Yellow Sale 2018: Springer Mathematik


Yellow Sale 2018: Springer Mathematik


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102 Combinatorial Problems : 978-0-8176-4317-1
Andreescu
From the Training of the USA IMO Team
Birkhäuser Boston
Seiten XII, 115 p. 2 illus.
Erscheinungsjahr 2003
ISBN 978-0-8176-4317-1
       9780817643171
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'102 Combinatorial Problems' consists of carefully selected problems that have been used in the training and testing of the USA International Mathematical Olympiad (IMO) team. Key features: * Provides in-depth enrichment in the important areas of combinatorics by reorganizing and enhancing problem-solving tactics and strategies * Topics include: combinatorial arguments and identities, generating functions, graph theory, recursive relations, sums and products, probability, number theory, polynomials, theory of equations, complex numbers in geometry, algorithmic proofs, combinatorial and advanced geometry, functional equations and classical inequalities The book is systematically organized, gradually building combinatorial skills and techniques and broadening the student's view of mathematics. Aside from its practical use in training teachers and students engaged in mathematical competitions, it is a source of enrichment that is bound to stimulate interest in a variety of mathematical areas that are tangential to combinatorics.

Inhaltsangaben

Preface and Introduction * Acknowledgments * Abbreviations and Notations * Introductory Problems * Advanced Problems * Solutions to Introductory Problems * Solutions to Advanced Problems * Glossary * Further Reading

103 Trigonometry Problems : 978-0-8176-4334-8
Andreescu
From the Training of the USA IMO Team
Birkhäuser Boston
Seiten XII, 214 p.
Erscheinungsjahr 2005
ISBN 978-0-8176-4334-8
       9780817643348
 (Softcover)
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Fachgebiet

Geometry

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103 Trigonometry Problems contains highly-selected problems and solutions used in the training and testing of the USA International Mathematical Olympiad (IMO) team. Though many problems may initially appear impenetrable to the novice, most can be solved using only elementary high school mathematics techniques. Key features: - Problem-solving tactics and strategies, along with practical test-taking techniques, provide in-depth enrichment and preparation for possible participation in various mathematical competitions - Comprehensive introduction (first chapter) to trigonometric functions, their relations and functional properties, and their applications in the Euclidean plane and solid geometry expose advanced students to college level material 103 Trigonometry Problems is a cogent problem-solving resource for advanced high school students, undergraduates, and mathematics teachers engaged in competition training.

Inhaltsangaben

Trigonometric Fundamentals.- Introductory Problems.- Advanced Problems.- Solutions to Introductory Problems.- Solutions to Advanced Problems.

104 Number Theory Problems : 978-0-8176-4527-4
Andreescu
From the Training of the USA IMO Team
Birkhäuser Boston
Seiten XII, 204 p.
Erscheinungsjahr 2007
ISBN 978-0-8176-4527-4
       9780817645274
 (Softcover)
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Number Theory

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This book contains 104 of the best problems used in the training and testing of the U. S. International Mathematical Olympiad (IMO) team. It is not a collection of very dif?cult, and impenetrable questions. Rather, the book gradually builds students’ number-theoretic skills and techniques. The ?rst chapter provides a comprehensive introduction to number theory and its mathematical structures. This chapter can serve as a textbook for a short course in number theory. This work aims to broaden students’ view of mathematics and better prepare them for possible participation in various mathematical competitions. It provides in-depth enrichment in important areas of number theory by reorganizing and enhancing students’ problem-solving tactics and strategies. The book further stimulates s- dents’ interest for the future study of mathematics. In the United States of America, the selection process leading to participation in the International Mathematical Olympiad (IMO) consists of a series of national contests called the American Mathematics Contest 10 (AMC 10), the American Mathematics Contest 12 (AMC 12), the American Invitational Mathematics - amination (AIME), and the United States of America Mathematical Olympiad (USAMO). Participation in the AIME and the USAMO is by invitation only, based on performance in the preceding exams of the sequence. The Mathematical Olympiad Summer Program (MOSP) is a four-week intensive training program for approximately ?fty very promising students who have risen to the top in the American Mathematics Competitions.

Inhaltsangaben

Foundations of Number Theory.- Introductory Problems.- Advanced Problems.- Solutions to Introductory Problems.- Solutions to Advanced Problems.

A Beginner's Guide to Discrete Mathematics : 978-0-8176-8285-9
Wallis
Birkhäuser Boston
Seiten XIII, 427 p. 219 illus.
Erscheinungsjahr 2012
ISBN 978-0-8176-8285-9
       9780817682859
 (Hardcover)
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Wallis's book on discrete mathematics is a resource for an introductory course in a subject fundamental to both mathematics and computer science, a course that is expected not only to cover certain specific topics but also to introduce students to important modes of thought specific to each discipline . . . Lower-division undergraduates through graduate students. —Choice reviews (Review of the First Edition)Very appropriately entitled as a 'beginner's guide', this textbook presents itself as the first exposure to discrete mathematics and rigorous proof for the mathematics or computer science student. —Zentralblatt Math (Review of the First Edition)This second edition of A Beginner’s Guide to Discrete Mathematics presents a detailed guide to discrete mathematics and its relationship to other mathematical subjects including set theory, probability, cryptography, graph theory, and number theory. This textbook has a distinctly applied orientation and explores a variety of applications. Key Features of the second edition: * Includes a new chapter on the theory of voting as well as numerous new examples and exercises throughout the book * Introduces functions, vectors, matrices, number systems, scientific notations, and the representation of numbers in computers * Provides examples which then lead into easy practice problems throughout the text and full exercise at the end of each chapter * Full solutions for practice problems are provided at the end of the bookThis text is intended for undergraduates in mathematics and computer science, however, featured special topics and applications may also interest graduate students.

Inhaltsangaben

Properties of Numbers.- Sets and Data Structures.- Boolean Algebras and Circuits.- Relations and Functions.- The Theory of Counting.- Probability.- Graph Theory.- Matrices.- Number Theory and Cryptography.- The Theory of Voting.- Solutions to Practic Exercises.- Answers to Selected Exercises.- Index.

A Concise Introduction to Analysis : 978-3-319-24467-9
Stroock
Springer International Publishing
Seiten XII, 218 p.
Erscheinungsjahr 2015
ISBN 978-3-319-24467-9
       9783319244679
 (Softcover)
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Fachgebiet

Functional Analysis

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This book provides an introduction to the basic ideas and tools used in mathematical analysis. It is a hybrid cross between an advanced calculus and a more advanced analysis text and covers topics in both real and complex variables. Considerable space is given to developing Riemann integration theory in higher dimensions, including a rigorous treatment of Fubini's theorem, polar coordinates and the divergence theorem. These are used in the final chapter to derive Cauchy's formula, which is then applied to prove some of the basic properties of analytic functions. Among the unusual features of this book is the treatment of analytic function theory as an application of ideas and results in real analysis. For instance, Cauchy's integral formula for analytic functions is derived as an application of the divergence theorem. The last section of each chapter is devoted to exercises that should be viewed as an integral part of the text. A Concise Introduction to Analysis should appeal to upper level undergraduate mathematics students, graduate students in fields where mathematics is used, as well as to those wishing to supplement their mathematical education on their own. Wherever possible, an attempt has been made to give interesting examples that demonstrate how the ideas are used and why it is important to have a rigorous grasp of them.

Inhaltsangaben

Analysis on The Real Line.- Elements of Complex Analysis.- Integration.- Higher Dimensions.- Integration in Higher Dimensions.- A Little Bit of Analytic Function Theory.

A Concrete Introduction to Higher Algebra : 978-0-387-74527-5
Childs
Springer New York
Seiten XIV, 604 p.
Erscheinungsjahr 2009
ISBN 978-0-387-74527-5
       9780387745275
 (Hardcover)
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Fachgebiet

Algebra

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Inhaltsangaben

Numbers.- Numbers.- Induction.- Euclid's Algorithm.- Unique Factorization.- Congruence.- Congruence classes and rings.- Congruence Classes.- Rings and Fields.- Matrices and Codes.- Congruences and Groups.- Fermat's and Euler's Theorems.- Applications of Euler's Theorem.- Groups.- The Chinese Remainder Theorem.- Polynomials.- Polynomials.- Unique Factorization.- The Fundamental Theorem of Algebra.- Polynomials in ?[x].- Congruences and the Chinese Remainder Theorem.- Fast Polynomial Multiplication.- Primitive Roots.- Cyclic Groups and Cryptography.- Carmichael Numbers.- Quadratic Reciprocity.- Quadratic Applications.- Finite Fields.- Congruence Classes Modulo a Polynomial.- Homomorphisms and Finite Fields.- BCH Codes.- Factoring Polynomials.- Factoring in ?[x].- Irreducible Polynomials.

A Course in Calculus and Real Analysis : 978-0-387-30530-1
Ghorpade
Springer New York
Seiten X, 432 p.
Erscheinungsjahr 2006
ISBN 978-0-387-30530-1
       9780387305301
 (Hardcover)
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Fachgebiet

Analysis

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Calculus is one of the triumphs of the human mind. It emerged from inv- tigations into such basic questions as ?nding areas, lengths and volumes. In the third century B. C. , Archimedes determined the area under the arc of a parabola. In the early seventeenth century, Fermat and Descartes studied the problem of ?nding tangents to curves. But the subject really came to life in the hands of Newton and Leibniz in the late seventeenth century. In part- ular, they showed that the geometric problems of ?nding the areas of planar regions and of ?nding the tangents to plane curves are intimately related to one another. In subsequent decades, the subject developed further through the work of several mathematicians, most notably Euler, Cauchy, Riemann, and Weierstrass. Today,calculus occupies a centralplacein mathematics and is an essential component of undergraduate education. It has an immense number of app- cations both within and outside mathematics. Judged by the sheer variety of the concepts and results it has generated, calculus can be rightly viewed as a fountainhead of ideas and disciplines in mathematics. Real analysis, often called mathematical analysis or simply analysis, may be regarded as a formidable counterpart of calculus. It is a subject where one revisits notionsencountered in calculus, but with greaterrigor and sometimes with greater generality. Nonetheless, the basic objects of study remain the same, namely, real-valued functions of one or several real variables. This book attempts to give a self-contained and rigorous introduction to calculusoffunctionsofonevariable.

Inhaltsangaben

Numbers and Functions.- Sequences.- Continuity and Limits.- Differentiation.- Applications of Differentiation.- Integration.- Elementary Transcendental Functions.- Applications and Approximations of Riemann Integrals.- Infinite Series and Improper Integrals.

A Course in Homological Algebra : 978-0-387-94823-2
Hilton
Springer New York
Seiten XII, 366 p.
Erscheinungsjahr 1997
ISBN 978-0-387-94823-2
       9780387948232
 (Hardcover)
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Fachgebiet

Algebra

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We have inserted, in this edition, an extra chapter (Chapter X) entitled 'Some Applications and Recent Developments.' The first section of this chapter describes how homological algebra arose by abstraction from algebraic topology and how it has contributed to the knowledge of topology. The other four sections describe applications of the methods and results of homological algebra to other parts of algebra. Most of the material presented in these four sections was not available when this text was first published. Naturally, the treatments in these five sections are somewhat cursory, the intention being to give the flavor of the homo­ logical methods rather than the details of the arguments and results. We would like to express our appreciation of help received in writing Chapter X; in particular, to Ross Geoghegan and Peter Kropholler (Section 3), and to Jacques Thevenaz (Sections 4 and 5). The only other changes consist of the correction of small errors and, of course, the enlargement of the Index. Peter Hilton Binghamton, New York, USA Urs Stammbach Zurich, Switzerland Contents Preface to the Second Edition vii Introduction. . I. Modules.

Inhaltsangaben

I. Modules.- 1. Modules.- 2. The Group of Homomorphisms.- 3. Sums and Products.- 4. Free and Projective Modules.- 5. Projective Modules over a Principal Ideal Domain.- 6. Dualization, Injective Modules.- 7 Injective Modules over a Principal Ideal Domain.- 8. Cofree Modules.- 9. Essential Extensions.- II. Categories and Functors.- 1. Categories.- 2. Functors.- 3. Duality.- 4. Natural Transformations.- 5. Products and Coproducts; Universal Constructions.- 6. Universal Constructions (Continued); Pull-backs and Push-outs.- 7. Adjoint Functors.- 8. Adjoint Functors and Universal Constructions.- 9. Abelian Categories.- 10. Projective, Injective, and Free Objects.- III. Extensions of Modules.- 1. Extensions.- 2. The Functor Ext.- 3. Ext Using Injectives.- 4. Computation of some Ext-Groups.- 5. Two Exact Sequences.- 6. A Theorem of Stein-Serre for Abelian Groups.- 7. The Tensor Product.- 8. The Functor Tor.- IV. Derived Functors.- 1. Complexes.- 2. The Long Exact (Co) Homology Sequence.- 3. Homotopy.- 4. Resolutions.- 5. Derived Functors.- 6. The Two Long Exact Sequences of Derived Functors.- 7. The Functors Extn? Using Projectives.- 8. The Functors % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqegm0B % 1jxALjharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY- % Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq % 0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaae % aaeaaakeaadaqdaaqaaGqaaiaa-veacaWF4bGaa8hDaaaadaqhaaWc % baacciGae43MdWeabaGaamOBaaaaaaa!40A3! $$ \overline {Ext} _\Lambda ^n $$ Using Injectives.- 9. Extn and n-Extensions.- 10. Another Characterization of Derived Functors.- 11. The Functor Torn?.- 12. Change of Rings.- V. The Kiinneth Formula.- 1. Double Complexes.- 2. The Künneth Theorem.- 3. The Dual Künneth Theorem.- 4. Applications of the Künneth Formulas.- VI. Cohomology of Groups.- 1. The Group Ring.- 2. Definition of (Co) Homology.- 3. H0, H0.- 4. H1, H1 with Trivial Coefficient Modules.- 5. The Augmentation Ideal, Derivations, and the Semi-Direct Product.- 6. A Short Exact Sequence.- 7. The (Co) Homology of Finite Cyclic Groups.- 8. The 5-Term Exact Sequences.- 9. H2, Hopf’s Formula, and the Lower Central Series.- 10. H2 and Extensions.- 11. Relative Projectives and Relative Injectives.- 12. Reduction Theorems.- 13. Resolutions.- 14. The (Co) Homology of a Coproduct.- 15. The Universal Coefficient Theorem and the (Co)Homology of a Product.- 16. Groups and Subgroups.- VII. Cohomology of Lie Algebras.- 1. Lie Algebras and their Universal Enveloping Algebra.- 2. Definition of Cohomology; H0, H1.- 3. H2 and Extensions.- 4. A Resolution of the Ground Field K.- 5. Semi-simple Lie Algebras.- 6. The two Whitehead Lemmas.- 7. Appendix : Hubert’s Chain-of-Syzygies Theorem.- VIII. Exact Couples and Spectral Sequences.- 1. Exact Couples and Spectral Sequences.- 2. Filtered Differential Objects.- 3. Finite Convergence Conditions for Filtered Chain Complexes.- 4. The Ladder of an Exact Couple.- 5. Limits.- 6. Rees Systems and Filtered Complexes.- 7. The Limit of a Rees System.- 8. Completions of Filtrations.- 9. The Grothendieck Spectral Sequence.- IX. Satellites and Homology.- 1. Projective Classes of Epimorphisms.- 2. ?-Derived Functors.- 3. ?-Satellites.- 4. The Adjoint Theorem and Examples.- 5. Kan Extensions and Homology.- 6. Applications: Homology of Small Categories, Spectral Sequences.- X. Some Applications and Recent Developments.- 1. Homological Algebra and Algebraic Topology.- 2. Nilpotent Groups.- 3. Finiteness Conditions on Groups.- 4. Modular Representation Theory.- 5. Stable and Derived Categories.

A Course in Multivariable Calculus and Analysis : 978-1-4419-1620-4
Ghorpade
Springer New York
Seiten XII, 475 p. 65 illus. in color.
Erscheinungsjahr 2010
ISBN 978-1-4419-1620-4
       9781441916204
 (Hardcover)
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Fachgebiet

Analysis

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Calculus of real-valued functions of several real variables, also known as m- tivariable calculus, is a rich and fascinating subject. On the one hand, it seeks to extend eminently useful and immensely successful notions in one-variable calculus such as limit, continuity, derivative, and integral to “higher dim- sions. ” On the other hand, the fact that there is much more room to move n about in the n-space R than on the real line R brings to the fore deeper geometric and topological notions that play a signi?cant role in the study of functions of two or more variables. Courses in multivariable calculus at an undergraduate level and even at an advanced level are often faced with the unenviable task of conveying the multifarious and multifaceted aspects of multivariable calculus to a student in the span of just about a semester or two. Ambitious courses and teachers would try to give some idea of the general Stokes’s theorem for di?erential forms on manifolds as a grand generalization of the fundamental theorem of calculus, and prove the change of variables formula in all its glory. They would also try to do justice to important results such as the implicit function theorem, which really have no counterpart in one-variable calculus. Most courses would require the student to develop a passing acquaintance with the theorems of Green, Gauss, and Stokes, never mind the tricky questions about orientability, simple connectedness, etc.

Inhaltsangaben

Vectors and Functions.- Sequences, Continuity, and Limits.- Partial and Total Differentiation.- Applications of Partial Differentiation.- Multiple Integration.- Applications and Approximations of Multiple Integrals.- Double Series and Improper Double Integrals.

A First Course in Bayesian Statistical Methods : 978-0-387-92299-7
Hoff
Springer New York
Seiten X, 272 p.
Erscheinungsjahr 2009
ISBN 978-0-387-92299-7
       9780387922997
 (Hardcover)
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Inhaltsangaben

and examples.- Belief, probability and exchangeability.- One-parameter models.- Monte Carlo approximation.- The normal model.- Posterior approximation with the Gibbs sampler.- The multivariate normal model.- Group comparisons and hierarchical modeling.- Linear regression.- Nonconjugate priors and Metropolis-Hastings algorithms.- Linear and generalized linear mixed effects models.- Latent variable methods for ordinal data.

A First Course in Discrete Mathematics : 978-1-85233-236-5
Anderson
Springer London
Seiten VIII, 200 p.
Erscheinungsjahr 2002
ISBN 978-1-85233-236-5
       9781852332365
 (Softcover)
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Discrete mathematics has now established its place in most undergraduate mathematics courses. This textbook provides a concise, readable and accessible introduction to a number of topics in this area, such as enumeration, graph theory, Latin squares and designs. It is aimed at second-year undergraduate mathematics students, and provides them with many of the basic techniques, ideas and results. It contains many worked examples, and each chapter ends with a large number of exercises, with hints or solutions provided for most of them. As well as including standard topics such as binomial coefficients, recurrence, the inclusion-exclusion principle, trees, Hamiltonian and Eulerian graphs, Latin squares and finite projective planes, the text also includes material on the ménage problem, magic squares, Catalan and Stirling numbers, and tournament schedules.

Inhaltsangaben

1. Counting and Binomial Coefficients.- 2. Recurrence.- 3. Introduction to Graphs.- 4. Travelling Round a Graph.- 5. Partitions and Colourings.- 6. The Inclusion Exclusion Principle.- 7. Latin Squares and Hall’s Theorem.- 8. Schedules and 1-Factorisations.- 9. Introduction to Designs.- Solutions.- Further Reading.

A Fixed-Point Farrago : 978-3-319-27976-3
Shapiro
Springer International Publishing
Seiten XIV, 221 p. 8 illus.
Erscheinungsjahr 2016
ISBN 978-3-319-27976-3
       9783319279763
 (Hardcover)
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Analysis

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This text provides an introduction to some of the best-known fixed-point theorems, with an emphasis on their interactions with topics in analysis. The level of exposition increases gradually throughout the book, building from a basic requirement of undergraduate proficiency to graduate-level sophistication. Appendices provide an introduction to (or refresher on) some of the prerequisite material and exercises are integrated into the text, contributing to the volume’s ability to be used as a self-contained text. Readers will find the presentation especially useful for independent study or as a supplement to a graduate course in fixed-point theory.The material is split into four parts: the first introduces the Banach Contraction-Mapping Principle and the Brouwer Fixed-Point Theorem, along with a selection of interesting applications; the second focuses on Brouwer’s theorem and its application to John Nash’s work; the third applies Brouwer’s theorem to spaces of infinite dimension; and the fourth rests on the work of Markov, Kakutani, and Ryll–Nardzewski surrounding fixed points for families of affine maps.

Inhaltsangaben

1. From Newton to Google.- 2. Brouwer in Dimension Two.- 3. Contraction Mappings.- 4. Brouwer in Higher Dimensions.- 5. Nash Equilibrium.- 6. Nash's 'one-page proof'.- 7. The Schauder Fixed-Point Theorem.- 8. The Invariant Subspace Problem.- 9. The Markov–Kakutani Theorem.- 10. The Meaning of Means.- 11. Paradoxical Decompositions.- 12. Fixed Points for Non-commuting Map Families.- 13. Beyond Markov–Kakutani.- A. Advanced Calculus.- B. Compact Metric Spaces.- C. Convex Sets and Normed Spaces.- D. Euclidean Isometries.- E. A Little Group Theory, a Little Set Theory.- References.- Index.- List of Symbols.

A Geometric Approach to Differential Forms : 978-0-8176-8303-0
Bachman
Birkhäuser Boston
Seiten XVI, 156 p. 43 illus.
Erscheinungsjahr 2012
ISBN 978-0-8176-8303-0
       9780817683030
 (Hardcover)
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The modern subject of differential forms subsumes classical vector calculus. This text presents differential forms from a geometric perspective accessible at the advanced undergraduate level.  The author approaches the subject with the idea that complex concepts can be built up by analogy from simpler cases, which, being inherently geometric, often can be best understood visually.Each new concept is presented with a natural picture that students can easily grasp; algebraic properties then follow. This facilitates the development of differential forms without assuming a background in linear algebra. Throughout the text, emphasis is placed on applications in 3 dimensions, but all definitions are given so as to be easily generalized to higher dimensions.The second edition includes a completely new chapter on differential geometry, as well as other new sections, new exercises and new examples. Additional solutions to selected exercises have also been included. The work is suitable for use as the primary textbook for a sophomore-level class in vector calculus, as well as for more upper-level courses in differential topology and differential geometry. 

Inhaltsangaben

Preface.- Guide to the Reader.-Multivariable Calculus.- Parameterizations.- Introduction to Forms.- Forms.- Differential Forms.- Differentiation of Forms.- Stokes' Theorem.- Applications.- Manifolds.- Non-linear Forms.- References.- Index.- Solutions.

A Mathematical Introduction to Compressive Sensing : 978-0-8176-4947-0
Foucart
Springer New York
Seiten XVIII, 625 p.
Erscheinungsjahr 2013
ISBN 978-0-8176-4947-0
       9780817649470
 (Hardcover)
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At the intersection of mathematics, engineering, and computer science sits the thriving field of compressive sensing. Based on the premise that data acquisition and compression can be performed simultaneously, compressive sensing finds applications in imaging, signal processing, and many other domains. In the areas of applied mathematics, electrical engineering, and theoretical computer science, an explosion of research activity has already followed the theoretical results that highlighted the efficiency of the basic principles. The elegant ideas behind these principles are also of independent interest to pure mathematicians.A Mathematical Introduction to Compressive Sensing gives a detailed account of the core theory upon which the field is build. With only moderate prerequisites, it is an excellent textbook for graduate courses in mathematics, engineering, and computer science. It also serves as a reliable resource for practitioners and researchers in these disciplines who want to acquire a careful understanding of the subject. A Mathematical Introduction to Compressive Sensing uses a mathematical perspective to present the core of the theory underlying compressive sensing.

Inhaltsangaben

1 An Invitation to Compressive Sensing.- 2 Sparse Solutions of Underdetermined Systems.- 3 Basic Algorithms.- 4 Basis Pursuit.- 5 Coherence.- 6 Restricted Isometry Property.- 7 Basic Tools from Probability Theory.- 8 Advanced Tools from Probability Theory.- 9 Sparse Recovery with Random Matrices.- 10 Gelfand Widths of l1-Balls.- 11 Instance Optimality and Quotient Property.- 12 Random Sampling in Bounded Orthonormal Systems.- 13 Lossless Expanders in Compressive Sensing.- 14 Recovery of Random Signals using Deterministic Matrices.- 15 Algorithms for l1-Minimization.- Appendix A Matrix Analysis.- Appendix B Convex Analysis.- Appendix C Miscellanea.- List of Symbols.- References.

A Modern Approach to Regression with R : 978-0-387-09607-0
Sheather
Springer New York
Seiten XIV, 393 p.
Erscheinungsjahr 2009
ISBN 978-0-387-09607-0
       9780387096070
 (Hardcover)
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This book focuses on tools and techniques for building regression models using real-world data and assessing their validity. A key theme throughout the book is that it makes sense to base inferences or conclusions only on valid models. Plots are shown to be an important tool for both building regression models and assessing their validity. We shall see that deciding what to plot and how each plot should be interpreted will be a major challenge. In order to overcome this challenge we shall need to understand the mathematical properties of the fitted regression models and associated diagnostic procedures. As such this will be an area of focus throughout the book. In particular, we shall carefully study the properties of resi- als in order to understand when patterns in residual plots provide direct information about model misspecification and when they do not. The regression output and plots that appear throughout the book have been gen- ated using R. The output from R that appears in this book has been edited in minor ways. On the book web site you will find the R code used in each example in the text.

Inhaltsangaben

Simple Linear Regression.- Diagnostics and Transformations for Simple Linear Regression.- Weighted Least Squares.- Multiple Linear Regression.- Diagnostics and Transformations for Multiple Linear Regression.- Variable Selection.- Logistic Regression.- Serially Correlated Errors.- Mixed Models.

A New Approach to Differential Geometry using Clifford's Geometric Algebra : 978-0-8176-8282-8
Snygg
Birkhäuser Boston
Seiten XVII, 465 p. 102 illus.
Erscheinungsjahr 2012
ISBN 978-0-8176-8282-8
       9780817682828
 (Hardcover)
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Differential geometry is the study of the curvature and calculus of curves and surfaces. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space. Complete with chapter-by-chapter exercises, an overview of general relativity, and brief biographies of historical figures, this comprehensive textbook presents a valuable introduction to differential geometry. It will serve as a useful resource for upper-level undergraduates,  beginning-level graduate students, and researchers in the algebra and physics communities. 

Inhaltsangaben

Preface.- Introduction.- Clifford Algebra in Euclidean 3-Space.- Clifford Algebra in Minkowski 4-Space.- Clifford Algebra in Flat n-Space.- Curved Spaces.- The Gauss-Bonnet Formula.- Non-Euclidean (Hyperbolic) Geometry.- Some Extrinsic Geometry in E^n.- Ruled Surfaces Continued.- Lines of Curvature.- Minimal Surfaces.- Some General Relativity.- Matrix Representation of a Clifford Algebra.- Construction of Coordinate Dirac Matrices.- A Few Terms of the Taylor's Series for the Urd?-Copernican Model for the Outer Planets.- A Few Terms of the Taylor's Series for Kepler's Orbits.- References.- Index.

A Panoramic View of Riemannian Geometry : 978-3-540-65317-2
Berger
Springer Berlin Heidelberg
Seiten XXIII, 824 p.
Erscheinungsjahr 2003
ISBN 978-3-540-65317-2
       9783540653172
 (Hardcover)
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Riemannian geometry has today become a vast and important subject. This new book of Marcel Berger sets out to introduce readers to most of the living topics of the field and convey them quickly to the main results known to date. These results are stated without detailed proofs but the main ideas involved are described and motivated. This enables the reader to obtain a sweeping panoramic view of almost the entirety of the field. However, since a Riemannian manifold is, even initially, a subtle object, appealing to highly non-natural concepts, the first three chapters devote themselves to introducing the various concepts and tools of Riemannian geometry in the most natural and motivating way, following in particular Gauss and Riemann.

Inhaltsangaben

1 Euclidean Geometry.- 2 Transition.- 3 Surfaces from Gauß to Today.- 4 Riemann’s Blueprints.- 5 A One Page Panorama.- 6 Metric Geometry and Curvature.- 7 Volumes and Inequalities on Volumes of Cycles.- 8 Transition: The Next Two Chapters.- 9 Spectrum of the Laplacian.- 10 Geodesic Dynamics.- 11 Best Metrics.- 12 From Curvature to Topology.- 13 Holonomy Groups and Kähler Manifolds.- 14 Some Other Important Topics.- 15 The Technical Chapter.- References.- Acknowledgements.- List of Notation.- List of Authors.

A Path to Combinatorics for Undergraduates : 978-0-8176-4288-4
Andreescu
Counting Strategies
Birkhäuser Boston
Seiten XIX, 228 p. 39 illus.
Erscheinungsjahr 2004
ISBN 978-0-8176-4288-4
       9780817642884
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Combinatorics

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The main goal of the two authors is to help undergraduate students understand the concepts and ideas of combinatorics, an important realm of mathematics, and to enable them to ultimately achieve excellence in this field. This goal is accomplished by familiariz­ ing students with typical examples illustrating central mathematical facts, and by challenging students with a number of carefully selected problems. It is essential that the student works through the exercises in order to build a bridge between ordinary high school permutation and combination exercises and more sophisticated, intricate, and abstract concepts and problems in undergraduate combinatorics. The extensive discussions of the solutions are a key part of the learning process. The concepts are not stacked at the beginning of each section in a blue box, as in many undergraduate textbooks. Instead, the key mathematical ideas are carefully worked into organized, challenging, and instructive examples. The authors are proud of their strength, their collection of beautiful problems, which they have accumulated through years of work preparing students for the International Math­ ematics Olympiads and other competitions. A good foundation in combinatorics is provided in the first six chapters of this book. While most of the problems in the first six chapters are real counting problems, it is in chapters seven and eight where readers are introduced to essay-type proofs. This is the place to develop significant problem-solving experience, and to learn when and how to use available skills to complete the proofs.

Inhaltsangaben

Preface.- Introduction.- Acknowledgments.- Abbreviations and Notations.- Addition on Multiplication?- Combinations.- Properties of Binomial Coefficients.- Bijections.- Inclusions and Exclusions.- Recursions.- Calculating in Two Ways – Fubini's Principle.- Generating Functions.- Review Exercises.- Glossary.- Further Reading.

A Primer on PDEs : 978-88-470-2861-6
Salsa
Models, Methods, Simulations
Springer Milan
Seiten XIV, 489 p.
Erscheinungsjahr 2013
ISBN 978-88-470-2861-6
       9788847028616
 (Softcover)
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This book is designed as an advanced undergraduate or a first-year graduate course for students from various disciplines like applied mathematics, physics, engineering. It has evolved while teaching courses on partial differential equations during the last decade at the Politecnico of Milan. The main purpose of these courses was twofold: on the one hand, to train the students to appreciate the interplay between theory and modelling in problems arising in the applied sciences and on the other hand to give them a solid background for numerical methods, such as finite differences and finite elements.

Inhaltsangaben

Introduction.- Scalar Conservation Laws.- Diffusion.- The Laplace Equation.- Reaction-diffusion models.- Waves and vibrations.- Elements of Functional Analysis.- Variational formulation of elliptic problems.- Weak formulation of evolution problems.- Solutions.- Fourier Series.- Notes on ordinary differential equations.- Finite difference approximation of time dependent problems.- Identities and Formulas.

A Royal Road to Algebraic Geometry : 978-3-642-19224-1
Holme
Springer Berlin Heidelberg
Seiten XIV, 366 p.
Erscheinungsjahr 2012
ISBN 978-3-642-19224-1
       9783642192241
 (Hardcover)
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Algebraic Geometry

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This book is about modern algebraic geometry. The title A Royal Road to Algebraic Geometry is inspired by the famous anecdote about the king asking Euclid if there really existed no simpler way for learning geometry, than to read all of his work Elements. Euclid is said to have answered: “There is no royal road to geometry!” The book starts by explaining this enigmatic answer, the aim of the book being to argue that indeed, in some sense there is a royal road to algebraic geometry. From a point of departure in algebraic curves, the exposition moves on to the present shape of the field, culminating with Alexander Grothendieck’s theory of schemes. Contemporary homological tools are explained. The reader will follow a directed path leading up to the main elements of modern algebraic geometry. When the road is completed, the reader is empowered to start navigating in this immense field, and to open up the door to a wonderful field of research. The greatest scientific experience of a lifetime!

Inhaltsangaben

Part I Curves: 1 Affine and Projective Space.- 2 Curves in A2 k and in P2.- 3 Higher Geometry in the Projective Plane.- 4 Plane Curves and Algebra.- 5 Projective Varieties in PNk.- Part II Introduction to Grothendieck’s Theory of Schemes: 6 Categories and Functors.- 7 Constructions and Representable Functors.- 8 Abelian Categories.- 9 The Concept of Spec(A).- 10 The Category of Schemes.- 11 Properties of Morphisms of Schemes.- 12 Modules, Algebras and Bundles on a Scheme.- 13 More Properties of Morphisms, Scheme Theoretic Image and the “Sorite”.- 14 Projective Schemes and Bundles.- 15 Further Properties of Morphisms.- 16 Conormal Sheaf and Projective Bundles.- 17 Cohomology Theory on Schemes.- 18 Intersection Theory.- 19 Characteristic Classes in Algebraic Geometry.- 20 The Riemann-Roch Theorem.- 21 Some Basic constructions in the category of projective kvarieties.- 22 More on Duality.- References.- Index

A Textbook on Ordinary Differential Equations : 978-3-319-16407-6
Ahmad
Springer International Publishing
Seiten XIV, 331 p.
Erscheinungsjahr 2015
ISBN 978-3-319-16407-6
       9783319164076
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This book offers readers a primer on the theory and applications of Ordinary Differential Equations. The style used is simple, yet thorough and rigorous. Each chapter ends with a broad set of exercises that range from the routine to the more challenging and thought-provoking. Solutions to selected exercises can be found at the end of the book. The book contains many interesting examples on topics such as electric circuits, the pendulum equation, the logistic equation, the Lotka-Volterra system, the Laplace Transform, etc., which introduce students to a number of interesting aspects of the theory and applications. The work is mainly intended for students of Mathematics, Physics, Engineering, Computer Science and other areas of the natural and social sciences that use ordinary differential equations, and who have a firm grasp of Calculus and a minimal understanding of the basic concepts used in Linear Algebra. It also studies a few more advanced topics, such as Stability Theory and Boundary Value Problems, which may be suitable for more advanced undergraduate or first-year graduate students. The second edition has been revised to correct minor errata, and features a number of carefully selected new exercises, together with more detailed explanations of some of the topics.A complete Solutions Manual, containing solutions to all the exercises published in the book, is available. Instructors who wish to adopt the book may request the manual by writing directly to one of the authors.

Inhaltsangaben

1 First order linear differential equations.- 2 Theory of first order differential equations.- 3 First order nonlinear differential equations.- 4 Existence and uniqueness for systems and higher order equations.- 5 Second order equations.- 6 Higher order linear equations.- 7 Systems of first order equations.- 8 Qualitative analysis of 2x2 systems and nonlinear second order equations.- 9 Sturm Liouville eigenvalue theory.- 10 Solutions by infinite series and Bessel functions.- 11 Laplace transform.- 12 Stability theory.- 13 Boundary value problems.- 14 Appendix A. Numerical methods.- 15 Answers to selected exercises.

Advanced Calculus of a Single Variable : 978-3-319-27806-3
Geveci
Springer International Publishing
Seiten XII, 382 p. 88 illus., 77 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-27806-3
       9783319278063
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This advanced undergraduate textbook is based on a one-semester course on single variable calculus that the author has been teaching at San Diego State University for many years. The aim of this classroom-tested book is to deliver a rigorous discussion of the concepts and theorems that are dealt with informally in the first two semesters of a beginning calculus course. As such, students are expected to gain a deeper understanding of the fundamental concepts of calculus, such as limits (with an emphasis on ?-? definitions), continuity (including an appreciation of the difference between mere pointwise and uniform continuity), the derivative (with rigorous proofs of various versions of L’Hôpital’s rule) and the Riemann integral (discussing improper integrals in-depth, including the comparison and Dirichlet tests). Success in this course is expected to prepare students for more advanced courses in real and complex analysis and this book will help to accomplish this. The first semester of advanced calculus can be followed by a rigorous course in multivariable calculus and an introductory real analysis course that treats the Lebesgue integral and metric spaces, with special emphasis on Banach and Hilbert spaces.

Inhaltsangaben

Preface.- Real Numbers, Sequences and Limits.- Limits and Continuity of Functions.- The Derivative.- The Riemann Integral.- Infinite Series.- Sequences and Series of Functions. Index.

Adventures in Stochastic Processes : 978-0-8176-3591-6
Resnick
Birkhäuser Boston
Seiten XII, 626 p.
Erscheinungsjahr 2002
ISBN 978-0-8176-3591-6
       9780817635916
 (Hardcover)
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Stochastic processes are necessary ingredients for building models of a wide variety of phenomena exhibiting time varying randomness. In a lively and imaginative presentation, studded with examples, exercises, and applications, and supported by inclusion of computational procedures, the author has created a textbook that provides easy access to this fundamental topic for many students of applied sciences at many levels. With its carefully modularized discussion and crystal clear differentiation between rigorous proof and plausibility argument, it is accessible to beginners but flexible enough to serve as well those who come to the course with strong backgrounds. The prerequisite background for reading the book is a graduate level pre-measure theoretic probability course. No knowledge of measure theory is presumed and advanced notions of conditioning are scrupulously avoided until the later chapters of the book. The tools of applied probability---discrete spaces, Markov chains, renewal theory, point processes, branching processes, random walks, Brownian motion---are presented to the reader in illuminating discussion. Applications include such topics as queuing, storage, risk analysis, genetics, inventory, choice, economics, sociology, and other. Because of the conviction that analysts who build models should know how to build them for each class of process studied, the author has included such constructions.

Inhaltsangaben

Preface * 1. Preliminaries: Discrete Index Sets and/or Discrete State Spaces * 2. Markov Chains * 3. Renewal Theory * 4. Point Processes * 5. Continuous Time Markov Chains * 6. Brownian Motion * 7. The General Random Walk * References * Index

Algebra for Applications : 978-3-319-21950-9
Slinko
Cryptography, Secret Sharing, Error-Correcting, Fingerprinting, Compression
Springer International Publishing
Seiten XIV, 328 p. 5 illus., 1 illus. in color.
Erscheinungsjahr 2015
ISBN 978-3-319-21950-9
       9783319219509
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Fachgebiet

Algebra

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This book examines the relationship between mathematics and data in the modern world. Indeed, modern societies are awash with data which must be manipulated in many different ways: encrypted, compressed, shared between users in a prescribed manner, protected from an unauthorised access and transmitted over unreliable channels. All of these operations can be understood only by a person with knowledge of basics in algebra and number theory.This book provides the necessary background in arithmetic, polynomials, groups, fields and elliptic curves that is sufficient to understand such real-life applications as cryptography, secret sharing, error-correcting, fingerprinting and compression of information. It is the first to cover many recent developments in these topics. Based on a lecture course given to third-year undergraduates, it is self-contained with numerous worked examples and exercises provided to test understanding. It can additionally be used for self-study.

Inhaltsangaben

Integers.- Cryptology.- Groups.- Fields.- Polynomials.- Secret Sharing.- Error-Correcting Codes.- Compression.- Appendix A: GAP.- Appendix B: Miscellanies.- Solutions to Exercises.

An Intermediate Course in Probability : 978-1-4419-0161-3
Gut
Springer New York
Seiten XV, 303 p.
Erscheinungsjahr 2009
ISBN 978-1-4419-0161-3
       9781441901613
 (Hardcover)
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The purpose of this book is to provide the reader with a solid background and understanding of the basic results and methods in probability theory before entering into more advanced courses (in probability and/or statistics). The presentation is fairly thorough and detailed with many solved examples. Several examples are solved with di erent methods in order to illustrate their di erent levels of sophistication, their pros, and their cons. The motivation for this style of exposition is that experience has proved that the hard part in courses of this kind usually is the application of the results and methods; to know how, when, and where to apply what; and then, technically, to solve a given problem once one knows how to proceed. Exercises are spread out along the way, and every chapter ends with a large selection of problems. Chapters 1 through 6 focus on some central areas of what might be called pure probability theory: multivariate random variables, conditioning, tra- forms, order variables, the multivariate normal distribution, and convergence.

Inhaltsangaben

Multivariate Random Variables.- Conditioning.- Transforms.- Order Statistics.- The Multivariate Normal Distribution.- Convergence.- An Outlook on Further Topics.- The Poisson Process.

An Introduction to Copulas : 978-0-387-28659-4
Nelsen
Springer New York
Seiten XIV, 272 p.
Erscheinungsjahr 2006
ISBN 978-0-387-28659-4
       9780387286594
 (Hardcover)
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Copulas are functions that join multivariate distribution functions to their one-dimensional margins. The study of copulas and their role in statistics is a new but vigorously growing field. In this book the student or practitioner of statistics and probability will find discussions of the fundamental properties of copulas and some of their primary applications. The applications include the study of dependence and measures of association, and the construction of families of bivariate distributions. With 116 examples, 54 figures, and 167 exercises, this book is suitable as a text or for self-study. The only prerequisite is an upper level undergraduate course in probability and mathematical statistics, although some familiarity with nonparametric statistics would be useful. Knowledge of measure-theoretic probability is not required. The revised second edition includes new sections on extreme value copulas, tail dependence, and quasi-copulas.

Inhaltsangaben

Definitions and Basic Properties.- Methods of Constructing Copulas.- Archimedean Copulas.- Dependence.- Additional Topics.

An Introduction to Frames and Riesz Bases : 978-3-319-25611-5
Christensen
Springer International Publishing
Seiten XXV, 704 p. 17 illus., 5 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-25611-5
       9783319256115
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Functional Analysis

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This revised and expanded monograph presents the general theory for frames and Riesz bases in Hilbert spaces as well as its concrete realizations within Gabor analysis, wavelet analysis, and generalized shift-invariant systems.  Compared with the first edition, more emphasis is put on explicit constructions with attractive properties.  Based on the exiting development of frame theory over the last decade, this second edition now includes new sections on the rapidly growing fields of LCA groups, generalized shift-invariant systems, duality theory for as well Gabor frames as wavelet frames, and open problems in the field. Key features include:*Elementary introduction to frame theory in finite-dimensional spaces * 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 and dual pairs of frames, with applications and connections to time-frequency analysis, wavelets, and generalized shift-invariant systems* Discussion of frames on LCA groups and the concrete realizations in terms of Gabor systems on the elementary groups; connections to sampling theory * Selected research topics presented with recommendations for more advanced topics and further reading* Open problems to stimulate further research An Introduction to Frames and Riesz Bases will be of interest to graduate students and researchers working in pure and applied mathematics, mathematical physics, and engineering. Professionals working in digital signal processing who wish to understand the theory behind many modern signal processing tools may also find this book a useful self-study reference. Review of the first edition:'Ole Christensen’s An Introduction to Frames and Riesz Bases is a first-rate introduction to the field … . The book provides an excellent exposition of these topics. The material is broad enough to pique the interest of many readers, the included exercises supply some interesting challenges, and the coverage provides enough background for those new to the subject to begin conducting original research.'   — Eric S. Weber, American Mathematical Monthly, Vol. 112, February, 2005 

Inhaltsangaben

Frames in Finite-dimensional Inner Product Spaces.- Infinite-dimensional Vector Spaces and Sequences.- Bases.- Bases and their Limitations.- Frames in Hilbert Spaces.- Tight Frames and Dual Frame Pairs.- Frames versus Riesz Bases.- Selected Topics in Frame Theory.- Frames of Translates.- Shift-Invariant Systems in l2(R).- Gabor Frames in L2(R).- Gabor Frames and Duality.- Selected Topics on Gabor Frames.- Gabor Frames in ?2(Z),L2(0,L),CL.- General Wavelet Frames in L2(R).- Dyadic Wavelet Frames for L2(R).- Frame Multiresolution Analysis.- Wavelet Frames via Extension Principles.- Selected Topics on Wavelet Frames.- Generalized Shift-Invariant Systems in L2(Rd).- Frames on Locally Compact Abelian Groups.- Perturbation of Frames.- Approximation of the Inverse Frame Operator.- Expansions in Banach Spaces. Appendix.

An Introduction to Homological Algebra : 978-0-387-24527-0
Rotman
Springer New York
Seiten XIV, 710 p. 11 illus.
Erscheinungsjahr 2009
ISBN 978-0-387-24527-0
       9780387245270
 (Softcover)
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Homological Algebra has grown in the nearly three decades since the rst e- tion of this book appeared in 1979. Two books discussing more recent results are Weibel, An Introduction to Homological Algebra, 1994, and Gelfand– Manin, Methods of Homological Algebra, 2003. In their Foreword, Gelfand and Manin divide the history of Homological Algebra into three periods: the rst period ended in the early 1960s, culminating in applications of Ho- logical Algebra to regular local rings. The second period, greatly in uenced by the work of A. Grothendieck and J. -P. Serre, continued through the 1980s; it involves abelian categories and sheaf cohomology. The third period, - volving derived categories and triangulated categories, is still ongoing. Both of these newer books discuss all three periods (see also Kashiwara–Schapira, Categories and Sheaves). The original version of this book discussed the rst period only; this new edition remains at the same introductory level, but it now introduces the second period as well. This change makes sense pe- gogically, for there has been a change in the mathematics population since 1979; today, virtually all mathematics graduate students have learned so- thing about functors and categories, and so I can now take the categorical viewpoint more seriously. When I was a graduate student, Homological Algebra was an unpopular subject. The general attitude was that it was a grotesque formalism, boring to learn, and not very useful once one had learned it.

Inhaltsangaben

Hom and Tensor.- Special Modules.- Specific Rings.- Setting the Stage.- Homology.- Tor and Ext.- Homology and Rings.- Homology and Groups.- Spectral Sequences.

An Introduction to Kolmogorov Complexity and Its Applications : 978-0-387-33998-6
Li
Springer New York
Seiten XXIV, 792 p. 21 illus.
Erscheinungsjahr 2008
ISBN 978-0-387-33998-6
       9780387339986
 (Hardcover)
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“The book is outstanding and admirable in many respects. ... is necessary reading for all kinds of readers from undergraduate students to top authorities in the field.” Journal of Symbolic Logic Written by two experts in the field, this is the only comprehensive and unified treatment of the central ideas and applications of Kolmogorov complexity. The book presents a thorough treatment of the subject with a wide range of illustrative applications. Such applications include the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and the thermodynamics of computing. It will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics. The book is self-contained in that it contains the basic requirements from mathematics and computer science. Included are also numerous problem sets, comments, source references, and hints to solutions of problems. New topics in this edition include Omega numbers, Kolmogorov–Loveland randomness, universal learning, communication complexity, Kolmogorov's random graphs, time-limited universal distribution, Shannon information and others.

Inhaltsangaben

Preliminaries.- Algorithmic Complexity.- Algorithmic Prefix Complexity.- Algorithmic Probability.- Inductive Reasoning.- The Incompressibility Method.- Resource-Bounded Complexity.- Physics, Information, and Computation.

An Introduction to Mathematical Cryptography : 978-1-4419-2674-6
Hoffstein
Springer New York
Seiten XVI, 524 p. 29 illus.
Erscheinungsjahr 2008
ISBN 978-1-4419-2674-6
       9781441926746
 (Softcover)
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Fachgebiet

Number Theory

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ThecreationofpublickeycryptographybyDi?eandHellmanin1976andthe subsequent invention of the RSA public key cryptosystem by Rivest, Shamir, and Adleman in 1978 are watershed events in the long history of secret c- munications. It is hard to overestimate the importance of public key cr- tosystems and their associated digital signature schemes in the modern world of computers and the Internet. This book provides an introduction to the theory of public key cryptography and to the mathematical ideas underlying that theory. Public key cryptography draws on many areas of mathematics, including number theory, abstract algebra, probability, and information theory. Each of these topics is introduced and developed in su?cient detail so that this book provides a self-contained course for the beginning student. The only prerequisite is a ?rst course in linear algebra. On the other hand, students with stronger mathematical backgrounds can move directly to cryptographic applications and still have time for advanced topics such as elliptic curve pairings and lattice-reduction algorithms. Amongthemanyfacetsofmoderncryptography,thisbookchoosestoc- centrate primarily on public key cryptosystems and digital signature schemes. This allows for an in-depth development of the necessary mathematics - quired for both the construction of these schemes and an analysis of their security. The reader who masters the material in this book will not only be well prepared for further study in cryptography, but will have acquired a real understanding of the underlying mathematical principles on which modern cryptography is based.

Inhaltsangaben

An Introduction to Cryptography.- Discrete Logarithms and Diffie Hellman.- Integer Factorization and RSA.- Combinatorics, Probability and Information Theory.- Elliptic Curves and Cryptography.- Lattices and Cryptography.- Digital Signatures.- Additional Topics in Cryptography.

An Introduction to Mathematical Finance with Applications : 978-1-4939-3781-3
Petters
Understanding and Building Financial Intuition
Springer New York
Seiten XVII, 483 p. 52 illus., 12 illus. in color.
Erscheinungsjahr 2016
ISBN 978-1-4939-3781-3
       9781493937813
 (Hardcover)
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This textbook aims to fill the gap between those that offer a theoretical treatment without many applications and those that present and apply formulas without appropriately deriving them. The balance achieved will give readers a fundamental understanding of key financial ideas and tools that form the basis for building realistic models, including those that may become proprietary. Numerous carefully chosen examples and exercises reinforce the student’s conceptual understanding and facility with applications.  The exercises are divided into conceptual, application-based, and theoretical problems, which probe the material deeper.The book is aimed toward advanced undergraduates and first-year graduate students who are new to finance or want a more rigorous treatment of the mathematical models used within. While no background in finance is assumed, prerequisite math courses include multivariable calculus, probability, and linear algebra. The authors introduce additional mathematical tools as needed. The entire textbook is appropriate for a single year-long course on introductory mathematical finance. The self-contained design of the text allows for instructor flexibility in topics courses and those focusing on financial derivatives. Moreover, the text is useful for mathematicians, physicists, and engineers who want to learn finance via an approach that builds their financial intuition and is explicit about model building, as well as business school students who want a treatment of finance that is deeper but not overly theoretical.

Inhaltsangaben

Preface.- 1. Preliminaries and Financial Markets.- 2. The Time Value of Money.- 3. Markowitz Portfolio Theory.- 4. Capital Market Theory and Portfolio Risk Measures.- 5. Binomial Trees and Security Pricing Modeling.- 6. Stochastic Calculus and Geometric Brownian Motion Model.-  7. Derivatives: Forwards, Futures, Swaps and Options.- 8. The BSM Model and European Option Pricing.- Index. 

An Introduction to Programming and Numerical Methods in MATLAB : 978-1-85233-919-7
Otto
Springer London
Seiten XVI, 464 p. 111 illus.
Erscheinungsjahr 2005
ISBN 978-1-85233-919-7
       9781852339197
 (Softcover)
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MATLAB is a powerful programme, which naturally lends itself to the rapid implementation of most numerical algorithms. This text, which uses MATLAB, gives a detailed overview of structured programming and numerical methods for the undergraduate student. The book covers numerical methods for solving a wide range of problems, from integration to the numerical solution of differential equations or the simulation of random processes. Each chapter includes extensive examples and tasks, at varying levels of complexity. For practice, the early chapters include programmes that require debugging by the reader, while full solutions are given for all the tasks. The book also includes: a glossary of MATLAB commands appendices of mathematical techniques used in numerical methods. Designed as a text for a first course in programming and algorithm design, as well as in numerical methods courses, the book will be of benefit to a wide range of students from mathematics and engineering, to commerce.

Inhaltsangaben

Simple Calculations with MATLAB.- Writing Scripts and Functions.- Loops and Conditional Statements.- Root Finding.- Interpolation and Extrapolation.- Matrices.- Numerical Integration.- Solving Differential Equations.- Simulations and Random Numbers.

An Introductory Course in Lebesgue Spaces : 978-3-319-30032-0
Castillo
Springer International Publishing
Seiten XII, 461 p. 14 illus.
Erscheinungsjahr 2016
ISBN 978-3-319-30032-0
       9783319300320
 (Hardcover)
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This book is devoted exclusively to Lebesgue spaces and their direct derived spaces. Unique in its sole dedication, this book explores Lebesgue spaces, distribution functions and nonincreasing rearrangement. Moreover, it also deals with weak, Lorentz and the more recent variable exponent and grand Lebesgue spaces with considerable detail to the proofs. The book also touches on basic harmonic analysis in the aforementioned spaces. An appendix is given at the end of the book giving it a self-contained character. This work is ideal for teachers, graduate students and researchers.

Inhaltsangaben

Convex Functions and Inequalities.- Lebesgue Sequence Spaces.- Lebesgue Spaces.- Distribution Function and Non-Increasing Rearrangement.- Weak Lebesgue Spaces.- Lorentz Spaces.- Non-Standard Lebesgue Spaces.- Interpolation of Operators.- Maximal Operator.- Integral Operators.- Convolution and Potentials.- Appendices.

An Invitation to Mathematics : 978-3-642-19532-7
Schleicher
From Competitions to Research
Springer Berlin Heidelberg
Seiten XIV, 220 p. 56 illus., 31 illus. in color.
Erscheinungsjahr 2011
ISBN 978-3-642-19532-7
       9783642195327
 (Softcover)
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This Invitation to Mathematics consists of 14 contributions, many from the world's leading mathematicians, which introduce the readers to exciting aspects of current mathematical research. The contributions are as varied as the personalities of active mathematicians, but together they show mathematics as a rich and lively field of research. The contributions are written for interested students at the age of transition between high school and university who know high school mathematics and perhaps competition mathematics and who want to find out what current research mathematics is about. We hope that it will also be of interest to teachers or more advanced mathematicians who would like to learn about exciting aspects of mathematics outside of their own work or specialization. Together with a team of young ``test readers'', editors and authors have taken great care, through a substantial ``active editing'' process, to make the contributions understandable by the intended readership.

Inhaltsangaben

Preface: What is Mathematics?.- Welcome.- Structure and Randomness in the Prime Numbers.- How to Solve a Diophantine Equation.- From Sex to Quadratic Forms.- Small Divisors: Number Theory in Dynamical Systems.- How do IMO Problems Compare with Research Problems?.- How do Research Problems Compare with IMO Problems?.- Graph Theory Over 45 Years.- Communication Complexity.- Ten Digit Problems.- The Ever-Elusive Blowup in the Mathematical Description of Fluids.- About the Hardy Inequality.- The Lion and the Christian, and Other Pursuit and Evasion Games.- Three Mathematics Competitions.- Complex Dynamics, the Mandelbrot Set, and Newton´s Method.

An R and S-Plus® Companion to Multivariate Analysis : 978-1-85233-882-4
Everitt
Springer London
Seiten XIII, 221 p.
Erscheinungsjahr 2005
ISBN 978-1-85233-882-4
       9781852338824
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Most data sets collected by researchers are multivariate, and in the majority of cases the variables need to be examined simultaneously to get the most informative results. This requires the use of one or other of the many methods of multivariate analysis, and the use of a suitable software package such as S-PLUS or R. In this book the core multivariate methodology is covered along with some basic theory for each method described. The necessary R and S-PLUS code is given for each analysis in the book, with any differences between the two highlighted. Graduate students, and advanced undergraduates on applied statistics courses, especially those in the social sciences, will find this book invaluable in their work, and it will also be useful to researchers outside of statistics who need to deal with the complexities of multivariate data in their work. From the reviews: 'This text is much more than just an R/S programming guide. Brian Everitt's expertise in multivariate data analysis shines through brilliantly.' Journal of the American Statistical Association, June 2006

Inhaltsangaben

Multivariate Data and Multivariate Analysis.- Looking at Multivariate Data.- Principal Components Analysis.- Exploratory Factor Analysis.- Multidimensional Scaling and Correspondence Analysis.- Cluster Analysis.- Grouped Multivariate Data: Multivariate Analysis of Variance and Discriminant Function Analysis.- Multiple Regression and Canonical Correlation.- Analysis of Repeated Measures Data.

Analyse mathématique I : 978-3-540-42057-6
Convergence, fonctions élémentaires
Springer Berlin Heidelberg
Seiten XX, 458 p.
Erscheinungsjahr 2001
ISBN 978-3-540-42057-6
       9783540420576
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Real Functions

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Les deux premiers volumes de cet ouvrage sont consacrés aux fonctions dans R ou C, y compris la théorie élémentaire des séries et intégrales de Fourier et une partie de celle des fonctions holomorphes. L'exposé, non strictement linéaire, combine indications historiques et raisonnements rigoureux. Il montre la diversité des voies d'accès aux principaux résultats afin de familiariser le lecteur avec les méthodes de raisonnement et idées fondamentales plutôt qu'avec les techniques de calcul, point de vue utile aussi aux personnes travaillant seules. Les volumes 3 et 4 traiteront principalement des fonctions analytiques (théorie de Cauchy, théorie analytique des nombres et fonctions modulaires), ainsi que du calcul différentiel sur les variétés, avec un court exposé de l'intégrale de Lebesgue, en suivant d'assez près le célèbre cours donné longtemps par l'auteur à l'Université Paris 7. On reconnaîtra dans ce nouvel ouvrage le style inimitable de l'auteur, et pas seulement par son refus de l'écriture condensée en usage dans de nombreux manuels.

Inhaltsangaben

Ensembles et fonctions.- Convergence: variables discrètes.- Convergence: variables continues.- Fonctions dérivables.- Puissances, exponentielles, logarithmes, fonctions trigonométriques. La table des matières détaillé est disponsible sur le serveur de Springer. Voir le catalogue sous: http://www.springer.de

Analyse mathématique II : 978-3-540-00655-8
Godement
Calcul différentiel et intégral, séries de Fourier, fonctions holomorphes
Springer Berlin Heidelberg
Seiten VIII, 490 p. 19 ill.
Erscheinungsjahr 2003
ISBN 978-3-540-00655-8
       9783540006558
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Real Functions

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Les deux premiers volumes sont consacrés aux fonctions dans R ou C, y compris la théorie élémentaire des séries et intégrales de Fourier et une partie de celle des fonctions holomorphes. L'exposé non strictement linéaire, combine indications historiques et raisonnements rigoureux. Il montre la diversité des voies d'accès aux principaux résultats afin de familiariser le lecteur avec les méthodes de raisonnement et idées fondamentales plutôt qu'avec les techniques de calcul, point de vue utile aussi aux personnes travaillant seules. Les volumes 3 et 4 traitent principalement des fonctions analytiques (théorie de Cauchy, théorie analytique des nombres et fonctions modulaires), ainsi que du calcul différentiel sur les variétés, avec un exposé de l'intégrale de Lebesgue, en suivant d'assez près le célèbre cours donné longtemps par l'auteur à l'Université Paris 7. On reconnaîtra dans ce nouvel ouvrage le style inimitable de l'auteur, et pas seulement par son refus de l'écriture condensée en usage dans ce nombreux manuels.

Inhaltsangaben

Calcul differentiel et intégral; Calculs asymptotiques; Analyse harmonique et fonctions holomorphes; Postface.

Analyse mathématique III : 978-3-540-66142-9
Godement
Fonctions analytiques, différentielles et variétés, surfaces de Riemann
Springer Berlin Heidelberg
Seiten IX, 338 p. 27 ill.
Erscheinungsjahr 2002
ISBN 978-3-540-66142-9
       9783540661429
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Real Functions

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Ce vol. III expose la théorie classique de Cauchy dans un esprit orienté bien davantage vers ses innombrables utilisations que vers une théorie plus ou moins complète des fonctions analytiques. On montre ensuite comment les intégrales curvilignes à la Cauchy se généralisent à un nombre quelconque de variables réelles (formes différentielles, formules de type Stokes). Les bases de la théorie des variétés sont ensuite exposées, principalement pour fournir au lecteur le langage 'canonique' et quelques théorèmes importants (changement de variables dans les intégrales, équations différentielles). Un dernier chapitre montre comment on peut utiliser ces théories pour construire la surface de Riemann compacte d'une fonction algébrique, sujet rarement traité dans la littérature non spécialisée bien que n'éxigeant que des techniques élémentaires. Un volume IV exposera, outre,l'intégrale de Lebesgue, un bloc de mathématiques spécialisées vers lequel convergera tout le contenu des volumes précédents: séries et produits infinis de Jacobi, Riemann, Dedekind, fonctions elliptiques, théorie classique des fonctions modulaires et la version moderne utilisant la structure de groupe de Lie de SL(2,R).

Inhaltsangaben

VIII. La théorie de Cauchy.- IX. Différentielles et intégrales.- X. La Surface de Riemann d'une fonction algébrique.

Analyse mathématique IV : 978-3-540-43841-0
Godement
Intégration et théorie spectrale, analyse harmonique, le jardin des délices modulaires
Springer Berlin Heidelberg
Seiten XII, 599 p. 4 ill.
Erscheinungsjahr 2003
ISBN 978-3-540-43841-0
       9783540438410
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Real Functions

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Ce 4ème volume de l'ouvrage Analyse mathématique initiera le lecteur à l'analyse fonctionnelle (intégration, espaces de Hilbert, analyse harmonique en théorie des groupes) et aux méthodes de la théorie des fonctions modulaires (séries L et theta, fonctions elliptiques, usage de l'algèbre de Lie de SL2). Tout comme pour les volumes 1 à 3, on reconnaîtra ici encore, le style inimitable de l'auteur et pas seulement par son refus de l'ecriture condensée en usage dans de nombreux manuels. Mariant judicieusement les mathématiques dites 'modernes' et' classiques', la première partie (Intégration) est d'utilité universelle tandis que la seconde oriente le lecteur vers un domaine de recherche spécialisé et très actif, avec de vastes généralisations possibles.

Inhaltsangaben

XI. Integration et transformation de Fourier.- XII. Le jardin des delices modulaires.

Analysis : 978-3-319-32409-8
Penot
From Concepts to Applications
Springer International Publishing
Seiten XXIII, 669 p. 26 illus.
Erscheinungsjahr 2016
ISBN 978-3-319-32409-8
       9783319324098
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Functional Analysis

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This textbook covers the main results and methods of real analysis in a single volume. Taking a progressive approach to equations and transformations, this book starts with the very foundations of real analysis (set theory, order, convergence, and measure theory) before presenting powerful results that can be applied to concrete problems. In addition to classical results of functional analysis, differential calculus and integration, Analysis discusses topics such as convex analysis, dissipative operators and semigroups which are often absent from classical treatises. Acknowledging that analysis has significantly contributed to the understanding and development of the present world, the book further elaborates on techniques which pervade modern civilization, including wavelets in information theory, the Radon transform in medical imaging and partial differential equations in various mechanical and physical phenomena. Advanced undergraduate and graduate students, engineers as well as practitioners wishing to familiarise themselves with concepts and applications of analysis will find this book useful. With its content split into several topics of interest, the book’s style and layout make it suitable for use in several courses, while its self-contained character makes it appropriate for self-study.

Inhaltsangaben

Sets, Orders, Relations and Measures.- Encounters With Limits.- Elements of Functional Analysis.- Hilbert Spaces.- The Power of Differential Calculus.- A Touch of Convex Analysis.- Integration.- Differentiation and Integration.- Partial Differential Equations.- Evolution Problems.- References.- Index.

Analytic Function Theory of Several Variables : 978-981-10-0289-2
Noguchi
Elements of Oka’s Coherence
Springer Singapore
Seiten XVIII, 397 p. 28 illus., 27 illus. in color.
Erscheinungsjahr 2016
ISBN 978-981-10-0289-2
       9789811002892
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The purpose of this book is to present the classical analytic function theory of several variables as a standard subject in a course of mathematics after learning the elementary materials (sets, general topology, algebra, one complex variable). This includes the essential parts of Grauert–Remmert's two volumes, GL227(236) (Theory of Stein spaces) and GL265 (Coherent analytic sheaves) with a lowering of the level for novice graduate students (here, Grauert's direct image theorem is limited to the case of finite maps).The core of the theory is 'Oka's Coherence', found and proved by Kiyoshi Oka. It is indispensable, not only in the study of complex analysis and complex geometry, but also in a large area of modern mathematics. In this book, just after an introductory chapter on holomorphic functions (Chap. 1), we prove Oka's First Coherence Theorem for holomorphic functions in Chap. 2. This defines a unique character of the book compared with other books on this subject, in which the notion of coherence appears much later.The present book, consisting of nine chapters, gives complete treatments of the following items: Coherence of sheaves of holomorphic functions (Chap. 2); Oka–Cartan's Fundamental Theorem (Chap. 4); Coherence of ideal sheaves of complex analytic subsets (Chap. 6); Coherence of the normalization sheaves of complex spaces (Chap. 6); Grauert's Finiteness Theorem (Chaps. 7, 8); Oka's Theorem for Riemann domains (Chap. 8). The theories of sheaf cohomology and domains of holomorphy are also presented (Chaps. 3, 5). Chapter 6 deals with the theory of complex analytic subsets. Chapter 8 is devoted to the applications of formerly obtained results, proving Cartan–Serre's Theorem and Kodaira's Embedding Theorem. In Chap. 9, we discuss the historical development of 'Coherence'.It is difficult to find a book at this level that treats all of the above subjects in a completely self-contained manner. In the present volume, a number of classical proofs are improved and simplified, so that the contents are easily accessible for beginning graduate students.

Inhaltsangaben

Holomorphic Functions.- Oka's First Coherence Theorem.- Sheaf Cohomology.- Holomorphically Convex Domains and Oka--Cartan's Fundamental Theorem.- Domains of Holomorphy.- Analytic Sets and Complex Spaces.- Pseudoconvex Domains and Oka's Theorem.- Cohomology of Coherent Sheaves and Kodaira's Embedding Theorem.- On Coherence.- Appendix.- References.- Index.- Symbols.

Approximation Algorithms and Semidefinite Programming : 978-3-642-22014-2
Gärtner
Springer Berlin Heidelberg
Seiten XI, 251 p.
Erscheinungsjahr 2012
ISBN 978-3-642-22014-2
       9783642220142
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Semidefinite programs constitute one of the largest classes of optimization problems that can be solved with reasonable efficiency - both in theory and practice. They play a key role in a variety of research areas, such as combinatorial optimization, approximation algorithms, computational complexity, graph theory, geometry, real algebraic geometry and quantum computing. This book is an introduction to selected aspects of semidefinite programming and its use in approximation algorithms. It covers the basics but also a significant amount of recent and more advanced material.  There are many computational problems, such as MAXCUT, for which one cannot reasonably expect to obtain an exact solution efficiently, and in such case, one has to settle for approximate solutions. For MAXCUT and its relatives, exciting recent results suggest that semidefinite programming is probably the ultimate tool. Indeed, assuming the Unique Games Conjecture, a plausible but as yet unproven hypothesis, it was shown that for these problems, known algorithms based on semidefinite programming deliver the best possible approximation ratios among all polynomial-time algorithms. This book follows the “semidefinite side” of these developments, presenting some of the main ideas behind approximation algorithms based on semidefinite programming. It develops the basic theory of semidefinite programming, presents one of the known efficient algorithms in detail, and describes the principles of some others. It also includes applications, focusing on approximation algorithms.

Inhaltsangaben

Part I (by Bernd Gärtner): 1 Introduction: MAXCUT via Semidefinite Programming.- 2 Semidefinite Programming.- 3 Shannon Capacity and Lovász Theta.-  4 Duality and Cone Programming.-  5 Approximately Solving Semidefinite Programs.- 6 An Interior-Point Algorithm for Semidefinite Programming.- 7 Compositive Programming.-  Part II (by Jiri Matousek): 8 Lower Bounds for the Goemans–Williamson MAXCUT Algorithm .- 9 Coloring 3-Chromatic Graphs.- 10 Maximizing a Quadratic Form on a Graph.- 11 Colorings With Low Discrepancy.- 12 Constraint Satisfaction Problems, and Relaxing Them Semidefinitely.- 13 Rounding Via Miniatures.- Summary.- References.- Index.

Approximation Methods in Probability Theory : 978-3-319-34071-5
?ekanavi?ius
Springer International Publishing
Seiten XII, 274 p.
Erscheinungsjahr 2016
ISBN 978-3-319-34071-5
       9783319340715
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This book presents a wide range of well-known and less common methods used for estimating the accuracy of probabilistic approximations, including the Esseen type inversion formulas, the Stein method as well as the methods of convolutions and triangle function. Emphasising the correct usage of the methods presented, each step required for the proofs is examined in detail. As a result, this textbook provides valuable tools for proving approximation theorems. While Approximation Methods in Probability Theory will appeal to everyone interested in limit theorems of probability theory, the book is particularly aimed at graduate students who have completed a standard intermediate course in probability theory. Furthermore, experienced researchers wanting to enlarge their toolkit will also find this book useful.

Inhaltsangaben

Definitions and preliminary facts.- The method of convolutions.- Local lattice estimates.- Uniform lattice estimates.- Total variation of lattice measures.- Non-uniform estimates for lattice measures.- Discrete non-lattice approximations.- Absolutely continuous approximations.- The Esseen type estimates.- Lower estimates.- The Stein method.- The triangle function method.- Heinrich's method for m-dependent variables.- Other methods.- Solutions to selected problems.- Bibliography.- Index.

Architecture of Systems Problem Solving : 978-0-306-47357-9
Klir
Springer US
Seiten XV, 349 p.
Erscheinungsjahr 2003
ISBN 978-0-306-47357-9
       9780306473579
 (Hardcover)
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One criterion for classifying books is whether they are written for a single pur­ pose or for multiple purposes. This book belongs to the category of multipurpose books, but one of its roles is predominant-it is primarily a textbook. As such, it can be used for a variety ofcourses at the first-year graduate or upper-division undergraduate level. A common characteristic of these courses is that they cover fundamental systems concepts, major categories of systems problems, and some selected methods for dealing with these problems at a rather general level. A unique feature of the book is that the concepts, problems, and methods are introduced in the context of an architectural formulation of an expert system­ referred to as the general systems problem solver or aSPS-whose aim is to provide users ofall kinds with computer-based systems knowledge and methodo­ logy. Theasps architecture,which is developed throughout the book, facilitates a framework that is conducive to acoherent, comprehensive, and pragmaticcoverage ofsystems fundamentals-concepts, problems, and methods. A course that covers systems fundamentals is now offered not only in sys­ tems science, information science, or systems engineering programs, but in many programs in other disciplines as well. Although the level ofcoverage for systems science or engineering students is surely different from that used for students in other disciplines, this book is designed to serve both of these needs.

Inhaltsangaben

1 Introduction.- 1.1 Systems Science.- 1.2 Systems Problem Solving.- 1.3 Hierarchy of Epistemological Levels of Systems.- 1.4 The Role of Mathematics.- 1.5 The Role of Computer Technology.- 1.6 Architecture of Systems Problem Solving.- 2 Source and Data Systems.- 2.1 Objects and Object Systems.- 2.2 Variables and Supports.- 2.3 Methodological Distinctions.- 2.4 Discrete versus Continuous.- 2.5 Image Systems and Source Systems.- 2.6 Data Systems.- 3 Generative Systems.- 3.1 Empirical Investigation.- 3.2 Behavior Systems.- 3.3 Methodological Distinctions.- 3.4 From Data Systems to Behavior Systems.- 3.5 Measures of Uncertainty.- 3.6 Search for Admissible Behavior Systems.- 3.7 State-Transition Systems.- 3.8 Generative Systems.- 3.9 Simplification of Generative Systems.- 3.10 Systems Inquiry and Systems Design.- 4 Structure Systems.- 4.1 Wholes and Parts.- 4.2 Systems, Subsystems, Supersystems.- 4.3 Structure Source Systems and Structure Data Systems.- 4.4 Structure Behavior Systems.- 4.5 Problems of Systems Design.- 4.6 Identification Problem.- 4.7 Reconstruction Problem.- 4.8 Reconstructability Analysis.- 4.9 Simulation Experiments.- 4.10 Inductive Reasoning.- 4.11 Inconsistent Structure Systems.- 5 Metasystems.- 5.1 Change versus Invariance.- 5.2 Primary and Secondary Systems Traits.- 5.3 Metasystems.- 5.4 Metasystems versus Structure Systems.- 5.5 Multilevel Metasystems.- 5.6 Identification of Change.- 6 GSPS: Architecture, Use, Evolution.- 6.1 Epistemological Hierarchy of Systems : Formal Definition.- 6.2 Methodological Distinctions: A Summary.- 6.3 Problem Requirements.- 6.4 Systems Problems.- 6.5 GSPS Conceptual Framework: Formal Definition.- 6.6 Overview of GSPS Architecture.- 6.7 GSPS Use: Some Case Studies.- 6.8 GSPS Evolution.- Author Index.

Automated Solution of Differential Equations by the Finite Element Method : 978-3-642-23098-1
Logg
The FEniCS Book
Springer Berlin Heidelberg
Seiten XIII, 723 p. 346 illus., 52 illus. in color.
Erscheinungsjahr 2012
ISBN 978-3-642-23098-1
       9783642230981
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This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Following are chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Inhaltsangaben


Basic Algebraic Topology and its Applications : 978-81-322-2841-7
Adhikari
Springer India
Seiten XXIX, 615 p. 176 illus.
Erscheinungsjahr 2016
ISBN 978-81-322-2841-7
       9788132228417
 (Hardcover)
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Algebraic Topology

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This book provides an accessible introduction to algebraic topology, a ?eld at the intersection of topology, geometry and algebra, together with its applications. Moreover, it covers several related topics that are in fact important in the overall scheme of algebraic topology. Comprising eighteen chapters and two appendices, the book integrates various concepts of algebraic topology, supported by examples, exercises, applications and historical notes. Primarily intended as a textbook, the book o?ers a valuable resource for undergraduate, postgraduate and advanced mathematics students alike. Focusing more on the geometric than on algebraic aspects of the subject, as well as its natural development, the book conveys the basic language of modern algebraic topology by exploring homotopy, homology and cohomology theories, and examines a variety of spaces: spheres, projective spaces, classical groups and their quotient spaces, function spaces, polyhedra, topological groups, Lie groups and cell complexes, etc. The book studies a variety of maps, which are continuous functions between spaces. It also reveals the importance of algebraic topology in contemporary mathematics, theoretical physics, computer science, chemistry, economics, and the biological and medical sciences, and encourages students to engage in further study.

Inhaltsangaben

Prerequisite Concepts and Notations.- Basic Homotopy.- The Fundamental Groups.-Covering Spaces.- Fibre Bundles, Vector Bundles and K-theory.- Geometry of Simplicial Complexes and Fundamental Groups.- Higher Homotopy Groups.- Products in Higher Homotopy Groups.- CW-complexes and Homotopy.- Eilenberg-MacLane Spaces.- Homology and Cohomology Theories.- Eilenberg-Steenrod Axioms for Homology and Cohomology Theories.- Consequences of the Eilenberg-Steenrod Axioms.- Some Applications of Homology Theory.- Spectral Homology and Cohomology Theories.- Obstruction Theory.- More Relations Between Homotopy and Homology Groups.- A Brief Historical Note.

Basic Linear Algebra : 978-1-85233-662-2
Blyth
Springer London
Seiten XI, 232 p.
Erscheinungsjahr 2002
ISBN 978-1-85233-662-2
       9781852336622
 (Softcover)
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Algebra

Information

Basic Linear Algebra is a text for first year students leading from concrete examples to abstract theorems, via tutorial-type exercises. More exercises (of the kind a student may expect in examination papers) are grouped at the end of each section. The book covers the most important basics of any first course on linear algebra, explaining the algebra of matrices with applications to analytic geometry, systems of linear equations, difference equations and complex numbers. Linear equations are treated via Hermite normal forms which provides a successful and concrete explanation of the notion of linear independence. Another important highlight is the connection between linear mappings and matrices leading to the change of basis theorem which opens the door to the notion of similarity. This new and revised edition features additional exercises and coverage of Cramer's rule (omitted from the first edition). However, it is the new, extra chapter on computer assistance that will be of particular interest to readers: this will take the form of a tutorial on the use of the 'LinearAlgebra' package in MAPLE 7 and will deal with all the aspects of linear algebra developed within the book.

Inhaltsangaben

1. The Algebra of Matrices.- 2. Some Applications of Matrices.- 3. Systems of Linear Equations.- 4. Invertible Matrices.- 5. Vector Spaces.- 6. Linear Mappings.- 7. The Matrix Connection.- 8. Determinants.- 9. Eigenvalues and Eigenvectors.- 10. The Minimum Polynomial.- 11. Computer Assistance.- 12. Solutions to the Exercises.

Basic Stochastic Processes : 978-3-540-76175-4
Brzezniak
A Course Through Exercises
Springer London
Seiten X, 226 p.
Erscheinungsjahr 1999
ISBN 978-3-540-76175-4
       9783540761754
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This book has been designed for a final year undergraduate course in stochastic processes. It will also be suitable for mathematics undergraduates and others with interest in probability and stochastic processes, who wish to study on their own. The main prerequisite is probability theory: probability measures, random variables, expectation, independence, conditional probability, and the laws of large numbers. The only other prerequisite is calculus. This covers limits, series, the notion of continuity, differentiation and the Riemann integral. Familiarity with the Lebesgue integral would be a bonus. A certain level of fundamental mathematical experience, such as elementary set theory, is assumed implicitly. Throughout the book the exposition is interlaced with numerous exercises, which form an integral part of the course. Complete solutions are provided at the end of each chapter. Also, each exercise is accompanied by a hint to guide the reader in an informal manner. This feature will be particularly useful for self-study and may be of help in tutorials. It also presents a challenge for the lecturer to involve the students as active participants in the course.

Inhaltsangaben

1. Review of Probability.- 1.1 Events and Probability.- 1.2 Random Variables.- 1.3 Conditional Probability and Independence.- 1.4 Solutions.- 2. Conditional Expectation.- 2.1 Conditioning on an Event.- 2.2 Conditioning on a Discrete Random Variable.- 2.3 Conditioning on an Arbitrary Random Variable.- 2.4 Conditioning on a ?-Field.- 2.5 General Properties.- 2.6 Various Exercises on Conditional Expectation.- 2.7 Solutions.- 3. Martingales in Discrete.- 3.1 Sequences of Random Variables.- 3.2 Filtrations.- 3.3 Martingales.- 3.4 Games of Chance.- 3.5 Stopping Times.- 3.6 Optional Stopping Theorem.- 3.7 Solutions.- 4. Martingale Inequalities and Convergence.- 4.1 Doob’s Martingale Inequalities.- 4.2 Doob’s Martingale Convergence Theorem.- 4.3 Uniform Integrability and L1 Convergence of Martingales.- 4.4 Solutions.- 5. Markov Chains.- 5.1 First Examples and Definitions.- 5.2 Classification of States.- 5.3 Long-Time Behaviour of Markov Chains: General Case.- 5.4 Long-Time Behaviour of Markov Chains with Finite State Space.- 5.5 Solutions.- 6. Stochastic Processes in Continuous Time.- 6.1 General Notions.- 6.2 Poisson Process.- 6.3 Brownian Motion.- 6.4 Solutions.- 7. Itô Stochastic Calculus.- 7.1 Itô Stochastic Integral: Definition.- 7.2 Examples.- 7.3 Properties of the Stochastic Integral.- 7.4 Stochastic Differential and Itô Formula.- 7.5 Stochastic Differential Equations.- 7.6 Solutions.

Brownian Motion, Martingales, and Stochastic Calculus  : 978-3-319-31088-6
Le Gall
Springer International Publishing
Seiten XIII, 273 p. 5 illus., 1 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-31088-6
       9783319310886
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This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Itô’s formula, the optional stopping theorem and Girsanov’s theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter.
Since its invention by Itô, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested in such developments.
Beginning graduate or advanced undergraduate students will benefit from this detailed approach to an essential area of probability theory. The emphasis is on concise and efficient presentation, without any concession to mathematical rigor. The material has been taught by the author for several years in graduate courses at two of the most prestigious French universities. The fact that proofs are given with full details makes the book particularly suitable for self-study. The numerous exercises help the reader to get acquainted with the tools of stochastic calculus.

Inhaltsangaben

Gaussian variables and Gaussian processes.- Brownian motion.- Filtrations and martingales.- Continuous semimartingales.- Stochastic integration.- General theory of Markov processes.- Brownian motion and partial differential equations.- Stochastic differential equations.- Local times.- The monotone class lemma.- Discrete martingales.- References.

Calcul Scientifique : 978-88-470-1675-0
Quarteroni
Cours, exercices corrigés et illustrations en Matlab et Octave
Springer Milan
Seiten 360 p.
Erscheinungsjahr 2010
ISBN 978-88-470-1675-0
       9788847016750
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Ce livre constitue une introduction au Calcul Scientifique. Son objectif est de présenter des méthodes numériques permettant de résoudre avec un ordinateur certains problèmes mathématiques qui ne peuvent être traités simplement avec un papier et un crayon. Les questions classiques du Calcul Scientifique sont abordées: la recherche des zéros ou le calcul d'intégrales de fonctions continues, la résolution de systèmes linéaires, l'approximation de fonctions par des polynômes, la résolution approchée d'équations différentielles. La présentation de ces méthodes est rendue vivante par le recours systématique aux environnements de programmation Matlab et Octave dont les principales commandes sont introduites progressivement. Tous les algorithmes sont présentés sous la forme de programmes. Ceci permet de vérifier très rapidement leurs propriétés théoriques, en particulier la stabilité, la précision et la complexité. La résolution de divers problèmes, souvent motivés par des applications concrètes, fait l'objet de nombreux exemples et exercices. À la fin de chaque chapitre, une section présente des aspects plus avancés et fournit des indications bibliographiques qui permettront au lecteur d'appronfondir les connaissances acquises. Le dernier chapitre est consacré à la correction des exercices proposés tout au long du livre

Inhaltsangaben

Ce qu’on ne peut ignorer.- Equations non linéaires.- Approximation de fonctions et de données.- Intégration et différentiation numérique.- Systèmes linéaires.- Valeurs propres et vecteurs propres.- Equations différentielles ordinaires.- Approximation numérique des problèmes.- Solutions des exercices.

Calculus Problems : 978-3-319-15427-5
Baronti
Springer International Publishing
Seiten XIII, 366 p. 132 illus.
Erscheinungsjahr 2016
ISBN 978-3-319-15427-5
       9783319154275
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Analysis

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This book, intended as a practical working guide for calculus students, includes 450 exercises. It is designed for undergraduate students in Engineering, Mathematics, Physics, or any other field where rigorous calculus is needed, and will greatly benefit anyone seeking a problem-solving approach to calculus. Each chapter starts with a summary of the main definitions and results, which is followed by a selection of solved exercises accompanied by brief, illustrative comments. A selection of problems with indicated solutions rounds out each chapter.A final chapter explores problems that are not designed with a single issue in mind but instead call for the combination of a variety of techniques, rounding out the book’s coverage. Though the book’s primary focus is on functions of one real variable, basic ordinary differential equations (separation of variables, linear first order and constant coefficients ODEs) are also discussed. The material is taken from actual written tests that have been delivered at the Engineering School of the University of Genoa. Literally thousands of students have worked on these problems, ensuring their real-world applicability.

Inhaltsangaben

1 Manipulation of Graphs.- 2 Invertible Mappings.- 3 Sequences.- 4 Maxima, minima, least upper bound, greatest lower bound.- 5 Limits of functions.- 6 Continuous Functions.- 7 Differentiable Functions.- 8 Taylor Expansions.- 9 Geometry of functions: extrema and convexity.- 10 Indefinite and Definite Integrals.- 11 Improper Integrals and Integral Functions.- 12 Numerical Series.- 13 Separation of Variables.- 14 First Order Linear Differential Equations.- 15 Constant Coefficient Differential Equations.

Classical Potential Theory and Its Probabilistic Counterpart : 978-3-540-41206-9
Doob
Springer Berlin Heidelberg
Seiten L, 1551 p.
Erscheinungsjahr 2001
ISBN 978-3-540-41206-9
       9783540412069
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Potential Theory

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From the reviews: 'This huge book written in several years by one of the few mathematicians able to do it, appears as a precise and impressive study (not very easy to read) of this bothsided question that replaces, in a coherent way, without being encyclopaedic, a large library of books and papers scattered without a uniform language. Instead of summarizing the author gives his own way of exposition with original complements. This requires no preliminary knowledge. ...The purpose which the author explains in his introduction, i.e. a deep probabilistic interpretation of potential theory and a link between two great theories, appears fulfilled in a masterly manner'. M. Brelot in Metrika (1986)

Inhaltsangaben

I Introduction to the Mathematical Background of Classical Potential Theory.- II Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions.- III Infima of Families of Superharmonic Functions.- IV Potentials on Special Open Sets.- V Polar Sets and Their Applications.- VI The Fundamental Convergence Theorem and the Reduction Operation.- VII Green Functions.- VIII The Dirichlet Problem for Relative Harmonic Functions.- IX Lattices and Related Classes of Functions.- X The Sweeping Operation.- XI The Fine Topology.- XII The Martin Boundary.- XIII Classical Energy and Capacity.- XIV One-Dimensional Potential Theory.- XV Parabolic Potential Theory: Basic Facts.- XVI Subparabolic, Superparabolic, and Parabolic Functions on a Slab.- XVII Parabolic Potential Theory (Continued).- XVIII The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets.- XIX The Martin Boundary in the Parabolic Context.- I Fundamental Concepts of Probability.- II Optional Times and Associated Concepts.- III Elements of Martingale Theory.- IV Basic Properties of Continuous Parameter Supermartingales.- V Lattices and Related Classes of Stochastic Processes.- VI Markov Processes.- VII Brownian Motion.- VIII The Itô Integral.- IX Brownian Motion and Martingale Theory.- X Conditional Brownian Motion.- I Lattices in Classical Potential Theory and Martingale Theory.- II Brownian Motion and the PWB Method.- III Brownian Motion on the Martin Space.- Appendixes.- Appendix I.- Analytic Sets.- 1. Pavings and Algebras of Sets.- 2. Suslin Schemes.- 3. Sets Analytic over a Product Paving.- 4. Analytic Extensions versus ? Algebra Extensions of Pavings.- 7. Projections of Sets in Product Pavings.- 8. Extension of a Measurability Concept to the Analytic Operation Context.- 10. Polish Spaces.- 11. The Baire Null Space.- 12. Analytic Sets.- 13. Analytic Subsets of Polish Spaces.- Appendix II.- Capacity Theory.- 1. Choquet Capacities.- 2. Sierpinski Lemma.- 3. Choquet Capacity Theorem.- 4. Lusin’s Theorem.- 5. A Fundamental Example of a Choquet Capacity.- 6. Strongly Subadditive Set Functions.- 7. Generation of a Choquet Capacity by a Positive Strongly Subadditive Set Function.- 8. Topological Precapacities.- 9. Universally Measurable Sets.- Appendix III.- Lattice Theory.- 1. Introduction.- 2. Lattice Definitions.- 3. Cones.- 4. The Specific Order Generated by a Cone.- 5. Vector Lattices.- 6. Decomposition Property of a Vector Lattice.- 7. Orthogonality in a Vector Lattice.- 8. Bands in a Vector Lattice.- 9. Projections on Bands.- 10. The Orthogonal Complement of a Set.- 11. The Band Generated by a Single Element.- 12. Order Convergence.- 13. Order Convergence on a Linearly Ordered Set.- Appendix IV.- Lattice Theoretic Concepts in Measure Theory.- 1. Lattices of Set Algebras.- 2. Measurable Spaces and Measurable Functions.- 3. Composition of Functions.- 4. The Measure Lattice of a Measurable Space.- 5. The ? Finite Measure Lattice of a Measurable Space (Notation of Section 4).- 6. The Hahn and Jordan Decompositions.- 8. Absolute Continuity and Singularity.- 9. Lattices of Measurable Functions on a Measure Space.- 10.Order Convergence of Families of Measurable Functions.- 11. Measures on Polish Spaces.- 12. Derivates of Measures.- Appendix V.- Uniform Integrability.- Appendix VI.- Kernels and Transition Functions.- 1. Kernels.- 2. Universally Measurable Extension of a Kernel.- 3. Transition Functions.- Appendix VII.- Integral Limit Theorems.- 1. An Elementary Limit Theorem.- 2. Ratio Integral Limit Theorems.- 3. A One-Dimensional Ratio Integral Limit Theorem.- 4. A Ratio Integral Limit Theorem Involving Convex Variational Derivates.- Appendix VIII.- Lower Semicontinuous Functions.- 1. The Lower Semicontinuous Smoothing of a Function.- 2. Suprema of Families of Lower Semicontinuous Functions.- 3. Choquet Topological Lemma.- Historical Notes.- 1.- 2.- 3.- Appendixes.- No

Combinatorics and Graph Theory : 978-0-387-79710-6
Harris
Springer New York
Seiten XV, 381 p.
Erscheinungsjahr 2008
ISBN 978-0-387-79710-6
       9780387797106
 (Hardcover)
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There are certain rules that one must abide by in order to create a successful sequel. — Randy Meeks, from the trailer to Scream 2 While we may not follow the precise rules that Mr. Meeks had in mind for s- cessful sequels, we have made a number of changes to the text in this second edition. In the new edition, we continue to introduce new topics with concrete - amples, we provide complete proofs of almost every result, and we preserve the book’sfriendlystyle andlivelypresentation,interspersingthetextwith occasional jokes and quotations. The rst two chapters, on graph theory and combinatorics, remain largely independent, and may be covered in either order. Chapter 3, on in nite combinatorics and graphs, may also be studied independently, although many readers will want to investigate trees, matchings, and Ramsey theory for nite sets before exploring these topics for in nite sets in the third chapter. Like the rst edition, this text is aimed at upper-division undergraduate students in mathematics, though others will nd much of interest as well. It assumes only familiarity with basic proof techniques, and some experience with matrices and in nite series. The second edition offersmany additionaltopics for use in the classroom or for independentstudy. Chapter 1 includesa new sectioncoveringdistance andrelated notions in graphs, following an expanded introductory section. This new section also introduces the adjacency matrix of a graph, and describes its connection to important features of the graph.

Inhaltsangaben

Graph Theory.- Combinatorics.- Infinite Combinatorics and Graphs.

Commodities, Energy and Environmental Finance : 978-1-4939-2732-6
Aïd
Springer New York
Seiten XII, 430 p. 121 illus., 90 illus. in color.
Erscheinungsjahr 2015
ISBN 978-1-4939-2732-6
       9781493927326
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This volume is a collection of chapters covering the latest developments in applications of financial mathematics and statistics to topics in energy, commodity financial markets and environmental economics. The research presented is based on the presentations and discussions that took place during the Fields Institute Focus Program on Commodities, Energy and Environmental Finance in August 2013. The authors include applied mathematicians, economists and industry practitioners, providing for a multi-disciplinary spectrum of perspectives on the subject. The volume consists of four sections: Electricity Markets; Real Options; Trading in Commodity Markets; and Oligopolistic Models for Energy Production. Taken together, the chapters give a comprehensive summary of the current state of the art in quantitative analysis of commodities and energy finance. The topics covered include structural models of electricity markets, financialization of commodities, valuation of commodity real options, game-theory analysis of exhaustible resource management and analysis of commodity ETFs. The volume also includes two survey articles that provide a source for new researchers interested in getting into these topics.

Inhaltsangaben

Financialization of the Commodity Markets (R. Carmona).- Understanding the Tracking Errors of Commodity Leveraged ETFs (K. Guo, T. Leung).- Systemic Risk in Commodity Markets: What Do Trees Tell Us About Crises? (D. Lautier, J. Ling, F. Raynaud).- Margrabe Revisited (H. Tuenter).- Cross-Commodity Modelling by Multivariate Ambit Fields (F. Benth, A. Veraart, O. Barndorff-Nielsen).- Hedging Expected Losses on Derivatives in Electricity Futures Market (A. Nguyen Huu, N. Oudjane).- Calibration of Electricity Price Models (O. Feron, E. Daboussi).- Measuring the Competitiveness Benefits of Transmission Expansions in Wholesale Electricity Markets (F. Wolak).- Incorporating Managerial Information into Real Option Valuation (S. Jaimungal, Y. Lawryshyn).- Real Options with Regulatory Policy Uncertainty (M. Davison, C. Maxwell).- A Hedged Monte Carlo Approach to Real Option Pricing (E. Brigatti, M. Souza, J. Zubelli).- Technological Transition to Electric Mobility (R. Aid, I. Ben-Tahar).- Game Theoretic Models for Energy Production (M. Ludkovski, R. Sircar).- Design Analysis of Carbon Auction Market, Through Electricity Market Coupling (M. Bossy, N. Maizi, O. Pourtalier).- Dynamic Cournot Duopolies Under Stochastic Demand (M. Ludkovski, X. Yang).- Oligopoly Markets with Renewable Resources, Exploration and Evolving Extraction Costs with an Application to Energy Policy (A. Dasarathy, R. Sircar).

Complex Analysis : 978-1-4419-7287-3
Bak
Springer New York
Seiten XII, 328 p. 77 illus.
Erscheinungsjahr 2010
ISBN 978-1-4419-7287-3
       9781441972873
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Analysis

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Beginning with the ?rst edition of Complex Analysis, we have attempted to present the classical and beautiful theory of complex variables in the clearest and most intuitive form possible. The changes inthisedition, which include additions to ten of the nineteen chapters, are intended to provide the additional insights that can be obtainedby seeing a little more of the “bigpicture”.This includesadditional related results and occasional generalizations that place the results inaslightly broader context. The Fundamental Theorem of Algebra is enhanced by three related results. Section 1.3 offers a detailed look at the solution of the cubic equation and its role in the acceptance of complex numbers. While there is no formula for determining the rootsof a generalpolynomial,we added a section on Newton’sMethod,a numerical technique for approximating the zeroes of any polynomial. And the Gauss-Lucas Theorem provides an insight into the location of the zeroes of a polynomial and those of its derivative. Aseries of new results relate to the mapping properties of analytic functions. Arevised proof of Theorem 6.15 leads naturally to a discussion of the connection between critical points and saddle points in the complex plane. The proof of the SchwarzRe?ectionPrinciplehasbeenexpandedtoincludere?ectionacrossanalytic arcs, which plays a key role in a new section (14.3) on the mapping properties of analytic functions on closed domains. And our treatment of special mappings has been enhanced by the inclusion of Schwarz-Christoffel transformations.

Inhaltsangaben

The Complex Numbers.- Functions of the Complex Variable z.- Analytic Functions.- Line Integrals and Entire Functions.- Properties of Entire Functions.- Properties of Analytic Functions.- Further Properties of Analytic Functions.- Simply Connected Domains.- Isolated Singularities of an Analytic Function.- The Residue Theorem.- Applications of the Residue Theorem to the Evaluation of Integrals and Sums.- Further Contour Integral Techniques.- to Conformal Mapping.- The Riemann Mapping Theorem.- Maximum-Modulus Theorems for Unbounded Domains.- Harmonic Functions.- Different Forms of Analytic Functions.- Analytic Continuation; The Gamma and Zeta Functions.- Applications to Other Areas of Mathematics.

Complex Analysis : 978-1-85233-733-9
Howie
Springer London
Seiten XI, 260 p.
Erscheinungsjahr 2003
ISBN 978-1-85233-733-9
       9781852337339
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Complex analysis is one of the most attractive of all the core topics in an undergraduate mathematics course. Its importance to applications means that it can be studied both from a very pure perspective and a very applied perspective. This book takes account of these varying needs and backgrounds and provides a self-study text for students in mathematics, science and engineering. Beginning with a summary of what the student needs to know at the outset, it covers all the topics likely to feature in a first course in the subject, including: complex numbers, differentiation, integration, Cauchy's theorem, and its consequences, Laurent series and the residue theorem, applications of contour integration, conformal mappings, and harmonic functions. A brief final chapter explains the Riemann hypothesis, the most celebrated of all the unsolved problems in mathematics, and ends with a short descriptive account of iteration, Julia sets and the Mandelbrot set. Clear and careful explanations are backed up with worked examples and more than 100 exercises, for which full solutions are provided.

Inhaltsangaben

1. What Do I Need to Know?.- 1.1 Set Theory.- 1.2 Numbers.- 1.3 Sequences and Series.- 1.4 Functions and Continuity.- 1.5 Differentiation.- 1.6 Integration.- 1.7 Infinite Integrals.- 1.8 Calculus of Two Variables.- 2. Complex Numbers.- 2.1 Are Complex Numbers Necessary?.- 2.2 Basic Properties of Complex Numbers.- 3. Prelude to Complex Analysis.- 3.1 Why is Complex Analysis Possible?.- 3.2 Some Useful Terminology.- 3.3 Functions and Continuity.- 3.4 The O and o Notations.- 4. Differentiation.- 4.1 Differentiability.- 4.2 Power Series.- 4.3 Logarithms.- 4.4 Cuts and Branch Points.- 4.5 Singularities.- 5. Complex Integration.- 5.1 The Heine-Borel Theorem.- 5.2 Parametric Representation.- 5.3 Integration.- 5.4 Estimation.- 5.5 Uniform Convergence.- 6. Cauchy’s Theorem.- 6.1 Cauchy’s Theorem: A First Approach.- 6.2 Cauchy’s Theorem: A More General Version.- 6.3 Deformation.- 7. Some Consequences of Cauchy’s Theorem.- 7.1 Cauchy’s Integral Formula.- 7.2 The Fundamental Theorem of Algebra.- 7.3 Logarithms.- 7.4 Taylor Series.- 8. Laurent Series and the Residue Theorem.- 8.1 Laurent Series.- 8.2 Classification of Singularities.- 8.3 The Residue Theorem.- 9. Applications of Contour Integration.- 9.1 Real Integrals: Semicircular Contours.- 9.2 Integrals Involving Circular Functions.- 9.3 Real Integrals: Jordan’s Lemma.- 9.4 Real Integrals: Some Special Contours.- 9.5 Infinite Series.- 10. Further Topics.- 10.1 Integration of f?/f; Rouché’s Theorem.- 10.2 The Open Mapping Theorem.- 10.3 Winding Numbers.- 11. Conformai Mappings.- 11.1 Preservation of Angles.- 11.2 Harmonic Functions.- 11.3 Möbius Transformations.- 11.4 Other Transformations.- 12. Final Remarks.- 12.1 Riemann’s Zeta function.- 12.2 Complex Iteration.- 13. Solutions to Exercises.- Subject IndexBibliography.- Subject IndexIndex.

Complex Analysis : 978-0-387-98592-3
Lang
Springer New York
Seiten XIV, 489 p. 85 illus.
Erscheinungsjahr 1999
ISBN 978-0-387-98592-3
       9780387985923
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Analysis

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The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. The first half, more or less, can be used for a one-semester course addressed to undergraduates. The second half can be used for a second semester, at either level. Somewhat more material has been included than can be covered at leisure in one or two terms, to give opportunities for the instructor to exercise individual taste, and to lead the course in whatever directions strikes the instructor's fancy at the time as well as extra read­ ing material for students on their own. A large number of routine exer­ cises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students. In some sense, I think the classical German prewar texts were the best (Hurwitz-Courant, Knopp, Bieberbach, etc. ) and I would recommend to anyone to look through them. More recent texts have emphasized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex analysis: the power series expansion, the uniqueness of analytic continuation, and the calculus of residues.

Inhaltsangaben

One Basic Theory.- I Complex Numbers and Functions.- II Power Series.- III Cauchy’s Theorem, First Part.- IV Winding Numbers and Cauchy’s Theorem.- V Applications of Cauchy’s integral Formula.- VI Calculus of Residues.- VII Conformal Mappings.- VIII Harmonic Functions.- Two Geometric Function Theory.- IX Schwarz Reflection.- X The Riemann Mapping Theorem.- XI Analytic Continuation Along Curves.- Three Various Analytic Topics.- XII Applications of the Maximum Modulus Principle and Jensen’s Formula.- XIII Entire and Meromorphic Functions.- XIV Elliptic Functions.- XV The Gamma and Zeta Functions.- XVI The Prime Number Theorem.- §1. Summation by Parts and Non-Absolute Convergence.- §2. Difference Equations.- §3. Analytic Differential Equations.- §4. Fixed Points of a Fractional Linear Transformation.- §6. Cauchy’s Theorem for Locally Integrable Vector Fields.- §7. More on Cauchy-Riemann.

Computational Linear and Commutative Algebra : 978-3-319-43599-2
Kreuzer
Springer International Publishing
Seiten XVIII, 321 p. 1 illus.
Erscheinungsjahr 2016
ISBN 978-3-319-43599-2
       9783319435992
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This book combines, in a novel and general way, an extensive development of the theory of families of commuting matrices with applications to zero-dimensional commutative rings, primary decompositions and polynomial system solving. It integrates the Linear Algebra of the Third Millennium, developed exclusively here, with classical algorithmic and algebraic techniques. Even the experienced reader will be pleasantly surprised to discover new and unexpected aspects in a variety of subjects including eigenvalues and eigenspaces of linear maps, joint eigenspaces of commuting families of endomorphisms, multiplication maps of zero-dimensional affine algebras, computation of primary decompositions and maximal ideals, and solution of polynomial systems. This book completes a trilogy initiated by the uncharacteristically witty books Computational Commutative Algebra 1 and 2 by the same authors. The material treated here is not available in book form, and much of it is not available at all. The authors continue to present it in their lively and humorous style, interspersing core content with funny quotations and tongue-in-cheek explanations.

Inhaltsangaben

Foreword.- Introduction.- 1 Endomorphisms.- 2 Families of Commuting Endomorphisms.- 3 Special Families of Endomorphisms.- 4 Zero-Dimensional Affine Algebras.- 5 Computing Primary and Maximal Components.- 6 Solving Zero-Dimensional Polynomial Systems.- Notation.- References.- Index.

Computational Methods for Fluid Dynamics : 978-3-540-42074-3
Ferziger
Springer Berlin Heidelberg
Seiten XIV, 426 p. 50 illus.
Erscheinungsjahr 2002
ISBN 978-3-540-42074-3
       9783540420743
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In its 3rd revised and extended edition the book offers an overview of the techniques used to solve problems in fluid mechanics on computers and describes in detail those most often used in practice. Included are advanced methods in computational fluid dynamics, like direct and large-eddy simulation of turbulence, multigrid methods, parallel computing, moving grids, structured, block-structured and unstructured boundary-fitted grids, free surface flows. The 3rd edition contains a new section dealing with grid quality and an extended description of discretization methods. The book shows common roots and basic principles for many different methods. The book also contains a great deal of practical advice for code developers and users; it is designed to be equally useful to beginners and experts.The issues of numerical accuracy, estimation and reduction of numerical errors are dealt with in detail, with many examples.

Inhaltsangaben

1. Basic Concepts of Fluid Flow.- 1.1 Introduction.- 1.2 Conservation Principles.- 1.3 Mass Conservation.- 1.4 Momentum Conservation.- 1.5 Conservation of Scalar Quantities.- 1.6 Dimensionless Form of Equations.- 1.7 Simplified Mathematical Models.- 1.8 Mathematical Classification of Flows.- 1.9 Plan of This Book.- 2. Introduction to Numerical Methods.- 2.1 Approaches to Fluid Dynamical Problems.- 2.2 What is CFD?.- 2.3 Possibilities and Limitations of Numerical Methods.- 2.4 Components of a Numerical Solution Method.- 2.5 Properties of Numerical Solution Methods.- 2.6 Discretization Approaches.- 3. Finite Difference Methods.- 3.1 Introduction.- 3.2 Basic Concept.- 3.3 Approximation of the First Derivative.- 3.4 Approximation of the Second Derivative.- 3.5 Approximation of Mixed Derivatives.- 3.6 Approximation of Other Terms.- 3.7 Implementation of Boundary Conditions.- 3.8 The Algebraic Equation System.- 3.9 Discretization Errors.- 3.10 An Introduction to Spectral Methods.- 3.11 Example.- 4. Finite Volume Methods.- 4.1 Introduction.- 4.2 Approximation of Surface Integrals.- 4.3 Approximation of Volume Integrals.- 4.4 Interpolation and Differentiation Practices.- 4.5 Implementation of Boundary Conditions.- 4.6 The Algebraic Equation System.- 4.7 Examples.- 5. Solution of Linear Equation Systems.- 5.1 Introduction.- 5.2 Direct Methods.- 5.3 Iterative Methods.- 5.4 Coupled Equations and Their Solution.- 5.5 Non-Linear Equations and their Solution.- 5.6 Deferred-Correction Approaches.- 5.7 Convergence Criteria and Iteration Errors.- 5.8 Examples.- 6. Methods for Unsteady Problems.- 6.1 Introduction.- 6.2 Methods for Initial Value Problems in ODEs.- 6.3 Application to the Generic Transport Equation.- 6.4 Examples.- 7. Solution of the Navier-Stokes Equations.- 7.1 Special Features of the Navier-Stokes Equations.- 7.2 Choice of Variable Arrangement on the Grid.- 7.3 Calculation of the Pressure.- 7.4 Other Methods.- 7.5 Solution Methods for the Navier-Stokes Equations.- 7.6 Note on Pressure and Incompressibility.- 7.7 Boundary Conditions for the Navier-Stokes Equations.- 7.8 Examples.- 8. Complex Geometries.- 8.1 The Choice of Grid.- 8.2 Grid Generation.- 8.3 The Choice of Velocity Components.- 8.4 The Choice of Variable Arrangement.- 8.5 Finite Difference Methods.- 8.6 Finite Volume Methods.- 8.7 Control-Volume-Based Finite Element Methods.- 8.8 Pressure-Correction Equation.- 8.9 Axi-Symmetric Problems.- 8.10 Implementation of Boundary Conditions.- 8.11 Examples.- 9. Turbulent Flows.- 9.1 Introduction.- 9.2 Direct Numerical Simulation (DNS).- 9.3 Large Eddy Simulation (LES).- 9.4 RANS Models.- 9.5 Reynolds Stress Models.- 9.6 Very Large Eddy Simulation.- 10. Compressible Flow.- 10.1 Introduction.- 10.2 Pressure-Correction Methods for Arbitrary Mach Number.- 10.3 Methods Designed for Compressible Flow.- 11. Efficiency and Accuracy Improvement.- 11.1 Error Analysis and Estimation.- 12. Special Topics.- 12.1 Introduction.- 12.2 Heat and Mass Transfer.- 12.3 Flows With Variable Fluid Properties.- 12.4 Moving Grids.- 12.5 Free-Surface Flows.- 12.6 Meteorological and Oceanographic Applications.- 12.7 Multiphase flows.- 12.8 Combustion.- A. Appendices.- A.l List of Computer Codes and How to Access Them.- A.2 List of Frequently Used Abbreviations.- References.

Conformal Symmetry Breaking Operators for Differential Forms on Spheres : 978-981-10-2656-0
Kobayashi
Springer Singapore
Seiten IX, 192 p.
Erscheinungsjahr 2016
ISBN 978-981-10-2656-0
       9789811026560
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This work is the first systematic study of all possible conformally covariant differential operators transforming differential forms on a Riemannian manifold X into those on a submanifold Y with focus on the model space (X, Y) = (Sn, Sn-1).The authors give a complete classification of all such conformally covariant differential operators, and find their explicit formulæ in the flat coordinates in terms of basic operators in differential geometry and classical hypergeometric polynomials. Resulting families of operators are natural generalizations of the Rankin–Cohen brackets for modular forms and Juhl's operators from conformal holography. The matrix-valued factorization identities among all possible combinations of conformally covariant differential operators are also established.The main machinery of the proof relies on the 'F-method' recently introduced and developed by the authors. It is a general method to construct intertwining operators between C?-induced representations or to find singular vectors of Verma modules in the context of branching rules, as solutions to differential equations on the Fourier transform side. The book gives a new extension of the F-method to the matrix-valued case in the general setting, which could be applied to other problems as well.This book offers a self-contained introduction to the analysis of symmetry breaking operators for infinite-dimensional representations of reductive Lie groups. This feature will be helpful for active scientists and accessible to graduate students and young researchers in differential geometry, representation theory, and theoretical physics.

Inhaltsangaben


Continuous-time Stochastic Control and Optimization with Financial Applications : 978-3-540-89499-5
Pham
Springer Berlin Heidelberg
Seiten XVII, 232 p.
Erscheinungsjahr 2009
ISBN 978-3-540-89499-5
       9783540894995
 (Hardcover)
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Stochastic optimization problems arise in decision-making problems under uncertainty, and find various applications in economics and finance. On the other hand, problems in finance have recently led to new developments in the theory of stochastic control. This volume provides a systematic treatment of stochastic optimization problems applied to finance by presenting the different existing methods: dynamic programming, viscosity solutions, backward stochastic differential equations, and martingale duality methods. The theory is discussed in the context of recent developments in this field, with complete and detailed proofs, and is illustrated by means of concrete examples from the world of finance: portfolio allocation, option hedging, real options, optimal investment, etc. This book is directed towards graduate students and researchers in mathematical finance, and will also benefit applied mathematicians interested in financial applications and practitioners wishing to know more about the use of stochastic optimization methods in finance.

Inhaltsangaben

Some elements of stochastic analysis.- Stochastic optimization problems. Examples in finance.- The classical PDE approach to dynamic programming.- The viscosity solutions approach to stochastic control problems.- Optimal switching and free boundary problems.- Backward stochastic differential equations and optimal control.- Martingale and convex duality methods.

Convex Analysis and Global Optimization : 978-3-319-31482-2
Tuy
Springer International Publishing
Seiten XVI, 505 p. 36 illus.
Erscheinungsjahr 2016
ISBN 978-3-319-31482-2
       9783319314822
 (Hardcover)
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This book presents state-of-the-art results and methodologies in modern global optimization, and has been a staple reference for researchers, engineers, advanced students (also in applied mathematics), and practitioners in various fields of engineering. The second edition has been brought up to date and continues to develop a coherent and rigorous theory of deterministic global optimization, highlighting the essential role of convex analysis. The text has been revised and expanded to meet the needs of research, education, and applications for many years to come.Updates for this new edition include:·        Discussion of modern approaches to minimax, fixed point, and equilibrium theorems, and to nonconvex optimization; ·        Increased focus on dealing more efficiently with ill-posed problems of global optimization, particularly those with hard constraints; ·        Important discussions of decomposition methods for specially structured problems; ·        A complete revision of the chapter on nonconvex quadratic programming, in order to encompass the advances made in quadratic optimization since publication of the first edition. ·        Additionally, this new edition contains entirely new chapters devoted to monotonic optimization, polynomial optimization and optimization under equilibrium constraints, including bilevel programming, multiobjective programming, and optimization with variational inequality constraint.From the reviews of the first edition: The book gives a good review of the topic. …The text is carefully constructed and well written, the exposition is clear. It leaves a remarkable impression of the concepts, tools and techniques in global optimization. It might also be used as a basis and guideline for lectures on this subject. Students as well as professionals will profitably read and use it.—Mathematical Methods of Operations Research, 49:3 (1999)

Inhaltsangaben

1. Convex Sets.- 2. Convex Functions.- 3. Fixed Point and Equilibrium.- 4. DC Functions and DC Sets.- 5. Motivation and Overview.- 6. General Methods.- 7. DC Optimization Problems.- 8. Special Methods.- 9. Parametric Decomposition.- 10. Nonconvex Quadratic Programming.- 11. Monotonic Optimization.- 12. Polynomial Optimization.- 13. Optimization Under Equilibrium Constraints.- References.- Index. 

Counting Surfaces : 978-3-7643-8796-9
Eynard
CRM Aisenstadt Chair lectures
Springer Basel
Seiten XVII, 414 p. 109 illus., 47 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-7643-8796-9
       9783764387969
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Algebraic Geometry

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The problem of enumerating maps (a map is a set of polygonal 'countries' on a world of a certain topology, not necessarily the plane or the sphere) is an important problem in mathematics and physics, and it has many applications ranging from statistical physics, geometry, particle physics, telecommunications, biology, ... etc. This problem has been studied by many communities of researchers, mostly combinatorists, probabilists, and physicists. Since 1978, physicists have invented a method called 'matrix models' to address that problem, and many results have been obtained.
Besides, another important problem in mathematics and physics (in particular string theory), is to count Riemann surfaces. Riemann surfaces of a given topology are parametrized by a finite number of real parameters (called moduli), and the moduli space is a finite dimensional compact manifold or orbifold of complicated topology. The number of Riemann surfaces is the volume of that moduli space. More generally, an important problem in algebraic geometry is to characterize the moduli spaces, by computing not only their volumes, but also other characteristic numbers called intersection numbers.
Witten's conjecture (which was first proved by Kontsevich), was the assertion that Riemann surfaces can be obtained as limits of polygonal surfaces (maps), made of a very large number of very small polygons. In other words, the number of maps in a certain limit, should give the intersection numbers of moduli spaces.
In this book, we show how that limit takes place. The goal of this book is to explain the 'matrix model' method, to show the main results obtained with it, and to compare it with methods used in combinatorics (bijective proofs, Tutte's equations), or algebraic geometry (Mirzakhani's recursions).
The book intends to be self-contained and accessible to graduate students, and provides comprehensive proofs, several examples, and gives the general formula for the enumeration of maps on surfaces of any topology. In the end, the link with more general topics such as algebraic geometry, string theory, is discussed, and in particular a proof of the Witten-Kontsevich conjecture is provided.

Inhaltsangaben

I Maps and discrete surfaces.- II Formal matrix integrals.- III Solution of Tutte-loop equations.- IV Multicut case.- V Counting large maps.- VI Counting Riemann surfaces.- VII Topological recursion and symplectic invariants.- VIII Ising model.- Index.- Bibliography.


Data-driven Modelling of Structured Populations : 978-3-319-28891-8
Ellner
A Practical Guide to the Integral Projection Model
Springer International Publishing
Seiten XIII, 329 p. 67 illus., 29 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-28891-8
       9783319288918
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This book is a “How To” guide for modeling population dynamics using Integral Projection Models (IPM) starting from observational data. It is written by a leading research team in this area and includes code in the R language (in the text and online) to carry out all computations. The intended audience are ecologists, evolutionary biologists, and mathematical biologists interested in developing data-driven models for animal and plant populations. IPMs may seem hard as they involve integrals. The aim of this book is to demystify IPMs, so they become the model of choice for populations structured by size or other continuously varying traits. The book uses real examples of increasing complexity to show how the life-cycle of the study organism naturally leads to the appropriate statistical analysis, which leads directly to the IPM itself. A wide range of model types and analyses are presented, including model construction, computational methods, and the underlying theory, with the more technical material in Boxes and Appendices. Self-contained R code which replicates all of the figures and calculations within the text is available to readers on GitHub.Stephen P. Ellner is Horace White Professor of Ecology and Evolutionary Biology at Cornell University, USA; Dylan Z. Childs is Lecturer and NERC Postdoctoral Fellow in the Department of Animal and Plant Sciences at The University of Sheffield, UK; Mark Rees is Professor in the Department of Animal and Plant Sciences at The University of Sheffield, UK.

Inhaltsangaben

Introduction.- Simple Deterministic IPM.- Basic Analysis 1: Demographic Measures and Events in the Life Cycle.- Basic Analysis 2: Prospective Perturbation Analysis.- Density Dependence.- General Deterministic IPM.- Environmental Stochasticity.- Spatial Models.- Evolutionary Demography.- Future Directions and Advanced Topics.

Design of Observational Studies : 978-1-4419-1212-1
Rosenbaum
Springer New York
Seiten XVIII, 384 p.
Erscheinungsjahr 2010
ISBN 978-1-4419-1212-1
       9781441912121
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An observational study is an empiric investigation of effects caused by treatments when randomized experimentation is unethical or infeasible. Observational studies are common in most fields that study the effects of treatments on people, including medicine, economics, epidemiology, education, psychology, political science and sociology. The quality and strength of evidence provided by an observational study is determined largely by its design. Design of Observational Studies is both an introduction to statistical inference in observational studies and a detailed discussion of the principles that guide the design of observational studies. Design of Observational Studies is divided into four parts. Chapters 2, 3, and 5 of Part I cover concisely, in about one hundred pages, many of the ideas discussed in Rosenbaum’s Observational Studies (also published by Springer) but in a less technical fashion. Part II discusses the practical aspects of using propensity scores and other tools to create a matched comparison that balances many covariates. Part II includes a chapter on matching in R. In Part III, the concept of design sensitivity is used to appraise the relative ability of competing designs to distinguish treatment effects from biases due to unmeasured covariates. Part IV discusses planning the analysis of an observational study, with particular reference to Sir Ronald Fisher’s striking advice for observational studies, 'make your theories elaborate.' The second edition of his book, Observational Studies, was published by Springer in 2002.

Inhaltsangaben

Beginnings.- Dilemmas and Craftsmanship.- Causal Inference in Randomized Experiments.- Two Simple Models for Observational Studies.- Competing Theories Structure Design.- Opportunities, Devices, and Instruments.- Transparency.- Matching.- A Matched Observational Study.- Basic Tools of Multivariate Matching.- Various Practical Issues in Matching.- Fine Balance.- Matching Without Groups.- Risk-Set Matching.- Matching in R.- Design Sensitivity.- The Power of a Sensitivity Analysis and Its Limit.- Heterogeneity and Causality.- Uncommon but Dramatic Responses to Treatment.- Anticipated Patterns of Response.- Planning Analysis.- After Matching, Before Analysis.- Planning the Analysis.

Dessins d'Enfants on Riemann Surfaces : 978-3-319-24709-0
Jones
Springer International Publishing
Seiten XIV, 259 p. 41 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-24709-0
       9783319247090
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Algebraic Geometry

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This volume provides an introduction to dessins d'enfants and embeddings of bipartite graphs in compact Riemann surfaces. The first part of the book presents basic material, guiding the reader through the current field of research. A key point of the second part is the interplay between the automorphism groups of dessins and their Riemann surfaces, and the action of the absolute Galois group on dessins and their algebraic curves. It concludes by showing the links between the theory of dessins and other areas of arithmetic and geometry, such as the abc conjecture, complex multiplication and Beauville surfaces.
Dessins d'Enfants on Riemann Surfaces will appeal to graduate students and all mathematicians interested in maps, hypermaps, Riemann surfaces, geometric group actions, and arithmetic.

Inhaltsangaben

Historical and introductory background.- Graph embeddings.- Dessins and triangle groups.- Galois actions.- Quasiplatonic surfaces, and automorphisms.- Regular maps.- Regular embeddings of complete graphs.- Wilson operations.- Further examples.- Arithmetic aspects.- Beauville surfaces.

Differential Equations and Dynamical Systems : 978-0-387-95116-4
Perko
Springer New York
Seiten XIV, 557 p. 11 illus.
Erscheinungsjahr 2001
ISBN 978-0-387-95116-4
       9780387951164
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Analysis

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Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas­ sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Math­ ematical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs.

Inhaltsangaben

Series Preface.- Preface to the Third Edition.- Linear Systems.- Nonlinear Systems: Local Theory.- Nonlinear Systems: Global Theory.- Nonlinear Systems: Bifurcation Theory.- References.- Additional References.- Index.

Differential Geometry of Curves and Surfaces : 978-3-319-39798-6
Tapp
Springer International Publishing
Seiten VIII, 366 p. 186 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-39798-6
       9783319397986
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This is a textbook on differential geometry well-suited to a variety of courses on this topic. For readers seeking an elementary text, the prerequisites are minimal and include plenty of examples and intermediate steps within proofs, while providing an invitation to more excursive applications and advanced topics. For readers bound for graduate school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems. For the benefit of all readers, the author employs various techniques to render the difficult abstract ideas herein more understandable and engaging.
Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical content. Color is even used within the text to highlight logical relationships.
Applications abound! The study of conformal and equiareal functions is grounded in its application to cartography. Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating story: in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later be applied to optics and gears. Clairaut’s Theorem is presented as a conservation law for angular momentum. Green’s Theorem makes possible a drafting tool called a planimeter. Foucault’s Pendulum helps one visualize a parallel vector field along a latitude of the earth. Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface.
In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book. The GPS in any car wouldn’t work without general relativity, formalized through the language of differential geometry. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it. 

Inhaltsangaben

Introduction.- Curves.- Additional topics in curves.- Surfaces.- The curvature of a surface.- Geodesics.- The Gauss–Bonnet theorem.- Appendix A: The topology of subsets of Rn.- Recommended excursions.- Index.

Divergent Series, Summability and Resurgence I : 978-3-319-28735-5
Mitschi
Monodromy and Resurgence
Springer International Publishing
Seiten XXI, 298 p. 24 illus., 19 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-28735-5
       9783319287355
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Providing an elementary introduction to analytic continuation and monodromy, the first part of this volume applies these notions to the local and global study of complex linear differential equations, their formal solutions at singular points, their monodromy and their differential Galois groups. The Riemann-Hilbert problem is discussed from Bolibrukh’s point of view. The second part expounds 1-summability and Ecalle’s theory of resurgence under fairly general conditions. It contains numerous examples and presents an analysis of the singularities in the Borel plane via “alien calculus”, which provides a full description of the Stokes phenomenon for linear or non-linear differential or difference equations. The first of a series of three, entitled Divergent Series, Summability and Resurgence, this volume is aimed at graduate students, mathematicians and theoretical physicists interested in geometric, algebraic or local analytic properties of dynamical systems. It includes useful exercises with solutions. The prerequisites are a working knowledge of elementary complex analysis and differential algebra.

Inhaltsangaben

Preface.-Preface to the three volumes.- Part I:Monodromy in Linear Differential Equations.- 1 analytic continuation and monodromy.- Differential Galois Theory.- Inverse Problems.- The Riemann-Hilbert problem.- Part II: Introduction to 1-Summability and Resurgence.- 5 Borel-Laplace Summation.- Resurgent Functions and Alien Calculus.- the Resurgent Viewpoint on Holomorphic Tangent-to-Identity Germs.- Acknowledgements.- Index.

Divergent Series, Summability and Resurgence II : 978-3-319-29074-4
Loday-Richaud
Simple and Multiple Summability
Springer International Publishing
Seiten XXIII, 272 p. 64 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-29074-4
       9783319290744
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Addressing the question how to “sum” a power series in one variable when it diverges, that is, how to attach to it analytic functions, the volume gives answers by presenting and comparing the various theories of k-summability and multisummability. These theories apply in particular to all solutions of ordinary differential equations.The volume includes applications, examples and revisits, from a cohomological point of view, the group of tangent-to-identity germs of diffeomorphisms of C studied in volume 1. With a view to applying the theories to solutions of differential equations, a detailed survey of linear ordinary differential equations is provided, which includes Gevrey asymptotic expansions, Newton polygons, index theorems and Sibuya’s proof of the meromorphic classification theorem that characterizes the Stokes phenomenon for linear differential equations.This volume is the second in a series of three, entitled Divergent Series, Summability and Resurgence. It is aimed at graduate students and researchers in mathematics and theoretical physics who are interested in divergent series, Although closely related to the other two volumes, it can be read independently.

Inhaltsangaben

Avant-propos.- Preface to the three volumes.- Introduction to this volume.- 1 Asymptotic Expansions in the Complex Domain.- 2 Sheaves and ?ech cohomology.- 3 Linear Ordinary Differential Equations.- 4 Irregularity and Gevrey Index Theorems.- 5 Four Equivalent Approaches to k-Summability.- 6 Tangent-to-Identity Diffeomorphisms.- 7 Six Equivalent Approaches to Multisummability.- Exercises.- Solutions to Exercises.- Index.- Glossary of Notations.- References.

Divergent Series, Summability and Resurgence III : 978-3-319-28999-1
Delabaere
Resurgent Methods and the First Painlevé Equation
Springer International Publishing
Seiten XXII, 230 p. 35 illus., 14 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-28999-1
       9783319289991
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The aim of this volume is two-fold. First, to show how the resurgent methods introduced in volume 1 can be applied efficiently in a non-linear setting; to this end further properties of the resurgence theory must be developed. Second, to analyze the fundamental example of the First Painlevé equation. The resurgent analysis of singularities is pushed all the way up to the so-called “bridge equation”, which concentrates all information about the non-linear Stokes phenomenon at infinity of the First Painlevé equation. The third in a series of three, entitled Divergent Series, Summability and Resurgence, this volume is aimed at graduate students, mathematicians and theoretical physicists who are interested in divergent power series and related problems, such as the Stokes phenomenon. The prerequisites are a working knowledge of complex analysis at the first-year graduate level and of the theory of resurgence, as presented in volume 1.

Inhaltsangaben

Avant-Propos.- Preface to the three volumes.- Preface to this volume.- Some elements about ordinary differential equations.- The first Painlevé equation.- Tritruncated solutions for the first Painlevé equation.- A step beyond Borel-Laplace summability.- Transseries and formal integral for the first Painlevé equation.- Truncated solutions for the first Painlevé equation.- Supplements to resurgence theory.- Resurgent structure for the first Painlevé equation.- Index.

Elementary Differential Geometry : 978-1-84882-890-2
Pressley
Springer London
Seiten XII, 474 p. 150 illus.
Erscheinungsjahr 2010
ISBN 978-1-84882-890-2
       9781848828902
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Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout. New features of this revised and expanded second edition include: a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book. Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature.Around 200 additional exercises, and a full solutions manual for instructors, available via www.springer.comul

Inhaltsangaben

Curves in the plane and in space.- How much does a curve curve?.- Global properties of curves.- Surfaces in three dimensions.- Examples of surfaces.- The first fundamental form.- Curvature of surfaces.- Gaussian, mean and principal curvatures.- Geodesics.- Gauss’ Theorema Egregium.- Hyperbolic geometry.- Minimal surfaces.- The Gauss–Bonnet theorem.

Elementary Number Theory: Primes, Congruences, and Secrets : 978-0-387-85524-0
Stein
A Computational Approach
Springer New York
Seiten X, 168 p. 45 illus.
Erscheinungsjahr 2009
ISBN 978-0-387-85524-0
       9780387855240
 (Hardcover)
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Number Theory

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This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. It grew out of undergr- uate courses that the author taught at Harvard, UC San Diego, and the University of Washington. The systematic study of number theory was initiated around 300B. C. when Euclid proved that there are in?nitely many prime numbers, and also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over a thousand years later (around 972A. D. ) Arab mathematicians formulated the congruent number problem that asks for a way to decide whether or not a given positive integer n is the area of a right triangle, all three of whose sides are rational numbers. Then another thousand years later (in 1976), Di?e and Hellman introduced the ?rst ever public-key cryptosystem, which enabled two people to communicate secretely over a public communications channel with no predetermined secret; this invention and the ones that followed it revolutionized the world of digital communication. In the 1980s and 1990s, elliptic curves revolutionized number theory, providing striking new insights into the congruent number problem, primality testing, publ- key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles’ resolution of Fermat’s Last Theorem.

Inhaltsangaben

Prime Numbers.- The Ring of Integers Modulo n.- Public-key Cryptography.- Quadratic Reciprocity.- Continued Fractions.- Elliptic Curves. 

Eléments d'histoire des mathématiques : 978-3-540-33938-0
Bourbaki
Springer Berlin Heidelberg
Seiten VII, 369 p.
Erscheinungsjahr 2007
ISBN 978-3-540-33938-0
       9783540339380
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Ce volume rassemble les notes historiques parues dans les différents livres des éléments de mathématique de l'auteur. Elles concernent donc l'ensemble des matières abordées dans ce traité : théorie des ensembles, algèbre, topologie, fonctions d'une variable réelle, espaces vectoriels topologiques, intégration, algèbre commutative, groupes et algèbres de Lie. Constitué d'études initialement séparées, cet ouvrage ne prétend pas esquisser une histoire suivie et complète du développement de la mathématique. L'entrelacement des différents thèmes et l'unité du point de vue en assurent cependant la cohérence profonde.

Inhaltsangaben

FONDEMENTS DES MATHÉMATIQUES LOGIQUE THÉORIE DES ENSEMBLES.- NUMÉRATION; ANALYSE COMBINATOIRE.- L'ÉVOLUTION DE L'ALGÈBRE.- ALGÈBRE LINÉAIRE ET ALGÈBRE MULTILINÉAIRE.- POLYNOMES ET CORPS COMMUTATIFS.- DIVISIBILITÉ; CORPS ORDONNÉS.- ALGÈBRE COMMUTATIVE THÉORIE DES NOMBRES ALGÉBRIQUES.- ALGÈBRE NON COMMUTATIVE.- FORMES QUADRATIQUES; GÉOMÉTRIE ÉLÉMENTAIRE.- ESPACES TOPOLOGIQUES.- ESPACES UNIFORMES.- NOMBRES RÉELS.- EXPONENTIELLES ET LOGARITHMES.- ESPACES A N DIMENSIONS.- NOMBRES COMPLEXES. MESURE DES ANGLES.- ESPACES MÉTRIQUES.- CALCUL INFINITÉSIMAL.- DÉVELOPPEMENTS ASYMPTOTIQUES.- LA FONCTION GAMMA.- ESPACES FONCTIONNELS.- ESPACES VECTORIELS TOPOLOGIQUES.- INTÉGRATION DANS LES ESPACES LOCALEMENT COMPACTS.- MESURE DE HAAR CONVOLUTION.- INTÉGRATION DANS LES ESPACES NON LOCALEMENT COMPACTS.- GROUPES DE LIE ET ALGÈBRES DE LIE.- GROUPES ENGENDRÉS PAR DES RÉFLEXIONS; SYSTÈMES DE RACINES.- Erratum.

Elliptic Regularity Theory : 978-3-319-27484-3
Beck
A First Course
Springer International Publishing
Seiten XII, 201 p.
Erscheinungsjahr 2016
ISBN 978-3-319-27484-3
       9783319274843
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These lecture notes provide a self-contained introduction to regularity theory for elliptic equations and systems in divergence form. After a short review of some classical results on everywhere regularity for scalar-valued weak solutions, the presentation focuses on vector-valued weak solutions to a system of several coupled equations. In the vectorial case, weak solutions may have discontinuities and so are expected, in general, to be regular only outside of a set of measure zero. Several methods are presented concerning the proof of such partial regularity results, and optimal regularity is discussed. Finally, a short overview is given on the current state of the art concerning the size of the singular set on which discontinuities may occur. The notes are intended for graduate and postgraduate students with a solid background in functional analysis and some familiarity with partial differential equations; they will also be of interest to researchers working on related topics.

Inhaltsangaben

Preliminaries.- Introduction to the Setting.- The Scalar Case.- Foundations for the Vectorial Case.- Partial Regularity Results for Quasilinear Systems.

Entropy, Large Deviations, and Statistical Mechanics : 978-3-540-29059-9
Ellis
Springer Berlin Heidelberg
Seiten XVIII, 367 p.
Erscheinungsjahr 2006
ISBN 978-3-540-29059-9
       9783540290599
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From the reviews: '... Besides the fact that the author's treatment of large deviations is a nice contribution to the literature on the subject, his book has the virue that it provides a beautifully unified and mathematically appealing account of certain aspects of statistical mechanics. ... Furthermore, he does not make the mistake of assuming that his mathematical audience will be familiar with the physics and has done an admireable job of explaining the necessary physical background. Finally, it is clear that the author's book is the product of many painstaking hours of work; and the reviewer is confident that its readers will benefit from his efforts.' D. Stroock in Mathematical Reviews 1985 '... Each chapter of the book is followed by a notes section and by a problems section. There are over 100 problems, many of which have hints. The book may be recommended as a text, it provides a completly self-contained reading ...' S. Pogosian in

Inhaltsangaben

Large Deviations and Statistical Mechanics.- to Large Deviations.- Large Deviation Property and Asymptotics of Integrals.- Large Deviations and the Discrete Ideal Gas.- Ferromagnetic Models on ?.- Magnetic Models on ?D and on the Circle.- Convexity and Proofs of Large Deviation Theorems.- Convex Functions and the Legendre-Fenchel Transform.- Large Deviations for Random Vectors.- Level-2 Large Deviations for I.I.D. Random Vectors.- Level-3 Large Deviations for I.I.D. Random Vectors.

Équations aux dérivées partielles elliptiques non linéaires : 978-3-642-36174-6
Le Dret
Springer Berlin Heidelberg
Seiten VIII, 225 p. 26 ill.
Erscheinungsjahr 2013
ISBN 978-3-642-36174-6
       9783642361746
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Cet ouvrage est issu d’un cours de Master 2 enseigné à l’UPMC entre 2004 et 2007. Nous y présentons une sélection de techniques mathématiques orientées vers la résolution des équations aux dérivées partielles elliptiques semi-linéaires et quasi-linéaires. Après un vade-mecum d'analyse réelle et d'analyse fonctionnelle de base pour les EDP, sans démonstrations pour les points les plus connus, nous parcourons ainsi les théorèmes de point fixe classiques, les opérateurs de superposition dans les espaces de Lebesgue et de Sobolev, la méthode de Galerkin, les principes du maximum et la régularité elliptique, nous faisons une excursion assez longue dans divers aspects du calcul des variations puis terminons par les opérateurs monotones et pseudo-monotones. Tout ceci est agrémenté d’exemples et chaque chapitre est complété d'un nombre d’exercices qui croît essentiellement avec le numéro du chapitre, au fur et à mesure que de nouveaux matériaux sont présentés.This book stems from lectures notes of a Master 2 class held at UPMC between 2004 and 2007. A selection of mathematical techniques geared towards the resolution of semilinear and quasilinear elliptic partial differential equations is presented. After a short survival guide in basic real and functional analysis for PDEs, without proofs for the most well-known results, we walk through the classical fixed point theorems, the superposition operators in Lebesgue and Sobolev spaces, the Galerkin method, the maximum principles and elliptic regularity, we make a rather long foray into various aspects of the calculus of variations, and conclude with monotone and pseudo-monotone operators, by way of numerous examples. Each chapter is complemented by a number of exercises that grows with the chapter number as more and more material is made available.  

Inhaltsangaben

Préface.- Table des matières.- Rappels d’analyse réelle et fonctionnelle.- Théorèmes de point fixe et applications.- Les opérateurs de superposition.- La méthode de Galerkin.- Principe du maximum, régularité elliptique et applications.- Calcul des variations et problèmes quasi-linéaires.- Calcul des variations et points critiques.- Opérateurs monotones et inéquations variationnelles.

Ergodic Theory and Dynamical Systems : 978-1-4471-7285-7
Coudène
Springer London
Seiten XIII, 190 p. 49 illus., 1 illus. in color.
Erscheinungsjahr 2016
ISBN 978-1-4471-7285-7
       9781447172857
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This textbook is a self-contained and easy-to-read introduction to ergodic theory and the theory of dynamical systems, with a particular emphasis on chaotic dynamics.
This book contains a broad selection of topics and explores the fundamental ideas of the subject. Starting with basic notions such as ergodicity, mixing, and isomorphisms of dynamical systems, the book then focuses on several chaotic transformations with hyperbolic dynamics, before moving on to topics such as entropy, information theory, ergodic decomposition and measurable partitions. Detailed explanations are accompanied by numerous examples, including interval maps, Bernoulli shifts, toral endomorphisms, geodesic flow on negatively curved manifolds, Morse-Smale systems, rational maps on the Riemann sphere and strange attractors.
Ergodic Theory and Dynamical Systems will appeal to graduate students as well as researchers looking for an introduction to the subject. While gentle on the beginning student, the book also contains a number of comments for the more advanced reader.

Inhaltsangaben

Introduction.- Part I Ergodic Theory.- The Mean Ergodic Theorem.- The Pointwise Ergodic Theorem.- Mixing.- The Hopf Argument.- Part II Dynamical Systems.- Topological Dynamics.- Nonwandering.- Conjugation.- Linearization.- A Strange Attractor.- Part III Entropy Theory.- Entropy.- Entropy and Information Theory.- Computing Entropy.- Part IV Ergodic Decomposition.- Lebesgue Spaces and Isomorphisms.- Ergodic Decomposition.- Measurable Partitions and -Algebras.- Part V Appendices.- Weak Convergence.- Conditional Expectation.- Topology and Measures.- References.

Essential Mathematical Biology : 978-1-85233-536-6
Britton
Springer London
Seiten XV, 335 p. 4 illus.
Erscheinungsjahr 2003
ISBN 978-1-85233-536-6
       9781852335366
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Essential Mathematical Biology is a self-contained introduction to the fast-growing field of mathematical biology. Written for students with a mathematical background, it sets the subject in its historical context and then guides the reader towards questions of current research interest, providing a comprehensive overview of the field and a solid foundation for interdisciplinary research in the biological sciences. A broad range of topics is covered including: Population dynamics, Infectious diseases, Population genetics and evolution, Dispersal, Molecular and cellular biology, Pattern formation, and Cancer modelling. This book will appeal to 3rd and 4th year undergraduate students studying mathematical biology. A background in calculus and differential equations is assumed, although the main results required are collected in the appendices. A dedicated website at www.springer.co.uk/britton/ accompanies the book and provides further exercises, more detailed solutions to exercises in the book, and links to other useful sites.

Inhaltsangaben

1. Single Species Population Dynamics.- 2. Population Dynamics of Interacting Species.- 3. Infectious Diseases.- 4. Population Genetics and Evolution.- 5. Biological Motion.- 6. Molecular and Cellular Biology.- 7. Pattern Formation.- 8. Tumour Modelling.- Further Reading.- A. Some Techniques for Difference Equations.- A.1 First-order Equations.- A.1.1 Graphical Analysis.- A.1.2 Linearisation.- A.2 Bifurcations and Chaos for First-order Equations.- A.2.1 Saddle-node Bifurcations.- A.2.2 Transcritical Bifurcations.- A.2.3 Pitchfork Bifurcations.- A.2.4 Period-doubling or Flip Bifurcations.- A.3 Systems of Linear Equations: Jury Conditions.- A.4 Systems of Nonlinear Difference Equations.- A.4.1 Linearisation of Systems.- A.4.2 Bifurcation for Systems.- B. Some Techniques for Ordinary Differential Equations.- B.1 First-order Ordinary Differential Equations.- B.1.1 Geometric Analysis.- B.1.2 Integration.- B.1.3 Linearisation.- B.2 Second-order Ordinary Differential Equations.- B.2.1 Geometric Analysis (Phase Plane).- B.2.2 Linearisation.- B.2.3 Poincaré-Bendixson Theory.- B.3 Some Results and Techniques for rath Order Systems.- B.3.1 Linearisation.- B.3.2 Lyapunov Functions.- B.3.3 Some Miscellaneous Facts.- B.4 Bifurcation Theory for Ordinary Differential Equations.- B.4.1 Bifurcations with Eigenvalue Zero.- B.4.2 Hopf Bifurcations.- C. Some Techniques for Partial Differential Equations.- C.1 First-order Partial Differential Equations and Characteristics.- C.2 Some Results and Techniques for the Diffusion Equation.- C.2.1 The Fundamental Solution.- C.2.2 Connection with Probabilities.- C.2.3 Other Coordinate Systems.- C.3 Some Spectral Theory for Laplace’s Equation.- C.4 Separation of Variables in Partial Differential Equations.- C.5 Systems of Diffusion Equations with Linear Kinetics.- C.6 Separating the Spatial Variables from Each Other.- D. Non-negative Matrices.- D.1 Perron-Frobenius Theory.- E. Hints for Exercises.

Essential Topology : 978-1-85233-782-7
Crossley
Springer London
Seiten X, 224 p. 110 illus.
Erscheinungsjahr 2005
ISBN 978-1-85233-782-7
       9781852337827
 (Softcover)
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Fachgebiet

Geometry

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Taking a direct route, 'Essential Topology' brings the most important aspects of modern topology within reach of a second-year undergraduate student. It begins with a discussion of continuity and, by way of many examples, leads to the celebrated 'Hairy Ball theorem' and on to homotopy and homology: the cornerstones of contemporary algebraic topology. While containing all the key results of basic topology, Essential Topology never allows itself to get mired in details. Instead, the focus throughout is on providing interesting examples that clarify the ideas and motivate the student. With chapters on: continuity and topological spaces; deconstructionist topology; the Euler number; homotopy groups including the fundamental group; simplicial and singular homology, and fibre bundles. 'Essential Topology' contains enough material for two semester-long courses, and offers a one-stop-shop for undergraduate-level topology, leaving students motivated for postgraduate study in the field, and well-prepared for it.

Inhaltsangaben

Continuous Functions.- Topological Spaces.- Topological Properties.- Deconstructionist Topology.- Homotopy.- The Euler Number.- Homotopy Groups.- Simplicial Homology.- Singular Homology.- More Deconstructionism.

Estimation and Testing Under Sparsity : 978-3-319-32773-0
van de Geer
École d'Été de Probabilités de Saint-Flour XLV – 2015
Springer International Publishing
Seiten XIII, 274 p.
Erscheinungsjahr 2016
ISBN 978-3-319-32773-0
       9783319327730
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Taking the Lasso method as its starting point, this book describes the main ingredients needed to study general loss functions and sparsity-inducing regularizers. It also provides a semi-parametric approach to establishing confidence intervals and tests. Sparsity-inducing methods have proven to be very useful in the analysis of high-dimensional data. Examples include the Lasso and group Lasso methods, and the least squares method with other norm-penalties, such as the nuclear norm. The illustrations provided include generalized linear models, density estimation, matrix completion and sparse principal components. Each chapter ends with a problem section. The book can be used as a textbook for a graduate or PhD course.

Inhaltsangaben

1 Introduction.- The Lasso.- 3 The square-root Lasso.- 4 The bias of the Lasso and worst possible sub-directions.- 5 Confidence intervals using the Lasso.- 6 Structured sparsity.- 7 General loss with norm-penalty.- 8 Empirical process theory for dual norms.- 9 Probability inequalities for matrices.- 10 Inequalities for the centred empirical risk and its derivative.- 11 The margin condition.- 12 Some worked-out examples.- 13 Brouwer’s fixed point theorem and sparsity.- 14 Asymptotically linear estimators of the precision matrix.- 15 Lower bounds for sparse quadratic forms.- 16 Symmetrization, contraction and concentration.- 17 Chaining including concentration.- 18 Metric structure of convex hulls.

Exercises in Analysis : 978-3-319-27815-5
Gasi?ski
Part 2: Nonlinear Analysis
Springer International Publishing
Seiten VIII, 1062 p.
Erscheinungsjahr 2016
ISBN 978-3-319-27815-5
       9783319278155
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Functional Analysis

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This second of two Exercises in Analysis volumes covers problems in five core topics of mathematical analysis: Function Spaces, Nonlinear and Multivalued Maps, Smooth and Nonsmooth Calculus, Degree Theory and Fixed Point Theory, and Variational and Topological Methods. Each of five topics corresponds to a different chapter with inclusion of the basic theory and accompanying main definitions and results,followed by suitable comments and remarks for better understanding of the material. Exercises/problems are presented for each topic, with solutions available at the end of each chapter. The entire collection of exercises offers a balanced and useful picture for the application surrounding each topic. This nearly encyclopedic coverage of exercises in mathematical analysis is the first of its kind and is accessible to a wide readership. Graduate students will find the collection of problems valuable in preparation for their preliminary or qualifying exams as well as for testing their deeper understanding of the material. Exercises are denoted by degree of difficulty. Instructors teaching courses that include one or all of the above-mentioned topics will find the exercises of great help in course preparation. Researchers in analysis may find this Work useful as a summary of analytic theories published in one accessible volume.

Inhaltsangaben

1. Function Spaces.- 2. Nonlinear and Multivalued Maps.- 3. Smooth and Nonsmooth Calculus.- 4. Degree Theory. Fixed Point Theory.- 5. Variational and Topological Methods.- Index.

Extensions of Positive Definite Functions : 978-3-319-39779-5
Jorgensen
Applications and Their Harmonic Analysis
Springer International Publishing
Seiten XXVI, 231 p. 48 illus., 9 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-39779-5
       9783319397795
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This monograph deals with the mathematics of extending given partial data-sets obtained from experiments; Experimentalists frequently gather spectral data when the observed data is limited, e.g., by the precision of instruments; or by other limiting external factors. Here the limited information is a restriction, and the extensions take the form of full positive definite function on some prescribed group. It is therefore both an art and a science to produce solid conclusions from restricted or limited data. While the theory of is important in many areas of pure and applied mathematics, it is difficult for students and for the novice to the field, to find accessible presentations which cover all relevant points of view, as well as stressing common ideas and interconnections. We have aimed at filling this gap, and we have stressed hands-on-examples.

Inhaltsangaben

Introduction.-Extensions of Continuous Positive Definite Functions.-The Case of More General Groups.-Examples.-Type I vs. Type II Extensions.-Spectral Theory for Mercer Operators, and Implications for Ext (F).-Green’s Functions.-Comparing the Different RKHSs HF and HK.-Comparing the Different RKHSs HF and HK.-Models for, and Spectral Representations of, Operator Extensions.-Overview and Open Questions.-List of Figures.-List of Tables.-References.-Index.

Fields and Galois Theory : 978-1-85233-986-9
Howie
Springer London
Seiten X, 226 p. 22 illus.
Erscheinungsjahr 2006
ISBN 978-1-85233-986-9
       9781852339869
 (Softcover)
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Fachgebiet

Algebra

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Fieldsaresetsinwhichallfouroftherationaloperations,memorablydescribed by the mathematician Lewis Carroll as “perdition, distraction, ugli?cation and derision”, can be carried out. They are assuredly the most natural of algebraic objects, since most of mathematics takes place in one ?eld or another, usually the rational ?eld Q, or the real ?eld R, or the complex ?eld C. This book sets out to exhibit the ways in which a systematic study of ?elds, while interesting in its own right, also throws light on several aspects of classical mathematics, notably on ancient geometrical problems such as “squaring the circle”, and on the solution of polynomial equations. The treatment is unashamedly unhistorical. When Galois and Abel dem- strated that a solution by radicals of a quintic equation is not possible, they dealt with permutations of roots. From sets of permutations closed under c- position came the idea of a permutation group, and only later the idea of an abstract group. In solving a long-standing problem of classical algebra, they laid the foundations of modern abstract algebra.

Inhaltsangaben

Rings and Fields.- Integral Domains and Polynomials.- Field Extensions.- Applications to Geometry.- Splitting Fields.- Finite Fields.- The Galois Group.- Equations and Groups.- Some Group Theory.- Groups and Equations.- Regular Polygons.- Solutions.

Finite Frames : 978-0-8176-8372-6
Casazza
Theory and Applications
Birkhäuser Boston
Seiten XVI, 485 p. 35 illus., 20 illus. in color.
Erscheinungsjahr 2013
ISBN 978-0-8176-8372-6
       9780817683726
 (Hardcover)
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Hilbert space frames have long served as a valuable tool for signal and image processing due to their resilience to additive noise, quantization, and erasures, as well as their ability to capture valuable signal characteristics.  More recently, finite frame theory has grown into an important research topic in its own right, with a myriad of applications to pure and applied mathematics, engineering, computer science, and other areas.  The number of research publications, conferences, and workshops on this topic has increased dramatically over the past few years, but no survey paper or monograph has yet appeared on the subject.Edited by two of the leading experts in the field, Finite Frames aims to fill this void in the literature by providing a comprehensive, systematic study of finite frame theory and applications.  With carefully selected contributions written by highly experienced researchers, it covers topics including:* Finite Frame Constructions; * Optimal Erasure Resilient Frames; * Quantization of Finite Frames; * Finite Frames and Compressed Sensing; * Group and Gabor Frames; * Fusion Frames.Despite the variety of its chapters' source and content, the book's notation and terminology are unified throughout and provide a definitive picture of the current state of frame theory.With a broad range of applications and a clear, full presentation, this book is a highly valuable resource for graduate students and researchers across disciplines such as applied harmonic analysis, electrical engineering, quantum computing, medicine, and more.  It is designed to be used as a supplemental textbook, self-study guide, or reference book.

Inhaltsangaben

Introduction.- Constructing Finite Frames with a Given Spectrum.-Spanning and Independence Properties of Finite.-Alegebraic Geometry and Finite Frames.- Group Frames.- Gabor Framses in Finite Dimensions.- Frames as Codes.- Quantization and Finite Frames.- Finite Frames for Sparse Signal Processing.- Finite Frames and Filter Banks.- Finite Frame theory in Pure Mathematics.- Probabilitstic Frames.- Fusion Frames.

Fonctions d'une variable réelle : 978-3-540-34036-2
Bourbaki
Théorie élémentaire
Springer Berlin Heidelberg
Seiten X, 314 p.
Erscheinungsjahr 2007
ISBN 978-3-540-34036-2
       9783540340362
 (Softcover)
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Fachgebiet

Real Functions

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Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce Livre est le quatrième du traité ; il est consacré aux bases de l’analyse réelle. Il comprend les chapitres: 1. Dérivées; 2. Primitives et intégrales; 3. Fonctions élémentaires; 4. Équations différentielles; 5. Étude locale des fonctions; 6. Développements tayloriens généralisés. 7. Formule sommatoire d’Euler-Maclaurin; 8. La function gamma. Il contient également des notes historiques. Ce volume est une réimpression de l’édition de 1976.

Inhaltsangaben

Dérivées.- Primitives et intégrales.- Fonctions élémentaires.- Équations diffentielles.- Etude locale des fonctions.- Développements tayloriens généralisés Formule sommatoire d'Euler-Maclaurin.- La fonction gamma.

Forecasting with Exponential Smoothing : 978-3-540-71916-8
Hyndman
The State Space Approach
Springer Berlin Heidelberg
Seiten XIII, 362 p.
Erscheinungsjahr 2008
ISBN 978-3-540-71916-8
       9783540719168
 (Softcover)
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Exponential smoothing methods have been around since the 1950s, and are still the most popular forecasting methods used in business and industry. However, a modeling framework incorporating stochastic models, likelihood calculation, prediction intervals and procedures for model selection, was not developed until recently. This book brings together all of the important new results on the state space framework for exponential smoothing. It will be of interest to people wanting to apply the methods in their own area of interest as well as for researchers wanting to take the ideas in new directions. Part 1 provides an introduction to exponential smoothing and the underlying models. The essential details are given in Part 2, which also provide links to the most important papers in the literature. More advanced topics are covered in Part 3, including the mathematical properties of the models and extensions of the models for specific problems. Applications to particular domains are discussed in Part 4.

Inhaltsangaben

Basic Concepts.- Getting Started.- Essentials.- Linear Innovations State Space Models.- Nonlinear and Heteroscedastic Innovations State Space Models.- Estimation of Innovations State Space Models.- Prediction Distributions and Intervals.- Selection of Models.- Further Topics.- Normalizing Seasonal Components.- Models with Regressor Variables.- Some Properties of Linear Models.- Reduced Forms and Relationships with ARIMA Models.- Linear Innovations State Space Models with Random Seed States.- Conventional State Space Models.- Time Series with Multiple Seasonal Patterns.- Nonlinear Models for Positive Data.- Models for Count Data.- Vector Exponential Smoothing.- Applications.- Inventory Control Applications.- Conditional Heteroscedasticity and Applications in Finance.- Economic Applications: The Beveridge–Nelson Decomposition.

Freedom in Mathematics : 978-81-322-2786-1
Cartier
Springer India
Seiten XVI, 117 p.
Erscheinungsjahr 2016
ISBN 978-81-322-2786-1
       9788132227861
 (Hardcover)
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This book challenges the views put forward by Pierre Cartier, one of the anchors of the famous Bourbaki group, and Cédric Villani, one of the most brilliant mathematicians of his generation, who received the Fields Medal in 2010. Jean Dhombres, mathematician and science historian, and Gerhard Heinzmann, philosopher of science and also a specialist in mathematics engage in a fruitful dialogue with the two mathematicians, prompting readers to reflect on mathematical activity and its social consequences in history as well as in the modern world. Cédric Villani’s popular success proves once again that a common awareness has developed, albeit in a very confused way, of the major role of mathematics in the construction and efficiency of natural sciences, which are at the origin of our technologies. Despite this, the idea that mathematics cannot be shared remains firmly entrenched, a perceived failing that has even been branded a lack of culture by vocal forces in the media as well as cultural and political establishment.  The authors explore three major directions in their dialogue: the highly complex relationship between mathematics and reality, the subject of many debates and opposing viewpoints; the freedom that the construction of mathematics has given humankind by enabling them to develop the natural sciences as well as mathematical research; and the responsibility with which the scientific community and governments should address the role of mathematics in research and education policies.

Inhaltsangaben

Presentation of the Authors (Pierre Cartier, Jean Dhombres, Gerhard Heinzmann, Cedric Villani).- On the Origins of Mathematics.- The Relations between Mathématics and Reality.- Social and Political History of Mathematics Education.- Nature and Issues of the Mathematics Research.- Conclusion.- Annexure.

From Arithmetic to Zeta-Functions : 978-3-319-28202-2
Sander
Number Theory in Memory of Wolfgang Schwarz
Springer International Publishing
Seiten XXXVII, 538 p. 67 illus., 45 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-28202-2
       9783319282022
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Fachgebiet

Number Theory

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This book collects more than thirty contributions in memory of Wolfgang Schwarz, most of which were presented at the seventh International Conference on Elementary and Analytic Number Theory (ELAZ), held July 2014 in Hildesheim, Germany. Ranging from the theory of arithmetical functions to diophantine problems, to analytic aspects of zeta-functions, the various research and survey articles cover the broad interests of the well-known number theorist and cherished colleague Wolfgang Schwarz (1934-2013), who contributed over one hundred articles on number theory, its history and related fields. Readers interested in elementary or analytic number theory and related fields will certainly find many fascinating topical results among the contributions from both respected mathematicians and up-and-coming young researchers. In addition, some biographical articles highlight the life and mathematical works of Wolfgang Schwarz.

Inhaltsangaben

Preface.- Forbidden Integer Ratios of Consecutive Power Sums, Ioulia N. Baoulina and Pieter Moree.- A Note on the Negative Pell Equation, Valentin Blomer.- Localisation Conditionnelle de Diviseurs, Régis de la Bretèche, Gérald Tenenbaum.- A Ternary Problem in Additive Prime Number Theory, Jörg Brüdern.- An Improvement of Liouville’s Inequality, Yann Bugeaud.- Guided by Schwarz’ Functions: a Walk Through the Garden of Mahlers Transcendence Method, Peter Bundschuh and Keijo Väänänen.- Sums of Two Squares and a Power, Rainer Dietmann and Christian Elsholtz.- Multiplicative Functions and the Sign of Maass Form Fourier Coefficients, P. D. T. A. Elliott.- On Error Sum Functions for Approximations with Arithmetic Conditions, Carsten Elsner.- Sum of the Lerch Zeta Function over Nontrivial Zeros of the Dirichlet L-Function, Ram?nas Garunkštis and Justas Kalpokas.- Schur-Weyl-Dualities - Old and New, Anne Henke.- Arithmetic functions - a Pivotal Topic in the Scientific Work of Wolfgang, Karl-Heinz Indlekofer.- On Some Selected Works of Wolfgang Schwarz, Aleksandar Ivi?.- Sums of Two Squares of Sums of Two Squares, Rebecca Jakob.- The Joint Discrete Universality of Periodic Zeta-Functions, Antanas Laurin?ikas.- Remembering Wolfgang Schwarz, his Life and Work, Lutz G. Lucht.- Dynamical Systems and Uniform Distribution of Sequences, Manfred G. Madritsch and Robert F. Tichy.- Asymptotics and Equidistribution of Cotangent Sums Associated to the Estermann and Riemann Zeta Functions, Helmut Maier and Michael Th. Rassias.- A Turán-Kubilius Inequality on Mappings of a Finite Set, Eugenijus Manstavi?ius.- Aspects of Zeta-Function Theory in the Mathematical Works of Adolf Hurwitz, Nicola M.R. Oswald and Jörn Steuding.- Selberg Sums a New Perspective, Samuel J. Patterson.- Polignac Numbers, Conjectures of Erd?s on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture, János Pintz.- Idempotents and Congruence ax=b (mod n), Štefan Porubský.- Recent Developments on the Edge Between Number Theory and Graph Theory, Jürgen Sander and Thorsten Sander.- The Leading Coefficients of Stern Polynomials, Andrzeij Schinzel.- The Non-Existence of Universal Carmichael Numbers, Jan-Christoph Schlage-Puchta.- Arithmetic Properties of Blocks of Consecutive Integers, Tarlok N. Shorey and Rob Tijdeman.- The GCD of the Shifted Fibonacci Sequence, Jürgen Spilker.- On Liouville Numbers - Yet Another Application of Functional Analysis to Number Theory, Jörn Steuding.- Natural Boundaries of Power Series with Multiplicative Coefficients in Algebraic Number Fields, Friedemann Tuttas.- A Minimal Result of Hardy, Eduard Wirsing.- Regular Dessins with Abelian Automorphism Groups, Jürgen Wolfart and Benjamin Mühlbauer.

Functional Analysis and the Feynman Operator Calculus : 978-3-319-27593-2
Gill
Springer International Publishing
Seiten XIX, 354 p. 3 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-27593-2
       9783319275932
 (Hardcover)
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Functional Analysis

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This book provides the mathematical foundations for Feynman's operator calculus and for the Feynman path integral formulation of quantum mechanics as a natural extension of analysis and functional analysis to the infinite-dimensional setting. In one application, the results are used to prove the last two remaining conjectures of Freeman Dyson for quantum electrodynamics. In another application, the results are used to unify methods and weaken domain requirements for non-autonomous evolution equations. Other applications include a general theory of Lebesgue measure on Banach spaces with a Schauder basis and a new approach to the structure theory of operators on uniformly convex Banach spaces. This book is intended for advanced graduate students and researchers.

Inhaltsangaben

Preliminary Background.- Integration on Infinite-dimensional Spaces.- HK-Integral and HK-Spaces.- Analysis on Hilbert Space.- Operators on Banach Space.- Spaces of von Neumann Type.- The Feynman Operator Calculus.- Applications of the Feynman Calculus.

Functional Analysis, Calculus of Variations and Optimal Control : 978-1-4471-4819-7
Clarke
Springer London
Seiten XIV, 591 p. 24 illus., 8 illus. in color.
Erscheinungsjahr 2013
ISBN 978-1-4471-4819-7
       9781447148197
 (Hardcover)
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Fachgebiet

Functional Analysis

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Functional analysis owes much of its early impetus to problems that arise in the calculus of variations. In turn, the methods developed there have been applied to optimal control, an area that also requires new tools, such as nonsmooth analysis. This self-contained textbook gives a complete course on all these topics. It is written by a leading specialist who is also a noted expositor.This book provides a thorough introduction to functional analysis and includes many novel elements as well as the standard topics. A short course on nonsmooth analysis and geometry completes the first half of the book whilst the second half concerns the calculus of variations and optimal control. The author provides a comprehensive course on these subjects, from their inception through to the present. A notable feature is the inclusion of recent, unifying developments on regularity, multiplier rules, and the Pontryagin maximum principle, which appear here for the first time in a textbook. Other major themes include existence and Hamilton-Jacobi methods.The many substantial examples, and the more than three hundred exercises, treat such topics as viscosity solutions, nonsmooth Lagrangians, the logarithmic Sobolev inequality, periodic trajectories, and systems theory. They also touch lightly upon several fields of application: mechanics, economics, resources, finance, control engineering.Functional Analysis, Calculus of Variations and Optimal Control is intended to support several different courses at the first-year or second-year graduate level, on functional analysis, on the calculus of variations and optimal control, or on some combination. For this reason, it has been organized with customization in mind. The text also has considerable value as a reference. Besides its advanced results in the calculus of variations and optimal control, its polished presentation of certain other topics (for example convex analysis, measurable selections, metric regularity, and nonsmooth analysis) will be appreciated by researchers in these and related fields.

Inhaltsangaben

Normed Spaces.- Convex sets and functions.- Weak topologies.- Convex analysis.- Banach spaces.- Lebesgue spaces.- Hilbert spaces.- Additional exercises for Part I.- Optimization and multipliers.- Generalized gradients.- Proximal analysis.- Invariance and monotonicity.- Additional exercises for Part II.- The classical theory.- Nonsmooth extremals.- Absolutely continuous solutions.- The multiplier rule.- Nonsmooth Lagrangians.- Hamilton-Jacobi methods.- Additional exercises for Part III.- Multiple integrals.- Necessary conditions.- Existence and regularity.- Inductive methods.- Differential inclusions.- Additional exercises for Part IV.

Fundamentals and Advanced Techniques in Derivatives Hedging : 978-3-319-38988-2
Bouchard
Springer International Publishing
Seiten XII, 280 p.
Erscheinungsjahr 2016
ISBN 978-3-319-38988-2
       9783319389882
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This book covers the theory of derivatives pricing and hedging as well as techniques used in mathematical finance. The authors use a top-down approach, starting with fundamentals before moving to applications, and present theoretical developments alongside various exercises, providing many examples of practical interest.A large spectrum of concepts and mathematical tools that are usually found in separate monographs are presented here. In addition to the no-arbitrage theory in full generality, this book also explores models and practical hedging and pricing issues. Fundamentals and Advanced Techniques in Derivatives Hedging further introduces advanced methods in probability and analysis, including Malliavin calculus and the theory of viscosity solutions, as well as the recent theory of stochastic targets and its use in risk management, making it the first textbook covering this topic.
Graduate students in applied mathematics with an understanding of probability theory and stochastic calculus will find this book useful to gain a deeper understanding of fundamental concepts and methods in mathematical finance.

Inhaltsangaben

Part A. Fundamental theorems.- Discrete time models.- Continuous time models.- Optimal management and price selection.- Part B. Markovian models and PDE approach.- Delta hedging in complete market.- Super-replication and its practical limits.- Hedging under loss contraints.- Part C. Practical implementation in local and stochastic volatility models.- Local volatility models.- Stochastic volatility models.- References.

Fundamentals of Functional Analysis : 978-3-319-45631-7
Farenick
Springer International Publishing
Seiten XIV, 451 p. 2 illus.
Erscheinungsjahr 2016
ISBN 978-3-319-45631-7
       9783319456317
 (Softcover)
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Fachgebiet

Functional Analysis

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This book provides a unique path for graduate or advanced undergraduate students to begin studying the rich subject of functional analysis with fewer prerequisites than is normally required. The text begins with a self-contained and highly efficient introduction to topology and measure theory, which focuses on the essential notions required for the study of functional analysis, and which are often buried within full-length overviews of the subjects. This is particularly useful for those in applied mathematics, engineering, or physics who need to have a firm grasp of functional analysis, but not necessarily some of the more abstruse aspects of topology and measure theory normally encountered. The reader is assumed to only have knowledge of basic real analysis, complex analysis, and algebra.The latter part of the text provides an outstanding treatment of Banach space theory and operator theory, covering topics not usually found together in other books on functional analysis. Written in a clear, concise manner, and equipped with a rich array of interesting and important exercises and examples, this book can be read for an independent study, used as a text for a two-semester course, or as a self-contained reference for the researcher.

Inhaltsangaben

Preface.- Topological Spaces.- Topological Spaces with Special Properties.- Measure Theory.- Integration.- Banach Spaces.- Dual Spaces.- Convexity.- Banach Space Operators.- Spectral Theory in Banach Algebras.- Hilbert Space Operators.- Algebras of Hilbert Space Operators.- References.- Index.

Generalized Principal Component Analysis : 978-0-387-87810-2
Vidal
Springer New York
Seiten XXXII, 566 p. 121 illus., 83 illus. in color.
Erscheinungsjahr 2016
ISBN 978-0-387-87810-2
       9780387878102
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This book provides a comprehensive introduction to the latest advances in the mathematical theory and computational tools for modeling high-dimensional data drawn from one or multiple low-dimensional subspaces (or manifolds) and potentially corrupted by noise, gross errors, or outliers. This challenging task requires the development of new algebraic, geometric, statistical, and computational methods for efficient and robust estimation and segmentation of one or multiple subspaces. The book also presents interesting real-world applications of these new methods in image processing, image and video segmentation, face recognition and clustering, and hybrid system identification etc. This book is intended to serve as a textbook for graduate students and beginning researchers in data science, machine learning, computer vision, image and signal processing, and systems theory. It contains ample illustrations, examples, and exercises and is made largely self-contained with three Appendices which survey basic concepts and principles from statistics, optimization, and algebraic-geometry used in this book.René Vidal is a Professor of Biomedical Engineering and Director of the Vision Dynamics and Learning Lab at The Johns Hopkins University. Yi Ma is Executive Dean and Professor at the School of Information Science and Technology at ShanghaiTech University. S. Shankar Sastry is Dean of the College of Engineering, Professor of Electrical Engineering and Computer Science and Professor of Bioengineering at the University of California, Berkeley.

Inhaltsangaben

Preface.- Acknowledgments.- Glossary of Notation.- Introduction.- I Modeling Data with Single Subspace.- Principal Component Analysis.- Robust Principal Component Analysis.- Nonlinear and Nonparametric Extensions.- II Modeling Data with Multiple Subspaces.- Algebraic-Geometric Methods.- Statistical Methods.- Spectral Methods.- Sparse and Low-Rank Methods.- III Applications.- Image Representation.- Image Segmentation.- Motion Segmentation.- Hybrid System Identification.- Final Words.- Appendices.- References.- Index.

Geometric Control of Mechanical Systems : 978-0-387-22195-3
Bullo
Modeling, Analysis, and Design for Simple Mechanical Control Systems
Springer New York
Seiten XXIV, 727 p.
Erscheinungsjahr 2005
ISBN 978-0-387-22195-3
       9780387221953
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The primary emphasis of this book is the modeling, analysis, and control of mechanical systems. The methods and results presented can be applied to a large class of mechanical control systems, including applications in robotics, autonomous vehicle control, and multi-body systems. The book is unique in that it presents a unified, rather than an inclusive, treatment of control theory for mechanical systems. A distinctive feature of the presentation is its reliance on techniques from differential and Riemannian geometry. The book contains extensive examples and exercises, and will be suitable for a growing number of courses in this area. It begins with the detailed mathematical background, proceeding through innovative approaches to physical modeling, analysis, and design techniques. Numerous examples illustrate the proposed methods and results, while the many exercises test basic knowledge and introduce topics not covered in the main body of the text. The audience of this book consists of two groups. The first group is comprised of graduate students in engineering or mathematical sciences who wish to learn the basics of geometric mechanics, nonlinear control theory, and control theory for mechanical systems. Readers will be able to immediately begin exploring the research literature on these subjects. The second group consists of researchers in mechanics and control theory. Nonlinear control theoreticians will find explicit links between concepts in geometric mechanics and nonlinear control theory. Researchers in mechanics will find an overview of topics in control theory that have relevance to mechanics.

Inhaltsangaben

Part I: Modeling of mechanical systems; Introductory examples and problems; Linear and multilinear algebra; Differential geometry; Simple mechanical control systems; Lie groups, systems on groups, and symmetries.- Part II: Analysis of mechanical control systems; Stability; Controllability; Low-order controllability and kinematic reduction ; Perturbation analysis.- Part III: A sampling of design methodologies; Linear and nonlinear potential shaping for stabilization; Stabilization and tracking for fully actuated systems; Stabilization and tracking using oscillatory controls; Motion planning for underactuated systems; Appendices; Time-dependent vector fields; Some proofs.

Geometry by Its History : 978-3-642-29162-3
Ostermann
Springer Berlin Heidelberg
Seiten XII, 440 p.
Erscheinungsjahr 2012
ISBN 978-3-642-29162-3
       9783642291623
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Geometry

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In this textbook the authors present first-year geometry roughly in the order in which it was discovered. The first five chapters show how the ancient Greeks established geometry, together with its numerous practical applications, while more recent findings on Euclidian geometry are discussed as well. The following three chapters explain the revolution in  geometry due to the progress made in the field of algebra by Descartes, Euler and Gauss. Spatial geometry, vector algebra and matrices are treated in chapters 9 and 10. The last chapter offers an introduction to projective geometry, which emerged in the 19thcentury.Complemented by numerous examples, exercises, figures and pictures, the book offers both motivation and insightful explanations, and provides stimulating and enjoyable reading for students and teachers alike.

Inhaltsangaben

Preface.- Part I: Classical Geometry.- Thales and Pythagoras.- The Elements of Euclid.- Conic Sections.- Further Results on Euclidean Geometry.- Trigonometry.- Part II: Analytic Geometry.- Descartes' Geometry.- Cartesian Coordinates.- To be Constructible, or not to be.- Spatial Geometry and Vector Algebra.- Matrices and Linear Mappings.- Projective Geometry.- Solutions to Exercises.-  References.- Figure Source and Copyright.- Index.

Graph Theory : 978-3-319-31938-4
Gera
Favorite Conjectures and Open Problems - 1
Springer International Publishing
Seiten XII, 291 p. 114 illus., 24 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-31938-4
       9783319319384
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Graph Theory

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This is the first in a series of volumes, which provide an extensive overview of conjectures and open problems in graph theory. The readership of each volume is geared toward graduate students who may be searching for research ideas. However, the well-established mathematician will find the overall exposition engaging and enlightening. Each chapter, presented in a story-telling style, includes more than a simple collection of results on a particular topic. Each contribution conveys the history, evolution, and techniques used to solve the authors’ favorite conjectures and open problems, enhancing the reader’s overall comprehension and enthusiasm. The editors were inspired to create these volumes by the popular and well attended special sessions, entitled “My Favorite Graph Theory Conjectures,' which were held at the winter AMS/MAA Joint Meeting in Boston (January, 2012), the SIAM Conference on Discrete Mathematics in Halifax (June,2012) and the winter AMS/MAA Joint meeting in Baltimore(January, 2014). In an effort to aid in the creation and dissemination of open problems, which is crucial to the growth and development of a field, the editors requested the speakers, as well as notable experts in graph theory, to contribute to these volumes.

Inhaltsangaben

Highly Irregular (G. Chartrand).- Hamiltonian Extension (P. Zhang).- On Some Open Questions for Ramsey and Folkman Numbers (S. Radziszowski and X. Xu).- All my favorite conjectures are critical(T. Haynes).- The local representation of graph conjecture(E. Scheinerman).- Some of My Favorite Coloring Problems for Graphs and Digraphs (J. Gimble).- My Top 10 Favorite Conjectures and Open Problems(S. Hedetniemi).- Chvátal's t0-tough conjecture (L. Lesniak).- What do Trees and Hypercubes have in Common (H. Mulder).- Two chromatic conjectures: one for vertices, one for edges (M. Kayll).- Some Conjectures and Questions in Chromatic Topological Graph Theory (J. Hutchinson).- Turan's Brick factory problem (L. Szekely). -It is all labeling (P. Slater).- My Favorite Domination Conjectures (M. Henning).- Circuit Double Covers of Graphs (C. Zhang).

Groupes et algèbres de Lie : 978-3-540-35335-5
Bourbaki
Chapitre 1
Springer Berlin Heidelberg
Seiten VI, 146 p.
Erscheinungsjahr 2007
ISBN 978-3-540-35335-5
       9783540353355
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Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes et algèbre de Lie, neuvième Livre du traité, est consacré aux concepts fondamentaux pour les algèbres de Lie. Il comprend les paragraphes: -§ 1 Définition des algèbres de Lie; §2 Algèbre enveloppante d’une algèbre de Lie; §3 Représentations; §4 Algèbres de Lie nilpotentes; §5 Algèbres de Lie résolubles; §6 Algèbres de Lie semi-simples; §7 Le théorème d’Ado. Ce volume est une réimpression de l’édition de 1971.

Inhaltsangaben

Algèbres de Lie.

Groupes et algèbres de Lie : 978-3-540-34392-9
Bourbaki
Chapitre 9 Groupes de Lie réels compacts
Springer Berlin Heidelberg
Seiten IV, 138 p.
Erscheinungsjahr 2007
ISBN 978-3-540-34392-9
       9783540343929
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Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce neuvième chapitre du Livre sur les Groupes et algèbres de Lie, neuvième Livre du traité, comprend les paragraphes: §1 Algèbres de Lie compactes; §2 Tores maximaux des groupes de Lie compacts; §3 Fromes compactes des algèbres de Lie semi-simples complexes; §4 Système de raciness associé à un groupe compact; §5 Classes de conjugaison; §6 Intégration dans les groupes de Lie compacts; §7 Représentations irréductibles des groupes de Lie compacts connexes; §8 Transformation de Fourier; §9 Opération des groupes de Lie compacts sur les variétés. Ce volume a été publié en 1982.

Inhaltsangaben

Groupes de Lie réels compacts.

Groupes et algèbres de Lie : 978-3-540-33940-3
Bourbaki
Chapitres 2 et 3
Springer Berlin Heidelberg
Seiten VI, 318 p.
Erscheinungsjahr 2006
ISBN 978-3-540-33940-3
       9783540339403
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Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce deuxième volume du Livre sur les Groupes et algèbres de Lie, neuvième Livre du traité, comprend les chapitres: 2. Algèbres de Lie libres; 3. Groupes de Lie. Le chapitre 2 poursuit la présentation des notions fondamentales des algèbres de Lie avec l’introduction des algèbres de Lie libres et de la série de Hausdorff. Le chapitre 3 est consacré aux concepts de base pour les groupes de Lies sur un corps archimédien ou ultramétrique. Ce volume contient également de notes historiques pour les chapitres 1 à 3. Ce volume est une réimpression de l’édition de 1972.

Inhaltsangaben

Algèbres de Lie libres.- Groupes de Lie.

Groupes et algèbres de Lie : 978-3-540-34490-2
Bourbaki
Chapitres 4, 5 et 6
Springer Berlin Heidelberg
Seiten VIII, 282 p.
Erscheinungsjahr 2007
ISBN 978-3-540-34490-2
       9783540344902
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Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce troisième volume du Livre sur les Groupes et algèbres de Lie, neuvième Livre du traité, est consacré aux structures de systèmes de racines, de groupes de Coxeter et de systèmes de Tits, qui apparaissent naturellement dans l’étude des groupes de Lie analytique ou algébriques. Il comprend les chapitres: -Groupes de Coxeter et systèmes de Tits, -Groupes engenders par des reflexions, -Systèmes de racines. Ce volume contient également des planches décrivant les différents types de systèmes de raciness et des notes historiques. Ce volume est une réimpression de l’édition de 1968.

Inhaltsangaben

aux chapitres IV, V et VI.- Groupes de Coxeter et systèmes de Tits.- Groupes engendrés par des réflexions.- Systèmes de racines.

Groupes et algèbres de Lie : 978-3-540-33939-7
Bourbaki
Chapitres 7 et 8
Springer Berlin Heidelberg
Seiten IV, 265 p.
Erscheinungsjahr 2006
ISBN 978-3-540-33939-7
       9783540339397
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Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce troisième volume du Livre sur les Groupes et algèbres de Lie, neuvième Livre du traité, poursuit l’étude des algèbres de Lie et leurs représentations. Il comprend les chapitres: 7. Sous-algèbres de Cartan, éléments réguliers; 8. Algèbres de Lie semi-simples déployées. Ce volume contient également un appendice sur la topologie de Zariski. Ce volume est une réimpression de l’édition de 1975.

Inhaltsangaben

Sous-algèbres de Cartan éléments réguliers.- Algèbres de Lie semi-simples déployées.

Groups, Rings and Fields : 978-3-540-76177-8
Wallace
Springer London
Seiten VIII, 248 p. 4 illus.
Erscheinungsjahr 1998
ISBN 978-3-540-76177-8
       9783540761778
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Algebra

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David Wallace has written a text on modern algebra which is suitable for a first course in the subject given to mathematics undergraduates. It aims to promote a feeling for the evolutionary and historical development of algebra. It assumes some familiarity with complex numbers, matrices and linear algebra which are commonly taught during the first year of an undergraduate course. Each chapter contains examples, exercises and solutions, perfectly suited to aid self-study. All arguments in the text are carefully crafted to promote understanding and enjoyment for the reader.

Inhaltsangaben

1 Sets and Mappings.- 2 The Integers.- 3 Introduction to Rings.- 4 Introduction to Groups.- 5 Rings.- 6 Topics in Group Theory.- Hints to Solutions.- Suggestions for Further Study.

Handbook of Floating-Point Arithmetic : 978-0-8176-4704-9
Muller
Birkhäuser Boston
Seiten XXIV, 572 p.
Erscheinungsjahr 2010
ISBN 978-0-8176-4704-9
       9780817647049
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Inhaltsangaben

Introduction, Basic Definitions, and Standards.- Definitions and Basic Notions.- Floating-Point Formats and Environment.- Cleverly Using Floating-Point Arithmetic.- Basic Properties and Algorithms.- The Fused Multiply-Add Instruction.- Enhanced Floating-Point Sums, Dot Products, and Polynomial Values.- Languages and Compilers.- Implementing Floating-Point Operators.- Algorithms for the Five Basic Operations.- Hardware Implementation of Floating-Point Arithmetic.- Software Implementation of Floating-Point Arithmetic.- Elementary Functions.- Evaluating Floating-Point Elementary Functions.- Solving the Table Maker’s Dilemma.- Extensions.- Formalisms for Certifying Floating-Point Algorithms.- Extending the Precision.- Perspectives and Appendix.- Conclusion and Perspectives.- Appendix: Number Theory Tools for Floating-Point Arithmetic.

Harmonic Analysis on Symmetric Spaces—Higher Rank Spaces, Positive Definite Matrix Space and Generalizations : 978-1-4939-3406-5
Terras
Springer New York
Seiten XV, 487 p. 41 illus., 21 illus. in color.
Erscheinungsjahr 2016
ISBN 978-1-4939-3406-5
       9781493934065
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This text is an introduction to harmonic analysis on symmetric spaces, focusing on advanced topics such as higher rank spaces, positive definite matrix space and generalizations. It is intended for beginning graduate students in mathematics or researchers in physics or engineering. As with the introductory book entitled 'Harmonic Analysis on Symmetric Spaces - Euclidean Space, the Sphere, and the Poincaré Upper Half Plane, the style is informal with an emphasis on motivation, concrete examples, history, and applications. The symmetric spaces considered here are quotients X=G/K, where G is a non-compact real Lie group, such as the general linear group GL(n,P) of all n x n non-singular real matrices, and K=O(n), the maximal compact subgroup of orthogonal matrices. Other examples are Siegel's upper half 'plane' and the quaternionic upper half 'plane'. In the case of the general linear group, one can identify X with the space Pn of n x n positive definite symmetric matrices.Many corrections and updates have been incorporated in this new edition. Updates include discussions of random matrix theory and quantum chaos, as well as recent research on modular forms and their corresponding L-functions in higher rank.  Many applications have been added, such as the solution of the heat equation on Pn, the central limit theorem of Donald St. P. Richards for Pn, results on densest lattice packing of spheres in Euclidean space, and GL(n)-analogs of the Weyl law for eigenvalues of the Laplacian in plane domains. Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, fundamental domains in X for discrete groups ? (such as the modular group GL(n,Z) of n x n matrices with integer entries and determinant ±1), connections with the problem of finding densest lattice packings of spheres in Euclidean space, automorphic forms, Hecke operators, L-functions, and the Selberg trace formula and its applications in spectral theory as well as number theory.

Inhaltsangaben

Part I: The Space Pn of Positive n x n Matrices.- Part II: The General Noncompact Symmetric Space.

Homotopical Topology : 978-3-319-23487-8
Fomenko
Springer International Publishing
Seiten XI, 627 p. 210 illus.
Erscheinungsjahr 2016
ISBN 978-3-319-23487-8
       9783319234878
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This textbook on algebraic topology updates a popular textbook from the golden era of the Moscow school of I. M. Gelfand. The first English translation, done many decades ago, remains very much in demand, although it has been long out-of-print and is difficult to obtain. Therefore, this updated English edition will be much welcomed by the mathematical community. Distinctive features of this book include: a concise but fully rigorous presentation, supplemented by a plethora of illustrations of a high technical and artistic caliber; a huge number of nontrivial examples and computations done in detail; a deeper and broader treatment of topics in comparison to most beginning books on algebraic topology; an extensive, and very concrete, treatment of the machinery of spectral sequences. The second edition contains an entirely new chapter on K-theory and the Riemann-Roch theorem (after Hirzebruch and Grothendieck).

Inhaltsangaben

Introduction.- Homotopy.- Homology.- Spectral Sequences of Fibrations.- Cohomology Operations.- The Adams Spectral Sequence.- K-Theory and Other Extraordinary Cohomology Theories.

Hyperbolic Conservation Laws in Continuum Physics : 978-3-662-49449-3
Dafermos
Springer Berlin Heidelberg
Seiten XXXVIII, 826 p. 52 illus.
Erscheinungsjahr 2016
ISBN 978-3-662-49449-3
       9783662494493
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This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics who may need analytical tools; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws.This new edition places increased emphasis on hyperbolic systems of balance laws with dissipative source, modeling relaxation phenomena. It also presents an account of recent developments on the Euler equations of compressible gas dynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised, expanded and brought up to date, and has been enriched with new applications to elasticity and differential geometry. The bibliography, also expanded and updated, now comprises close to two thousand titles.From the reviews of the 3rd edition:'This is the third edition of the famous book by C.M. Dafermos. His masterly written book is, surely, the most complete exposition in the subject.' Evgeniy Panov, Zentralblatt MATH 'A monumental book encompassing all aspects of the mathematical theory of hyperbolic conservation laws, widely recognized as the 'Bible' on the subject.' Philippe G. LeFloch, Math. Reviews

Inhaltsangaben

I Balance Laws. - II Introduction to Continuum Physics. - III Hyperbolic Systems of Balance Laws. - IV The Cauchy Problem. - V Entropy and the Stability of Classical Solutions. - VI The L1 Theory for Scalar Conservation Laws. - VII Hyperbolic Systems of Balance Laws in One-Space Dimension. - VIII Admissible Shocks. - IX Admissible Wave Fans and the Riemann Problem. - X Generalized Characteristics. - XI Scalar Conservation Laws in One Space Dimension. - XII Genuinely Nonlinear Systems of Two Conservation Laws. - XIII The Random Choice Method. - XIV The Front Tracking Method and Standard Riemann Semigroups. - XV Construction of BV Solutions by the Vanishing Viscosity Method. - XVI BV Solutions for Systems of Balance Laws. - XVII Compensated Compactness. - XVIII Steady and Self-similar Solutions in Multi-Space Dimensions. - Bibliography. - Author Index. - Subject Index.

Hyperbolic Geometry : 978-1-85233-934-0
Anderson
Springer London
Seiten XII, 276 p. 21 illus.
Erscheinungsjahr 2005
ISBN 978-1-85233-934-0
       9781852339340
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Fachgebiet

Geometry

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The geometry of the hyperbolic plane has been an active and fascinating field of mathematical inquiry for most of the past two centuries. This book provides a self-contained introduction to the subject, providing the reader with a firm grasp of the concepts and techniques of this beautiful area of mathematics. Topics covered include the upper half-space model of the hyperbolic plane, Möbius transformations, the general Möbius group and the subgroup preserving path length in the upper half-space model, arc-length and distance, the Poincaré disc model, convex subsets of the hyperbolic plane, and the Gauss-Bonnet formula for the area of a hyperbolic polygon and its applications. This updated second edition also features: an expanded discussion of planar models of the hyperbolic plane arising from complex analysis; the hyperboloid model of the hyperbolic plane; a brief discussion of generalizations to higher dimensions; many new exercises.

Inhaltsangaben

The Basic Spaces.- The General Möbius Group.- Length and Distance in ?.- Planar Models of the Hyperbolic Plane.- Convexity, Area, and Trigonometry.- Nonplanar models.

Indiscrete Thoughts : 978-0-8176-4780-3
Palombi
Birkhäuser Boston
Seiten XXIII, 280 p.
Erscheinungsjahr 1997
ISBN 978-0-8176-4780-3
       9780817647803
 (Softcover)
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Indiscrete Thoughts gives a glimpse into a world that has seldom been described that of science and technology as seen through the eyes of a mathematician. The era covered by this book, 1950 to 1990, was surely one of the golden ages of science as well as the American university. Cherished myths are debunked along the way as Gian-Carlo Rota takes pleasure in portraying, warts and all, some of the great scientific personalities of the period —Stanislav Ulam (who, together with Edward Teller, signed the patent application for the hydrogen bomb), Solomon Lefschetz (Chairman in the 50s of the Princeton mathematics department), William Feller (one of the founders of modern probability theory), Jack Schwartz (one of the founders of computer science), and many others. Rota is not afraid of controversy. Some readers may even consider these essays indiscreet. After the publication of the essay “The Pernicious Influence of Mathematics upon Philosophy” (reprinted six times in five languages) the author was blacklisted in analytical philosophy circles. Indiscrete Thoughts should become an instant classic and the subject of debate for decades to come.

Inhaltsangaben

Persons and Places.- Fine Hall in its Golden Age.- Light Shadows.- Combinatorics, Representation Theory and Invariant Theory.- The Barrier of Meaning.- Stan Ulam.- The Lost Café.- Philosophy.- The Pernicious Influence of Mathematics Upon Philosophy.- Philosophy and Computer Science.- The Phenomenology of Mathematical Truth.- The Phenomenology of Mathematical Beauty.- The Phenomenology of Mathematical Proof.- Syntax, Semantics, and the Problem of the Identity of Mathematical Items.- The Barber of Seville, or The Useless Precaution.- Kant and Husserl.- Fundierung as a Logical Concept.- The Primacy of Identity.- Three Senses of “A is B” in Heidegger.- Indiscrete Thoughts.- Ten Lessons I wish I Had Been Taught.- Ten Lessons for the Survival of a Mathematics Department.- A Mathematician’s Gossip.- Book Reviews.

Inequalities : 978-3-642-23791-1
Cvetkovski
Theorems, Techniques and Selected Problems
Springer Berlin Heidelberg
Seiten X, 444 p.
Erscheinungsjahr 2012
ISBN 978-3-642-23791-1
       9783642237911
 (Softcover)
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Algebra

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This work is about inequalities which play an important role in mathematical Olympiads. It contains 175 solved problems in the form of exercises and, in addition, 310 solved problems. The book also covers the theoretical background of the most important theorems and techniques required for solving inequalities. It is written for all middle and high-school students, as well as for graduate and undergraduate students. School teachers and trainers for mathematical competitions will also gain benefit from this book.

Inhaltsangaben

'Basic (elementary) inequalities and their application.- Inequalities between means, (with two and three variables).- Geometric (triangle) inequalities.- Bernoulli’s inequality, the Cauchy–Schwarz inequality, Chebishev’s inequality, Surányi’s inequality.- Inequalities between means (general case).- Points of incidence in applications of the AM–GM inequality.- The rearrangement inequality.- Convexity, Jensen’s inequality.- Trigonometric substitutions and their application for proving algebraic inequalities.- The most usual forms of trigonometric substitutions.- Characteristic examples, using trigonometric substitutions.- Hölder’s inequality, Minkowski’s inequality and their generalizations.- Generalizations of the Cauchy–Schwarz inequality, Chebishev’s inequality and the mean inequalities.- Newton’s inequality, Maclaurin’s inequality.- Schur’s inequality, Muirhead’s inequality.- Two theorems from differential calculus, and their applications for proving inequalities.- One method of proving symmetric inequalities with three variables.- Method for proving symmetric inequalities with three variables defined on set of real numbers.- Abstract concreteness method (ABC method).- Sum of Squares (S.O.S - method).- Strong mixing variables method (S.M.V Theorem).- Lagrange multipliers method.

Integers, Polynomials, and Rings : 978-0-387-20172-6
Irving
A Course in Algebra
Springer New York
Seiten XVI, 288 p.
Erscheinungsjahr 2004
ISBN 978-0-387-20172-6
       9780387201726
 (Softcover)
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Fachgebiet

Algebra

Information

This book began life as a set of notes that I developed for a course at the University of Washington entitled Introduction to Modern Algebra for Tea- ers. Originally conceived as a text for future secondary-school mathematics teachers, it has developed into a book that could serve well as a text in an - dergraduatecourseinabstractalgebraoracoursedesignedasanintroduction to higher mathematics. This book di?ers from many undergraduate algebra texts in fundamental ways; the reasons lie in the book’s origin and the goals I set for the course. The course is a two-quarter sequence required of students intending to f- ?ll the requirements of the teacher preparation option for our B.A. degree in mathematics, or of the teacher preparation minor. It is required as well of those intending to matriculate in our university’s Master’s in Teaching p- gram for secondary mathematics teachers. This is the principal course they take involving abstraction and proof, and they come to it with perhaps as little background as a year of calculus and a quarter of linear algebra. The mathematical ability of the students varies widely, as does their level of ma- ematical interest.

Inhaltsangaben

Introduction: The McNugget Problem.- Introduction: The McNugget Problem.- Integers.- Induction and the Division Theorem.- The Euclidean Algorithm.- Congruences.- Prime Numbers.- Rings.- Euler’ Theorem.- Binomial Coefficients.- Polynomials.- Polynomials and Roots.- Polynomials with Real Coefficients.- Polynomials with Rational Coefficients.- Polynomial Rings.- Quadratic Polynomials.- Polynomial Congruence Rings.- All Together Now.- Euclidean Rings.- The Ring of Gaussian Integers.- Finite Fields.

Integral and Discrete Inequalities and Their Applications : 978-3-319-33300-7
Qin
Volume I: Linear Inequalities
Springer International Publishing
Seiten XVI, 989 p. 9 illus.
Erscheinungsjahr 2016
ISBN 978-3-319-33300-7
       9783319333007
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Real Functions

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This book focuses on one- and multi-dimensional linear integral and discrete Gronwall-Bellman type inequalities. It provides a useful collection and systematic presentation of known and new results, as well as many applications to differential (ODE and PDE), difference, and integral equations. With this work the author fills a gap in the literature on inequalities, offering an ideal source for researchers in these topics.The present volume is part 1 of the author’s two-volume work on inequalities. Integral and discrete inequalities are a very important tool in classical analysis and play a crucial role in establishing the well-posedness of the related equations, i.e., differential, difference and integral equations.

Inhaltsangaben

Preface.- Linear One-Dimensional Continuous Integral Inequalities.- Linear One-Dimensional Discrete (Difference) Inequalities.- Linear One-Dimensional Discontinuous Integral Inequalities.- Applications of Linear One-Dimensional Inequalities.- Linear Multi-Dimensional Continuous Integral Inequalities.- Linear Multi-Dimensional Discrete (Difference) Inequalities.- Linear Multi-Dimensional Discontinuous Integral Inequalities.- Applications of Linear Multi-Dimensional Integral and Difference Inequalities.

Integral and Discrete Inequalities and Their Applications : 978-3-319-33303-8
Qin
Volume II: Nonlinear Inequalities
Springer International Publishing
Seiten XVI, 1072 p.
Erscheinungsjahr 2016
ISBN 978-3-319-33303-8
       9783319333038
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Real Functions

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This book concentrates on one- and multi-dimensional nonlinear integral and discrete Gronwall-Bellman type inequalities. It complements the author’s book on linear inequalities and serves as an essential tool for researchers interested in differential (ODE and PDE), difference, and integral equations.The present volume is part 2 of the author’s two-volume work on inequalities. Integral and discrete inequalities are a very important tool in classical analysis and play a crucial role in establishing the well-posedness of the related equations, i.e., differential, difference and integral equations.

Inhaltsangaben

Preface.- Nonlinear One-Dimensional Continuous Integral Inequalities.- Nonlinear One-Dimensional Discrete (Difference) Inequalities.- Nonlinear One-Dimensional Discontinuous Integral Inequalities.- Applications of Nonlinear One-Dimensional Continuous, Discontinuous Integral Inequalities and Discrete Inequalities.- Nonlinear Multi-Dimensional Continuous Integral Inequalities.- Nonlinear Multi-Dimensional Discrete (Difference) Inequalities.- Nonlinear Multi-Dimensional Discontinuous Inequalities.- Applications of Nonlinear Multi-Dimensional Continuous, Discontinuous Integral Inequalities and Discrete Inequalities.

Integral Operators in Non-Standard Function Spaces : 978-3-319-21014-8
Kokilashvili
Volume 1: Variable Exponent Lebesgue and Amalgam Spaces
Springer International Publishing
Seiten XX, 567 p.
Erscheinungsjahr 2016
ISBN 978-3-319-21014-8
       9783319210148
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Operator Theory

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This book, the result of the authors' long and fruitful collaboration, focuses on integral operators in new, non-standard function spaces and presents a systematic study of the boundedness and compactness properties of basic, harmonic analysis integral operators in the following function spaces, among others: variable exponent Lebesgue and amalgam spaces, variable Hölder spaces, variable exponent Campanato, Morrey and Herz spaces, Iwaniec-Sbordone (grand Lebesgue) spaces, grand variable exponent Lebesgue spaces unifying the two spaces mentioned above, grand Morrey spaces, generalized grand Morrey spaces, and weighted analogues of some of them.The results obtained are widely applied to non-linear PDEs, singular integrals and PDO theory. One of the book's most distinctive features is that the majority of the statements proved here are in the form of criteria.The book is intended for a broad audience, ranging from researchers in the area to experts in applied mathematics and prospective students.

Inhaltsangaben

Preface.- I: Variable Exponent Lebesgue and Amalgam spaces.- 1 Hardy Type Operators.- 2 Oscillating weights.- 3 Kernel Integral Operators.- 4 Two-Weight Estimates.- 5 One-sided Operators.- 6 Two-weight Inequalities for Fractional Maximal Functions.- 7 Hypersingular Integrals.- 8 Description of the Range of Potentials 213.- 9 More on Compactness.- 10 Applications to Singular Integral Equations.- II: Hölder Spaces of Variable Order.- 11 Variable Order Hölder Spaces.- III: Variable Exponent Morrey-Campanato and Herz Spaces.- 12 Morrey Type Spaces; Constant Exponents.- 13 Morrey Type Spaces; Variable Exponents.- Bibliography.- Symbol Index.- Subject Index.

Integral Operators in Non-Standard Function Spaces : 978-3-319-21017-9
Kokilashvili
Volume 2: Variable Exponent Hölder, Morrey–Campanato and Grand Spaces
Springer International Publishing
Seiten XXIII, 1003 p.
Erscheinungsjahr 2016
ISBN 978-3-319-21017-9
       9783319210179
 (Hardcover)
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Operator Theory

Information

This book, the result of the authors’ long and fruitful collaboration, focuses on integral operators in new, non-standard function spaces and presents a systematic study of the boundedness and compactness properties of basic, harmonic analysis integral operators in the following function spaces, among others: variable exponent Lebesgue and amalgam spaces, variable Hölder spaces, variable exponent Campanato, Morrey and Herz spaces, Iwaniec-Sbordone (grand Lebesgue) spaces, grand variable exponent Lebesgue spaces unifying the two spaces mentioned above, grand Morrey spaces, generalized grand Morrey spaces, and weighted analogues of some of them.The results obtained are widely applied to non-linear PDEs, singular integrals and PDO theory. One of the book’s most distinctive features is that the majority of the statements proved here are in the form of criteria.The book is intended for a broad audience, ranging from researchers in the area to experts in applied mathematics and prospective students.

Inhaltsangaben

IV: Grand Lebesgue Spaces.- 14 Maximal Functions and Potentials.- 15 Grand Lebesgue Spaces on Sets with Infinite Measure.- V: Grand Morrey Spaces.- 16 Maximal Functions, Fractional and Singular Integrals.- 17 Multiple Operators on the Cone of Decreasing Functions.- A: Grand Bochner Spaces.- Bibliography.- Symbol Index.- Subject Index.IV: Grand Lebesgue Spaces.- 14 Maximal Functions and Potentials.- 15 Grand Lebesgue Spaces on Sets with Infinite Measure.- V: Grand Morrey Spaces.- 16 Maximal Functions, Fractional and Singular Integrals.- 17 Multiple Operators on the Cone of Decreasing Functions.- A: Grand Bochner Spaces.- Bibliography.- Symbol Index.- Subject Index.

Interactive Theorem Proving and Program Development : 978-3-540-20854-9
Bertot
Coq’Art: The Calculus of Inductive Constructions
Springer Berlin Heidelberg
Seiten XXV, 472 p. 1 illus.
Erscheinungsjahr 2004
ISBN 978-3-540-20854-9
       9783540208549
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Coq is an interactive proof assistant for the development of mathematical theories and formally certified software. It is based on a theory called the calculus of inductive constructions, a variant of type theory. This book provides a pragmatic introduction to the development of proofs and certified programs using Coq. With its large collection of examples and exercises it is an invaluable tool for researchers, students, and engineers interested in formal methods and the development of zero-fault software.

Inhaltsangaben

1 A Brief Overview.- 2 Types and Expressions.- 3 Propositions and Proofs.- 4 Dependent Products, or Pandora’s Box.- 5 Everyday Logic.- 6 Inductive Data Types.- 7 Tactics and Automation.- 8 Inductive Predicates.- 9* Functions and Their Specifications.- 10 * Extraction and Imperative Programming.- 11 * A Case Study.- 12 * The Module System.- 13 ** Infinite Objects and Proofs.- 14 ** Foundations of Inductive Types.- 15 * General Recursion.- 16 * Proof by Reflection.- Insertion Sort.- References.- Coq and Its Libraries.- Examples from the Book.

Introduction au traitement mathématique des images - méthodes déterministes : 978-3-662-46538-7
Bergounioux
Springer Berlin Heidelberg
Seiten XVII, 241 p. 168 ill.
Erscheinungsjahr 2015
ISBN 978-3-662-46538-7
       9783662465387
 (Softcover)
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Ce cours est une introduction au traitement d'image mathématique déterministe. Les principales problématiques en traitement et analyse d’image y sont présentées: débruitage/filtrage/restauration, segmentation, rehaussement/défloutage, ainsi qu’un aperçu de quelques techniques d’acquisition. Les méthodes mathématiques utilisées ont essentiellement déterministes : transformation  de Fourier, ondelettes,  équations aux dérivées partielles, morphologie mathématique et méthodes variationnelles. Quelques applications y sont brièvement présentées pour illustrer le propos : la stéganographie, la compression et l’inpainting (ou désocclusion). Le livre comprend également  un rappel des principales notions mathématiques utilisées (il se veut auto-suffisant) et la bibliographie abondante doit permettre au lecteur d’approfondir les techniques qui l’intéressent. Cet ouvrage s’adresse à des étudiants de MASTER, élèves-ingénieurs ou chercheurs désireux  de comprendre ou d’approfondir les techniques mathématiques  de base  en traitement et analyse d’image.This course is an introduction to deterministic mathematical image processing. The main issues in processing and image analysis are presented: denoising, filtering, restoration, segmentation, enhancement and deblurring.There is also an overview of some acquisition techniques. Mathematical methods are essentially deterministic: Fourier transform, wavelets, partial differential equations, mathematical morphology and variational methods. Some applications are briefly presented to illustrate the topic, such as steganography, compression and inpainting. This self-contained book also includes a recap of the basic mathematical concepts used, and the extensive bibliography will enable readers to develop their skills.This book is intended for masters students, engineering students and researchers wanting to comprehend or deepen  their understanding of thebasic mathematical techniques in processing and image analysis.

Inhaltsangaben

Introduction.- Traitement ponctuel des images numériques.- Débruitage par fitrage linéaire.- Débruitage par méthodes non linéaires.- Segmentation.- Morphologie mathématique.- Applications.- Quelques outils mathématiques.- Récapitulatif des différents procédés de traitement.- Littérature.- Index. 

Introduction to Calculus and Analysis I : 978-3-540-65058-4
Courant
Springer Berlin Heidelberg
Seiten XXIII, 661 p. 184 illus.
Erscheinungsjahr 1999
ISBN 978-3-540-65058-4
       9783540650584
 (Softcover)
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Real Functions

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From the reviews: 'Volume 1 covers a basic course in real analysis of one variable and Fourier series. It is well-illustrated, well-motivated and very well-provided with a multitude of unusually useful and accessible exercises. (...) There are three aspects of Courant and John in which it outshines (some) contemporaries: (i) the extensive historical references, (ii) the chapter on numerical methods, and (iii) the two chapters on physics and geometry. The exercises in Courant and John are put together purposefully, and either look numerically interesting, or are intuitively significant, or lead to applications. It is the best text known to the reviewer for anyone trying to make an analysis course less abstract. (...)' The Mathematical Gazette (75.1991.471)

Inhaltsangaben

1 Introduction.- 2 The Fundamental Ideas of the Integral and Differential Calculus.- 3 The Techniques of Calculus.- 4 Applications in Physics and Geometry.- 5 Taylor’s Expansion.- 6 Numerical Methods.- 7 Infinite Sums and Products.- 8 Trigonometric Series.- 9 Differential Equations for the Simplest Types of Vibration.- List of Biographical Dates.

Introduction to Calculus and Analysis II/1 : 978-3-540-66569-4
Courant
Springer Berlin Heidelberg
Seiten XXV, 556 p. 157 illus.
Erscheinungsjahr 2000
ISBN 978-3-540-66569-4
       9783540665694
 (Softcover)
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Fachgebiet

Analysis

Information

From the reviews: 'These books (Introduction to Calculus and Analysis Vol. I/II) are very well written. The mathematics are rigorous but the many examples that are given and the applications that are treated make the books extremely readable and the arguments easy to understand. These books are ideally suited for an undergraduate calculus course. Each chapter is followed by a number of interesting exercises. More difficult parts are marked with an asterisk. There are many illuminating figures...Of interest to students, mathematicians, scientists and engineers. Even more than that.' Newsletter on Computational and Applied Mathematics, 1991 '...one of the best textbooks introducing several generations of mathematicians to higher mathematics. ... This excellent book is highly recommended both to instructors and students.' Acta Scientiarum Mathematicarum, 1991

Inhaltsangaben

1 Functions of Several Variables and Their Derivatives.- 1.1 Points and Points Sets in the Plane and in Space.- 1.2 Functions of Several Independent Variables.- 1.3 Continuity.- 1.4 The Partial Derivatives of a Function.- 1.5 The Differential of a Function and Its Geometrical Meaning.- 1.6 Functions of Functions (Compound Functions) and the Introduction of New In-dependent Variables.- 1.7 The Mean Value Theorem and Taylor’s Theorem for Functions of Several Variables.- 1.8 Integrals of a Function Depending on a Parameter.- 1.9 Differentials and Line Integrals.- 1.10 The Fundamental Theorem on Integrability of Linear Differential Forms.- Appendix A.1. The Principle of the Point of Accumulation in Several Dimensions and Its Applications.- A.2. Basic Properties of Continuous Functions.- A.3. Basic Notions of the Theory of Point Sets.- A.4. Homogeneous functions..- 2 Vectors, Matrices, Linear Transformations.- 2.1 Operations with Vectors.- 2.2 Matrices and Linear Transformations.- 2.3 Determinants.- 2.4 Geometrical Interpretation of Determinants.- 2.5 Vector Notions in Analysis.- 3 Developments and Applications of the Differential Calculus.- 3.1 Implicit Functions.- 3.2 Curves and Surfaces in Implicit Form.- 3.3 Systems of Functions, Transformations, and Mappings.- 3.4 Applications.- 3.5 Families of Curves, Families of Surfaces, and Their Envelopes.- 3.6 Alternating Differential Forms.- 3.7 Maxima and Minima.- Appendix A.1 Sufficient Conditions for Extreme Values.- A.2 Numbers of Critical Points Related to Indices of a Vector Field.- A.3 Singular Points of Plane Curves 360 A.4 Singular Points of Surfaces.- A.5 Connection Between Euler’s and Lagrange’s Representation of the motion of a Fluid.- A.6 Tangential Representation of a Closed Curve and the Isoperi-metric Inequality.- 4 Multiple Integrals.- 4.1 Areas in the Plane.- 4.2 Double Integrals.- 4.3 Integrals over Regions in three and more Dimensions.- 4.4 Space Differentiation. Mass and Density.- 4.5 Reduction of the Multiple Integral to Repeated Single Integrals.- 4.6 Transformation of Multiple Integrals.- 4.7 Improper Multiple Integrals.- 4.8 Geometrical Applications.- 4.9 Physical Applications.- 4.10 Multiple Integrals in Curvilinear Coordinates.- 4.11 Volumes and Surface Areas in Any Number of Dimensions.- 4.12 Improper Single Integrals as Functions of a Parameter.- 4.13 The Fourier Integral.- 4.14 The Eulerian Integrals (Gamma Function).- Appendix: Detailed Analysis of the Process Of Integration A.1 Area.- A.2 Integrals of Functions of Several Variables.- A.3 Transformation of Areas and Integrals.- A.4 Note on the Definition of the Area of a Curved Surface.- 5 Relations Between Surface and Volume Integrals.- 5.1 Connection Between Line Integrals and Double Integrals in the Plane (The Integral Theorems of Gauss, Stokes, and Green).- 5.2 Vector Form of the Divergence Theorem. Stokes’s Theorem.- 5.3 Formula for Integration by Parts in Two Dimensions. Green’s Theorem.- 5.4 The Divergence Theorem Applied to the Transformation of Double Integrals.- 5.5 Area Differentiation. Transformation of Au to Polar Coordinates.- 5.6 Interpretation of the Formulae of Gauss and Stokes by Two-Dimensional Flows.- 5.7 Orientation of Surfaces.- 5.8 Integrals of Differential Forms and of Scalars over Surfaces.- 5.9 Gauss’s and Green’s Theorems in Space.- 5.10 Stokes’s Theorem in Space.- 5.11 Integral Identities in Higher Dimensions.- Appendix: General Theory Of Surfaces And Of Surface Integals A.I Surfaces and Surface Integrals in Three dimensions.- A.2 The Divergence Theorem.- A.3 Stokes’s Theorem.- A.4 Surfaces and Surface Integrals in Euclidean Spaces of Higher Dimensions.- A.5 Integrals over Simple Surfaces, Gauss’s Divergence Theorem, and the General Stokes Formula in Higher Dimensions.- 6 Differential Equations.- 6.1 The Differential Equations for the Motion of a Particle in Three Dimensions.- 6.2

Introduction to Calculus and Analysis II/2 : 978-3-540-66570-0
Courant
Chapters 5 - 8
Springer Berlin Heidelberg
Seiten XXV, 412 p. 68 illus.
Erscheinungsjahr 2000
ISBN 978-3-540-66570-0
       9783540665700
 (Softcover)
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Fachgebiet

Analysis

Information

From the reviews: 'These books (Introduction to Calculus and Analysis Vol. I/II) are very well written. The mathematics are rigorous but the many examples that are given and the applications that are treated make the books extremely readable and the arguments easy to understand. These books are ideally suited for an undergraduate calculus course. Each chapter is followed by a number of interesting exercises. More difficult parts are marked with an asterisk. There are many illuminating figures...Of interest to students, mathematicians, scientists and engineers. Even more than that.' Newsletter on Computational and Applied Mathematics, 1991 '...one of the best textbooks introducing several generations of mathematicians to higher mathematics. ... This excellent book is highly recommended both to instructors and students. Acta Scientiarum Mathematicarum, 1991

Inhaltsangaben

5 Relations Between Surface and Volume Integrals.- 6 Differential Equations.- 7 Calculus of VariationsMultipliers.- 8 Functions of a Complex Variable.- List of Biographical Dates.- Index.

Introduction to Calculus and Classical Analysis : 978-3-319-28399-9
Hijab
Springer International Publishing
Seiten XIII, 427 p. 69 illus., 1 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-28399-9
       9783319283999
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Involving rigorous analysis, computational dexterity, and a breadth of applications, this text is ideal for an undergraduate honors calculus course or for an introduction to analysis. This fourth 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;• self-contained treatment of the fundamental theorems of calculus in the general case using the Sunrise Lemma• There are 450 problems with all the solutions at the back of the text.Reviews from previous editions:'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 Newspaper'Chapter 5 is... an astonishing tour de force'—Steven G. Krantz, The American Mathematical Monthly'For a treatment [of infinite products and the Bernoulli series] that is very close to Euler’s and even more elementary...' — V.S. Varadarajan, Bulletin of the American Mathematical Society

Inhaltsangaben

Preface.- A Note to the Reader.- 1. The Set of Real Numbers.- 2. Continuity.- 3. Differentiation.-4. Integration.- 5. Applications.- 6. Generalizations.- A. Solutions.- References.- Index.

Introduction to Global Optimization Exploiting Space-Filling Curves : 978-1-4614-8041-9
Sergeyev
Springer New York
Seiten X, 125 p. 32 illus., 30 illus. in color.
Erscheinungsjahr 2013
ISBN 978-1-4614-8041-9
       9781461480419
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Introduction to Global Optimization Exploiting Space-Filling Curves provides an overview of classical and new results pertaining to the usage of space-filling curves in global optimization. The authors look at a family of derivative-free numerical algorithms applying space-filling curves to reduce the dimensionality of the global optimization problem; along with a number of unconventional ideas, such as adaptive strategies for estimating Lipschitz constant, balancing global and local information to accelerate the search. Convergence conditions of the described algorithms are studied in depth and theoretical considerations are illustrated through numerical examples. This work also contains a code for implementing space-filling curves that can be used for constructing new global optimization algorithms. Basic ideas from this text can be applied to a number of problems including problems with multiextremal and partially defined constraints and non-redundant parallel computations can be organized. Professors, students, researchers, engineers, and other professionals in the fields of pure mathematics, nonlinear sciences studying fractals, operations research, management science, industrial and applied mathematics, computer science, engineering, economics, and the environmental sciences will find this title useful . ?

Inhaltsangaben

1. Introduction.- 2. Approximations to Peano curves.- 3. Global optimization algorithms using curves to reduce dimensionality of the problem.- 4. Ideas for acceleration.- 5. A brief conclusion.- References.

Introduction to Lie Algebras : 978-1-84628-040-5
Erdmann
Springer London
Seiten XII, 251 p.
Erscheinungsjahr 2006
ISBN 978-1-84628-040-5
       9781846280405
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Algebra

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Lie groups and Lie algebras have become essential to many parts of mathematics and theoretical physics, with Lie algebras a central object of interest in their own right. This book provides an elementary introduction to Lie algebras based on a lecture course given to fourth-year undergraduates. The only prerequisite is some linear algebra and an appendix summarizes the main facts that are needed. The treatment is kept as simple as possible with no attempt at full generality. Numerous worked examples and exercises are provided to test understanding, along with more demanding problems, several of which have solutions. Introduction to Lie Algebras covers the core material required for almost all other work in Lie theory and provides a self-study guide suitable for undergraduate students in their final year and graduate students and researchers in mathematics and theoretical physics.

Inhaltsangaben

Ideals and Homomorphisms.- Low-Dimensional Lie Algebras.- Solvable Lie Algebras and a Rough Classification.- Subalgebras of gl(V).- Engel’s Theorem and Lie’s Theorem.- Some Representation Theory.- Representations of sl(2, C).- Cartan’s Criteria.- The Root Space Decomposition.- Root Systems.- The Classical Lie Algebras.- The Classification of Root Systems.- Simple Lie Algebras.- Further Directions.- Appendix A: Linear Algebra.- Appendix B: Weyl’s Theorem.- Appendix C: Cartan Subalgebras.- Appendix D: Weyl Groups.- Appendix E: Answers to Selected Exercises.

Introduction to Mathematical Biology : 978-3-319-29636-4
Chou
Modeling, Analysis, and Simulations
Springer International Publishing
Seiten VII, 172 p. 49 illus., 38 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-29636-4
       9783319296364
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This book is based on a one semester course that the authors have been teaching for several years, and includes two sets of case studies. The first includes chemostat models, predator-prey interaction, competition among species, the spread of infectious diseases, and oscillations arising from bifurcations. In developing these topics, readers will also be introduced to the basic theory of ordinary differential equations, and how to work with MATLAB without having any prior programming experience. The second set of case studies were adapted from recent and current research papers to the level of the students. Topics have been selected based on public health interest. This includes the risk of atherosclerosis associated with high cholesterol levels, cancer and immune interactions, cancer therapy, and tuberculosis. Readers will experience how mathematical models and their numerical simulations can provide explanations that guide biological and biomedical research. Considered to be the undergraduate companion to the more advanced book 'Mathematical Modeling of Biological Processes' (A. Friedman, C.-Y. Kao, Springer – 2014), this book is geared towards undergraduate students with little background in mathematics and no biological background.

Inhaltsangaben

Introduction.- Bacterial Growth in Chemostat.- System of Two Linear Differential Equations.- System of Two Differential Equations.- Predator-Prey Models.- Two Competing Populations.- General Systems of Differential Equations.- The Chemostat Model Revisited.- Spread of Disease.- Enzyme Dynamics.- Bifurcation Theory.- Atherosclerosis: The Risk of High Cholesterol.- Cancer-Immune Interaction. Cancer Therapy.- Tuberculosis.- Solutions.

Introduction to Probability with Statistical Applications : 978-3-319-30618-6
Schay
Springer International Publishing
Seiten XII, 385 p. 49 illus.
Erscheinungsjahr 2016
ISBN 978-3-319-30618-6
       9783319306186
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Now in its second edition, this textbook serves as an introduction to probability and statistics for non-mathematics majors who do not need the exhaustive detail and mathematical depth provided in more comprehensive treatments of the subject. The presentation covers the mathematical laws of random phenomena, including discrete and continuous random variables, expectation and variance, and common probability distributions such as the binomial, Poisson, and normal distributions. More classical examples such as Montmort's problem, the ballot problem, and Bertrand’s paradox are now included, along with applications such as the Maxwell-Boltzmann and Bose-Einstein distributions in physics.Key features in new edition:* 35 new exercises* Expanded section on the algebra of sets * Expanded chapters on probabilities to include more classical examples* New section on regression* Online instructors' manual containing solutions to all exercises

Inhaltsangaben

Introduction.- The Algebra of Events.- Combinatorial Problems.- Probabilities.- Random Variables.- Expectation, Variance, Moments.- Some Special Distributions.- The Elements of Mathematical Statistics.

Introduction to Smooth Manifolds : 978-1-4419-9981-8
Lee
Springer New York
Seiten XVI, 708 p.
Erscheinungsjahr 2012
ISBN 978-1-4419-9981-8
       9781441999818
 (Hardcover)
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This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer.This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard’s theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.

Inhaltsangaben

Preface.- 1 Smooth Manifolds.- 2 Smooth Maps.- 3 Tangent Vectors.- 4 Submersions, Immersions, and Embeddings.- 5 Submanifolds.- 6 Sard's Theorem.- 7 Lie Groups.- 8 Vector Fields.- 9 Integral Curves and Flows.- 10 Vector Bundles.- 11 The Cotangent Bundle.- 12 Tensors.- 13 Riemannian Metrics.- 14 Differential Forms.- 15 Orientations.- 16 Integration on Manifolds.- 17 De Rham Cohomology.- 18 The de Rham Theorem.- 19 Distributions and Foliations.- 20 The Exponential Map.- 21 Quotient Manifolds.-  22 Symplectic Manifolds.- Appendix A: Review of Topology.- Appendix B: Review of Linear Algebra.- Appendix C: Review of Calculus.- Appendix D: Review of Differential Equations.- References.- Notation Index.- Subject Index.

Introduction to Topological Manifolds : 978-1-4419-7939-1
Lee
Springer New York
Seiten XVII, 433 p.
Erscheinungsjahr 2011
ISBN 978-1-4419-7939-1
       9781441979391
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This book is an introduction to manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition.Although this second edition has the same basic structure as the first edition, it has been extensively revised and clarified; not a single page has been left untouched. The major changes include a new introduction to CW complexes (replacing most of the material on simplicial complexes in Chapter 5); expanded treatments of manifolds with boundary, local compactness, group actions, and proper maps; and a new section on paracompactness.This text is designed to be used for an introductory graduate course on the geometry and topology of manifolds. It should be accessible to any student who has completed a solid undergraduate degree in mathematics. The author’s book Introduction to Smooth Manifolds is meant to act as a sequel to this book.

Inhaltsangaben

Preface.- 1 Introduction.- 2 Topological Spaces.- 3 New Spaces from Old.- 4 Connectedness and Compactness.- 5 Cell Complexes.- 6 Compact Surfaces.- 7 Homotopy and the Fundamental Group.- 8 The Circle.- 9 Some Group Theory.- 10 The Seifert-Van Kampen Theorem.- 11 Covering Maps.- 12 Group Actions and Covering Maps.- 13 Homology.- Appendix A: Review of Set Theory.- Appendix B: Review of Metric Spaces.- Appendix C: Review of Group Theory.- References.- Notation Index.- Subject Index.

Introductory Lectures on Convex Optimization : 978-1-4613-4691-3
Nesterov
A Basic Course
Springer US
Seiten XVIII, 236 p.
Erscheinungsjahr 2004
ISBN 978-1-4613-4691-3
       9781461346913
 (Softcover)
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Optimization

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It was in the middle of the 1980s, when the seminal paper by Kar­ markar opened a new epoch in nonlinear optimization. The importance of this paper, containing a new polynomial-time algorithm for linear op­ timization problems, was not only in its complexity bound. At that time, the most surprising feature of this algorithm was that the theoretical pre­ diction of its high efficiency was supported by excellent computational results. This unusual fact dramatically changed the style and direc­ tions of the research in nonlinear optimization. Thereafter it became more and more common that the new methods were provided with a complexity analysis, which was considered a better justification of their efficiency than computational experiments. In a new rapidly develop­ ing field, which got the name 'polynomial-time interior-point methods', such a justification was obligatory. Afteralmost fifteen years of intensive research, the main results of this development started to appear in monographs [12, 14, 16, 17, 18, 19]. Approximately at that time the author was asked to prepare a new course on nonlinear optimization for graduate students. The idea was to create a course which would reflect the new developments in the field. Actually, this was a major challenge. At the time only the theory of interior-point methods for linear optimization was polished enough to be explained to students. The general theory of self-concordant functions had appeared in print only once in the form of research monograph [12].

Inhaltsangaben


Inverse Problems in Ordinary Differential Equations and Applications : 978-3-319-26337-3
Llibre
Springer International Publishing
Seiten XII, 266 p. 9 illus., 8 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-26337-3
       9783319263373
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This book is dedicated to study the inverse problem of ordinary differential equations, that is it focuses in finding all ordinary differential equations that satisfy a given set of properties. The Nambu bracket is the central tool in developing this approach. The authors start characterizing the ordinary differential equations in R^N which have a given set of partial integrals or first integrals. The results obtained are applied first to planar polynomial differential systems with a given set of such integrals, second to solve the 16th Hilbert problem restricted to generic algebraic limit cycles, third for solving the inverse problem for constrained Lagrangian and Hamiltonian mechanical systems, fourth for studying the integrability of a constrained rigid body. Finally the authors conclude with an analysis on nonholonomic mechanics, a generalization of the Hamiltonian principle, and the statement an solution of the inverse problem in vakonomic mechanics.

Inhaltsangaben

Preface.- 1.Differential Equations with Given Partial and First Integrals.- 2.Polynomial Vector Fields with Given Partial and First Integrals.- 3.16th Hilbert Problem for Algebraic Limit Cycles.- 4.Inverse Problem for Constrained Lagrangian Systems.- 5.Inverse Problem for Constrained Hamiltonian Systems.- 6.Integrability of the Constrained Rigid Body.- 7.Inverse Problem in the Vakonomic Mechanics.- Index.- Bibliography.

Iterative Solution of Large Sparse Systems of Equations : 978-3-319-28481-1
Hackbusch
Springer International Publishing
Seiten XXIII, 509 p. 26 illus., 11 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-28481-1
       9783319284811
 (Hardcover)
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Numerical Analysis

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In the second edition of this classic monograph, complete with four new chapters and updated references, readers will now have access to content describing and analysing classical and modern methods with emphasis on the algebraic structure of linear iteration, which is usually ignored in other literature. The necessary amount of work increases dramatically with the size of systems, so one has to search for algorithms that most efficiently and accurately solve systems of, e.g., several million equations. The choice of algorithms depends on the special properties the matrices in practice have. An important class of large systems arises from the discretization of partial differential equations. In this case, the matrices are sparse (i.e., they contain mostly zeroes) and well-suited to iterative algorithms. The first edition of this book grew out of a series of lectures given by the author at the Christian-Albrecht University of Kiel to students of mathematics. The second edition includes quite novel approaches.

Inhaltsangaben

Part I: Linear Iterations.- Introduction.- Iterative Methods.- Classical Linear Iterations in the Positive Definite Case.- Analysis of Classical Iterations Under Special Structural Conditions.- Algebra of Linear Iterations.- Analysis of Positive Definite Iterations.- Generation of Iterations. Part II: Semi-Iterations and Krylov Methods.- Semi-Iterative Methods.- Gradient Methods.- Conjugate Gradient Methods and Generalizations.- Part III: Special Iterations.- Multigrid Iterations.- Domain Decomposition and Subspace Methods.- H-LU Iteration.- Tensor-based Methods.- Appendices.

L'analyse au fil de l'histoire : 978-3-540-67463-4
Hairer
Springer Berlin Heidelberg
Seiten X, 372 p. 121 ill., 1 ill. en couleurs.
Erscheinungsjahr 2001
ISBN 978-3-540-67463-4
       9783540674634
 (Softcover)
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Analysis

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Cet ouvrage traite des théorèmes fondamentaux de l'analyse (convergence, continuité, calcul différentiel et intégral à une et plusieurs variables) avec grand soin pédagogique, une centaine de dessins, d'exemples et de contre-exemples. Pour une meilleure compréhension du sujet, il commence avec des calculs anciens de problèmes géometriques et mécaniques, qui ont conduit aux séries infinies, dérivées, intégrales, équations différentielles. Ainsi ce volume nous fait découvrir 'l'analyse au fil de l'histoire'. De nombreuses motivations et remarques historiques enrichissent le texte.

Inhaltsangaben

Introduction à l'analyse infinitésimale.- Calcul différentiel et intégral.- Les fondements de l'analyse classique.- Calcul différentiel et intégral à plusieurs variables.- Bibliographie.- Index des notations.- Index

Lattice Theory: Special Topics and Applications : 978-3-319-44235-8
Grätzer
Volume 2
Springer International Publishing
Seiten XV, 616 p. 136 illus.
Erscheinungsjahr 2016
ISBN 978-3-319-44235-8
       9783319442358
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George Grätzer's Lattice Theory: Foundation is his third book on lattice theory (General Lattice Theory, 1978, second edition, 1998). In 2009, Grätzer considered updating the second edition to reflect some exciting and deep developments. He soon realized that to lay the foundation, to survey the contemporary field, to pose research problems, would require more than one volume and more than one person. So Lattice Theory: Foundation provided the foundation. Now we complete this project with Lattice Theory: Special Topics and Applications, in two volumes, written by a distinguished group of experts, to cover some of the vast areas not in Foundation. This second volume is divided into ten chapters contributed by K. Adaricheva, N. Caspard, R. Freese, P. Jipsen, J.B. Nation, N. Reading, H. Rose, L. Santocanale, and F. Wehrung.

Inhaltsangaben

Varieties of Lattices.- Free and Finitely Presented Lattices.- Classes of Semidistributive Lattices.- Lattices of Algebraic Subsets and Implicational Classes.- Convex Geometries.- Bases of Closure Systems.- Permutohedra and Associahedra.- Generalizations of the Permutohedron.- Lattice Theory of the Poset of Regions.- Finite Coxeter Groups and the Weak Order.

Lectures on Functional Analysis and the Lebesgue Integral : 978-1-4471-6810-2
Komornik
Springer London
Seiten XX, 403 p. 46 illus.
Erscheinungsjahr 2016
ISBN 978-1-4471-6810-2
       9781447168102
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Functional Analysis

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This textbook, based on three series of lectures held by the author at the University of Strasbourg, presents functional analysis in a non-traditional way by generalizing elementary theorems of plane geometry to spaces of arbitrary dimension. This approach leads naturally to the basic notions and theorems. Most results are illustrated by the small ?p spaces. The Lebesgue integral, meanwhile, is treated via the direct approach of Frigyes Riesz, whose constructive definition of measurable functions leads to optimal, clear-cut versions of the classical theorems of Fubini-Tonelli and Radon-Nikodým. Lectures on Functional Analysis and the Lebesgue Integral presents the most important topics for students, with short, elegant proofs. The exposition style follows the Hungarian mathematical tradition of Paul Erd?s and others. The order of the first two parts, functional analysis and the Lebesgue integral, may be reversed. In the third and final part they are combined to study various spaces of continuous and integrable functions. Several beautiful, but almost forgotten, classical theorems are also included. Both undergraduate and graduate students in pure and applied mathematics, physics and engineering will find this textbook useful. Only basic topological notions and results are used and various simple but pertinent examples and exercises illustrate the usefulness and optimality of most theorems. Many of these examples are new or difficult to localize in the literature, and the original sources of most notions and results are indicated to help the reader understand the genesis and development of the field.

Inhaltsangaben

Some papers of general interest.- Topological prerequisites.- Part 1 Functional analysis.- Hilbert spaces.- Banach spaces.- Locally convex spaces.- Part 2 The Lebesgue integral.- Monotone functions.- The Lebesgue integral in R.- Generalized Newton-Leibniz formula.- Integrals on measure spaces.- Part 3 Function spaces.- Spaces of continuous functions.- Spaces of integrable functions.- Almost everywhere convergence.- Hints and solutions to some exercises.- Bibliography.- Teaching remarks.- Subject index.- Name index.


Linear Algebraic Groups : 978-0-8176-4839-8
Springer
Birkhäuser Boston
Seiten XII, 334 p.
Erscheinungsjahr 1998
ISBN 978-0-8176-4839-8
       9780817648398
 (Softcover)
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The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. The second edition, thoroughly revised and expanded, extends the theory over arbitrary fields, which are not necessarily algebraically closed. It thus represents a higher aim. As in the first edition, the book includes a self-contained treatment of the prerequisites from algebraic geometry and commutative algebra, as well as basic results on reductive groups. As a result, the first part of the book can well serve as a text for an introductory graduate course on linear algebraic groups.

Inhaltsangaben

Some Algebraic Geometry.- Linear Algebraic Groups, First Properties.- Commutative Algebraic Groups.- Derivations, Differentials, Lie Algebras.- Topological Properties of Morphisms, Applications.- Parabolic Subgroups, Borel Subgroups, Solvable Groups.- Weyl Group, Roots, Root Datum.- Reductive Groups.- The Isomorphism Theorem.- The Existence Theorem.- More Algebraic Geometry.- F-groups: General Results.- F-tori.- Solvable F-groups.- Freductive Groups.- Reductive F-Groups.- Classification.

Linear Functional Analysis : 978-1-4471-7279-6
Alt
An Application-Oriented Introduction
Springer London
Seiten XII, 435 p. 19 illus.
Erscheinungsjahr 2016
ISBN 978-1-4471-7279-6
       9781447172796
 (Softcover)
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Functional Analysis

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This book gives an introduction to Linear Functional Analysis, which is a synthesis of algebra, topology, and analysis. In addition to the basic theory it explains operator theory, distributions, Sobolev spaces, and many other things. The text is self-contained and includes all proofs, as well as many exercises, most of them with solutions. Moreover, there are a number of appendices, for example on Lebesgue integration theory. A complete introduction to the subject, Linear Functional Analysis will be particularly useful to readers who want to quickly get to the key statements and who are interested in applications to differential equations.

Inhaltsangaben

Introduction.- Preliminaries.- Function spaces.- Subsets of function spaces.- Linear operators.- Linear functionals.- Uniform boundedness principle.- Weak convergence.- Finite-dimensional approximation.- Compact operators.- Spectrum of compact operators.- Self-adjoint operators.- References.- Symbols.- Index.

Linear Functional Analysis : 978-1-84800-004-9
Rynne
Springer London
Seiten X, 324 p. 6 illus.
Erscheinungsjahr 2008
ISBN 978-1-84800-004-9
       9781848000049
 (Softcover)
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Functional Analysis

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This book provides an introduction to the ideas and methods of linear fu- tional analysis at a level appropriate to the ?nal year of an undergraduate course at a British university. The prerequisites for reading it are a standard undergraduate knowledge of linear algebra and real analysis (including the t- ory of metric spaces). Part of the development of functional analysis can be traced to attempts to ?nd a suitable framework in which to discuss di?erential and integral equations. Often, the appropriate setting turned out to be a vector space of real or complex-valued functions de?ned on some set. In general, such a v- tor space is in?nite-dimensional. This leads to di?culties in that, although many of the elementary properties of ?nite-dimensional vector spaces hold in in?nite-dimensional vector spaces, many others do not. For example, in general in?nite-dimensionalvectorspacesthereisnoframeworkinwhichtomakesense of analytic concepts such as convergence and continuity. Nevertheless, on the spaces of most interest to us there is often a norm (which extends the idea of the length of a vector to a somewhat more abstract setting). Since a norm on a vector space gives rise to a metric on the space, it is now possible to do analysis in the space. As real or complex-valued functions are often called functionals, the term functional analysis came to be used for this topic. We now brie?y outline the contents of the book.

Inhaltsangaben

Preliminaries.- Normed Spaces.- Inner Product Spaces, Hilbert Spaces.- Linear Operators.- Duality and the Hahn—Banach Theorem.- Linear Operators on Hilbert Spaces.- Compact Operators.- Integral and Differential Equations.- Solutions to Exercises.

Linear Functional Analysis for Scientists and Engineers : 978-981-10-0970-9
Limaye
Springer Singapore
Seiten XIV, 254 p. 15 illus.
Erscheinungsjahr 2016
ISBN 978-981-10-0970-9
       9789811009709
 (Hardcover)
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Functional Analysis

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This book provides a concise and meticulous introduction to functional analysis. Since the topic draws heavily on the interplay between the algebraic structure of a linear space and the distance structure of a metric space, functional analysis is increasingly gaining the attention of not only mathematicians but also scientists and engineers. The purpose of the text is to present the basic aspects of functional analysis to this varied audience, keeping in mind the considerations of applicability. A novelty of this book is the inclusion of a result by Zabreiko, which states that every countably subadditive seminorm on a Banach space is continuous. Several major theorems in functional analysis are easy consequences of this result. The entire book can be used as a textbook for an introductory course in functional analysis without having to make any specific selection from the topics presented here. Basic notions in the setting of a metric space are defined in terms of sequences. These include total boundedness, compactness, continuity and uniform continuity. Offering concise and to-the-point treatment of each topic in the framework of a normed space and of an inner product space, the book represents a valuable resource for advanced undergraduate students in mathematics, and will also appeal to graduate students and faculty in the natural sciences and engineering. The book is accessible to anyone who is familiar with linear algebra and real analysis.

Inhaltsangaben

Chapter 1. Preliminaries.- Chapter 2. Basic Framework.- Chapter 3. Bounded Linear Maps.- Chapter 4. Dual Spaces, Transposes and Adjoints.- Chapter 5. Spectral Theory.

Linear Partial Differential Equations for Scientists and Engineers : 978-0-8176-4393-5
Myint-U
Birkhäuser Boston
Seiten XXII, 778 p.
Erscheinungsjahr 2007
ISBN 978-0-8176-4393-5
       9780817643935
 (Hardcover)
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Analysis

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One of the most fundamental and active areas in mathematics, the theory of partial differential equations (PDEs) is essential in the modeling of natural phenomena. PDEs have a wide range of interesting and important applications in every branch of applied mathematics, physics, and engineering, including fluid dynamics, elasticity, and optics. This significantly expanded fourth edition is designed as an introduction to the theory and applications of linear PDEs. The authors provide fundamental concepts, underlying principles, a wide range of applications, and various methods of solutions to PDEs. In addition to essential standard material on the subject, the book contains new material that is not usually covered in similar texts and reference books, including conservation laws, the spherical wave equation, the cylindrical wave equation, higher-dimensional boundary-value problems, the finite element method, fractional partial differential equations, and nonlinear partial differential equations with applications. Linear Partial Differential Equations for Scientists and Engineers, Fourth Edition will primarily serve as a textbook for the first two courses in PDEs, or in a course on advanced engineering mathematics. The book may also be used as a reference for graduate students, researchers, and professionals in modern applied mathematics, mathematical physics, and engineering. Readers will gain a solid mathematical background in PDEs, sufficient to start interdisciplinary collaborative research in a variety of fields. Also by L. Debnath: Nonlinear Partial Differential Equations for Scientists and Engineers, Second Edition, ISBN 0-8176-4323-0.

Inhaltsangaben

First-Order, Quasi-Linear Equations and Method of Characteristics.- Mathematical Models.- Classification of Second-Order Linear Equations.- The Cauchy Problem and Wave Equations.- Fourier Series and Integrals with Applications.- Method of Separation of Variables.- Eigenvalue Problems and Special Functions.- Boundary-Value Problems and Applications.- Higher-Dimensional Boundary-Value Problems.- Green’s Functions and Boundary-Value Problems.- Integral Transform Methods with Applications.- Nonlinear Partial Differential Equations with Applications.- Numerical and Approximation Methods.- Tables of Integral Transforms.

Local Features in Natural Images via Singularity Theory : 978-3-319-41470-6
Damon
Springer International Publishing
Seiten X, 255 p. 107 illus., 50 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-41470-6
       9783319414706
 (Softcover)
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This monograph considers a basic problem in the computer analysis of natural images, which are images of scenes involving multiple objects that are obtained by a camera lens or a viewer’s eye. The goal is to detect geometric features of objects in the image and to separate regions of the objects with distinct visual properties. When the scene is illuminated by a single principal light source, we further include the visual clues resulting from the interaction of the geometric features of objects, the shade/shadow regions on the objects, and the “apparent contours”. We do so by a mathematical analysis using a repertoire of methods in singularity theory. This is applied for generic light directions of both the “stable configurations” for these interactions, whose features remain unchanged under small viewer movement, and the generic changes which occur under changes of view directions. These may then be used to differentiate between objects and determine their shapes and positions.

Inhaltsangaben

Introduction.- Overview.- Part I-Mathematical Basis for Analysis of Feature-Shade/Shadow- Contours.- Abstract Classification of Singularities Preserving Features.- Singularity Equivalence Groups Capturing Interactions.- Methods for Classification of Singularities.- Methods for Topological Classification of Singularities.- Part II-The Classification of Interactions Involving Feature– Shade/Shadow–Contours.- Stratifications of Generically Illuminated Surfaces with Geometric Features.- Realizations of Abstract Mappings Representing Projection Singularities.- Statements of the Main Classification Results.- Part III-Classifications of Interactions of Pairs of Feature– Shade/Shadow–Contours.- Stable View Projections and Transitions involving Shade/Shadow Curves on a Smooth Surface (SC).- Transitions involving Views of Geometric Features (FC).- Part IV-Classifications of Multiple Interactions.- Transitions involving Geometric Features and Shade/Shadow Curves (SFC).- Classifications of Stable Multilocal Configurations and Their Generic Transitions.- Bibliography.

Logic and Structure : 978-1-4471-4557-8
van Dalen
Springer London
Seiten X, 263 p. 27 illus.
Erscheinungsjahr 2013
ISBN 978-1-4471-4557-8
       9781447145578
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Dirk van Dalen’s popular textbook Logic and Structure, now in its fifth edition, provides a comprehensive introduction to the basics of classical and intuitionistic logic, model theory and Gödel’s famous incompleteness theorem. Propositional and predicate logic are presented in an easy-to-read style using Gentzen’s natural deduction. The book proceeds with some basic concepts and facts of model theory: a discussion on compactness, Skolem-Löwenheim, non-standard models and quantifier elimination. The discussion of classical logic is concluded with a concise exposition of second-order logic. In view of the growing recognition of constructive methods and principles, intuitionistic logic and Kripke semantics is carefully explored. A number of specific constructive features, such as apartness and equality, the Gödel translation, the disjunction and existence property are also included. The last chapter on Gödel's first incompleteness theorem is self-contained and provides a systematic exposition of the necessary recursion theory. This new edition has been properly revised and contains a new section on ultra-products.

Inhaltsangaben

Introduction.- Propositional Logic.- Predicate Logic.- Completeness and Applications.- Second Order Logic.- Intuitionistic Logic.- Normalization.- Gödel's theorem.

Mathematical Biology : 978-0-387-95223-9
Murray
I. An Introduction
Springer New York
Seiten XXIII, 553 p. 64 illus.
Erscheinungsjahr 2002
ISBN 978-0-387-95223-9
       9780387952239
 (Hardcover)
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It has been over a decade since the release of the now classic original edition of Murray's Mathematical Biology. Since then mathematical biology has grown at an astonishing rate and is well established as a distinct discipline. Mathematical modeling is now being applied in every major discipline in the biomedical sciences. Though the field has become increasingly large and specialized, this book remains important as a text that introduces some of the exciting problems that arise in biology and gives some indication of the wide spectrum of questions that modeling can address. Due to the tremendous development in the field this book is being published in two volumes. This first volume is an introduction to the field, the mathematics mainly involves ordinary differential equations that are suitable for undergraduate and graduate courses at different levels. For this new edition Murray is covering certain items in depth, giving new applications such as modeling marital interactions and temperature dependence sex determination. SIAM, 2004: 'Murray's Mathematical Biology is a classic that belongs on the shelf of any serious student or researcher in the field. Together the two volumes contain well over 1000 references, a rich source of material, together with an excellent index to help readers quickly find key words. ... I recommend the new and expanded third edition to any serious young student interested in mathematical biology who already has a solid basis in applied mathematics.'

Inhaltsangaben

Continuous Population Models for Single Species.- Discrete Population Models for a Single Species.- Models for Interacting Populations.- Temperature-Dependent Sex Determination (TSD): Crocodilian Survivorship.- Modelling the Dynamics of Marital Interaction: Divorce Prediction and Marriage Repair.- Reaction Kinetics.- Biological Oscillators and Switches.- BZ Oscillating Reactions.- Perturbed and Coupled Oscillators and Black Holes.- Dynamics of Infectious Diseases: Epidemic Models and AIDS.- Reaction Diffusion, Chemotaxis, and Nonlocal Mechanisms.- Oscillator-Generated Wave Phenomena and Central Pattern Generators.- Biological Waves: Single-Species Models.- Use and Abuse of Fractals.

Mathematical Biology II : 978-0-387-95228-4
Murray
Spatial Models and Biomedical Applications
Springer New York
Seiten XXV, 814 p.
Erscheinungsjahr 2003
ISBN 978-0-387-95228-4
       9780387952284
 (Hardcover)
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It has been over a decade since the release first edition of the now classic original edition of Murray's Mathematical Biology. Since then mathematical biology and medicine has grown at an astonishing rate and has established itself as a distinct discipline. Mathematical modelling is now being applied in every major discipline in the biomedical sciences. Though the field has become increasingly large and specialized, this book remains important as a text that introduces some of the exciting problems which arise in the biomedical sciences and gives some indication of the wide spectrum of questions that modelling can address. Due to the tremendous development in recent years, this new edition is being published in two volumes. This second volume covers spatial models and biomedical applications. For this new edition, Murray covers certain items in depth, introducing new applications such as modelling growth and control of brain tumours, bacterial patterns, wound healing and wolf territoriality. In other areas, he discusses basic modelling concepts and provides further references as needed. He also provides even closer links between models and experimental data throughout the text. Graduate students and researchers will find this book invaluable as it gives an excellent background from which to begin genuinely practical interdisciplinary research in the biomedical sciences.

Inhaltsangaben

Multi-Species Waves and Practical Applications.- Spatial Pattern Formation with Reaction Diffusion Systems.- Animal Coat Patterns and Other Practical Applications of Reaction Diffusion Mechanisms.- Pattern Formation on Growing Domains: Alligators and Snakes.- Bacterial Patterns and Chemotaxis.- Mechanical Theory for Generating Pattern and Form in Development.- Evolution, Morphogenetic Laws, Developmental Constraints and Teratologies.- A Mechanical Theory of Vascular Network Formation.- Epidermal Wound Healing.- Dermal Wound Healing.- Growth and Control of Brain Tumours.- Neural Models of Pattern Formation.- Geographic Spread and Control of Epidemics.- Wolf Territoriality, Wolf-Deer Interaction and Survival.

Mathematical Modelling : 978-3-319-27834-6
Pohjolainen
Springer International Publishing
Seiten XI, 242 p. 56 illus., 13 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-27834-6
       9783319278346
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This book provides a thorough introduction to the challenge of applying mathematics in real-world scenarios. Modelling tasks rarely involve well-defined categories, and they often require multidisciplinary input from mathematics, physics, computer sciences, or engineering. In keeping with this spirit of modelling, the book includes a wealth of cross-references between the chapters and frequently points to the real-world context.The book combines classical approaches to modelling with novel areas such as soft computing methods, inverse problems, and model uncertainty. Attention is also paid to the interaction between models, data and the use of mathematical software. The reader will find a broad selection of theoretical tools for practicing industrial mathematics, including the analysis of continuum models, probabilistic and discrete phenomena, and asymptotic and sensitivity analysis.

Inhaltsangaben

1 Introduction by S. Pohjolainen and M. Heiliö.- 2 Models and Applications by M. Heiliö.- 3 Viewpoints on Systems and Models by E. Laitinen.- 4 Integer Models by R. Silvennoinen and J. Merikoski.- 5 Data Based Models by J. Merikoski et al.- 6 Soft Computing Methods by E. Turunen et al.- 7 Dimensional Analysis by T. Tiihonen.- 8 Modelling With Differential Equations by J. Tuomela.- 9 Continuum Models by T. Tiihonen.- 10 Simplification of Models by T. Tiihonen.- 11 Acoustic Modelling by S. Pohjolainen and A. Suutala.- 12 Inverse Problems by M. Vauhkonen et al.- 13 Project Titles by M. Heiliö.

Mathematical Models and Methods for Living Systems : 978-3-319-42678-5
Ciarletta
Levico Terme, Italy 2014
Springer International Publishing
Seiten XI, 321 p. 111 illus., 78 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-42678-5
       9783319426785
 (Softcover)
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The aim of these lecture notes is to give an introduction to several mathematical models and methods that can be used to describe the behaviour of living systems. This emerging field of application intrinsically requires the handling of phenomena occurring at different spatial scales and hence the use of multiscale methods.Modelling and simulating the mechanisms that cells use to move, self-organise and develop in tissues is not only fundamental to an understanding of embryonic development, but is also relevant in tissue engineering and in other environmental and industrial processes involving the growth and homeostasis of biological systems. Growth and organization processes are also important in many tissue degeneration and regeneration processes, such as tumour growth, tissue vascularization, heart and muscle functionality, and cardio-vascular diseases.

Inhaltsangaben

Preface.- Cell-based, continuum and hybrid models of tissue dynamics.- The Diffusion Limit of Transport Equations in Biology.- Mathematical Models of the Interaction of Cells and Cell Aggregates with the Extracellular Matrix.- Mathematical modeling of morphogenesis in living materials.- Multiscale computational modelling and analysis of cancer invasion.

Mathematical Statistics for Economics and Business : 978-1-4614-5021-4
Mittelhammer
Springer New York
Seiten XXIX, 755 p.
Erscheinungsjahr 2013
ISBN 978-1-4614-5021-4
       9781461450214
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Mathematical Statistics for Economics and Business, Second Edition, provides a comprehensive introduction to the principles of mathematical statistics which underpin statistical analyses in the fields of economics, business, and econometrics.  The selection of topics in this textbook is designed to provide students with a conceptual foundation that will facilitate a substantial understanding of statistical applications in these subjects.  This new edition has been updated throughout and now also includes a downloadable Student Answer Manual containing detailed solutions to half of the over 300 end-of-chapter problems. After introducing the concepts of probability, random variables, and probability density functions, the author develops the key concepts of mathematical statistics, most notably: expectation, sampling, asymptotics, and the main families of distributions.  The latter half of the book is then devoted to the theories of estimation and hypothesis testing with associated examples and problems that indicate their wide applicability in economics and business.  Features of the new edition include: a reorganization of topic flow and presentation to facilitate reading and understanding; inclusion of additional topics of relevance to statistics and econometric applications; a more streamlined and simple-to-understand notation for multiple integration and multiple summation over general sets or vector arguments; updated examples; new end-of-chapter problems; a solution manual for students; a comprehensive answer manual for instructors; and a theorem and definition map.This book has evolved from numerous graduate courses in mathematical statistics and econometrics taught by the author, and will be ideal for students beginning graduate study as well as for advanced undergraduates. 

Inhaltsangaben

Elements of Probability Theory.- Random Variables, Densities, and Cumulative Distribution Functions.- Expectations and Moments of Random Variables.- Parametric Families of Density Functions.- Basic Asymptotics.- Sampling, Sample Moments, Sampling Distributions, and Simulation.- Point Estimation Theory.- Point Estimation Methods.- Hypothesis Testing Theory.- Hypothesis Testing Methods and Confidence Regions.- Appendix.

Mathematical Tapas : 978-3-319-42185-8
Hiriart-Urruty
Volume 1 (for Undergraduates)
Springer International Publishing
Seiten X, 159 p.
Erscheinungsjahr 2016
ISBN 978-3-319-42185-8
       9783319421858
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This book contains a collection of exercises (called “tapas”) at undergraduate level, mainly from the fields of real analysis, calculus, matrices, convexity, and optimization. Most of the problems presented here are non-standard and some require broad knowledge of different mathematical subjects in order to be solved. The author provides some hints and (partial) answers and also puts these carefully chosen exercises into context, presents information on their origins, and comments on possible extensions. With stars marking the levels of difficulty, these tapas show or prove something interesting, challenge the reader to solve and learn, and may have surprising results. This first volume of Mathematical Tapas will appeal to mathematicians, motivated undergraduate students from science-based areas, and those generally interested in mathematics.

Inhaltsangaben

Chapter 1 Proposals.- Chapter 2 Hint(s).- Chapter 3 Answer(s), Comment(s).

Mathematics and Its History : 978-1-4419-6052-8
Stillwell
Springer New York
Seiten XXII, 662 p.
Erscheinungsjahr 2010
ISBN 978-1-4419-6052-8
       9781441960528
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From the reviews of the second edition:'This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations. The fact that the topics are introduced in their historical contexts will enable students to better appreciate and understand the mathematical ideas involved...If one constructs a list of topics central to a history course, then they would closely resemble those chosen here.'(David Parrott, Australian Mathematical Society)This third edition includes new chapters on simple groups and combinatorics, and new sections on several topics, including the Poincare conjecture. The book has also been enriched by added exercises.

Inhaltsangaben

The Theorem of Pythagoras.- Greek Geometry.- Greek Number Theory.- Infinity in Greek Mathematics.- Number Theory in Asia.- Polynomial Equations.- Analytic Geometry.- Projective Geometry.- Calculus.- Infinite Series.- The Number Theory Revival.- Elliptic Functions.- Mechanics.- Complex Numbers in Algebra.- Complex Numbers and Curves.- Complex Numbers and Functions.- Differential Geometry.- Non-Euclidean Geometry.- Group Theory.- Hypercomplex Numbers.- Algebraic Number Theory.- Topology.- Simple Groups.- Sets, Logic, and Computation.- Combinatorics.

Mathematics Education : 978-3-319-44949-4
Dewar
A Spectrum of Work in Mathematical Sciences Departments
Springer International Publishing
Seiten XI, 391 p. 35 illus., 22 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-44949-4
       9783319449494
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Many in the mathematics community in the U.S. are involved in mathematics education in various capacities.  This book highlights the breadth of the work in K-16 mathematics education done by members of US departments of mathematical sciences. It contains contributions by mathematicians and mathematics educators who do work in areas such as teacher education, quantitative literacy, informal education, writing and communication, social justice, outreach and mentoring, tactile learning, art and mathematics, ethnomathematics, scholarship of teaching and learning, and mathematics education research.  Contributors describe their work, its impact, and how it is perceived and valued.  In addition, there is a chapter, co-authored by two mathematicians who have become administrators, on the challenges of supporting, evaluating, and rewarding work in mathematics education in departments of mathematical sciences.This book is intended to inform the readership of the breadth of the work and to encourage discussion of its value in the mathematical community. The writing is expository, not technical, and should be accessible and informative to a diverse audience. The primary readership includes all those in departments of mathematical sciences in two or four year colleges and universities, and their administrators, as well as graduate students. Researchers in education may also find topics of interest.  Other potential readers include those doing work in mathematics education in schools of education, and teachers of secondary or middle school mathematics as well as those involved in their professional development.

Inhaltsangaben

Tentative ToC

Part I, Benefitting the readers of this volume: Introduction (Jacqueline M. Dewar, Pao-sheng Hsu, Harriet Pollatsek).- Communication, culture, and work in mathematics education in the mathematics department (Shandy Hauk, University of Northern Colorado and WestEd).- Valuing and supporting work in mathematics education: A view from administration (Minerva Cordero-Epperson, University of Texas, Arlington & Maura Mast, Fordham University).- Part II, Benefitting pre-service and in-service teachers and graduate student instructors: A scholarly investigation of engaging future secondary teachers with authentic mathematics capstone projects: Impacts on the instructor and the students (Curtis Bennett, Loyola Marymount University).- By definition: The interplay between research and teaching future teachers about mathematical definitions (Elizabeth A. Burroughs and Maurice J. Burke, Montana State University).- Preparing PK-12 mathematics teachers: How one department used mathematics education research to shape mathematics students’ knowledge (Yvonne Lai, Wendy M. Smith, and Nathan Wakefield, University of Nebraska, Lincoln).- Lessons Learned from a Math Teachers’ Circle (Katherine Morrison, University of Northern Colorado).- Transforming practices in mathematics teaching and learning through effective partnerships (Padmanabhan Seshaiyer, George Mason University & Kris Kappmeyer, Arlington Public School District).- Developing true collaborations between teachers, mathematicians, and mathematics educators (Kristin Umland, University of New Mexico).- Part III, Benefitting STEM majors: Finding synergy among research, teaching, and service: An example from mathematics education research (Megan Wawro, Virginia Tech).- Communicating mathematics through interdisciplinary writing and speaking assignments (Suzanne Sumner, University of Mary Washington).- Using data-enhanced classroom experiences to engage student learning (Talithia D. Williams and Susan E. Martonosi, Harvey Mudd College).- Helping students gain abstract understanding through concrete work using GeoGebra (Kathy Tomlinson, University of Wisconsin - River Falls).- A discovery approach to abstract algebra using laboratory experiences (Ellen J. Maycock, Depauw University).- A scholarly investigation of a department-wide effort to connect mathematics and the world (Melanie Pivarski, Barbara Gonzalez and Steve Cohen, Roosevelt University).- Building a community of women in mathematics: The Summer Mathematics Program for women undergraduates at Carleton College (Pamela Richardson, Westminster College).- Part IV, Benefitting students in general education courses: Developing a quantitative reasoning course: a case study (Kathleen Lopez, University of Louisiana, Lafayette).- Discovering the art of mathematics (Christine von Renesse, Westfield State University).- Ethno-mathematics: A study of mathematics in pre-Columbian cultures (Ximena Catepillán, Millersville University).- Writing for quantitative literacy (Maria Fung, Worcester State University).- Tactile mathematics: Helping students learn (Carolyn Yackel, Mercer University).- Incorporating write-to-learn activities in Statistics classes (Katherine G. Johnson, Metropolitan State University).- Mathematics for all:  Mathematics of social justice (Priscilla Bremser, Middlebury College).- Part V, Benefitting the public and the larger mathematical community: The Simpsons, Futurama, and more (Sarah Greenwald, Appalachian State University).- Transforming post-secondary education in mathematics (Tara Holm, Cornell University).- Epilogue: Reflections by the Editors.

Mathematics for Finance : 978-0-85729-081-6
Capi?ski
An Introduction to Financial Engineering
Springer London
Seiten XIII, 336 p. 66 illus.
Erscheinungsjahr 2011
ISBN 978-0-85729-081-6
       9780857290816
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Mathematics for Finance: An Introduction to Financial Engineering combines financial motivation with mathematical style. Assuming only basic knowledge of probability and calculus, it presents three major areas of mathematical finance, namely Option pricing based on the no-arbitrage principle in discrete and continuous time setting, Markowitz portfolio optimisation and Capital Asset Pricing Model, and basic stochastic interest rate models in discrete setting.

Inhaltsangaben

A Simple Market Model.- Risk-Free Assets.- Portfolio Management.- Forward and Futures Contracts.- Options: General Properties.- Binomial Model.- General Discrete Time Models.- Continuous Time Model.- Interest Rates.

Mathematics for the Life Sciences : 978-1-4614-7275-9
Ledder
Calculus, Modeling, Probability, and Dynamical Systems
Springer New York
Seiten XX, 431 p. 109 illus.
Erscheinungsjahr 2013
ISBN 978-1-4614-7275-9
       9781461472759
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?? ?? Mathematics for the Life Sciences  provides  present and future biologists with the mathematical concepts and tools needed to understand and use mathematical models and read advanced mathematical biology books.  It presents mathematics in biological contexts, focusing on the central mathematical ideas, and providing detailed explanations.  The author assumes no mathematics background beyond algebra and precalculus.  Calculus is presented as a one-chapter primer that is suitable for readers who have not studied the subject before, as well as readers who have taken a calculus course and need a review.  This primer is followed by a novel chapter on mathematical modeling that begins with discussions of biological data and the basic principles of modeling.  The remainder of the chapter introduces the reader to topics in mechanistic modeling (deriving models from biological assumptions) and empirical modeling (using data to parameterize and select models).  The modeling chapter contains a thorough treatment of key ideas and techniques that are often neglected in mathematics books.  It also provides the reader with a sophisticated viewpoint and the essential background needed to make full use of the remainder of the book, which includes two chapters on probability and its applications to inferential statistics and three chapters on discrete and continuous dynamical systems.  The biological content of the book is self-contained and includes many basic biology topics such as the genetic code, Mendelian genetics, population dynamics, predator-prey relationships, epidemiology, and immunology.  The large number of problem sets include some drill problems along with a large number of case studies.  The latter are divided into step-by-step problems and sorted into the appropriate section, allowing readers to gradually develop complete investigations from understanding the biological assumptions to a complete analysis.

Inhaltsangaben

A Brief Summary of Calculus.- Mathematical Modeling.- Probability Distributions.- Working with Probability.- Dynamics of Single Populations.- Discrete Dynamical Systems.- Continuous Dynamical Systems.?- Index.

Mathematics in Everyday Life : 978-3-319-27937-4
Haigh
Springer International Publishing
Seiten IX, 159 p. 16 illus., 14 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-27937-4
       9783319279374
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How does mathematics impact everyday events? The purpose of this book is to show a range of examples where mathematics can be seen at work in everyday life. From money (APR, mortgage repayments, personal finance), simple first and second order ODEs, sport and games (tennis, rugby, athletics, darts, tournament design, soccer, snooker), business (stock control, linear programming, check digits, promotion policies, investment), the social sciences (voting methods, Simpson’s Paradox, drug testing, measurements of inequality) to TV game shows and even gambling (lotteries, roulette, poker, horse racing), the mathematics behind commonplace events is explored. Fully worked examples illustrate the ideas discussed and each chapter ends with a collection of exercises. Everyday Mathematics supports other first year modules by giving students extra practice in working with calculus, linear algebra, geometry, trigonometry and probability. Secondary/high school level mathematics is all that is required for students to understand the material. Those students whose degree course includes writing an extended mathematical essay will find many suitable topics here, with pointers to extend and develop the material.

Inhaltsangaben

Money.- Differential Equations.- Sport and Games.- Business Applications.- Social Sciences.- TV Game Shows.

Mathématiques et Technologie : 978-0-387-69212-8
Rousseau
Springer New York
Seiten XVIII, 594 p. 214 ill.
Erscheinungsjahr 2009
ISBN 978-0-387-69212-8
       9780387692128
 (Hardcover)
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Les auteurs présentent des concepts mathématiques de base, comme les chaînes de Markov, les fonctions itérées, la structure de corps fini, et les développent dans le contexte de problèmes réels tels que la navigation internet, la compression de données et les codes correcteurs d'erreurs. Un des leitmotivs du texte est de montrer comment la modélisation mathématique, la puissance des outils mathématiques et l'abstraction sont toutes cruciales pour l'innovation technologique. Les divers sujets sont présentés avec clarté et les mathématiques toujours discutées à partir de connaissances de base. Chaque chapitre suggère de nombreux exercices, certains élémentaires pour renforcer la compréhension, d'autres plus avancés pour explorer de nouvelles problématiques. Une mise en contexte historique de certains concepts mathématiques ou de l'évolution d'une technologique enrichit le texte.

Inhaltsangaben

Le positionnement sur la Terre et dans l'espace.- Frises et mosaïques.- Les mouvements d'un robot.- Squelette et chirurgie aux rayons gamma.- Épargner et emprunter.- Codes correcteurs.- La cryptographie à cle publique: le code RSA (1978).- Générateurs de nombres aléatoires.- Google et l'algorithme PageRank.- Pourquoi 44 100 nombres à la seconde?.- La compression d'images: les systèmes de fonctions itérées.- La compression d'images: le standard JPEG.- L'ordinateur à ADN1.- Le calcul des variations et ses applications.- Flashs-science.

Maximum Principles and Geometric Applications : 978-3-319-24335-1
Alías
Springer International Publishing
Seiten XXVII, 570 p.
Erscheinungsjahr 2016
ISBN 978-3-319-24335-1
       9783319243351
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This monograph presents an introduction to some geometric and analytic aspects of the maximum principle. In doing so, it analyses with great detail the mathematical tools and geometric foundations needed to develop the various new forms that are presented in the first chapters of the book. In particular, a generalization of the Omori-Yau maximum principle to a wide class of differential operators is given, as well as a corresponding weak maximum principle and its equivalent open form and parabolicity as a special stronger formulation of the latter. In the second part, the attention focuses on a wide range of applications, mainly to geometric problems, but also on some analytic (especially PDEs) questions including: the geometry of submanifolds, hypersurfaces in Riemannian and Lorentzian targets, Ricci solitons, Liouville theorems, uniqueness of solutions of Lichnerowicz-type PDEs and so on. Maximum Principles and Geometric Applications is written in an easy style making it accessible to beginners. The reader is guided with a detailed presentation of some topics of Riemannian geometry that are usually not covered in textbooks. Furthermore, many of the results and even proofs of known results are new and lead to the frontiers of a contemporary and active field of research.

Inhaltsangaben

A crash course in Riemannian geometry.- The Omori-Yau maximum principle.- New forms of the maximum principle.- Sufficient conditions for the validity of the weak maximum principle.- Miscellany results for submanifolds.- Applications to hypersurfaces.- Hypersurfaces in warped products.- Applications to Ricci Solitons.- Spacelike hypersurfaces in Lorentzian spacetimes.

Measure Theory : 978-1-4614-6955-1
Cohn
Second Edition
Springer New York
Seiten XXI, 457 p.
Erscheinungsjahr 2013
ISBN 978-1-4614-6955-1
       9781461469551
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Intended as a self-contained introduction to measure theory, this textbook also includes a comprehensive treatment of integration on locally compact Hausdorff spaces, the analytic and Borel subsets of Polish spaces, and Haar measures on locally compact groups. This second edition includes a chapter on measure-theoretic probability theory, plus brief treatments of the Banach-Tarski paradox, the Henstock-Kurzweil integral, the Daniell integral, and the existence of liftings. Measure Theory provides a solid background for study in both functional analysis and probability theory and is an excellent resource for advanced undergraduate and graduate students in mathematics. The prerequisites for this book are basic courses in point-set topology and in analysis, and the appendices present a thorough review of essential background material.

Inhaltsangaben

1. Measures.- Algebras and sigma-algebras.- Measures.- Outer measures.- Lebesgue measure.- Completeness and regularity.- Dynkin classes.- 2. Functions and Integrals.- Measurable functions.- Properties that hold almost everywhere.- The integral.- Limit theorems.- The Riemann integral.- Measurable functions again, complex-valued functions, and image measures.- 3. Convergence.- Modes of Convergence.- Normed spaces.- Definition of L^p and L^p.- Properties of L^p and L-p.- Dual spaces.- 4. Signed and Complex Measures.- Signed and complex measures.- Absolute continuity.- Singularity.- Functions of bounded variation.- The duals of the L^p spaces.- 5. Product Measures.- Constructions.- Fubini’s theorem.- Applications.- 6. Differentiation.- Change of variable in R^d.- Differentiation of measures.- Differentiation of functions.- 7. Measures on Locally Compact Spaces.- Locally compact spaces.- The Riesz representation theorem.- Signed and complex measures; duality.- Additional properties of regular measures.- The µ^*-measurable sets and the dual of L^1.- Products of locally compact spaces.- 8. Polish Spaces and Analytic Sets.- Polish spaces.- Analytic sets.- The separation theorem and its consequences.- The measurability of analytic sets.- Cross sections.- Standard, analytic, Lusin, and Souslin spaces.- 9. Haar Measure.- Topological groups.- The existence and uniqueness of Haar measure.- The algebras L^1 (G) and M (G).- Appendices.- A. Notation and set theory.- B. Algebra.- C. Calculus and topology in R^d.- D. Topological spaces and metric spaces.- E. The Bochner integral.- F Liftings.- G The Banach-Tarski paradox.- H The Henstock-Kurzweil and McShane integralsBibliography.- Index of notation.- Index.

Méthodes Numériques : 978-88-470-0495-5
Quarteroni
Algorithmes, analyse et applications
Springer Milan
Seiten XIV, 538 p.
Erscheinungsjahr 2007
ISBN 978-88-470-0495-5
       9788847004955
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Ce livre a pour but de présenter les fondements théoriques et méthodologiques de l'analyse numérique. Une attention toute particulière est portée sur les concepts de stabilité, précision et complexité des algorithmes. Les méthodes modernes relatives aux thèmes suivants sont presentées et analysées en détail : résolution des systèmes lineaires et non linéaires, approximation polynomiale, optimisation, intégration numérique, polynômes orthogonaux, transformations rapides, équations différentielles ordinaires. Les techniques presentées sont illustrées par de nombreux tableaux et figures. Beaucoup d'exemples et de contre-exemples sont proposés pour permettre au lecteur de développer son sens critique. Une caractéristique principale du livre réside dans l'abondance des programmes MATLAB qui accompagnent toutes les méthodes numériques présentées et qui les illustrent par des applications concrètes. Le lecteur détient ainsi tous les outils pour acquérir de solides connaissances théoriques et les appliquer directement sur ordinateur. Cet ouvrage s'adresse aux étudiants du second cycle des universités, aux élèves des écoles d'ingénieurs et, plus généralement, à toutes les personnes qui pratiquent le calcul scientifique.

Inhaltsangaben

Notions de base.- Eléments d’analyse matricielle.- Les fondements du calcul scientifique.- Algèbre linéaire numérique.- Méthodes directes pour la résolution des systèmes linéaires.- Méthodes itératives pour la résolution des systèmes linéaires.- Approximation des valeurs propres et des vecteurs propres.- Sur les fonctions et les fonctionnelles.- Résolution des équations et des systèmes non linéaires.- Interpolation polynomiale.- Intégration numérique.- Transformations, dérivations et discrétisations.- Polynômes orthogonaux en théorie de l’approximation.- Résolution numérique des équations différentielles ordinaires.- Problèmes aux limites en dimension un.- Problèmes transitoires paraboliques et hyperboliques.- Applications.

Methods of Fourier Analysis and Approximation Theory : 978-3-319-27465-2
Ruzhansky
Springer International Publishing
Seiten VIII, 258 p. 8 illus.
Erscheinungsjahr 2016
ISBN 978-3-319-27465-2
       9783319274652
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Fourier Analysis

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Different facets of interplay between harmonic analysis and approximation theory are covered in this volume. The topics included are Fourier analysis, function spaces, optimization theory, partial differential equations, and their links to modern developments in the approximation theory. The articles of this collection were originated from two events. The first event took place during the 9th ISAAC Congress in Krakow, Poland, 5th-9th August 2013, at the section “Approximation Theory and Fourier Analysis”. The second event was the conference on Fourier Analysis and Approximation Theory in the Centre de Recerca Matemàtica (CRM), Barcelona, during 4th-8th November 2013, organized by the editors of this volume. All articles selected to be part of this collection were carefully reviewed.

Inhaltsangaben

1. Introduction.- 2. Fourier analysis.- 2.1. Parseval frames.- 2.2. Hyperbolic Hardy classes and logarithmic Bloch spaces.- 2.3. Logan's and Bohman's extremal problems.- 2.4. Weighted estimates for the Hilbert transform.- 2.5. Q-Measures and uniqueness sets for Haar series.- 2.6. O-diagonal estimates for Calderón-Zygmund operators.- 3. Function spaces of radial functions.- 3.1. Potential spaces of radial functions.-
3.2. On Leray's formula.- 4. Approximation theory.- 4.1. Approximation order of Besov classes.- 4.2. Ulyanov inequalities for moduli of smoothness.- 4.3. Approximation order of Besov classes.- 5. Optimization theory and related topics.- 5.1. The Laplace-Borel transform.- 5.2. Optimization control problems.- 2 Michael Ruzhansky and Sergey Tikhonov.-5.3. Optimization control problems for parabolic equation.- 5.4. Numerical modeling of the linear filtration.- References.

Metric Spaces : 978-1-84628-369-7
O'Searcoid
Springer London
Seiten XX, 304 p. 102 illus.
Erscheinungsjahr 2007
ISBN 978-1-84628-369-7
       9781846283697
 (Softcover)
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Fachgebiet

Analysis

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The abstract concepts of metric spaces are often perceived as difficult. This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of distance as far as possible, illustrating the text with examples and naturally arising questions. The book provides a thorough exposition of all the standard necessary results of the theory and, in addition, includes selected topics not normally found in introductory books, such as: the Tietze Extension Theorem; the Hausdorff metric and its completeness; and the existence of curves of minimum length. With its many examples, careful illustrations, and full solutions to selected exercises, this book provides a gentle introduction that is ideal for self-study and an excellent preparation for applications.

Inhaltsangaben

Metrics.- Distance.- Boundary.- Open, Closed and Dense Subsets.- Balls.- Convergence.- Bounds.- Continuity.- Uniform Continuity.- Completeness.- Connectedness.- Compactness.- Equivalence.

Minimal Free Resolutions over Complete Intersections  : 978-3-319-26436-3
Eisenbud
Springer International Publishing
Seiten X, 107 p.
Erscheinungsjahr 2016
ISBN 978-3-319-26436-3
       9783319264363
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This book introduces a theory of higher matrix factorizations for regular sequences and uses it to describe the minimal free resolutions of high syzygy modules over complete intersections. Such resolutions have attracted attention ever since the elegant construction of the minimal free resolution of the residue field by Tate in 1957. The theory extends the theory of matrix factorizations of a non-zero divisor, initiated by Eisenbud in 1980, which yields a description of the eventual structure of minimal free resolutions over a hypersurface ring. Matrix factorizations have had many other uses in a wide range of mathematical fields, from singularity theory to mathematical physics.

Inhaltsangaben


Modeling Survival Data: Extending the Cox Model : 978-0-387-98784-2
Therneau
Springer New York
Seiten XIV, 350 p.
Erscheinungsjahr 2000
ISBN 978-0-387-98784-2
       9780387987842
 (Hardcover)
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Extending the Cox Model is aimed at researchers, practitioners, and graduate students who have some exposure to traditional methods of survival analysis. The emphasis is on semiparametric methods based on the proportional hazards model. The inclusion of examples with SAS and S-PLUS code will make the book accessible to most working statisticians.

Inhaltsangaben

Estimating the Survival and Hazard Functions.- The Cox Model.- Residuals.- Functional Form.- Testing Proportional Hazards.- Influence.- Multiple Events per Subject.- Frailty Models.- Expected Survival.

Modern Multivariate Statistical Techniques : 978-0-387-78188-4
Izenman
Regression, Classification, and Manifold Learning
Springer New York
Seiten XXV, 733 p.
Erscheinungsjahr 2008
ISBN 978-0-387-78188-4
       9780387781884
 (Hardcover)
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Remarkable advances in computation and data storage and the ready availability of huge data sets have been the keys to the growth of the new disciplines of data mining and machine learning, while the enormous success of the Human Genome Project has opened up the field of bioinformatics. These exciting developments, which led to the introduction of many innovative statistical tools for high-dimensional data analysis, are described here in detail. The author takes a broad perspective; for the first time in a book on multivariate analysis, nonlinear methods are discussed in detail as well as linear methods. Techniques covered range from traditional multivariate methods, such as multiple regression, principal components, canonical variates, linear discriminant analysis, factor analysis, clustering, multidimensional scaling, and correspondence analysis, to the newer methods of density estimation, projection pursuit, neural networks, multivariate reduced-rank regression, nonlinear manifold learning, bagging, boosting, random forests, independent component analysis, support vector machines, and classification and regression trees. Another unique feature of this book is the discussion of database management systems. This book is appropriate for advanced undergraduate students, graduate students, and researchers in statistics, computer science, artificial intelligence, psychology, cognitive sciences, business, medicine, bioinformatics, and engineering. Familiarity with multivariable calculus, linear algebra, and probability and statistics is required. The book presents a carefully-integrated mixture of theory and applications, and of classical and modern multivariate statistical techniques, including Bayesian methods. There are over 60 interesting data sets used as examples in the book, over 200 exercises, and many color illustrations and photographs.

Inhaltsangaben

and Preview.- Data and Databases.- Random Vectors and Matrices.- Nonparametric Density Estimation.- Model Assessment and Selection in Multiple Regression.- Multivariate Regression.- Linear Dimensionality Reduction.- Linear Discriminant Analysis.- Recursive Partitioning and Tree-Based Methods.- Artificial Neural Networks.- Support Vector Machines.- Cluster Analysis.- Multidimensional Scaling and Distance Geometry.- Committee Machines.- Latent Variable Models for Blind Source Separation.- Nonlinear Dimensionality Reduction and Manifold Learning.- Correspondence Analysis.

Moduli Spaces of Riemannian Metrics : 978-3-0348-0947-4
Tuschmann
Springer Basel
Seiten X, 123 p. 3 illus.
Erscheinungsjahr 2015
ISBN 978-3-0348-0947-4
       9783034809474
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This book studies certain spaces of Riemannian metrics on both compact and non-compact manifolds. These spaces are defined by various sign-based curvature conditions, with special attention paid to positive scalar curvature and non-negative sectional curvature, though we also consider positive Ricci and non-positive sectional curvature. If we form the quotient of such a space of metrics under the action of the diffeomorphism group (or possibly a subgroup) we obtain a moduli space. Understanding the topology of both the original space of metrics and the corresponding moduli space form the central theme of this book. For example, what can be said about the connectedness or the various homotopy groups of such spaces? We explore the major results in the area, but provide sufficient background so that a non-expert with a grounding in Riemannian geometry can access this growing area of research.

Inhaltsangaben

Part I: Positive scalar curvature.- The (moduli) space of all Riemannian metrics.- Clifford algebras and spin.- Dirac operators and index theorems.- Early results on the space of positive scalar curvature metrics.- Kreck-Stolz invariants.- Applications of Kreck-Stolz invariants.- The eta invariant and applications.- The case of dimensions 2 and 3.- The observer moduli space and applications.- Other topological structures.- Negative scalar and Ricci curvature.- Part II: Sectional curvature.- Moduli spaces of compact manifolds with positive or non-negative sectional curvature.- Moduli spaces of compact manifolds with negative and non-positive sectional curvature.- Moduli spaces of non-compact manifolds with non-negative sectional curvature.- Positive pinching and the Klingenberg-Sakai conjecture.

Moral Calculations : 978-0-387-98419-3
Mero
Game Theory, Logic, and Human Frailty
Springer New York
Seiten XI, 276 p.
Erscheinungsjahr 1998
ISBN 978-0-387-98419-3
       9780387984193
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Are people ever rational? Consider this: You auction off a one-dollar bill to the highest bidder, but you set the rules so that the second highest bidder also has to pay the amount of his last bid, even though he gets nothing. Would people ever enter such an auction? Not only do they, but according to Martin Shubik, the game's inventor, the average winning bid (for a dollar, remember) is $3.40. Many winners report that they bid so high only because their opponent 'went completely crazy.' This game lies at the intersection of three subjects of eternal fascination: human psychology, morality, and John von Neumann's game theory. Hungarian game-theorist Laszlo Mero introduces us to the basics of game theory, including such concepts as zero-sum games, Prisoner's Dilemma and the origins of altruism; shows how game theory is applicable to fields ranging from physics to politics; and explores the role of rational thinking in the context of many different kinds of thinking. This fascinating, urbane book will interest everyone who wonders what mathematics can tell us about the human condition.

Inhaltsangaben

Games of Morality.- 1. Auction and Posing.- 2. The Brute as Hero.- 3. The Prisoner’s Dilemma.- 4. The Golden Rule.- 5. The Bluff.- The Sources of Diversity.- 6. John von Neumann’s Game Theory.- 7. Competition for a Common Goal.- 8. Hawks and Doves.- 9. Socialism and Free Enterprise.- 10. Games Elementary Particles Play.- The Psychology of Rationality.- 11. Loves Me, Loves Me Not.- 12. Rational Irrationality.- 13. Collective Rationality.- 14. The Heterogeneity of Human Thought.- 15. There Are Many Ways to Nirvana.- References and Further Reading.

Moving Interfaces and Quasilinear Parabolic Evolution Equations : 978-3-319-27697-7
Prüss
Springer International Publishing
Seiten XIX, 609 p. 7 illus.
Erscheinungsjahr 2016
ISBN 978-3-319-27697-7
       9783319276977
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In this monograph, the authors develop a comprehensive approach for the mathematical analysis of a wide array of problems involving moving interfaces. It includes an in-depth study of abstract quasilinear parabolic evolution equations, elliptic and parabolic boundary value problems, transmission problems, one- and two-phase Stokes problems, and the equations of incompressible viscous one- and two-phase fluid flows. The theory of maximal regularity, an essential element, is also fully developed. The authors present a modern approach based on powerful tools in classical analysis, functional analysis, and vector-valued harmonic analysis.The theory is applied to problems in two-phase fluid dynamics and phase transitions, one-phase generalized Newtonian fluids, nematic liquid crystal flows, Maxwell-Stefan diffusion, and a variety of geometric evolution equations. The book also includes a discussion of the underlying physical and thermodynamic principles governing the equations of fluid flows and phase transitions, and an exposition of the geometry of moving hypersurfaces.

Inhaltsangaben

Preface.- Basic Notations.- General References.- Part I Background.- 1Problems and Strategies.- 2.Tools from Differential Geometry.- Part II Abstract Theory.- 3Operator Theory and Semigroups.- 4.Vector-Valued Harmonic Analysis.- 5.Quasilinear Parabolic Evolution Equations.- Part III Linear Theory.- 6.Elliptic and Parabolic Problems.- 7.Generalized Stokes Problems.- 8.Two-Phase Stokes Problems.- Part IV Nonlinear Problems.- 9.Local Well-Posedness and Regularity.- 10.Linear Stability of Equilibria.- 11.Qualitative Behaviour of the Semiows.- 12.Further Parabolic Evolution Problems.- Biographical Comments.- Outlook and Future Challenges.- References.- List of Figures.- List of Symbols.- Subject Index.

Multidimensional Diffusion Processes : 978-3-662-22201-0
Stroock
Springer Berlin Heidelberg
Seiten XII, 338 p.
Erscheinungsjahr 2006
ISBN 978-3-662-22201-0
       9783662222010
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'This book is an excellent presentation of the application of martingale theory to the theory of Markov processes, especially multidimensional diffusions. This approach was initiated by Stroock and Varadhan in their famous papers. (...) The proofs and techniques are presented in such a way that an adaptation in other contexts can be easily done. (...) The reader must be familiar with standard probability theory and measure theory which are summarized at the beginning of the book. This monograph can be recommended to graduate students and research workers but also to all interested in Markov processes from a more theoretical point of view.' Mathematische Operationsforschung und Statistik, 1981

Inhaltsangaben

Preliminary Material: Extension Theorems, Martingales, and Compactness.- Markov Processes, Regularity of Their Sample Paths, and the Wiener Measure.- Parabolic Partial Differential Equations.- The Stochastic Calculus of Diffusion Theory.- Stochastic Differential Equations.- The Martingale Formulation.- Uniqueness.- Itô’s Uniqueness and Uniqueness to the Martingale Problem.- Some Estimates on the Transition Probability Functions.- Explosion.- Limit Theorems.- The Non-unique Case.

Neoclassical Theory of Electromagnetic Interactions : 978-1-4471-7282-6
Babin
A Single Theory for Macroscopic and Microscopic Scales
Springer London
Seiten XXIII, 696 p.
Erscheinungsjahr 2016
ISBN 978-1-4471-7282-6
       9781447172826
 (Hardcover)
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In this monograph, the authors present their recently developed theory of electromagnetic interactions. This neoclassical approach extends the classical electromagnetic theory down to atomic scales and allows the explanation of various non-classical phenomena in the same framework. While the classical Maxwell–Lorentz electromagnetism theory succeeds in describing the physical reality at macroscopic scales, it struggles at atomic scales. Here, quantum mechanics traditionally takes over to describe non-classical phenomena such as the hydrogen spectrum and de Broglie waves. By means of modifying the classical theory, the approach presented here is able to consistently explain quantum-mechanical effects, and while similar to quantum mechanics in some respects, this neoclassical theory also differs markedly from it. In particular, the newly developed framework omits probabilistic interpretations of the wave function and features a new fundamental spatial scale which, at the size of the free electron, is much larger than the classical electron radius and is relevant to plasmonics and emission physics. This book will appeal to researchers interested in advanced aspects of electromagnetic theory. Treating the classical approach in detail, including non-relativistic aspects and the Lagrangian framework, and comparing the neoclassical theory with quantum mechanics and the de Broglie–Bohm theory, this work is completely self-contained.

Inhaltsangaben

Introduction.- The History of Views on Charges, Currents and the Electromagnetic Field.- The Neoclassical Field Theory of Charged Matter: A Concise Presentation.- Part I. Classical Electromagnetic Theory and Special Relativity.- Part II. Classical Field Theory.- Part III. The Neoclassical Theory of Charges.- Part IV. The Neoclassical Theory of Charges with Spin.- Part V. Mathematical Aspects of the Theory of Distributed Elementary Charges.- Part VI. Appendices.

Néron Models and Base Change : 978-3-319-26637-4
Halle
Springer International Publishing
Seiten X, 151 p.
Erscheinungsjahr 2016
ISBN 978-3-319-26637-4
       9783319266374
 (Softcover)
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Algebraic Geometry

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Presenting the first systematic treatment of the behavior of Néron models under ramified base change, this book can be read as an introduction to various subtle invariants and constructions related to Néron models of semi-abelian varieties, motivated by concrete research problems and complemented with explicit examples. Néron models of abelian and semi-abelian varieties have become an indispensable tool in algebraic and arithmetic geometry since Néron introduced them in his seminal 1964 paper. Applications range from the theory of heights in Diophantine geometry to Hodge theory. We focus specifically on Néron component groups, Edixhoven’s filtration and the base change conductor of Chai and Yu, and we study these invariants using various techniques such as models of curves, sheaves on Grothendieck sites and non-archimedean uniformization. We then apply our results to the study of motivic zeta functions of abelian varieties. The final chapter contains a list of challenging open questions. This book is aimed towards researchers with a background in algebraic and arithmetic geometry.

Inhaltsangaben

Normal 0 false false false EN-US X-NONE X-NONE MicrosoftInternetExplorer4 Introduction.- Preliminaries.- Models of curves and the Neron component series of a Jacobian.- Component groups and non-archimedean uniformization.- The base change conductor and Edixhoven's ltration.- The base change conductor and the Artin conductor.- Motivic zeta functions of semi-abelian varieties.- Cohomological interpretation of the motivic zeta function. /* Style Definitions */ table.MsoNormalTable {mso-style-name:'Table Normal'; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:''; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin-top:0in; mso-para-margin-right:0in; mso-para-margin-bottom:10.0pt; mso-para-margin-left:0in; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:'Calibri','sans-serif'; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:'Times New Roman'; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:'Times New Roman'; mso-bidi-theme-font:minor-bidi;}

Networked Control Systems : 978-0-85729-032-8
Bemporad
Springer London
Seiten XIV, 366 p. 112 illus., 9 illus. in color.
Erscheinungsjahr 2010
ISBN 978-0-85729-032-8
       9780857290328
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This book nds its origin in the WIDE PhD School on Networked Control Systems, which we organized in July 2009 in Siena, Italy. Having gathered experts on all the aspects of networked control systems, it was a small step to go from the summer school to the book, certainly given the enthusiasm of the lecturers at the school. We felt that a book collecting overviewson the important developmentsand open pr- lems in the eld of networked control systems could stimulate and support future research in this appealing area. Given the tremendouscurrentinterests in distributed control exploiting wired and wireless communication networks, the time seemed to be right for the book that lies now in front of you. The goal of the book is to set out the core techniques and tools that are ava- able for the modeling, analysis and design of networked control systems. Roughly speaking, the book consists of three parts. The rst part presents architectures for distributed control systems and models of wired and wireless communication n- works. In particular, in the rst chapter important technological and architectural aspects on distributed control systems are discussed. The second chapter provides insight in the behavior of communication channels in terms of delays, packet loss and information constraints leading to suitable modeling paradigms for commu- cation networks.

Inhaltsangaben

The Importance, Design and Implementation of a Middleware for Networked Control Systems.- Wireless Networking for Control: Technologies and Models.- A Survey on Distributed Estimation and Control Applications Using Linear Consensus Algorithms.- Distributed Optimization and Games: A Tutorial Overview.- Decentralized Model Predictive Control.- Decentralized Control.- Stability and Stabilization of Networked Control Systems.- Feedback Control over Limited Capacity Channels.- Event-Triggered Feedback in Control, Estimation, and Optimization.

Number Theory : 978-3-319-43873-3
Fine
An Introduction via the Density of Primes
Springer International Publishing
Seiten XIII, 413 p. 12 illus., 1 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-43873-3
       9783319438733
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Number Theory

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Now in its second edition, this textbook provides an introduction and overview of number theory based on the density and properties of the prime numbers. This unique approach offers both a firm background in the standard material of number theory, as well as an overview of the entire discipline. All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. New in this edition are coverage of p-adic numbers, Hensel's lemma, multiple zeta-values, and elliptic curve methods in primality testing. Key topics and features include:A solid introduction to analytic number theory, including full proofs of Dirichlet's Theorem and the Prime Number Theorem Concise treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of idealsDiscussion of the AKS algorithm, which shows that primality testing is one of polynomial time, a topic not usually included in such texts Many interesting ancillary topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbersThe user-friendly style, historical context, and wide range of exercises that range from simple to quite difficult (with solutions and hints provided for select exercises) make Number Theory: An Introduction via the Density of Primes ideal for both self-study and classroom use. Intended for upper level undergraduates and beginning graduates, the only prerequisites are a basic knowledge of calculus, multivariable calculus, and some linear algebra. All necessary concepts from abstract algebra and complex analysis are introduced where needed.

Inhaltsangaben

Introduction and Historical Remarks.- Basic Number Theory.- The Infinitude of Primes.- The Density of Primes.- Primality Testing: An Overview.- Primes and Algebraic Number Theory.- The Fields Q_p of p-adic Numbers: Hensel's Lemma.- References.- Index.

Numerical Approximation of Partial Differential Equations : 978-3-319-32353-4
Bartels
Springer International Publishing
Seiten XV, 535 p. 170 illus. With online files/update.
Erscheinungsjahr 2016
ISBN 978-3-319-32353-4
       9783319323534
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Numerical Analysis

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Finite element methods for approximating partial differential equations have reached a high degree of maturity, and are an indispensible tool in science and technology. This textbook aims at providing a thorough introduction to the construction, analysis, and implementation of finite element methods for model problems arising in continuum mechanics. The first part of the book discusses elementary properties of linear partial differential equations along with their basic numerical approximation, the functional-analytical framework for rigorously establishing existence of solutions, and the construction and analysis of basic finite element methods. The second part is devoted to the optimal adaptive approximation of singularities and the fast iterative solution of linear systems of equations arising from finite element discretizations. In the third part, the mathematical framework for analyzing and discretizing saddle-point problems is formulated, corresponding finte element methods are analyzed, and particular applications including incompressible elasticity, thin elastic objects, electromagnetism, and fluid mechanics are addressed. The book includes theoretical problems and practical projects for all chapters, and an introduction to the implementation of finite element methods.

Inhaltsangaben

Preface.- Part I Finite differences and finite elements.- Elliptic partial differential equations.- Finite Element Method.- Part II Local resolution and iterative solution.- Local Resolution Techniques.- Iterative Solution Methods.- Part III Constrained and singularly perturbed problems.- Saddled-point Problems.- Mixed and Nonstandard methods.- Applications.- Problems and Projects.- Implementation aspects.- Notations, inequalities, guidelines.- Index

On the Hypotheses Which Lie at the Bases of Geometry : 978-3-319-26040-2
Jost
Springer International Publishing
Seiten X, 172 p. 1 illus.
Erscheinungsjahr 2016
ISBN 978-3-319-26040-2
       9783319260402
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This book presents William Clifford’s English translation of Bernhard Riemann’s classic text together with detailed mathematical, historical and philosophical commentary. The basic concepts and ideas, as well as their mathematical background, are provided, putting Riemann’s reasoning into the more general and systematic perspective achieved by later mathematicians and physicists (including Helmholtz, Ricci, Weyl, and Einstein) on the basis of his seminal ideas. Following a historical introduction that positions Riemann’s work in the context of his times, the history of the concept of space in philosophy, physics and mathematics is systematically presented. A subsequent chapter on the reception and influence of the text accompanies the reader from Riemann’s times to contemporary research. Not only mathematicians and historians of the mathematical sciences, but also readers from other disciplines or those with an interest in physics or philosophy will find this work both appealing and insightful.

Inhaltsangaben

Introduction.- Historical background.- Riemann's habilitation.- Presentation of the texts.- Reception and impact on history.- Modern research.- Bibliography.

Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE : 978-1-4614-4285-1
Touzi
Springer New York
Seiten X, 214 p.
Erscheinungsjahr 2013
ISBN 978-1-4614-4285-1
       9781461442851
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?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.?

Inhaltsangaben

Preface.- 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.

Optimal Transport : 978-3-540-71049-3
Villani
Old and New
Springer Berlin Heidelberg
Seiten XXII, 976 p.
Erscheinungsjahr 2009
ISBN 978-3-540-71049-3
       9783540710493
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At the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and John Mather launched a revolution in the venerable field of optimal transport founded by G. Monge in the 18th century, which has made breathtaking forays into various other domains of mathematics ever since. The author presents a broad overview of this area, supplying complete and self-contained proofs of all the fundamental results of the theory of optimal transport at the appropriate level of generality. Thus, the book encompasses the broad spectrum ranging from basic theory to the most recent research results. PhD students or researchers can read the entire book without any prior knowledge of the field. A comprehensive bibliography with notes that extensively discuss the existing literature underlines the book’s value as a most welcome reference text on this subject.  

Inhaltsangaben

Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge—Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Optimisation et contrôle stochastique appliqués à la finance : 978-3-540-73736-0
Pham
Springer Berlin Heidelberg
Seiten XVI, 188 p.
Erscheinungsjahr 2007
ISBN 978-3-540-73736-0
       9783540737360
 (Softcover)
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L'objectif et l'originalité de ce livre est de présenter les différents aspects et méthodes utilisés dans la résolution des problèmes d'optimisation stochastique avec en vue des applications plus spécifiques à la finance: gestion de portefeuille, couverture d'options, investissement optimal. Nous avons inclus certains développements récents sur le sujet sans chercher a priori la plus grande généralité. Nous avons voulu une exposition graduelle des méthodes mathématiques en présentant d'abord les idées intuitives puis en énonçant précisément les résultats avec des démonstrations complètes et détaillées. Nous avons aussi pris soin d'illustrer chacune des méthodes de résolution sur de nombreux exemples issus de la finance. Nous espérons ainsi que ce livre puisse être utile aussi bien pour des étudiants que pour des chercheurs du monde académique ou professionnel intéressés par l'optimisation et le contrôle stochastique appliqués à la finance.

Inhaltsangaben

Quelques éléments d'analyse stochastique.- Probl`emes d'optimisation stochastique. Exemples en finance.- Approche EDP classique de la programmation dynamique.- Approche des équations de Bellman par les solutions de viscosité.- Méthodes d'équations différentielles stochastiques rétrogrades.- Méthodes martingales de dualité convexe.

Ordered Sets : 978-3-319-29786-6
Schröder
An Introduction with Connections from Combinatorics to Topology
Springer International Publishing
Seiten XVI, 420 p. 53 illus., 10 illus. in color.
Erscheinungsjahr 2016
ISBN 978-3-319-29786-6
       9783319297866
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The second edition of this highly praised textbook provides an expanded introduction to the theory of ordered sets and its connections to various subjects. Utilizing a modular presentation, the core material is purposely kept brief, allowing for the benefits of a broad exposure to the subject without the risk of overloading the reader with too much information all at once. The remaining chapters can then be read in almost any order, giving the text a greater depth and flexibility of use. Most topics are introduced by examining how they relate to research problems, some of them still open, allowing for continuity among diverse topics and encouraging readers to explore these problems further with research of their own.A wide range of material is presented, from classical results such as Dilworth's, Szpilrajn's, and Hashimoto's Theorems to more recent results such as the Li-Milner Structure Theorem. Major topics covered include chains and antichains, lowest upper and greatest lower bounds, retractions, algorithmic approaches, lattices, the dimension of ordered sets, interval orders, lexicographic sums, products, enumeration, and the role of algebraic topology. This new edition shifts the primary focus to finite ordered sets, with results on infinite ordered sets presented toward the end of each chapter whenever possible. Also new are Chapter 6 on graphs and homomorphisms, which serves to separate the fixed clique property from the more fundamental fixed simplex property as well as to discuss the connections and differences between graph homomorphisms and order-preserving maps, and an appendix on discrete Morse functions and their use for the fixed point property for ordered sets.Rich in examples, diagrams, and exercises, the second edition of Ordered Sets will be an excellent text for undergraduate and graduate students and a valuable resource for interested researchers. It will be especially useful to those looking for an introduction to the theory of ordered sets and its connections to such areas as algebraic topology, analysis, and computer science.PRAISE FOR THE FIRST EDITION 'The author has done the field a service by producing an excellent text strong in the presentation of certain topological aspects of the underlying diagrams, e.g., which should serve the developing community and field well and which can be recommended without reservations as one of the volumes which should grace a poseteer's library whether she is interested only or mainly in the theory of these objects or has directed her gaze towards applications. As a sourcebook of ideas and understanding it will make its mark. And deservedly so.' — ZENTRALBLATT MATH“…A gem. Undergraduate mathematics and computer science majors will find the first chapters offering background that will serve them well in many courses. The rest of the book, which features many open problems, constitutes an accessible and stimulating invitation to research . . . Highly recommended.' — CHOICE“The author presents the field of ordered sets in an attractive way and the many open problems presented in the book are invaluable...[T]he book is a success, it presents an in depth and up to date carefully written coverage of ordered sets.' —SIGACT NEWS<

Inhaltsangaben

Preface.- Basics.- Chains, Antichains, and Fences.- Upper and Lower Bounds.- Retractions.- Constraint Satisfaction Problems.- Graphs and Homomorphisms.- Lexicographic Sums.- Lattices.- Truncated Lattices.- Dimension.- Interval Orders.- Sets P^Q = Hom (Q, P) and Products.- Enumeration of Ordered Sets.- Appendix A: Some Algebraic Topology.- Appendix B: Some Discrete Morse Theory.- References.- Index.

p-adic Numbers : 978-3-540-62911-5
Gouvea
An Introduction
Springer Berlin Heidelberg
Seiten VI, 306 p. 1 illus. in color.
Erscheinungsjahr 1997
ISBN 978-3-540-62911-5
       9783540629115
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Fachgebiet

Number Theory

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In the course of their undergraduate careers, most mathematics majors see little beyond 'standard mathematics:' basic real and complex analysis, ab­ stract algebra, some differential geometry, etc. There are few adventures in other territories, and few opportunities to visit some of the more exotic cor­ ners of mathematics. The goal of this book is to offer such an opportunity, by way of a visit to the p-adic universe. Such a visit offers a glimpse of a part of mathematics which is both important and fun, and which also is something of a meeting point between algebra and analysis. Over the last century, p-adic numbers and p-adic analysis have come to playa central role in modern number theory. This importance comes from the fact that they afford a natural and powerful language for talking about congruences between integers, and allow the use of methods borrowed from calculus and analysis for studying such problems. More recently, p-adic num­ bers have shown up in other areas of mathematics, and even in physics.

Inhaltsangaben

1 Apéritif.- 1 Apéritif.- 1.1 Hensel’s Analogy.- 1.2 Solving Congruences Modulopn.- 1.3 Other Examples.- 2 Foundations.- 2.1 Absolute Values on a Field.- 2.2 Basic Properties.- 2.3 Topology.- 2.4 Algebra.- 3 p-adic Numbers.- 3.1 Absolute Values on ?.- 3.2 Completions.- 3.3 Exploring ?p.- 3.4 Hensel’s Lemma.- 3.5 Local and Global.- 4 Elementary Analysis in ?p.- 4.1 Sequences and Series.- 4.2 Functions, Continuity, Derivatives.- 4.3 Power Series.- 4.4 Functions Defined by Power Series.- 4.5 Some Elementary Functions.- 4.6 Interpolation.- 5 Vector Spaces and Field Extensions.- 5.1 Normed Vector Spaces over Complete Valued Fields.- 5.2 Finite-dimensional Normed Vector Spaces.- 5.3 Finite Field Extensions.- 5.4 Properties of Finite Extensions.- 5.5 Analysis.- 5.6 Example: Adjoining a p-th Root of Unity.- 5.7 On to ?.- 6 Analysis in ?p.- 6.1 Almost Everything Extends.- 6.2 Deeper Results on Polynomials and Power Series.- 6.3 Entire Functions.- 6.4 Newton Polygons.- 6.5 Problems.- A Hints and Comments on the Problems.- B A Brief Glance at the Literature.- B.1 Texts.- B.2 Software.- B.3 Other Books.

Partitions, q-Series, and Modular Forms : 978-1-4614-0027-1
Alladi
Springer New York
Seiten XII, 224 p.
Erscheinungsjahr 2012
ISBN 978-1-4614-0027-1
       9781461400271
 (Hardcover)
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Number Theory

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Partitions, q-Series, and Modular Forms contains a collection of research and survey papers that grew out of a Conference on Partitions, q-Series and Modular Forms at the University of Florida, Gainesville in March 2008. It will be of interest to researchers and graduate students that would like to learn of recent developments in the theory of q-series and modular and how it relates to number theory, combinatorics and special functions.

Inhaltsangaben

-Preface (K. Alladi and F. Garvan).- 1. MacMahon's dream (G. E. Andrews and P. Paule).- 2. Ramanujan's elementary method in partition congruences (B. Berndt, C. Gugg, and S. Kim).- 3. Coefficients of harmonic Maass forms (K. Bringmann and K. Ono).- 4. On the growth of restricted partition functions (E. R. Canfield and H. Wilf).- 5. On applications of roots of unity to product identities (Z. Cao).- 6. Lecture hall sequences, q-series, and asymmetric partition identities (S. Corteel, C. Savage and A. Sills).- 7. Generalizations of Hutchinson's curve and the Thomae formula (H. Farkas).- 8. On the parity of the Rogers-Ramanujan coefficients (B. Gordon).- 9. A survey of the classical mock theta functions (B. Gordon and R. McIntosh).- 10. An application of the Cauchy-Sylvester theorem on compound determinants to a BC_n Jackson integral (M. Ito and S. Okada).- 11. Multiple generalizations of q-series identities found in Ramanujan's Lost Notebook (Y. Kajihara).- 12. Non-terminating q-Whipple transformations for basic hypergeometric series in U(n) (S. C. Milne and J. W. Newcomb).      

Pi: The Next Generation : 978-3-319-32375-6
Bailey
A Sourcebook on the Recent History of Pi and Its Computation
Springer International Publishing
Seiten XIV, 507 p.
Erscheinungsjahr 2016
ISBN 978-3-319-32375-6
       9783319323756
 (Hardcover)
Regular Price (Blue Sale Expired)
74,89
(inkl. 7% Mwst)
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