopics, including Cayley's formula, blocking flows, fasterb-matching separation, multidimensional knapsack, multicommoditymax-flow min-cut ratio, and sparsest cut. Thus, this book represents the state of the art of combinatorial optimization.Well-written, populartextbook on combinatorial optimization

One of very few textbooks on this topic

Subject area has manifold applications

Offers complete but concise proofs, making it an invaluable practical tool for students

Updated fifth edition

978-3-642-42767-1XIX, 659 p. 77 illus.978-3-642-30903-8Kranakis;Evangelos Kranakis, Carleton University, Ottawa, ON, Canada1Advances in Network Analysis and its ApplicationsMathematics in Industry*XVI, 409 p. 95 illus., 39 illus. in color.SCM13090Complex SystemsSCI13022Computer Communication NetworksUKNFINANCIAL NETWORKS: 1. Mathematical modeling of systemic risk: H. Amini, A. Minca.- 2. Systemic risk in banking networks without Monte Carlo simulation: J. P. Gleeson, T. R. Hurd, S. Melnik, A. Hackett.- 3. Systemic Valuation of Banks Interbank Equilibrium and Contagion: G. HaBaj.- 4. An Open Problem: J. B. Walsh.- II SECURITY NETWORKS: 5. Dynamic Trust Management: Network Profiling for High Assurance Resilience: M. Burmester , W. O. Redwood.- 6. Security Issues in Link State Routing Protocols for MANETs: G.Cervera, M.Barbeau, J. Garcia-Alfaro, E.Kranakis.-7. TCHo: a Code-based Cryptosystem: A. Duc , S. Vaudenay.- 8. Formal Method for (k)-Neighborhood Discovery Protocols: R.Jamet, P.Lafourcade.- 9. A Tutorial on White-box AES: J. A. Muir.- 10. Efficient 1-Round Almost-Perfect Secure Message Transmission Protocols with Flexible Connectivity: R. Safavi-Naini, M. Ashraful Alam Tuhin.- III SOCIAL NETWORKS: 11. Mathematical modelling to evaluate measures and control the spread of illicit drug use: A. Bakhtiari, A. Rutherford.- 12. Complex Networks and Social Networks: A. Bonato, A. Tian.- 13. NAVEL Gazing: Studying a Networked Scholarly Organization: D. Dimitrova, A. Gruzd, Z.Hayat, G. Ying Mo, D.Mok, Th. Robbins, B.Wellman, X. Zhuo.- 14. How Al Qaeda can use order theory to evade or defeat U.S. Forces: J.D.Farley.- 15. The ABCs of Designing Social Networks for Health Behaviour Change: The VivoSpace Social Network: N. Kamal, S. Fels, M. Blackstock, K. Ho.- 16. Evolution of an Open Source Community Network: N. Saraf, A. Seary, D. Chandrasekaran, P.Monge.- 17. SociQL: A Query Language for the SocialWeb: D. Serrano, E. Stroulia, D. Barbosa, V. Guana.;As well as highlighting potentially useful applications for network analysis, this volume identifies new targets for mathematical research that promise to provide insights into network systems theo< ry as well as facilitating the cross-fertilization of ideas between sectors. Focusing on financial, security and social aspects of networking, the volume adds to the growing body of evidence showing that network analysis has applications to transportation, communication, health, finance, and social policy more broadly. It provides powerful models for understanding the behavior of complex systems that, in turn, will impact numerous cutting-edge sectors in science and engineering, such as wireless communication, network security, distributed computing and social networking, financial analysis, and cyber warfare.The volume offers an insider s view of cutting-edge research in network systems, including methodologies with immense potential for interdisciplinary application. The contributors have all presented material at a series of workshops organized on behalf of Canada s MITACS initiative, which funds projects and study grants in mathematics for information technology and complex systems . These proceedings include papers from workshops on financial networks, network security and cryptography, and social networks. MITACS has shown that the partly ghettoized nature of network systems research has led to duplicated work in discrete fields, and thus this initiative has the potential to save time and accelerate the pace of research in a number of areas of network systems research.Information on the state-of-the-art research

Unifying diverse disciplines based on common technology

Highlighting diverse applications of network analysis to new problems

978-3-642-43391-7978-0-8176-4264-8KrantzSteven G. Krantz, Washington University in St. Louis, St. Louis, MO, USA; Harold R. Parks, Oregon State University, Corvallis, OR, USA#A Primer of Real Analytic FunctionsXIII, 209 p.l1 Elementary Properties.- 1.1 Basic Properties of Power Series.- 1.2 Analytic Continuation.- 1.3 The Formula of Fa di Bruno.- 1.4 Composition of Real Analytic Functions.- 1.5 Inverse Functions.- 2 Multivariable Calculus of Real Analytic Functions.- 2.1 Power Series in Several Variables.- 2.2 Real Analytic Functions of Several Variables.- 2.3 The Implicit function Theorem.- 2.4 A Special Case of the Cauchy-Kowalewsky Theorem.- 2.5 The Inverse function Theorem.- 2.6 Topologies on the Space of Real Analytic Functions.- 2.7 Real Analytic Submanifolds.- 2.8 The General Cauchy-Kowalewsky Theorem.- 3 Classical Topics.- 3.0 Introductory Remarks.- 3.1 The Theorem ofPringsheim and Boas.- 3.2 Besicovitch s Theorem.- 3.3 Whitney s Extension and Approximation Theorems.- 3.4 The Theorem of S. Bernstein.- 4 Some Questions of Hard Analysis.- 4.1 Quasi-analytic and Gevrey Classes.- 4.2 Puiseux Series.- 4.3 Separate Real Analyticity.- 5 Results Motivated by Partial Differential Equations.- 5.1 Division of Distributions I.- 5.2 Division of Distributions II.- 5.3 The FBI Transform.- 5.4 The Paley-Wiener Theorem.- 6 Topics in Geometry.- 6.1 The Weierstrass Preparation Theorem.- 6.2 Resolution of Singularities.- 6.3 Lojasiewicz s Structure Theorem for Real Analytic Varieties.- 6.4 The Embedding of Real Analytic Manifolds.- 6.5 Semianalytic and Subanalytic Sets.- 6.5.1 Basic Definitions.978-0-8176-4668-4HSteven G. Krantz, Washington University in St. Louis, St. Louis, MO, USA!Explorations in Harmonic AnalysisEWith Applications to Complex Function Theory and the Heisenberg GroupSCM12015Abstract Harmonic AnalysisOntology and History of Real Analysis.- The Central Idea: The Hilbert Transform.- Essentials of the Fourier Transform.- Fractional and Singular Integrals.- A Crash Course in Several Complex Variables.- Pseudoconvexity and Domains of Holomorphy.- Canonical Complex Integral Operators.- Hardy Spaces Old and New.- to the Heisenberg Group.- Analysis on the Heisenberg Group.- A Coda on Domains of Finite Type.sThis self-contained text provides an introduction to modern harmonic analysis in the context in which it is actually applied, in particular, through complex function theory and partial differential equations. It takes the novice mathematical reader from the rudiments of harmonic analysis (Fourier series) to the Fourier transform, pseudodifferential operators, and finally to Heisenberg analysis. Within the textbook, the new ideas on the Heisenberg group are applied to the study of estimates for both the Szeg and Poisson Szeg integrals on the unit ball in complex space. Thus the main theme of the book is also tied into complex analysis of several variables. With a rigorous but well-paced exposition, this text provides all the necessary background in singular and fraction< al integrals, as well as Hardy spaces and the function theory of several complex variables, needed to understand Heisenberg analysis. Explorations in Harmonic Analysis is ideal for graduate students in mathematics, physics, and engineering. Prerequisites include a fundamental background in real and complex analysis and some exposure to functional analysis.Provides an introduction to a particular direction in modern harmonic analysis

Self-contained text on analysis of integral operators

Presents both fundamentals and applications of harmonic analysis, especially to important concepts about the Heisenberg group

978-0-8176-4676-9Geometric Integration TheoryBasics.- Carathodory s Construction and Lower-Dimensional Measures.- Invariant Measures and the Construction of Haar Measure..- Covering Theorems and the Differentiation of Integrals.- Analytical Tools: The Area Formula, the Coarea Formula, and Poincar Inequalities..- The Calculus of Differential Forms and Stokes s Theorem.- to Currents.- Currents and the Calculus of Variations.- Regularity of Mass-Minimizing Currents.This textbook introduces geometric measure theory through the notion of currents. Currents, continuous linear functionals on spaces of differential forms, are a natural language in which to formulate types of extremal problems arising in geometry, and can be used to study generalized versions of the Plateau problem and related questions in geometric analysis. The text provides considerable background for the student and discusses techniques that are applicable to complex geometry, partial differential equations, harmonic analysis, differential geometry, and many other parts of mathematics. Topics include the deformation theorem, the area and coareas formulas, the compactness theorem, the slicing theorem and applications to minimal surfaces. Motivating key ideas with examples and figures, Geometric Integration Theory is a comprehensive introduction ideal for both use in the classroom and for self-study. The exposition demands minimal background, is self-contained and accessible, and thus is ideal for both graduate students and researchers.Self-contained, inclusive, and accessible for both the graduate students and researchers

Motivates the key ideas with examples and figures

Includes considerable background material and complete proofs

978-0-387-98408-7KressRaimer KressXII, 326 p.S1 Introduction.- 2 Linear Systems.- 2.1 Examples for Systems of Equations.- 2.2 Gaussian Elimination.- 2.3 LR Decomposition.- 2.4 QR Decomposition.- Problems.- 3 Basic Functional Analysis.- 3.1 Normed Spaces.- 3.2 Scalar Products.- 3.3 Bounded Linear Operators.- 3.4 Matrix Norms.- 3.5 Completeness.- 3.6 The Banach Fixed Point Theorem.- 3.7 Best Approximation.- Problems.- 4 Iterative Methods for Linear Systems.- 4.1 Jacobi and Gauss Seidel Iterations.- 4.2 Relaxation Methods.- 4.3 Two-Grid Methods.- Problems.- 5 Ill-Conditioned Linear Systems.- 5.1 Condition Number.- 5.2 Singular Value Decomposition.- 5.3 Tikhonov Regularization.- Problems.- 6 Iterative Methods for Nonlinear Systems.- 6.1 Successive Approximations.- 6.2 Newton s Method.- 6.3 Zeros of Polynomials.- 6.4 Least Squares Problems.- Problems.- 7 Matrix Eigenvalue Problems.- 7.1 Examples.- 7.2 Estimates for the Eigenvalues.- 7.3 The Jacobi Method.- 7.4 The QR Algorithm.- 7.5 Hessenberg Matrices.- Problems.- 8 Interpolation.- 8.1 Polynomial Interpolation.- 8.2 Trigonometric Interpolation.- 8.3 Spline Interpolation.- 8.4 Bzier Polynomials.- Problems.- 9 Numerical Integration.- 9.1 Interpolatory Quadratures.- 9.2 Convergence of Quadrature Formulae.- 9.3 Gaussian Quadrature Formulae.- 9.4 Quadrature of Periodic Functions.- 9.5 Romberg Integration.- 9.6 Improper Integrals.- Problems.- 10 Initial Value Problems.- 10.1 The Picard Lindelf Theorem.- 10.2 Euler s Method.- 10.3 Single-Step Methods.- 10.4 Multistep Methods.- Problems.- 11 Boundary Value Problems.- 11.1 Shooting Methods.- 11.2 Finite Difference Methods.- 11.3 The Riesz and Lax-Milgram Theorems.- 11.4 Weak Solutions.- 11.5 The Finite Element Method.- Problems.- 12 Integral Equations.- 12.1 The Riesz Theory.- 12.2 Operator Approximations.- 12.3 Nystrm s Method.- 12.4 The Collocation Method.- 12.5 Stability.- Problems.- References.5This volume is intended as an introduction into numerical analysis for students in mathematics, physics, and engineering. Instead of attempting to exhaustively cover all parts of numerical analysis, the goal is to guide the reader towards the basic ideas and general principles by way of considering main and important numerical methods. Given the rapid development of numerical algorithms, a reasonable introduction to numerical analysis has to confine itself to presenting a solid foundation by restricting the presentation to the basic principles and procedures. The book includes the necessary basic functional analytic tools for the solid mathematical foundation of numerical analysis. These are indispensable for any deeper study and understanding of numerical methods, in particular, for differential equations and integral equations. Particular emphasis will be given to the question of stability--especially to well-posedness and ill-posedness. The text is presented in a concise and easily understandable fashion and can be successfully mastered in a one-year course.* Good introduction to an area experiencing rapid development for those in math, physics, and engineering. * Gives a solid foundation by restricting the presentation to the basic principles and procedures, as well as the primary numerical algorithms. * Includes the necessary functional analytic framework for a solid mathematical foundation in the subject. * Gives particular emphasis to the question of stability. * Presented in a concise and easily understandable fashion.978-1-4419-7019-0< Kristensson2Gerhard Kristensson, Lund University, Lund, Sweden#Second Order Differential Equations*Special Functions and Their ClassificationXIV, 216p. 31 illus..SCM1221XSpecial FunctionsBasic properties of the solutions.- Equations of Fuchsian type.- Equations with one to four regular singular points.- The hypergeometric differential equation.- Legendre functions and related functions.- Confluent hypergeometric functions.- Heun s differential equation.Second Order Differential Equations presents a classical piece of theory concerning hypergeometric special functions as solutions of second-order linear differential equations. The theory is presented in an entirely self-contained way, starting with an introduction of the solution of the second-order differential equations and then focusingon the systematic treatment and classification of these solutions. Each chapter contains a set of problems which help reinforce the theory. Some of the preliminaries are covered in appendices at the end of the book, one of which provides an introduction to Poincar-Perron theory, and the appendix also contains a new way of analyzing the asymptomatic behavior of solutions of differential equations. This textbook is appropriate for advanced undergraduate and graduate students in Mathematics, Physics, and Engineering interested in Ordinary and Partial Differntial Equations. A solutions manual is available online.Contains problems at the end of each chapter which reinforce the material

Features solutions of the Heun equation, usually only found in more advanced monographs

Includes useful appendices on background material, including the Poincar-Perron theory

978-1-4939-0177-7XIV, 216 p. 31 illus.978-3-540-70913-8KrylovQN. V. Krylov, University of Minnesota School of Mathematics, Minneapolis, MN, USAControlled Diffusion Processesto the Theory of Controlled Diffusion Processes.- Auxiliary Propositions.- General Properties of a Payoff Function.- The Bellman Equation.- The Construction of ?-Optimal Strategies.- Controlled Processes with Unbounded Coefficients: The Normed Bellman Equation. This book deals with the optimal control of solutions of fully observable It-type stochastic differential equations. The validity of the Bellman differential equation for payoff functions is proved and rules for optimal control strategies are developed. Topics include optimal stopping; one dimensional controlled diffusion; the Lp-estimates of stochastic integral distributions; the existence theorem for stochastic equations; the It formula for functions; and the Bellman principle, equation, and normalized equation.978-0-387-40510-0KurzweilHans Kurzweil, Universitt Erlangen-Nrnberg Mathematisches Institut, Erlangen, Germany; Bernd Stellmacher, Universitt Kiel Mathematisches Seminar, Kiel, GermanyThe Theory of Finite GroupsXII, 387 p.!Basic Concepts.- Abelian Groups.- Action and Conjugation.- Permutation Groups.- p-Groups and Nilpotent Groups.- Normal and Subnormal Structure.- Transfer and p-Factor Groups.- Groups Acting on Groups.- Quadratic Action.- The Embedding of p-Local Subgroups.- Signalizer Functors.- N-Groups.iFrom reviews of the German edition: 'This is an exciting text and a refreshing contribution to an area in which challenges continue to flourish and to captivate the viewer. Even though representation theory and constructions of simple groups have been omitted, the text serves as a springboard for deeper study in many directions. One who completes this text not only gains an appreciation of both the depth and the breadth of the theory of finite groups, but also witnesses the evolutionary development of concepts that form a basis for current investigations. This is accomplished by providing a thread that permits a natural flow from one concept to another rather than compartmentalizing. Operators on sets and groups are introduced early and used effectively throughout. The bibliography provides excellent supplemental support...The text is tight; there is no fluff. The format builds on concepts essential for later expansion and associated reading. On occasion, results are stated without proof; continuity is maintained. Several proofs are provided free of representation theory on which the originals were based. More generally the proofs are direct, perhaps at times brief. The focus is on the underlying structural components, with selected details left to the reader. As a result the reader develops the maturity required for approaching the literature with confidence. The first eight chapters have an abundance of exercises, not prorated, and some of the more challenging are addressed later in the text. Due to the nature of the material, fewer exercises appear in the remaining chapters.' (H. Bechtell, Mathematical Reviews)978-1-4419-2340-0978-1-4614-1128-4LagariasQJeffrey C. Lagarias, University of Michigan Dept. Mathematics, Ann Arbor, MI, USAThe Kepler ConjectureThe Hales-Ferguson Proof*XIV, 456 p. 93 illus., 11 illus. in color.SCM21014Convex and Discrete GeometrySCM131202Mathematical Applications in the Physical SciencesCommemorative publicationPreface.- Part I, Introduction and Survey.- 1 The Kepler Conjecture and Its Proof, by J. C. Lagarias.- 2 Bounds for Local Density of Sphere Packings and the Kepler Conjecture, by J. C. Lagarias.- Part II, Proof of the Kepler Conjecture.- Guest Editor's Foreword.- 3 Historical Overview of the Kepler Conjecture, by T. C. Hales.- 4 A Formulation of the Kepler Conjecture, by T. C. Hales and S. P. Ferguson.- 5 Sphere Packings III. Extremal Cases, by T. C. Hales.- 6 Sphere Packings IV. Detailed Bounds, by T. C. Hales.- 7 Sphere Packings V. Pentahedral Prisms, by S. P. Ferguson.- 8 Sphere Packings VI. Tame Graphs and Linear Programs, by T. C. Hales.- Part III, A Revision to the Proof of the Kepler Conjecture.- 9 A Revision of the Proof of the Kepler Conjecture, by T. C. Hales, J. Harrison, S. McLaughlin, T. Nipkow, S. Obua, and R. Zumkeller.- Part IV, Initial Papers of the Hales Program.- 10 Sphere Packings I, by T. C. Hales.- 11 Sphere Packings II, by T. C. Hales.- Index of Symbols.- Index of Subjects.The Kepler conjecture, one of geometry's oldest unsolved problems, was formulated in 1611 by Johannes Kepler and mentioned by Hilbert in his famous 1900 problem list. The Kepler conjecture states that the densest packing of three-dimensional Euclidean space by equal spheres is attained by the cannonball' packing. In a landmark result, this was proved by Thomas C. Hales and Samuel P. Ferguson, using an analytic argument completed with extensive use of computers.This book centers around six papers, presenting the detailed proof of the Kepler conjecture given by Hales and Ferguson, published in 2006 in a special issue of Discrete & Computational Geometry. Further supporting material is also presented: a follow-up paper of Hales et al (2010) revising the proof, and describing progress towards a formal proof of the Kepler conjecture. For histor< ical reasons, this book also includes two early papers of Hales that indicate his original approach to the conjecture.The editor's two introductory chapters situate the conjecture in a broader historical and mathematical context. These chapters provide a valuable perspective and are a key feature of this work.Complete solution of a four hundred year old geometry problem

A fundamental achievement in discrete geometry and mathematical physics

Provides history and summary of approaches to the problem

978-1-4614-5316-1LakatosLaszlo Lakatos, Eotvos Lorant University, Budapest, Hungary; Laszlo Szeidl, Obuda University, Budapest, Hungary; Miklos Telek, Technical University of Budapest, Budapest, HungaryDIntroduction to Queueing Systems with Telecommunication ApplicationsXI, 385 p. 91 illus.Preface.- Introduction to probability theory.- Introduction to stochastic processes.- Markov chains.- Renewal and regenerative processes.- Markov chains with special structures.- Introduction to queueing systems.- Markovian queueing systems.- Non-Markovian queueing systems.- Queueing systems with structured Markov chains.- Queueing networks.- Applied queueing systems.- Functions and transforms.- Exercises.- References.-The book is composed of two main parts: mathematical background and queueing systems with applications. The mathematical background is a self containing introduction to the stochastic processes of the later studies queueing systems. It starts with a quick introduction to probability theory and stochastic processes and continues with chapters on Markov chains and regenerative processes. More recent advances of queueing systems are based on phase type distributions, Markov arrival processes and quasy birth death processes, which are introduced in the last chapter of the first part.The second part is devoted to queueing models and their applications. After the introduction of the basic Markovian (from M/M/1 to M/M/1//N) and non-Markovian (M/G/1, G/M/1) queueing systems, a chapter presents the analysis of queues with phase type distributions, Markov arrival processes (from PH/M/1 to MAP/PH/1/K). The next chapter presents the classical queueing network results and the rest of this part is devoted to the application examples. There are queueing models for bandwidth charing with different traffic classes, slotted multiplexers, ATM switches, media access protocols like Aloha and IEEE 802.11b, priority systems and retrial systems.An appendix supplements the technical content with Laplace and z transformation rules, Bessel functions and a list of notations. The book contains examples and exercises throughout and could be used for graduate students in engineering, mathematics and sciences.Examples from modern communication system

Accurately established mathematical background (part 1), with intuitive explanations for queueing applications (part 2 Guidelines for application of queueing theory in practice.

978-1-4899-7335-1978-0-387-00500-3Lam[T.Y. Lam, Department of Mathematics University of California at Berkeley, Berkeley, CA, USA"Exercises in Classical Ring TheoryXIX, 361 p.SCM11027Associative Rings and AlgebrasWedderburn-Artin Theory.- Jacobson Radical Theory.- to Representation Theory.- Prime and Primitive Rings.- to Division Rings.- Ordered Structures in Rings.- Local Rings, Semilocal Rings, and Idempotents.- Perfect and Semiperfect Rings.' This useful book, which grew out of the author's lectures at Berkeley, presents some 400 exercises of varying degrees of difficulty in classical ring theory, together with complete solutions, background information, historical commentary, bibliographic details, and indications of possible improvements or generalizations. The book should be especially helpful to graduate students as a model of the problem-solving process and an illustration of the applications of different theorems in ring theory. The author also discusses 'the folklore of the subject: the 'tricks of the trade' in ring theory, which are well known to the experts in the field but may not be familiar to others, and for which there is usually no good reference'. The problems are from the following areas: the Wedderburn-Artin theory of semisimple rings, the Jacobson radical, representation theory of groups and algebras, (semi)prime rings, (semi)primitive rings, division rings, ordered rings, (semi)local rings, the theory of idempotents, and (semi)perfect rings. Problems in the areas of module theory, category theory, and rings of quotients are not included, since they will appear in a later book. ' (T. W. Hungerford, Mathematical Reviews)978-1-4419-1829-1978-0-387-96405-8LangASerge Lang, Yale University Dept. Mathematics, New Haven, CT, USACalculus of Several VariablesXII, 619 p. One Basic Material.- I Vectors.- II Differentiation of Vectors.- III Functions of Several Variables.- IV The Chain Rule and the Gradient.- Two Maxima, Minima, and Taylor s Formula.- V Maximum and Minimum.- VI Higher Derivatives.- Three Curve Integrals and Double Integrals.- VII Potential Functions.- VIII Curve Integrals.- IX Double Integrals.- X Green s Theorem.- Four Triple and Surface Integrals.- XI Triple Integrals.- XII Surface Integrals.- Five Mappings, Inverse Mappings, and Change of Variables Formula..- XIII Matrices.- XIV Linear Mappings.- XV Determinants.- XVI Applications to Functions of Several Variables.- XVII The Change of Variables Formula.- Appendix Fourier Series.- 1. General Scalar Products.- 2. Computation of Fourier Series.- Answers to Exercises.$This is a new, revised edition of this widely known text. All of the basic topics in calculus of several variables are covered, including vectors, curves, functions of several variables, gradient, tangent plane, maxima and minima, potential functions, curve integrals, Green's theorem, multiple integrals, surface integrals, Stokes' theorem, and the inverse mapping theorem and its consequences. The presentation is self-contained, assuming only a knowledge of basic calculus in one variable. Many completely worked-out problems have been included.978-0-387-96671-7Cyclotomic Fields I and IIXVII, 436 p.<1 Character Sums.- 1. Character Sums over Finite Fields.- 2. Stickelberger s Theorem.- 3. Relations in the Ideal Classes.- 4. Jacobi Sums as Hecke Characters.- 5. Gauss Sums over Extension Fields.- 6. Application to the Fermat Curve.- 2 Stickelberger Ideals and Bernoulli Distributions.- 1. The Index of the First Stickelberger Ideal.- 2. Bernoulli Numbers.- 3. Integral Stickelberger Ideals.- 4. General Comments on Indices.- 5. The Index for k Even.- 6. The Index for k Odd.- 7. Twistings and Stickelberger Ideals.- 8. Stickelberger Elements as Distributions.- 9. Universal Distributions.< - 10. The Davenport-Hasse Distribution.- Appendix. Distributions.- 3 Complex Analytic Class Number Formulas.- 1. Gauss Sums on Z/mZ.- 2. Primitive L-series.- 3. Decomposition of L-series.- 4. The ( 1)-eigenspaces.- 5. Cyclotomic Units.- 6. The Dedekind Determinant.- 7. Bounds for Class Numbers.- 4 The p-adic L-function.- 1. Measures and Power Series.- 2. Operations on Measures and Power Series.- 3. The Mellin Transform and p-adic L-function.- Appendix. The p-adic Logarithm.- 4. The p-adic Regulator.- 5. The Formal Leopoldt Transform.- 6. The p-adic Leopoldt Transform.- 5 Iwasawa Theory and Ideal Class Groups.- 1. The Iwasawa Algebra.- 2. Weierstrass Preparation Theorem.- 3. Modules over ZP[[X]].- 4. Zp-extensions and Ideal Class Groups.- 5. The Maximal p-abelian p-ramified Extension.- 6. The Galois Group as Module over the Iwasawa Algebra.- 6 Kummer Theory over Cyclotomic Zp-extensions.- 1. The Cyclotomic Zp-extension.- 2. The Maximal p-abelian p-ramified Extension of the Cyclotomic Zp-extension.- 3. Cyclotomic Units as a Universal Distribution.- 4. The Iwasawa-Leopoldt Theorem and the Kummer-Vandiver Conjecture.- 7 Iwasawa Theory of Local Units.- 1. The Kummer-Takagi Exponents.- 2. Projective Limit of the Unit Groups.- 3. A Basis for U(x) over A.- 4. The Coates-Wiles Homomorphism.- 5. The Closure of the Cyclotomic Units.- 8 Lubin-Tate Theory.- 1. Lubin-Tate Groups.- 2. Formal p-adic Multiplication.- 3. Changing the Prime.- 4. The Reciprocity Law.- 5. The Kummer Pairing.- 6. The Logarithm.- 7. Application of the Logarithm to the Local Symbol.- 9 Explicit Reciprocity Laws.- 1. Statement of the Reciprocity Laws.- 2. The Logarithmic Derivative.- 3. A Local Pairing with the Logarithmic Derivative.- 4. The Main Lemma for Highly Divisible x and ? = xn.- 5. The Main Theorem for the Symbol ?x, xn?n.- 6. The Main Theorem for Divisible x and ? = unit.- 7. End of the Proof of the Main Theorems.- 10 Measures and Iwasawa Power Series.- 1. Iwasawa Invariants for Measures.- 2. Application to the Bernoulli Distributions.- 3. Class Numbers as Products of Bernoulli Numbers.- Appendix by L. Washington: Probabilities.- 4. Divisibility by l Prime to p: Washington s Theorem.- 11 The Ferrero-Washington Theorems.- 1. Basic Lemma and Applications.- 2. Equidistribution and Normal Families.- 3. An Approximation Lemma.- 4. Proof of the Basic Lemma.- 12 Measures in the Composite Case.- 1. Measures and Power Series in the Composite Case.- 2. The Associated Analytic Function on the Formal Multiplicative Group.- 3. Computation of Lp(1, x) in the Composite Case.- 13 Divisibility of Ideal Class Numbers.- 1. Iwasawa Invariants in Zp-extensions.- 2. CM Fields, Real Subfields, and Rank Inequalities.- 3. The l-primary Part in an Extension of Degree Prime to l.- 4. A Relation between Certain Invariants in a Cyclic Extension.- 5. Examples of Iwasawa.- 6. A Lemma of Kummer.- 14 P-adic Preliminaries.- 1. The p-adic Gamma Function.- 2. The Artin-Hasse Power Series.- 3. Analytic Representation of Roots of Unity.- Appendix: Barsky s Existence Proof for the p-adic Gamma Function.- 15 The Gamma Function and Gauss Sums.- 1. The Basic Spaces.- 2. The Frobenius Endomorphism.- 3. The Dwork Trace Formula and Gauss Sums.- 4. Eigenvalues of the Frobenius Endomorphism and the p-adic Gamma Function.- 5. eThis book is a combined edition of the books previously published as Cyclotomic Fields, Vol. I and II. It continues to provide a basic introduction to the theory of these number fields, which are of great interest in classical number theory, as well as in other areas, such as K-theory. Cyclotomic Fields begins with basic material on character sums, and proceeds to treat class number formulas, p-adic L-functions, Iwasawa theory, Lubin-Tate theory, and explicit reciprocity laws, and the Ferrero-Washington theorems, which prove Iwasawa's conjecture on the growth of the p-primary part of the ideal class group.978-0-387-94338-1%Differential and Riemannian ManifoldsXIII, 364 pp. 20 illus.BI Differential Calculus.- 1. Categories.- 2. Topological Vector Spaces.- 3. Derivatives and Composition of Maps.- 4. Integration and Taylor s Formula.- 5. The Inverse Mapping Theorem.- II Manifolds.- 1. Atlases, Charts, Morphisms.- 2. Submanifolds, Immersions, Submersions.- 3. Partitions of Unity.- 4. Manifolds with Boundary.- III Vector Bundles.- 1. Definition, Pull Backs.- 2. The Tangent Bundle.- 3. Exact Sequences of Bundles.- 4.< Operations on Vector Bundles.- 5. Splitting of Vector Bundles.- IV Vector Fields and Differential Equations.- 1. Existence Theorem for Differential Equations.- 2. Vector Fields, Curves, and Flows.- 3. Sprays.- 4. The Flow of a Spray and the Exponential Map.- 5. Existence of Tubular Neighborhoods.- 6. Uniqueness of Tubular Neighborhoods.- V Operations on Vector Fields and Differential Forms.- 1. Vector Fields, Differential Operators, Brackets.- 2. Lie Derivative.- $3. Exterior Derivative.- 4. The Poincar Lemma.- 5. Contractions and Lie Derivative.- 6. Vector Fields and 1-Forms Under Self Duality.- 7. The Canonical 2-Form.- 8. Darboux s Theorem.- VI The Theorem of Frobenius.- 1. Statement of the Theorem.- 2. Differential Equations Depending on a Parameter.- 3. Proof of the Theorem.- 4. The Global Formulation.- 5. Lie Groups and Subgroups.- VII Metrics.- 1. Definition and Functoriality.- 2. The Hilbert Group.- 3. Reduction to the Hilbert Group.- 4. Hilbertian Tubular Neighborhoods.- 5. The Morse Palais Lemma.- 6. The Riemannian Distance.- 7. The Canonical Spray.- VIII Covariant Derivatives and Geodesics.- 1. Basic Properties.- 2. Sprays and Covariant Derivatives.- 3. Derivative Along a Curve and Parallelism.- 4. The Metric Derivative.- 5. More Local Results on the Exponential Map.- 6. Riemannian Geodesic Length and Completeness.- IX Curvature.- 1. The Riemann Tensor.- 2. Jacobi Lifts.- 3. Application of Jacobi Lifts to dexpx.- 4. The Index Form, Variations, and the Second Variation Formula.- 5. Taylor Expansions.- X Volume Forms.- 1. The Riemannian Volume Form.- 2. Covariant Derivatives.- 3. The Jacobian Determinant of the Exponential Map.- 4. The Hodge Star on Forms.- 5. Hodge Decomposition of Differential Forms.- XI Integration of Differential Forms.- 1. Sets of Measure 0.- 2. Change of Variables Formula.- 3. Orientation.- 4. The Measure Associated with a Differential Form.- XII Stokes Theorem.- 1. Stokes Theorem for a Rectangular Simplex.- 2. Stokes Theorem on a Manifold.- 3. Stokes Theorem with Singularities.- XIII Applications of Stokes Theorem.- 1. The Maximal de Rham Cohomology.- 2. Moser s Theorem.- 3. The Divergence Theorem.- 4. The Adjoint of d for Higher Degree Forms.- 5. Cauchy s Theorem.- 6. The Residue Theorem.- Appendix The Spectral Theorem.- 1. Hilbert Space.- 2. Functionals and Operators.- 3. Hermitian Operators.OThis is the third version of a book on Differential Manifolds; in this latest expansion three chapters have been added on Riemannian and pseudo-Riemannian geometry, and the section on sprays and Stokes' theorem have been rewritten. This text provides an introduction to basic concepts in differential topology, differential geometry and differential equations. In differential topology one studies classes of maps and the possibility of finding differentiable maps in them, and one uses differentiable structures on manifolds to determine their topological structure. In differential geometry one adds structures to the manifold (vector fields, sprays, a metric, and so forth) and studies their properties. In differential equations one studies vector fields and their integral curves, singular points, stable and unstable manifolds, and the like.978-0-387-95477-6(Introduction to Differentiable Manifolds XI, 250 p.Differential Calculus.- Manifolds.- Vector Bundles.- Vector Fields and Differential Equations.- Operations on Vector Fields and Differential Forms.- The Theorem of Frobenius.- Metrics.- Integration of Differential Forms.- Stokes Theorem.- Applications of Stokes Theorem.AThis book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. A certain number of concepts are essential for all three of these areas, and are so basic and elementary, that it is worthwhile to collect them together so that more advanced expositions can be given without having to start from the very beginning. The concepts are concerned with the general basic theory of differential manifolds. As a result, this book can be viewed as a prerequisite to Fundamentals of Differential Geometry. Since this book is intended as a text to follow advanced calculus, manifolds are assumed finite dimensional. In the new edition of this book, the author has made numerous corrections to the text and he has added a chapter on applications of Stokes' Theorem.978-1-4419-3019-4978-0-387-94001-4Real and Functional AnalysisXIV, 580 p.1I Sets.- II Topological Spaces.- III Continuous Functions on Compact Sets.- IV Banach Spaces.- V Hilbert Space.- VI The General Integral.- VII Duality and Representation Theorems.- VIII Some Applications of Integration.- IX Integration and Measures on Locally Compact Spaces.- X Riemann-Stieltjes Integral and Measure.- XI Distributions.- XII Integration on Locally Compact Groups.- XIII Differential Calculus.- XIV Inverse Mappings and Differential Equations.- XV The Open Mapping Theorem, Factor Spaces, and Duality.- XVI The Spectrum.- XVII Compact and Fredholm Operators.- XVIII Spectral Theorem for Bounded Hermltian Operators.- XIX Further Spectral Theorems.- XX Spectral Measures.- XXI Local Integration off Differential Forms.- XXII Manifolds.- XXIII Integration and Measures on Manifolds.- Table of Notation.}This book is meant as a text for a first-year graduate course in analysis. In a sense, the subject matter covers the same topics as elementa< ry calculus - linear algebra, differentiation, integration - but treated in a manner suitable for people who will be using it in further mathematical investigations. The book begins with point-set topology, essential for all analysis. The second part deals with the two basic spaces of analysis, Banach and Hilbert spaces. The book then turns to the subject of integration and measure. After a general introduction, it covers duality and representation theorems, some applications (such as Dirac sequences and Fourier transforms), integration and measures on locally compact spaces, the Riemann-Stjeltes integral, distributions, and integration on locally compact groups. Part four deals with differential calculus (with values in a Banach space). The next part deals with functional analysis. It includes several major spectral theorems of analysis, showing how one can extend to infinite dimensions certain results from finite-dimensional linear algebra; a discussion of compact and Fredholm operators; and spectral theorems for Hermitian operators. The final part, on global analysis, provides an introduction to differentiable manifolds. The text includes worked examples and numerous exercises, which should be viewed as an integral part of the book. The organization of the book avoids long chains of logical interdependence, so that chapters are as independent as possible. This allows a course using the book to omit material from some chapters without compromising the exposition of material from later chapters.978-0-387-96198-9S. LangSLThe author has also included a chapter on groups of matrices which is unique in a book at this level

He also includes Noah Snyder's beautiful proof of the Mason-Stothers polynomial abc theorem

978-1-4419-1959-5978-0-387-94049-6Lasota!Andrzej Lasota; Michael C. MackeyChaos, Fractals, and NoiseStochastic Aspects of DynamicsXIV, 474 p.SCP330005Statistical Physics, Dynamical Systems and ComplexityPHSdThis book gives a unified treatment of a variety of mathematical systems generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integro-partial-differential equations. Examples have been drawn from a variety of the sciences to illustrate the utility of the techniques presented. This material was organized and written to be accessible to scientists with knowledge of advanced calculus and differential equations. In various concepts from measure theory, ergodic theory, the geometry of manifolds, partial differential equations, probability theory and Markov processes, and chastic integrals and differential equations are introduced. The past few years have witnessed an explosive growth in interest in physical, biological, and economic systems that could be profitably studied using densities. Due to the general inaccessibility of the mathematical literature to the non-mathematician, there has been little diffusion of the concepts and techniques from ergodic theory into the study of these 'chaotic' systems. This book intends to bridge that gap.978-3-642-31089-8Laurent-GengouxCamille Laurent-Gengoux, Universit de Lorraine CNRS UMR 7122, Metz, France; Anne Pichereau, Universit de Lyon CNRS UMR 5208, Saint Etienne, France; Pol Vanhaecke, Universit de Poitiers CNRS UMR 7348, Futuroscope Chasseneuil, FrancePoisson StructuresXXIV, 461 p. 16 illus. Part I Theoretical Background:1.Poisson Structures: Basic Definitions.- 2.Poisson Structures: Basic Constructions.- 3.Multi-Derivations and Khler Forms.- 4.Poisson (Co)Homology.- 5.Reduction.- Part II Examples:6.Constant Poisson Structures, Regular and Symplectic Manifolds.- 7.Linear Poisson Structures and Lie Algebras.- 8.Higher Degree Poisson Structures.- 9.Poisson Structures in Dimensions Two and Three.- 10.R-Brackets and r-Brackets.- 11.Poisson Lie Groups.- Part III Applications:12.Liouville Integrable Systems.- 13.Deformation Quantization.- A Multilinear Algebra.- B Real and Complex Differential Geometry.- References.- Index.- List of Notations.Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures.The first part coverssolid foundations, the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the final part is devoted to the two main applications of Poisson structures (integrable systems and deformation quantization). The clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers whoare interested in anintroduction to the many facets and applications of Poisson structures. <p>First book about Poisson structures giving a comprehensive introduction as well as solid foundations to the theory </p><p>Unique structure of thevolumetailoredto graduate students or advanced researchers </p><p>Provides examples and exercises </p>978-3-642-43283-5978-0-387-96965-7LawdenDerek F. Lawden#Elliptic Functions and ApplicationsXIV, 335 p.1 Theta Functions.- 2 Jacobi s Elliptic Functions.- 3 Elliptic Integrals.- 4 Geometrical Applications.- 5 Physical Applications.- 6 Weierstrass s Elliptic Function.- 7 Applications of the Weierstrass F< unctions.- 8 Complex Variable Analysis.- 9 Modular Transformations..- Appendix A Fourier Series for a Periodic Analytic Function.- Appendix B Calculation of a Definite Integral.- Appendix C BASIC Program for Reduction of Elliptic Integral to Standard Form.- Appendix D Computation of Tables.- Table A. Theta Functions.- Table B. Nome and Complete Integrals of the First and Second Kinds as Functions of the Squared Modulus.- Table D. Legendre s Incomplete Integrals of First and Second Kinds.- Table E. Jacobi s Zeta and Epsilon Functions.- Table F. Sigma Functions.[This book develops the fundamental properties of elliptic functions and illustrates them by applications in geometry, mathematical physics and engineering. Its purpose is to provide an introductory text for private study by students and research workers who wish to be able to use elliptic functions in the solution of both pure and applied mathematical problems. In the first half of the book, a knowledge of no more than first year university mathematics is assumed of the reader. In the later chapters, the theory of functions of a complex variable is increasingly employed as an analytical tool. Accordingly, the book should prove helpful to mathematicians at all stages of an undergraduate or post-graduate course. The book is liberally supplied with sets of exercises (over 180 total) with which the reader can gain practice in the use of the functions.978-1-4419-3090-3978-1-4614-5971-2Lawler:Gregory F. Lawler, University of Chicago, Chicago, IL, USAIntersections of Random WalksXVI, 223 p.Simple Random Walk.- Harmonic Measure.- Intersection Probabilities.- Four Dimensions.- Two and Three Dimensions.-Self-Avoiding Walks.-Loop-Erased walk.- Recent Results.A central study in Probability Theory is the behavior of fluctuation phenomena of partial sums of different types of random variable. One of the most useful concepts for this purpose is that of the random walk which has applications in many areas, particularly in statistical physics and statistical chemistry. Originally published in 1991,Intersections of Random Walks focuses on and explores a number of problems dealing primarily with the nonintersection of random walks and the self-avoiding walk. Many of these problems arise in studying statistical physics and other critical phenomena. Topics include: discrete harmonic measure, including an introduction to diffusion limited aggregation (DLA); the probability that independent random walks do not intersect; and properties of walks without self-intersections. The presentsoftcover reprint includes corrections andaddenda fromthe1996 printing, andmakesthis classic monographavailable to a wider audience. With a self-contained introduction to the properties of simple random walks, and an emphasis on rigorous results, the book will be useful to researchers in probability and statistical physics and to graduate students interested in basic properties of random walks.#<p> Affordable reprint of a classic monograph</p><p>Topics covered include: discrete harmonic measure; the probability that independent random walks do not intersect; and properties of walks without self-intersections </p><p>Includes thecorrections and addendumfrom the second printing</p>978-1-4614-5867-8LebedevLeonid P. Lebedev, Rostov State University Dept of Mathematics & Mechanics, Rostov on Don, Russia; Iosif I. Vorovich, Southern Federal University Dept. Mathematics & Mechanics, Rostov-on-Don, Russia; Michael J. Cloud, Lawrence Technological University, Southfield, MI, USA Functional Analysis in MechanicsIX, 308 p. 1 illus.?Introduction.- Metric, Banach, and Hilbert Spaces.- Mechanics Problems from the Functional Analysis Viewpoint.- Some Spectral Problems of Mechanics.- Elements of Nonlinear Functional Analysis.- Summary of Inequalities and Imbeddings.- Hints for Selected Problems.- References.- In Memoriam: Iosif I. Vorovich.- Index.- This book offers a brief, practically complete, and relatively simple introduction to functional analysis. It also illustrates the application of functional analytic methods to the science of continuum mechanics. Abstract but powerful mathematical notions are tightly interwoven with physical ideas in the treatment of nontrivial boundary value problems for mechanical objects.This second edition includes more extended coverage of the classical andabstract portions of functional analysis. Taken together, the first three chapters now constitute a regular text on applied functional analysis. This potential use of the book is supported by a significantly extended set of exercises with hints and solutions. A new appendix, providing a convenient listing of essential inequalities and imbedding results, has been added.The book should appeal to graduate students and researchers in physics, engineering, and applied mathematics.Reviews of first edition:'This book covers functional analysis and its applications to continuum mechanics. The presentation is concise but complete, and is intended for readers in continuum mechanics who wish to understand the mathematical underpinnings of the discipline. & Detailed solutions of the exercises are provided in an appendix.' (L Enseignment Mathematique, Vol. 49 (1-2), 2003)'The reader comes away with a profound appreciation both of the physics and its importance, and of the beauty of the functional analytic method, which, in skillful hands, has the power to dissolve and clarify these difficult problems as peroxide does clotted blood. Numerous exercises & test the reader s comprehension at every stage. Summing Up: Recommended.' (F. E. J< . Linton, Choice, September, 2003)The mathematical material is treated in a non-abstract manner and is fully illuminated by the underlying mechanical ideas

The presentation is concise but complete, and is intended for specialists in continuum mechanics who wish to understand the mathematical underpinnings of the discipline

Exercises and examples are included throughout with detailed solutions provided in the appendix

978-1-4899-9756-2978-0-387-34171-2Lefebvre_Mario Lefebvre, Ecole Polytechnique de Montral Dpt. Mathmatiques et de, Montreal, QC, CanadaApplied Stochastic ProcessesXIII, 382 p.Review of Probability Theory.- Stochastic Processes.- Markov Chains.- Diffusion Processes.- Poisson Processes.- Queueing Theory.Applied Stochastic Processes uses a distinctly applied framework to present the most important topics in the field of stochastic processes. Key features: -Presents carefully chosen topics such as Gaussian and Markovian processes, Markov chains, Poisson processes, Brownian motion, and queueing theory -Examines in detail special diffusion processes, with implications for finance, various generalizations of Poisson processes, and renewal processes -Serves graduate students in a variety of disciplines such as applied mathematics, operations research, engineering, finance, and business administration -Contains numerous examples and approximately 350 advanced problems, reinforcing both concepts and applications -Includes entertaining mini-biographies of mathematicians, giving an enriching historical context -Covers basic results in probability Two appendices with statistical tables and solutions to the even-numbered problems are included at the end. This textbook is for graduate students in applied mathematics, operations research, and engineering. Pure mathematics students interested in the applications of probability and stochastic processes and students in business administration will also find this book useful.Presents carefully chosen topics such as Gaussian and Markovian processes, Markov Chains, Poisson processes, Brownian motion and queueing theory

Examines in detail special diffusion processes, with implications for finance, various generalizations of Poisson processes and renewal processes

Serves graduate students in a variety of disciplines such as applied mathematics, operations research, engineering, finance and business administration

Contains numerous examples and approximately 350 advanced problems, reinforcing both concepts and applications

Includes entertaining minibiographies of mathematicians, giving an enriching historical context

Covers basic results in probability to make the text self-contained

978-0-387-74994-5*Basic Probability Theory with ApplicationsPreface.- Review of Differential Calculus.- Elementary Probability.- Random Variables.- Random Vectors.- Reliability.- Queueing.- Time Series.- Appendix A: List of Symbols and Abbreviations.- Appendix B: Statistical Tables.- Appendix C: Solutions to 'Solved Exercises'.- Appendix D: Answers to Even-Numbered Exercises.- Appendix E: Answers to Multiple-Choice Questions.- References.- Index.This book presents elementary probability theory with interesting and well-chosen applications that illustrate the theory. An introductory chapter reviews the basic elements of differential calculus which are used in the material to follow. The theory is presented systematically, beginning with the main results in elementary probability theory. This is followed by material on random variables. Random vectors, including the all important central limit theorem, are treated next. The last three chapters concentrate on applications of this theory in the areas of reliability theory, basic queuing models, and time series. Examples are elegantly woven into the text and over 400 exercises reinforce the material and provide students with ample practice. This textbook can be used by undergraduate students in pure and applied sciences such as mathematics, engineering, computer science, finance and economics. A separate solutions manual is available to instructors who adopt the text for their course.&Presents elementary probability theory with interesting and well-chosen applications that illustrate the theory

Main results in elementary probability, random variables, random vectors and the central limit theorem are covered

Applications in reliability theory, basic queuing models, and time series are presented

Over 400 exercises reinforce the material and provide students with ample practice

Introductory chapter reviews the basic elements of differential calculus which are used in the material to follow

978-1-4614-2923-4XVI, 340p. 50 illus..978-0-8176-3408-7LepowskyqJames Lepowsky, Piscataway, NJ, USA; Haisheng Li, Rutgers University Dept. Mathematical Sciences, Camden, NJ, USABIntroduction to Vertex Operator Algebras and Their RepresentationsXIII, 318 p. 1 Introduction.- 1.1 Motivation.- 1.2 Example of a vertex operator.- 1.3 The notion of vertex operator algebra.- 1.4 Simplification of the definition.- 1.5 Representations and modules.- 1.6 Construction of families of examples.- 1.7 Some further developments.- 2 Formal Calculus.- 2.1 Formal series and the formal delta function.- 2.2 Derivations and the formal Taylor Theorem.- 2.3 Expansions of zero and applications.- 3 Vertex Operator Algebras: The Axiomatic Basics.- 3.1 Definitions and some fundamental properties.- 3.2 Commutativity properties.- 3.3 Associativity properties.- 3.4 The Jacobi identity from commutativity and associativity.- 3.5 The Jacobi identity from commutativity.- 3.6 The Jacobi identity from skew symmetry and associativity.- 3.7 S3-symmetry of the Jacobi identity.- 3.8 The iterate formula and normal-ordered products.- 3.9 Further elementary notions.- 3.10 Weak nilpotence and nilpotence.- 3.11 Centralizers and the center.- 3.12 Direct product and tensor product vertex algebras.- 4 Modules.- 4.1 Definition and some consequences.- 4.2 Commutativity properties.- 4.3 Associativity properties.- 4.4 The Jacobi identity as a consequence of associativity and commutativity properties.- 4.5 Further elementary notions.- 4.6 Tensor product modules for tensor product vertex algebras.- 4.7 Vacuum-like vectors.- 4.8 Adjoining a module to a vertex algebra.- 5 Representations of Vertex Algebras and the Construction of Vertex Algebras and Modules.- 5.1 Weak vertex operators.- 5.2 The action of weak vertex operators on the space of weak vertex operator< s.- 5.3 The canonical weak vertex algebra ?(W) and the equivalence between modules and representations.- 5.4 Subalgebras of ?(W).- 5.5 Local subalgebras and vertex subalgebras of ?(W).- 5.6 Vertex subalgebras of ?(W) associated with the Virasoro algebra.- 5.7 General construction theorems for vertex algebras and modules.- 6 Construction of Families of Vertex Operator Algebras and Modules.- 6.1 Vertex operator algebras and modules associated to the Virasoro algebra.- 6.2 Vertex operator algebras and modules associated to affine Lie algebras.- 6.3 Vertex operator algebras and modules associated to Heisenberg algebras.- 6.4 Vertex operator algebras and modules associated to even lattices the setting.- 6.5 Vertex operator algebras and modules associated to even lattices the main results.- 6.6 Classification of the irreducible L?(?, O)-modules for g finite-dimensional simple and ? a positive integer.- References.The deep and relatively new field of vertex operator algebras is intimately related to a variety of areas in mathematics and physics: for example, the concepts of 'monstrous moonshine,' infinite-dimensional Lie theory, string theory, and conformal field theory. This book introduces the reader to the fundamental theory of vertex operator algebras and its basic techniques and examples. Beginning with a detailed presentation of the theoretical foundations and proceeding to a range of applications, the text includes a number of new, original results and also highlights and brings fresh perspective to important works of many researchers.Introduces the fundamental theory of vertex operator algebras and its basic techniques and examples

Begins with a detailed presentation of the theoretical foundations and proceeds to a range of applications

Includes a number of new, original results and brings fresh perspective to important works of many other researchers in algebra, lie theory, representation theory, string theory, quantum field theory, and other areas of math and physics

978-3-540-36822-9LevineMarc Levine, Northeastern University Dept. Mathematics, Boston, MA, USA; Fabien Morel, Universitt Mnchen Institut fr Mathematik, Mnchen, GermanyAlgebraic CobordismXII, 246 p.Introduction.- I. Cobordism and oriented cohomology.- 1.1. Oriented cohomology theories. 1.2. Algebraic cobordism. 1.3. Relations with complex cobordism. - II. The definition of algebraic cobordism. 2.1. Oriented Borel-Moore functions. 2.2. Oriented functors of geometric type. 2.3. Some elementary properties. 2.4. The construction of algebraic cobordism. 2.5. Some computations in algebraic cobordism.- III. Fundamental properties of algebraic cobordism. 3.1. Divisor classes. 3.2. Localization. 3.3. Transversality. 3.4. Homotopy invariance. 3.5. The projective bundle formula. 3.6. The extended homotopy property. IV. Algebraic cobordism and the Lazard ring. 4.1. Weak homology and Chern classes. 4.2. Algebraic cobordism and K-theory. 4.3. The cobordism ring of a point. 4.4. Degree formulas. 4.5. Comparison with the Chow groups. V. Oriented Borel-Moore homology. 5.1. Oriented Borel-Moore homology theories. 5.2. Other oriented theories.- VI. Functoriality. 6.1. Refined cobordism. 6.2. Intersection with a pseudo-divisor. 6.3. Intersection with a pseudo-divisor II. 6.4. A moving lemma. 6.5. Pull-back for l.c.i. morphisms. 6.6. Refined pull-back and refined intersections. VII. The universality of algebraic cobordism. 7.1. Statement of results. 7.2. Pull-back in Borel-Moore homology theories. 7.3. Universality 7.4. Some applications.- Appendix A: Resolution of singularities.- References.- Index.- Glossary of Notation.Following Quillen's approach to complex cobordism, the authors introduce the notion of oriented cohomology theory on the category of smooth varieties over a fixed field. They prove the existence of a universal such theory (in characteristic 0) called Algebraic Cobordism. Surprisingly, this theory satisfies the analogues of Quillen's theorems: the cobordism of the base field is the Lazard ring and the cobordism of a smooth variety is generated over the Lazard ring by the elements of positive degrees. This implies in particular the generalized degree formula conjectured by Rost. The book also contains some examples of computations and applications.978-3-642-07191-1978-3-642-15003-6LigZenghu Li, Beijing Normal University School of Mathematical Sciences, Beijing, China, People's Republic)Measure-Valued Branching Markov Processes XI, 350 p.(Preface.- 1. Random Measures on Metric Spaces.- 2. Measure-valued Branching Processes.- 3. One-dimensional Branching Processes.- 4. Branching Particle Systems.- 5. Basic Regularities of Superprocesses.- 6. Constructions by Transformations.- 7. Martingale Problems of Superprocesses.- 8. Entrance Laws and Excursion Laws.- 9. Structures of Independent Immigration.- 10. State-dependent Immigration Structures.- 11. Generalized Ornstein-Uhlenbeck Processes.- 12. Small Branching Fluctuation Limits.- 13. Appendix: Markov Processes.- Bibliography.- Index.Measure-valued branching processes arise as high density limits of branching particle systems. The Dawson-Watanabe superprocess is a special class of those. The author constructs superprocesses with Borel right underlying motions and general branching mechanisms and shows the existence of their Borel right realizations. He then uses transformations to derive the existence and regularity of several different forms of the superprocesses. This treatment simplifies the constructions and gives useful perspectives. Martingale problems of superprocesses are discussed under Feller type assumptions. The most important feature of the book is the systematic treatment of immigration superprocesses and generalized Ornstein--Uhlenbeck processes based on skew convolution semigroups. The volume addresses researchers in measure-valued processes, branching processes, stochastic analysis, biological and genetic models, and graduate students in probability theory and stochastic processes.BFor the first time a book in this research area

Gives treatments of the basic regularities of Dawson-Watanabe superprocesses with general branching mechanisms and spatial motions

Studies immigration superprocesses and generalized Ornstein-Uhlenbeck processes

Designed as a study text for graduates

978-3-642-26620-1978-3-642-65163-2LionsEJacques Louis Lions, Collge de France, Paris, France; Enrico Magenes8Non-Homogeneous Boundary Value Problems and ApplicationsVol. 1XVI, 360 p. 0.- 13. Intersection Interpolation.- 14. Holomorphic Interpolation.- 15. Another Intrinsic Definition of the Spaces [X, Y]0.- 16. Compactness Properties.- 17. Comments.- 18. Problems.- 2 Elliptic Operators. Hilbert Theory.- 1. Elliptic Operators and Regular Boundary Value Problems.- 2. Green s Formula and Adjoint Boundary Value Problems.-< 3. The Regularity of Solutions of Elliptic Equations in the Interior of ?.- 4. A priori Estimates in the Half-Space.- 5. A priori Estimates in the Open Set ? and the Existence of Solutions in Hs(?)-Spaces, with Real s ? 2m.- 6. Application of Transposition: Existence of Solutions in Hs(?)-Spaces, with Real s ? 0.- 7. Application of Interpolation: Existence of Solutions in Hs(?)-Spaces, with Real s, 0 < s < 2m.- 8. Complements and Generalizations.- 9. Variational Theory of Boundary Value Problems.- 10. Comments.- 11. Problems.- 3 Variational Evolution Equations.- 1. An Isomorphism Theorem.- 2. Transposition.- 3. Interpolation.- 4. Example: Abstract Parabolic Equations, Initial Condition Problem (I).- 5. Example: Abstract Parabolic Equations, Initial Condition Problem (II).- 6. Example: Abstract Parabolic Equations, Periodic Solutions.- 7. Elliptic Regularization.- 8. Equations of the Second Order in t.- 9. Equations of the Second Order in t; Transposition.- 10. Schroedinger Type Equations.- 11. Schroedinger Type Equations; Transposition.- 12. Comments.- 13. Problems.978-3-540-63929-9LiptsernRobert Liptser; Albert N. Shiryaev, Russian Academy of Sciences Steklov Mathematical Institute, Moscow, RussiaStatistics of Random ProcessesI. General Theory XV, 427 p.SCS11001Statistical Theory and MethodsI1. Essentials of Probability Theory and Mathematical Statistics.- 2. Martingales and Related Processes: Discrete Time.- 3. Martingales and Related Processes: Continuous Time.- 4. The Wiener Process, the Stochastic Integral over the Wiener Process, and Stochastic Differential Equations.- 5. Square Integrable Martingales and Structure of the Functionals on a Wiener Process.- 6. Nonnegative Supermartingales and Martingales, and the Girsanov Theorem.- 7. Absolute Continuity of Measures corresponding to the It Processes and Processes of the Diffusion Type.- 8. General Equations of Optimal Nonlinear Filtering, Interpolation and Extrapolation of Partially Observable Random Processes.- 9. Optimal Filtering, Interpolation and Extrapolation of Markov Processes with a Countable Number of States.- 10. Optimal Linear Nonstationary Filtering.The subject of these two volumes is non-linear filtering (prediction and smoothing) theory and its application to the problem of optimal estimation, control with incomplete data, information theory, and sequential testing of hypothesis. The book is not only addressed to mathematicians but should also serve the interests of other scientists who apply probabilistic and statistical methods in their work. The theory of martingales presented in the book has an independent interest in connection with problems from financial mathematics. In the second edition, the authors have made numerous corrections, updating every chapter, adding two new subsections devoted to the Kalman filter under wrong initial conditions, as well as a new chapter devoted to asymptotically optimal filtering under diffusion approximation. Moreover, in each chapter a comment is added about the progress of recent years.In the second edition, two new subsections devoted to the Kalman filter under wrong initial conditions, and a new chapter on asymptotically optimal filtering under diffusion approximation have been addedIn each chapter a comment is added about the progress of recent years978-3-642-08366-2978-0-387-90210-4LoeveM. LoeveProbability Theory I XVII, 425 pp.Lof Volume I.- Introductory Part: Elementary Probability Theory.- I. Intuitive Background.- II. Axioms; Independence and the Bernoulli Case.- III. Dependence and Chains.- Complements and Details.- One: Notions of Measure Theory.- I: Sets, Spaces, and Measures.- II: Measurable Functions and Integration.- Two: General Concepts and Tools of Probability Theory.- III: Probability Concepts.- IV: Distribution Functions and Characteristic Functions.- Three: Independence.- V: Sums of Independent Random Variables.- VI: Central Limit Problem.- VII: Independent Identically Distributed Summands.978-0-387-90262-3Probability Theory IIXVI, 416 p.This book is intended as a text for graduate students and as a reference for workers in probability and statistics. The prerequisite is honest calculus. The material covered in Parts Two to Five inclusive requires about three to four semesters of graduate study. The introductory part may serve as a text for an undergraduate course in elementary probability theory. Numerous historical marks about results, methods, and the evolution of various fields are an intrinsic part of the text. About a third of the second volume is devoted to conditioning and properties of sequences of various types of dependence. The other two thirds are devoted to random functions; the last Part on Elements of random analysis is more sophisticated.978-0-387-34432-4LovelockDavid Lovelock, University of Arizona Dept. Mathematics, Tucson, AZ, USA; Marilou Mendel, University of Arizona Dept. Mathematics, Tucson, AZ, USA; Arthur L. Wright, University of Arizona Dept. Mathematics, Tucson, AZ, USA+An Introduction to the Mathematics of MoneySaving and Investing XI, 294 p.SCW23008$Business/Management Science, generalKJSimple Interest.- Compound Interest.- Infiation and Taxes.- Annuities.- Loans and Risks.- Amortization.- Credit Cards.- Bonds.- Stocks and Stock Markets.- Stock Market Indexes, Pricing, and Risk.- Options.This is an undergraduate textbook on the basic aspects of personal savings and investing with a balanced mix of mathematical rigor and economic intuition. It uses routine financial calculations as the motivation and basis for tools of elementary real analysis rather than taking the latter as given. Proofs using induction, recurrence relations and proofs by contradiction are covered. Inequalities such as the Arithmetic-Geometric Mean Inequality and the Cauchy-Schwarz Inequality are used. Basic topics in probability and statistics are presented. The student is introduced to elements of saving and investing that are of life-long practical use. These include savings and checking accounts, certificates of deposit, student loans, credit cards, mortgages, buying and selling bonds, and buying and selling stocks. The book is self contained and accessible. The authors follow a systematic pattern for each chapter including a variety of examples and exercises ensuring that the student deals with realities, rather than theoretical idealizations. It is suitable for courses in mathematics, investing, banking, financial engineering, and related topics.

Many illustrations included throughout the text

Exercises at the end of each chapter

An only undergraduate text providing an introduction to the mathematics of savings

978-1-4419-2232-8978-3-0348-0556-8Lunardi5Alessandra Lunardi, University of Parma, Parma, Italy@Analytic Semigroups and Optimal Regularity in Parabolic ProblemsXVII, 424 p.Introduction.- 0 Preliminary material: spaces of continuous and Hlder continuous functions.- 1 Interpolation theory.- Analytic semigroups and intermediate spaces.- 3 Generation of analytic semigroups by elliptic operators.- 4 Nonhomogeneous equations.- 5 Linear parabolic problems.- 6 Linear nonautonomous equations.- 7 Semilinearequations.- 8 Fully nonlinear equations.- 9 Asymptotic behavior in fully nonlinear equations.- Appendix: Spectrum and resolvent.- Bibliography.- Index. 9The book shows how the abstract methods of analytic semigroups and evolution equations in Banach spaces can be fruitfully applied to the study of parabolic problems. Particular attention is paid to optimal regularity results in linear equations. Furthermore, these re< sults are used to study several other problems, especially fully nonlinear ones. Owing to the new unified approach chosen, known theorems are presented from a novel perspective and new results are derived. The book is self-contained. It is addressed to PhD students and researchers interested in abstract evolution equations and in parabolic partial differential equations and systems. It gives a comprehensive overview on the present state of the art in the field, teaching at the same time how to exploit its basic techniques. - - - This very interesting book provides a systematic treatment of the basic theory of analytic semigroups and abstract parabolic equations in general Banach spaces, and how this theory may be used in the study of parabolic partial differential equations; it takes into account the developments of the theory during the last fifteen years. (...) For instance, optimal regularity results are a typical feature of abstract parabolic equations; they are comprehensively studied in this book, and yield new and old regularity results for parabolic partial differential equations and systems. (Mathematical Reviews) Motivated by applications to fully nonlinear problems the approach is focused on classical solutions with continuous or Hlder continuous derivatives. (Zentralblatt MATH)"<p>Systematic treatment of the basic theory of analytic semigroups and abstract parabolic equations in general Banach spaces </p><p>Known Theorems are presented from a novel perspective </p><p>Teacheshow to exploitbasic techniques </p><p>Addresses PhD studentsas well asresearchers </p>978-0-8176-4915-9LuongBao Luong, Columbia, MD, USA)Fourier Analysis on Finite Abelian GroupsSCM12058Fourier AnalysisPreface.- Overview.- Chapter 1: Foundation Material.- Results from Group Theory.- Quadratic Congruences.- Chebyshev Systems of Functions.- Chapter 2: The Fourier Transform.- A Special Class of Linear Operators.- Characters.- The Orthogonal Relations for Characters.- The Fourier Transform.- The Fourier Transform of Periodic Functions.- The Inverse Fourier Transform.- The Inversion Formula.- Matrices of the Fourier Transform.- Iterated Fourier Transform.- Is the Fourier Transform a Self-Adjoint Operator?.- The Convolutions Operator.- Banach Algebra.- The Uncertainty Principle.- The Tensor Decomposition.- The Tensor Decomposition of Vector Spaces.- The Fourier Transform and Isometries.- Reduction to Finite Cyclic Groups.- Symmetric and Antisymmetric Functions.- Eigenvalues and Eigenvectors.- Spectrak Theorem.- Ergodic Theorem.- Multiplicities of Eigenvalues.- The Quantum Fourier Transform.- Chapter 3: Quadratic Sums.- 1. The Number G_n(1).- Reduction Formulas.]Fourier analysis has been the inspiration for a technological wave of advances in fields such as imaging processing, financial modeling, algorithms and sequence design. This unified, self-contained book examines the mathematical tools used for decomposing and analyzing functions, specifically, the application of the [discrete] Fourier transform to finite Abelian groups. With countless examples and unique exercise sets at the end of each section, Fourier Analysis on Finite Abelian Groups is a perfect companion to a first course in Fourier analysis. The first chapter provides the fundamental material that is a strong foundation for all subsequent chapters. Special topics including: * Computing Eigenvalues of the Fourier transform * Applications to Banach algebras * Tensor decompositions of the Fourier transform * Quadratic Gaussian sums. This book introduces mathematics students to subjects that are within their reach, but it also has powerful applications that may appeal to advanced researchers and mathematicians. The only prerequisites necessary are group theory, linear algebra, and complex analysis.Examines the mathematical tools used for decomposing and analyzing functions, specifically, the application of the [discrete] Fourier transform to finite Abelian groups

Provides countless examples and unique exercise sets at the end of each section

A perfect companion to a first course in Fourier analysis

Includes special topics such as computing Eigenvalues of the Fourier transform, applications to Banach algebras, tensor decompositions of the Fourier transform and quadratic Gaussian sums

Introduces mathematics students to subjects that are within their reach but have powerful applications that also appeal to advanced researchers and mathematicians. The only prerequisites are group theory, linear algebra, and complex analysis

978-0-8176-4321-8Lynch_Stephen Lynch, Manchester Metropolitan University Dept. Computing & Mathematics, Manchester, UK1Dynamical Systems with Applications using MATLABXVII, 459 p.Preface.- A Tutorial Introduction to MATLAB and the Symbolic Math Toolbox.- Linear Discrete Dynamical Systems.- Nonlinear Discrete Dynamical Systems.- Complex Iterative Maps.- Electromagnetic Waves and Optical Resonators.- Fractals and Multifractals.- Controlling Chaos.- Differential Equations.- Planar Systems.- Interacting Species.- Limit Cycles.- Hamiltonian Systems, Liapunov Functions, and Stability.- Bifurcation Theory.- Three Dimensional Autonomous Systems and Chaos.- Poincar Maps and Nonautonomous Systems in the Plane.- Local and Global Bifurcations.- The Second Part of David Hilbert's Sixteenth Problem.- Neural Networks.- SIMULINK.- Solutions to Exercises.- References.- MATLAB Program File Index.- SIMULINK Model File Index.- Index.SBeginning with a tutorial guide to MATLAB, the text thereafter is divided into two main areas. In Part I, both real and complex discrete dynamical systems are considered, with examples presented from population dynamics, nonlinear optics, and materials science. Part II includes examples from mechanical systems, chemical kinetics, electric circuits, economics, population dynamics, epidemiology, and neural networks. Common themes such as bifurcation, bistability, chaos, fractals, instability, multistability, periodicity, and quasiperiodicity run through several chapters. Chaos control and multifractal theories are also included along with an example of chaos synchronization. Some material deals with cutting-edge published research articles and provides a useful resource for open problems in nonlinear dynamical systems. Readers are guided through theory via example, and the graphical MATLAB interface. The Simulink accessory is used to simulate real-world dynamical processes. Examples from: mechanics, electric circuits, economics, population dynamics, epidemiology, nonlinear optics, materials science, and neural networks. Over 330 illustrations, 300 examples, and exercises with solutions. Aimed at senior undergraduates, graduate students, and working scientists in various branches of engineering, applied mathematics, and the natural sciences.No existing text at this level integrates MATLAB^{}/SIMULINK^{}

Examples from many diverse disciplines, including mechanics, optics, electronics, economics, population dynamics, epidemiology, neural networks

For broad audience of students and researchers in applied mathematics, physics, engineering, and the natural sciences

Hands-on examples and the MATLAB^{} graphical interface guide readers through the theory

SIMULINK^{} allows for the treatment of more modeling/simulation topics geared toward the engineer

Presentation is driven by many examples, illustrations, exercises, and solutions

978-0-387-28178-0MacherasPanos Macheras, University of Athens School of Pharmacy, Zographou, Greece; Athanassios Iliadis, University of Mditerrane Faculty of Pharmacy, Marseille CX 07, FranceCModeling in Biopharmaceutics, Pharmacokinetics and Pharmacodynamics(Homogeneous and Heterogene< ous ApproachesXX, 442 p. 131 illus.SCB21007Pharmacology/ToxicologyMMGBasic Concepts.- The Geometry of Nature.- Diffusion and Kinetics.- Nonlinear Dynamics.- Modeling in Biopharmaceutics.- Drug Release.- Drug Dissolution.- Oral Drug Absorption.- Modeling in Pharmacokinetics.- Empirical Models.- Deterministic Compartmental Models.- Stochastic Compartmental Models.- Modeling in Pharmacodynamics.- Classical Pharmacodynamics.- Nonclassical Pharmacodynamics.#The state of the art in Biopharmaceutics, Pharmacokinetics, and Pharmacodynamics Modeling is presented in this book. It shows how advanced physical and mathematical methods can expand classical models in order to cover heterogeneous drug-biological processes and therapeutic effects in the body. The book is divided into four parts; the first deals with the fundamental principles of fractals, diffusion and nonlinear dynamics; the second with drug dissolution, release, and absorption; the third with empirical, compartmental, and stochastic pharmacokinetic models, and the fourth mainly with nonclassical aspects of pharmacodynamics. The classical models that have relevance and application to these sciences are also considered throughout. Many examples are used to illustrate the intrinsic complexity of drug administration related phenomena in the human, justifying the use of advanced modeling methods. This timely and useful book will appeal to graduate students and researchers in pharmacology, pharmaceutical sciences, bioengineering, and physiology.Brings together the disciplines of Biopharmaceutics, Pharmacokinetics and Pharmacodynamics (BPP)

Coveres the theory with specific examples behind all three under the perspective of heterogeneous processes while also treating the current state of the art in applied and theoretical BPP

Can serve as a text for graduate courses in bioengineering or BPP and as working tool for the empirical scientist who wants to expand into theory or the mathematician working in the biopharmaceutical field

978-1-4419-2100-0XX, 442p. 131 illus..978-0-387-98386-8 MaclachlanColin Maclachlan; Alan W. Reid(The Arithmetic of Hyperbolic 3-ManifoldsXIII, 467 p.0 Number-Theoretic Menagerie.- 1 Kleinian Groups and Hyperbolic Manifolds.- 2 Quaternion Algebras I.- 3 Invariant Trace Fields.- 4 Examples.- 5 Applications.- 6 Orders in Quaternion Algebras.- 7 Quaternion Algebras II.- 8 Arithmetic Kleinian Groups.- 9 Arithmetic Hyperbolic 3-Manifolds and Orbifolds.- 10 Discrete Arithmetic Groups.- 11 Commensurable Arithmetic Groups and Volumes.- 12 Length and Torsion in Arithmetic Hyperbolic Orbifolds.- 13 Appendices.VFor the past 25 years, the Geometrization Program of Thurston has been a driving force for research in 3-manifold topology. This has inspired a surge of activity investigating hyperbolic 3-manifolds (and Kleinian groups), as these manifolds form the largest and least well- understood class of compact 3-manifolds. Familiar and new tools from diverse areas of mathematics have been utilized in these investigations, from topology, geometry, analysis, group theory, and from the point of view of this book, algebra and number theory. This book is aimed at readers already familiar with the basics of hyperbolic 3-manifolds or Kleinian groups, and it is intended to introduce them to the interesting connections with number theory and the tools that will be required to pursue them. While there are a number of texts which cover the topological, geometric and analytical aspects of hyperbolic 3-manifolds, this book is unique in that it deals exclusively with the arithmetic aspects, which are not covered in other texts. Colin Maclachlan is a Reader in the Department of Mathematical Sciences at the University of Aberdeen in Scotland where he has served since 1968. He is a former President of the Edinburgh Mathematical Society. Alan Reid is a Professor in the Department of Mathematics at The University of Texas at Austin. He is a former Royal Society University Research Fellow, Alfred P. Sloan Fellow and winner of the Sir Edmund Whittaker Prize from The Edinburgh Mathematical Society. Both authors have published extensively in the general area of discrete groups, hyperbolic manifolds and low-dimensional topology.Brings together much of the existing literature on arithmetic Kleinan groups in a clear and concise wayNo such presentation currently exists

Contains many examples and lots of problems978-1-4419-3122-1978-1-4612-9340-8MacLane%Saunders MacLane, Heidelberg, GermanyMathematics Form and Function476p.<I Origins of Formal Structure.- 1. The Natural Numbers.- 2. Infinite Sets.- 3. Permutations.- 4. Time and Order.- 5. Space and Motion.- 6. Symmetry.- 7. Transformation Groups.- 8. Groups.- 9. Boolean Algebra.- 10. Calculus, Continuity, and Topology.- 11. Human Activity and Ideas.- 12. Mathematical Activities.- 13. Axiomatic Structure.- II From Whole Numbers to Rational Numbers.- 1. Properties of Natural Numbers.- 2. The Peano Postulates.- 3. Natural Numbers Described by Recursion.- 4. Number Theory.- 5. Integers.- 6. Rational Numbers.- 7. Congruence.- 8. Cardinal Numbers.- 9. Ordinal Numbers.- 10. What Are Numbers?.- III Geometry.- 1. Spatial Activities.- 2. Proofs without Figures.- 3. The Parallel Axiom.- 4. Hyperbolic Geometry.- 5. Elliptic Geometry.- 6. Geometric Magnitude.- 7. Geometry by Motion.- 8. Orientation.- 9. Groups in Geometry.- 10. Geometry by Groups.- 11. Solid Geometry.- 12. Is Geometry a Science?.- IV Real Numbers.- 1. Measures of Magnitude.- 2. Magnitude as a Geometric Measure.- 3. Manipulations of Magnitudes.- 4. Comparison of Magnitudes.- 5. Axioms for the Reals.- 6. Arithmetic Construction of the Reals.- 7. Vector Geometry.- 8. Analytic Geometry.- 9. Trigonometry.- 10. Complex Numbers.- 11. Stereographic Projection and Infinity.- 12. Are Imaginary Numbers Real?.- 13. Abstract Algebra Revealed.- 14. The Quaternions and Beyond.- 15. Summary.- V Functions, Transformations, and Groups.- 1. Types of Functions.- 2. Maps.- 3. What Is a Function?.- 4. Functions as Sets of Pairs.- 5. Transformation Groups.- 6. Groups.- 7. Galois Theory.- 8. Constructions of Groups.- 9. Simple Groups.- 10. Summary: Ideas of Image and Composition.- VI Concepts of Calculus.- 1. Origins.- 2. Integration.- 3. Derivatives.- 4. The Fundamental Theorem of the Integral Calculus.- < 5. Kepler s Laws and Newton s Laws.- 6. Differential Equations.- 7. Foundations of Calculus.- 8. Approximations and Taylor s Series.- 9. Partial Derivatives.- 10. Differential Forms.- 11. Calculus Becomes Analysis.- 12. Interconnections of the Concepts.- VII Linear Algebra.- 1. Sources of Linearity.- 2. Transformations versus Matrices.- 3. Eigenvalues.- 4. Dual Spaces.- 5. Inner Product Spaces.- 6. Orthogonal Matrices.- 7. Adjoints.- 8. The Principal Axis Theorem.- 9. Bilinearity and Tensor Products.- 10. Collapse by Quotients.- 11. Exterior Algebra and Differential Forms.- 12. Similarity and Sums.- 13. Summary.- VIII Forms of Space.- 1. Curvature.- 2. Gaussian Curvature for Surfaces.- 3. Arc Length and Intrinsic Geometry.- 4. Many-Valued Functions and Riemann Surfaces.- 5. Examples of Manifolds.- 6. Intrinsic Surfaces and Topological Spaces.- 7. Manifolds.- 8. Smooth Manifolds.- 9. Paths and Quantities.- 10. Riemann Metrics.- 11. Sheaves.- 12. What Is Geometry?.- IX Mechanics.- 1. Kepler s Laws.- 2. Momentum, Work, and Energy.- 3. Lagrange s Equations.- 4. Velocities and Tangent Bundles.- 5. Mechanics in Mathematics.- 6. Hamilton s Principle.- 7. Hamilton s Equations.- 8. Tricks versus Ideas.- 9. The Principal Function.- 10. The Hamilton Jacobi Equation.- 11. The Spinning Top.- 12. The Form of Mechanics.- 13. Quantum Mechanics.- X Complex Analysis and Topology.- 1. Functions of a Complex Variable.- 2. Pathological Functions.- 3. Complex Derivatives.- 4. Complex Integration.- 5. Paths in the Plane.- 6. The Cauchy Theorem.- 7. Uniform Convergence.- 8. Power Series.- 9. The Cauchy Integral Formula.- 10. Singularities.- 11. Riemann Surfaces.- 12. Germs and Sheaves.- 13. Analysis, Geometry, and Topology.- XI Sets, Logic, and Categories.- 1. The Hierarchy of Sets.- 2. Axiomatic Set Theory.- 3. The Propositional Calculus.- 4. First Order Language.- 5. The Predicate Calculus.- 6. Precision and Understanding.- 7. Gdel Incompleteness Theorems.- 8. Independence Results.- 9. Categories and Functions.- 10. Natural Transformations.- 11. Universals.- 12. Axioms on Functions.- 13.978-3-540-27264-9 MalevergneYannick Malevergne, Universit St.-Etienne Inst. Suprieur d'Economie de, St.-Etienne CX 2, France; Didier Sornette, ETH Zrich Dept. Management,, Zrich, SwitzerlandExtreme Financial Risks"From Dependence to Risk ManagementXVI, 312 p.On the Origin of Risks and Extremes.- Marginal Distributions of Returns.- Notions of Copulas.- Measures of Dependences.- Description of Financial Dependences with Copulas.- Measuring Extreme Dependences.- Summary and Outlook.Portfolio analysis and optimization, together with the associated risk assessment and management, require knowledge of the likely distributions of returns at different time scales and insights into the nature and properties of dependences between the different assets. This book offers an original and thorough treatment of these two domains, focusing mainly on the concepts and tools that remain valid for large and extreme price moves. Strong emphasis is placed on the theory of copulas and their empirical testing and calibration, because they offer intrinsic and complete measures of dependences. Extreme Financial Risks will be useful to: students looking for a general and in-depth introduction to the field; financial engineers, economists, econometricians, actuarial professionals; researchers and mathematicians looking for a synoptic view comparing the pros and cons of different modelling strategies; and quantitative practitioners for the insights offered on the subtleties and the many dimensional components of both risk and dependence. In toto, the content of this book will also be useful to a broader scientific community interested in quantifying the complexity of many natural and artificial processes in which a growing emphasis is on the role and importance of extreme phenomena.

This is the first book to offer an in-depth introduction to the field to a broad range of graduate students, scientists and professionals such as econophysicists, financial engineers, economists, econometricians and quantitative practitioners.

978-0-387-94409-8 Malliavin Paul Malliavin, Paris CX, FranceIntegration and ProbabilityXXI, 325 p.I Measurable Spaces and Integrable Functions.- 1 ?-algebras.- 2 Measurable Spaces.- 3 Measures and Measure Spaces.- 4 Negligible Sets and Classes of Measurable Mappings.- 5 Convergence in M ((X,A);(Y,BY)).- 6 The Space of Integrable Functions.- 7 Theorems on Passage to the Limit under the Integral Sign.- 8 Product Measures and the Fubini-Lebesgue Theorem.- 9 The Lp Spaces.- II Borel Measures and Radon Measures.- 1 Locally Compact Spaces and Partitions of Unity.- 2 Positive Linear Functionals onCK(X) and Positive Radon Measures.- 3 Regularity of Borel Measures and Lusin s Theorem.- 4 The Lebesgue Integral on R and on Rn.- 5 Linear Functionals on CK(X) and Signed Radon Measures.- 6 Measures and Duality with Respect to Spaces of Continuous Functions on a Locally Compact Space.- III Fourier Analysis.- 1 Convolutions and Spectral Analysis on Locally Compact Abelian Groups.- 2 Spectral Synthesis on Tn and Rn.- 3< Vector Differentiation and Sobolev Spaces.- 4 Fourier Transform of Tempered Distributions.- 5 Pseudo-differential Operators.- IV Hilbert Space Methods and Limit Theorems in Probability Theory.- 1 Foundations of Probability Theory.- 2 Conditional Expectation.- 3 Independence and Orthogonality.- 4 Characteristic Functions and Theorems on Convergence in Distribution.- 5 Theorems on Convergence of Martingales.- 6 Theory of Differentiation.- V Gaussian Sobolev Spaces and Stochastic Calculus of Variations.- 1 Gaussian Probability Spaces.- 2 Gaussian Sobolev Spaces.- 3 Absolute Continuity of Distributions.- Appendix I. Hilbert Spectral Analysis.- 1 Functions of Positive Type.- 2 Bochner s Theorem.- 3 Spectral Measures for a Unitary Operator.- 4 Spectral Decomposition Associated with a Unitary Operator.- 5 Spectral Decomposition for Several Unitary Operators.- Appendix II. Infinitesimal and Integrated Forms of the Change-of-Variables Formula.- 1 Notation.- 2 Velocity Fields and Densities.- Exercises for Chapter I.- Exercises for Chapter II.- Exercises for Chapter III.- Exercises for Chapter IV.- Exercises for Chapter V.1This book is designed to be an introduction to analysis with the proper mix of abstract theories and concrete problems. It starts with general measure theory, treats Borel and Radon measures (with particular attention paid to Lebesgue measure) and introduces the reader to Fourier analysis in Euclidean spaces with a treatment of Sobolev spaces, distributions, and the Fourier analysis of such. It continues with a Hilbertian treatment of the basic laws of probability including Doob's martingale convergence theorem and finishes with Malliavin's 'stochastic calculus of variations' developed in the context of Gaussian measure spaces. This invaluable contribution to the existing literature gives the reader a taste of the fact that analysis is not a collection of independent theories but can be treated as a whole.978-3-642-33405-4 MalyarenkoYAnatoliy Malyarenko, Mlardalen University School of Education, Culture, Vsters, Sweden5Invariant Random Fields on Spaces with a Group ActionXVII, 261 p.1.Introduction.- 2.Spectral Expansions.- 3.L2 Theory of Invariant Random Fields.- 4.Sample Path Properties of Gaussian Invariant Random Fields.- 5.Applications.- A.Mathematical Background.- References.- Index. The author describes the current state of the art in the theory of invariant random fields. This theory is based on several different areas of mathematics, including probability theory, differential geometry, harmonic analysis, and special functions. The present volume unifies many results scattered throughout the mathematical, physical, and engineering literature, as well as itintroduces new results from this area first proved by the author. The book also presents many practical applications, in particular in such highly interesting areas as approximation theory, cosmology and earthquake engineering. It is intended for researchers and specialists working in the fields of stochastic processes, statistics, functional analysis, astronomy, and engineering.<p>Highly interdisciplinary nature </p><p>Fills a gap in the literature</p><p>Many new results, andpractical applications asfor examplein cosmology and earthquake engineering </p>978-3-642-44450-0978-0-387-95225-3MartinGeorge E. Martin.Counting: The Art of Enumerative Combinatorics XI, 252 p.SCI17001Mathematics of ComputingUYA1. Elementary Enumeration.- 2. The Principle of Inclusion and Exclusion.- 3. Generating Functions.- 4. Groups.- 5. Actions.- 6. Recurrence Relations.- 7. Mathematical Induction.- 8. Graphs.- The Back of the Book.;Counting is hard. 'Counting' is short for 'Enumerative Combinatorics,' which certainly doesn't sound easy. This book provides an introduction to discrete mathematics that addresses questions that begin, How many ways are there to... . At the end of the book the reader should be able to answer such nontrivial counting questions as, How many ways are there to stack n poker chips, each of which can be red, white, blue, or green, such that each red chip is adjacent to at least 1 green chip? There are no prerequisites for this course beyond mathematical maturity. The book can be used for a semester course at the sophomore level as introduction to discrete mathematics for mathematics, computer science, and statistics students. The first five chapters can also serve as a basis for a graduate course for in-service teachers.978-1-4419-2915-0978-0-387-90636-2Transformation GeometryAn Introduction to SymmetryXII, 237 p. 222 illus.1 Introduction.- 1.1 Transformations and Collineations.- 1.2 Geometric Notation.- 1.3 Exercises.- 2 Properties of Transformations.- 2.1 Groups of Transformations.- 2.2 Involutions.- 2.3 Exercises.- 3 Translations and Halfturns.- 3.1 Translations.- 3.2 Halfturns.- 3.3 Exercises.- 4 Reflections.- 4.1 Equations for a Reflection.- 4.2 Properties of a Reflection.- 4.3 Exercises.- 5 Congruence.- 5.1 Isometries as Products of Reflections.- 5.2 Paper Folding Experiments and Rotations.- 5.3 Exercises.- 6 The Product of Two Reflections.- 6.1 Translations and Rotations.- 6.2 Fixed Points and Involutions.- 6.3 Exercises.- 7 Even Isometries.- 7.1 Parity.- 7.2 The Dihedral Groups.- 7.3 Exercises.- 8 Classification of Plane Isometries.- 8.1 Glide Reflections.- 8.2 Leonardo s Theorem.- 8.3 Exercises.- 9 Equations for Isometries.- 9.1 Equations.- 9.2 Supplementary Exercises (Chapter 1 8).- 9.3 Exercises.- 10 The Seven Frieze Groups.- 10.1 Frieze Groups.- 10.2 Frieze Patterns.- 10.3 Exercises.- 11 The Seventeen Wallpaper Grou< ps.- 11.1 The Crystallographic Restriction.- 11.2 Wallpaper Groups and Patterns.- 11.3 Exercises.- 12 Tessellations.- 12.1 Tiles.- 12.2 Reptiles.- 12.3 Exercises.- 13 Similarities on the Plane.- 13.1 Classification of Similarities.- 13.2 Equations for Similarities.- 13.3 Exercises.- 14 Classical Theorems.- 14.1 Menelaus, Ceva, Desargues, Pappus, Pascal.- 14.2 Euler, Brianchon, Poncelet, Feuerbach.- 14.3 Exercises.- 15 Affine Transformations.- 15.1 Collineations.- 15.2 Linear Transformations.- 15.3 Exercises.- 16 Transformations on Three-space.- 16.1 Isometries on Space.- 16.2 Similarities on Space.- 16.3 Exercises.- 17 Space and Symmetry.- 17.1 The Platonic Solids.- 17.2 Finite Symmetry Groups on Space.- 17.3 Exercises.- Hints and Answers.- Notation Index.Transformation Geometry: An Introduction to Symmetry is a modern approach to Euclidean Geometry. This study of the automorphism groups of the plane and space gives the classical concrete examples that serve as a meaningful preparation for the standard undergraduate course in abstract algebra. The detailed development of the isometries of the plane is based on only the most elementary geometry and is appropriate for graduate courses for secondary teachers.978-0-387-95344-1Martins4Lina Martins, Universit di Bologna, Martinez, Italy8An Introduction to Semiclassical and Microlocal AnalysisVIII, 191 p.1 Introduction.- 2 Semiclassical Pseudodifferential Calculus.- 3 Microlocalization.- 4 Applications to the Solutions of Analytic Linear PDEs.- 5 Complements: Symplectic Aspects.- 6 Appendix: List of Formulae.- List of Notation.This book presents the techniques used in the microlocal treatment of semiclassical problems coming from quantum physics in a pedagogical way and is mainly addressed to non-specialists in the subject. Both the standard C pseudodifferential calculus and the analytic microlocal analysis are developed, while the author replaces the delicate local theory with a more straightforward global theory. The book is based on lectures taught by the author over several years. Many exercises provide outlines of useful applications of the semi-classical theory and the only prerequisite required is a basic knowledge of the theory of distributions.- Addressed to non-specialists- detailed exercises at the end of the main chapters

- proofs presented have been made as simple as possible

- introduces tools useful in the study of phase space tunneling978-1-4419-2961-7978-1-4614-5340-6 MashreghiJavad Mashreghi, Universit Laval, Quebec, QC, Canada; Emmanuel Fricain, Universit Claude Bernard Lyon 1, Villeurbanne cedex, France(Blaschke Products and Their ApplicationsFields Institute Communications&X, 319 p. 5 illus., 2 illus. in color. -Preface. - Applications of Blaschke products to the spectral theory of Toeplitz operators (Grudsky, Shargorodsky). -A survey on Blaschke-oscillatory differential equations, with updates (Heittokangas.).- Bi-orthogonal expansions in the space L2(0,1) ( Boivin, Zhu). - Blaschke products as solutions of a functional equation (Mashreghi.). - Cauchy Transforms and Univalent Functions( Cima, Pfaltzgraff). - Critical points, the Gauss curvature equation and Blaschke products (Kraus, Roth). - Growth, zero distribution and factorization of analytic functions of moderate growth in the unit disc, (Chyzhykov, Skaskiv). - Hardy means of a finite Blaschke product and its derivative ( Gluchoff, Hartmann). -Hyperbolic derivatives determine a function uniquely (Baribeau). - Hyperbolic wavelets and multiresolution in the Hardy space of the upper half plane (Feichtinger, Pap). - Norm of composition operators induced by finite Blaschke products on Mobius invariant spaces (Martin, Vukotic). - On the computable theory of bounded analytic functions (McNicholl). - Polynomials versus finite Blaschke products ( Tuen Wai Ng, Yin Tsang). -Recent progress on truncated Toeplitz operators (Garcia, Ross). Blaschke Products and Their Applications presents a collection of survey articles that examine Blaschke products and several of its applications to fields such as approximation theory, differential equations, dynamical systems, harmonic analysis, to name a few. Additionally, this volume illustrates the historical roots of Blaschke products and highlights key research on this topic. For nearly a century, Blaschke products have been researched. Their boundary behaviour, the asymptomatic growth of various integral means and their derivatives, their applications within several branches of mathematics, and their membership in different function spaces and their dynamics, are a few examples of where Blaschke products have shown to be important. The contributions written by experts from various fields of mathematical research will engage graduate students and researches alike, bringing the reader to the forefront of research in the topic. The readers will also discover the various open problems, enabling them to better pursue their own research.<p>Aids graduate students as well as researchers to get a thorough understanding of Blaschke products</p><p>Topics such as computability or applications in differential equations are examined for the first time</p><p>Presents a collection ofsurvey articles that examine Blaschke products and several of itsapplications to fields such as approximation theory, differentialequations, dynamical systems, harmonic analysis, to name a few </p>978-1-4899-9082-2978-1-4614-5610-0< 5Javad Mashreghi, Universit Laval, Quebec, QC, CanadaDerivatives of Inner FunctionsFields Institute MonographsX, 169 p. 2 illus.B .-Preface.-1. Inner Functions.-2. The Exceptional Set of an Inner Function.-3. The Derivative of Finite Blaschke Products.-4. Angular Derivative.-5. Hp-Means of S'.-6. Bp-Means of S'.-7. The Derivative of a Blaschke Product.-8. Hp-Means of B'.-9. Bp-Means of B'.-10. The Growth of Integral Means of B'.-References.-Index. Inner functions form an important subclass of bounded analytic functions. Since they have unimodular boundary values, they appear in many extremal problems of complex analysis. They have been extensively studied since early last century, and the literature on this topic is vast. Therefore, this book is devoted to a concise study of derivatives of these objects, and confined to treating the integral means of derivatives and presenting a comprehensive list of results on Hardy and Bergman means. The goal is to provide rapid access to the frontiers of research in this field. This monograph will allow researchers to get acquainted with essentials on inner functions, and it is self-contained, which makes it accessible to graduate students.

Includes a comprehensive list of results on integral means taken from several research papers

Text is concise and self-contained, making it easily accessible to graduate students

Provides rapid access to the frontiers of research in this field

978-1-4899-8941-3978-0-387-90271-5MasseyWilliam S. Massey#Algebraic Topology: An IntroductionXXI, 261 pp.L1: Two-Dimensional Manifolds. 2: The Fundamental Group. 3: Free Groups and Free Products of Groups. 4: Seifert and Van Kampen Theorem of the Fundamental Groups of the Union of Two Spaces. 5: Covering Spaces. 6: The Fundamental Group and Covering Spaces of a Graph. 7: The Fundamental Group of Higher Dimensional Spaces. 8: Epilogue.William S. Massey Professor Massey, born in Illinois in 1920, received his bachelor's degree from the University of Chicago and then served for four years in the U.S. Navy during World War II. After the War he received his Ph.D. from Princeton University and spent two additional years there as a post-doctoral research assistant. He then taught for ten years on the faculty of Brown University, and moved to his present position at Yale in 1960. He is the author of numerous research articles on algebraic topology and related topics. This book developed from lecture notes of courses taught to Yale undergraduate and graduate students over a period of several years.978-1-4614-9822-3XXI, 292 p.978-0-387-95373-1Lectures on Discrete GeometryXVI, 481 p. 206 illus. 1 Convexity.- 1.1 Linear and Affine Subspaces, General Position.- 1.2 Convex Sets, Convex Combinations, Separation.- 1.3 Radon s Lemma and Helly s Theorem.- 1.4 Centerpoint and Harn Sandwich.- 2 Lattices and Minkowski s Theorem.- 2.1 Minkowski s Theorem.- 2.2 General Lattices.- 2.3 An Application in Number Theory.- 3 Convex Independent Subsets.- 3.1 The Erd?s-Szekeres Theorem.- 3.2 Horton Sets.- 4 Incidence Problems.- 4.1 Formulation.- 4.2 Lower Bounds: Incidences and Unit Distances.- 4.3 Point-Line Incidences via Crossing Numbers.- 4.4 Distinct Distances via Crossing Numbers.- 4.5 Point-Line Incidences via Cuttings.- 4.6 A Weaker Cutting Lemma.- 4.7 The Cutting Lemma: A Tight Bound.- 5 Convex Polytopes.- 5.1 Geometric Duality.- 5.2 H-Polytopes and V-Polytopes.- 5.3 Faces of a Convex Polytope.- 5.4 Many Faces: The Cyclic Polytopes.- 5.5 The Upper Bound Theorem.- 5.6 The Gale Transform.- 5.7 Voronoi Diagrams.- 6 Number of Faces in Arrangements.- 6.1 Arrangements of Hyperplanes.- 6.2 Arrangements of Other Geometric Objects.- 6.3 Number of Vertices of Level at Most k.- 6.4 The Zone Theorem.- 6.5 The Cutting Lemma Revisited.- 7 Lower Envelopes.- 7.1 Segments and Davenport-Schinzel Sequences.- 7.2 Segments: Superlinear Complexity of the Lower Envelope.- 7.3 More on Davenport-Schinzel Sequences.- 7.4 Towards the Tight Upper Bound for Segments.- 7.5 Up to Higher Dimension: Triangles in Space.- 7.6 Curves in the Plane.- 7.7 Algebraic Surface Patches.- 8 Intersection Patterns of Convex Sets.- 8.1 The Fractional Helly Theorem.- 8.2 The Colorful Carathodory Theorem.- 8.3 Tverberg s Theorem.- 9 Geometric Selection Theorems.- 9.1 A Point in Many Simplices: The First Selection Lemma.- 9.2 The Second Selection Lemma.- 9.3 Order Types and the Same-Type Lemma.- 9.4 A Hypergraph Regularity Lemma.- 9.5 A Positive-Fraction Selection Lemma.- 10 Transversals and Epsilon Nets.- 10.1 General Preliminaries: Transversals and Matchings.- 10.2 Epsilon Nets and VC-Dimension.- 10.3 Bounding the VC-Dimension and Applications.- 10.4 Weak Epsilon Nets for Convex Sets.- 10.5 The Hadwiger-Debrunner (p, q)-Problem.- 10.6 A (p, q)-Theorem for Hyperplane Transversals.- 11 Attempts to Count k-Sets.- 11.1 Definitions < and First Estimates.- 11.2 Sets with Many Halving Edges.- 11.3 The Lovsz Lemma and Upper Bounds in All Dimensions.- 11.4 A Better Upper Bound in the Plane.- 12 Two Applications of High-Dimensional Polytopes.- 12.1 The Weak Perfect Graph Conjecture.- 12.2 The Brunn-Minkowski Inequality.- 12.3 Sorting Partially Ordered Sets.- 13 Volumes in High Dimension.- 13.1 Volumes, Paradoxes of High Dimension, and Nets.- 13.2 Hardness of Volume Approximation.- 13.3 Constructing Polytopes of Large Volume.- 13.4 Approximating Convex Bodies by Ellipsoids.- 14 Measure Concentration and Almost Spherical Sections.- 14.1 Measure Concentration on the Sphere.- 14.2 Isoperimetric Inequalities and More on Concentration.- 14.3 Concentration of Lipschitz Functions.- 14.4 Almost Spherical Sections: The First Steps.- 14.5 Many Faces of Symmetric Polytopes.- 14.6 Dvoretzky s Theorem.- 15 Embedding Finite Metric Spaces into Normed Spaces.- 15.1 Introduction: Approximate Embeddings.- 15.2 The Johnson-Lindenstrauss Flattening Lemma.- 15.3 Lower Bounds By Counting.- 15.4 A Lower Bound for the Hamming Cube.- 15.5 A Tight Lower Bound via Expanders.- 15.6 Upper Bounds for ??-Embeddings.- 15.7 Upper Bounds for Euclidean Embeddings.- What Was It About? An Informal Summary.- Hints to Selected Exercises.Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization. This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces. Jiri Matousek is Professor of Computer Science at Charles University in Prague. His research has contributed to several of the considered areas and to their algorithmic applications. This is his third book.- Quickly leads the reader to the edge of current research- Introduces many important oncepts and techniques on carefully chosen results where technicalities are used

- Book does not require any special background beyond undergraduate mathematics978-1-4614-5388-8MelnikRoderick Melnik, Wilfrid Laurier University, Waterloo, ON, Canada; Ilias S. Kotsireas, Wilfrid Laurier University Dept. of Physics and Computer Science, Waterloo, ON, CanadaDAdvances in Applied Mathematics, Modeling, and Computational Science)IX, 242 p. 63 illus., 37 illus. in color.Preface.-Interconnected Challenges and New Perspectives in Applied Mathematical and Computational Sciences (Melnik, Korsireas).-Dynamic Blocking Problems for a Model of Fire Propagation (Bressan).-Inverse Lax-Wendroff Procedure for Numerical Boundary Conditions of Hyperbolic Equations: Survey and New Developments (Tan, Shu).-Illiptic Curves Over Finite Fields (Shparlinski).-Random Matrix Theory and its Innovative (Edelman, Wang).-Boundary Closures for Sixth-Order Energy-Stable Weighted Essentially Non-Oscillatory Finite-Difference Schemes (Carpenter, Yamaleev, Fisher).-A Multiscale Method Coupling Network and Continuum Models in Porous Media II Single- and Two-Phase Flows (Chu, Engquist, Prodanovic, Tsai).-Statistical Geometry and Topology of the Human Placenta (Seong, Getreuer, Li, Girardi, Salafia, Vvedensky).-Illustrating Optimal Control Applications with Discrete and Continuous Features (Lenhart, Bodine, Zhong, Joshi).The volume presents a selection of in-depth studies and state-of-the-art surveys of several challenging topics that are at the forefront of modern applied mathematics, mathematical modeling, and computational science. These three areas represent the foundation upon which the methodology of mathematical modeling and computational experiment is built as a ubiquitous tool in all areas of mathematical applications. This book covers both fundamental and applied research, ranging from studies of elliptic curves over finite fields with their applications to cryptography, to dynamic blocking problems, to random matrix theory with its innovative applications. The book provides the reader with state-of-the-art achievements in the development and application of new theories at the interface of applied mathematics, modeling, and computational science.This book aims at fostering interdisciplinary collaborations required to meet the modern challenges of applied mathematics, modeling, and computational science. At the same time, the contributions combine rigorous mathematical and computational procedures and examples from applications ranging from engineering to life sciences, providing a rich ground for graduate student projects.<p> Provides state-of-the-art original works and surveys on several key areas in applied mathematics</p><p>Combines rigorous mathematical procedures and important examples from applications </p><p>With its many examples, the book is accessible to graduate students and can serve as a source for graduate student projects </p><p>Has a strong multidisciplinary focus, promoting the < methodology of mathematical modeling and computational experiment as a ubiquitous tool in applications978-1-4899-8987-1978-0-387-09723-7MeyerKenneth Meyer, University of Cincinnati Dept. Mathematics, Cincinnati, OH, USA; Glen Hall, Boston University Dept. Mathematics & Statistics, Boston, MA, USA; Dan Offin, Queen's University, Kingston Dept. Mathematics & Statistics, Kingston, ON, CanadaDIntroduction to Hamiltonian Dynamical Systems and the N-Body ProblemXIII, 399p. 67 illus..Hamiltonian Systems.- Equations of Celestial Mechanics.- Linear Hamiltonian Systems.- Topics in Linear Theory.- Exterior Algebra and Differential Forms.- Symplectic Transformations.- Special Coordinates.- Geometric Theory.- Continuation of Solutions.- Normal Forms.- Bifurcations of Periodic Orbits.- Variational Techniques.- Stability and KAM Theory.- Twist Maps and Invariant Circle.:This text grew out of notes from a graduate course taught to students in mathematics and mechanical engineering. The goal was to take students who had some basic knowledge of differential equations and lead them through a systematic grounding in the theory of Hamiltonian systems, an introduction to the theory of integrals and reduction. Poincar s continuation of periodic solution, normal forms, and applications of KAM theory. There is a special chapter devoted to the theory of twist maps and various extensions of the classic Poincar-Birkhoff fixed point theorem.978-1-4419-1886-4978-0-8176-8402-0Michel4Volker Michel, University of Siegen, Siegen, Germany&Lectures on Constructive ApproximationOFourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball(XVI, 326 p. 7 illus., 5 illus. in color.Introduction: the Problem to be Solved.- Part I Basics.- Basic Fundamentals What You Need to Know.- Approximation of Functions on the Real Line.- Part II Approximation on the Sphere.- Basic Aspects.- Fourier Analysis.- Spherical Splines.- Spherical Wavelet Analysis.- Spherical Slepian Functions.- Part III Approximation on the 3D Ball.- Orthonormal Bases.- Splines.- Wavelets for Inverse Problems on the 3D Ball.- The Regularized Functional Matching Pursuit (RFMP).- References.- Index. Lectures on Constructive Approximation: Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball focuses on spherical problems as they occur in the geosciences and medical imaging. It comprises the author s lectures on classical approximation methods based on orthogonal polynomials and selected modern tools such as splines and wavelets.Methods for approximating functions on the real line are treated first, as they provide the foundations for the methods on the sphere and the ball and are useful for the analysis of time-dependent (spherical) problems. The author then examines the transfer of these spherical methods to problems on the ball, such as the modeling of the Earth s or the brain s interior. Specific topics covered include:* the advantages and disadvantages of Fourier, spline, and wavelet methods* theory and numerics of orthogonal polynomials on intervals, spheres, and balls* cubic splines and splines based on reproducing kernels* multiresolution analysis using wavelets and scaling functionsThis textbook is written for students in mathematics, physics, engineering, and the geosciences who have a basic background in analysis and linear algebra. The work may also be suitable as a self-study resource for researchers in the above-mentioned fields.

Combines an explanation of classical and modern approximation methods for Euclidean and spherical geometries

Detailed explanations and illustrations included to optimize the understanding of topics

Concentrates on the essentials for a course

Uses examples of data sets to explain the tasks, challenges, advantages, and disadvantages of the methods presented

First work that explicitly treats approximation methods on the ball

978-1-4614-4231-8MigrskiStanisBaw Migrski, Jagiellonian University, Krakw, Poland; Anna Ochal, Jagiellonian University, Krakw, Poland; Mircea Sofonea, Universit de Perpignan, Perpignan, France5Nonlinear Inclusions and Hemivariational Inequalities'Models and Analysis of Contact Problems%Advances in Mechanics and Mathematics+XVI, 285 p. 105 illus., 69 illus. in color.Preface.- List of Symbols.- 1. Preliminaries.- 2. Function Spaces.- 3. Elements of Nonlinear Analysis.- 4. Stationary Inclusions and Hemivariational Inequalities.- 5. Evolutionary Inclusions and Hemivarational Inequalities.- 6. Modeling of Contact Problems.- 7. Analysis of Static Contact Problems.- 8. Analysis of Dynamic Contact Problems.- Bibliographic Notes.- References.- Index.Thisbook introduces the reader the theory of nonlinear inclusions and hemivariational inequalities with emphasison the study of contact mechanics. The work covers both abstract results in thearea of nonlinear inclusions, hemivariational inequalities as well as the study of specific contact problems, including their modelling and their variational analysis. Provided results are based on original research on the existence, uniqueness, regularity and behavior of the solution for various classes of nonlinear stationary and evolutionary inclusions. In carrying out the variational analysis of various contact models, onesystematically uses results of hemivariational inequalities and, in this way, illustrates the applications of nonlinear analysis in contact mechanics. New mathematical methods are introduced and applied in the study of nonlinear problems, which describe the contact between a deformable body and a foundation. Co< ntact problems arise in industry, engineering and geophysics. Their variational analysis presented in this book lies the background for their numerical analysis. This volume will interest mathematicians, applied mathematicians, engineers, and scientists as well as advanced graduate students.!Gathers new results on nonlinear inclusions and hemivariational inequalities and provides a uniqueoverviewof thistopic

Deals with new and nonstandard models of contact involving subdifferential of nonconvex functions, including models for the contact of piezoelectric materials

Intendsto represent a bridge between the functional analysis and the mechanics of continua

Provides the reader an example of cross fertilization between modelling and applications on one hand, and nonsmooth nonlinear analysis on the other

978-1-4899-9561-2978-3-540-88232-9MikoschVThomas Mikosch, University Copenhagen Inst. Mathematical Sciences, Copenhagen, DenmarkNon-Life Insurance Mathematics(An Introduction with the Poisson ProcessCollective Risk Models.- The Basic Model.- Models for the Claim Number Process.- The Total Claim Amount.- Ruin Theory.- Experience Rating.- Bayes Estimation.- Linear Bayes Estimation.- A Point Process Approach to Collective Risk Theory.- The General Poisson Process.- Poisson Random Measures in Collective Risk Theory.- Weak Convergence of Point Processes.- Special Topics.- An Excursion to Lévy Processes.- Cluster Point Processes.^The volume offers a mathematical introduction to non-life insurance and, at the same time, to a multitude of applied stochastic processes. It includes detailed discussions of the fundamental models regarding claim sizes, claim arrivals, the total claim amount, and their probabilistic properties. Throughout the volume the language of stochastic processes is used for describing the dynamics of an insurance portfolio in claim size, space and time. Special emphasis is given to the phenomena which are caused by large claims in these models. The reader learns how the underlying probabilistic structures allow determining premiums in a portfolio or in an individual policy. The second edition contains various new chapters that illustrate the use of point process techniques in non-life insurance mathematics. Poisson processes play a central role. Detailed discussions show how Poisson processes can be used to describe complex aspects in an insurance business such as delays in reporting, the settlement of claims and claims reserving. Also the chain ladder method is explained in detail. More than 150 figures and tables illustrate and visualize the theory. Every section ends with numerous exercises. An extensive bibliography, annotated with various comments sections with references to more advanced relevant literature, makes the volume broadly and easily accessible. <P>Rigorous mathematical introduction and as such quite a unique textbook which can be used not only by a specialised audience</P> <P>More than 100 figures and tables illustrating and visualizing the theory</P> <P>Every section ends with extensive exercises</P> <P>Book's content is in agreement with the European Group Consultatif standards</P> <P>An extensive bibliography, annotated by various comments sections with references to more advanced relevant literature, make the book broadly and easily accessible</P>978-0-8176-8396-2MitreaDorina Mitrea, University of Missouri, Columbia, MO, USA; Irina Mitrea, Temple University Department of Mathematics, Philadelphia, PA, USA; Marius Mitrea, University of Missouri, Columbia, MO, USA; Sylvie Monniaux, Universite Aix-Marseille III, Marseille, FranceGroupoid Metrization TheoryLWith Applications to Analysis on Quasi-Metric Spaces and Functional AnalysisXII, 479 p. 1 illus. Introduction.- Semigroupoids and Groupoids.- Quantitative Metrization Theory.- Applications to Analysis on Quasi-Metric Spaces.- Non-Locally Convex Functional Analysis.- Functional Analysis on Quasi-Pseudonormed Groups.- References.- Symbol Index.- Subject Index.- Author Index.zThe topics in this research monograph are at the interface of several areas of mathematics such as harmonic analysis, functional analysis, analysis on spaces of homogeneous type, topology, and quasi-metric geometry. The presentation is self-contained with complete, detailed proofs, and a large number of examples and counterexamples are provided.Unique features of Metrization Theory for Groupoids: With Applications to Analysis on Quasi-Metric Spaces and Functional Analysis include:* treatment of metrization from a wide, interdisciplinary perspective, with accompanying applications ranging across diverse fields;* coverage of topics applicable to a variety of scientific areas within pure mathematics;* useful techniques and extensive reference material; * includes sharp results in the field of metrization. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties. * coverage of topics applicable to a variety of scientific areas within pure mathematics;* useful techniques and extensive reference material; * includes sharp results in the field of metrization. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties. * useful techniques and extensive reference material; * includes sharp results in the field of metrization. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties. * includes sharp results in the field of metrization. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties. <Provides treatment of metrization from a wide, interdisciplinary perspective, with accompanying applications rang< ing across diverse fields Replete withextensive reference material and useful techniques Includes cutting-edge results in the field of metrization978-3-540-42139-9MolloyIMichael Molloy; Bruce Reed, Universit Paris VI CNRS, Paris CX 05, France,Graph Colouring and the Probabilistic MethodXIV, 326 p.1. Colouring Preliminaries.- 2. Probabilistic Preliminaries.- 3. The First Moment Method.- 4. The Lovsz Local Lemma.- 5. The Chernoff Bound.- 6. Hadwiger s Conjecture.- 7. A First Glimpse of Total Colouring.- 8. The Strong Chromatic Number.- 9. Total Colouring Revisited.- 10. Talagrand s Inequality and Colouring Sparse Graphs.- 11. Azuma s Inequality and a Strengthening of Brooks Theorem.- 12. Graphs with Girth at Least Five.- 13. Triangle-Free Graphs.- 14. The List Colouring Conjecture.- 15. The Structural Decomposition.- 16. ?, ? and ?.- 17. Near Optimal Total Colouring I: Sparse Graphs.- 18. Near Optimal Total Colouring II: General Graphs.- 19. Generalizations of the Local Lemma.- 20. A Closer Look at Talagrand s Inequality.- 21. Finding Fractional Colourings and Large Stable Sets.- 22. Hard-Core Distributions on Matchings.- 23. The Asymptotics of Edge Colouring Multigraphs.- 24. The Method of Conditional Expectations.- 25. Algorithmic Aspects of the Local Lemma.- References.Over the past decade, many major advances have been made in the field of graph colouring via the probabilistic method. This monograph provides an accessible and unified treatment of these results, using tools such as the Lovasz Local Lemma and Talagrand's concentration inequality. The topics covered include: Kahn's proofs that the Goldberg-Seymour and List Colouring Conjectures hold asymptotically; a proof that for some absolute constant C, every graph of maximum degree Delta has a Delta+C total colouring; Johansson's proof that a triangle free graph has a O(Delta over log Delta) colouring; algorithmic variants of the Local Lemma which permit the efficient construction of many optimal and near-optimal colourings. This begins with a gentle introduction to the probabilistic method and will be useful to researchers and graduate students in graph theory, discrete mathematics, theoretical computer science and probability.yThe book covers a topic of central interest to discrete mathematics.- The authors are two of the very best on this topic.978-0-387-94284-1MorrisPeter MorrisIntroduction to Game TheoryXXVI, 225 p. 44 illus.1. Games in Extensive Form.- 1.1. Trees.- 1.2. Game Trees.- 1.3. Choice Functions and Strategies.- 1.4. Games with Chance Moves.- 1.5. Equilibrium N-tuples of Strategies.- 1.6. Normal Forms.- 2. Two-Person Zero-Sum Games.- 2.1. Saddle Points.- 2.2. Mixed Strategies.- 2.3. Small Games.- 2.4. Symmetric Games.- 3. Linear Programming.- 3.1. Primal and Dual Problems.- 3.2. Basic Forms and Pivots.- 3.3. The Simplex Algorithm.- 3.4. Avoiding Cycles and Achieving Feasibility.- 3.5. Duality.- 4. Solving Matrix Games.- 4.1. The Minimax Theorem.- 4.2. Some Examples.- 5. Non-Zero-Sum Games.- 5.1. Noncooperative Games.- 5.2. Solution Concepts for Noncooperative Games.- 5.3. Cooperative Games.- 6. N-Person Cooperative Games.- 6.1. Coalitions.- 6.2. Imputations.- 6.3. Strategic Equivalence.- 6.4. Two Solution Concepts.- 7. Game-Playing Programs.- 7.1. Three Algorithms.- 7.2. Evaluation Functions.- Appendix. Solutions.>This is a textbook for a course in the theory of games. It is intended for advanced undergraduates and graduate students in mathematics and other quantitative disciplines, e.g., statistics, operations research, etc. It treats the central topics in game theory and is meant to give students a basis from which they can go on to more advanced topics. The subject matter is approached in a mathematically rigorous way, but , within this constraint, an effort is made to keep it interesting and lively. New definitions and topics are motivated as thoroughly as possible. The mathematical prerequisites for understanding the book are modest: basic probability together with a little calculus and linear algebra. Among others, two topics of great current interest are discussed in this book. The idea of iterated Prisoner's Dilemma (super games) is considered. It is specially of great interest to biologists, sociologists and others who use it in studying the evolution of cooperative behavior both in nature and in human society. Also covered are challenging game-playing computer programs.978-0-387-88614-5 MucherinoAntonio Mucherino, University of Florida Information Technology Office, Gainesville, FL, USA; Petraq Papajorgji, National Agency for Exams, Tirana, Albania; Panos Pardalos, University of Florida, Gainesville, FL, USAData Mining in AgricultureXVIII, 274p. 92 illus..SCU24005$Math. Appl. in Environmental ScienceRNto Data Mining.- Statistical Based Approaches.- Clustering by -means.- -Nearest Neighbor Classification.- Artificial Neural Networks.- Support Vector Machines.- Biclustering.- Validation.- Data Mining in a Parallel Environment.- Solutions to Exercises.Data Mining in Agriculture represents a comprehensive effort to provide graduate students and researchers with an analytical text on data mining techniques applied to agriculture and environmental related fields. This book presents both theoretical and practical insights with a focus on presenting the context of each data mining technique rather intuitively with ample concrete examples represented graphically and with algorithms written in Matlab. Examples and exercises with solutions are provided at the end of each chapter to facilitate the comprehension of the material. For each data mining technique described in the book variants and improvements of the basic algorithm are also given.MFirst textbook in data mining in agriculture

Presentation suitable for students, researchers, and professionals, in the classroom or as a self-study

Explores examples in agriculture/environmental fields

Provides Matlab codes to illustrate examples

Includes numerous exercises and some solutions

978-1-4614-2935-7978-1-4614-5127-3&Antonio Mucherino, IRISA, University of Rennes 1, Rennes, France; Carlile Lavor, State University of Campinas, Campinas, Brazil; Leo Liberti, Ecole Polytechnique, Palaiseau, France; Nelson Maculan, Universidade Federal do Rio de Janeiro Instituto Alberto Luiz Coimbra de, Rio de Janeiro, BrazilDistance Geometry!Theory, Methods, and Applications*XVI, 420 p. 74 illus., 29 illus. in color.Preface.- 1. Universal Rigidity of Bar Frameworks in General Position (A. Alfakih).- 2. Mixed Volume and Distance Geometry Techniques for Counting Euclidean Embeddings of Rigid Graphs (I. Emiris, E. Tsigaridas, A. Varvitsiotis).- 3. (The discretizable molecular distance Geometry Problem Seems Easier on Proteins (L. Liberti, C. Lavor, A. Mucherino).- 4. Spheres Unions and Intersections and Some of Their Applications in Molecular Modeling (M. Petitjean).- 5. Is the Distance Geometry Problem in NP? (N. < Beeker, S. Gaubert, C. Glusa, L. Liberti).- 6. Solving Spatial Constraints with Generalized Distance Geometry (L. Yang).- 7. A Topological Interpretation of the Walk Distances (P. Chebotarev, M. Deza).- 8. Distance Geometry Methods for Protein Structure Determination (Z. Voller, Z. Wu).- 9. Solving the discretizable molecular distance geometry problem by multiple realization trees (P. Nucci, L. Nogueira, C. Lavor).- 10.-ASAP - An Eigenvector Synchronization Algorithm for the Graph Realization Problem (M. Cucuringu).- 11. Global Optimization for Atomic Cluster Distance Geometry Problems (M. Locatelli, F. Schoen).- 12. Solving molecular distance geometry problems using a continuous optimization approach (R. Lima, J.M. Martinez).- 13. DC Programming Approaches for Distance Geometry Problems (H. Thi, T. Dinh).- 14. Stochastic Proximity Embedding (D. Agrafiotis, D. Bandyopadhyay, E. Yang).- 15. Distance Geometry for Realistic Molecular Conformations.- 16. Distance Geometry in Structural Biology (T. Malliavin, A. Mucherino, M. Nilges).- 17. Using a Distributed SDP Approach to Solve Simulated Protein Molecular Conformation Problems (X. Fang, K-C. Toh).- 18. An Overview on Protein Structure Determintion by NMR - Historical and Future Perspectives of the Use of Distance Geometry Methods.-Index.This volume is a collection of research surveys on the Distance Geometry Problem (DGP) and its applications. It will be divided into three parts: Theory, Methods and Applications. Each part will contain at least one survey and several research papers.The first part, Theory, will deal with theoretical aspects of the DGP, including a new class of problems and the study of its complexities as well as the relation between DGP and other related topics, such as: distance matrix theory, Euclidean distance matrix completion problem, multispherical structure of distance matrices, distance geometry and geometric algebra, algebraic distance geometry theory, visualization of K-dimensional structures in the plane, graph rigidity, and theory of discretizable DGP: symmetry and complexity.The second part, Methods, will discuss mathematical and computational properties of methods developed to the problems considered in the first chapter including continuous methods (based on Gaussian and hyperbolic smoothing, difference of convex functions, semidefinite programming, branch-and-bound), discrete methods (based on branch-and-prune, geometric build-up, graph rigidity), and also heuristics methods (based on simulated annealing, genetic algorithms, tabu search, variable neighborhood search).Applications will comprise the third part and will consider applications of DGP to NMR structure calculation, rational drug design, molecular dynamics simulations, graph drawing and sensor network localization.This volume will be the firstedited book on distance geometry and applications. The editors are in correspondence with the major contributors to the field of distance geometry, including important research centers in molecular biology such as Institut Pasteur in Paris.'Presents new information on thesubject of distance geometry and applications, whichis an untapped area of research with little in the way of printed information

Discusses theoretical aspects of the distance geometry problem, including a new class of problems and the study of its complexities as well as the relation between distance geometry problem and other related subjects

Coversapplications including structure calculation, rational drug design, molecular dynamics simulations, graph drawing and sensor network localization

978-1-4899-8578-1978-0-387-22182-3MurtyM. Ram Murty, Queen's University Jeffery Hall, Kingston, ON, Canada; Jody (Indigo) Esmonde, McGill University Dept. Mathematics & Statistics, Montreal, QC, Canada#Problems in Algebraic Number TheoryXVI, 352 p.Problems.- Elementary Number Theory.- Euclidean Rings.- Algebraic Numbers and Integers.- Integral Bases.- Dedekind Domains.- The Ideal Class Group.- Quadratic Reciprocity.- The Structure of Units.- Higher Reciprocity Laws.- Analytic Methods.- Density Theorems.- Solutions.- Elementary Number Theory.- Euclidean Rings.- Algebraic Numbers and Integers.- Integral Bases.- Dedekind Domains.- The Ideal Class Group.- Quadratic Reciprocity.- The Structure of Units.- Higher Reciprocity Laws.- Analytic Methods.- Density Theorems.Asking how one does mathematical research is like asking how a composer creates a masterpiece. No one really knows. However, it is a recognized fact that problem solving plays an important role in training the mind of a researcher. It would not be an exaggeration to say that the ability to do mathematical research lies essentially asking 'well-posed' questions. The approach taken by the authors in Problems in Algebraic Number Theory is based on the principle that questions focus and orient the mind. The book is a collection of about 500 problems in algebraic number theory, systematically arranged to reveal ideas and concepts in the evolution of the subject. While some problems are easy and straightforward, others are more difficult. For this new edition the authors added a chapter and revised several sections. The text is suitable for a first course in algebraic number theory with minimal supervision by the instructor. The exposition facilitates independent study, and students having taken a basic course in calculus, linear algebra, and abstract algebra will find these problems interesting and challenging. For the same reasons, it is ideal for non-specialists in acquiring a quick introduction to the subject.{Theory is quickly translated into experience with the problem-solving approach. The problems are systematically arranged to reveal the evolution of concepts and ideas of the subject. Includes various levels of problems - some are easy and straightforward, while others are more challenging. All problems are elegantly solved

An ideal book for self-study

978-1-4419-1967-0978-81-322-0769-6M. Ram Murty, Queen's University Department of Mathematics, Kingston, ON, Canada; V. Kumar Murty, University of Toronto Department of Mathematics, Toronto, ON, Canada.The Mathematical Legacy of Srinivasa Ramanujan XI, 186 p.Springer IndiaPreface.- Chapter 1. The Legacy of Srinivasa Ramanujan.- Chapter 2. The Ramanujan tau function.- Chapter 3. Ramanujan s conjecture and l-adic representations.- Chapter 4. The Ramanujan conjecture from GL(2) to GL(n).- Chapter 5. The circle method.- Chapter 6. Ramanujan and transcendence.- Chapter 7. Arithmetic of the partition function.- Chapter 8. Some nonlinear identities for divisor functions.- Chapter 9. Mock theta functions and mock modular forms.- Chapter 10. Prime numbers and highly composite numbers.- Chapter 11. Probabilistic number theory.- Chapter 12. The Sato-Tate conjecture for the Ramanujan tau-function.- Bibliography.- Index.Srinivasa Ramanujan was a mathematician brilliant beyond comparison who inspired many great mathematicians. There is extensive literature available on the work of Ramanujan. But what is missing in the literature is an analysis that would place his mathematics in context and interpret it in terms < of modern developments. The 12 lectures by Hardy, delivered in 1936, served this purpose at the time they were given. This book presents Ramanujan s essential mathematical contributions and gives an informal account of some of the major developments that emanated from his work in the 20th and 21st centuries. It contends that his work still has an impact on many different fields of mathematical research. This book examines some of these themes in the landscape of 21st-century mathematics. These essays, based on the lectures given by the authors focus on a subset of Ramanujan s significant papers and show how these papers shaped the course of modern mathematics.Gives a panoramic view of the main contributions of Srinivasa Ramanujan

Presents a major theme of Ramanujan's work in non-technical language

Provides an excellent introduction to Ramanujan's life and work

978-81-322-1743-5978-1-4419-7837-0NaberTGregory L. Naber, Drexel University Department of Mathematics, Philadelphia, PA, USA#The Geometry of Minkowski SpacetimeFAn Introduction to the Mathematics of the Special Theory of RelativityXVI, 324p. 67 illus..SCP190704Classical and Quantum Gravitation, Relativity TheoryPHDVL This book offers a presentation of the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. It treats, in addition to the usual menu of topics one is accustomed to finding in introductions to special relativity, a wide variety of results of more contemporary origin. These include Zeeman s characterization of the causal automorphisms of Minkowski spacetime, the Penrose theorem on the apparent shape of a relativistically moving sphere, a detailed introduction to the theory of spinors, a Petrov-type classification of electromagnetic fields in both tensor and spinor form, a topology for Minkowski spacetime whose homeomorphism group is essentially the Lorentz group, and a careful discussion of Dirac s famous Scissors Problem and its relation to the notion of a two-valued representation of the Lorentz group. This second edition includes a new chapter on the de Sitter universe which is intended to serve two purposes. The first is to provide a gentle prologue to the steps one must take to move beyond special relativity and adapt to the presence of gravitational fields that cannot be considered negligible. The second is to understand some of the basic features of a model of the empty universe that differs markedly from Minkowski spacetime, but may be recommended by recent astronomical observations suggesting that the expansion of our own universe is accelerating rather than slowing down. The treatment presumes only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and, in two appendices, some elementary point-set topology. The first edition of the book received the 1993 CHOICE award for Outstanding Academic Title.Reviews of first edition: & a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics. (American Mathematical Society, 1993) Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations. (CHOICE, 1993) & his talent in choosing the most significant results and ordering them within the book can t be denied. The reading of the book is, really, a pleasure. (Dutch Mathematical Society, 1993)3Mathematically rigorous treatment of special relativity with precise statement of the physical interpretation

Detailed introduction to the the theory of spinors in Minkowski spacetime

Thorough treatments of numerous topics not generally discussed at the introductory level

978-1-4939-0241-5XVI, 324 p. 67 illus.978-1-4419-7253-8#Topology, Geometry and Gauge fieldsFoundations XX, 437 p.Contents: Preface.- Physical and geometrical motivation 1 Topological spaces.- Homotopy groups.- Principal bundles.- Differentiable manifolds and matrix Lie groups.- Gauge fields and Instantons. Appendix. References. Index.This is a book on topology and geometry and, like any books on subjects as vast as these, it has a point-of-view that guided the selection of topics. Naber takes the view that the rekindled interest that mathematics and physics have shown in each other of late should be fostered and that this is best accomplished by allowing them to cohabit. The book weaves together rudimentary notions from the classical gauge theory of physics with the topological and geometrical concepts that became the mathematical models of these notions. We ask the reader to come to us with some vague notion of what an electromagnetic field might be, a willingness to accept a few of the more elementary pronouncements of quantum mechanics, a solid background in real analysis and linear algebra and some of the vocabulary of modern algebra. To such a reader we offer an excursion that begins with the defi< nition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2) connections on S4 with instanton number -1. Iwould go over both volumes thoroughly and make some minor changes in terminology and notation and correct any errors I find. In this new edition, a chapter on Singular Homology will be added as well as minor changes in notation and terminology throughout and some sections have been rewritten or omitted. Reviews of First Edition: It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics...Naber combines a knowledge of his subject with an excellent informal writing style.' (NZMS Newletter) '...this book should be very interesting for mathematicians and physicists (as well as other scientists) who are concerned with gauge theories.' (Zentralblatt Fuer Mathematik)Detailed calculations of a number of concrete examples Written for both mathematicians who want to see something of the applications of topology and geometry to modern physics Written for physicists who want to see the foundations of their subject treated with mathematical rigor978-1-4419-7894-3InteractionsXII, 420 p.Preface.- Acknowledgements.- Geometrical Background.- Physical Motivation.- Frame Bundles and Spacetime.- Differential Forms and Integration Introduction.- de Rham Cohomology Introduction.- Characteristic Classes.- Appendix.- References.- Symbols.- Index.A study of topology and geometry, beginning with a comprehensible account of the extraordinary and rather mysterious impact of mathematical physics, and especially gauge theory, on the study of the geometry and topology of manifolds. The focus of the book is the Yang-Mills-Higgs field and some considerable effort is expended to make clear its origin and significance in physics. Much of the mathematics developed here to study these fields is standard, but the treatment always keeps one eye on the physics and sacrifices generality in favor of clarity. This second edition has replaced a brief appendix in the first on the Seiberg-Witten equations with a much more detailed survey of Donaldson-Witten Theory and the Witten Conjecture regarding the relationship between Donaldson and Seiberg-Witten invariants.A large number of exercises are included to encourage active participation on the part of the reader.Review from first edition:'It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics...Naber combines a knowledge of his subject with an excellent informal writing style.'SIAM REVIEWDetailed survey of Donaldson-Witten Theory and the Witten Conjecture

Chapter and section summaries

Detailed illustrations

Exercises at the end of chapters

978-1-4614-2682-0978-1-4614-2838-1978-0-8176-4692-9NapierTerrence Napier, Lehigh University, Bethlehem, PA, USA; Mohan Ramachandran, State University New York at Buffalo, Buffalo, NY, USA#An Introduction to Riemann SurfacesXVII, 560p. 42 illus..ePreface.- Introduction.- Complex analysis in C.- Riemann Surfaces and the L2 \delta-Method for Scalar-Valued Forms.- The L2 \delta-Method in a Holomorphic Line Bundle.- Compact Riemann Surfaces.- Uniformization and Embedding ofRiemann Surfaces.-Holomorphic Structures on Topological Surfaces.- Background Material on Analysis in Rnand Hilbert Space Theory.- Background Material on Linear Algebra.- Background Material on Manifolds.- Background Material on Fundamental Groups, Covering Spaces, and (Co)homology.- Background Material on Sobolev Spaces and Regularity.- References.- Notation Index.- Subject Index.This textbook presents a unified approach to compact and noncompact Riemann surfaces from the point of view of the so-called L2 $\bar{\delta}$-method. This method is a powerful technique from the theory of several complex variables, and provides for a unique approach to the fundamentally different characteristics of compact and noncompact Riemann surfaces.The inclusion of continuing exercises running throughout the book, which lead to generalizations of the main theorems, as well as the exercises included in each chapter make this text ideal for a one- or two-semester graduate course. The prerequisites are a working knowledge of standard topics in graduate level real and complex analysis, and some familiarity of manifolds and differential forms..<p>Presents a unified and competitive approach to compact and noncompact Riemann surfaces</p><p>Includes continuing exercises that run throughout the book and lead to generalizations of the main theorems</p><p>Will help expand and reinforce a student s knowledge of analysis, geometry, and topology</p>978-0-8176-4164-1 Narasimhan{Raghavan Narasimhan, Chicago, IL, USA; Yves Nievergelt, Eastern Washington University Dept. of Mathematics, Cheney, WA, USA Complex Analysis in One VariableXIV, 381 p.EI Complex Analysis in One Variable.- 1 Elementary Theory of Holomorphic Functions.- 2 Covering Spaces and the Monodromy Theorem.- 3 The Winding Number and the Residue Theorem.- 4 Picard s Theorem.- 5 Inhomogeneous Cauchy-Riemann Equation and Runge s Theorem.- 6 Applications of Runge s Theorem.- 7 Riemann Mapping Theorem and Simple Connectedness in the Plane.- 8 Functions of Several Complex Variables.- 9 Compact Riemann Surfaces.- 10 The Corona Theorem.- 11 Subharmonic Functions and the Dirichlet Problem.- II Exercises.- 0 Review of Complex Numbers.- 1 Elementary Theory of Holomorphic Functions.- 2 Covering Spaces and the Monodromy Theorem.- 3 The Winding Number and the Residue Theorem.- 4 Picard s Theorem.- 5 The Inhomogeneous Cauchy Riemann Equation and Runge s Theorem.- 6 Applications of Runge s Theorem.- 7 The Riemann Mapping Theorem and Simple Connectedness in the Plane.- 8 Functions of Several Complex Variables.- 9 Compact Riemann Surfaces.- 10 The Corona Theorem.- 11 Subharmonic Functions and the Dirichlet Pro< blem.- Notes for the exercises.- References for the exercises.This book presents complex analysis in one variable in the context of modern mathematics, with clear connections to several complex variables, de Rham theory, real analysis, and other branches of mathematics. Thus, covering spaces are used explicitly in dealing with Cauchy's theorem, real variable methods are illustrated in the Loman-Menchoff theorem and in the corona theorem, and the algebraic structure of the ring of holomorphic functions is studied. Using the unique position of complex analysis, a field drawing on many disciplines, the book also illustrates powerful mathematical ideas and tools, and requires minimal background material. Cohomological methods are introduced, both in connection with the existence of primitives and in the study of meromorphic functionas on a compact Riemann surface. The proof of Picard's theorem given here illustrates the strong restrictions on holomorphic mappings imposed by curvature conditions. 978-0-387-94655-9 NathansonOMelvyn B. Nathanson, City University of New York Lehman College, Bronx, NY, USADAdditive Number Theory: Inverse Problems and the Geometry of SumsetsXIV, 293 pp.AMany classical problems in additive number theory are direct problems, in which one starts with a set A of natural numbers and an integer H -> 2, and tries to describe the structure of the sumset hA consisting of all sums of h elements of A. By contrast, in an inverse problem, one starts with a sumset hA, and attempts to describe the structure of the underlying set A. In recent years there has been ramrkable progress in the study of inverse problems for finite sets of integers. In particular, there are important and beautiful inverse theorems due to Freiman, Kneser, Plnnecke, Vosper, and others. This volume includes their results, and culminates with an elegant proof by Ruzsa of the deep theorem of Freiman that a finite set of integers with a small sumset must be a large subset of an n-dimensional arithmetic progression.978-1-4614-9754-7XIV, 296 p.978-0-387-95155-3NedelecJean-Claude Nedelec&Acoustic and Electromagnetic Equations.Integral Representations for Harmonic Problems IX, 318 p.1 Some Wave Equations.- 2 The Helmholtz Equation.- 3 Integral Representations and Integral Equations.- 4 Singular Integral Operators.- 5 Maxwell Equations and Electromagnetic Waves.- References.:This self-contained book is devoted to the study of the acoustic wave equations and the Maxwell system, the two most common waves equations that are encountered in physics or engineering. It presents a detailed analysis of their mathematical and physical properties. In particular the author focuses on the study of the harmonic exterior problems, building a mathematical framework which provides the existence and uniqueness of the solutions. This book will serve as a useful introduction to wave problems for graduate students in mathematics, physics, and engineering.978-1-4419-2889-4978-3-642-27874-7NeaetYilJaroslav NeaetYil, Charles University Department of Applied Mathematics, Prague, Czech Republic; Patrice Ossona de Mendez, CAMS - CNRS UMR 8557 cole des Hautes tudes, Paris, FranceSparsity"Graphs, Structures, and Algorithms-XXIII, 457p. 132 illus., 100 illus. in color.SCI17028(Discrete Mathematics in Computer SciencePart I Presentation: 1. Introduction.- 2. A Few Problems.- 3. Commented Contents.- Part II. The Theory: 4. Prolegomena.- 5. Measuring Sparsity.- 6. Classes and their Classification.- 7. Bounded Height Trees and Tree-Depth.- 8. Decomposition.- 9. Independence.- 10. First-Order Constraint Satisfaction Problems and Homomorphism Dualities.- 11. Restricted Homomorphism Dualities.- 12. Counting.- 13. Back to Classes.- Part III Applications: 14. Classes with Bounded Expansion Examples.- 15. Property Testing, Hyperfiniteness and Separators.- 16. Algorithmic Applications.- 17. Other Applications.- 18. Conclusion.- Bibliography.- Index.- List of Symbols .BThis is the first book devoted to the systematic study of sparse graphs and sparse finite structures. Although the notion of sparsity appears in various contexts and is a typical example of a hard to define notion, the authors devised an unifying classification of general classes of structures. This approach is very robust and it has many remarkable properties. For example the classification is expressible in many different ways involving most extremal combinatorial invariants.This study of sparse structures found applications in such diverse areas as algorithmic graph theory, complexity of algorithms, property testing, descriptive complexity and mathematical logic (homomorphism preservation,fixed parameter tractability and constraint satisfaction problems). It should be stressed that despite of its generality this approach leads to linear (and nearly linear) algorithms. Jaroslav NeaetYil is a professor at Charles University, Prague; Patrice Ossona de Mendez is a CNRS researcher et EHESS, Paris.This book is related to the material presented by the first author at ICM 2010.One of the first textbooks on the topic of sparsity - a core area of current discrete mathematics

Extremely useful to both mathematicians and computer scientists

Pedagogically excellent

978-3-642-42776-3.XXIII, 457 p. 132 illus., 100 illus. in color.978-1-4020-7553-7NesterovbYurii Nesterov, Universit Catholique de Louvain Center for Operations Reserach &, Leuven, Belgium,Introductory Lectures on Convex OptimizationA Basic CourseApplied Optimization XVIII, 236 p.This is the first elementary exposition of the main ideas of complexity theory for convex optimization. Up to now, most of the material can be found only in special journals and research monographs. The book covers optimal methods and lower complexity bounds for smooth and non-smooth convex optimization. A separate chapter is devoted to polynomial-time interior-point methods. Audience: The book is suitable for industrial engineers and economists.978-0-387-95543-8NestruevJet Nestruev Smooth Manifolds and ObservablesXIV, 225 p.TCutoff and Other Special Smooth Functions on ?n.- Algebras and Points.- Smooth Man< ifolds (Algebraic Definition).- Charts and Atlases.- Smooth Maps.- Equivalence of Coordinate and Algebraic Definitions.- Spectra and Ghosts.- The Differential Calculus as a Part of Commutative Algebra.- Smooth Bundles.- Vector Bundles and Projective Modules.This book is a self-contained introduction to fiber spaces and differential operators on smooth manifolds that is accessible to graduate students specializing in mathematics and physics. Over the last 20 years the authors developed an algebraic approach to the subject and they explain in this book why differential calculus on manifolds can be considered as an aspect of commutative algebra. This new approach is based on the fundamential notion of 'observable' which is used by physicists and it will further the understanding of the mathematics underlying quantum field theory. The prerequisites for this book are a standard advanced calculus course as well as courses in linear algebra and algebraic structures.3Completely new approach to the subject978-1-4419-3047-7978-1-4614-1104-8 NicolaescuQLiviu Nicolaescu, University of Notre Dame Dept. Mathematics, Notre Dame, IN, USAAn Invitation to Morse TheoryXVI, 353p. 47 illus..Preface.- Notations and Conventions.- 1 Morse Functions.- 2 The Topology of Morse Functions.- 3 Applications.- 4 Morse-Smale Flows and Whitney Stratifications.- 5 Basics of Complex Morse Theory.- 6 Exercises and Solutions.- References.- IndexThis self-contained treatment of Morse theory focuses on applications and is intended for a graduate course on differential or algebraic topology. The book is divided into three conceptually distinct parts. The first part contains the foundations of Morse theory. The second part consists of applications of Morse theory over the reals, while the last part describes the basics and some applications of complex Morse theory, a.k.a. Picard-Lefschetz theory. This is the first textbook to include topics such as Morse-Smale flows, Floer homology, min-max theory, moment maps and equivariant cohomology, and complex Morse theory. The exposition is enhanced with examples, problems, and illustrations, and will be of interest to graduate students as well as researchers. The reader is expected to have some familiarity with cohomology theory and with the differential and integral calculus on smooth manifolds. Some features of the second edition include added applications, such as Morse theory and the curvature of knots, the cohomology of the moduli space of planar polygons, and the Duistermaat-Heckman formula. The second edition also includes a new chapter on Morse-Smale flows and Whitney stratifications, many new exercises, and various corrections from the first edition.

New edition extensively revised and updated with three new applications and a new chapter on Morse-Smale flows and Whitney stratifications

Provides a useful introduction to Morse Theory

Covers many of the most important topics in Morse theory along with a broad range of applications

Contains manyexcellent exercises

Far more up-to-date and less specialized than any other book on Morse Theory

978-3-211-75539-6NierhausMGerhard Nierhaus, Universitt fr Musik und darstellende Kunst, Graz, AustriaAlgorithmic Composition'Paradigms of Automated Music GenerationX, 287 p. With 20 tables.SCI23036%Computer Appl. in Arts and HumanitiesHHistorical Development of Algorithmic Procedures.- Markov Models.- Generative Grammars.- Transition Networks.- Chaos and Self-Similarity.- Genetic Algorithms.- Cellular Automata.- Artificial Neural Networks.- Artificial Intelligence.- Final Synopsis.Algorithmic composition composing by means of formalizable methods has a century old tradition not only in occidental music history. This is the first book to provide a detailed overview of prominent procedures of algorithmic composition in a pragmatic way rather than by treating formalizable aspects in single works. In addition to an historic overview, each chapter presents a specific class of algorithm in a compositional context by providing a general introduction to its development and theoretical basis and describes different musical applications. Each chapter outlines the strengths, weaknesses and possible aesthetical implications resulting from the application of the treated approaches. Topics covered are: markov models, generative grammars, transition networks, chaos and self-similarity, genetic algorithms, cellular automata, neural networks and artificial intelligence are covered. The comprehensive bibliography makes this work ideal for the musician and the researcher alike. 978-3-211-99915-8 X, 287 p.978-0-387-98676-0OkuboAkira Okubo; Smon A. Levin6Diffusion and Ecological Problems: Modern PerspectivesXXI, 469 p.SCL19147Theoretical Ecology/StatisticsPSAF1. Introduction: The Mathematics of Ecological Diffusion.- 2. The Basics of Diffusion.- 3. Passive Diffusion in Ecosystems.- 4. Diffusion of Smell and Taste : Chemical Communication.- 5. Mathematical Treatment of Biological Diffusion.- 6. Some Examples of Animal Diffusion.- 7. The Dynamics of Animal Grouping.- 8. Animal Movements in Home Range.- 9. Patchy Distribution and Diffusion.- 10. Population Dynamics in Temporal and Spatial Domains.- References.- Author Index.This book surveys a wide variety of mathematical models of diffusion in the ecological context. It is written with the primary intent of providing scientists, particularly physicists but also biologists, with some background of the mathematics and physics of diffusion and how they can be applied to ecological problems. The secondary intent is to provide a specialized text book for graduate students, who are interested in mathematical ecology. The reader is assumed to have a basic knowledge of probability and differential equations. Each chapter in this new edition is substantially updated by appopriate leading researchers in the field and contains much new material covering developments in the field in the last 20 years.b- 2nd edition of a classic title- developments in mathematical ecology since 1980 included

978-1-4419-3151-1978-3-642-16028-8OlivierimAnnamaria Olivieri, Universit di Parma, Parma, Italy; Ermanno Pitacco, Universit di Trieste, Trieste, Italy%Introduction to Insurance Mathematics2Technical and Financial Features of Risk TransfersXV, 475p. 180 illus..SCW23032Finance/Investment/BankingKFFvRisks and insurance.- Managing a portfolio of risks.- Life insurance: modelling the lifetime.- Life insurance: pricing.- Life insurance: reserving.- Reserves and profits in a life insurance portfo< lio.- Finance in life insurance: linking benefits to the investment performance.- Pension plans: technical and financial perspectives.- Non-life insurance: pricing and reserving.JThe book aims at presenting technical and financial features of life insurance, non-life insurance, pension plans. The book has been planned assuming non-actuarial readers as its natural target, namely - advanced undergraduate and graduate students in Economics, Business and Finance; - professionals and technicians operating in Insurance and pension areas, whose job may regard investments, risk analysis, financial reporting, etc, and hence implies a communication with actuarial professionals and managers. Given the assumed target, the book focuses on technical and financial aspects of insurance, however avoiding the use of complex mathematical tools. In this sense, the book can be placed at some midpoint of the existing literature, part of which adopts more formal approaches to insurance problems implying the use of non-elementary mathematics, whereas another part addresses practical questions totally avoiding even simple mathematical tools (which, in our opinion, can conversely provide effective tools for presenting technical and financial features of the insurance business).<p>Wide range of topics dealt with (life insurance, non-life insurance, and pensions included) <br>Careful attention to practical issues <br>A less mathematical and formal perspective, if compared to other textbooks</p>978-1-4419-6359-8O'ReillyfOliver M. O'Reilly, University of California, Berkeley Dept. Mechanical Engineering, Berkeley, CA, USAEngineering DynamicsA PrimerX, 240p.SCT1501X.Continuum Mechanics and Mechanics of MaterialsTGYDynamics of a Single Particle.- Elementary Particle Dynamics.- Particles and Cylindrical Polar Coordinates.- Particles and Space Curves.- Friction Forces and Spring Forces.- Power, Work, and Energy.- Dynamics of a System of Particles.- Momenta, Impulses, and Collisions.- Dynamics of Systems of Particles.- Dynamics of a Single Rigid Bodies.- Planar Kinematics of Rigid Bodies.- Kinetics of a Rigid Body.- Dynamics of Systems of Particles and Rigid Bodies.- Systems of Particles and Rigid Bodies.- Appendices.- Preliminaries on Vectors and Calculus.- Weekly Course Content and Notation in Other Texts.This primer is intended to provide the theoretical background for the standard undergraduate, mechanical engineering course in dynamics. It grew out of the author's desire to provide an affordable compliment to the standard texts on the subject in which the gap between the theory presented and the problems to be solved is oftentimes too large. The book contains several worked examples and at the end of each chapters summaries and exercises to aid the student in their understanding of the chapter. Teachers who wish to have a source of more detailed theory for the course, as well as graduate students who need a refresher course on undergraduate dynamics when preparing for certain first year graduate school examinations, and students taking the course will find the work very helpful.rMany useful and important examples are provided Provides a good introduction to the subject Comprehensive in scope978-0-387-98934-1OsborneM. Scott OsborneBasic Homological Algebra X, 398 p.1 Categories.- 2 Modules.- 2.1 Generalities.- 2.2 Tensor Products.- 2.3 Exactness of Functors.- 2.4 Projectives, Injectives, and Flats.- 3 Ext and Tor.- 3.1 Complexes and Projective Resolutions.- 3.2 Long Exact Sequences.- 3.3 Flat Resolutions and Injective Resolutions.- 3.4 Consequences.- 4 Dimension Theory.- 4.1 Dimension Shifting.- 4.2 When Flats are Projective.- 4.3 Dimension Zero.- 4.4 An Example.- 5 Change of Rings.- 5.1 Computational Considerations.- 5.2 Matrix Rings.- 5.3 Polynomials.- 5.4 Quotients and Localization.- 6 Derived Functors.- 6.1 Additive Functors.- 6.2 Derived Functors.- 6.3 Long Exact Sequences I. Existence.- 6.4 Long Exact Sequences II. Naturality.- 6.5 Long Exact Sequences III. Weirdness.- 6.6 Universality of Ext.- 7 Abstract Homologieal Algebra.- 7.1 Living Without Elements.- 7.2 Additive Categories.- 7.3 Kernels and Cokernels.- 7.4 Cheating with Projectives.- 7.5 (Interlude) Arrow Categories.- 7.6 Homology in Abelian Categories.- 7.7 Long Exact Sequences.- 7.8 An Alternative for Unbalanced Categories.- 8 Colimits and Tor.- 8.1 Limits and Colimits.- 8.2 Adjoint Functors.- 8.3 Directed Colimits, ?, and Tor.- 8.4 Lazard s Theorem.- 8.5 Weak Dimension Revisited.- 9 Odds and Ends.- 9.1 Injective Envelopes.- 9.2 Universal Coefficients.- 9.3 The Knneth Theorems.- 9.4 Do Connecting Homomorphisms Commute?.- 9.5 The Ext Product.- 9.6 The Jacobson Radical, Nakayama s Lemma, and Quasilocal Rings.- 9.7 Local Rings and Localization Revisited (Expository).- A GCDs, LCMs, PIDs, and UFDs.- B The Ring of Entire Functions.- C The Mitchell Freyd Theorem and Cheating in Abelian Categories.- D Noether Correspondences in Abelian Categories.- Solution Outlines.- References.- Symbol Index.<This book is intended for one-quarter or one semester-courses in homological algebra. The aim is to cov< er Ext and Tor early and without distraction. It includes several further topics, which can be pursued independently of each other. Many of these, such as Lazard's theorem, long exact sequences in Abelian categories with no cheating, or the relation between Krull dimension and global dimension, are hard to find elsewhere. The intended audience is second or third year graduate students in algebra, algebraic topology, or any other field that uses homological algebra.978-0-387-95482-0OsherStanley Osher; Ronald Fedkiw/Level Set Methods and Dynamic Implicit Surfaces%XIII, 273 p. 109 illus., 24 in color.SCP2100XClassical Continuum Physics;Implicit Surfaces.- Implicit Functions.- Signed Distance Functions.- Level Set Methods.- Motion in an Externally Generated Velocity Field.- Motion Involving Mean Curvature.- Hamilton-Jacobi Equations.- Motion in the Normal Direction.- Constructing Signed Distance Functions.- Extrapolation in the Normal Direction.- Particle Level Set Method.- Codimension-Two Objects.- Image Processing and Computer Vision.- Image Restoration.- Snakes, Active Contours, and Segmentation.- Reconstruction of Surfaces from Unorganized Data Points.- Computational Physics.- Hyperbolic Conservation Laws and Compressible Flow.- Two-Phase Compressible Flow.- Shocks, Detonations, and Deflagrations.- Solid-Fluid Coupling.- Incompressible Flow.- Free Surfaces.- Liquid-Gas Interactions.- Two-Phase Incompressible Flow.- Low-Speed Flames.- Heat Flow.The goal of this book is to promote the use of level set methods by the many scientists and engineers working on moving interface problems. The authors provide some motivational/intuitive background on the scope and variety of moving interface problems and their mathematical formulation. They also review the pros and cons of traditional numerical techniques. The bulk of the book addresses the foundation of essential mathematical and numerical methods necessary for applying the level set method, with particular emphasis on problems in which the interface is just one part of a more complicated physical system. The book concludes with a few select example applications drawn from the authors' research. The book is intended for students and researchers interested in computation in the physical sciences, i.e. engineers, computational fluid dynamicists and applied mathematicians who want to use these methods for their own computations. It is suitable for use in a graduate level course on numerical methods, as a users guide to applying the methods, and as a general reference for mathematics related to level sets and the numerical solution of equations of ''Hamiltonian-Jacobi'' type.978-1-4614-0796-6Ovchinnikov\Sergei Ovchinnikov, San Francisco State University Dept. Mathematics, San Francisco, CA, USAGraphs and CubesXIII, 287p. 172 illus..SCM29020Graph TheoryPreface.- 1 Graphs.- 2 Bipartite Graphs.- 3 Cubes.- 4 Cubical Graphs.- 5 Partial Cubes.- 6 Lattice Embeddings.- 7 Hyperplane Arrangements.- 8 Token Systems.- Notation.- References.- Index.This introductory text in graph theory focuses on partial cubes, which are graphs that are isometrically embeddable into hypercubes of an arbitrary dimension, as well as bipartite graphs, and cubical graphs. This branch of graph theory has developed rapidly during the past three decades, producing exciting results and establishing links to other branches of mathematics. Currently, Graphs and Cubes is the only book available on the market that presents a comprehensive coverage of cubical graph and partial cube theories. Many exercises, along with historical notes, are included at the end of every chapter, and readers are encouraged to explore the exercises fully, and use them as a basis for research projects. The prerequisites for this text include familiarity with basic mathematical concepts and methods on the level of undergraduate courses in discrete mathematics, linear algebra, group theory, and topology of Euclidean spaces. While the book is intended for lower-division graduate students in mathematics, it will be of interest to a much wider audience; because of their rich structural properties, partial cubes appear in theoretical computer science, coding theory, genetics, and even the political and social sciences.C

This is the only book available on the market that presents a comprehensive coverage of cubical graph and partial cube theories

Aimed towards lower-division graduate students in mathematics, but could be of interest to researchers using cubical graph and partial cube theories in mathematics and applications

Has applications in theoretical computer science, coding theory, data transmission, genetics, and even the political and social sciences

With its numerous illustrations and exercises, this presentation of partial cubes theory is student-friendly

978-1-4614-5057-3PachlHJan Pachl, The Fields Institute for Research in Mat, Toronto, ON, CanadaUniform Spaces and Measures IX, 209 p.Prerequisites.- 1. Uniformities and Topologies.- 2. Induced Uniform Structures.- 3. Uniform Structures on Semigroups.- 4. Some Notable Classes of Uniform Spaces.- 5. Measures on Complete Metric Spaces.- 6. Uniform Measures.- 7. Uniform Measures as Measures.- 8. Instances of Uniform Measures.- 9. Direct Product and Convolution.- 10. Free Uniform Measures.- 11. Approximation of Probability Distributions.- 12. Measurable Functionals.- Hints to Excercises.- References.- Notation Index.- Author Index.- Subject Index. This book addresses the need for an accessible comprehensive exposition of the theory of uniform measures; the need that became more critical when recently uniform measures reemerged in new results in abstract harmonic analysis. Until now, results about uniform measures have been scattered through many papers written by a number of authors, some unpublished, written using a variety of definitions and notations. Uniform measures are certain functionals on the space of bounded uniformly continuous functions on a uniform space. They are a common generalization of several classes of measures and measure-like functionals studied in abstract and topological measure theory, probability theory, and abstract harmonic analysis. They offer a natural framework for results about topologies on spaces of measures and about the continuity of convolution of measures on topological groups and semitopological semigroups. The book is a reference for the theory of uniform measures. It includes a self-contained development of the theory with complete proofs, starting with the necessary parts of the theory of uniform spaces. It presents diverse results from many sources organized in a logical whole, and includes several new results. The book is also suitable for graduate or advanced undergraduate courses on selected topics in< topology and functional analysis. The text contains a number of exercises with solution hints, and four problems with suggestions for further research. <p> Presents a self-contained development of selected topics in the theory of uniform spaces using pseudometrics rather than the more common approach via entourages </p><p>Contains exercises and research problems and can be used as supplementary text in graduate and advanced undergraduate courses </p><p>Details the history of core concepts and links to key references to help the reader understand connections to related areas and explore other sources </p>978-1-4899-9258-1978-3-642-29074-9Palm-Gnther Palm, University of Ulm, Ulm, Germany!Novelty, Information and SurpriseXXIII, 248 p. 28 illus.SCI21017(Artificial Intelligence (incl. Robotics)Part I Surprise and Information of Descriptions: Prerequisites.- Improbability and Novelty of Descriptions.- Conditional Novelty and Information.- Part II Coding and Information Transmission: On Guessing and Coding.- Information Transmission.- Part III Information Rate and Channel Capacity: Stationary Processes and Information Rate.- Channel Capacity.- Shannon's Theorem.- Part IV Repertoires and Covers: Repertoires and Descriptions.- Novelty, Information and Surprise of Repertoires.- Conditioning, Mutual Information and Information Gain.- Part V Information, Novelty and Surprise in Science: Information, Novelty and Surprise in Brain Theory.- Surprise from Repetitions and Combination of Surprises.- Entropy in Physics.- Part VI Generalized Information Theory: Order- and Lattice-Structures.- Three Orderings on Repertoires.- Information Theory on Lattices of Covers.- Appendices: A. Fuzzy Repertoires and Descriptions.- A.1 Basic Definitions.- A.2 Definition and Properties of Fuzzy Repertoires.- Glossary.- Bibliography.- Index.The book offers a new approach to information theory that is more general then the classical approach by Shannon. The classical definition of information is given for an alphabet of symbols or for a set of mutually exclusive propositions (a partition of the probability space ) with corresponding probabilities adding up to 1. The new definition is given for an arbitrary cover of , i.e. for a set of possibly overlapping propositions. The generalized information concept is called novelty and it is accompanied by two new concepts derived from it, designated as information and surprise, which describe 'opposite' versions of novelty, information being related more to classical information theory and surprise being related more to the classical concept of statistical significance. In the discussion of these three concepts and their interrelations several properties or classes of covers are defined, which turn out to be lattices. The book also presents applications of these new concepts, mostly in statistics and in neuroscience.<p> Definition of useful new concepts: description, novelty, surprise, template </p><p>New viewpoint on information theory in relation to (natural) science </p><p>New method to analyse neuronal spike trains (burst surprise)</p>978-3-642-43583-6978-3-0348-0565-0 ParthasarathyEK.R. Parthasarathy, Indian National Science Academy, New Delhi, India.An Introduction to Quantum Stochastic Calculus XI, 290 p.Preface.- I Events, Observables and States.- II Observables and States in Tensor Products of Hilbert Spaces.- III Stochastic Integration and Quantum Ito s Formula.- References.- Index.- Author Index. $An Introduction to Quantum Stochastic Calculus aims to deepen our understanding of the dynamics of systems subject to the laws of chance both from the classical and the quantum points of view and stimulate further research in their unification. This is probably the first systematic attempt to weave classical probability theory into the quantum framework and provides a wealth of interesting features: The origin of Ito s correction formulae for Brownian motion and the Poisson process can be traced to commutation relations or, equivalently, the uncertainty principle. Quantum stochastic integration enables the possibility of seeing new relationships between fermion and boson fields. Many quantum dynamical semigroups as well as classical Markov semigroups are realised through unitary operator evolutions. The text is almost self-contained and requires only an elementary knowledge of operator theory and probability theory at the graduate level. - - - This is an excellent volume which will be a valuable companion bothto those who are already active in the field and those who are new to it. Furthermore there are a large number of stimulating exercises scattered through the text which will be invaluable to students. (Mathematical Reviews) This monograph gives a systematic and self-< contained introduction to the Fock space quantum stochastic calculus in its basic form (...) by making emphasis on the mathematical aspects of quantum formalism and its connections with classical probability and by extensive presentation of carefully selected functional analytic material. This makes the book very convenient for a reader with the probability-theoretic orientation, wishing to make acquaintance with wonders of the noncommutative probability, and, more specifcally, for a mathematics student studying this field. (Zentralblatt MATH) Elegantly written, with obvious appreciation for fine points of higher mathematics (...) most notable is [the] author's effort to weave classical probability theory into [a] quantum framework. (The American Mathematical Monthly)$<p>One of the first systematic attempts to weave classical probability theory into the quantum framework </p><p>Self-contained introduction to the Fock space quantum stochastic calculus in its basic form </p><p>Alarge number of stimulating exercisesmake the text invaluable to students </p>978-0-387-40398-4PedregalPablo PedregalIntroduction to OptimizationX, 245 p. 41 illus.Linear Programming.- Nonlinear Programming.- Approximation Techniques.- Variational Problems and Dynamic Programming.- Optimal Control.This undergraduate textbook introduces students of science and engineering to the fascinating field of optimization. It is a unique book that brings together the subfields of mathematical programming, variational calculus, and optimal control, thus giving students an overall view of all aspects of optimization in a single reference. As a primer on optimization, its main goal is to provide a succinct and accessible introduction to linear programming, nonlinear programming, numerical optimization algorithms, variational problems, dynamic programming, and optimal control. Prerequisites have been kept to a minimum, although a basic knowledge of calculus, linear algebra, and differential equations is assumed. There are numerous examples, illustrations, and exercises throughout the text, making it an ideal book for self-study. Applied mathematicians, physicists, engineers, and scientists will all find this introduction to optimization extremely useful.978-1-4419-2334-9978-1-4614-4537-1PenotZJean-Paul Penot, Universit de Pau et des Pays de l'Adour Btiment IPRA, PAU CEDEX, FranceCalculus Without Derivatives XX, 524 p.9Preface.- 1 Metric and Topological Tools.- 2 Elements of Differential Calculus.- 3 Elements of Convex Analysis.- 4 Elementary and Viscosity Subdifferentials.- 5 Circa-Subdifferentials, Clarke Subdifferentials.- 6 Limiting Subdifferentials.- 7 Graded Subdifferentials, Ioffe Subdifferentials.- References.- Index .Calculus Without Derivatives expounds the foundations and recent advances in nonsmooth analysis, a powerful compound of mathematical tools that obviates the usual smoothness assumptions. This textbook also provides significant tools and methods towards applications, in particular optimization problems. Whereas most books on this subject focus on a particular theory, this text takes a general approach including all main theories. In order to be self-contained, the book includes three chapters of preliminary material, each of which can be used as anindependent course if needed. The first chapter deals with metric properties, variational principles, decrease principles, methods of error bounds, calmness and metric regularity. The second one presents the classical tools of differential calculus and includes a section about the calculus of variations. The third contains a clearexposition of convex analysis.Includes all necessary preliminary material

Introduces fundamental aspects of nonsmooth analysis that impact many applications

Presents a balanced picture of the most elementary attempts to replace a derivative with a one-sided generalized derivative called a subdifferential

Includes references, notes, exercises and supplements that will give the reader a thorough insight into the subject

978-1-4899-8942-0978-0-8176-8363-4PisanskiTomaz Pisanski, IMFM Oddelek za Teoreti no Racunalniatvo, Ljubljana, Slovenia; Brigitte Servatius, Mathematics Department Worcester Polytechnic Institute, Worcester, MA, USA)Configurations from a Graphical Viewpoint,XIII, 279 p. 274 illus., 45 illus. in color.Preface.- Introduction.- Graphs.- Groups, Actions, and Symmetry.- Maps.- Combinatorial Configurations.- Geometric Configurations.- Index.- Bibliography.Configurations can be studied from a graph-theoretical viewpoint via the so-called Levi graphs and lie at the heart of graphs, groups, surfaces, and geometries, all of which are very active areas of mathematical exploration. In this self-contained textbook, algebraic graph theory is used to introduce groups; topological graph theory is used to explore surfaces; and geometric graph theory is implemented to analyze incidence geometries.After a preview of configurations in Chapter 1, a concise introduction to graph theory is presented in Chapter 2, followed by a geometric introduction to groups in Chapter 3. Maps and surfaces are combinatorially treated in Chapter 4. Chapter 5 introduces the concept of incidence structure through vertex colored graphs, and the combinatorial aspects of classical configurations are studied.Geometric aspects, some historical remarks, references, and applicationsof classical configurationsappear in the last chapter.With over two hundred illustrations, challenging exercises at the end of each chapter, a comprehensive bibliography, and a set of open problems, Configurations from a Graphical Viewpoint is well suited for a graduate graph theory course, an advanced undergraduate seminar, or a self-contained reference for mathematicians and researchers.Among the oldest combinatorial structures, configurations are explored from a graphical viewpoint for the first time in this book

Explores configurations from a graph-theoretical viewpoint

Includes over two hundred illustrations, challenging exercises at the end of each chapter, a comprehensive bibliography, and a set of open problems

Well suited for a graduate graph theory course, advanced undergraduate seminar, or self-contained reference for mathematicians and researchers

978-1-4614-4480-0Plakhov9Alexander Plakhov, University of Aveiro, Aveiro, PortugalExterior Billiards,Systems with Impacts Outside Bounded Domains,XIII, 284 p. 108 illus., 61 illus. in color.k -Notation and synopsis of main results. -Problem of minimum resistance to translational motion of bodies. -Newton s problem in media with positive temperature. -Scattering in billiards. -Problems of optimal mass transportation. -Problems on optimization of mean resistance. -Magnus effect and dynamics of a rough disc. -On invisible bodies. Retroreflectors. A billiard is a dynamical system in which a point particle alternates between free motion and specular refle< ctions fromthe boundaryof a domain.Exterior Billiards presents billiards in the complement of domains and their applications in aerodynamics and geometrical optics. This book distinguishes itself from existing literature by presenting billiard dynamics outside bounded domains, including scattering, resistance, invisibility and retro-reflection. It begins with an overview of the mathematical notations used throughout the book and a brief review of the main results. Chapters 2 and 3 are focused on problems of minimal resistance and Newton s problem in media with positive temperature. In chapters 4 and 5, scattering of billiards bynonconvex and rough domains is characterized and some related special problems of optimal mass transportation are studied. Applications in aerodynamics are addressed next and problems of invisibility and retro-reflection within the framework of geometric optics conclude the text.The book will appeal to mathematicians working in dynamical systems and calculus of variations. Specialists working in the areas of applications discussed will also find it useful.<p> Examines properties of billiard dynamics outside bounded domains </p><p>Includes topics such as scattering, resistance, invisibility and retro-reflection </p><p>Useful for mathematicians working in dynamical systems</p>978-1-4899-9913-9978-3-642-23839-0PostLOlaf Post, Durham University Department of Mathematical Sciences, Durham, UK&Spectral Analysis on Graph-like SpacesXV, 431p. 28 illus..11 Introduction.- 2 Graphs and associated Laplacians.- 3 Scales of Hilbert space and boundary triples.- 4 Two operators in different Hilbert spaces.- 5 Manifolds, tubular neighbourhoods and their perturbations.- 6 Plumber s shop: Estimates for star graphs and related spaces.- 7 Global convergence results.Small-radius tubular structures have attracted considerable attention in the last few years, and are frequently used in different areas such as Mathematical Physics, Spectral Geometry and Global Analysis. In this monograph, we analyse Laplace-like operators on thin tubular structures ('graph-like spaces''), and their natural limits on metric graphs. In particular, we explore norm resolvent convergence, convergence of the spectra and resonances. Since the underlying spaces in the thin radius limit change, and become singular in the limit, we develop new tools such as norm convergence of operators acting in different Hilbert spaces, an extension of the concept of boundary triples to partial differential operators, and an abstract definition of resonances via boundary triples. These tools are formulated in an abstract framework, independent of the original problem of graph-like spaces, so that they can be applied in many other situations where the spaces are perturbed.RA thorough analysis of quantum graphs and their approximations (graph-like spaces)

A self-contained explanation of the tools needed (convergence of operators in different spaces, boundary triples)

The book is accessible for a graduate student with some knowledge in functional analysis and operators on Hilbert spaces

978-3-0348-0498-1PrssMJan Prss, Martin-Luther-Universitt Halle-Wittenberg, Halle (Saale), Germany0Evolutionary Integral Equations and ApplicationsXXVI, 366 p. 8 illus.+Preface.- Introduction.- Preliminaries.- I Equations of Scalar Type.- 1 Resolvents.- 2 Analytic Resolvents.- 3 Parabolic Equations.- 4 Subordination.- 5 Linear Viscoelasticity.- II Nonscalar Equations.- 6 Hyperbolic Equations of Nonscalar Type.- 7 Nonscalar Parabolic Equations.- 8 Parabolic Problems in Lp-Spaces.- 9 Viscoelasticity and Electrodynamics with Memory.- III Equations on the Line.- 10 Integrability of Resolvents.- 11 Limiting Equations.- 12 Admissibility of Function Spaces.- 13 Further Applications and Complements.- Bibliography.- Index. ZThis book deals with evolutionary systems whose equation of state can be formulated as a linear Volterra equation in a Banach space. The main feature of the kernels involved is that they consist of unbounded linear operators. The aim is a coherentpresentation of the state of art of the theory including detailed proofs and its applications to problems from mathematical physics, such as viscoelasticity, heat conduction, and electrodynamics with memory. The importance of evolutionary integral equations which form a larger class than do evolution equations stems from such applications and therefore special emphasis is placed on these. A number of models are derived and, by means of the developed theory, discussed thoroughly. An annotated bibliography containing 450 entries increases the book s value as an incisive reference text. --- This excellent book presents a general approach to linear evolutionary systems, with an emphasis on infinite-dimensional systems with time delays, such as those occurring in linear viscoelasticity with or without < thermal effects. It gives a very natural and mature extension of the usual semigroup approach to a more general class of infinite-dimensional evolutionary systems. This is the first appearance in the form of a monograph of this recently developed theory. A substantial part of the results are due to the author, or are even new. (& ) It is not a book that one reads in a few days. Rather, it should be considered as an investment with lasting value. (Zentralblatt MATH) In this book, the author, who has been at the forefront of research on these problems for the last decade, has collected, and in many places extended, the known theory for these equations. In addition, he has provided a framework that allows one to relate and evaluate diverse results in the literature. (Mathematical Reviews) This book constitutes a highly valuable addition to the existing literature on the theory of Volterra (evolutionary) integral equations and their applications in physics and engineering. (& ) and for the first time the stress is on the infinite-dimensional case. (SIAM Reviews)<p>Presents a general approach to linear evolutionary systems</p><p><p>Clearly written and of lasting value</p><p><p>Asubstantial part of the results presented originate from the author </p>978-3-7643-8144-8PucciPatrizia Pucci, Universit Perugia Dipto. Matematica, Perugia, Italy; J. B. Serrin, University of Minnesota School of Mathematics, Minneapolis, MN, USAThe Maximum Principle X, 234 p.and Preliminaries.- Tangency and Comparison Theorems for Elliptic Inequalities.- Maximum Principles for Divergence Structure Elliptic Differential Inequalities.- Boundary Value Problems for Nonlinear Ordinary Differential Equations.- The Strong Maximum Principle and the Compact Support Principle.- Non-homogeneous Divergence Structure Inequalities.- The Harnack Inequality.- Applications.Maximum principles are bedrock results in the theory of second order elliptic equations. This principle, simple enough in essence, lends itself to a quite remarkable number of subtle uses when combined appropriately with other notions. Intended for a wide audience, the book provides a clear and comprehensive explanation of the various maximum principles available in elliptic theory, from their beginning for linear equations to recent work on nonlinear and singular equations.MThe only book containing a detailed description of modern work on the maximum principle for nonlinear elliptic differential equations

Contains applications to the celebrated symmetry question, to elliptic dead core phenomena, uniqueness theorems, the Harnack inequality and the compact support principle

978-3-642-11211-9 QuefflecaMartine Quefflec, Universit des Sciences et Technologies de Lille, Villeneuve d'Ascq CX, France2Substitution Dynamical Systems - Spectral Analysis XV, 351p.The Banach Algebra (T).- Spectral Theory of Unitary Operators.- Spectral Theory of Dynamical Systems.- Dynamical Systems Associated with Sequences.- Dynamical Systems Arising from Substitutions.- Eigenvalues of Substitution Dynamical Systems.- Matrices of Measures.- Matrix Riesz Products.- Bijective Automata.- Maximal Spectral Type of General Automata.- Spectral Multiplicity of General Automata.- Compact Automata.&This volume mainly deals with the dynamics of finitely valued sequences, and more specifically, of sequences generated by substitutions and automata. Those sequences demonstrate fairly simple combinatorical and arithmetical properties and naturally appear in various domains. As the title suggests, the aim of the initial version of this book was the spectral study of the associated dynamical systems: the first chapters consisted in a detailed introduction to the mathematical notions involved, and the description of the spectral invariants followed in the closing chapters. This approach, combined with new material added to the new edition, results in a nearly self-contained book on the subject. New tools - which have also proven helpful in other contexts - had to be developed for this study. Moreover, its findings can be concretely applied, the method providing an algorithm to exhibit the spectral measures and the spectral multiplicity, as is demonstrated in several examples. Beyond this advanced analysis, many readers will benefit from the introductory chapters on the spectral theory of dynamical systems; others will find complements on the spectral study of bounded sequences; finally, a very basic presentation of substitutions, together with some recent findings and questions, rounds out the book.978-0-387-77378-0 Radulescu%Teodora-Liliana Radulescu, University Craiova """Fratii Buzesti"" College", Craiova, Romania; Vicentiu D. Radulescu, University of Craiova Fac. Mathematics & Computer Science, Craiova, Romania; Titu Andreescu, University of Texas at Dallas Natural Sciences and Mathematics, Richardson, TX, USAProblems in Real Analysis"Advanced Calculus on the Real AxisSequences, Series, and Limits.- Sequences.- Series.- Limits of Functions.- Qualitative Properties of Continuous and Differentiable Functions.- Continuity.- Differentiability.- Applications to Convex Functions and Optimization.- Convex Functions.- Inequalities and Extremum Problems.- Antiderivatives, Riemann Integrability, and Applications.- Antiderivatives.- Riemann Integrability.- Applications of the Integral Calculus.- Basic Elements of Set Theory.Problems in Real Analysis: Advanced Calculus on the Real Axis features a comprehensive collection of challenging problems in mathematical analysis that aim to promote creative, non-standard techniques for solving problems. This self-contained text offers a host of new mathematical tools and strategies which develop a connection between analysis and other mathematical disciplines, such as physics and engineering. A broad view of mathematics is presented throughout; the text is excellent for the classroom or self-study. It is intended for undergraduate and graduate students in mathematics, as well as for researchers engaged in the interplay between applied analysis, mathematical physics, and numerical analysis. Key features: *Uses competition-inspired problems as a platform for training typical inventive skills; *Develops basic valuable techniques for solving problems in mathematical analysis on the real axis and provides solid preparation for deeper study of real analysis; *Includes numerous examples and interesting, valuable historical accounts of ideas and methods in analysis; *Offers a systematic path to organizing a natural transition that bridges elementary problem-solving activity to independent exploration of new results and properties.

Contains a collection of challenging problems in elementary mathematical analysis

Uses competition-inspired problems as a platform for training typical inventive skills

Develops bas< ic valuable techniques for solving problems in mathematical analysis on the real axis

Assumes only a basic knowledge of the topic but opens the path to competitive research in the field

Includes interesting and valuable historical accounts of ideas and methods in analysis

Presents a connection between analysis and other mathematical disciplines, such as physics and engineering

May be applied in the classroom or as a self-study

978-1-4419-0494-2Rassias4Michael Th. Rassias, ETH Zurich, Zurich, Switzerland4Problem-Solving and Selected Topics in Number Theory+In the Spirit of the Mathematical OlympiadsXV, 324p. 3 illus..- Introduction.- The Fundamental Theorem of Arithmetic.- Arithmetic functions.- Perfect numbers, Fermat numbers.- Basic theory of congruences.- Quadratic residues and the Law of Quadratic Reciprocity.- The functions p(x) and li(x).- The Riemann zeta function.- Dirichlet series.- Partitions of integers.- Generating functions.- Solved exercises and problems.- The harmonic series of prime numbers.- Lagrange four-square theorem.- Bertrand postulate.- An inequality for the function p(n).- An elementary proof of the Prime Number Theorem.- Historical remarks on Fermat s Last Theorem.- Author index.- Subject index.- Bibliography and Cited References.The book provides a self-contained introduction to classical Number Theory. All the proofs of the individual theorems and the solutions of the exercises are being presented step by step. Some historical remarks are also presented. The book will be directed to advanced undergraduate, beginning graduate students as well as to students who prepare for mathematical competitions (ex. Mathematical Olympiads and Putnam Mathematical competition).XPresents the historical backgroundof varioustopics innumber theory;

Provides a self-contained introduction to classicalnumber theory;

Includes step-by-step proofs oftheorems andsolutionsto exercises;

Designed forundergraduate students, particularlythose who would like toprepare for mathematical competitions.

978-1-4899-8194-3XVI, 326 p. 3 illus.978-0-387-97472-9Rauch Jeffrey Rauch X, 266 p. 1 Power Series Methods.- 1.1. The Simplest Partial Differential Equation.- 1.2. The Initial Value Problem for Ordinary Differential Equations.- 1.3. Power Series and the Initial Value Problem for Partial Differential Equations.- 1.4. The Fully Nonlinear Cauchy Kowaleskaya Theorem.- 1.5. Cauchy Kowaleskaya with General Initial Surfaces.- 1.6. The Symbol of a Differential Operator.- 1.7. Holmgren s Uniqueness Theorem.- 1.8. Fritz John s Global Holmgren Theorem.- 1.9. Characteristics and Singular Solutions.- 2 Some Harmonic Analysis.- 2.1. The Schwartz Space $$\mathcal{J}({\mathbb{R}^d})$$.- 2.2. The Fourier Transform on $$\mathcal{J}({\mathbb{R}^d})$$.- 2.3. The Fourier Transform onLp$${\mathbb{R}^d}$$d):1 ?p?2.- 2.4. Tempered Distributions.- 2.5. Convolution in $$\mathcal{J}({\mathbb{R}^d})$$ and $$\mathcal{J}'({\mathbb{R}^d})$$.- 2.6. L2Derivatives and Sobolev Spaces.- 3 Solution of Initial Value Problems by Fourier Synthesis.- 3.1. Introduction.- 3.2. Schrdinger s Equation.- 3.3. Solutions of Schrdinger s Equation with Data in $$\mathcal{J}({\mathbb{R}^d})$$.- 3.4. Generalized Solutions of Schrdinger s Equation.- 3.5. Alternate Characterizations of the Generalized Solution.- 3.6. Fourier Synthesis for the Heat Equation.- 3.7. Fourier Synthesis for the Wave Equation.- 3.8. Fourier Synthesis for the Cauchy Riemann Operator.- 3.9. The Sideways Heat Equation and Null Solutions.- 3.10. The Hadamard Petrowsky Dichotomy.- 3.11. Inhomogeneous Equations, Duhamel s Principle.- 4 Propagators andx-Space Methods.- 4.1. Introduction.- 4.2. Solution Formulas in x Space.- 4.3. Applications of the Heat Propagator.- 4.4. Applications of the Schrdinger Propagator.- 4.5. The Wave Equation Propagator ford = 1.- 4.6. Rotation-Invariant Smooth Solutions of $${\square _{1 + 3}}\mu = 0$$.- 4.7. The Wave Equation Propagator ford =3.- 4.8. The Method of Descent.- 4.9. Radiation Problems.- 5 The Dirichlet Problem.- 5.1. Introduction.- 5.2. Dirichlet s Principle.- 5.3. The Direct Method of the Calculus of Variations.- 5.4. Variations on the Theme.- 5.5.H1 the Dirichlet Boundary Condition.- 5.6. The Fredholm Alternative.- 5.7. Eigenfunctions and the Method of Separation of Variables.- 5.8. Tangential Regularity for the Dirichlet Problem.- 5.9. Standard Elliptic Regularity Theorems.- 5.10. Maximum Principles from Potential Theory.- 5.11. E. Hopf s Strong Maximum Principles.- APPEND.- A Crash Course in Distribution Theory.- References.The objective of this book is to present an introduction to the ideas, phenomena, and methods of partial differential equations. This material can be presented in one semester and requires no previous knowledge of differential equations, but assumes the reader to be familiar with advanced calculus, real analysis, the rudiments of complex analysis, a< nd thelanguage of functional analysis. Topics discussed in the text include elliptic, hyperbolic, and parabolic equations, the energy method, maximum principle, and the Fourier Transform. The text features many historical and scientific motivations and applications. Included throughout are exercises, hints, and discussions which form an important and integral part of the course.978-0-387-98307-3Reddy B.D. Reddy Introductory Functional Analysis@With Applications to Boundary Value Problems and Finite ElementsXIV, 471 p.Contents.- Introduction.- Linear Functional Analysis.- Sets.- The algebra of sets.- Sets of numbers.- Rn and its subsets.- Relations, equivalence classes and Zorn's lemma.- Theorem-proving.- Bibliographical remarks.- Exercises.- Sets of functions and Lebesgue integration.- Continuous functions.- Meansure of sets in Rn.- Lebesgue integration and the space Lp(_).- Bibliographical remarks .- Exercises.- Vector spaces, normed and inner product spaces.THIS IS BOTH PROMO COPY AND BACK COVER COPY!!!!! This book provides an introduction to functional analysis and treats in detail its application to boundary-value problems and finite elements. The book is intended for use by senior undergraduate and graduate students in mathematics, the physical sciences and engineering, who may not have been exposed to the conventional prerequisites for a course in functional analysis, such as real analysis. Mature researchers wishing to learn the basic ideas of functional analysis would also find the text useful. The text is distinguished by the fact that abstract concepts are motivated and illustrated wherever possible. Readers of this book can expect to obtain a good grounding in those aspects of functional analysis which are most relevant to a proper understanding and appreciation of the mathematical aspects of boundary-value problems and the finite element method.WAssumes only elementary knowledge of linear algebra, vector analysis, and differential equations

New concepts made more accessible by copious use of concrete worked examples

Descriptive approach favored over detailed mathematical argument

Chapter end with exercises to consolidate the material

Solutions included

978-0-387-98221-2RemmertFReinhold Remmert, Diakonie Wohnstift am Westerberg, Osnabrck, Germany+Classical Topics in Complex Function TheoryXIX, 352 p.X1 Infinite Products of Holomorphic Functions.- 2 The Gamma Function.- 3 Entire Functions with Prescribed Zeros.- 4* Holomorphic Functions with Prescribed Zeros.- 5 Iss sa s Theorem. Domains of Holomorphy.- 6 Functions with Prescribed Principal Parts.- 7 The Theorems of Montel and Vitali.- 8 The Riemann Mapping Theorem.- 9 Automorphisms and Finite Inner Maps.- 10 The Theorems of Bloch, Picard, and Schottky.- 11 Boundary Behavior of Power Series.- 12 Runge Theory for Compact Sets.- 13 Runge Theory for Regions.- 14 Invariance of the Number of Holes.- Short Biographies.- Symbol Index.- Name Index.This book is an ideal text for an advanced course in the theory of complex functions. The author leads the reader to experience function theory personally and to participate in the work of the creative mathematician. The book contains numerous glimpses of the function theory of several complex variables, which illustrate how autonomous this discipline has become. Topics covered include Weierstrass's product theorem, Mittag-Leffler's theorem, the Riemann mapping theorem, and Runge's theorems on approximation of analytic functions. In addition to these standard topics, the reader will find Eisenstein's proof of Euler's product formula for the sine function; Wielandt's uniqueness theorem for the gamma function; a detailed discussion of Stirling's formula; Iss'sa's theorem; Besse's proof that all domains in C are domains of holomorphy; Wedderburn's lemma and the ideal theory of rings of holomorphic functions; Estermann's proofs of the overconvergence theorem and Bloch's theorem; a holomorphic imbedding of the unit disc in C3; and Gauss's expert opinion of November 1851 on Riemann's dissertation. Remmert elegantly presents the material in short clear sections, with compact proofs and historical comments interwoven throughout the text. The abundance of examples, exercises, and historical remarks, as well as the extensive bibliography, will make this book an invaluable source for students and teachers.978-1-4419-3114-6978-0-387-97195-7Theory of Complex FunctionsXIX, 458 p._Historical Introduction.- Chronological Table.- A. Elements of Function Theory.- 0. Complex Numbers and Continuous Functions.- 1. Complex-Differential Calculus.- 2. Holomorphy and Conformality. Biholomorphic Mappings...- 3. Modes of Convergence in Function Theory.- 4. Power Series.- 5. Elementary Transcendental Functions.- B. The Cauchy Theory.- 6. Complex Integral Calculus.- 7. The Integral Theorem, Integral Formula and Power Series Development.- C. Cauchy-Weierstrass-Riemann Function Theory.- 8. Fundamental Theorems about Holomorphic Functions.- 9. Miscellany.- 10. Isolated Singularities. Meromorphic Functions.- 11. Convergent Series of Meromorphic Functions.- 12. Laurent Series and Fourier Series.- 13. The Residue Calculus.- 14. Definite Integrals and the Residue Calculus.- Short Biographies o/Abel, Cauchy, Eisenstein, Euler, Riemann and Weierstrass.- Photograph of Riemann s gravestone.- Literature.- Classical Literature on Function Theory Textbooks on Function Theory Literature on the History of Function Theory and of Mathematics Symbol Index.- Name Index.- Portraits of famous mathematicians 3.&The material from function theory, up to the residue calculus, is developed in a lively and vivid style, well motivated throughout by examples and practice exercises. Additionally, there is ample discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations (original language together with English translation) from their classical works. Yet the book is far from being a mere history of function theory. Even experts will find here few new or long forgotten gems, like Eisenstein's novel approach to the circular functions. This book is destined to accompany many students making their way into a classical area of mathematics which represents the most fruitful example to date of the intimate connection between algebra and analysis. For exam preparation it offers quick access to the essential< results and an abundance of interesting inducements. Teachers and interested mathematicians in finance, industry and science will also find reading it profitable, again and again referring to it with pleasure.978-1-4419-1820-8Renardy!Michael Renardy; Robert C. Rogers1An Introduction to Partial Differential EquationsCharacteristics.- Conservation Laws and Shocks.- Maximum Principles.- Distributions.- Function Spaces.- Sobolev Spaces.- Operator Theory.- Linear Elliptic Equations.- Nonlinear Elliptic Equations.- Energy Methods for Evolution Problems.- Semigroup Methods.Partial differential equations are fundamental to the modeling of natural phenomena, arising in every field of science. Consequently, the desire to understand the solutions of these equations has always had a prominent place in the efforts of mathematicians; it has inspired such diverse fields as complex function theory, functional analysis and algebraic topology. Like algebra, topology, and rational mechanics, partial differential equations are a core area of mathematics. This book aims to provide the background necessary to initiate work on a Ph.D. thesis in PDEs for beginning graduate students. Prerequisites include a truly advanced calculus course and basic complex variables. Lebesgue integration is needed only in Chapter 10, and the necessary tools from functional analysis are developed within the course. The book can be used to teach a variety of different courses. This new edition features new problems throughout and the problems have been rearranged in each section from simplest to most difficult. New examples have also been added. The material on Sobolev spaces has been rearranged and expanded. A new section on nonlinear variational problems with 'Young-measure' solutions appears. The reference section has also been expanded.978-3-540-64325-8RevuzDaniel Revuz, Universit Paris VII U.E.R. Mathmatiques, Paris CX 05, France; Marc Yor, Universit Paris VI CNRS UMR 7599, Paris CX 05, France*Continuous Martingales and Brownian Motion XI, 606 p.0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.- VIII. Girsanov s Theorem and First Applications.- IX. Stochastic Differential Equations.- X. Additive Functionals of Brownian Motion.- XI. Bessel Processes and Ray-Knight Theorems.- XII. Excursions.- XIII. Limit Theorems in Distribution.- 1. Gronwall s Lemma.- 2. Distributions.- 3. Convex Functions.- 4. Hausdorff Measures and Dimension.- 5. Ergodic Theory.- 6. Probabilities on Function Spaces.- 7. Bessel Functions.- 8. Sturm-Liouville Equation.- Index of Notation.- Index of Terms.- Catalogue.|From the reviews: 'This is a magnificent book! Its purpose is to describe in considerable detail a variety of techniques used by probabilists in the investigation of problems concerning Brownian motion. The great strength of Revuz and Yor is the enormous variety of calculations carried out both in the main text and also (by implication) in the exercises. ... This is THE book for a capable graduate student starting out on research in probability: the effect of working through it is as if the authors are sitting beside one, enthusiastically explaining the theory, presenting further developments as exercises, and throwing out challenging remarks about areas awaiting further research...' Bull.L.M.S. 24, 4 (1992) Since the first edition in 1991, an impressive variety of advances has been made in relation to the material of this book, and these are reflected in the successive editions.978-3-642-08400-3978-0-8176-8297-2RieselAHans Riesel, The Royal Institute of Technology, Stockholm, Sweden4Prime Numbers and Computer Methods for FactorizationXVIII, 464p. 20 illus..Preface.- The Number of Primes Below a Given Limit.- The Primes Viewed at Large.- Subtleties in the Distribution of Primes.- The Recognition of Primes.- Classical Methods of Factorization.- Modern Factorization Methods.- Prime Numbers and Cryptography.- Appendix 1. Basic Concepts in Higher Algebra.- Appendix 2. Basic concepts in Higher Arithmetic.- Appendix 3. Quadratic Residues.- Appendix 4. The Arithmetic of Quadratic Fields.- Appendix 5. Higher Algebraic Number Fields.- Appendix 6. Algebraic Factors.- Appendix 7. Elliptic Curves.- Appendix 8. Continued Fractions.- Appendix 9. Multiple-Precision Arithmetic.- Appendix 10. Fast Multiplication of Large Integers.- Appendix 11. The Stieltjes Integral.- Tables.- List of Textbooks.- Index. From the original hard cover edition:In the modern age of almost universal computer usage, practically every individual in a technologically developed society has routine access to the most up-to-date cryptographic technology that exists, the so-called RSA public-key cryptosystem. A major component of this system is the factorization of large numbers into their primes. Thus an ancient number-theory concept now plays a crucial role in communication among millions of people who may have little or no knowledge of even elementary mathematics. Hans Riesel s highly successful first edition of this book has now been enlarged and updated with the goal of satisfying the needs of researchers, students, practitioners of cryptography, and non-scientific readers with a mathematical inclination. It includes important advances in computational prime number theory and in factorization as well as re-computed and enlarged tables, accompanied by new tables reflecting current research by both the author and his coworkers and by independent researchers. The book treats four fun< damental problems: the number of primes below a given limit, the approximate number of primes, the recognition of primes and the factorization of large numbers. The author provides explicit algorithms and computer programs, and has attempted to discuss as many of the classically important results as possible, as well as the most recent discoveries. The programs include are written in PASCAL to allow readers to translate the programs into the language of their own computers. The independent structure of each chapter of the book makes it highly readable for a wide variety of mathematicians, students of applied number theory, and others interested in both study and research in number theory and cryptography. 4Affordablereprint ofthesuccessful, expanded second edition of auniquemonograph

Provides a unique perspective on the study of prime numbers and their importance in modern technology

Applications remain relevant across the fields of modern computer science and information transmission

978-0-85729-182-0RochSteffen Roch, Technische Universitt Darmstadt Fachbereich Mathematik, Darmstadt, Germany; Pedro A. Santos, Universidade Tcnica de Lisboa Instituto Superior Tcnico, Lisboa, Portugal; Bernd Silbermann, Technische Universitt Chemnitz, Chemnitz, Germany Non-commutative Gelfand Theories8A Tool-kit for Operator Theorists and Numerical Analysts(XIV, 383p. 14 illus., 2 illus. in color.Banach algebras.- Local principles.- Banach algebras generated by idempotents.- Singular integral operators.- Convolution operators.- Algebras of operator sequences.Written as a hybrid between a research monograph and a textbook the first half of this book is concerned with basic concepts for the study of Banach algebras that, in a sense, are not too far from being commutative. Essentially, the algebra under consideration either has a sufficiently large center or is subject to a higher order commutator property (an algebra with a so-called polynomial identity or in short: Pl-algebra). In the second half of the book, a number of selected examples are used to demonstrate how this theory can be successfully applied to problems in operator theory and numerical analysis.

Distinguished by the consequent use of local principles (non-commutative Gelfand theories), PI-algebras, Mellin techniques and limit operator techniques, each one of the applications presented in chapters 4, 5 and 6 forms a theory that is up to modern standards and interesting in its own right.

Written in a way that can be worked through by the reader with fundamental knowledge of analysis, functional analysis and algebra, this book will be accessible to 4th year students of mathematics or physics whilst also being of interest to researchers in the areas of operator theory, numerical analysis, and the general theory of Banach algebras.

Provides an overview of the theoretical developments of the last thirty years in addition to a number of concrete applications in operator theory and numerical analysis

Focuses on non-C*-algebras

Contains new results not yet published

Written in a way that it can be worked through by a reader with fundamental knowledge of analysis, functional analysis and algebra

978-1-4419-7322-1 RodriguezRubi Rodriguez, Pontificia Universidad Catlica de Chile Facultad de Matemticas, Santiago, Chile; Irwin Kra, University of Stony Brook, Stony Brook, NY, USA; Jane P. Gilman, Rutgers University Dept. Mathematics and Comp. Science, Newark, NJ, USAComplex AnalysisIn the Spirit of Lipman BersXVIII, 306 p. 27 illus.-Preface to Second Edition.- Preface to First Edition.- Standard Notation and Commonly Used Symbols.- 1 The Fundamental Theorem in Complex Function Theory.- 2 Foundations.- 3 Power Series.- 4 The Cauchy Theory - A Fundamental Theorem.- 5 The Cauchy Theory - Key Consequences.- 6 Cauchy Theory: Local Behavior and Singularities of Holomorphic Functions.- 7 Sequences and Series of Holomorphic Functions.- 8 Conformal Equivalence and Hyperbolic Geometry.- 9 Harmonic Functions.- 10 Zeros of Holomorphic Functions.- Bibliographical Notes.- Bibliography.- Index.@This book is intended for a graduate course in complex analysis, where the main focus is the theory of complex-valued functions of a single complex variable. This theory is a prerequisite for the study of many areas of mathematics, including the theory of several finitely and infinitely many complex variables, hyperbolic geometry, two- and three-manifolds, and number theory. Complex analysis has connections and applications to many other subjects in mathematics and to other sciences. Thus this material will also be of interest to computer scientists, physicists, and engineers.The book covers most, if not all, of the material contained in Lipman Bers s courses on first year complex analysis. In addition, topics of current interest, such as zeros of holomorphic functions and the connection between hyperbolic geometry and complex analysis, are explored.In addition to many new exercises, this second edition introduces a variety of new and interesting topics. New features include a section on Bers's theorem on isomorphisms between rings of holomorphic functions on plane domains; necessary and sufficient conditions for the existence of a bounded analytic function on the disc with prescribed zeros; sections on subharmonic functions and Perron's principle; and a section on the ring of holomorphic functions on a plane domain. There are three new appendices: the first is a contribution by Ranjan Roy on the history of complex analysis, the second contains background material on exterior differential calculus, and the third appendix includes an alternate approach to the Cauchy theory.New edition extensively revised and updated

New topics include Bers's theorem on isomo< rphisms between rings of holomorphic functions on plane domains, necessary and sufficient conditions for the existence of a bounded analytic function on the disc, subharmonic functions, Perron's principle, the ring of holomorphic functions on a plane domain, solutions to the Dirichlet problem, Green's function and its relation to the Riemann Mapping Theorem

Covers vast majority of the material needed for a beginning graduate level course on complex analysis

Material may be of interest to mathematicians, computer scientists, physicists, and engineers

978-1-4899-9908-5978-0-8176-8300-9RomanSteven Roman, Irvine, CA, USAFundamentals of Group TheoryAn Advanced ApproachXII, 380p. 21 illus..mPreliminaries.- Groups and Subgroups.- Cosets, Index and Normal Subgroups.- Homomorphisms.- Chain Conditions and Subnormality.- Direct and Semidirect Products.- Permutation Groups.- Group Actions.- The Structure of Groups.- Sylow Theory.- The Classification Problem for Groups.- Finiteness Conditions.- Free Groups and Presentations.- Abelian Groups.- References.This book will be suitable for graduate courses in group theory and abstract algebra, and will also have appeal to advanced undergraduates. In addition it will serve as a valuable resource for those pursuing independent study. *Group Theory *is a timely and fundamental addition to literature in the study of groups.

Integrates classic material and new concepts

Gives a comprehensive account of the basic theory of groups

Content is written in a clear, precise and easy to read style

Provides an introduction for those new to the theory of groups

978-0-387-94248-3 RosenbergJonathan Rosenberg'Algebraic K-Theory and Its ApplicationsX, 392 p. 2 illus.1. K0 of Rings.- 1. Defining K0.- 2. K0 from idempotents.- 3. K0 of PIDs and local rings.- 4. K0 of Dedekind domains.- 5. Relative K0 and excision.- 6. An application: Swan s Theorem and topological K- theory.- 7. Another application: Euler characteristics and the Wall finiteness obstruction.- 2.K1 of Rings.- 1. Defining K1.- 2. K1 of division rings and local rings.- 3. 1 of PIDs and Dedekind domains.- 4. Whitehead groups and Whitehead torsion.- 5. Relative K1 and the exact sequence.- 3. K0 and K1 of Categories, Negative K-Theory.- 1. K0 and K1 of categories, Go and G1 of rings.- 2. The Grothendieck and Bass-Heller-Swan Theorems.- 3. Negative K-theory.- 4. Milnor s K2.- 1. Universal central extensions and H2.- 2. The Steinberg group.- 3. Milnor s K2.- 4. Applications of K2.- 5. The +?Construction and Quillen K-Theory.- 1. An introduction to classifying spaces.- 2. Quillen s +?construction and its basic properties.- 3. A survey of higher K-theory.- 6. Cyclic homology and its relation to K-Theory.- 1. Basics of cyclic homology.- 2. The Chern character.- 3. Some applications.- References.- Books and Monographs on Related Areas of Algebra, Analysis, Number Theory, and Topology.- Books and Monographs on Algebraic K-Theory.- Specialized References.- Notational Index.Algebraic K-Theory plays an important role in many areas of modern mathematics: most notably algebraic topology, number theory, and algebraic geometry, but even including operator theory. The broad range of these topics has tended to give the subject an aura of inapproachability. This book, based on a course at the University of Maryland in the fall of 1990, is intended to enable graduate students or mathematicians working in other areas not only to learn the basics of algebraic K-Theory, but also to get a feel for its many applications. The required prerequisites are only the standard one-year graduate algebra course and the standard introductory graduate course on algebraic and geometric topology. Many topics from algebraic topology, homological algebra, and algebraic number theory are developed as needed. The final chapter gives a concise introduction to cyclic homology and its interrelationship with K-Theory.978-0-387-21284-5RossNClay C. Ross, University of the South Mathematics Department, Sewanee, TN, USADifferential Equations!An Introduction with MathematicaXIII, 433 p.1 About Differential Equations.- 2 Linear Algebra.- 3 First-Order Differential Equations.- 4 Applications of First-Order Equations.- 5 Higher-Order Linear Differential Equations.- 6 Applications of Second-Order Equations.- 7 The Laplace Transform.- 8 Higher-Order Differential Equations with Variable Coefficients.- 9 Differential Systems: Theory.- 10 Differential Systems: Applications.- References.This introductory differential equations textbook presents a convenient way for professors to integrate symbolic computing into the study of differential equations and linear algebra. Mathematica provides the necessary computational power and is employed from the very beginning of the text. Each new idea is interactively developed using it. After first learning about the fundamentals of differential equations and linear algebra, the student is immediately given an opportunity to examine each new concept using Mathematica. All ideas are explored utilizing Mathematica, and though the computer eases the computational burden, the student is encouraged to think about what the computations reveal, how they are consistent with the mathematics, what any conclusions mean, and how they may be applied. This new edition updates the text to Mathematica 5.0 and offers a more extensive treatment of linear algebra. It has been thoroughly revised and corrected throughout.Offers a huge variety of examples and exercises

Lots of Mathematica code, which is well integrated into the main discussion

978-1-4419-1941-0978-0-387-96678-6RotmanZJoseph Rotman, University of Illinois, Urbana-Champaign Dept. Mathematics, Urbana, IL, USA%An Introduction< to Algebraic TopologyXIII, 433 p. 95 illus.>0 Introduction.- Notation.- Brouwer Fixed Point Theorem.- Categories and Functors.- 1.Some Basic Topological Notions.- Homotopy.- Convexity, Contractibility, and Cones.- Paths and Path Connectedness.- 2 Simplexes.- Affine Spaces.- Affine Maps.- 3 The Fundamental Group.- The Fundamental Groupoid.- The Functor ?1.- ?1(S1).- 4 Singular Homology.- Holes and Green s Theorem.- Free Abelian Groups.- The Singular Complex and Homology Functors.- Dimension Axiom and Compact Supports.- The Homotopy Axiom.- The Hurewicz Theorem.- 5 Long Exact Sequences.- The Category Comp.- Exact Homology Sequences.- Reduced Homology.- 6 Excision and Applications.- Excision and Mayer-Vietoris.- Homology of Spheres and Some Applications.- Barycentric Subdivision and the Proof of Excision.- More Applications to Euclidean Space.- 7 Simplicial Complexes.- Definitions.- Simplicial Approximation.- Abstract Simplicial Complexes.- Simplicial Homology.- Comparison with Singular Homology.- Calculations.- Fundamental Groups of Polyhedra.- The Seifert-van Kampen Theorem.- 8 CW Complexes.- Hausdorff Quotient Spaces.- Attaching Cells.- Homology and Attaching Cells.- CW Complexes.- Cellular Homology.- 9 Natural Transformations.- Definitions and Examples.- Eilenberg-Steenrod Axioms.- Chain Equivalences.- Acyclic Models.- Lefschetz Fixed Point Theorem.- Tensor Products.- Universal Coefficients.- Eilenberg-Zilber Theorem and the Knneth Formula.- 10 Covering Spaces.- Basic Properties.- Covering Transformations.- Existence.- Orbit Spaces.- 11 Homotopy Groups.- Function Spaces.- Group Objects and Cogroup Objects.- Loop Space and Suspension.- Homotopy Groups.- Exact Sequences.- Fibrations.- A Glimpse Ahead.- 12 Cohomology.- Differential Forms.- Cohomology Groups.- Universal Coefficients Theorems for Cohomology.- Cohomology Rings.- Computations and Applications.- Notation.GThis book is a clear exposition, with exercises, of the basic ideas of algebraic topology: homology (singular, simplicial, and cellular), homotopy groups, and cohomology rings. It is suitable for a two- semester course at the beginning graduate level, requiring as a prerequisite a knowledge of point set topology and basic algebra. Although categories and functors are introduced early in the text, excessive generality is avoided, and the author explains the geometric or analytic origins of abstract concepts as they are introduced, making this book of great value to the student.978-1-84996-872-0RyanRaymond A. Ryan0Introduction to Tensor Products of Banach SpacesG1 Tensor Products.- 2 The Projective Tensor Product.- 3 The Injective Tensor Product.- 4 The Approximation Property.- 5 The Radon-Nikodm Property.- 6 The Chevet-Saphar Tensor Products.- 7 Tensor Norms.- 8 Operator Ideals.- A Suggestions for Further Reading.- B Summability in Banach Spaces.- C Spaces of Measures.- References.LThis volume provides a self-contained introduction to the theory of tensor products of Banach spaces. It is written for graduate students in analysis or for researchers in other fields who wish to become acquainted with this area. The only prerequisites are a basic knowledge of functional analysis and measure theory. Features of particular interest include: - A full treatment of the Grothendieck theory of tensor norms; - Coverage of the Chevet-Saphar norms and their duals, along with the associated classes of nuclear, integral and summing operators; - Chapters on the approximation property and the Radon-Nikodym property; - Topics such as the Bochner and Pettis integrals, the principle of local reflexivity and the Grothendieck inequality placed in a natural setting; - The classes of operators generated by a tensor norm and connections with the theory of operator ideals. Each chapter is accompanied by worked examples and a set of exercises, and two appendices provide essential material on summability in Banach spaces and properties of spaces of measures that may be new to the beginner.The first truly introductory book in this field - first graduate courses everywhere are looking for such a textIncludes worked examples and many exercises

The Radon-Nikodym Property is central to the book

Presents many new, interesting and revealing results for the first time978-1-85233-437-6978-1-4614-5350-5Thomas Rylander, Chalmers University of Technology Dept. Electromagnetics, Gteborg, Sweden; Pr Ingelstrm, Chalmers University of Technology Dept. Electromagnetics, Gteborg, Sweden; Anders Bondeson, Chalmers University of Technology Dept. Electromagnetics, Gteborg, SwedenComputational ElectromagneticsXIX, 286 p. 87 illus.SCT24000Electrical EngineeringTHRIntroduction.- Convergence.- Finite Differences.- Eigenvalues.- The Finite-Difference Time-Domain Method.- The Finite Element Method.- The Method of Moments.- Summary and Overview.-Large Linear Systems.- Krylov Methods.Computational Electromagnetics is a young and growing discipline, expanding as a result of the steadily increasing demand for software for the design and analysis of electrical devices. This book introduces three of the most popular numerical methods for simulating electromagnetic fields: the finite difference method, the finite element method and the method of moments. In particular it focuses on how these methods are used to obtain valid approximations to the solutions of Maxwell's equations, using, for example, 'staggered grids' and 'edge elements.' The main goal of the book is to make the reader aware of different sources of errors in numerical computations, and also to provide the tools for assessing the accuracy of numerical methods and their solutions. To reach this goal, convergence analysis, extrapolation, von Neumann stability analysis, and dispersion analysis are introduced and used frequently throughout the book. Another major goal of the book is to provide students with enough practical understanding of the methods so they are able to write simple programs on their own. To achieve this, the book contains several MATLAB programs and detailed description of practical issues such as assembly of finite element matrices and handling of unstructured meshes. Finally, the book aims at making the students well-aware of the strengths and weaknesses of the different methods, so they can decid< e which method is best for each problem. In thissecond edition, extensive computer projects are added as well as new material throughout.Reviews of previous edition:'The well-written monograph is devoted to students at the undergraduate level, but is also useful for practising engineers.' (Zentralblatt MATH, 2007)

Describes most popular computational methods used to solve problems in electromagnetics

Matlab code is included throughout, so that the reader can implement the various techniques discussed

Extensive computer projects included in this new edition

978-1-4899-8602-3978-1-4471-2980-6Sauvigny`Friedrich Sauvigny, Brandenburgian Technical University Mathematical Institute, Cottbus, Germany Partial Differential Equations 1(Foundations and Integral RepresentationsXV, 447p. 16 illus..Differentiation and Integration on Manifolds.- Foundations of Functional Analysis.- Brouwer s Degree of Mapping.- Generalized Analytic Functions.- Potential Theory and Spherical Harmonics.- Linear Partial Differential Equations in Rn. This two-volume textbook provides comprehensive coverage of partial differential equations, spanning elliptic, parabolic, and hyperbolic types in two and several variables.In this first volume, special emphasis is placed on geometric and complex variable methods involving integral representations. The following topics are treated: " integration and differentiation on manifolds" foundations of functional analysis" Brouwer's mapping degree" generalized analytic functions " potential theory and spherical harmonics " linear partial differential equationsThis new second edition of this volume has been thoroughly revised and a new section on the boundary behavior of Cauchy s integral has been added.The second volume will present functional analytic methods and applications to problems in differential geometry.This textbook will be of particular use to graduate and postgraduate students interested in this field and will be of interest to advanced undergraduate students. It may also be used for independent study.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

978-1-4471-2983-7 Partial Differential Equations 2Functional Analytic MethodsXVI, 453p. 11 illus..Operators in Banach Spaces.- Linear Operators in Hilbert Spaces.- Linear Elliptic Differential Equations.- Weak Solutions of Elliptic Differential Equations.- Nonlinear Partial Differential Equations.- Nonlinear Elliptic Systems.- Boundary Value Problems from Differential Geometry.xThis 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 spaceslinear operators in Hilbert spaces and spectral theory Schauder's theory of linear elliptic differential equations weak solutions of differential equationsnonlinear partial differential equations and characteristicsnonlinear elliptic systemsboundary value problems from differential geometryThis new second edition of this volume has been thoroughly revised and a new chapter on boundary value problems from differential geometry has been added.In the first volume, partial differential equations by integral representations are treated in a classical way.This textbook will be of particular use to graduate and postgraduate students interested in this field and will be of interest to advanced undergraduate students. It may also be used for independent study.978-0-387-30931-6ScherzerdOtmar Scherzer, University of Vienna, Vienna, Austria; Markus Grasmair, Universitt Innsbruck Abt. Informatik, Innsbruck, Austria; Harald Grossauer, Universitt Innsbruck Abt. Informatik, Innsbruck, Austria; Markus Haltmeier, Universitt Innsbruck Abt. Informatik, Innsbruck, Austria; Frank Lenzen, Universitt Innsbruck Abt. Informatik, Innsbruck, AustriaVariational Methods in ImagingSCI22021$Image Processing and Computer VisionUYTFundamentals of Imaging.- Case Examples of Imaging.- Image and Noise Models.- Regularization.- Variational Regularization Methods for the Solution of Inverse Problems.- Convex Regularization Methods for Denoising.- Variational Calculus for Non-convex Regularization.- Semi-group Theory and Scale Spaces.- Inverse Scale Spaces.- Mathematical Foundations.- Functional Analysis.- Weakly Differentiable Functions.- Convex Analysis and Calculus of Variations.This book is devoted to the study of variational methods in imaging. The presentation is mathematically rigorous and covers a detailed treatment of the approach from an inverse problems point of view. Key Features: (1) Introduces variational methods with motivation from the deterministic, geometric, and stochastic point of view. (2) Bridges the gap between regularization theory in image analysis and in inverse problems. (3) Presents case examples in imaging to illustrate the use of variational methods e.g. denoising, thermoacoustics, computerized tomography. (4) Discusses link between non-convex calculus of variations, morphological analysis, and level set methods. (5) Analyses variational methods containing classical analysis of variational methods, modern analysis such as G-norm properties, and non-convex calculus of variations. (6) Uses numerical examples to enhance the theory. This book is geared towards graduate students and researchers in applied mathematics. It can serve as a main text for graduate courses in image processing and inverse problems or as a supplemental text for courses on regularization. Researchers and computer scientists in the area of imaging science will also find this book useful.RIntroduces variational methods with motivation from the deterministic, geometric and stochastic point of view

Presents case examples in imaging to illustrate the use of variational methods e.g. denoising, thermoacoustics, computerized tomography

Discusses link between noncovex calculus of variations, morphological analysis and level set methods

Analyses variational methods containing classical analysis of variational methods, modern analysis such as G-norm properties and nonconvex calculus of variations

Includes additional material and images online

978-1-4419-2166-6978-1-4419-6350-5SchlickYTamar Schlick, New York University Courant Inst. Mathematical Sciences, New York, NY, USA=Molecular Modeling and Simulation: An Interdisciplinary GuideAn Interdisciplinary Guide XLV, 723p.SCC12029Biochemical Enginee< ringTCBiomolecular Structure and Modeling: Historical Perspective.- Biomolecular Structure and Modeling: Problem and Application Perspective.- Protein Structure Introduction.- Protein Structure Hierarchy.- Nucleic Acids Structure Minitutorial.- Topics in Nucleic Acids Structure: DNA Interactions and Folding.- Topics in Nucleic Acids Structure: Noncanonical Helices and RNA Structure.- Theoretical and Computational Approaches to Biomolecular Structure.- Force Fields.- Nonbonded Computations.- Multivariate Minimization in Computational Chemistry.- Monte Carlo Techniques.- Molecular Dynamics: Basics.- Molecular Dynamics: Further Topics.- Similarity and Diversity in Chemical Design.^ This book evolved from an interdisciplinary graduate course entitled Molecular Modeling developed at New York University. Its primary goal is to stimulate excitement for molecular modeling research while introducing readers to the wide range of biomolecular problems being solved by computational techniques and to those computational tools. The book is intended for beginning graduate students in medical schools and scientific fields such as biology, chemistry, physics, mathematics, and computer science. Other scientists who wish to enter, or become familiar, with the field of biomolecular modeling and simulation may also benefit from the broad coverage of problems and approaches. The book surveys three broad areas: biomolecular structure and modeling: current problems and state of computations; molecular mechanics: force field origin, composition, and evaluation techniques; and simulation methods: geometry optimization, Monte Carlo, and molecular dynamics approaches. Besides small additions and revisions made throughout the text and displayed materials to reflect the latest literature and field developments, some chapters have undergone more extensive revisions for this second edition. The book has been updated throughout, in particularly changes include: Chapters 1 and 2 that provide a historical perspective and an overview of current applications to biomolecular systems have been substantially updated; Chapter 4 which reflects modified protein classification with new protein examples and sequence statistics; the chapter Topics in Nucleic Acids (now expanded into two chapters, 6 and 7, which includes recent developments in RNA structure and function; the force field chapters 4--6, which contain new sections on enhanced sampling methods; Chapter 15 which includes an update on pharmacogenomics developments. 'Molecular modeling & is now an important branch of modern biochemistry. & Schlick has brought her unique interdisciplinary expertise to the subject. & One of the most distinguished characteristics of the book is that it makes the reading really fun & and the material accessible. & a crystal clear logical presentation & . Schlick has added a unique title to the collection of mathematical biology textbooks & . a valuable introduction to the field of computational molecular modeling. It is a unique textbook & .' (Hong Qian, SIAM Reviews, Vol. 47 (4), 2005).Very broad overview of the field intended for an interdisciplinary audience Lively discussion of current challenges written in a colloquial style Author is a rising star in this discipline Suitably accessible for beginners and suitably rigorous for experts Features extensive four-color illustrations Appendices featuring homework assignments and reading lists complement the material in the main text978-1-4614-2650-9XLV, 723 p.978-3-0348-0276-5Schmidt.Klaus Schmidt, Universitt Wien, Wien, Austria%Dynamical Systems of Algebraic OriginXVIII, 310p. 1 illus..I Group actions by automorphisms of compact groups.- II -actions on compact abelian groups.- III Expansive automorphisms of compact groups.- IV Periodic points.- V Entropy.- VI Positive entropy.- VII Zero entropy.- VIII Mixing.- IX Rigidity.Although much of classical ergodic theory is concerned with single transformations and one-parameter flows, the subject inherits from statistical mechanics not only its name, but also an obligation to analyze spatially extended systems with multidimensional symmetry groups. However, the wealth of concrete and natural examples which has contributed so much to the appeal and development of classical dynamics, is noticeably absent in this more general theory. The purpose of this book is to help remedy this scarcity of explicit examples by introducing a class of continuous Zd-actions diverse enough to exhibit many of the new phenomena encountered in the transition from Z to Zd, but which nevertheless lends itself to systematic study: the Zd-actions by automorphisms of compact, abelian groups. One aspect of these actions, not surprising in itself but quite striking in its extent and depth nonetheless, is the connection with commutative algeb< ra and arithmetical algebraic geometry. The algebraic framework resulting from this connection allows the construction of examples with a variety of specified dynamical properties, and by combining algebraic and dynamical tools one obtains a quite detailed understanding of this class of Zd-actions.<p> Beautifully written monograph on an interesting topic in ergodic theory </p><p>First systematic account of the ergodic theory of algebraic Zd-actions </p><p>Valuable to researchers and graduate students of ergodic theory</p>978-1-4614-0486-6SchussQZeev Schuss, Tel Aviv University School of Mathematical Science, Tel Aviv, Israel.Nonlinear Filtering and Optimal Phase Tracking XVIII, 262 p.IDiffusion and Stochastic Differential Equations.- Euler's Simulation Scheme and Wiener's Measure.- Nonlinear Filtering and Smoothing of Diffusions.- Small Noise Analysis of Zakai's Equation.- Loss of Lock in Phase Trackers.- Loss of Lock in RADAR and Synchronization.- Phase Tracking with Optimal Lock Time.- Bibliography.- IndexSThis book offers an analytical rather than measure-theoretical approach to the derivation of the partial differential equations of nonlinear filtering theory. The basis for this approach is the discrete numerical scheme used in Monte-Carlo simulations of stochastic differential equations and Wiener's associated path integral representation of the transition probability density.Furthermore, it presents analytical methods for constructing asymptotic approximations to their solution and for synthesizing asymptotically optimal filters. It also offers a new approach to the phase tracking problem, based on optimizing the mean time to loss of lock. The bookis based on lecture notes from a one-semester special topics course on stochastic processes and their applications that the author taught many times to graduate students of mathematics, applied mathematics, physics, chemistry, computer science, electrical engineering, and other disciplines. The book contains exercises and worked-out examples aimed at illustrating the methods of mathematical modeling and performance analysis of phase trackers.Many exercises and examples included

Balance between mathematical rigor and physical intuition

An analytical rather than measure-theoretical approach to the derivation and solution of the partial differential equations of nonlinear filltering theory

978-1-4899-7381-8978-3-540-73724-7SendraJ. Rafael Sendra, Universidad de Alcal Depto. Matemticas, Alcal de Henares, Spain; Franz Winkler, Universitt Linz Research Institute for Symbolic, Linz, Austria; Sonia Prez-Diaz, Universidad de Alcal Depto. Matemticas, Alcal de Henares, SpainRational Algebraic CurvesA Computer Algebra Approachand Motivation.- Plane Algebraic Curves.- The Genus of a Curve.- Rational Parametrization.- Algebraically Optimal Parametrization.- Rational Reparametrization.- Real Curves.The central problem considered in this book is the determination of rational parametrizability of an algebraic curve, and, in the positive case, the computation of a good rational parametrization. This amounts to determining the genus of a curve, i.e. its complete singularity structure, computing regular points of the curve in small coordinate fields, and constructing linear systems of curves with prescribed intersection multiplicities. Various optimality criteria for rational parametrizations of algebraic curves are discussed. This book is mainly intended for graduate students and researchers in constructive algebraic curve geometry.0Nice and easy introduction to topic for students978-3-642-09291-6978-1-4614-1230-4SenguptaCAmbar N. Sengupta, Louisiana State University, Baton Rouge, LA, USARepresenting Finite GroupsA Semisimple IntroductionXIV, 369p. 12 illus..Concepts and Constructs. -Basic Examples. - The Group Algebra. - More Group Algebra. - Simply Semisimple . -Representations of Sn. - Characters. - Induced Representations. - Commutant Duality. -Character Duality. -Representations of U(N). - PS: Algebra. -Bibliography.pThis graduate textbook presents the basics of representation theory for finite groups from the point of view of semisimple algebras and modules over them. The presentation interweaves insights from specific examples with development of general and powerful tools based on the notion of semisimplicity. The elegant ideas of commutant duality are introduced, along with an introduction to representations of unitary groups.The text progresses systematically and the presentation is friendly and inviting. Central concepts are revisited and exploredfrom multiple viewpoints. Exercises at the end of the chapter help reinforce the material.Representing Finite Groups: A Semisimple Introduction would serve as a textbook for graduate and some advanced undergraduate courses in mathematics. Prerequisites include acquaintance with elementary group theory and some familiarity with rings and modules. A final chapter presents a self-contained account of notions and results in algebra that are used. Researchers in mathematics and mathematical physics will also find this book useful.A separate solutions manual is available for instructors.Presents basics of representation theory of finite groups from the point of view of semisimple algebras and modules over them

Progresses systematically with a gentle and inviting pace

Exercises at the end of the chapter help reinforce the material

978-1-4899-9808-8XIV, 369 p. 12 illus.978-3-540-89331-8SerfozoYRichard Serfozo, Georgia Institute of Technology School of Industrial &, Atlanta, GA, USA&Basics of Applied Stochastic ProcessesXIV, 443p. 11 illus..yMarkov Chains.- Renewal and Regenerative Processes.- Poisson Processes.- Continuous-Time Markov Chains.- Brownian Motion.Stochastic processes are mathematical models of random phenomena that evolve according to prescribed dynamics. Processes commonly used in applications are Markov chains in discrete and continuous time, renewal and regenerative processes, Poisson processes, and Brownian motion. This volume gives an in-depth description of the structure and basic properties of these stochastic processes. A main focus is on equilibrium distributions, strong laws of large numbers, and ordinary and functional central limit theorems for cost and performance parameters. Although these results differ for various processes, they have a common trait of being limit theorems for processes with regenerative increments. Extensive examples and exercises show how to formulate stochastic models of systems as functions of a system s data and dynamics, and how to represent and analyze cost and performance measures. Topics include stochastic networks, spatial and space-time Poisson processes, queueing, reversible processes, simulation, Brownian approximations, and varied < Markovian models. The technical level of the volume is between that of introductory texts that focus on highlights of applied stochastic processes, and advanced texts that focus on theoretical aspects of processes.978-3-642-43043-5XIV, 443 p. 11 illus.978-3-540-30608-5SernesiCEdoardo Sernesi, Universit Roma Tre Dipto. Matematica, Roma, Italy!Deformations of Algebraic Schemes XI, 339 p.Introduction.- Infinitesimal Deformations: Extensions. Locally Trivial Deformations.- Formal Deformation Theory: Obstructions. Extensions of Schemes. Functors of Artin Rings. The Theorem of Schlessinger. The Local Moduli Functors.- Formal Versus Algebraic Deformations. Automorphisms and Prorepresentability.- Examples of Deformation Functors: Affine Schemes. Closed Subschemes. Invertible Sheaves. Morphisms.- Hilbert and Quot Schemes: Castelnuovo-Mumford Regularity. Flatness in the Projective Case. Hilbert Schemes. Quot Schemes. Flag Hilbert Schemes. Examples and Applications. Plane Curves.- Appendices: Flatness. Differentials. Smoothness. Complete Intersections. Functorial Language.- List of Symbols.- Bibliography.The study of small and local deformations of algebraic varieties originates in the classical work of Kodaira and Spencer and its formalization by Grothendieck in the late 1950's. It has become increasingly important in algebraic geometry in every context where variational phenomena come into play, and in classification theory. Today deformation theory is highly formalized and has ramified widely. This self-contained account of deformation theory in classical algebraic geometry (over an algebraically closed field) brings together for the first time some results previously scattered in the literature, with relatively little known proofs, yet of everyday relevance to algebraic geometers. It also includes applications to the construction and properties of Severi varieties of families of plane nodal curves, space curves, deformations of quotient singularities, Hilbert schemes of points, local Picard functors, etc. The exposition, amenable at graduate student level, includes many examples.- This book brings together for the first time some results so far scattered across a vast literature

- Fills a long-standing gap in the literatur, where no reference book existed and the results had become "folklore"

978-3-642-06787-7978-0-387-96648-9Serre]Jean-Pierre Serre, College de France Paris Chaire d'Algebre et Geometrie, Paris CX 05, France!Algebraic Groups and Class Fields IX, 207 p.I Summary of Main Results.- II Algebraic Curves.- III Maps From a Curve to a Commutative Group.- IV Singular Algebraic Curves.- V Generalized Jacobians.- VI Class Field Theory.- VII Group Extension and Cohomology.- Supplementary Bibliography.Reprint of a successful book

Written by the author of the classic, best-selling title Galois Cohomology

Presents local class field theory from the cohomological point of view

978-0-387-90424-5Local FieldsVIII, 245 p.GOne Local Fields (Basic Facts).- I Discrete Valuation Rings and Dedekind Domains.- II Completion.- Two Ramification.- III Discriminant and Different.- IV Ramification Groups.- V The Norm.- VI Artin Representation.- Three Group Cohomology.- VII Basic Facts.- VIII Cohomology of Finite Groups.- IX Theorems of Tate and Nakayama.- X Galois Cohomology.- XI Class Formations.- Four Local Class Field Theory.- XII Brauer Group of a Local Field.- XIII Local Class Field Theory.- XIV Local Symbols and Existence Theorem.- XV Ramification.- Supplementary Bibliography for the English Edition.978-1-4471-2992-9SeydelJRdiger Seydel, Universitt zu Kln Mathematisches Institut, Kln, GermanyTools for Computational FinanceXVII, 429p. 98 illus..Modeling Tools for Financial Options.- Generating Random Numbers with Specified Distributions.- Monte Carlo Simulation with Stochastic Differential Equations.- Standard Methods for Standard Options.- Finite-Element Methods.- Pricing of Exotic Options.- Beyond Black and Scholes.The disciplines of financial engineering and numerical computation differ greatly, however computational methods are used in a number of ways across the field of finance. It is the aim of this book to explain how such methods work in financial engineering; specifically the use of numerical methods as tools for computational finance. By concentrating on the field of option pricing, a core task of financial engineering and risk analysis, this book explores a wide range of computational tools in a coherent and focused manner and will be of use to the entire field of computational finance. Starting with an introductory chapter that presents the financial and stochastic background, the remainder of the book goes on to detail computational methods using both stochastic and deterministic approaches.Now in its fifth edition, Tools for Computational Finance has been significantly revised and contains:Anew chapter on incomplete markets which links to new appendices on Viscosity solutions and the Dupire equation;Several new parts throughout the book such as that on the calculation of sensitivities (Sect. 3.7) and the introduction of penalty methods and their application to a two-factor model (Sect. 6.7)Additional material in the field of analytical methods including Kim s integral representation and its computationGuidelines for comparing algorithms and judging their efficiencyAn extended chapter on finite elements that now includes a discussion of two-asset optionsAdditional exercises, figures and referencesWritten from the perspective of an applied mathematician, methods are introduced as tools within the book for immediate and straightforward application. A learning by calculating approach is adopted throughout this book enabling readers to explore several areas of the financial world.Interdisciplinary in nature, this book will appeal to advanced undergraduate students in mathematics, engineering and other scientific disciplines as well as professionals in financial engineering.KProvides exercises at the end of each chapter that range from simple tasks to more challenging projects

Co< vers on an introductory level the very important issue of computational aspects of derivative pricing

People with a background of stochastics, numerics, and derivative pricing will gain an immediate profit

978-0-387-98235-9 ShakarchiRami Shakarchi1Problems and Solutions for Undergraduate AnalysisXII, 368 p.^0 Sets and Mappings.- I Real Numbers.- II Limits and Continuous Functions.- III Differentiation.- IV Elementary Functions.- V The Elementary Real Integral.- VI Normed Vector Spaces.- VII Limits.- VIII Compactness.- IX Series.- X The Integral in One Variable.- XI Approximation with Convolutions.- XII Fourier Series.- XIII Improper Integrals.- XIV The Fourier Integral.- XV Functions on n-Space.- XVI The Winding Number and Global Potential Functions.- XVII Derivatives in Vector Spaces.- XVIII Inverse Mapping Theorem.- XIX Ordinary Differential Equations.- XX Multiple Integrals.- XXI Differential Forms.This volume contains all the exercises and their solutions for Lang's second edition of UNDERGRADUATE ANALYSIS. The wide variety of exercises, which range from computational to more conceptual and which are of varying difficulty, cover the following subjects and more: real numbers, limits, continuous functions, differentiation and elementary integration, normed vector spaces, compactness, series, integration in one variable, improper integrals, convolutions, Fourier series and the Fourier integral, functions in n-space, derivatives in vector spaces, inverse and implicit mapping theorem, ordinary differential equations, multiple integrals and differential forms. This volume also serves as an independent source of problems with detailed answers beneficial for anyone interested in learning analysis. Intermediary steps and original drawings provided by the author assists students in their mastery of problem solving techniques and increases their overall comprehension of the subject matter.978-0-387-94732-7SharpeR.W. Sharpe3Cartan's Generalization of Klein's Erlangen ProgramXIX, 421 p. 104 illus.*In the Ashes of the Ether: Differential Topology.- Looking for the Forest in the Leaves: Folations.- The Fundamental Theorem of Calculus.- Shapes Fantastic: Klein Geometries.- Shapes High Fantastical: Cartan Geometries.- Riemannian Geometry.- Mbius Geometry.- Projective Geometry.- Appendix A - E.]Cartan geometries were the first examples of connections on a principal bundle. They seem to be almost unknown these days, in spite of the great beauty and conceptual power they confer on geometry. The aim of the present book is to fill the gap in the literature on differential geometry by the missing notion of Cartan connections. Although the author had in mind a book accessible to graduate students, potential readers would also include working differential geometers who would like to know more about what Cartan did, which was to give a notion of 'espaces gnraliss' (= Cartan geometries) generalizing homogeneous spaces (= Klein geometries) in the same way that Riemannian geometry generalizes Euclidean geometry. In addition, physicists will be interested to see the fully satisfying way in which their gauge theory can be truly regarded as geometry.978-1-4614-9757-8XIX, 452 p. 104 illus.978-0-85729-191-2ShiraliSatish Shirali, Indian Inst. of Science Educ. & Research, Panchkula, India; Harkrishan Lal Vasudeva, Indian Inst. of Science Educ. & Research, Chandigarh, IndiaMultivariable AnalysisV, 393p. 18 illus..Preliminaries.- Functions between Euclidean Spaces.- Differentiation.- Inverse and Implicit Function Theorems.- Extrema.- Riemann Integration in Euclidean Space.- The General Stokes Theorem.- Solutions.This book provides a rigorous treatment of multivariable differential and integral calculus. Implicit function theorem and the inverse function theorem based on total derivatives is explained along with the results and the connection to solving systems of equations. There is an extensive treatment of extrema, including constrained extrema and Lagrange multipliers, covering both first order necessary conditions and second order sufficient conditions. The material on Riemann integration in n dimensions, being delicate by its very nature, is discussed in detail. Differential forms and the general Stokes' Theorem are expounded in the last chapter. With a focus on clarity rather than brevity, this text gives clear motivation, definitions and examples with transparent proofs. Much of the material included is published for the first time in textbook form, for example Schwarz' Theorem in Chapter 2 and double sequences and sufficient conditions for constrained extrema in Chapter 4. A wide selection of problems, ranging from simple to more challenging, are included with carefully formed solutions. Ideal as a classroom text or a self study resource for students, this book will appeal to higher level undergraduates in Mathematics.ZContains a number of examples as well as section-by section problems, ranging from simple to complex

Problems illustrate the application of concepts introduced as well as further developments

Extensive hints and solutions are provided

Full chapter on extrema, including examples when the Lagrange method appears to "fail"

978-3-540-74010-0Shiryaev^Albert N. Shiryaev, Russian Academy of Sciences Steklov Mathematical Institute, Moscow, RussiaOptimal Stopping RulesRandom Processes: Markov Times.- Optimal Stopping of Markov Sequences.- Optimal Stopping of Markov Processes.- Some Applications to Problems of Mathematical Statistics.MAlthough three decades have passed since first publication of this book reprinted now as a result of popular demand, the content remains up-to-date and interesting for many researchers as is shown by the many references to it in current publications

The area of application of the optimal stopping theory is very broad

978-1-4614-3687-4Problems in ProbabilityXII, 427 p. With Original Russian Edition of "Probability": Shir\t{ia}ev, Al bert Nikolaevich, Zadachi po Teorii Vero\t{ia}tnoste\u{i}, Moscow: MCCME, 2006..GPreface.- 1. Elementary Probability Theory.- 2. Mathematical Foundations of Probability Theory.- 3. Convergence of Probability Measures.- 4. Independent Random Variables.- 5. Stationary Random Sequences in Strict Sense.- 6. Stationary Random Sequences in Broad Sense.- 7. Martingales.- 8. Markov Chains.- Appendix.- References.For the first two editions of the book Probability (GTM 95), each chapter included a comprehensive and diverse set of relevant exercises. While the work on the third edition was still in progress, it was decided that it would be more appropriate to publish a separate book that would comprise all of the exercises from previous editions,in addition tomany new exercises.Most of the material in this book consists of exercises created by Shiryaev, collected and compiled over the course of many years while working on many interesting topics.Many of the exercises resulted from discussions that took place during special seminars for graduate and undergraduate students. Many of the exercises included in the book contain helpful hints and other relevant information.Lastly, the author has included an appendix at the end of the book that contains a summary of the main results, notation and terminology from Probability Theory that are used throughout the present book. This Appendix also contains additional material from Combinatorics, Potential Theory and Markov Chains, which is not covered in the book, but is nevertheless needed for many of the exercises included here.NProvides more than 1500 exercises and problems for professors using GTM 95 as a course text

Volume is self-contained, although it can be used along with GTM 95

Covers traditional areas of probability theory, as well as recent developments

Author is an experienced writer and a well-known expert in the field

978-1-4899-9941-2978-0-387-87< 836-2 ShonkwilerRonald W. Shonkwiler, Georgia Institute of Technology, Palmetto, GA, USA; Franklin Mendivil, Acadia University Dept. Mathematics, Wolfville, NS, Canada#Explorations in Monte Carlo MethodsXII, 244p. 50 illus..SCI17036.Probability and Statistics in Computer ScienceUYAMto Monte Carlo Methods.- Some Probability Distributions and Their Uses.- Markov Chain Monte Carlo.- Optimization by Monte Carlo Methods.- Random Walks.DMonte Carlo methods are among the most used and useful computational tools available today, providing efficient and practical algorithims to solve a wide range of scientific and engineering problems. Explorations in Monte Carlo Methods provides a hands-on approach to learning this subject. Each new idea is carefully motivated by a realistic problem, thus leading from questions to theory via examples and numerical simulations. Programming exercises are integrated throughout the text as the primary vehicle for learning the material. Each chapter ends with a large collection of problems illustrating and directing the material. This book is suitable as a textbook for students of engineering and the sciences, as well as mathematics. The problem-oriented approach makes it ideal for an applied course in basic probability and for a more specialized course in Monte Carlo methods. Topics include probability distributions, counting combinatorial objects, simulated annealing, genetic algorithms, option pricing, gamblers ruin, statistical mechanics, sampling, and random number generation.Applications covered: optimization, finance, statistical mechanics, birth and death processes, and gambling systems

Hands-on approach is used via realistic problems demonstrated with examples and numerical simulations

A wealth of completely solved example problems provide the reader with a sourcebook to follow toward the solution of their own computational problems.

Each chapter ends with a large collection of homework problems illustrating and directing the material

978-1-4899-8379-4XII, 244 p. 50 illus.978-3-540-41195-6ShubinM.A. Shubin0Pseudodifferential Operators and Spectral TheoryXII, 288 pp.KI. Foundations of ?DO Theory.- 1. Oscillatory Integrals.- 2. Fourier Integral Operators (Preliminaries).- 3. The Algebra of Pseudodifferential Operators and Their Symbols.- 4. Change of Variables and Pseudodifferential Operators on Manifolds.- 5. Hypoellipticity and Ellipticity.- 6. Theorems on Boundedness and Compactness of Pseudodifferential Operators.- 7. The Sobolev Spaces.- 8. The Fredholm Property, Index and Spectrum.- II. Complex Powers of Elliptic Operators.- 9. Pseudodifferential Operators with Parameter. The Resolvent.- 10. Definition and Basic Properties of the Complex Powers of an Elliptic Operator.- 11. The Structure of the Complex Powers of an Elliptic Operator.- 12. Analytic Continuation of the Kernels of Complex Powers.- 13. The ?-Function of an Elliptic Operator and Formal Asymptotic Behaviour of the Spectrum.- 14. The Tauberian Theorem of Ikehara.- 15. Asymptotic Behaviour of the Spectral Function and the Eigenvalues (Rough Theorem).- III. Asymptotic Behaviour of the Spectral Function.- 16. Formulation of the Hormander Theorem and Comments.- 17. Non-linear First Order Equations.- 18. The Action of a Pseudodifferential Operator on an Exponent.- 19. Phase Functions Defining the Class of Pseudodifferential Operators.- 20. The Operator exp( it A).- 2l. Precise Formulation and Proof of the Hormander Theorem.- 22. The Laplace Operator on the Sphere.- IV. Pseudodifferential Operators in ?n.- 23. An Algebra of Pseudodifferential Operators in ?n.- 24. The Anti-Wick Symbol. Theorems on Boundedness and Compactness.- 25. Hypoellipticity and Parametrix. Sobolev Spaces. The Fredholm Property.- 26. Essential Self-Adjointness. Discreteness of the Spectrum.- 27. Trace and Trace Class Norm.- 28. The Approximate Spectral Projection.- 29. Operators with Parameter.- 30. Asymptotic Behaviour ofthe Eigenvalues.- Appendix 1. Wave Fronts and Propagation of Singularities.- Appendix 2. Quasiclassical Asymptotics of Eigenvalues.- Appendix 3. Hilbert-Schmidt and Trace Class Operators.- A Short Guide to the Literature.- Index of Notation.-This is the second edition of Shubin's classical book. It provides an introduction to the theory of pseudodifferential operators and Fourier integral operators from the very basics. The applications discussed include complex powers of elliptic operators, Hrmander asymptotics of the spectral function and eigenvalues, and methods of approximate spectral projection. Exercises and problems are included to help the reader master the essential techniques. The book is written for a wide audience of mathematicians, be they interested students or researchers.Second editon of Shubin's classic from 1987 as softcoverIntroduction to the theory of pseudodifferential operators and Fourier integral operators from the very basics

Contains numerous exercises and problems

For interested students or researchers978-0-387-30526-4SinclairNathalie Sinclair, Simon Fraser University Fac. Education, Burnaby, BC, Canada; William Higginson, Queen's University Faculty of Education, Kingston, ON, CanadaMathematics and the Aesthetic%New Approaches to an Ancient Affinity XV, 288 p.6A Historical Gaze at the Mathematical Aesthetic.- A Historical Gaze at the Mathematical Aesthetic.- The Mathematician s Art.- Aesthetics for the Working Mathematician.- Beauty and Truth in Mathematics.- Experiencing Meanings in Geometry.- A Sense for Mathematics.- The Aesthetic Sensibilities of Mathematicians.- The Meaning of Patt< ern.- Mathematics, Aesthetics and Being Human.- Mathematical Agency.- Mechanism and Magic in the Psychology of Dynamic Geometry.- Drawing on the Image in Mathematics and Art.- Sensible Objects.- Aesthetics and the Mathematical Mind .The essays in this book explore the ancient affinity between the mathematical and the aesthetic, focusing on the fundamental connections between these two modes of reasoning and communicating. From historical, philosophical and psychological perspectives, with particular attention to certain mathematical areas such as geometry and analysis, the authors examine the ways in which the aesthetic is ever present in mathematical thinking and contributes to the growth and value of mathematical knowledge.

Focuses on affinities between mathematics and the arts

Articulates common strains between the mathematical and the aesthetic

Shows how the fundamentals are deeply related to human sense-making and learning

Offers new possibilities for mathematics education

978-1-4419-2144-4978-0-8176-8384-9SobczykDGarret Sobczyk, Universitad de Las Amricas, Cholula, Puebla, MexicoNew Foundations in MathematicsThe Geometric Concept of Number*XIV, 370 p. 55 illus., 32 illus. in color.u1 Modular Number Systems.- 2 Complex and Hyperbolic Numbers.- 3 Geometric Algebra.- 4 Vector Spaces and Matrices.- 5 Outer Product and Determinants.- 6 Systems of Linear Equations.- 7 Linear Transformations on R^n.- 8 Structure of a Linear Operator.- 9 Linear and Bilinear Forms.- 10 Hermitian Inner Product Spaces.- 11 Geometry of Moving Planes.- 12 Representations of the Symmetric Group.- 13 Calculus on m-Surfaces.- 14 Differential Geometry of Curves.- 15 Differential Geometry of k-Surfaces.- 16 Mappings Between Surfaces.- 17 Non-Euclidean and Projective Geometries.- 18 Lie Groups and Lie Algebras.- References.- Symbols. The first book of its kind, New Foundations in Mathematics: The Geometric Concept of Number uses geometric algebra to present an innovative approach to elementary and advanced mathematics. Geometric algebra offers a simple and robust means of expressing a wide range of ideas in mathematics, physics, and engineering. In particular, geometric algebra extends the real number system to include the concept of direction, which underpins much of modern mathematics and physics. Much of the material presented has been developed from undergraduate courses taught by the author over the years in linear algebra, theory of numbers, advanced calculus and vector calculus, numerical analysis, modern abstract algebra, and differential geometry. The principal aim of this book is to present these ideas in a freshly coherent and accessible manner.New Foundations in Mathematics will be of interest to undergraduate and graduate students of mathematics and physics who are looking for a unified treatment of many important geometric ideas arising in these subjects at all levels. The material can also serve as a supplemental textbook in some or all of the areas mentioned above and as a reference book for professionals who apply mathematics to engineering and computational areas of mathematics and physics.Geometric algebra, the central topic of the book, provides a uniquewayto unify, streamline, and simplifyteaching and applyingmany different areas of mathematics

Presents and casts new light onan impressively broad range of math from elementary level

Features many exercises and problems to enhance the reader's skills and understanding

Topics and applicationsselected to encourage further, more specific research by both graduates and undergraduates

978-0-387-75469-7SoiferWAlexander Soifer, University of Colorado at Colorado Springs, Colorado Springs, CO, USA-Geometric Etudes in Combinatorial MathematicsXXX, 348p. 332 illus..sORIGINAL ETUDES.- Tiling a Checker Rectangle.- Proofs of Existence.- A Word About Graphs.- Ideas of Combinatorial Geometry.- NEW LANDSCAPE, OR THE VIEW 18 YEARS LATER.- Mitya Karabash and a Tiling Conjecture.- Norton Starr s 3-Dimensional Tromino Tiling.- Large Progress in Small Ramsey Numbers.- The Borsuk Problem Conquered.- Etude on the Chromatic Number of the Plane.AThis second edition of Alexander Sofier s Geometric Etudes in Combinatorial Mathematics provides supplementary reading materials to students of all levels interested in pursuing mathematics, especially in algebra, geometry, and combinatorial geometry. Within the text, the author outlines an introduction to graph theory and combinatorics while exploring topics such as the pigeonhole principle, Borsuk problem, and theorems of Helly and Szokefalvi Nagy. The book introduces these ideas along with practical applications that will prepare young readers for the mathematical world. Geometric Etudes in Combinatorial Mathematics is not only educational; it is inspirational. This distinguished mathematician captivates his readers, propelling them to search for solutions of life s problems- problems that previously seemed hopeless.Appeals to talented students from various levels

Provides insights into combinatorial theory, geometry and graph theory

Explores practical applications of combinatorial geometry

Engages a general audience

978-0-8176-8091-6 Ramsey TheoryYesterday, Today, and TomorrowXIV, 190p. 28 illus..How This Book Came into Being.- Table of Contents.- Ramsey Theory before Ramsey, Prehistory and Early History: An Essay in 13 Parts.- Eighty Years of Ramsey R(3, k). . . and Counting!.- Ramsey Numbers Involving Cycles.- On the function of ErdEs and Rogers.- Large Monochromatic Components in Edge Colorings of Graphs.- Szlam s Lemma: Mutant Offspring of a Euclidean Ramsey Problem: From 1973, with Numerous Applications.- Open Problems in Euclidean Ramsey Theory.- Chromatic Number of the Plane and Its Relatives, History, Problems and Results: An Essay in 11 Parts.- Euclidean Distance Graphs on the Rational Points.- Open Problems Session.Ramsey theory is a relatively new, approx< imately 100 year-old direction of fascinating mathematical thought that touches on many classic fields of mathematics such as combinatorics, number theory, geometry, ergodic theory, topology, combinatorial geometry, set theory, and measure theory. Ramsey theory possesses its own unifying ideas, and some of its results are among the most beautiful theorems of mathematics. The underlying theme of Ramsey theory can be formulated as: any finite coloring of a large enough system contains a monochromatic subsystem of higher degree of organization than the system itself, or as T.S. Motzkin famously put it, absolute disorder is impossible. Ramsey Theory: Yesterday, Today, and Tomorrow explores the theory s history, recent developments, and some promising future directions through invited surveys written by prominent researchers in the field. The first three surveys provide historical background on the subject; the last three address Euclidean Ramsey theory and related coloring problems. In addition, open problems posed throughout the volume and in the concluding open problem chapter will appeal to graduate students and mathematicians alike.hExplores Ramsey theory s history, recent developments, and some promising future directions through invited surveys written by prominent researchers in the field Provides historical background on the subject Addresses Euclidean Ramsey theory and related coloring problems Open problems are posed throughout the volume and in the concluding open problem chapter978-0-387-75471-0;The Colorado Mathematical Olympiad and Further Explorations:From the Mountains of Colorado to the Peaks of Mathematics,XXXIX, 408p. 185 illus., 18 illus. in color.Preface.- Olympiad History: What it is and How it Started.- Three Celebrated Ideas.- Year 1.- Year 2.- Year 3.- Year 4.- Year 5.- Year 6.- Year 7.- Year 8.- Year 9.- Year 10.- Further Explorations.- Rooks in Space.- Chromatic Number of the Plane.- Polygons in a Colored Circle, Polyhedra in a colored Sphere.- How Does one Cut a Triangle?.- Points in Convex Figures.- Triangles in a Colored Plane.- Rectangles in a Colored Plane.- Colored Polygons.- Infinite-Finite.- Schur Theorem.- Bibliography.- Year 11.- Year 12.- Year 13.- Year 14.- Year 15.- Year 16.- Year 17.- Year 18.- Year 19.- Year 20.- Further Explorations.- Chromatic Number of a Grid.- Stone Age Entertainment.- The Erds Problem.- Squares in a Square.- Washington Recangles.- Olde Victorian Map Colouring.- More Stone Age Entertainment.- The 1-10-100 Problem.- King Arthur and the Knights of the Round Table.- A Map Coloring 'Game'.- Bibliography.vOver the past two decades, the once small local Colorado Springs Mathematics Olympiad, founded by the author himself, has now become an annual state-wide competition, hosting over one-thousand high school contenders each year. This updated printing of the first edition of Colorado Mathematical Olympiad: the First Twenty Years and Further Explorations offers an interesting history of the competition as well as an outline of all the problems and solutions that have been a part of the contest over the years. Many of the essay problems were inspired by Russian mathematical folklore and written to suit the young audience; for example, the 1989 Sugar problem was writtenas a pleasant Lewis Carroll-like story. Some other entertaining problems involve old Victorian map colorings, King Arthur and the knights of the round table, rooks in space, Santa Claus and his elves painting planes, football for 23, and even the Colorado Springs subway system.The book is more than just problems, their solutions, and event statistics; it tells a compelling story involving the lives of those who have been part of the Olympiad from every perspective.W<p>Builds bridges between Olympiads and real mathematics by showing how a solved Olympiad problem gives birth to deeper problems and leads to the forefront of mathematical research</p> <p>Appeals to both serious and recreational mathematicians on all levels of expertise</p> <p>Pairs excellent mathematical content with artful exposition</p>978-0-387-97970-0CJohn Stillwell, University of San Francisco, San Francisco, CA, USA1Classical Topology and Combinatorial Group TheoryXII, 334 p. 312 illus.50 Introduction and Foundations.- 0.1 The Fundamental Concepts and Problems of Topology.- 0.2 Simplicial Complexes.- 0.3 The Jordan Curve Theorem.- 0.4 Algorithms.- 0.5 Combinatorial Group Theory.- 1 Complex Analysis and Surface Topology.- 1.1 Riemann Surfaces.- 1.2 Nonorientable Surfaces.- 1.3 The Classification Theorem for Surfaces.- 1.4 Covering Surfaces.- 2 Graphs and Free Groups.- 2.1 Realization of Free Groups by Graphs.- 2.2 Realization of Subgroups.- 3 Foundations for the Fundamental Group.- 3.1 The Fundamental Group.- 3.2 The Fundamental Group of the Circle.- 3.3 Deformation Retracts.- 3.4 The Seifert Van Kampen Theorem.- 3.5 Direct Products.- 4 Fundamental Groups of Complexes.- 4.1 Poincar s Method for Computing Presentations.- 4.2 Examples.- 4.3 Surface Complexes and Subgroup Theorems.- 5 Homology Theory and Abelianization.- 5.1 Homology Theory.- 5.2 The Structure Theorem for Finitely Generated Abelian Groups.- 5.3 Abelianization.- 6 Curves on Surfaces.- 6.1 Dehn s Algorithm.- 6.2 Simple Curves on Surfaces.- 6.3 Simpl< ification of Simple Curves by Homeomorphisms.- 6.4 The Mapping Class Group of the Torus.- 7 Knots and Braids.- 7.1 Dehn and Schreier s Analysis of the Torus Knot Groups.- 7.2 Cyclic Coverings.- 7.3 Braids.- 8 Three-Dimensional Manifolds.- 8.1 Open Problems in Three-Dimensional Topology.- 8.2 Polyhedral Schemata.- 8.3 Heegaard Splittings.- 8.4 Surgery.- 8.5 Branched Coverings.- 9 Unsolvable Problems.- 9.1 Computation.- 9.2 HNN Extensions.- 9.3 Unsolvable Problems in Group Theory.- 9.4 The Homeomorphism Problem.- Bibliography and Chronology.This is a well-balanced introduction to topology that stresses geometric aspects. Focusing on historical background and visual interpretation of results, it emphasizes spaces with few dimensions, where visualization is possible, and interaction with combinatorial group theory via the fundamental group. It also present algorithms for topological problems. Most of the results and proofs are known, but some have been simplified or placed in a new perspective. Over 300 illustrations, many interesting exercises, and challenging open problems are included. New in this edition is a chapter on unsolvable problems, which includes the first textbook proof that the main problem of topology, the homeomorphism problem, is unsolvable.978-0-8176-4197-9 StojanovicRSrdjan Stojanovic, University of Cincinnati Dept. Mathematics, Cincinnati, OH, USA6Computational Financial Mathematics using MATHEMATICA%Optimal Trading in Stocks and Options$XI, 481 p. With online files/update.SCI23001Computer ApplicationsUB0 Introduction.- 0.1 Audience, Highlights, Agenda.- 0.2 Software Installation.- 0.3 Acknowledgments.- Chapter1 Cash Account Evolution.- 1.1 Symbolic Solutions of ODEs.- 1.2 Numerical Solutions of ODEs.- 2 Stock Price Evolution.- 2.1 What are Stocks?.- 2.2 Stock Price Modeling: Stochastic Differential Equations.- 2.3 It Calculus.- 2.4 Multivariate and Symbolic It Calculus.- 2.5 Relationship Between SDEs and PDEs.- 3 European Style Stock Options.- 3.1 What Are Stock Options?.- 3.2 Black-Scholes PDE and Hedging.- 3.3 Solving Black-Scholes PDE Symbolically.- 3.4 Generalized Black-Scholes Formulas: Time-Dependent Data.- 4 Stock Market Statistics.- 4.1 Remarks.- 4.2 Stock Market Data Import and Manipulation.- 4.3 Volatility Estimates: Scalar Case.- 4.4 Appreciation Rate Estimates: Scalar Case.- 4.5 Statistical Experiments: Bayesian and Non-Bayesian.- 4.6 Vector Basic Price Model Statistics.- 4.7 Dynamic Statistics: Filtering of Conditional Gaussian Processes.- 5 Implied Volatility for European Options.- 5.1 Remarks.- 5.2 Option Market Data.- 5.3 Black-Scholes Theory vs. Market Data: Implied Volatility.- 5.4 Numerical PDEs, Optimal Control, and Implied Volatility.- 6 American Style Stock Options.- 6.1 Remarks.- 6.2 American Options and Obstacle Problems.- 6.3 General Implied Volatility for American Options.- 7 Optimal Portfolio Rules.- 7.1 Remarks.- 7.2 Utility of Wealth.- 7.3 Merton s Optimal Portfolio Rule Derived and Implemented.- 7.4 Portfolio Rules under Appreciation Rate Uncertainty.- 7.5 Portfolio Optimization under Equality Constraints.- 7.6 Portfolio Optimization under Inequality Constraints.- 8 Advanced Trading Strategies.- 8.1 Remarks.- 8.2 Reduced Monge Ampere PDEs of Advanced Portfolio Hedging.- 8.3 Hypoelliptic Obstacle Problems in Optimal Momentum Trading.Given the explosion of interest in mathematical methods for solving problems in finance and trading, a great deal of research and development is taking place in universities, large brokerage firms, and in the supporting trading software industry. Mathematical advances have been made both analytically and numerically in finding practical solutions. This book provides a comprehensive overview of existing and original material, about what mathematics when allied with Mathematica can do for finance. Sophisticated theories are presented systematically in a user-friendly style, and a powerful combination of mathematical rigor and Mathematica programming. Three kinds of solution methods are emphasized: symbolic, numerical, and Monte-- Carlo. Nowadays, only good personal computers are required to handle the symbolic and numerical methods that are developed in this book. Key features: * No previous knowledge of Mathematica programming is required * The symbolic, numeric, data management and graphic capabilities of Mathematica are fully utilized * Monte--Carlo solutions of scalar and multivariable SDEs are developed and utilized heavily in discussing trading issues such as Black--Scholes hedging * Black--Scholes and Dupire PDEs are solved symbolically and numerically * Fast numerical solutions to free boundary problems with details of their Mathematica realizations are provided * Comprehensive study of optimal portfolio diversification, including an original theory of optimal portfolio hedging under non-Log-Normal asset price dynamics is presented The book is designed for the academic community of instructors and students, and most importantly, will meet the everyday trading needs of quantitatively inclined professional and individual investors.978-1-4614-1134-5StroockLDaniel W. Stroock, Massachusetts Institute of Technology, Cambridge, MA, USA-Essentials of Integration Theory for AnalysisXII, 244 p.8-Preface.-1. The Classical Theory.-2. Measures. -3. Lebesgue Integration.-4. Products of Measures.-5. Changes of Variable.-6. Basic Inequalities and Lebesgue Spaces.-7. Hilbert Space and Elements of Fourier Analysis.-8. The Radon-Nikodym Theorem, Daniell Integration, and Carathodory's Extension Theorem.-Index. A Concise Introduction to the Theory of Integration was once a best-selling Birkhuser title which published 3 editions. This< manuscript is a substantial revision of the material. Chapter one now includes a section about the rate of convergence of Riemann sums. The second chapter now covers both Lebesgue and Bernoulli measures, whose relation to one another is discussed. The third chapter now includes a proof of Lebesgue's differential theorem for all monotone functions. This is a beautiful topic which is not often covered. The treatment of surface measure and the divergence theorem in the fifth chapter has been improved. Loose ends from the discussion of the Euler-MacLauren in Chapter I are tied together in Chapter seven. Chapter eight has been expanded to include a proof of Carathory's method for constructing measures; his result is applied to the construction of Hausdorff measures. The new material is complemented by the addition of several new problems based on that material.2Refocus and substantial revision of previous successful publication "A Concise Introduction to the Theory of Integration" by D.W. Stroock (Birkhauser)

Separate solutions manual available to those who adopt the textbook

New material is complemented by the addition of several new problems

978-1-4614-2988-3978-0-387-90357-6ThorpeJohn A. Thorpe*Elementary Topics in Differential GeometryXIII, 267 pp. 100 illus.I Graphs and Level Sets.- 2 Vector Fields.- 3 The Tangent Space.- 4 Surfaces.- 5 Vector Fields on Surfaces; Orientation.- 6 The Gauss Map.- 7 Geodesics.- 8 Parallel Transport.- 9 The Weingarten Map.- 10 Curvature of Plane Curves.- 11 Arc Length and Line Integrals.- 12 Curvature of Surfaces.- 13 Convex Surfaces.- 14 Parametrized Surfaces.- 15 Local Equivalence of Surfaces and Parametrized Surfaces.- 16 Focal Points.- 17 Surface Area and Volume.- 18 Minimal Surfaces.- 19 The Exponential Map.- 20 Surfaces with Boundary.- 21 The Gauss-Bonnet Theorem.- 22 Rigid Motions and Congruence.- 23 Isometries.- 24 Riemannian Metrics.- Notational Index.This introductory text develops the geometry of n-dimensional oriented surfaces in Rn+1. By viewing such surfaces as level sets of smooth functions, the author is able to introduce global ideas early without the need for preliminary chapters developing sophisticated machinery. the calculus of vector fields is used as the primary tool in developing the theory. Coordinate patches are introduced only after preliminary discussions of geodesics, parallel transport, curvature, and convexity. Differential forms are introduced only as needed for use in integration. The text, which draws significantly on students' prior knowledge of linear algebra, multivariate calculus, and differential equations, is designed for a one-semester course at the junior/senior level.978-1-4614-4285-1Touzi9Nizar Touzi, cole Polytechnique, Palaiseau Cedex, FranceHOptimal Stochastic Control, Stochastic Target Problems, and Backward SDEX, 214 p. 1 illus.9Preface.- 1. Conditional Expectation and Linear Parabolic PDEs.- 2. Stochastic Control and Dynamic Programming.- 3. Optimal Stopping and Dynamic Programming.- 4. Solving Control Problems by Verification.- 5. Introduction to Viscosity Solutions.- 6. Dynamic Programming Equation in the Viscosity Sense.- 7. Stochastic Target Problems.- 8. Second Order Stochastic Target Problems.- 9. Backward SDEs and Stochastic Control.- 10. Quadratic Backward SDEs.- 11. Probabilistic Numerical Methods for Nonlinear PDEs.- 12. Introduction to Finite Differences Methods.- References. This book collects some recent developments in stochastic control theory with applications to financial mathematics. We first address standard stochastic control problems from the viewpoint of the recently developed weak dynamic programming principle. A special emphasis is put on the regularity issues and, in particular, on the behavior of the value function near the boundary. We then provide a quick review of the main tools from viscosity solutions which allow to overcome all regularity problems. We next address the class of stochastic target problems which extends in a nontrivial way the standard stochastic control problems. Here the theory of viscosity solutions plays a crucial role in the derivation of the dynamic programming equation as the infinitesimal counterpart of the corresponding geometric dynamic programming equation. The various developments of this theory have been stimulated by applications in finance and by relevant connections with geometric flows. Namely, the second order extension was motivated by illiquidity modeling, and the controlled loss version was introduced following the problem of quantile hedging. The third part specializes to an overview of Backward stochastic differential equations, and their extensions to the quadratic case. <p> Provides a self-contained presentation of the recent developments in Stochastic target problems which cannot be found in any other monograph</p><p>Approaches quadratic backward stochastic differential equations following the point of view of Tevzadze and presented in a way to maximize the ease of understanding </p><p>Contains relevant examples from finance, including the Nash equilibrium example </p>978-1-4939-0042-8978-3-0348-0568-1Triebel?Hans Triebel, Friedrich-Schiller-University Jena, Jena, GermanyThe Structure of FunctionsXII, 425 p.Preface.- I Decompositions of Functions.- II Sharp Inequalities.- III Fractal Elliptic Operators.- IV Truncations and Semi-linear Equations.- References.- Symbols.- Index. This book deals with the constructive Weierstrassian approach to the theory of function spaces and various applications. The first chapter is devoted to a detailed study of quarkonial (subatomic) decompositions of functions and distributions on euclidean spaces, domains, manifolds and fractals. This approach combines the advantages of atomic and wavelet representations. It paves the way to sharp inequalities and embeddings in function spaces, spectral theory of fractal elliptic operators, and a regularity theory of some semi-linear equations. The book is self-contained, although some parts may be considered as a continuation of the author's book Fractals and Spectra. It is directed to mathematicians and (theoretical) physicists interested in the topics indicated and, in particular, how they are interrelated. - - - The book un< der review can be regarded as a continuation of [his book on 'Fractals and spectra', 1997] (...) There are many sections named: comments, preparations, motivations, discussions and so on. These parts of the book seem to be very interesting and valuable. They help the reader to deal with the main course. (Mathematical Reviews)~<p>Self-contained, i.e. the main ideas can be understood independently of the existing literature </p><p>Summarizesthe results of the author and his co-workers in recent years </p><p>The material is presented in such a way that the main ideas can be understood independently of the existing literature </p><p>Addresses mathematicians and (theoretical) physicists </p>978-0-387-94511-8John L. Troutman(Variational Calculus and Optimal Control&Optimization with Elementary Convexity XV, 462 p.a0 Review of Optimization in ?d.- Problems.- One Basic Theory.- 1 Standard Optimization Problems.- 2 Linear Spaces and Gteaux Variations.- 3 Minimization of Convex Functions.- 4 The Lemmas of Lagrange and Du Bois-Reymond.- 5 Local Extrema in Normed Linear Spaces.- 6 The Euler-Lagrange Equations.- Two Advanced Topics.- 7 Piecewise C1 Extremal Functions.- 8 Variational Principles in Mechanics.- 9 Sufficient Conditions for a Minimum.- Three Optimal Control.- 10 Control Problems and Sufficiency Considerations.- 11 Necessary Conditions for Optimality.- A.1. The Intermediate and Mean Value Theorems.- A.2. The Fundamental Theorem of Calculus.- A.3. Partial Integrals: Leibniz Formula.- A.4. An Open Mapping Theorem.- A.5. Families of Solutions to a System of Differential Equations.- A.6. The Rayleigh Ratio.- Historical References.- Answers to Selected Problems.*The text provides an introduction to the variational methods used to formulate and solve mathematical and physical problems and gives the reader an insight into the systematic use of elementary (partial) convexity of differentiable functions in Euclidian space. By helping students directly characterize then the solutions for many minimization problems, the text serves as a prelude to the field theory for sufficiency. It lays the groundwork for further explorations in mathematics, physics, mechanical and electrical engineering, and computer science.978-1-4614-0477-4UllahMukhtar Ullah, Universitt Rostock Institut of Computer Science, Rostock, Germany; Olaf Wolkenhauer, Universitt Rostock Institut of Computer Science, Rostock, Germany)Stochastic Approaches for Systems BiologyXXXII, 290p. 73 illus..SCL15010Systems BiologyPSAWPreface.-Acknowledgements.- Acronyms, notation.- Matlab functions, revisited examples.- Introduction.- Biochemical reaction networks.- Randomness.- Probability and random variables.- Stochastic modeling of biochemical networks.- The 2MA approach.- The 2MA cell cycle model.- Hybrid Markov processes.- Wet-lab experiments and noise.- Glossary.This textbook focuses on stochastic analysis in systems biology containing both the theory and application. While the authors provide a review of probability and random variables, subsequent notions of biochemical reaction systems and the relevant concepts of probability theory are introduced side by side. This leads to an intuitive and easy-to-follow presentation of stochastic framework for modeling subcellular biochemical systems. In particular, the authors make an effort to show how the notion of propensity, the chemical master equation and the stochastic simulation algorithm arise as consequences of the Markov property.The text contains many illustrations, examples and exercises to illustrate the ideas and methods that are introduced. Matlab code is also provided where appropriate. Additionally, the cell cycle is introduced as a more complex case study. Senior undergraduate and graduate students in mathematics and physics as well as researchers working in the area of systems biology, bioinformatics and related areas will find this text useful.Focuses on both analytical and numericalapproaches

Key Concepts in Stochastic Processes

Matlab examples provided

Many illustrations and examples included

978-1-4899-9491-2XXXII, 290 p. 73 illus.978-0-387-72765-3 UnderwoodIRobert G. Underwood, Auburn University at Montgomery, Montgomery, AL, USA An Introduction to Hopf AlgebrasXIV, 273p. 3 illus..Preface.- Some Notation.- 1. The Spectrum of a Ring.-2. The Zariski Topology on the Spectrum.-3. Representable Group Functors.-4. Hopf Algebras. -5. Larson Orders.-6. Formal Group Hopf Orders.-7. Hopf Orders in KC_p.-8. Hopf Orders in KC_{p^2}.-9. Hopf Orders in KC_{p^3}.-10. Hopf Orders and Galois Module Theory.-11. The Class Group of a Hopf Order.-12. Open Questions and Research Problems.-Bibliography.-Index.<With wide-ranging connections to fields from theoretical physics to computer science, Hopf algebras offer students a glimpse at the applications of abstract mathematics. This book is unique in making this engaging subject accessible to advanced undergraduate and beginning graduate students. After providing a self-contained introduction to group and ring theory, the book thoroughly treats the concept of the spectrum of a ring and the Zariski topology. In this way the student transitions smoothly from basic abstract algebra to Hopf algebras. The importance of Hopf orders is underscored with applications to algebraic number theory, Galois module theory and the theory of formal groups. By the end of the book, readers will be familiar with established results in the field and ready to pose research questions of their own.1Offers the only self-contained treatment of Hopf algebras, and makes Hopf algebras accessible to advanced undergraduates and beginning graduate students

Underscores importance of Hopf orders with applications to algebraic number theory, Galois module theory and the theory of formal groups

978-1-4899-9784-5XIV, 273 p. 3 illus.978-3-0348-0165-2Unterberger[Andr Unterberger, Universit Reims Dpt. Mathmatiques et Informatique, Reims CX 2, FranceUPseudodifferential Analysis, Automorphic Distributions in the Plane and Modular FormsVIII, 300p.Introduction.- The Weyl calculus.- The Radon transformation and applications.- Automorphic functions and automorphic distributions.- A class of Poincar series.- Spectral decomposition of the Poincar summation process.- The totally radial Weyl calculus and arithmetic.- Should one generalize the Weyl calculus to an adelic setting?.- Index of notation.- Subject Index.- Bibliography._Pseudodifferential analysis, introduced in this book in a way adapted to the needs of number theorists, relates automorphic function theory in the hyperbolic half-plane to< automorphic distribution theory in the plane. Spectral-theoretic questions are discussed in one or the other environment: in the latter one, the problem of decomposing automorphic functions in according to the spectral decomposition of the modular Laplacian gives way to the simpler one of decomposing automorphic distributions in R2 into homogeneous components. The Poincar summation process, which consists in building automorphic distributions as series of g-transforms, for g E SL(2;Z), of some initial function, say in S(R2), is analyzed in detail. On , a large class of new automorphic functions or measures is built in the same way: one of its features lies in an interpretation, as a spectral density, of the restriction of the zeta function to any line within the critical strip.The book is addressed to a wide audience of advanced graduate students and researchers working in analytic number theory or pseudo-differential analysis..Presents pseudodifferential analysis tailored to the needs of number theorists

Besides containing novel features of the Weil calculus, it may constitute an approach to non-holomorphic modular form theory, suitable for analysts

Gives hints to possible generalizations of the construction of new classes of automorphic functions

Explains how and why pseudodifferential analysis should be developed in the adelic setting

Series of Kloosterman sums, some of which are of a novel kind, play a major

role in several parts of the book978-0-8176-8318-4Wallis>W.D. Wallis, Southern Illinois University, Evansville, IN, USA(A Beginner's Guide to Finite Mathematics1For Business, Management, and the Social SciencesXIII, 483p. 186 illus..Preface.-Numbers and Sets.-Counting.-Probability.-Graph Theory.- Linear Equations and Matrices.- Linear Programming.-Theory of Games.-Financial Mathematics.-Your Turn Solutions.- Answers to Exercises A.- Index.This second edition of A Beginner s Guide to Finite Mathematics: For Business, Management, and the Social Sciencestakes a distinctly applied approach to finite mathematics at the freshman and sophomore level. Topics are presented sequentially: the book opens with a brief review of sets and numbers, followed by an introduction to data sets, histograms, means and medians. Counting techniques and the Binomial Theorem are covered, which provide the foundation for elementary probability theory; this, in turn, leads to basic statistics. This new edition includes chapters on game theory and financial mathematics.Requiring little mathematical background beyond high school algebra, the text will be especially useful for business and liberal arts majors for study in the classroom or for self-study. Its straightforward treatment of the essential concepts in finite mathematics will appeal to a wide audience of students and teachers.Covers the topics of counting, discrete probability, graph theory, linear equations, and linear programming

Includes many commercial applications, including linear programming, the theory of games, and financial mathematics

Especially useful for business and liberal arts majors for study in the classroom or self-study

Ample examples and illustrations are provided throughout

This new edition includes chapters on game theory and financial mathematics

978-0-387-98459-9WalterTWolfgang Walter, Universitt Karlsruhe Mathematisches Institut I, Karlsruhe, Germany XI, 384 p.1I. First Order Equations: Some Integrable Cases.- 1. Explicit First Order Equations.- 2. The Linear Differential Equation. Related Equations.- 3. Differential Equations for Families of Curves. Exact Equations.- 4. Implicit First Order Differential Equations.- II: Theory of First Order Differential Equations.- 5. Tools from Functional Analysis.- 6. An Existence and Uniqueness Theorem.- 7. The Peano Existence Theorem.- 8. Complex Differential Equations. Power Series Expansions.- 9. Upper and Lower Solutions. Maximal and Minimal Integrals.- III: First Order Systems. Equations of Higher Order.- 10. The Initial Value Problem for a System of First Order.- 11. Initial Value Problems for Equations of Higher Order.- 12. Continuous Dependence of Solutions.- 13. Dependence of Solutions on Initial Values and Parameters.- IV: Linear Differential Equations.- 14. Linear Systems.- 15. Homogeneous Linear Systems.- 16. Inhomogeneous Systems.- 17. Systems with Constant Coefficients.- 18. Matrix Functions. Inhomogeneous Systems.- 19. Linear Differential Equations of Order n.- 20. Linear Equations of Order nwith Constant Coefficients.- V: Complex Linear Systems.- 21. Homogeneous Linear Systems in the Regular Case.- 22. Isolated Singularities.- 23. Weakly Singular Points. Equations of Fuchsian Type.- 24. Series Expansion of Solutions.- 25. Second Order Linear Equations.- VI:< Boundary Value and Eigenvalue Problems.- 26. Boundary Value Problems.- 27. The Sturm Liouville Eigenvalue Problem.- 28. Compact Self-Adjoint Operators in Hilbert Space.- VII: Stability and Asymptotic Behavior.- 29. Stability.- 30. The Method of Lyapunov.- A. Topology.- B. Real Analysis.- C. C0111plex Analysis.- D. Functional Analysis.- Solutions and Hints for Selected Exercises.- Literature.- Notation.Develops the theory of initial-, boundary-, and eigenvalue problems, real and complex linear systems, asymptotic behavior and stability. Using novel approaches to many subjects, the book emphasizes differential inequalities and treats more advanced topics such as Caratheodory theory, nonlinear boundary value problems and radially symmetric elliptic problems. New proofs are given which use concepts and methods from functional analysis. Applications from mechanics, physics, and biology are included, and exercises, which range from routine to demanding, are dispersed throughout the text. Solutions for selected exercises are included at the end of the book. All required material from functional analysis is developed in the book and is accessible to students with a sound knowledge of calculus and familiarity with notions from linear algebra. This text would be an excellent choice for a course for beginning graduate or advanced undergraduate students.978-0-387-90421-4 WaterhouseW.C. Waterhouse$Introduction to Affine Group Schemes164p.3I The Basic Subject Matter.- 1 Affine Group Schemes.- 2 Affine Group Schemes: Examples.- 3 Representations.- 4 Algebraic Matrix Groups.- II Decomposition Theorems.- 5 Irreducible and Connected Components.- 6 Connected Components and Separable Algebras.- 7 Groups of Multiplicative Type.- 8 Unipotent Groups.- 9 Jordan Decomposition.- 10 Nilpotent and Solvable Groups.- III The Infinitesimal Theory.- 11 Differentials.- 12 Lie Algebras.- IV Faithful Flatness and Quotients.- 13 Faithful Flatness.- 14 Faithful Flatness of Hopf Algebras.- 15 Quotient Maps.- 16 Construction of Quotients.- V Descent Theory.- 17 Descent Theory Formalism.- 18 Descent Theory Computations.- Appendix: Subsidiary Information.- A.1 Directed Sets and Limits.- A.2 Exterior Powers.- A.3 Localization. Primes, and Nilpotents.- A.4 Noetherian Rings.- A.5 The Hilbert Basis Theorem.- A.6 The Krull Intersection Theorem.- A.7 The Nocthcr Normalization Lemma.- A.8 The Hilbert Nullstellensatz.- A.9 Separably Generated Fields.- A.10 Rudimentary Topological Terminology.- Further Reading.- Index of Symbols.978-3-0348-0138-6WilliamsKim Williams, Torino, Italy.Crossroads: History of Science, History of Art Essays by David Speiser, vol. II*XII, 154 p. 37 illus., 13 illus. in color.SCK12008 Architectural History and TheoryAMAForeword.- Editor s Note.- The Symmetry of the Ornament on a Jewel of the Treasure of Mycenae.- Arab and Pisan Mathematics in the Piazza dei Miracoli.- Architecture, Mathematics and Theology in Raphael's Paintings.- What can the Historian of Science Learn from the Historian of the Fine Arts?- The Importance of Concepts for Science.- Remarks on Space and Time in Newton, Leibniz, Euler and in Modern Physics.- Gruppentheorie und Quantenmechanik: The Book and its Position in Weyl's Work.- Clifford A. Truesdell's Contributions to the Euler and the Bernoulli Editions.- Publishing Complete Works of the Great Scientists: An International Undertaking.- The History of Science at the Crossroads of the Pathways towards Philosophy and History.- Index of Names.A follow-up to the volume 'Discovering the Principles of Mechanics 1600-1800. Essays by David Speiser' (Birkhuser 2008), this volume contains the essays of David Speiser on relationships between science, history of science, history of art and philosophy.RRounds out the collection of essays published previously by well-known scientist and historian David Speiser

Multidisciplinary essays provide a broad panorama on the relationships between science and the arts 3 out of 10 essays translated for the first time into English

Hard-to-find essays collected in a single work

978-3-0348-0743-2978-1-4471-2527-3Wilson:Robert Wilson, Queen Mary University of London, London, UKThe Finite Simple Groups XV, 298p.]The alternating groups.- The classical groups.- The exceptional groups.- The sporadic groups.The finite simple groups are the building blocks from which all the finite groups are made and as such they are objects of fundamental importance throughout mathematics. The classification of the finite simple groups was one of the great mathematical achievements of the twentieth century, yet these groups remain difficult to study which hinders applications of the classification. This textbook brings the finite simple groups to life by giving concrete constructions of most of them, sufficient to illuminate their structure and permit real calculations both in the groups themselves and in the underlying geometrical or algebraic structures. This is the first time that all the finite simple groups have been treated together in this way and the book points out their connections, for example between exceptional behaviour of generic groups and the existence of sporadic groups, and discusses a number of new approaches to some of the groups. Many exercises of varying difficulty are provided. ;Written by a world-renowned expert, this textbook is the first to offer a comprehensive introduction to all the finite simple groups at a level accessible to students.

Gives the facts about the finite simple groups and also useful insights into how to work with them, either by hand or on a computer.

978-1-84800-987-5 xv, 298 pp978-3-540-66973-9Wolf-GladrowDieter A. Wolf-Gladrow:Lattice-Gas Cellular Automata and Lattice Boltzmann Models X, 314 p.From the contents: Introduction: Preface; Overview.- The basic idea of lattice-gas cellular automata and lattice Boltzmann models. Cellular Automata: What are cellular automata?- A short history of cellular automata.- One-dimensional cellular automata.- Two-dimensional cellular automata.- Lattice-gas cellular automata: The HPP lattice-gas cellular automata.- The FHP lattice-gas cellular automata.- Lattice tensors and isotropy in the macroscopic limit.- Desperately seeking a lattice for simulations in three dimensions.- 5 FCHC.- The pair interaction (PI) lattice-gas cellular automata.- Multi-speed and thermal lattice-gas cellular automata.- Zanetti (staggered) invariants.- Lattice-gas cellular automata: What else? Some statistical mechanics: The Boltzmann equation.- Chapman-Enskog: From Boltzmann to Navier-Stokes.- The maximum entropy principle. Lattice Boltzmann Models: .... Appendix.Lattice-gas cellular automata (LGCA) and lattice Boltzmann models (LBM) are relatively new and promising methods for the numerical solution of nonlinear partial differential equations. The book provides an introduction for graduate students and researchers. Wo< rking knowledge of calculus is required and experience in PDEs and fluid dynamics is recommended. Some peculiarities of cellular automata are outlined in Chapter 2. The properties of various LGCA and special coding techniques are discussed in Chapter 3. Concepts from statistical mechanics (Chapter 4) provide the necessary theoretical background for LGCA and LBM. The properties of lattice Boltzmann models and a method for their construction are presented in Chapter 5.978-0-8176-8261-3YeungDavid W.K. Yeung, Hong Kong Shue Yan University, Hong Kong, China, People's Republic; Leon A. Petrosyan, St. Petersburg State University Faculty of Applied Mathematics and Contr, Saint Petersburg, Russia(Subgame Consistent Economic Optimization-An Advanced Cooperative Dynamic Game AnalysisXVI, 395p. 2 illus..Preface.- Introduction.-1 Dynamic Strategic Interactions in Economic System.-2 Dynamic Economic Optimization: Group Optimality and Individual Rationality.- 3 Time Consistency and Optimal-Trajectory-Subgame Consistent Economic Optimization.- 4 Dynamically Stable Cost-saving Joint Venture.- 5 Collaborative Environmental Management.- 6 Dynamically Stable Dormant Firm Cartel.- 7 Subgame Consistent Economic Optimization Under Uncertainty.- 8 Cost-saving Joint Venture Under Uncertainty.- 9 Collaborative Environmental Management Under Uncertainty.- 10Subgame Consistent Dormant Firm Cartel.- 11 Dynamic Consistency in Discrete-time Cooperative Games.- 12 Discrete-time Cooperative Games Under Uncertainty.- Technical Appendices.- References.- Index.Various imperfections in existing market systems prevent the free market from serving as a truly efficient allocation mechanism, but optimization of economic activities provides an effective remedial measure. Cooperative optimization claims that socially optimal and individually rational solutions to decision problems involving strategic action over time exist. To ensure that cooperation will last throughout the agreement period, however, the stringent condition of subgame consistency is required.This textbook presents a study of subgame consistent economic optimization, developing game-theoretic optimization techniques to establish the foundation for an effective policy menu to tackle the suboptimal behavior that the conventional market mechanism fails to resolve.The first title on the topic of subgame consistent economic optimization

Illustrates the potential ofcooperative dynamic gamesto helpformulate new policy on previously unsolvable market problems

Valuable as both a research reference and as a textbook, with exercises included throughout the book

Suitable for an interdisciplinary audience of game theorists, economists, mathematicians, policy makers, and corporate planners

978-1-4614-4345-2YinG. George Yin, Wayne State University Department of Mathematics, Detroit, MI, USA; Qing Zhang, University of Georgia Dept. Mathematics, Athens, GA, USA.Continuous-Time Markov Chains and ApplicationsA Two-Time-Scale ApproachXXI, 427 p. 13 illus.2 Prologue and Preliminaries: Introduction and overview- Mathematical preliminaries.- Markovian models.- Two-Time-Scale Markov Chains: Asymptotic Expansions of Solutions for Forward Equations.- Occupation Measures: Asymptotic Properties and Ramification.- Asymptotic Expansions of Solutions for Backward Equations.- Applications:MDPs, Near-optimal Controls, Numerical Methods, and LQG with Switching: Markov Decision Problems.- Stochastic Control of Dynamical Systems.- Numerical Methods for Control and Optimization.- Hybrid LQG Problems.- References.- Index.- 0This book gives a systematic treatment of singularly perturbed systems that naturally arise in control and optimization, queueing networks, manufacturing systems, and financial engineering. It presents results on asymptotic expansions of solutions of Komogorov forward and backward equations, properties of functional occupation measures, exponential upper bounds, and functional limit results for Markov chains with weak and strong interactions. To bridge the gap between theory and applications, a large portion of the book is devoted to applications in controlled dynamic systems, production planning, and numerical methods for controlled Markovian systems with large-scale and complex structures in the real-world problems. This second edition has been updated throughout and includes two new chapters on asymptotic expansions of solutions for backward equations and hybrid LQG problems. The chapters on analytic and probabilistic properties of two-time-scale Markov chains have been almost completely rewritten and the notation has been streamlined and simplified. This book is written for applied mathematicians, engineers, operations researchers, and applied scientists. Selected material from the book can also be used for a one semester advanced graduate-level course in applied probability and stochastic processes.New chapters added on backward equations and LQG control problems

Bridges the gap between theory and applications

Presents results on asymptotic expansions of the corresponding probability distributions

978-1-4899-9118-8978-0-387-94422-7ZeidleraEberhard Zeidler, Max-Planck-Institut fr Mathematik in den Naturwissenschaften, Leipzig, GermanyApplied Functional Analysis&Main Principles and Their ApplicationsXVI, 406 p.1 The Hahn-Banach Theorem Optimization Problems.- 1.1 The Hahn-Banach Theorem.- 1.2 Applications to the Separation of Convex Sets.- 1.3 The Dual Space C[a,b]*.- 1.4 Applications to the Moment Problem.- 1.5 Minimum Norm Problems and Duality Theory.- 1.6 Applications to ?ebyaev Approximation.- 1.7 Applications to the Optimal Control of Rockets.- 2 Variational Principles and Weak Convergence.- 2.1 The nth Variation.- 2.2 Necessary and Sufficient Conditions for Local Extrema and the Classical Calculus of Variations.- 2.3 The Lack of Compactness in Infinite-Dimensional Banach Spaces.- 2.4 Weak Convergence.- 2.5 The Generalized Weierstrass Existence Theorem.- 2.6 Applications to the Calculus of Variations.- 2.7 Applications to Nonlinear Eigenvalue Problems.- 2.8 Reflexive Banach Spaces.- 2.9 Applications to Convex Minimum Problems and Variational Inequalities.- 2.10 Applications to Obstacle Problems in Elasticity.- 2.11 Saddle Points.- 2.12 Applications to Duality Theory.- 2.13 The von Neumann Minimax Theorem on the Existence of Saddle Points.- 2.14 Applications to Game Theory.- 2.15 The Ekeland P< rinciple about Quasi-Minimal Points.- 2.16 Applications to a General Minimum Principle via the Palais-Smale Condition.- 2.17 Applications to the Mountain Pass Theorem.- 2.18 The Galerkin Method and Nonlinear Monotone Operators.- 2.19 Symmetries and Conservation Laws (The Noether Theorem).- 2.20 The Basic Ideas of Gauge Field Theory.- 2.21 Representations of Lie Algebras.- 2.22 Applications to Elementary Particles.- 3 Principles of Linear Functional Analysis.- 3.1 The Baire Theorem.- 3.2 Application to the Existence of Nondifferentiable Continuous Functions.- 3.3 The Uniform Boundedness Theorem.- 3.4 Applications to Cubature Formulas.- 3.5 The Open Mapping Theorem.- 3.6 Product Spaces.- 3.7 The Closed Graph Theorem.- 3.8 Applications to Factor Spaces.- 3.9 Applications to Direct Sums and Projections.- 3.10 Dual Operators.- 3.11 The Exactness of the Duality Functor.- 3.12 Applications to the Closed Range Theorem and to Fredholm Alternatives.- 4 The Implicit Function Theorem.- 4.1 m-Linear Bounded Operators.- 4.2 The Differential of Operators and the Frchet Derivative.- 4.3 Applications to Analytic Operators.- 4.4 Integration.- 4.5 Applications to the Taylor Theorem.- 4.6 Iterated Derivatives.- 4.7 The Chain Rule.- 4.8 The Implicit Function Theorem.- 4.9 Applications to Differential Equations.- 4.10 Diffeomorphisms and the Local Inverse Mapping Theorem.- 4.11 Equivalent Maps and the Linearization Principle.- 4.12 The Local Normal Form for Nonlinear Double Splitting Maps.- 4.13 The Surjective Implicit Function Theorem.- 4.14 Applications to the Lagrange Multiplier Rule.- 5 Fredholm Operators.- 5.1 Duality for Linear Compact Operators.- 5.2 The Riesz-Schauder Theory on Hilbert Spaces.- 5.3 Applications to Integral Equations.- 5.4 Linear Fredholm Operators.- 5.5 The Riesz-Schauder Theory on Banach Spaces.- 5.6 Applications to the Spectrum of Linear Compact Operators.- 5.7 The Parametrix.- 5.8 Applications to the Perturbation of Fredholm Operators.- 5.9 Applications to the Product Index Theorem.- 5.10 Fredholm Alternatives via Dual Pairs.- 5.11 Applications to Integral Equations and Boundary-Value Problems.- 5.12 Bifurcation Theory.- 5.13 Applications to Nonlinear Integral Equations.- 5.14 Applications to Nonlinear Boundary-Value Problems.- 5.15 Nonlinear Fredholm Operators.- 5.16 Interpolation Inequalities.- 5.17 Applications to the Navier-Stokes Equations.- References.- List of Symbols.- List of Theorems.- List of Most Important Definitions.This is the second part of an elementary textbook which combines linear functional analysis, nonlinear functional analysis, and their substantial applications with each other. The book addresses undergraduate students and beginning graduate students of mathematics, physics, and engineering who want to learn how functional analysis elegantly solves mathematical problems which relate to our real world and which play an important role in the history of mathematics. The book's approach begins with the question 'what are the most important applications' and proceeds to try to answer this question. The applications concern integral equations, differential equations, bifurcation theory, the moment problem, Cebysev approximation, the optimal control of rockets, game theory, symmetries and conservation laws (the Noether theorem), the quark model, and gauge theory in elementary particle physics. The presentation is self-contained. As for prerequisites, the reader should be familiar with some basic facts of calculus. The first part of this textbook has been published under the title Applied Functional Analysis: Applications to Mathematical Physics.978-0-387-96802-5 E. Zeidler2Nonlinear Functional Analysis and Its Applications II/ A: Linear Monotone Operators XVIII, 467 p.Jto the Subject.- 18 Variational Problems, the Ritz Method, and the Idea of Orthogonality.- 19 The Galerkin Method for Differential and Integral Equations, the Friedrichs Extension, and the Idea of Self-Adjointness.- 20 Difference Methods and Stability.- Linear Monotone Problems.- 21 Auxiliary Tools and the Convergence of the Galerkin Method for Linear Operator Equations.- 22 Hilbert Space Methods and Linear Elliptic Differential Equations.- 23 Hilbert Space Methods and Linear Parabolic Differential Equations.- 24 Hilbert Space Methods and Linear Hyperbolic Differential Equations.This is the second of a five-volume exposition of the main principles of nonlinear functional analysis and its applications to the natural sciences, economics, and numerical analysis. The presentation is self -contained and accessible to the nonspecialist. Part II concerns the theory of monotone operators. It is divided into two subvolumes, II/A and II/B, which form a unit. The present Part II/A is devoted to linear monotone operators. It serves as an elementary introduction to the modern functional analytic treatment of variational problems, integral equations, and partial differential equations of elliptic, parabolic and hyperbolic type. This book also represents an introduction to numerical functional analysis with applications to the Ritz method along with the method of finite elements, the Galerkin methods, and the difference method. Many exercises complement the text. The theory of monotone operators is closely related to Hilbert's rigorous justification of the Dirichlet principle, and to the 19th and 20th problems of Hilbert which he formulated in his famous Paris lecture in 1900, and which strongly influenced the development of analysis in the twentieth century.978-0-8176-8348-1ZemyanNStephen M. Zemyan, Pennsylvania State University Mont Alto, Mont Alto, PA, USA*The Classical Theory of Integral EquationsA Concise TreatmentXIII, 344 p.Preface.- Introduction.- Fredholm Integral Equations of the Second Kind(Separable Kernel).- Fredholm Integral Equations of the Second Kind (General Kernel).- Volterra Integral Equations.- Differential and Integrodifferential Equations.- Nonlinear Integral Equations.- Singular Integral Equations.- Systems of Integral Equations.- Appendix A 2010 Mathematics Subject Classification 45-XX Integral Equations.- Appendix B Specialized Vocabularies and Sample Translations.- Bibliography.- Index._The Classical Theory of Integral Equations is a thorough, concise, and rigorous treatment of the essential aspects of the theory of integral equations. The book provides the background and insight necessary to facilitate a complete understanding of the fundamental results in the field. With a firm foundation for the theory in their grasp, students will be well prepared and motivated for further study.Included in the presentation are:A section entitled Tools of the Trade at the beginning of each chapter, providing necessary background information for comprehension of the results presented in that chapter;Thorough discussions of the analytical methods used to solve many types of integral equations;An introduction to the numerical methods that are commonly used to produce approximate solutions to integral equations;Over 80 illustrative examples that are explained in meticulous detail; Nearly 300 exercises specifically constructed to enhance the understanding of both routine and challenging concepts; Guides to Computation to assist the student with particularly complicated algorithmic procedures.This unique textbook offers a comprehensive and balanced treatment of material needed for a general understanding of the theory of integral equations by using only the mathematical background that a typical undergraduate senior should have. The self-contained book will serve as a valuable resource for advanced undergraduate and beginning graduate-level students as well as for independent study. Scientists and engineers who are working in the field will also find this text to be user friendly and informative.cProvides complete and straightforward coverage of topics central to the theory of integral equations Includes chapter-by-chapter exercises and illustrative examples Incl< udes Tools of the Trade sections thatpresent essential review material Features computation guides for especially complicated computational procedures Useful as a self-study guide978-3-642-30708-9ZhangpZhitao Zhang, Chinese Academy of Science Academy of Mathem. & Systems Science, Beijing, China, People's RepublicKVariational, Topological, and Partial Order Methods with Their ApplicationsDevelopments in MathematicsXI, 332 p. 3 illus.1 Preliminaries.- Sobolev spaces and embedding theorems.- Critical point.- Cone and partial order.- Brouwer Degree.- Compact map and Leray-Schauder Degree.- Fredholm operators.- Fixed point index.- Banach's Contract Theorem, Implicit Functions Theorem.- Krein-Rutman theorem.- Bifurcation theory.- Rearrangements of sets and functions.- Genus and Category.- Maximum principles and symmetry of solution.- Comparison theorems.- 2 Cone and Partial Order Methods.- Increasing operators.- Decreasing operators.- Mixed monotone operators.- Applications of mixed monotone operators.- Further results on cones and partial order methods.- 3 Minimax Methods.- Mountain Pass Theorem and Minimax Principle.- Linking Methods.- Local linking Methods.- 4 Bifurcation and Critical Point.- Introduction.- Main results with parameter.- Equations without the parameter.- 5 Solutions of a Class of Monge-Ampre Equations.- Introduction.- Moving plane argument.- Existence and non-existence results.- Bifurcation and the equation with a parameter.- Appendix.- 6 Topological Methods and Applications.- Superlinear system of integral equations and applications.- Existence of positive solutions for a semilinear elliptic system.- 7 Dancer-Fu ik Spectrum.- The spectrum of a self-adjoint operator.- Dancer-Fu ik Spectrum on bounded domains.- Dancer-Fu ik point spectrum on RN.- Dancer-Fu ik spectrum and asymptotically linear elliptic problems.- 8 Sign-changing Solutions.- Sign-changing solutions for superlinear Dirichlet problems.- Sign-changing solutions for jumping nonlinear problems.- 9Extension of Brezis-Nirenberg's Results and Quasilinear Problems.- Introduction.- W01,p() versus C01() local minimizers.- Multiplicity results for the quasilinear problems.- Uniqueness results.- 10 Nonlocal Kirchhoff Elliptic Problems.- Introduction.- Yang index and critical groups to nonlocal problems.- Variational methods and invariant sets of descent flow.- Uniqueness of solution for a class of Kirchhoff-type equations.- 11 Free Boundary Problems, System of equations for Bose-Einstein Condensate and Competing Species.- Competing system with many species.- Optimal partition problems.- Schrdinger systems from Bose-Einstein condensate.- Bibliography.Nonlinear functional analysis is an important branch of contemporary mathematics. It's related to topology, ordinary differential equations, partial differential equations, groups, dynamical systems, differential geometry, measure theory, and more. In this book, the author presents some new and interesting results on fundamental methods in nonlinear functional analysis, namely variational, topological and partial order methods, which have been used extensively to solve existence of solutions for elliptic equations, wave equations, Schrdinger equations, Hamiltonian systems etc., and are also used to study the existence of multiple solutions and properties of solutions. This book is useful for researchers and graduate students in the field of nonlinear functional analysis.wFront research in this field

New results about this topic

Theory and applications are shown together

978-3-642-42715-2978-3-0348-0011-2AbbesOAhmed Abbes, Institut des Hautes tudes Scientifiques, Bures-sur-Yvette, Francelments de Gomtrie Rigide?Volume I. Construction et tude Gomtrique des Espaces Rigides XV, 496p.FrenchQPrface par Michel Raynaud.- Avant-propos.- Introduction.- Chapitre 1. Prliminaires.- Chapitre 2. Gomtrie formelle.- Chapitre 3. clatements admissibles.- Chapitre 4. Gomtrie rigide.- Chapitre 5. Platitude.- Chapitre 6. Invariants diffrentiels. Morphismes lisses.- Chapitre 7. Espaces rigides quasi-spars.- Bibliographie.- Index.}La gomtrie rigide est devenue, au fil des ans, un outil indispensable dans un grand nombre de questions en gomtrie arithmtique. Depuis ses premires fondations, poses par J. Tate en 1961, la thorie s'est dveloppe dans des directions varies. Ce livre est le premier volume d'un trait qui expose un dveloppement systmatique de la gomtrie rigide suivant l'approche de M. Raynaud, base sur les schmas formels clatements admissibles prs. Ce volume est consacr la construction des espaces rigides dans une situation relative et l'tude de leurs proprits gomtriques. L'accent est particulirement mis sur l'tude de la topologie admissible d'un espace rigide cohrent, analogue de la topologie de Zariski d'un schma. Parmi les sujets traits figurent l'tude des faisceaux cohrents et de leur cohomologie, le thorme de platification par clatements admissibles qui gnralise au cadre formel-rigide un thorme de Raynaud-Gruson dans le cadre algbrique, et le thorme de comparaison du type GAGA pour les faisceaux cohrents. Ce volume contient aussi de larges rappels et complments de la thorie des schmas formels de Grothendieck. Ce trait est destin tout autant aux tudiants ayant une bonne connaissance de la gomtrie algbrique et souhaitant apprendre la gomtrie rigide qu'aux experts en gomtrie algbrique et en thorie des nombres comme source de rfrences. First book to give a systematic development of Raynaud's rigid geometry Extensive review and complements on Grothendieck's formal geometry Thorough study of the topological aspects of rigid spaces Detailed exp< osition of the flattening theorem by admissible blow-ups of Raynaud-Gruson978-3-642-00445-2Fatou, Julia, Montel,=le grand prix des sciences mathmatiques de 1918, et aprs... VI, 276p.Le Grand Prix, le cadre.- Le Grand Prix des Sciences Mathématiques.- Les mémoires.- Suites de l'itération.- Sur Pierre Fatou.- Cicatrices de l’histoire — Une polémique en 1965.Comment Fatou et Julia ont invent ce que l on appelle aujourd hui les ensembles de Julia, avant, pendant et aprs la premire guerre mondiale? L histoire est raconte, avec ses mathmatiques, ses conflits, ses personnalits. Elle est traite partir de sources nouvelles, et avec rigueur. On pourra s y initier l itration des fractions rationnelles et la dynamique complexe (ensembles de Julia, de Mandelbrot, ensembles-limites). Qui taient Pierre Fatou, Gaston Julia, Paul Montel? On y trouvera en particulier des informations sur un mathmaticien mal connu, Pierre Fatou. On dcouvrira aussi quelques incidences de la blessure reue par Julia pendant la guerre sur la vie mathmatique en France au vingtime sicle.How did Pierre Fatou and Gaston Julia create what we now call Complex Dynamics, in the context of the early twentieth century and especially of the First World War? The book isbased partly on new, unpublished sources.Who were Pierre Fatou, Gaston Julia, Paul Montel? New biographical information is given on the little known mathematician that was Pierre Fatou. How did the seriousinjury of Julia during WWI influence mathematical life in France?978-3-540-19351-7Azema]Jaques Azema; Paul A. Meyer; Marc Yor, Universit Paris VI CNRS UMR 7599, Paris CX 05, FranceSeminaire de Probabilites XXIIIV, 600 p. (196 p. en Anglais)8 La proprit de sous-harmonicit des diffusions dans les varits.- Chaos de Wiener et integrale de Feynman.- Sur les integrales multiples de Stratonovitch.- Un nouvel exemple de distribution de Hida.- Une remarque sur les processus de Dirichlet forts.- Quasimartingales hilbertiennes d'aprs Enchev.- A perturbation theorem for semigroups of linear operators.- A formula for densities of transition functions.- Elments de Probabilits quantiques. IX Calculs Antisymtriques Et Supersymtriques En Probabilits.- Elements de probabilites quantiques. X Calculs avec des noyaux discrets.- Integration stochastique et geometrie des espaces de Banach.- Une surmartingale limite de martingales continues.- Sur un theoreme de B. Rajeev.- A propos d'une conjecture de meyer.- En cherchant une caractrisation variationnelle des martingales.- A note on approximation for stochastic differential equations.- Extending Lvy's characterisation of Brownian motion.- Penetration times and skorohod stopping.- The statistical equilibrium of an isotropic stochastic flow with negative lyapounov exponents is trivial.- Sur les fonctions polaires pour le mouvement brownien.- Sur un calcul de F. Knight.- Operateurs filtres et chaines de tribus invariantes sur un espace probabilise denombrable.- A simple proof of a theorem of blackwell & dubins on the maximum of a uniformly integrable martingale.- Remarques sur certaines constructions des mouvements browniens fractionnaires.- Le mouvement brownien de Levy index par ?3 comme limite centrale de temps locaux d'intersection.- Ingalits isoprimtriques et calcul stochastique.- Remarks on absolute continuity, contiguity and convergence in variation of probability measures.- Integration by parts for jump processes.- Diffusion semigroups corresponding to uniformly elliptic divergence form operators.- Sur le theoreme de l'indice des familles.- Calcul des variations sur un brownien subordonne.- Une condition necessaire et suffisante pour la convergence en pseudo-loi des processus.- Systeme de particules et mesures-martingales: Un theoreme de propagation du chaos.- Un exemple de processus mesurable adapte non progressif.- Sur la loi des temps locaux browniens pris en un temps exponentiel.- Distributions, noyaux, symboles d'aprs kree.- Calcul stochastique non adapte par rapport a la mesure aleatoire de poisson.- Riesz transforms: A simpler analytic proof of P.A. Meyer's inequality.- Brownian excursions from extremes.- Controle de processus de Markov.- Pathwise approximations of processes based on the fine structure of their filtrations.- Erratum au Seminaire XX.978-3-540-33849-9BourbakiN. BourbakiAlgbreChapitres 1 3XIII, 634 p.nStructures algbriques.- Algbre linaire.- Algbres tensorielles, algbres extrieures, algbres symtriques.978-3-540-34398-1Chapitre 4 7VII, 422 p.yPolynmes et fractions rationnelles.- Corps commutatifs.- Groupes et corps ordonns.- Modules sur les anneaux principaux.<Les lments de mathmatique de Nicolas Bourbaki ont pour objet une prsentation rigoureuse, systmatique et sans prrequis des mathmatiques depuis leurs fondements. Ce deuxime volume du Livre d Algbre, deuxime Livre des lments de mathmatique, traite notamment des extensions de corps et de la thorie de Galois. Il comprend les chapitres: 4. Polynmes et fractions rationnelles; 5. Corps commutatifs; 6. Groupes et corps ordonns; 7. Modules sur les anneaux principaux. Il contient galement des notes historiques. Ce volume est une nouvelle dition parue en 1981.978-3-540-35315-7=N. Bourbaki, Ecole normale suprieure, Paris Cedex 05, France Chapitre 8 X, 489 p.Introduction.- Chapitre VIII. Modules et anneaux semi-simples.- 1. Modules artiniens et modules noethriens.- 2. Structure des modules de longueur finie.- 3. Modules simples.- 4. Modules semi-simples.- 5. Commutation.- 6. quivalence de Morita des modules et des algbres.- 7. Anneaux simples.- 8. Anneaux semi-simples.- 9. Radical.- 10. Modules sur un anneau artinien.- 11. Groupes de Grothendieck.- 12. Produit tensoriel de modules semi-simples.- 13. Algbres absolument semi-simples.- 14. Algbres cen< trales et simples.- 15. Groupes de Brauer.- 16. Autres descriptions du groupe de Brauer.- 17. Normes et traces rduites.- 18. Algbres simples sur un corps fini.- 20. Reprsentations linaires des algbres.- 21. Reprsentations linaires des groupes finis.- Appendice 1. Algbres sans lment unit.- Appendice 2. Dterminants sur un corps non commutatif.- Appendice 3. Le thorme des zros de Hilbert.- Appendice 4. Trace d un endomorphisme de rang fini.- Note Historique.- Bibliographie.- Index des notations.- Index terminologiqueCe huitime chapitre du Livre d'Algbre, deuxime Livre des lments de mathmatique, est consacr l'tude de certaines classes d'anneaux et des modules sur ces anneaux.Il couvre les notions de moduleet d'anneau noethrien et artinien, ainsi que celle de radical. Ce chapitre dcrit galement la structure des anneaux semi-simples. Nous y donnons aussi la dfinition de divers groupes de Grothendieck qui jouent un rle universel pour les invariants de modules et plusieurs descriptions du groupe de Brauer qui intervient dans la classification des anneaux simples.Une note historique en fin de volume, reprise de l'dition prcdente, retrace l'mergence d'une grande partie des notions dveloppes.Ce volume est une deuxime dition entirement refondue de l'dition de 1958.Nouvelle dition entirement refondue /p>

Prsente des notions absentes dans les ditions antrieures

Note historique en fin de volume

978-3-540-33939-7Groupes et algbres de LieChapitres 7 et 8271 p.SCM11116"Non-associative Rings and AlgebrasTSous-algbres de Cartan lments rguliers.- Algbres de Lie semi-simples dployes.6 Les lments de mathmatique de Nicolas Bourbaki ont pour objet une prsentation rigoureuse, systmatique et sans prrequis des mathmatiques depuis leurs fondements. Ce troisime volume du Livre sur les Groupes et algbres de Lie, neuvime Livre du trait, poursuit l tude des algbres de Lie et leurs reprsentations. Il comprend les chapitres: 7. Sous-algbres de Cartan, lments rguliers; 8. Algbres de Lie semi-simples dployes. Ce volume contient galement un appendice sur la topologie de Zariski. Ce volume est une rimpression de l dition de 1975.978-3-540-33940-3Chapitres 2 et 3320 p.(Algbres de Lie libres.- Groupes de Lie.Les lments de mathmatique de Nicolas Bourbaki ont pour objet une prsentation rigoureuse, systmatique et sans prrequis des mathmatiques depuis leurs fondements. Ce deuxime volume du Livre sur les Groupes et algbres de Lie, neuvime Livre du trait, comprend les chapitres: 2. Algbres de Lie libres; 3. Groupes de Lie. Le chapitre 2 poursuit la prsentation des notions fondamentales des algbres de Lie avec l introduction des algbres de Lie libres et de la srie de Hausdorff. Le chapitre 3 est consacr aux concepts de base pour les groupes de Lies sur un corps archimdien ou ultramtrique. Ce volume contient galement de notes historiques pour les chapitres 1 3. Ce volume est une rimpression de l dition de 1972.978-3-540-08241-5 BrezinskiZClaude Brezinski, Universite Lille 1 UFR IEEA/Lab. d'Analyse, Villeneuve d'Ascq CX, France3Acceleration de la convergence en analyse numerique IV, 313p.Comparaison de Suites Convergentes.- Les Procedes de Sommation.- L ?-Algorithme.- Etude de Divers Algorithmes d Acceleration de la Convergence.- Transformation de Suites Non Scalaires.- Algorithmes de Prediction Continue.- Les Fractions Continues.978-3-540-73755-1CaspardSNathalie Caspard; Bruno Leclerc, EHESS CAMS, Paris CX 06, France; Bernard Monjardet8Ensembles ordonns finis : concepts, rsultats et usagesMathmatiques et ApplicationsSCM11124-Order, Lattices, Ordered Algebraic StructuresConcepts et exemples.- Classes particulires d'ensembles ordonns.- Morphismes d'ensembles ordonns.- Chanes et antichanes.- Ensembles ordonns et treillis distributifs.- Codages et dimensions des ordres.- Quelques usages.6Les notions d'ordre, de classement, de rangement sont prsents dans de multiples activits et situations humaines. La formalisation mathmatique de ces notions a permis d'abord le grand dveloppement de la thorie des treillis, puis celui de structures ordonnes plus gnrales, notamment celles relevant des mathmatiques discrtes. Les buts principaux de cet ouvrage qui comble un vide sont donc de: - donner les concepts et rsultats fondamentaux sur les ensembles ordonns finis, - prsenter leurs usages dans des domaines varis (de la RO ou l IA la micro-conomie), - signaler un certain nombre de rsultats et de recherches en cours. Le lecteur sera ainsi mme de trouver tout ce qu'il a besoin de connatre sur ces structures sans devoir le rechercher dans de multiples revues relevant de disciplines varies.978-3-540-08052-7CaubetJ.-P. Caubet!Le mouvement brownien relativiste IX,212 pages.Probabilites et Esperance.- Mesure de Wiener et Mouvement Brownien.- Differentielles et Integrales Stochastiques.- Diffusions.- Le Mouvement Brownien Relativiste.- Equations de Lagrange et de Hamilton.978-3-540-33707-2 Chenciner[Alain Chenciner, IMCCE - Observatoire de Paris Institut de Mcanique Cleste, Paris, FranceCourbes Algbriques Planes8Ensembles algbriques a< ffines.- Courbes planes affines.- Ensembles algbriques projectifs.- Courbes projectives planes : le thorme de Bzout.- Le rsultant.- Point de vue local: anneaux de sries formelles.- Anneaux de sries convergentes.- Le thorme de Puiseux.- Thorie locale des intersections de courbes.vIssu d un cours de matrise de l Universit Paris VII, ce texte est rdit tel qu il tait paru en 1978. A propos du thorme de Bzout sont introduits divers outils ncessaires au dveloppement de la notion de multiplicit d intersection de deux courbes algbriques dans le plan projectif complexe. Partant des notions lmentaires sur les sous-ensembles algbriques affines et projectifs, on dfinit les multiplicits d intersection et interprte leur somme entermes du rsultant de deux polynmes. L tude locale est prtexte l introduction des anneaux de srie formelles ou convergentes ; elle culmine dans le thorme de Puiseux dont la convergence est ramene par des clatements celle du thorme des fonctions implicites. Diverses figures clairent le texte: on y 'voit' en particulier que l quation homogne x3+y3+z3 = 0 dfinit un tore dans le plan projectif complexe. 978-3-540-30995-6DedieuTJean-Pierre Dedieu, Universit Toulouse III Dept. Mathmatiques, Toulouse CX, France+Points fixes, zros et la mthode de NewtonXII, 196 p.Points fixes.- La mthode de Newton.- La mthode de Newton pour des systmes.- La mthode de Newton-Gauss pour des systmes sur-dtermins.- Appendices.Cet ouvrage est consacr aux points fixes d'applications diffrentiables, aux zros de systmes non-linaires et la mthode de Newton. Il s'adresse des tudiants de mastre ou prparant l'agrgation de mathmatique et des chercheurs confirms. La premire partie est consacre la mthode des approximations successives et confronte un point de vue systmes dynamiques (thormes de Grobman-Hartman, de la varit stable) des exemples issus de l'analyse numrique. La seconde partie de cet ouvrage expose la mthode de Newton et ses dveloppements les plus rcents (thorie alpha de Smale, systmes sous ou sur-dtermins). Elle prsente une nouvelle approche de ce sujet et un ensemble de rsultats originaux publis pour la premire fois dans un ouvrage de langue franaise.978-3-540-12669-0Dies J.-E. Dies&Chaines de Markov sur les Permutations IX,226 pages.Structures De Permutation.- Librairies, Librairies Stationnaires.- Mesures Stationnaires.- Recurrence Positive Des Librairies (e,T ? ? ,p) Et (e,M ? ? ,p).- Transience Des Librairies (e,T ? ? ,p)..- Variantes Mixtes Finies Des Librairies De Tsetlin Transience Des Librairies (e,M ? ? ,p).- Structures Recurrences, Structures Transientes.- Recurrence Positive Des Librairies Mixtes.- Classification Des Librairies Et Des Structures Mixtes.- Optimalite De La Police De Transposition.978-3-540-42335-5Gianella)H. Gianella; R. Krust; F. Taieb; N. Tosel.Problmes choisis de mathmatiques suprieuresSCOPOSVIII, 267 pp. 19 fig.Nombres rels et complexes, suites.- Continuit et drivabilit.- Calcul intgral.- Structures algbriques usuelles, arithmtique.- Polynmes.- Algbre linare.- Gomtrie.kCe livre rassemble des noncs de problmes de mathmatiques proposs par les auteurs leurs tudiants en classe prparatoire MPSI au Lyce Louis-Le-Grand Paris. Il se divise en sept chapitres, correspondants aux principaux thmes gnralement abords dans une premire anne d'tudes scientifiques. Les problmes sont de difficult progressive, pour la plupart originaux, et parfois tablissent des rsultats mathmatiques rcents. Une brve introduction permet de les situer dans un contexte mathmatique plus vaste. Tous les noncs sont suivis de corrigs dtaills, et complts s'il y a lieu d'indications bibliographiques permettant d'engager une tude plus pousse du sujet. L'tudiant de premire anne trouvera dans ce livre un complment l'enseignement qui lui est dispens; en seconde anne, il trouvera matire rvision en vue des examens ou des concours.978-3-540-26211-4HenrotAntoine Henrot, Nancy Universit Institut lie Cartan IECN, Vandoeuvre-les-Nancy CX, France; Michel Pierre, ENS de Cachan Antenne de Bretagne, Bruz, France#Variation et optimisation de formesUne analyse gomtriqueXII, 334 p.Introduction, Exemples.- Topologies sur les domaines de ?N.- Continuit par rapport au domaine.- Existence de formes optimales.- Drivation par rapport au domaine.- Proprits gomtriques de l optimum.- Relaxation, homognisation.`Ce livre est une initiation aux approches modernes de l optimisation mathmatique de formes. On y dveloppe la mthodologie ainsi que les outils d analyse mathmatique et de gomtrie ncessaires l tude des variations de domaines. On y trouve une tude systmatique des questions gomtriques associes l oprateur de Laplace, de la capacit classique, de la drivation par rapport une forme, ainsi qu un FAQ sur les topologies usuelles sur les domaines et sur les proprits gomtriques des formes optimales avec ce qui se passe quand elles n existent pas, le tout avec une importante bibliographie.978-3-642-30734-8PJean-Baptiste Hiriart-Urruty, Universit Paul Sabatier, Toulouse Cedex 9, France8Bases, outils et principes pour l'analyse variationnelleXIII, 171 p. 36 ill.L tude mathmatique des problmes d optimisation, ou de ceux dits variationnels de manire gnrale (c est--dire, toute situation o il y a quelque chose minimiser sous des contraintes ), requiert en pralable qu on en matrise les bases, les outils fondamentaux et quelques principes. Le prsent ouvrage est un cours rpondant en partie cette demande, il est principalement destin des tudiants de Master en formation, et restreint l essentiel. Sont abords successivement : La semicontinuit infrieure, les topologies faibles, les rsult< ats fondamentaux d existence en optimisation ; Les conditions d optimalit approche ; Des dveloppements sur la projection sur un convexe ferm, notamment sur un cne convexe ferm ; L analyse convexe dans son rle opratoire ; Quelques schmas de dualisation dans des problmes d optimisation non convexe structurs ; Une introduction aux sous-diffrentiels gnraliss de fonctions non diffrentiables.Prsentation pdagogique des points essentiels connatre en dbutant des recherches en analyse variationnelle

Prsentation condense mais rigoureuse de plusieurs choses en un seul document

Cours expriment et qui a fait ses preuves sur trois ans

978-3-642-31897-9Le GallJJean-Francois Le Gall, Universit Paris-Sud, Campus d'Orsay, Orsay, France6Mouvement brownien, martingales et calcul stochastiqueVIII, 176 p. 2 ill.Cet ouvrage propose une approche concise mais complte de la thorie de l'intgrale stochastique dans le cadre gnral des semimartingales continues. Aprs une introduction au mouvement brownien et ses principales proprits, les martingales et les semimartingales continues sont prsentes en dtail avant la construction de l'intgrale stochastique. Les outils du calcul stochastique, incluant la formule d'It, le thorme d'arrt et de nombreuses applications, sont traits de manire rigoureuse. Le livre contient aussi un chapitre sur les processus de Markov et un autre sur les quations diffrentielles stochastiques, avec une preuve dtaille des proprits markoviennes des solutions. De nombreux exercices permettent au lecteur de se familiariser avec les techniques du calcul stochastique. This book offers a rigorous and self-contained approach to the theory of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including It's formula, the optional stopping theorem and the Girsanov theorem are treated in detail including many important applications. Two chapters are devoted to general Markov processes and to stochastic differential equations, with a complete derivation of Markovian properties of solutions in the Lipschitz case. Numerous exercises help the reader to get acquainted with the techniques of stochastic calculus.kPrsentation concise et rigoureuse de la thorie du calcul stochastique Traitement de cette thorie dans un cadre gnral adapt la majeure partie des applications Chapitres d'introduction aux processus de Markov et aux quations diffrentielles stochastiques donnant les ides importantes de ces thories Nombreux exercices de niveau adapt proposs au lecteur978-3-540-06665-1Poenaru V. PoenaruAnalyse differentielle228p.>Le theoreme de division de Mather.- Le theoreme de preparation de Weierstrass-Malgrange-Mather.- Appendice au chapitre II.- Le theoreme d'extension de Whitney.- Le theoreme 'de recollement' de ?ojasiewicz.- Le theoreme de synthese spectrale de Whitney.- La division des distributions a partir de la resolution des singularites de Hironaka.- Sur la stabilite des applications differentiables (stabilite infinitesimale ? stabilite).- Germes d'applications C? . (Stabilite, jets k-suffisants, e.a.d.s.).- Caracterisation des applications (et des germes d'applications) stables.978-3-540-07028-3Algbre Locale, Multiplicits'Cours au Collge de France, 1957 - 1958 X, 160 p.Idaux Premiers et Localisation.- Outils et Sorites.- Thorie de la Dimension.- Dimension et Codimension Homologiques.- Les Multiplicits.This edition reproduces the 2nd corrected printing of the 3rd edition of the now classic notes by Professor Serre, long established as one of the standard introductory texts on local algebra. Referring for background notions to Bourbaki's 'Commutative Algebra' (English edition Springer-Verlag 1988), the book focusses on the various dimension theories and theorems on mulitplicities of intersections with the Cartan-Eilenberg functor Tor as the central concept. The main results are the decomposition theorems, theorems of Cohen-Seidenberg, the normalisation of rings of polynomials, dimension (in the sense of Krull) and characteristic polynomials (in the sense of Hilbert-Samuel).978-3-540-58002-7Cohomologie GaloisienneIX, 181 pp.uCohomologie des groupes profinis.- Cohomologie galoisieme cas commutatif.- Cohomologie galoisienne non commutative.From the reviews: 'This book surveys an elegant new subject which has developed out of the cohomological treatment of class field theory by E. Artin and J. Tate. The bulk of the early contributions were by Tate, and we are greatly indebted to the author for publishing them in his very lucid style. Many others have made impressive discoveries in the field science. [...] An Appendix by J.-L. Verier covers duality in profinite groups.' M. Greenberg in Mathematical Reviews, 1966 The current edition includes a survey (mostly without proofs) of the main results obtained in the 30 years following original publication. It also incorporates newer material, e.g. two 'rsums de cours' at the Collge de France (1990 - 1991 and 1991 - 1992), and an updated bibliography.978-3-540-65504-6Tricot Claude TricotCourbes et dimension fractale XX, 377 p.Partie I: Ensemble de mesure nulle sur la droite.- Partie II: Courbes rectifiables.- Partie III: Courbes non rectifiables.- Partie IV: Annexes, rfrences et index.kCe livre fait une revue de diverses techniques d'analyse des c