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978-1-0716-1417-4

Gareth James, University of Southern California, Los Angeles, CA, USA; Daniela Witten, University of Washington, Seattle, WA, USA; Trevor Hastie, Stanford University, Stanford, CA, USA; Robert Tibshirani, Stanford University, Stanford, CA, USA

,978-1-4614-7137-0,978-1-4614-7139-4,978-1-4614-7138-7,978-1-0716-1305-4

Preface.- 1 Introduction.- 2 Statistical Learning.- 3 Linear Regression.- 4 Classification.- 5 Resampling Methods.- 6 Linear Model Selection and Regularization.- 7 Moving Beyond Linearity.- 8 Tree-Based Methods.- 9 Support Vector Machines.- 10 Deep Learning.- 11 Survival Analysis and Censored Data.- 12 Unsupervised Learning.- 13 Multiple Testing.- Index.

An Introduction to Statistical Learning provides an accessible overview of the field of statistical learning, an essential toolset for making sense of the vast and complex data sets that have emerged in fields ranging from biology to finance to marketing to astrophysics in the past twenty years. This book presents some of the most important modeling and prediction techniques, along with relevant applications. Topics include linear regression, classification, resampling methods, shrinkage approaches, tree-based methods, support vector machines, clustering, deep learning, survival analysis, multiple testing, and more. Color graphics and real-world examples are used to illustrate the methods presented. Since the goal of this textbook is to facilitate the use of these statistical learning techniques by practitioners in science, industry, and other fields, each chapter contains a tutorial on implementing the analyses and methods presented in R, an extremely popular open source statistical software platform.Two of the authors co-wrote The Elements of Statistical Learning (Hastie, Tibshirani and Friedman, 2nd edition 2009), a popular reference book for statistics and machine learning researchers. An Introduction to Statistical Learning covers many of the same topics, but at a level accessible to a much broader audience. This book is targeted at statisticians and non-statisticians alike who wish to use cutting-edge statistical learning techniques to analyze their data. The text assumes only a previous course in linear regression and no knowledge of matrix algebra.This Second Edition features new chapters on deep learning, survival analysis, and multiple testing, as well as expanded treatments of naïve Bayes, generalized linear models, Bayesian additive regression trees, and matrix completion. R code has been updated throughout to ensure compatibility.

An Introduction to Statistical Learning provides an accessible overview of the field of statistical learning, an essential toolset for making sense of the vast and complex data sets that have emerged in fields ranging from biology to finance to marketing to astrophysics in the past twenty years. This book presents some of the most important modeling and prediction techniques, along with relevant applications. Topics include linear regression, classification, resampling methods, shrinkage approaches, tree-based methods, support vector machines, clustering, deep learning, survival analysis, multiple testing, and more. Color graphics and real-world examples are used to illustrate the methods presented. Since the goal of this textbook is to facilitate the use of these statistical learning techniques by practitioners in science, industry, and other fields, each chapter contains a tutorial on implementing the analyses and methods presented in R, an extremely popular open source statistical software platform.Two of the authors co-wrote The Elements of Statistical Learning (Hastie, Tibshirani and Friedman, 2nd edition 2009), a popular reference book for statistics and machine learning researchers. An Introduction to Statistical Learning covers many of the same topics, but at a level accessible to a much broader audience. This book is targeted at statisticians and non-statisticians alike who wish to use cutting-edge statistical learning techniques to analyze their data. The text assumes only a previous course in linear regression and no knowledge of matrix algebra.This Second Edition features new chapters on deep learning, survival analysis, and multiple testing, as well as expanded treatments of naïve Bayes, generalized linear models, Bayesian additive regression trees, and matrix completion. R code has been updated throughout to ensure compatibility.

Presents an essential statistical learning toolkit for practitioners in science, industry, and other fieldsDemonstrates application of the statistical learning methods in RIncludes new chapters on deep learning, survival analysis, and multiple testingCovers a range of topics, such as linear regression, classification, resampling methods, shrinkage approaches, tree-based methods, support vector machines, clustering, and deep learningFeatures extensive color graphics for a dynamic learning experienceIncludes supplementary material: sn.pub/extras

Gareth James is a professor of data sciences and operations, and the E. Morgan Stanley Chair in Business Administration, at the University of Southern California. He has published an extensive body of methodological work in the domain of statistical learning with particular emphasis on high-dimensional and functional data. The conceptual framework for this book grew out of his MBA elective courses in this area.Daniela Witten is a professor of statistics and biostatistics, and the Dorothy Gilford Endowed Chair, at the University of Washington. Her research focuses largely on statistical machine learning techniques for the analysis of complex, messy, and large-scale data, with an emphasis on unsupervised learning.Trevor Hastie and Robert Tibshirani are professors of statistics at Stanford University, and are co-authors of the successful textbook Elements of Statistical Learning. Hastie and Tibshirani developed generalized additive models and wrote a popular book of that title. Hastie co-developed much of the statistical modeling software and environment in R/S-PLUS and invented principal curves and surfaces. Tibshirani proposed the lasso and is co-author of the very successful An Introduction to the Bootstrap.

9781071614174

/Statistical Theory and Methods/Statistics/Mathematics and Computing/

/Statistical Theory and Methods/Statistics/Mathematics and Computing//Statistics and Computing/Statistics/Mathematics and Computing//Artificial Intelligence/Computer Science/Mathematics and Computing//Statistics/Mathematics and Computing///

,978-3-319-11079-0,978-3-319-11081-3,978-3-319-11080-6,978-3-319-30765-7,978-3-319-93902-5

Preface for the Instructor-Preface for the Student-Acknowledgments-1. Vector Spaces.- 2. Finite-Dimensional Vector Spaces.- 3. Linear Maps.- 4. Polynomials.- 5. Eigenvalues, Eigenvectors, and Invariant Subspaces.- 6. Inner Product Spaces.- 7. Operators on Inner Product Spaces.- 8. Operators on Complex Vector Spaces.- 9. Operators on Real Vector Spaces.- 10. Trace and Determinant-Photo Credits-Symbol Index-Index.

Now available in Open Access, this best-selling textbook for a second course in linear algebra is aimed at undergraduate math majors and graduate students. The fourth edition gives an expanded treatment of the singular value decomposition and its consequences. It includes a new chapter on multilinear algebra, treating bilinear forms, quadratic forms, tensor products, and an approach to determinants via alternating multilinear forms. This new edition also increases the use of the minimal polynomial to provide cleaner proofs of multiple results. Also, over 250 new exercises have been added.The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. Beautiful formatting creates pages with an unusually student-friendly appearance in both print and electronic versions.No prerequisites are assumed other than the usual demand for suitable mathematical maturity. The text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.From reviews of previous editions:
Altogether, the text is a didactic masterpiece. — zbMATH
The determinant-free proofs are elegant and intuitive. — American Mathematical Monthly
The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library — CHOICE

Now available in Open Access, this best-selling textbook for a second course in linear algebra is aimed at undergraduate math majors and graduate students. The fourth edition gives an expanded treatment of the singular value decomposition and its consequences. It includes a new chapter on multilinear algebra, treating bilinear forms, quadratic forms, tensor products, and an approach to determinants via alternating multilinear forms. This new edition also increases the use of the minimal polynomial to provide cleaner proofs of multiple results. Also, over 250 new exercises have been added.The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. Beautiful formatting creates pages with an unusually student-friendly appearance in both print and electronic versions. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. The text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.From the reviews of previous editions:
Altogether, the text is a didactic masterpiece. — zbMATHThe determinant-free proofs are elegant and intuitive. — American Mathematical Monthly
The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library — CHOICE

This book is open access, meaning free unlimited access to the eBookMaintains the classic approach of previous editionsFourth edition gives an expanded treatment of the singular value decomposition and its consequencesIncludes 250 additional exercises and several new topics

Sheldon Axler, Professor Emeritus of the Mathematics Department at San Francisco State University, has authored many well-received books including <div><div>Linear Algebra Done Right (in four editions) Measure, Integration & Real Analysis (Open Access) Precalculus: A Prelude to Calculus, Algebra & Trigonometry (in three editions)College Algebra Harmonic Function Theory (in two editions).</div><div><div>Axler has served as Editor-in-Chief of the Mathematical Intelligencer and Associate Editor of the American Mathematical Monthly. He has been a member of the Council of the American Mathematical Society and a member of the Board of Trustees of the Mathematical Sciences Research Institute. He is a Fellow of the American Mathematical Society and has been a recipient of numerous grants from the National Science Foundation.
</div><div>
</div></div></div>

9783031410253

/Linear Algebra/Algebra/Mathematics and Computing/Mathematics/

/Linear Algebra/Algebra/Mathematics and Computing/Mathematics//////

978-0-387-40272-7

Larry Wasserman, Carnegie Mellon University Dept. Statistics, Pittsburgh, PA, USA

Probability.- Random Variables.- Expectation.- Inequalities.- Convergence of Random Variables.- Models, Statistical Inference and Learning.- Estimating the CDF and Statistical Functionals.- The Bootstrap.- Parametric Inference.- Hypothesis Testing and p-values.- Bayesian Inference.- Statistical Decision Theory.- Linear and Logistic Regression.- Multivariate Models.- Inference about Independence.- Causal Inference.- Directed Graphs and Conditional Independence.- Undirected Graphs.- Loglinear Models.- Nonparametric Curve Estimation.- Smoothing Using Orthogonal Functions.- Classification.- Probability Redux: Stochastic Processes.- Simulation Methods.

This book is for people who want to learn probability and statistics quickly. It brings together many of the main ideas in modern statistics in one place. The book is suitable for students and researchers in statistics, computer science, data mining and machine learning.

This book covers a much wider range of topics than a typical introductory text on mathematical statistics. It includes modern topics like nonparametric curve estimation, bootstrapping and classification, topics that are usually relegated to follow-up courses. The reader is assumed to know calculus and a little linear algebra. No previous knowledge of probability and statistics is required. The text can be used at the advanced undergraduate and graduate level.

Larry Wasserman is Professor of Statistics at Carnegie Mellon University. He is also a member of the Center for Automated Learning and Discovery in the School of Computer Science. His research areas include nonparametric inference,asymptotic theory, causality, and applications to astrophysics, bioinformatics, and genetics. He is the 1999 winner of the Committee of Presidents of Statistical Societies Presidents' Award and the 2002 winner of the Centre de recherches mathematiques de Montreal–Statistical Society of Canada Prize in Statistics. He is Associate Editor of The Journal of the American Statistical Association and The Annals of Statistics. He is a fellow of the American Statistical Association and of the Institute of Mathematical Statistics.

Taken literally, the title 'All of Statistics' is an exaggeration. But in spirit, the title is apt, as the book does cover a much broader range of topics than a typical introductory book on mathematical statistics. This book is for people who want to learn probability and statistics quickly. It is suitable for graduate or advanced undergraduate students in computer science, mathematics, statistics, and related disciplines. <div>
</div><div>The book includes modern topics like non-parametric curve estimation, bootstrapping, and classification, topics that are usually relegated to follow-up courses. The reader is presumed to know calculus and a little linear algebra. No previous knowledge of probability and statistics is required. Statistics, data mining, and machine learning are all concerned with collecting and analysing data. </div>

Provides a concise introduction to a larger number of topics than are usually included in a graduate-level mathematical statistics class

Larry Wasserman is Professor of Statistics at Carnegie Mellon University. He is also a member of the Center for Automated Learning and Discovery in the School of Computer Science. His research areas include nonparametric inference, asymptotic theory, causality, and applications to astrophysics, bioinformatics, and genetics. He is the 1999 winner of the Committee of Presidents of Statistical Societies Presidents' Award and the 2002 winner of the Centre de recherches mathematiques de Montreal–Statistical Society of Canada Prize in Statistics. He is Associate Editor of The Journal of the American Statistical Association and The Annals of Statistics. He is a fellow of the American Statistical Association and of the Institute of Mathematical Statistics.

9780387402727

Statistics in Engineering, Physics, Computer Science, Chemistry and Earth Sciences

/Computational Mathematics and Numerical Analysis/Mathematics and Computing/Mathematics/

/Computational Mathematics and Numerical Analysis/Mathematics and Computing/Mathematics//Mathematics and Computing/Probability Theory/Mathematics//Complex Systems/Theoretical, Mathematical and Computational Physics/Physics and Astronomy/Physical Sciences/Applications of Mathematics/Mathematics and Computing/Mathematics//Statistical Theory and Methods/Statistics/Mathematics and Computing//Probability and Statistics in Computer Science/Mathematics of Computing/Computer Science/Mathematics and Computing//Statistics in Engineering, Physics, Computer Science, Chemistry and Earth Sciences/Applied Statistics/Statistics/Mathematics and Computing/

978-0-387-96787-5

Serge Lang, Yale University Dept. Mathematics, New Haven, CT, USA

1 Numbers.- 2 Linear Equations.- 3 Real Numbers.- 4 Quadratic Equations Interlude On Logic and Mathematical Expressions.- Interlude On Logic and Mathematical Expressions.- 5 Distance and Angles.- 6 Isometries.- 7 Area and Applications.- 8 Coordinates and Geometry.- 9 Operations on Points.- 10 Segments, Rays, and Lines.- 11 Trigonometry.- 12 Some Analytic Geometry.- 13 Functions.- 14 Mappings.- 15 Complex Numbers.- 16 Induction and Summations.- 17 Determinants.

This is a text in basic mathematics with multiple uses for either high school or college level courses. Readers will get a firm foundation in basic principles of mathematics which are necessary to know in order to go ahead in calculus, linear algebra or other topics. The subject matter is clearly covered and the author develops concepts so the reader can see how one subject matter can relate and grow into another.

9780387967875

10.1007/978-1-4612-1027-6

978-1-4419-9981-8

,978-0-387-95448-6,978-0-387-95495-0,978-1-4757-5601-2,978-0-387-21752-9

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.

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 newtopics 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.

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 fewnew 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.

New edition extensively revised and clarified, and topics have been substantially rearrangedIntroduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier in the textAdded topics include 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 structuresIncludes supplementary material: sn.pub/extrasIncludes supplementary material: sn.pub/extras

John M. Lee is Professor of Mathematics at the University of Washington in Seattle, where he regularly teaches graduate courses on the topology and geometry of manifolds. He was the recipient of the American Mathematical Society's Centennial Research Fellowship and he is the author of four previous Springer books: the first edition (2003) of Introduction to Smooth Manifolds, the first edition (2000) and second edition (2010) of Introduction to Topological Manifolds, and Riemannian Manifolds: An Introduction to Curvature (1997).

9781441999818

/Differential Geometry/Geometry/Mathematics and Computing/Mathematics/

/Differential Geometry/Geometry/Mathematics and Computing/Mathematics//////

10.1007/978-1-4419-9982-5

978-0-387-40101-0

1 General Probability Theory.- 2 Information and Conditioning.- 3 Brownian Motion.- 4 Stochastic Calculus.- 5 Risk-Neutral Pricing.- 6 Connections with Partial Differential Equations.- 7 Exotic Options.- 8 American Derivative Securities.- 9 Change of Numéraire.- 10 Term-Structure Models.- 11 Introduction to Jump Processes.- A Advanced Topics in Probability Theory.- A.1 Countable Additivity.- A.3 Random Variable with Neither Density nor Probability Mass Function.- B Existence of Conditional Expectations.- C Completion of the Proof of the Second Fundamental Theorem of Asset Pricing.- References.

Stochastic Calculus for Finance evolved from the first ten years of the Carnegie Mellon Professional Master's program in Computational Finance. The content of this book has been used successfully with students whose mathematics background consists of calculus and calculus-based probability. The text gives both precise statements of results, plausibility arguments, and even some proofs, but more importantly intuitive explanations developed and refine through classroom experience with this material are provided. The book includes a self-contained treatment of the probability theory needed for stochastic calculus, including Brownian motion and its properties. Advanced topics include foreign exchange models, forward measures, and jump-diffusion processes. This book is being published in two volumes. This second volume develops stochastic calculus, martingales, risk-neutral pricing, exotic options and term structure models, all in continuous time. Masters level students and researchers in mathematical finance and financial engineering will find this book useful. Steven E. Shreve is Co-Founder of the Carnegie Mellon MS Program in Computational Finance and winner of the Carnegie Mellon Doherty Prize for sustained contributions to education.

Stochastic Calculus for Finance evolved from the first ten years of the Carnegie Mellon Professional Master's program in Computational Finance. The content of this book has been used successfully with students whose mathematics background consists of calculus and calculus-based probability. The text gives both precise statements of results, plausibility arguments, and even some proofs, but more importantly intuitive explanations developed and refine through classroom experience with this material are provided. The book includes a self-contained treatment of the probability theory needed for stochastic calculus, including Brownian motion and its properties. Advanced topics include foreign exchange models, forward measures, and jump-diffusion processes. This book is being published in two volumes. This second volume develops stochastic calculus, martingales, risk-neutral pricing, exotic options and term structure models, all in continuous time. Master's level students and researchers in mathematical finance and financial engineering will find this book useful.

<p>Developed for the professional Master's program in Computational Finance at Carnegie Mellon, the leading financial engineering program in the U.S.</p><p>Tested in the classroom and revised over a period of several years</p><p>Includes supplementary material: sn.pub/extras</p>

Steven E. Shreve is Co-Founder of the Carnegie Mellon MS Program in Computational Finance and winner of the Carnegie Mellon Doherty Prize for sustained contributions to education.

9780387401010

/Mathematics in Business, Economics and Finance/Applications of Mathematics/Mathematics and Computing/Mathematics/

/Mathematics in Business, Economics and Finance/Applications of Mathematics/Mathematics and Computing/Mathematics//Applications of Mathematics/Mathematics and Computing/Mathematics//Mathematics and Computing/Probability Theory/Mathematics//Public Economics/Economics/Humanities and Social Sciences//Economics/Humanities and Social Sciences/Financial Economics/

10.1007/978-1-4757-4296-1

978-0-387-95385-4

Serge Lang, Yale University Dept. Mathematics, New Haven, CT, USA

One The Basic Objects of Algebra.- I Groups.- II Rings.- III Modules.- IV Polynomials.- Two Algebraic Equations.- V Algebraic Extensions.- VI Galois Theory.- VII Extensions of Rings.- VIII Transcendental Extensions.- IX Algebraic Spaces.- X Noetherian Rings and Modules.- XI Real Fields.- XII Absolute Values.- Three Linear Algebra and Representations.- XIII Matrices and Linear Maps.- XIV Representation of One Endomorphism.- XV Structure of Bilinear Forms.- XVI The Tensor Product.- XVII Semisimplicity.- XVIII Representations of Finite Groups.- XIX The Alternating Product.- Four Homological Algebra.- XX General Homology Theory.- XXI Finite Free Resolutions.- Appendix 2 Some Set Theory.

This book is intended as a basic text for a one-year course in Algebra at the graduate level, or as a useful reference for mathematicians and professionals who use higher-level algebra. It successfully addresses the basic concepts of algebra. For the revised third edition, the author has added exercises and made numerous corrections to the text.

Comments on Serge Lang's Algebra:
Lang's Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books.
April 1999 Notices of the AMS, announcing that the author was awarded the Leroy P. Steele Prize for Mathematical Exposition for his many mathematics books.

The author has an impressive knack for presenting the important and interesting ideas of algebra in just the 'right' way, and he never gets bogged down in the dry formalism which pervades some parts of algebra.
MathSciNet's review of the first edition

From April 1999 Notices of the AMS, announcing that the author was awarded the Leroy P. Steele Prize for Mathematical Exposition for his many mathematics books: 'Lang's Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books.'
From MathSciNet's review of the first edition:
'The author has an impressive knack for presenting the important and interesting ideas of algebra in just the 'right' way, and he never gets bogged down in the dry formalism which pervades some parts of algebra.'
This book is intended as a basic text for a one-year course in Algebra at the graduate level, or as a useful reference for mathematicians and professionals who use higher-level algebra. This book successfully addresses all of the basic concepts of algebra. For the new edition, the author has added exercises and made numerous corrections to the text.

This book is considered a classic, which "changed the way algebra was taught" (Notices of the AMS)

9780387953854

/Algebra/Mathematics and Computing/Mathematics//Commutative Rings and Algebras/Algebra/Mathematics and Computing/Mathematics//Linear Algebra/Algebra/Mathematics and Computing/Mathematics//Associative Rings and Algebras/Algebra/Mathematics and Computing/Mathematics//Group Theory and Generalizations/Algebra/Mathematics and Computing/Mathematics//

10.1007/978-1-4613-0041-0

978-3-662-57264-1

Martin Aigner, Freie Universität Berlin, Berlin, Germany; Günter M. Ziegler, Freie Universität Berlin, Berlin, Germany

,978-3-662-44204-3,978-3-662-44206-7,978-3-662-44205-0,978-3-662-49592-6

Number Theory: 1. Six proofs of the infinity of primes.- 2. Bertrand’s postulate.- 3. Binomial coefficients are (almost) never powers.- 4. Representing numbers as sums of two squares.- 5. The law of quadratic reciprocity.- 6. Every finite division ring is a field.- 7. The spectral theorem and Hadamard’s determinant problem.- 8. Some irrational numbers.- 9. Three times π2/6.- Geometry: 10. Hilbert’s third problem: decomposing polyhedral.- 11. Lines in the plane and decompositions of graphs.- 12. The slope problem.- 13. Three applications of Euler’s formula.- 14. Cauchy’s rigidity theorem.- 15. The Borromean rings don’t exist.- 16. Touching simplices.- 17. Every large point set has an obtuse angle.- 18. Borsuk’s conjecture.- Analysis: 19. Sets, functions, and the continuum hypothesis.- 20. In praise of inequalities.- 21. The fundamental theorem of algebra.- 22. One square and an odd number of triangles.- 23. A theorem of Pólya on polynomials.- 24. Van der Waerden's permanent conjecture.- 25. On a lemma of Littlewood and Offord.- 26. Cotangent and the Herglotz trick.- 27. Buffon’s needle problem.- Combinatorics: 28. Pigeon-hole and double counting.- 29. Tiling rectangles.- 30. Three famous theorems on finite sets.- 31. Shuffling cards.- 32. Lattice paths and determinants.- 33. Cayley’s formula for the number of trees.- 34. Identities versus bijections.- 35. The finite Kakeya problem.- 36. Completing Latin squares.- Graph Theory: 37. Permanents and the power of entropy.- 38. The Dinitz problem.- 39. Five-coloring plane graphs.- 40. How to guard a museum.- 41. Turán’s graph theorem.- 42. Communicating without errors.- 43. The chromatic number of Kneser graphs.- 44. Of friends and politicians.- 45. Probability makes counting (sometimes) easy.- About the Illustrations.- Index.

This revised and enlarged sixth edition of Proofs from THE BOOK features an entirely new chapter on Van der Waerden’s permanent conjecture, as well as additional, highly original and delightful proofs in other chapters.

From the citation on the occasion of the 2018 'Steele Prize for Mathematical Exposition'

“… It is almost impossible to write a mathematics book that can be read and enjoyed by people of all levels and backgrounds, yet Aigner and Ziegler accomplish this feat of exposition with virtuoso style. […] This book does an invaluable service to mathematics, by illustrating for non-mathematicians what it is that mathematicians mean when they speak about beauty.”

From the Reviews

'... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. ... Aigner and Ziegler... write: '... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations.' I do. ... '

Notices of the AMS, August 1999

'... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures and beautiful drawings ... It is a pleasure to read as well: the style is clear and entertaining, the level is close to elementary, the necessary background is given separately and the proofs are brilliant. ...'

LMS Newsletter, January 1999

'Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdös. The theorems are so fundamental, their proofs so elegant and the remaining open questions so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. ... '

SIGACT News, December 2011

This revised and enlarged sixth edition of Proofs from THE BOOK features an entirely new chapter on Van der Waerden’s permanent conjecture, as well as additional, highly original and delightful proofs in other chapters.From the citation on the occasion of the 2018 'Steele Prize for Mathematical Exposition' “… It is almost impossible to write a mathematics book that can be read and enjoyed by people of all levels and backgrounds, yet Aigner and Ziegler accomplish this feat of exposition with virtuoso style. […] This book does an invaluable service to mathematics, by illustrating for non-mathematicians what it is that mathematicians mean when they speak about beauty.”From the Reviews'... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. ... Aigner and Ziegler... write: '... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations.' I do. ... 'Notices of the AMS, August 1999'... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures and beautiful drawings ... It is a pleasure to read as well: the style is clear and entertaining, the level is close to elementary, the necessary background is given separately and the proofs are brilliant. ...'LMS Newsletter, January 1999'Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdös. The theorems are so fundamental, their proofs so elegant and the remaining open questions so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. ... ' SIGACT News, December 2011

<p>Revised and enlarged sixth edition</p><p>New chapter on Van der Waerden’s permanent conjecture</p><p>New sections on the asymptotics for the number of Latin squares</p><p>New proof for the Basel problem</p><p>Geometric explanation for the involution proof for Fermat's two squares theorem</p><p>Presents some recent jewels and surprises</p>

Martin Aigner received his Ph.D. from the University of Vienna and has been professor of mathematics at the Freie Universität Berlin since 1974. He has published in various fields of combinatorics and graph theory and is the author of several monographs on discrete mathematics, among them the Springer books Combinatorial Theory and A Course on Enumeration. Martin Aigner is a recipient of the 1996 Lester R. Ford Award for mathematical exposition of the Mathematical Association of America MAA.Günter M. Ziegler received his Ph.D. from M.I.T. and has been professor of mathematics in Berlin – first at TU Berlin, now at Freie Universität – since 1995. He has published in discrete mathematics, geometry, topology, and optimization, including the Lectures on Polytopes with Springer, as well as „Do I Count? Stories from Mathematics“. Günter M. Ziegler is a recipient of the 2006 Chauvenet Prize of the MAA for his expository writing and the 2008 Communicator award of the German Science Foundation.Martin Aigner and Günter M. Ziegler have started their work on Proofs from THE BOOK in 1995 together with Paul Erdös. The first edition of this book appeared in 1998 – it has since been translated into 13 languages: Brazilian, Chinese, German, Farsi, French, Hungarian, Italian, Japanese, Korean, Polish, Russian, Spanish, and Turkish.

9783662572641

/Number Theory/Mathematics and Computing/Mathematics//Geometry/Mathematics and Computing/Mathematics//Analysis/Mathematics and Computing/Mathematics//Discrete Mathematics/Mathematics and Computing/Mathematics//Graph Theory/Discrete Mathematics/Mathematics and Computing/Mathematics//Mathematics of Computing/Computer Science/Mathematics and Computing/

978-0-387-96890-2

,978-1-4757-1695-5,978-1-4757-1694-8,978-0-387-90314-9,978-1-4757-1693-1

I Newtonian Mechanics.- 1 Experimental facts.- 2 Investigation of the equations of motion.- II Lagrangian Mechanics.- 3 Variational principles.- 4 Lagrangian mechanics on manifolds.- 5 Oscillations.- 6 Rigid bodies.- III Hamiltonian Mechanics.- 7 Differential forms.- 8 Symplectic manifolds.- 9 Canonical formalism.- 10 Introduction to perturbation theory.- Appendix 1 Riemannian curvature.- Appendix 2 Geodesics of left-invariant metrics on Lie groups and the hydrodynamics of ideal fluids.- Appendix 3 Symplectic structures on algebraic manifolds.- Appendix 4 Contact structures.- Appendix 5 Dynamical systems with symmetries.- Appendix 6 Normal forms of quadratic hamiltonians.- Appendix 7 Normal forms of hamiltonian systems near stationary points and closed trajectories.- Appendix 8 Theory of perturbations of conditionally periodic motion, and Kolmogorov’s theorem.- Appendix 9 Poincaré’s geometric theorem, its generalizations and applications.- Appendix 10 Multiplicities of characteristic frequencies, and ellipsoids depending on parameters.- Appendix 11 Short wave asymptotics.- Appendix 12 Lagrangian singularities.- Appendix 13 The Korteweg-de Vries equation.- Appendix 14 Poisson structures.- Appendix 15 On elliptic coordinates.- Appendix 16 Singularities of ray systems.

In this text, the author constructs the mathematical apparatus of classical mechanics from the beginning, examining all the basic problems in dynamics, including the theory of oscillations, the theory of rigid body motion, and the Hamiltonian formalism. This modern approch, based on the theory of the geometry of manifolds, distinguishes iteself from the traditional approach of standard textbooks. Geometrical considerations are emphasized throughout and include phase spaces and flows, vector fields, and Lie groups. The work includes a detailed discussion of qualitative methods of the theory of dynamical systems and of asymptotic methods like perturbation techniques, averaging, and adiabatic invariance.

9780387968902

/Analysis/Mathematics and Computing/Mathematics//Theoretical, Mathematical and Computational Physics/Physics and Astronomy/Physical Sciences/////

978-0-387-98403-2

,978-0-387-90036-0,978-0-387-90035-3,978-1-4612-9840-3,978-1-4612-9839-7

I. Categories, Functors, and Natural Transformations.- II. Constructions on Categories.- III. Universals and Limits.- IV. Adjoints.- V Limits.- VI. Monads and Algebras.- VII. Monoids.- VIII. Abelian Categories.- IX. Special Limits.- X. Kan Extensions.- XI. Symmetry and Braiding in Monoidal Categories.- XII. Structures in Categories.- Appendix. Foundations.- Table of Standard Categories: Objects and Arrows.- Table of Terminology.

Categories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. The book then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterized by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including two new chapters on topics of active interest. One is on symmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories.

9780387984032

10.1007/978-1-4757-4721-8

978-1-4419-7939-1

,978-0-387-95026-6,978-0-387-98759-0,978-1-4757-7431-3,978-0-387-22727-6

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.

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.

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.

<p>New edition extensively revised and updated</p><p>New introduction to CW complexes (along with a brief and streamlined introduction to simplicial complexes)</p><p>Expanded treatments of manifolds with boundary, local compactness, group actions, proper maps, and a new section on paracompactness</p>

John M. Lee is a professor of mathematics at the University of Washington. His previous Springer textbooks in the Graduate Texts in Mathematics series include the first edition of Introduction to Topological Manifolds, Introduction to Smooth Manifolds, and Riemannian Manifolds: An Introduction.

9781441979391

/Manifolds and Cell Complexes/Topology/Mathematics and Computing/Mathematics/

/Manifolds and Cell Complexes/Topology/Mathematics and Computing/Mathematics//Algebraic Topology/Topology/Mathematics and Computing/Mathematics/////

10.1007/978-1-4419-7940-7

Sheldon Axler, San Francisco State University, San Francisco, CA, USA

About the Author.- Preface for Students.- Preface for Instructors.- Acknowledgments.- 1. Riemann Integration.- 2. Measures.- 3. Integration.- 4. Differentiation.- 5. Product Measures.- 6. Banach Spaces.- 7. L^p Spaces.- 8. Hilbert Spaces.- 9. Real and Complex Measures.- 10. Linear Maps on Hilbert Spaces.- 11. Fourier Analysis.- 12. Probability Measures.- Photo Credits.- Bibliography.- Notation Index.- Index.- Colophon: Notes on Typesetting.

This open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics.Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn.Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn–Banach Theorem, Hölder’s Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability.Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysis that isfreely available online.

This open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics.Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn.Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn–Banach Theorem, Hölder’s Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability.Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysisthat is freely available online. For errata and updates, visit https://measure.axler.net/

Electronic version is free to the world via Springer’s Open Access programProvides student-friendly explanations with ample examples and exercises throughoutIncludes chapters on Hilbert space operators, Fourier analysis, and probability measuresPrepares students for further graduate studies by promoting a deep understanding of key conceptsIncludes supplementary material: sn.pub/extras

Sheldon Axler is Professor of Mathematics at San Francisco State University. He has won teaching awards at MIT and Michigan State University. His career achievements include the Mathematical Association of America’s Lester R. Ford Award for expository writing, election as Fellow of the American Mathematical Society, over a decade as Dean of the College of Science & Engineering at San Francisco State University, member of the Council of the American Mathematical Society, member of the Board of Trustees of the Mathematical Sciences Research Institute, and Editor-in-Chief of the Mathematical Intelligencer. His previous publications include the widely used textbook Linear Algebra Done Right.

9783030331429

/Measure and Integration/Analysis/Mathematics and Computing/Mathematics/

/Measure and Integration/Analysis/Mathematics and Computing/Mathematics//////

978-3-540-25484-3

Leonid Koralov, University of Maryland Dept. Mathematics, College Park, MD, USA; Yakov G. Sinai, Princeton University Dept. Mathematics, Princeton, NJ, USA

Originally published as Springer Textbook: Probability Theory. An Introductory Course

Probability Theory.- Random Variables and Their Distributions.- Sequences of Independent Trials.- Lebesgue Integral and Mathematical Expectation.- Conditional Probabilities and Independence.- Markov Chains with a Finite Number of States.- Random Walks on the Lattice ?d.- Laws of Large Numbers.- Weak Convergence of Measures.- Characteristic Functions.- Limit Theorems.- Several Interesting Problems.- Random Processes.- Basic Concepts.- Conditional Expectations and Martingales.- Markov Processes with a Finite State Space.- Wide-Sense Stationary Random Processes.- Strictly Stationary Random Processes.- Generalized Random Processes.- Brownian Motion.- Markov Processes and Markov Families.- Stochastic Integral and the Ito Formula.- Stochastic Differential Equations.- Gibbs Random Fields.

A one-year course in probability theory and the theory of random processes, taught at Princeton University to undergraduate and graduate students, forms the core of the content of this book
It is structured in two parts: the first part providing a detailed discussion of Lebesgue integration, Markov chains, random walks, laws of large numbers, limit theorems, and their relation to Renormalization Group theory. The second part includes the theory of stationary random processes, martingales, generalized random processes, Brownian motion, stochastic integrals, and stochastic differential equations. One section is devoted to the theory of Gibbs random fields.
This material is essential to many undergraduate and graduate courses. The book can also serve as a reference for scientists using modern probability theory in their research.

Comprehensive, self-contained exposition of classical probability theory and the theory of random processesDwells on a number of modern topics, not addressed in most textbooksAuthor Ya. G. Sinai is one of the world's leading probabilists and mathematical physicistsIncludes supplementary material: sn.pub/extras

YAKOV SINAI has been a professor at Princeton University since 1993. He was educated at Moscow State University, and was a professor there till 1993.
Since 1971 he has also held the position of senior researcher at the Landau Institute of Theoretical Physics. He is known for fundamental work on dynamical systems, probability theory, mathematical physics, and statistical mechanics. He has been awarded, among other honors, the Boltzmann Medal (in 1986) and Wolf Prize in Mathematics (in 1997). He is a member of Russian and American Academies of Sciences.LEONID KORALOV is an assistant professor at the University of Maryland. From 2000 till 2006 he was an assistant professor at Princeton University, prior to which he worked at the Institute for Advanced Study in Princeton. He did his undergraduate work at Moscow State University, and got his PhD from SUNY at Stony Brook in 1998. He works on problems in the areas of homogenization, diffusion processes, and partial differential equations.

9783540254843

/Mathematics and Computing/Probability Theory/Mathematics//////

10.1007/978-3-540-68829-7

978-3-540-04758-2

Some Mathematical Preliminaries.- Itô Integrals.- The Itô Formula and the Martingale Representation Theorem.- Stochastic Differential Equations.- The Filtering Problem.- Diffusions: Basic Properties.- Other Topics in Diffusion Theory.- Applications to Boundary Value Problems.- Application to Optimal Stopping.- Application to Stochastic Control.- Application to Mathematical Finance.

This well-established textbook on stochastic differential equations has turned out to be very useful to non-specialists of the subject and has sold steadily in 5 editions, both in the EU and US marketIncludes supplementary material: sn.pub/extras

9783540047582

/Analysis/Mathematics and Computing/Mathematics//Mathematics and Computing/Probability Theory/Mathematics//Theoretical, Mathematical and Computational Physics/Physics and Astronomy/Physical Sciences//Systems Theory, Control /Optimization/Mathematics and Computing/Mathematics//Calculus of Variations and Optimization/Optimization/Mathematics and Computing/Mathematics//Differential Equations/Analysis/Mathematics and Computing/Mathematics/

10.1007/978-3-642-14394-6

978-0-85729-081-6

Marek Capiński, AGH University of Science and Technology Faculty of Applied Mathematics, Kraków, Poland; Tomasz Zastawniak, University of York Department of Mathematics, Heslington, York, UK

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.

As with the first edition, 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 the Capital Asset Pricing Model, and basic stochastic interest rate models in discrete setting.In this second edition, the material has been thoroughly revised and rearranged. New features include:• A case study to begin each chapter – a real-life situation motivating the development of theoretical tools;• A detailed discussion of the case study at the end of each chapter;• A new chapter on time-continuous models with intuitive outlines of the mathematical arguments and constructions;• Complete proofs of the two fundamental theorems of mathematical finance in discrete setting.From the reviews of the first edition:”This text is an excellent introduction to Mathematical Finance. Armed with a knowledge of basic calculus and probability a student can use this book to learn about derivatives, interest rates and their term structure and portfolio management.”(Zentralblatt MATH)”Given these basic tools, it is surprising how high a level of sophistication the authors achieve, covering such topics as arbitrage-free valuation, binomial trees, and risk-neutral valuation.” (www.riskbook.com)”The reviewer can only congratulate the authors with successful completion of a difficult task of writing a useful textbook on a traditionally hard topic.” (K. Borovkov, The Australian Mathematical Society Gazette, Vol. 31 (4), 2004)

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.

A case study to begin each chapter – a real-life situation motivating the development of theoretical toolsA detailed discussion of the case study at the end of each chapterA new chapter on time-continuous models with intuitive outlines of the mathematical arguments and constructionsComplete proofs of the two fundamental theorems of mathematical finance in discrete settingIncludes supplementary material: sn.pub/extras

​Marek Capinski is Professor of Mathematics at AGH University of Science and Technology, Poland. <div>
</div><div>Tomasz Zastawniak is Professor of Mathematics at the University of York, UK.
</div>

9780857290816

/Mathematics in Business, Economics and Finance/Applications of Mathematics/Mathematics and Computing/Mathematics/

/Mathematics in Business, Economics and Finance/Applications of Mathematics/Mathematics and Computing/Mathematics//Economics/Humanities and Social Sciences/Financial Economics/////

10.1007/978-0-85729-082-3

Charles Chapman Pugh, University of California Dept of Mathematics, Berkeley, CA, USA

,978-1-4419-2941-9,978-0-387-95297-0,978-1-4684-9541-6,978-0-387-21684-3,978-1-4939-7071-1

Real Numbers.- A Taste of Topology.- Functions of a Real Variable.- Function Spaces.- Multivariable Calculus.- Lebesgue Theory.

Based on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.<br/><br/>New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri’s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points — which are rarely treated in books at this level — and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.

Based on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.<br/><br/>New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri’s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points — which are rarely treated in books at this level — and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.

Elucidates abstract concepts and salient points in proofs with over 150 detailed illustrationsTreats the rigorous foundations of both single and multivariable CalculusGives an intuitive presentation of Lebesgue integration using the undergraph approach of BurkillIncludes over 500 exercises that are interesting and thought-provoking, not merely routine

Charles C. Pugh is Professor Emeritus at the University of California, Berkeley. His research interests include geometry and topology, dynamical systems, and normal hyperbolicity.

9783319177700

/Measure and Integration/Analysis/Mathematics and Computing/Mathematics/

/Measure and Integration/Analysis/Mathematics and Computing/Mathematics//Real Functions/Analysis/Mathematics and Computing/Mathematics//Sequences, Series, Summability/Analysis/Mathematics and Computing/Mathematics////

,978-1-4614-3614-0,978-1-4614-3616-4,978-1-4899-8967-3,978-1-4614-3615-7

1) Markov Chains.- 2) Poisson Processes.- 3) Renewal Processes.- 4) Continuous Time Markov Chains.- 5) Martingales.- 6) Mathematical Finance.- 7) A Review of Probability.

Building upon the previous editions, this textbook is a first course in stochastic processes taken by undergraduate and graduate students (MS and PhD students from math, statistics, economics, computer science, engineering, and finance departments) who have had a course in probability theory. It covers Markov chains in discrete and continuous time, Poisson processes, renewal processes, martingales, and option pricing. One can only learn a subject by seeing it in action, so there are a large number of examples and more than 300 carefully chosen exercises to deepen the reader’s understanding.

Drawing from teaching experience and student feedback, there are many new examples and problems with solutions that use TI-83 to eliminate the tedious details of solving linear equations by hand, and the collection of exercises is much improved, with many more biological examples. Originally included in previous editions, material too advanced for this first course in stochastic processes has been eliminated while treatment of other topics useful for applications has been expanded. In addition, the ordering of topics has been improved; for example, the difficult subject of martingales is delayed until its usefulness can be applied in the treatment of mathematical finance.
• A concise treatment and textbook on the most important topics in Stochastic Processes
• Illustrates all concepts with examples and presents more than 300 carefully chosen exercises for effective learning
• New edition includes added and revised exercises, including many biological exercises, in addition to restructured and rewritten sections with a goal toward clarity and simplicity

Richard Durrett received his Ph.D. in Operations Research from Stanford in 1976. He taught at the UCLA mathematics department for 9 years and at Cornell for 25 years before moving to Duke in 2010. He is author of 8 books and more than 200 journal articles and has supervised more that 45 Ph.D. students. He is a member of the National Academy of Science. Most of his current research concerns the applications of probability to biology: ecology, genetics, and cancer modeling.

Building upon the previous editions, this textbook is a first course in stochastic processes taken by undergraduate and graduate students (MS and PhD students from math, statistics, economics, computer science, engineering, and finance departments) who have had a course in probability theory. It covers Markov chains in discrete and continuous time, Poisson processes, renewal processes, martingales, and option pricing. One can only learn a subject by seeing it in action, so there are a large number of examples and more than 300 carefully chosen exercises to deepen the reader’s understanding.

Drawing from teaching experience and student feedback, there are many new examples and problems with solutions that use TI-83 to eliminate the tedious details of solving linear equations by hand, and the collection of exercises is much improved, with many more biological examples. Originally included in previous editions, material too advanced for this first course in stochastic processes has been eliminated while treatment of other topics useful for applications has been expanded. In addition, the ordering of topics has been improved; for example, the difficult subject of martingales is delayed until its usefulness can be applied in the treatment of mathematical finance.

A concise treatment and textbook on the most important topics in Stochastic ProcessesAll concepts illustrated by examples and more than 300 carefully chosen exercises for effective learningNew edition includes added and revised exercises, including many biological exercises, in addition to restructured and rewritten sections with a goal toward clarity and simplicityIncludes supplementary material: sn.pub/extras

Richard Durrett received his Ph.D. in Operations Research from Stanford in 1976. He taught at the UCLA mathematics department for 9 years and at Cornell for 25 years before moving to Duke in 2010. He is author of 8 books and more than 200 journal articles and has supervised more that 45 Ph.D. students. He is a member of the National Academy of Science. Most of his current research concerns the applications of probability to biology: ecology, genetics, and most recently cancer.

9783319456133

/Statistical Theory and Methods/Statistics/Mathematics and Computing/

/Statistical Theory and Methods/Statistics/Mathematics and Computing//Mathematics and Computing/Probability Theory/Mathematics//Statistics in Business, Management, Economics, Finance, Insurance/Applied Statistics/Statistics/Mathematics and Computing//Operations Research, Management Science /Optimization/Mathematics and Computing/Mathematics//Population Genetics/Biological Sciences/Life Sciences/Genetics and Genomics//Quantitative Economics/Economics/Humanities and Social Sciences/

978-1-4419-6052-8

,978-1-4419-2955-6,978-0-387-95336-6,978-1-4684-9282-8,978-1-4684-9281-1

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.

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)'The book...is presented in a lively style without unnecessary detail. It is very stimulating and will be appreciated not only by students. Much attention is paid to problems and to the development of mathematics before the end of the nineteenth century... This book brings to the non-specialist interested in mathematics many interesting results. It can be recommended for seminars and willbe enjoyed by the broad mathematical community.' (European Mathematical Society)'Since Stillwell treats many topics, most mathematicians will learn a lot from this book as well as they will find pleasant and rather clear expositions of custom materials. The book is accessible to students that have already experienced calculus, algebra and geometry and will give them a good account of how the different branches of mathematics interact.'(Denis Bonheure, Bulletin of the Belgian 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.

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.

New edition extensively revised and updated The author’s style and exposition are uniqueFeatures new exercises throughout the book Contains a new section on the Poincare conjecture Includes new chapters on simple groups and combinatorics

John Stillwell is a professor of mathematics at the University of San Francisco. He is also an accomplished author, having published several books with Springer, including The Four Pillars of Geometry; Elements of Algebra; Numbers and Geometry; and many more.

9781441960528

/History of Mathematical Sciences/Mathematics and Computing/Mathematics/

/History of Mathematical Sciences/Mathematics and Computing/Mathematics//Geometry/Mathematics and Computing/Mathematics//Number Theory/Mathematics and Computing/Mathematics//Analysis/Mathematics and Computing/Mathematics///

978-1-4614-7115-8

1 The Experimental Origins of Quantum Mechanics.- 2 A First Approach to Classical Mechanics.- 3 A First Approach to Quantum Mechanics.- 4 The Free Schrödinger Equation.- 5 A Particle in a Square Well.- 6 Perspectives on the Spectral Theorem.- 7 The Spectral Theorem for Bounded Self-Adjoint Operators: Statements.- 8 The Spectral Theorem for Bounded Sef-Adjoint Operators: Proofs.- 9 Unbounded Self-Adjoint Operators.- 10 The Spectral Theorem for Unbounded Self-Adjoint Operators.- 11 The Harmonic Oscillator.- 12 The Uncertainty Principle.- 13 Quantization Schemes for Euclidean Space.- 14 The Stone–von Neumann Theorem.- 15 The WKB Approximation.- 16 Lie Groups, Lie Algebras, and Representations.- 17 Angular Momentum and Spin.- 18 Radial Potentials and the Hydrogen Atom.- 19 Systems and Subsystems, Multiple Particles.- V Advanced Topics in Classical and Quantum Mechanics.- 20 The Path-Integral Formulation of Quantum Mechanics.- 21 Hamiltonian Mechanics on Manifolds.- 22 Geometric Quantization on Euclidean Space.- 23 Geometric Quantization on Manifolds.- A Review of Basic Material.- References.​- Index.

Although ideas from quantum physics play an important role in many parts of modern mathematics, there are few books about quantum mechanics aimed at mathematicians. This book introduces the main ideas of quantum mechanics in language familiar to mathematicians. Readers with little prior exposure to physics will enjoy the book's conversational tone as they delve into such topics as the Hilbert space approach to quantum theory; the Schrödinger equation in one space dimension; the Spectral Theorem for bounded and unbounded self-adjoint operators; the Stone–von Neumann Theorem; the Wentzel–Kramers–Brillouin approximation; the role of Lie groups and Lie algebras in quantum mechanics; and the path-integral approach to quantum mechanics.The numerous exercises at the end of each chapter make the book suitable for both graduate courses and independent study. Most of the text is accessible to graduate students in mathematics who have had a first course in real analysis, covering the basics of L2 spaces and Hilbert spaces.  The final chapters introduce readers who are familiar with the theory of manifolds to more advanced topics, including geometric quantization.

Although ideas from quantum physics play an important role in many parts of modern mathematics, there are few books about quantum mechanics aimed at mathematicians. This book introduces the main ideas of quantum mechanics in language familiar to mathematicians. Readers with little prior exposure to physics will enjoy the book's conversational tone as they delve into such topics as the Hilbert space approach to quantum theory; the Schrödinger equation in one space dimension; the Spectral Theorem for bounded and unbounded self-adjoint operators; the Stone–von Neumann Theorem; the Wentzel–Kramers–Brillouin approximation; the role of Lie groups and Lie algebras in quantum mechanics; and the path-integral approach to quantum mechanics.The numerous exercises at the end of each chapter make the book suitable for both graduate courses and independent study. Most of the text is accessible to graduate students in mathematics who have had a first course in real analysis, covering the basics of L2 spaces and Hilbert spaces.  The final chapters introduce readers who are familiar with the theory of manifolds to more advanced topics, including geometric quantization.

<p>Explains physical ideas in the language of mathematics</p><p>Provides a self-contained treatment of the necessary mathematics, including spectral theory and Lie theory</p><p>Contains many exercises that will appeal to graduate students</p>

Brian C. Hall is a Professor of Mathematics at the University of Notre Dame.

9781461471158

/Mathematical Physics/Theoretical, Mathematical and Computational Physics/Physics and Astronomy/Physical Sciences/Applications of Mathematics/Mathematics and Computing/Mathematics/

/Mathematical Physics/Theoretical, Mathematical and Computational Physics/Physics and Astronomy/Physical Sciences/Applications of Mathematics/Mathematics and Computing/Mathematics//Quantum Physics/Physics and Astronomy/Physical Sciences//Functional Analysis/Analysis/Mathematics and Computing/Mathematics//Topological Groups and Lie Groups/Algebra/Mathematics and Computing/Mathematics//Mathematical Methods in Physics/Theoretical, Mathematical and Computational Physics/Physics and Astronomy/Physical Sciences//

Why probability and statistics?.- Outcomes, events, and probability.- Conditional probability and independence.- Discrete random variables.- Continuous random variables.- Simulation.- Expectation and variance.- Computations with random variables.- Joint distributions and independence.- Covariance and correlation.- More computations with more random variables.- The Poisson process.- The law of large numbers.- The central limit theorem.- Exploratory data analysis: graphical summaries.- Exploratory data analysis: numerical summaries.- Basic statistical models.- The bootstrap.- Unbiased estimators.- Efficiency and mean squared error.- Maximum likelihood.- The method of least squares.- Confidence intervals for the mean.- More on confidence intervals.- Testing hypotheses: essentials.- Testing hypotheses: elaboration.- The t-test.- Comparing two samples.

Probability and Statistics are studied by most science students, usually as a second- or third-year course. Many current texts in the area are just cookbooks and, as a result, students do not know why they perform the methods they are taught, or why the methods work. The strength of this book is that it readdresses these shortcomings; by using examples, often from real-life and using real data, the authors can show how the fundamentals of probabilistic and statistical theories arise intuitively. It provides a tried and tested, self-contained course, that can also be used for self-study.

A Modern Introduction to Probability and Statistics has numerous quick exercises to give direct feedback to the students. In addition the book contains over 350 exercises, half of which have answers, of which half have full solutions. A website at www.springeronline.com/1-85233-896-2 gives access to the data files used in the text, and, for instructors, the remaining solutions. The only pre-requisite for the book is a first course in calculus; the text covers standard statistics and probability material, and develops beyond traditional parametric models to the Poisson process, and on to useful modern methods such as the bootstrap.

This will be a key text for undergraduates in Computer Science, Physics, Mathematics, Chemistry, Biology and Business Studies who are studying a mathematical statistics course, and also for more intensive engineering statistics courses for undergraduates in all engineering subjects.

Many current texts in the area are just cookbooks and, as a result, students do not know why they perform the methods they are taught, or why the methods work. The strength of this book is that it readdresses these shortcomings; by using examples, often from real life and using real data, the authors show how the fundamentals of probabilistic and statistical theories arise intuitively. A Modern Introduction to Probability and Statistics has numerous quick exercises to give direct feedback to students. In addition there are over 350 exercises, half of which have answers, of which half have full solutions. A website gives access to the data files used in the text, and, for instructors, the remaining solutions. The only pre-requisite is a first course in calculus; the text covers standard statistics and probability material, and develops beyond traditional parametric models to the Poisson process, and on to modern methods such as the bootstrap.

Developed from tried and tested course material, this book provides a self-contained course that is also suitable for self-studyUses real examples and real data sets that will be familiar to studentsFeatures quick exercises to give direct feedback to the student, and over 350 exercisesIncludes an introduction to the bootstrap, a modern method that is often missing in other booksIncludes full solutions to half the exercises given in the book; solutions to the rest are provided on an accompanying websiteIncludes supplementary material: sn.pub/extrasRequest lecturer material: sn.pub/lecturer-material

Michel Dekking, Cor Kraaikamp, Rik Lopuhaä and Ludolf Meester are professors in the Department of Applied Mathematics at TU Delft, The Netherlands. The material in this book has been successfully taught there for several years, and at the University of Leiden, The Netherlands, and Wesleyan University, USA, since 2003.

9781852338961

Statistics in Engineering, Physics, Computer Science, Chemistry and Earth Sciences

/Mathematics and Computing/Probability Theory/Mathematics//Statistics in Engineering, Physics, Computer Science, Chemistry and Earth Sciences/Applied Statistics/Statistics/Mathematics and Computing//Mathematical and Computational Engineering Applications/Technology and Engineering////

978-0-387-94001-4

Serge Lang, Yale University Dept. Mathematics, New Haven, CT, USA

I Sets.- II Topological Spaces.- III Continuous Functions on Compact Sets.- IV Banach Spaces.- V Hilbert Space.- VI The General Integral.- VII Duality and Representation Theorems.- VIII Some Applications of Integration.- IX Integration and Measures on Locally Compact Spaces.- X Riemann-Stieltjes Integral and Measure.- XI Distributions.- XII Integration on Locally Compact Groups.- XIII Differential Calculus.- XIV Inverse Mappings and Differential Equations.- XV The Open Mapping Theorem, Factor Spaces, and Duality.- XVI The Spectrum.- XVII Compact and Fredholm Operators.- XVIII Spectral Theorem for Bounded Hermltian Operators.- XIX Further Spectral Theorems.- XX Spectral Measures.- XXI Local Integration off Differential Forms.- XXII Manifolds.- XXIII Integration and Measures on Manifolds.- Table of Notation.

This book is meant as a text for a first year graduate course in analysis. Any standard course in undergraduate analysis will constitute sufficient preparation for its understanding, for instance, my Undergraduate Anal­ ysis. I assume that the reader is acquainted with notions of uniform con­ vergence and the like. In this third edition, I have reorganized the book by covering inte­ gration before functional analysis. Such a rearrangement fits the way courses are taught in all the places I know of. I have added a number of examples and exercises, as well as some material about integration on the real line (e.g. on Dirac sequence approximation and on Fourier analysis), and some material on functional analysis (e.g. the theory of the Gelfand transform in Chapter XVI). These upgrade previous exercises to sections in the text. In a sense, the subject matter covers the same topics as elementary calculus, viz. linear algebra, differentiation and integration. This time, however, these subjects are treated in a manner suitable for the training of professionals, i.e. people who will use the tools in further investiga­ tions, be it in mathematics, or physics, or what have you. In the first part, we begin with point set topology, essential for all analysis, and we cover the most important results.

9780387940014

/Analysis/Mathematics and Computing/Mathematics//Real Functions/Analysis/Mathematics and Computing/Mathematics/////

10.1007/978-1-4612-0897-6

978-0-387-09493-9

Joseph H. Silverman, Brown University Department of Mathematics, Providence, RI, USA

,978-1-4757-1922-2,978-0-387-96203-0,978-1-4757-1921-5,978-1-4757-1920-8

Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of an Elliptic Curve.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.- Computing the Mordell#x2013;Weil Group.- Algorithmic Aspects of Elliptic Curves.

The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, and elliptic curves over finite fields, the complex numbers, local fields, and global fields. Included are proofs of the Mordell–Weil theorem giving finite generation of the group of rational points and Siegel's theorem on finiteness of integral points.

For this second edition of The Arithmetic of Elliptic Curves, there is a new chapter entitled Algorithmic Aspects of Elliptic Curves, with an emphasis on algorithms over finite fields which have cryptographic applications. These include Lenstra's factorization algorithm, Schoof's point counting algorithm, Miller's algorithm to compute the Tate and Weil pairings, and a description of aspects of elliptic curve cryptography. There is also a new section on Szpiro's conjecture and ABC, as well as expanded and updated accounts of recent developments and numerous new exercises.

The book contains three appendices: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and a third appendix giving an overview of more advanced topics.

<p>Second Edition of highly successful introductory textbook, with new content, from acclaimed author</p><p>Thorough introduction to arithmetic theory of elliptic curves</p><p>Many exercises to hone the reader's knowledge</p><p>Text enlightens proofs through general principles, rather than line-by-line algebraic proof</p><p>Ideal for students to learn the basics of the subject and as a reference for researchers</p><p>Includes supplementary material: sn.pub/extras</p>

Dr. Joseph Silverman is a professor at Brown University and has been an instructor or professors since 1982. He was the Chair of the Brown Mathematics department from 2001-2004. He has received numerous fellowships, grants and awards, as well as being a frequently invited lecturer. He is currently a member of the Council of the American Mathematical Society. His research areas of interest are number theory, arithmetic geometry, elliptic curves, dynamical systems and cryptography. He has co-authored over 120 publications and has had over 20 doctoral students under his tutelage. He has published 9 highly successful books with Springer, including the recently released, An Introduction to Mathematical Cryptography, for Undergraduate Texts in Mathematics.

9780387094939

/Algebraic Geometry/Algebra/Mathematics and Computing/Mathematics/

/Algebraic Geometry/Algebra/Mathematics and Computing/Mathematics//Algebra/Mathematics and Computing/Mathematics//Number Theory/Mathematics and Computing/Mathematics////

10.1007/978-0-387-09494-6

David A. Cox, Amherst College, Amherst, MA, USA; John Little, College of the Holy Cross, Worcester, MA, USA; Donal O'Shea, New College of Florida, Sarasota, FL, USA

An Introduction to Computational Algebraic Geometry and Commutative Algebra

,978-0-387-51485-7,978-1-4419-2257-1,978-0-387-35650-1,978-0-387-35651-8

Preface.- Notation for Sets and Functions.- 1. Geometry, Algebra, and Algorithms.- 2. Groebner Bases.- 3. Elimination Theory.- 4.The Algebra-Geometry Dictionary.- 5. Polynomial and Rational Functions on a Variety.- 6. Robotics and Automatic Geometric Theorem Proving.- 7. Invariant Theory of Finite Groups.- 8. Projective Algebraic Geometry.- 9. The Dimension of a Variety.- 10. Additional Groebner Basis Algorithms.- Appendix A. Some Concepts from Algebra.- Appendix B. Pseudocode.- Appendix C. Computer Algebra Systems.- Appendix D. Independent Projects.- References.- Index.

This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry—the elimination theorem, the extension theorem, the closure theorem, and the Nullstellensatz—this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new chapter (ten), which presents some of the essentials of progress made over the last decades in computing Gröbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D). The book may serve as a first or second course in undergraduate abstract algebra and, with some supplementation perhaps, for beginning graduate level courses in algebraic geometry or computational algebra. Prerequisites for the reader include linear algebra and a proof-oriented course. It is assumed that the reader has access to a computer algebra system. Appendix C describes features of Maple™, Mathematica®, and Sage, as well as other systems that are most relevant to the text. Pseudocode is used in the text; Appendix B carefully describes the pseudocode used.From the reviews of previous editions:“…The book gives an introduction to Buchberger’s algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations, and elimination theory. …The book is well-written. …The reviewer is sure that it will be an excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry.” —Peter Schenzel, zbMATH, 2007“I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures, and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry.” —The American Mathematical Monthly

This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry—the elimination theorem, the extension theorem, the closure theorem and the Nullstellensatz—this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new Chapter (ten), which presents some of the essentials of progress made over the last decades in computing Gröbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D).The book may serve as a first or second course in undergraduate abstract algebra and with some supplementation perhaps, for beginning graduate levelcourses in algebraic geometry or computational algebra. Prerequisites for the reader include linear algebra and a proof-oriented course. It is assumed that the reader has access to a computer algebra system. Appendix C describes features of Maple™, Mathematica® and Sage, as well as other systems that are most relevant to the text. Pseudocode is used in the text; Appendix B carefully describes the pseudocode used.Readers who are teaching from Ideals, Varieties, and Algorithms, or are studying the book on their own, may obtain a copy of the solutions manual by sending an email to jlittle@holycross.edu.From the reviews of previous editions: “…The book gives an introduction to Buchberger’s algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations and elimination theory. …The book is well-written. …The reviewer is sure that it will be an excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry.” —Peter Schenzel, zbMATH, 2007 “I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry.” —The American Mathematical Monthly

New edition extensively revised and updatedCovers important topics such as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry and dimension theoryFourth edition includes updates on the computer algebra and independent projects appendicesFeatures new central theoretical results such as the elimination theorem, the extension theorem, the closure theorem and the nullstellensatzDiscusses some of the newer approaches to computing Groebner bases

David A. Cox is currently Professor of Mathematics at Amherst College. John Little is currently Professor of Mathematics at College of the Holy Cross. Donal O'Shea is currently President and Professor of Mathematics at New College of Florida.

9783319167206

/Algebraic Geometry/Algebra/Mathematics and Computing/Mathematics/

/Algebraic Geometry/Algebra/Mathematics and Computing/Mathematics//Commutative Rings and Algebras/Algebra/Mathematics and Computing/Mathematics//Mathematical Logic and Foundations/Mathematics and Computing/Mathematics//Mathematical Software/Computational Mathematics and Numerical Analysis/Mathematics and Computing/Mathematics///

,978-1-4419-2313-4,978-0-387-40122-5,978-1-4684-9515-7,978-0-387-21554-9

Part I: General Theory.-Matrix Lie Groups.- The Matrix Exponential.- Lie Algebras.- Basic Representation Theory.- The Baker–Campbell–Hausdorff Formula and its Consequences.- Part II: Semisimple Lie Algebras.- The Representations of sl(3;C).-Semisimple Lie Algebras.- Root Systems.- Representations of Semisimple Lie Algebras.- Further Properties of the Representations.- Part III: Compact lie Groups.- Compact Lie Groups and Maximal Tori.- The Compact Group Approach to Representation Theory.- Fundamental Groups of Compact Lie Groups.- Appendices.

This textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject.In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including:<br/><br/><br/>a treatment of the Baker–Campbell–Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebrasmotivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C)an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebrasa self-contained construction of the representations of compact groups, independent of Lie-algebraic argumentsThe second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them: an entirely new part devoted to the structure and representation theory of compact Lie groups; a complete derivation of the main properties of root systems; the construction of finite-dimensional representations of semisimple Lie algebras has been elaborated; a treatment of universal enveloping algebras, including a proof of the Poincaré–Birkhoff–Witt theorem and the existence of Verma modules; complete proofs of the Weyl character formula, the Weyl dimension formula and the Kostant multiplicity formula.Review of the first edition:“This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory ... an importantaddition to the textbook literature ... it is highly recommended.”— The Mathematical Gazette

This textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject.In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including:a treatment of the Baker–Campbell–Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebrasmotivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C)an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebrasa self-contained construction of the representations of compact groups, independent of Lie-algebraic argumentsThe second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them: an entirely new part devoted to the structure and representation theory of compact Lie groups; a complete derivation of the main properties of root systems; the construction of finite-dimensional representations of semisimple Lie algebras has been elaborated; a treatment of universal enveloping algebras, including a proof of the Poincaré–Birkhoff–Witt theorem and the existence of Verma modules; complete proofs of the Weyl character formula, the Weyl dimension formula and the Kostant multiplicity formula.Review of the first edition:This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory ... an important addition tothe textbook literature ... it is highly recommended.— The Mathematical Gazette

New edition extensively revised and updatedCovers the core topics of Lie theory from an elementary point of viewIncludes many new exercises

Brian Hall is Professor of Mathematics at the University of Notre Dame, IN.

9783319134666

/Topological Groups and Lie Groups/Algebra/Mathematics and Computing/Mathematics/

/Topological Groups and Lie Groups/Algebra/Mathematics and Computing/Mathematics//Non-associative Rings and Algebras/Algebra/Mathematics and Computing/Mathematics//Manifolds and Cell Complexes/Topology/Mathematics and Computing/Mathematics///

,978-1-4419-7591-1,978-1-4614-2722-3,978-1-4419-7593-5,978-1-4419-7592-8

Preface to the Third Edition.-1. First-Order Differential Equations.- 2. Second-Order Linear Equations.- 3. Laplace Transforms.- 4. Linear Systems.- 5. Nonlinear Systems.- 6. Computation of Solutions.- Appendix A. Review and Supplementary Exercises.-Appendix B. Matlab(R) Supplement.- References.- Index.

The third edition of this concise, popular textbook on elementary differential equations gives instructors an alternative to the many voluminous texts on the market. It presents a thorough treatment of the standard topics in an accessible, easy-to-read, format. The overarching perspective of the text conveys that differential equations are about applications. This book illuminates the mathematical theory in the text with a wide variety of applications that will appeal to students in physics, engineering, the biosciences, economics and mathematics. Instructors are likely to find that the first four or five chapters are suitable for a first course in the subject.This edition contains a healthy increase over earlier editions in the number of worked examples and exercises, particularly those routine in nature. Two appendices include a review with practice problems, and a MATLAB® supplement that gives basic codes and commands for solving differential equations. MATLAB® is not required; students are encouraged to utilize available software to plot many of their solutions. Solutions to even-numbered problems are available on springer.com. From the reviews of the second edition:“The coverage of linear systems in the plane is nicely detailed and illustrated. …Simple numerical methods are illustrated and the use of Maple and MATLAB is encouraged. …select Dave Logan’s new and improved text for my course.”—Robert E. O’Malley, Jr., SIAM Review, Vol. 53 (2), 2011 “Aims to provide material for a one-semester course that emphasizes the basic ideas, solution methods, and an introduction to modeling. …The book that results offers a concise introduction to the subject for students of mathematics, science and engineering who have completed the introductory calculus sequence. …This book is worth a careful look as a candidate text for the next differential equations course you teach.” —William J. Satzer,MAA Reviews, January, 2011

The third edition of this concise, popular textbook on elementary differential equations gives instructors an alternative to the many voluminous texts on the market. It presents a thorough treatment of the standard topics in an accessible, easy-to-read, format. The overarching perspective of the text conveys that differential equations are about applications. This book illuminates the mathematical theory in the text with a wide variety of applications that will appeal to students in physics, engineering, the biosciences, economics and mathematics. Instructors are likely to find that the first four or five chapters are suitable for a first course in the subject.This edition contains a healthy increase over earlier editions in the number of worked examples and exercises, particularly those routine in nature. Two appendices include a review with practice problems, and a MATLAB® supplement that gives basic codes and commands for solving differential equations. MATLAB® is not required; students are encouraged to utilize available software to plot many of their solutions. Solutions to even-numbered problems are available on springer.com.

Presents a thorough treatment of the standard topics in an accessible, easy-to-read, formatIlluminates mathematical theory with a wide variety of applications, appealing to students in physics, engineering, the biosciences, economics and mathematicsProvides many more exercises and worked examples in third editionReview with practice problems and Matlab(R) supplement included as AppendicesSolutions to even-numbered exercises available on springer.comIncludes supplementary material: sn.pub/extrasRequest lecturer material: sn.pub/lecturer-material

J. David Logan is Willa Cather Professor of Mathematics at the University of Nebraska Lincoln. He received his PhD from The Ohio State University and has served on the faculties at the University of Arizona, Kansas State University, and Rensselaer Polytechnic Institute. For many years he served as a visiting scientist at Los Alamos and Lawrence Livermore National Laboratories. He has published widely in differential equations, mathematical physics, fluid and gas dynamics, hydrogeology, and mathematical biology. Dr. Logan has authored 7 books, among them Applied Partial Differential Equations, now in its 3rd edition, published by Springer.

9783319178516

/Differential Equations/Analysis/Mathematics and Computing/Mathematics/

/Differential Equations/Analysis/Mathematics and Computing/Mathematics//Mathematical Modeling and Industrial Mathematics/Computational Mathematics and Numerical Analysis/Mathematics and Computing/Mathematics//Applications of Mathematics/Mathematics and Computing/Mathematics////

978-3-319-74072-0

Peter D. Lax, New York University, New York, NY, USA; Maria Shea Terrell, Cornell University, Ithaca, NY, USA

1. Vectors and matrices.- 2. Functions.- 3. Differentiation.- 4. More about differentiation.- 5. Applications to motion.- 6. Integration.- 7. Line and surface integrals.- 8. Divergence and Stokes’ Theorems and conservation laws.- 9. Partial differential equations.- Answers to selected problems.- Index.

This text in multivariable calculus fosters comprehension through meaningful explanations. Written with students in mathematics, the physical sciences, and engineering in mind, it extends concepts from single variable calculus such as derivative, integral, and important theorems to partial derivatives, multiple integrals, Stokes’ and divergence theorems. Students with a background in single variable calculus are guided through a variety of problem solving techniques and practice problems. Examples from the physical sciences are utilized to highlight the essential relationship between calculus and modern science. The symbiotic relationship between science and mathematics is shown by deriving and discussing several conservation laws, and vector calculus is utilized to describe a number of physical theories via partial differential equations. Students will learn that mathematics is the language that enables scientific ideas to be precisely formulated and that science is a source for the development of mathematics.

This text in multivariable calculus fosters comprehension through meaningful explanations. Written with students in mathematics, the physical sciences, and engineering in mind, it extends concepts from single variable calculus such as derivative, integral, and important theorems to partial derivatives, multiple integrals, Stokes’ and divergence theorems. Students with a background in single variable calculus are guided through a variety of problem solving techniques and practice problems. Examples from the physical sciences are utilized to highlight the essential relationship between calculus and modern science. The symbiotic relationship between science and mathematics is shown by deriving and discussing several conservation laws, and vector calculus is utilized to describe a number of physical theories via partial differential equations. Students will learn that mathematics is the language that enables scientific ideas to be precisely formulated and that science is a source for the development of mathematics.

Describes theoretical as well as practical aspects of multivariable calculusCorrelates concepts and results of multivariable calculus with their counterparts in one-variable calculusPresents a wealth of problems varying degrees of difficultyIncludes supplementary material: sn.pub/extras

Peter D. Lax is currently an Emeritus Professor of Mathematics at the Courant Institute of Mathematical Sciences.
Maria Shea Terrell is currently a retired Senior Lecturer in Mathematics at Cornell University.

9783319740720

/Analysis/Mathematics and Computing/Mathematics//Applications of Mathematics/Mathematics and Computing/Mathematics/////

Raoul Bott, Harvard University Dept. Mathematics, Cambridge, MA, USA; Loring W. Tu, Tufts University, Medford, MA, USA

I De Rham Theory.- II The ?ech-de Rham Complex.- III Spectral Sequences and Applications.- IV Characteristic Classes.- References.- List of Notations.

The guiding principle in this book is to use differential forms as an aid in exploring some of the less digestible aspects of algebraic topology. Accord­ ingly, we move primarily in the realm of smooth manifolds and use the de Rham theory as a prototype of all of cohomology. For applications to homotopy theory we also discuss by way of analogy cohomology with arbitrary coefficients. Although we have in mind an audience with prior exposure to algebraic or differential topology, for the most part a good knowledge of linear algebra, advanced calculus, and point-set topology should suffice. Some acquaintance with manifolds, simplicial complexes, singular homology and cohomology, and homotopy groups is helpful, but not really necessary. Within the text itself we have stated with care the more advanced results that are needed, so that a mathematically mature reader who accepts these background materials on faith should be able to read the entire book with the minimal prerequisites. There are more materials here than can be reasonably covered in a one-semester course. Certain sections may be omitted at first reading with­ out loss of continuity. We have indicated these in the schematic diagram that follows. This book is not intended to be foundational; rather, it is only meant to open some of the doors to the formidable edifice of modern algebraic topology. We offer it in the hope that such an informal account of the subject at a semi-introductory level fills a gap in the literature.

9780387906133

/Algebraic Topology/Topology/Mathematics and Computing/Mathematics/

/Algebraic Topology/Topology/Mathematics and Computing/Mathematics//////

978-0-387-95093-8

First Part.- I The Complex Plane and Elementary Functions.- II Analytic Functions.- III Line Integrals and Harmonic Functions.- IV Complex Integration and Analyticity.- V Power Series.- VI Laurent Series and Isolated Singularities.- VII The Residue Calculus.- Second Part.- VIII The Logarithmic Integral.- IX The Schwarz Lemma and Hyperbolic Geometry.- X Harmonic Functions and the Reflection Principle.- XI Conformal Mapping.- Third Part.- XII Compact Families of Meromorphic Functions.- XIII Approximation Theorems.- XIV Some Special Functions.- XV The Dirichlet Problem.- XVI Riemann Surfaces.- Hints and Solutions for Selected Exercises.- References.- List of Symbols.

The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. It conists of sixteen chapters. The first eleven chapters are aimed at an Upper Division undergraduate audience. The remaining five chapters are designed to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis. Topics studied in the book include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces. The three geometries, spherical, euclidean, and hyperbolic, are stressed. Exercises range from the very simple to the quite challenging, in all chapters. The book is based on lectures given over the years by the author at several places, including UCLA, Brown University, the universities at La Plata and Buenos Aires, Argentina; and the Universidad Autonomo de Valencia, Spain.

9780387950938

10.1007/978-0-387-21607-2

978-0-387-97527-6

I: Finite Groups.- 1. Representations of Finite Groups.- 2. Characters.- 3. Examples; Induced Representations; Group Algebras; Real Representations.- 4. Representations of:
$$
{\mathfrak{S}_d}$$
Young Diagrams and Frobenius’s Character Formula.- 5. Representations of
$$
{\mathfrak{A}_d}$$
and
$$
G{L_2}\left( {{\mathbb{F}_q}} \right)$$.- 6. Weyl’s Construction.- II: Lie Groups and Lie Algebras.- 7. Lie Groups.- 8. Lie Algebras and Lie Groups.- 9. Initial Classification of Lie Algebras.- 10. Lie Algebras in Dimensions One, Two, and Three.- 11. Representations of
$$
\mathfrak{s}{\mathfrak{l}_2}\mathbb{C}$$.- 12. Representations of
$$
\mathfrak{s}{\mathfrak{l}_3}\mathbb{C},$$
Part I.- 13. Representations of
$$
\mathfrak{s}{\mathfrak{l}_3}\mathbb{C},$$
Part II: Mainly Lots of Examples.- III: The Classical Lie Algebras and Their Representations.- 14. The General Set-up: Analyzing the Structure and Representations of an Arbitrary Semisimple Lie Algebra.- 15.
$$
\mathfrak{s}{\mathfrak{l}_4}\mathbb{C}$$
and
$$
\mathfrak{s}{\mathfrak{l}_n}\mathbb{C}$$.- 16. Symplectic Lie Algebras.- 17.
$$
\mathfrak{s}{\mathfrak{p}_6}\mathbb{C}$$
and
$$
\mathfrak{s}{\mathfrak{p}_2n}\mathbb{C}$$.- 18. Orthogonal Lie Algebras.- 19.
$$
\mathfrak{s}{\mathfrak{o}_6}\mathbb{C},$$$$
\mathfrak{s}{\mathfrak{o}_7}\mathbb{C},$$
and
$$
\mathfrak{s}{\mathfrak{o}_m}\mathbb{C}$$.- 20. Spin Representations of
$$
\mathfrak{s}{\mathfrak{o}_m}\mathbb{C}$$.- IV: Lie Theory.- 21. The Classification of Complex Simple Lie Algebras.- 22. $$
{g_2}$$and Other Exceptional Lie Algebras.- 23. Complex Lie Groups; Characters.- 24. Weyl Character Formula.- 25. More Character Formulas.- 26. Real Lie Algebras and Lie Groups.- Appendices.- A. On Symmetric Functions.- §A.1: Basic Symmetric Polynomials and Relations among Them.- §A.2: Proofs of the Determinantal Identities.- §A.3: Other Determinantal Identities.- B. On Multilinear Algebra.- §B.1: Tensor Products.- §B.2: Exterior and Symmetric Powers.- §B.3: Duals and Contractions.- C. On Semisimplicity.- §C.1: The Killing Form and Caftan’s Criterion.- §C.2: Complete Reducibility and the Jordan Decomposition.- §C.3: On Derivations.- D. Cartan Subalgebras.- §D.1: The Existence of Cartan Subalgebras.- §D.2: On the Structure of Semisimple Lie Algebras.- §D.3: The Conjugacy of Cartan Subalgebras.- §D.4: On the Weyl Group.- E. Ado’s and Levi’s Theorems.- §E.1: Levi’s Theorem.- §E.2: Ado’s Theorem.- F. Invariant Theory for the Classical Groups.- §F.1: The Polynomial Invariants.- §F.2: Applications to Symplectic and Orthogonal Groups.- §F.3: Proof of Capelli’s Identity.- Hints, Answers, and References.- Index of Symbols.

The primary goal of these lectures is to introduce a beginner to the finite­ dimensional representations of Lie groups and Lie algebras. Since this goal is shared by quite a few other books, we should explain in this Preface how our approach differs, although the potential reader can probably see this better by a quick browse through the book. Representation theory is simple to define: it is the study of the ways in which a given group may act on vector spaces. It is almost certainly unique, however, among such clearly delineated subjects, in the breadth of its interest to mathematicians. This is not surprising: group actions are ubiquitous in 20th century mathematics, and where the object on which a group acts is not a vector space, we have learned to replace it by one that is {e. g. , a cohomology group, tangent space, etc. }. As a consequence, many mathematicians other than specialists in the field {or even those who think they might want to be} come in contact with the subject in various ways. It is for such people that this text is designed. To put it another way, we intend this as a book for beginners to learn from and not as a reference. This idea essentially determines the choice of material covered here. As simple as is the definition of representation theory given above, it fragments considerably when we try to get more specific.

9780387975276

/Topological Groups and Lie Groups/Algebra/Mathematics and Computing/Mathematics/

/Topological Groups and Lie Groups/Algebra/Mathematics and Computing/Mathematics//////

10.1007/978-1-4612-0979-9

978-3-319-19424-0

Frank E. Harrell , Jr., School of Medicine, Vanderbilt University, Nashville, TN, USA

With Applications to Linear Models, Logistic and Ordinal Regression, and Survival Analysis

,978-1-4419-2918-1,978-0-387-95232-1,978-1-4757-3463-8,978-1-4757-3462-1

Introduction.- General Aspects of Fitting Regression Models.- Missing Data.- Multivariable Modeling Strategies.- Describing, Resampling, Validating and Simplifying the Model.- R Software.- Modeling Longitudinal Responses using Generalized Least Squares.- Case Study in Data Reduction.- Overview of Maximum Likelihood Estimation.- Binary Logistic Regression.- Binary Logistic Regression Case Study 1.- Logistic Model Case Study 2: Survival of Titanic Passengers.- Ordinal Logistic Regression.- Case Study in Ordinal Regression, Data Reduction and Penalization.- Regression Models for Continuous Y and Case Study in Ordinal Regression.- Transform-Both-Sides Regression.- Introduction to Survival Analysis.- Parametric Survival Models.- Case Study in Parametric Survival Modeling and Model Approximation.- Cox Proportional Hazards Regression Model.- Case Study in Cox Regression.- Appendix.   

This highly anticipated second edition features new chapters and sections, 225 new references, and comprehensive R software. In keeping with the previous edition, this book is about the art and science of data analysis and predictive modeling, which entails choosing and using multiple tools. Instead of presenting isolated techniques, this text emphasizes problem solving strategies that address the many issues arising when developing multivariable models using real data and not standard textbook examples. It includes imputation methods for dealing with missing data effectively, methods for fitting nonlinear relationships and for making the estimation of transformations a formal part of the modeling process, methods for dealing with 'too many variables to analyze and not enough observations,' and powerful model validation techniques based on the bootstrap.  The reader will gain a keen understanding of predictive accuracy, and the harm of categorizing continuous predictors or outcomes.  This text realistically deals with model uncertainty, and its effects on inference, to achieve 'safe data mining.' It also presents many graphical methods for communicating complex regression models to non-statisticians.Regression Modeling Strategies presents full-scale case studies of non-trivial datasets instead of over-simplified illustrations of each method. These case studies use freely available R functions that make the multiple imputation, model building, validation, and interpretation tasks described in the book relatively easy to do. Most of the methods in this text apply to all regression models, but special emphasis is given to multiple regression using generalized least squares for longitudinal data, the binary logistic model, models for ordinal responses, parametric survival regression models, and the Cox semiparametric survival model.  A new emphasis is given to the robust analysis of continuous dependent variables using ordinal regression.As in the first edition, this text is intended for Masters' or Ph.D. level graduate students who have had a general introductory probability and statistics course and who are well versed in ordinary multiple regression and intermediate algebra. The book will also serve as a reference for data analysts and statistical methodologists, as it contains an up-to-date survey and bibliography of modern statistical modeling techniques. Examples used in the text mostly come from biomedical research, but the methods are applicable anywhere predictive models ('analytics') are useful, including economics, epidemiology, sociology, psychology, engineering, and marketing.

This highly anticipated second edition features new chapters and sections, 225 new references, and comprehensive R software. In keeping with the previous edition, this book is about the art and science of data analysis and predictive modelling, which entails choosing and using multiple tools. Instead of presenting isolated techniques, this text emphasises problem solving strategies that address the many issues arising when developing multi-variable models using real data and not standard textbook examples. Regression Modelling Strategies presents full-scale case studies of non-trivial data-sets instead of over-simplified illustrations of each method. These case studies use freely available R functions that make the multiple imputation, model building, validation and interpretation tasks described in the book relatively easy to do. Most of the methods in this text apply to all regression models, but special emphasis is given to multiple regression using generalised least squares for longitudinal data, the binary logistic model, models for ordinal responses, parametric survival regression models and the Cox semi parametric survival model. A new emphasis is given to the robust analysis of continuous dependent variables using ordinal regression.As in the first edition, this text is intended for Masters' or PhD. level graduate students who have had a general introductory probability and statistics course and who are well versed in ordinary multiple regression and intermediate algebra. The book will also serve as a reference for data analysts and statistical methodologists, as it contains an up-to-date survey and bibliography of modern statistical modelling techniques. 

Fully revised new edition features new material and color figuresPublished with mature, supplementary R package: rmsNew chapters and sections on generalized least squares for analysis of serial response data, redundancy analysis, bootstrap confidence intervals for rankings of predictors, expanded material on multiple imputation and predictive mean matching and moreIncludes supplementary material: sn.pub/extras

Frank E. Harrell, Jr. is Professor of Biostatistics and Chair, Department of Biostatistics, Vanderbilt University School of Medicine, Nashville. He has developed numerous methods for predictive modeling, quantifying predictive accuracy and model validation and has published numerous predictive models and articles on applied statistics, medical research and clinical trials. He is on the editorial board for several biomedical and methodologic journals. He is a Fellow of the American Statistical Association (ASA) and a consultant to the U.S. Food and Drug Administration and to the pharmaceutical industry. He teaches a graduate course in regression modeling strategies and a course in biostatistics for medical researchers. In 2014 he was chosen to receive the WJ Dixon Award for Excellence in Statistical Consulting by the ASA. 

9783319194240

/Statistical Theory and Methods/Statistics/Mathematics and Computing/

/Statistical Theory and Methods/Statistics/Mathematics and Computing//Biostatistics/Applied Statistics/Statistics/Mathematics and Computing//Statistics and Computing/Statistics/Mathematics and Computing////

978-0-387-72828-5

Basic Linear Algebra.- Vector Spaces.- Linear Transformations.- The Isomorphism Theorems.- Modules I: Basic Properties.- Modules II: Free and Noetherian Modules.- Modules over a Principal Ideal Domain.- The Structure of a Linear Operator.- Eigenvalues and Eigenvectors.- Real and Complex Inner Product Spaces.- Structure Theory for Normal Operators.- Topics.- Metric Vector Spaces: The Theory of Bilinear Forms.- Metric Spaces.- Hilbert Spaces.- Tensor Products.- Positive Solutions to Linear Systems: Convexity and Separation.- Affine Geometry.- Singular Values and the Moore–Penrose Inverse.- An Introduction to Algebras.- The Umbral Calculus.

For the third edition, the author has added a new chapter on associative algebras that includes the well known characterizations of the finite-dimensional division algebras over the real field (a theorem of Frobenius) and over a finite field (Wedderburn's theorem); polished and refined some arguments (such as the discussion of reflexivity, the rational canonical form, best approximations and the definitions of tensor products); upgraded some proofs that were originally done only for finite-dimensional/rank cases; added new theorems, including the spectral mapping theorem; considerably expanded the reference section with over a hundred references to books on linear algebra. From the reviews of the second edition: 'In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials....As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields...the exercises are rewritten and expanded....Overall, I found the book a very useful one....It is a suitable choice as a graduate text or as a reference book.' Ali-Akbar Jafarian, ZentralblattMATH 'This is a formidable volume, a compendium of linear algebra theory, classical and modern... The development of the subject is elegant...The proofs are neat...The exercise sets are good, with occasional hints given for the solution of trickier problems...It represents linear algebra and does so comprehensively.' Henry Ricardo, MAA Online

For the third edition, the author has added a new chapter on associative algebras that includes the well known characterizations of the finite-dimensional division algebras over the real field (a theorem of Frobenius) and over a finite field (Wedderburn's theorem); polished and refined some arguments (such as the discussion of reflexivity, the rational canonical form, best approximations and the definitions of tensor products); upgraded some proofs that were originally done only for finite-dimensional/rank cases; added new theorems, including the spectral mapping theorem; corrected all known errors; the reference section has been enlarged considerably, with over a hundred references to books on linear algebra. From the reviews of the second edition: “In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials. … As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields. … the exercises are rewritten and expanded. … Overall, I found the book a very useful one. … It is a suitable choice as a graduate text or as a reference book.” Ali-Akbar Jafarian, ZentralblattMATH “This is a formidable volume, a compendium of linear algebra theory, classical and modern … . The development of the subject is elegant … . The proofs are neat … . The exercise sets are good, with occasional hints given for the solution of trickier problems. … It represents linear algebra and does so comprehensively.” Henry Ricardo, MathDL

Contains topics that are not generally found in linear algebra booksCoverage is especially broadAn extensive bibliography has been addedEncyclopedic treatment of linear algebra theory, both classical and modern

Dr. Roman has authored 32 books, including a number of books on mathematics, such as Introduction to the Finance of Mathematics, Coding and Information Theory, and Field Theory, published by Springer-Verlag. He has also written Modules in Mathematics, a series of 15 small books designed for the general college-level liberal arts student. Besides his books for O'Reilly, Dr. Roman has written two other computer books, both published by Springer-Verlag.

9780387728285

/Linear Algebra/Algebra/Mathematics and Computing/Mathematics/

/Linear Algebra/Algebra/Mathematics and Computing/Mathematics//////

978-0-387-96412-6

Serge Lang, Yale University Dept. Mathematics, New Haven, CT, USA

I Vector Spaces.- II Matrices.- III Linear Mappings.- IV Linear Maps and Matrices.- V Scalar Products and Orthogonality.- VI Determinants.- VII Symmetric, Hermitian, and Unitary Operators.- VIII Eigenvectors and Eigenvalues.- IX Polynomials and Matrices.- X Triangulation of Matrices and Linear Maps.- XI Polynomials and Primary Decomposition.- XII Convex Sets.- Appendix I Complex Numbers.- Appendix II Iwasawa Decomposition and Others.

Linear Algebra is intended for a one-term course at the junior or senior level. It begins with an exposition of the basic theory of vector spaces and proceeds to explain the fundamental structure theorems for linear maps, including eigenvectors and eigenvalues, quadric and hermitian forms, diagonalization of symmetric, hermitian, and unitary linear maps and matrices, triangulation, and Jordan canonical form. The book also includes a useful chapter on convex sets and the finite-dimensional Krein-Milman theorem. The presentation is aimed at the student who has already had some exposure to the elementary theory of matrices, determinants, and linear maps. However, the book is logically self-contained. In this new edition, many parts of the book have been rewritten and reorganized, and new exercises have been added.

9780387964126

/Linear Algebra/Algebra/Mathematics and Computing/Mathematics/

/Linear Algebra/Algebra/Mathematics and Computing/Mathematics//////

10.1007/978-1-4757-1949-9

978-3-540-76197-6

Gareth A. Jones, University of Southampton School of Mathematics, Southampton, UK; Josephine M. Jones

1. Divisibility.- 1.1 Divisors.- 1.2 Bezout’s identity.- 1.3 Least common multiples.- 1.4 Linear Diophantine equations.- 1.5 Supplementary exercises.- 2. Prime Numbers.- 2.1 Prime numbers and prime-power factorisations.- 2.2 Distribution of primes.- 2.3 Fermat and Mersenne primes.- 2.4 Primality-testing and factorisation.- 2.5 Supplementary exercises.- 3. Congruences.- 3.1 Modular arithmetic.- 3.2 Linear congruences.- 3.3 Simultaneous linear congruences.- 3.4 Simultaneous non-linear congruences.- 3.5 An extension of the Chinese Remainder Theorem.- 3.6 Supplementary exercises.- 4. Congruences with a Prime-power Modulus.- 4.1 The arithmetic of ?p.- 4.2 Pseudoprimes and Carmichael numbers.- 4.3 Solving congruences mod (pe).- 4.4 Supplementary exercises.- 5. Euler’s Function.- 5.1 Units.- 5.2 Euler’s function.- 5.3 Applications of Euler’s function.- 5.4 Supplementary exercises.- 6. The Group of Units.- 6.1 The group Un.- 6.2 Primitive roots.- 6.3 The group Une, where p is an odd prime.- 6.4 The group U2e.- 6.5 The existence of primitive roots.- 6.6 Applications of primitive roots.- 6.7 The algebraic structure of Un.- 6.8 The universal exponent.- 6.9 Supplementary exercises.- 7. Quadratic Residues.- 7.1 Quadratic congruences.- 7.2 The group of quadratic residues.- 7.3 The Legendre symbol.- 7.4 Quadratic reciprocity.- 7.5 Quadratic residues for prime-power moduli.- 7.6 Quadratic residues for arbitrary moduli.- 7.7 Supplementary exercises.- 8. Arithmetic Functions.- 8.1 Definition and examples.- 8.2 Perfect numbers.- 8.3 The Mobius Inversion Formula.- 8.4 An application of the Mobius Inversion Formula.- 8.5 Properties of the Mobius function.- 8.6 The Dirichlet product.- 8.7 Supplementary exercises.- 9. The Riemann Zeta Function.- 9.1 Historical background.- 9.2 Convergence.- 9.3 Applications to prime numbers.- 9.4 Random integers.- 9.5 Evaluating ?(2).- 9.6 Evaluating ?(2k).- 9.7 Dirichlet series.- 9.8 Euler products.- 9.9 Complex variables.- 9.10 Supplementary exercises.-10. Sums of Squares.- 10.1 Sums of two squares.- 10.2 The Gaussian integers.- 10.3 Sums of three squares.- 10.4 Sums of four squares.- 10.5 Digression on quaternions.- 10.6 Minkowski’s Theorem.- 10.7 Supplementary exercises.- 11. Fermat’s Last Theorem.- 11.1 The problem.- 11.2 Pythagoras’s Theorem.- 11.3 Pythagorean triples.- 11.4 Isosceles triangles and irrationality.- 11.5 The classification of Pythagorean triples.- 11.6 Fermat.- 11.7 The case n = 4.- 11.8 Odd prime exponents.- 11.9 Lame and Kummer.- 11.10 Modern developments.- 11.11 Further reading.- Solutions to Exercises.- Index of symbols.- Index of names.

Our intention in writing this book is to give an elementary introduction to number theory which does not demand a great deal of mathematical back­ ground or maturity from the reader, and which can be read and understood with no extra assistance. Our first three chapters are based almost entirely on A-level mathematics, while the next five require little else beyond some el­ ementary group theory. It is only in the last three chapters, where we treat more advanced topics, including recent developments, that we require greater mathematical background; here we use some basic ideas which students would expect to meet in the first year or so of a typical undergraduate course in math­ ematics. Throughout the book, we have attempted to explain our arguments as fully and as clearly as possible, with plenty of worked examples and with outline solutions for all the exercises. There are several good reasons for choosing number theory as a subject. It has a long and interesting history, ranging from the earliest recorded times to the present day (see Chapter 11, for instance, on Fermat's Last Theorem), and its problems have attracted many of the greatest mathematicians; consequently the study of number theory is an excellent introduction to the development and achievements of mathematics (and, indeed, some of its failures). In particular, the explicit nature of many of its problems, concerning basic properties of inte­ gers, makes number theory a particularly suitable subject in which to present modern mathematics in elementary terms.

The essential guide to number theory for undergraduatesDistinguishing features include discussions of the Riemann Zeta Function and Riemann HypothesisIncludes supplementary material: sn.pub/extras

9783540761976

Lawrence M. Friedman; Curt D. Furberg; David L. DeMets; David M. Reboussin; Christopher B. Granger

Lawrence M. Friedman, North Bethesda, MD, USA; Curt D. Furberg, Wake Forest School of Medicine, Winston-Salem, NC, USA; David L. DeMets, University of Wisconsin, Madison, WI, USA; David M. Reboussin, Wake Forest School of Medicine, Winston-Salem, NC, USA; Christopher B. Granger, Duke University, Durham, NC, USA

Introduction to Clinical Trials.- Ethical Issues.- What is the Question?.- Study Population.- Basic Study Design.- The Randomization Process.- Blinding.- Sample Size.- Baseline Assessment.- Recruitment of Study Participants.- Data Collection and Quality Control.- Assessment and Reporting of Harm.- Assessment of Health Related Quality of Life.- Participant Adherence.- Survival Analysis.- Monitoring Committee Structure & Function.- Statistical Methods Used in Interim Monitoring.- Issues in Data Analysis.- Closeout.- Reporting and Interpreting of Results.- Multicenter Trials.- Regulatory Issues.

This is the fifth edition of a very successful textbook on clinical trials methodology, written by recognized leaders who have long and extensive experience in all areas of clinical trials. The three authors of the first four editions have been joined by two others who add great expertise. Most chapters have been revised considerably from the fourth edition. A chapter on regulatory issues has been included and the chapter on data monitoring has been split into two and expanded. Many contemporary clinical trial examples have been added. There is much new material on adverse events, adherence, issues in analysis, electronic data, data sharing, and international trials. This book is intended for the clinical researcher who is interested in designing a clinical trial and developing a protocol. It is also of value to researchers and practitioners who must critically evaluate the literature of published clinical trials and assess the merits of each trial and the implications for the care and treatment of patients. The authors use numerous examples of published clinical trials to illustrate the fundamentals. The text is organized sequentially from defining the question to trial closeout. One chapter is devoted to each of the critical areas to aid the clinical trial researcher. These areas include pre-specifying the scientific questions to be tested and appropriate outcome measures, determining the organizational structure, estimating an adequate sample size, specifying the randomization procedure, implementing the intervention and visit schedules for participant evaluation, establishing an interim data and safety monitoring plan, detailing the final analysis plan, and reporting the trial results according to the pre-specified objectives.Although a basic introductory statistics course is helpful in maximizing the benefit of this book, a researcher or practitioner with limited statistical background would still find most if not all the chapters understandable and helpful. While the technical material has been kept to a minimum, the statistician may still find the principles and fundamentals presented in this text useful. This book has been successfully used for teaching courses in clinical trial methodology.

This is the fifth edition of a very successful textbook on clinical trials methodology, written by recognized leaders who have long and extensive experience in all areas of clinical trials. The three authors of the first four editions have been joined by two others who add great expertise. A chapter on regulatory issues has been included and the chapter on data monitoring has been split into two and expanded. Many contemporary clinical trial examples have been added. There is much new material on adverse events, adherence, issues in analysis, electronic data, data sharing and international trials.This book is intended for the clinical researcher who is interested in designing a clinical trial and developing a protocol. It is also of value to researchers and practitioners who must critically evaluate the literature of published clinical trials and assess the merits of each trial and the implications for the care and treatment of patients. The authors use numerous examplesof published clinical trials to illustrate the fundamentals.The text is organized sequentially from defining the question to trial closeout. One chapter is devoted to each of the critical areas to aid the clinical trial researcher. These areas include pre-specifying the scientific questions to be tested and appropriate outcome measures, determining the organizational structure, estimating an adequate sample size, specifying the randomization procedure, implementing the intervention and visit schedules for participant evaluation, establishing an interim data and safety monitoring plan, detailing the final analysis plan and reporting the trial results according to the pre-specified objectives.Although a basic introductory statistics course is helpful in maximizing the benefit of this book, a researcher or practitioner with limited statistical background would still find most if not all the chapters understandable and helpful. While the technical material has been kept to aminimum, the statistician may still find the principles and fundamentals presented in this text useful.

New chapter covers current regulatory issues and data monitoring is now covered in two chaptersAn essential, up-to-date reference for researchers and students involved with clinical trialsIncludes numerous examples of published clinical trials from a variety of medical disciplinesTechnical details are kept to a minimum through the use of graphs and tablesThe authors are active researchers leading clinical trials in a broad range of subjects

Lawrence M. Friedman received his M.D. from the University of Pittsburgh. After training in internal medicine, he went to the National Heart, Lung and Blood Institute of the National Institutes of Health. During his many years there, Dr. Friedman was involved in numerous clinical trials and epidemiology studies, having major roles in their design, management and monitoring. While at the NIH and subsequently, he served as a consultant on clinical trials to various NIH institutes and to other governmental and nongovernmental organizations. Dr. Friedman has been a member of many data monitoring and other safety committees. Curt D. Furberg is Professor Emeritus of the Division of Public Health Sciences of the Wake Forest University School of Medicine. He received his M.D. and Ph.D. at the University of Umea, Sweden, and is a former chief, Clinical Trials Branch and Associate Director, Clinical Applications and Prevention Program, National Heart, Lung, and Blood Institute. Dr. Furberg established the Department of Public Health Sciences and served as its chair from 1986 to 1999. He has played major scientific and administrative roles in numerous multicenter clinical trials and has served in a consultative or advisory capacity on others. Dr. Furberg’s research activities include the areas of clinical trials methodology and cardiovascular epidemiology. David L. DeMets, PhD is currently the Max Halperin Professor of Biostatistics and former Chair of the Department of Biostatistics and Medical Informatics at the University of Wisconsin – Madison He has co-authored numerous papers on statistical methods and four texts on clinical trials, two specifically on data monitoring. He has served on many NIH and industry-sponsored data monitoring committees for clinical trials in diverse disciplines. He served on the Board of Directors of the American Statistical Association, as well as having been President of the Society for Clinical Trialsand President of the Eastern North American Region (ENAR) of the Biometric Society. In addition he was Elected Fellow of the International Statistics Institute, the American Statistical Association, the Association for the Advancement of Science, the Society for Clinical Trials and the American Medical Informatics Association. In 2013, he was elected as a member of the Institute of Medicine. Christopher B. Granger is Professor of Medicine at Duke University, where he is an active clinical cardiologist and a clinical trialist at the Duke Clinical Research Institute. He received his M.D. at University of Connecticut and his residency training at the University of Colorado. He has had Steering Committee, academic leadership, and operational responsibilities for many clinical trials in cardiology. He has been on numerous Data Monitoring Committees. He serves on the National Heart, Lung, and Blood Institute Board of External Experts. He works with the Clinical Trials Transformation Initiative, a partnership between the U.S. Food and Drug Administration and Duke aiming to increase the quality and efficiency of clinical trials. He is a founding member of the Sensible Guidelines for the Conduct of Clinical Trials group, a collaboration between McMaster, Oxford, and Duke Universities. David M. Reboussin is a Professor in the Department of Biostatistical Science at the Wake Forest University School of Medicine, where he has worked since 1992. He has a master’s degree in Statistics from the University of Chicago and received his doctorate in Statistics from the University of Wisconsin at Madison. He is currently Principle Investigator for the Systolic Blood Pressure Intervention Trial Coordinating Center and has been a co-investigator in the coordinating centers for several NIH and industry funded clinical trials including Action to Control Cardiovasc

9783319185385

/Biostatistics/Applied Statistics/Statistics/Mathematics and Computing/

/Biostatistics/Applied Statistics/Statistics/Mathematics and Computing//Public Health/Life Sciences/Health Sciences//Epidemiology/Biomedical Research/Life Sciences/Health Sciences//Cancer Biology/Biological Sciences/Life Sciences//Oncology/Life Sciences/Clinical Medicine/Health Sciences//

Heinz-Dieter Ebbinghaus, University of Freiburg, Freiburg, Germany; Jörg Flum, University of Freiburg, Freiburg, Germany; Wolfgang Thomas, RWTH Aachen University, Aachen, Germany

,978-1-4757-2357-1,978-0-387-94258-2,978-1-4757-2356-4,978-1-4757-2355-7

A.- I Introduction.- II Syntax of First-Order Languages.- III Semantics of First-Order Languages.- IV A Sequent Calculus.- V The Completeness Theorem.- VI The Löwenheim–Skolem and the Compactness Theorem.- VII The Scope of First-Order Logic.- VIII Syntactic Interpretations and Normal Forms.- B.- IX Extensions of First-Order Logic.- X Computability and Its Limitations.- XI Free Models and Logic Programming.- XII An Algebraic Characterization of Elementary Equivalence.- XIII Lindström’s Theorems.- References.- List of Symbols.- Subject Index.

This textbook introduces first-order logic and its role in the foundations of mathematics by examining fundamental questions. What is a mathematical proof? How can mathematical proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical proofs? In answering these questions, this textbook explores the capabilities and limitations of algorithms and proof methods in mathematics and computer science.

The chapters are carefully organized, featuring complete proofs and numerous examples throughout. Beginning with motivating examples, the book goes on to present the syntax and semantics of first-order logic. After providing a sequent calculus for this logic, a Henkin-type proof of the completeness theorem is given. These introductory chapters prepare the reader for the advanced topics that follow, such as Gödel's Incompleteness Theorems, Trakhtenbrot's undecidability theorem, Lindström's theorems on the maximality of first-order logic, and results linking logic with automata theory. This new edition features many modernizations, as well as two additional important results: The decidability of Presburger arithmetic, and the decidability of the weak monadic theory of the successor function.

Mathematical Logic is ideal for students beginning their studies in logic and the foundations of mathematics. Although the primary audience for this textbook will be graduate students or advanced undergraduates in mathematics or computer science, in fact the book has few formal prerequisites. It demands of the reader only mathematical maturity and experience with basic abstract structures, such as those encountered in discrete mathematics or algebra.

This textbook introduces first-order logic and its role in the foundations of mathematics by examining fundamental questions. What is a mathematical proof? How can mathematical proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical proofs? In answering these questions, this textbook explores the capabilities and limitations of algorithms and proof methods in mathematics and computer science.

The chapters are carefully organized, featuring complete proofs and numerous examples throughout. Beginning with motivating examples, the book goes on to present the syntax and semantics of first-order logic. After providing a sequent calculus for this logic, a Henkin-type proof of the completeness theorem is given. These introductory chapters prepare the reader for the advanced topics that follow, such as Gödel's Incompleteness Theorems, Trakhtenbrot's undecidability theorem, Lindström's theorems on the maximality of first-order logic, and results linking logic with automata theory. This new edition features many modernizations, as well as two additional important results: The decidability of Presburger arithmetic, and the decidability of the weak monadic theory of the successor function.

Mathematical Logic is ideal for students beginning their studies in logic and the foundations of mathematics. Although the primary audience for this textbook will be graduate students or advanced undergraduates in mathematics or computer science, in fact the book has few formal prerequisites. It demands of the reader only mathematical maturity and experience with basic abstract structures, such as those encountered in discrete mathematics or algebra.

Explores additional important decidability results in this thoroughly updated new editionIntroduces mathematical logic by analyzing foundational questions on proofs and provability in mathematicsHighlights the capabilities and limitations of algorithms and proof methods both in mathematics and computer scienceExamines advanced topics, such as linking logic with computability and automata theory, as well as the unique role first-order logic plays in logical systems

Heinz-Dieter Ebbinghaus is Professor Emeritus at the Mathematical Institute of the University of Freiburg. His work spans fields in logic, such as model theory and set theory, and includes historical aspects.Jörg Flum is Professor Emeritus at the Mathematical Institute of the University of Freiburg. His research interests include mathematical logic, finite model theory, and parameterized complexity theory.

Wolfgang Thomas is Professor Emeritus at the Computer Science Department of RWTH Aachen University. His research interests focus on logic in computer science, in particular logical aspects of automata theory.

9783030738389

/Mathematical Logic and Foundations/Mathematics and Computing/Mathematics/

/Mathematical Logic and Foundations/Mathematics and Computing/Mathematics//Mathematics of Computing/Computer Science/Mathematics and Computing/////

978-0-8176-3490-2

Manfredo P. do Carmo, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil

0-Differentiable Manifolds.- 1-Riemannian Metrics.- 2-Affine Connections; Riemannian Connections.- 3-Geodesics; Convex Neighborhoods.- 4-Curvature.- 5-Jacobi Fields.- 6-Isometric Immersions.- 7-Complete Manifolds; Hopf-Rinow and Hadamard Theorems.- 8-Spaces of Constant Curvature.- 9-Variations of Energy.- 10-The Rauch Comparison Theorem.- 11-The Morse Index Theorem.- 12-The Fundamental Group of Manifolds of Negative Curvature.- 13-The Sphere Theorem.- References.

Riemannian Geometry is an expanded edition of a highly acclaimed and successful textbook (originally published in Portuguese) for first-year graduate students in mathematics and physics. The author's treatment goes very directly to the basic language of Riemannian geometry and immediately presents some of its most fundamental theorems. It is elementary, assuming only a modest background from readers, making it suitable for a wide variety of students and course structures. Its selection of topics has been deemed 'superb' by teachers who have used the text. A significant feature of the book is its powerful and revealing structure, beginning simply with the definition of a differentiable manifold and ending with one of the most important results in Riemannian geometry, a proof of the Sphere Theorem. The text abounds with basic definitions and theorems, examples, applications, and numerous exercises to test the student's understanding and extend knowledge and insight intothe subject. Instructors and students alike will find the work to be a significant contribution to this highly applicable and stimulating subject.

9780817634902

/Differential Geometry/Geometry/Mathematics and Computing/Mathematics/

/Differential Geometry/Geometry/Mathematics and Computing/Mathematics//Mathematical Methods in Physics/Theoretical, Mathematical and Computational Physics/Physics and Astronomy/Physical Sciences/

10.1007/978-1-4757-2201-7

978-3-540-40448-4

Vladimir I. Arnold, Russian Academy of Sciences Steklov Mathematical Institute, Moscow, Russia

Jointly published with PHASIS. Original Russian edition by PHASIS, Moscow, Russia

1. The General Theory for One First-Order Equation.- 2. The General Theory for One First-Order Equation (Continued).- 3. Huygens’ Principle in the Theory of Wave Propagation.- 4. The Vibrating String (d’Alembert’s Method).- 5. The Fourier Method (for the Vibrating String).- 6. The Theory of Oscillations. The Variational Principle.- 7. The Theory of Oscillations. The Variational Principle (Continued).- 8. Properties of Harmonic Functions.- 9. The Fundamental Solution for the Laplacian. Potentials.- 10. The Double-Layer Potential.- 11. Spherical Functions. Maxwell’s Theorem. The Removable Singularities Theorem.- 12. Boundary-Value Problems for Laplace’s Equation. Theory of Linear Equations and Systems.- A. The Topological Content of Maxwell’s Theorem on the Multifield Representation of Spherical Functions.- A.1. The Basic Spaces and Groups.- A.2. Some Theorems of Real Algebraic Geometry.- A.3. From Algebraic Geometry to Spherical Functions.- A.4. Explicit Formulas.- A.6. The History of Maxwell’s Theorem.- Literature.- B. Problems.- B.1. Material from the Seminars.- B.2. Written Examination Problems.

Choice Outstanding Title! (January 2006) Like all of Vladimir Arnold's books, this book is full of geometric insight. Arnold illustrates every principle with a figure. This book aims to cover the most basic parts of the subject and confines itself largely to the Cauchy and Neumann problems for the classical linear equations of mathematical physics, especially Laplace's equation and the wave equation, although the heat equation and the Korteweg-de Vries equation are also discussed. Physical intuition is emphasized. A large number of problems are sprinkled throughout the book, and a full set of problems from examinations given in Moscow are included at the end. Some of these problems are quite challenging! What makes the book unique is Arnold's particular talent at holding a topic up for examination from a new and fresh perspective. He likes to blow away the fog of generality that obscures so much mathematical writing and reveal the essentially simple intuitive ideas underlying the subject. No other mathematical writer does this quite so well as Arnold.

A top textbook written by one of the most famous living mathematiciansIncludes supplementary material: sn.pub/extras

9783540404484

/Differential Equations/Analysis/Mathematics and Computing/Mathematics/

/Differential Equations/Analysis/Mathematics and Computing/Mathematics//Theoretical, Mathematical and Computational Physics/Physics and Astronomy/Physical Sciences/////

10.1007/978-3-662-05441-3

978-1-84628-969-9

Adrian Bondy, Université Lyon I LaPCS-Domaine de Gerland, Lyon CX, France; U.S.R. Murty, University of Waterloo Department of Pure Mathematics, Waterloo, ON, Canada

Graphs.- Subgraphs.- Connected Graphs.- Trees.- Nonseparable Graphs.- Tree-Search Algorithms.- Flows in Networks.- Complexity of Algorithms.- Connectivity.- Planar Graphs.- The Four-Colour Problem.- Stable Sets and Cliques.- The Probabilistic Method.- Vertex Colourings.- Colourings of Maps.- Matchings.- Edge Colourings.- Hamilton Cycles.- Coverings and Packings in Directed Graphs.- Electrical Networks.- Integer Flows and Coverings.

Graph theory is a flourishing discipline containing a body of beautiful and powerful theorems of wide applicability. Its explosive growth in recent years is mainly due to its role as an essential structure underpinning modern applied mathematics – computer science, combinatorial optimization, and operations research in particular – but also to its increasing application in the more applied sciences. The versatility of graphs makes them indispensable tools in the design and analysis of communication networks, for instance. The primary aim of this book is to present a coherent introduction to the subject, suitable as a textbook for advanced undergraduate and beginning graduate students in mathematics and computer science. It provides a systematic treatment of the theory of graphs without sacrificing its intuitive and aesthetic appeal. Commonly used proof techniques are described and illustrated, and a wealth of exercises - of varying levels of difficulty - are providedto help the reader master the techniques and reinforce their grasp of the material. A second objective is to serve as an introduction to research in graph theory. To this end, sections on more advanced topics are included, and a number of interesting and challenging open problems are highlighted and discussed in some detail. Despite this more advanced material, the book has been organized in such a way that an introductory course on graph theory can be based on the first few sections of selected chapters.

Graph theory is a flourishing discipline containing a body of beautiful and powerful theorems of wide applicability. Its explosive growth in recent years is mainly due to its role as an essential structure underpinning modern applied mathematics – computer science, combinatorial optimization, and operations research in particular – but also to its increasing application in the more applied sciences. The versatility of graphs makes them indispensable tools in the design and analysis of communication networks, for instance.The primary aim of this book is to present a coherent introduction to the subject, suitable as a textbook for advanced undergraduate and beginning graduate students in mathematics and computer science. It provides a systematic treatment of the theory of graphs without sacrificing its intuitive and aesthetic appeal. Commonly used proof techniques are described and illustrated, and a wealth of exercises - of varying levels of difficulty - are provided tohelp the reader master the techniques and reinforce their grasp of the material.A second objective is to serve as an introduction to research in graph theory. To this end, sections on more advanced topics are included, and a number of interesting and challenging open problems are highlighted and discussed in some detail. Despite this more advanced material, the book has been organized in such a way that an introductory course on graph theory can be based on the first few sections of selected chapters.

By the authors of the classic text, Graph Theory with ApplicationsServes as both a textbook and an introduction to graph theory research, suitable for both mathematicians and computer scientistsFeatures many new exercises of varying levels of difficulty to help the reader master the techniquesAn accompanying website/blog at blogs.springer.com/bondyandmurty provides a forum for further discussion and a wealth of supplementary materialIncludes supplementary material: sn.pub/extras

9781846289699

/Discrete Mathematics/Mathematics and Computing/Mathematics/

/Discrete Mathematics/Mathematics and Computing/Mathematics//Discrete Mathematics in Computer Science/Mathematics of Computing/Computer Science/Mathematics and Computing//Algorithms/Computational Mathematics and Numerical Analysis/Mathematics and Computing/Mathematics//Optimization/Mathematics and Computing/Mathematics//Mathematics of Computing/Computer Science/Mathematics and Computing//

10.1007/978-1-84628-970-5

978-0-387-97655-6

Ioannis Karatzas, Columbia University Dept. Statistics, New York, NY, USA; Steven Shreve, Carnegie Mellon University, Pittsburgh, PA, USA

,978-1-4684-0304-6,978-0-387-96535-2,978-1-4684-0303-9,978-1-4684-0302-2

1 Martingales, Stopping Times, and Filtrations.- 1.1. Stochastic Processes and ?-Fields.- 1.2. Stopping Times.- 1.3. Continuous-Time Martingales.- 1.4. The Doob—Meyer Decomposition.- 1.5. Continuous, Square-Integrable Martingales.- 1.6. Solutions to Selected Problems.- 1.7. Notes.- 2 Brownian Motion.- 2.1. Introduction.- 2.2. First Construction of Brownian Motion.- 2.3. Second Construction of Brownian Motion.- 2.4. The SpaceC[0, ?), Weak Convergence, and Wiener Measure.- 2.5. The Markov Property.- 2.6. The Strong Markov Property and the Reflection Principle.- 2.7. Brownian Filtrations.- 2.8. Computations Based on Passage Times.- 2.9. The Brownian Sample Paths.- 2.10. Solutions to Selected Problems.- 2.11. Notes.- 3 Stochastic Integration.- 3.1. Introduction.- 3.2. Construction of the Stochastic Integral.- 3.3. The Change-of-Variable Formula.- 3.4. Representations of Continuous Martingales in Terms of Brownian Motion.- 3.5. The Girsanov Theorem.- 3.6. Local Time and a Generalized Itô Rule for Brownian Motion.- 3.7. Local Time for Continuous Semimartingales.- 3.8. Solutions to Selected Problems.- 3.9. Notes.- 4 Brownian Motion and Partial Differential Equations.- 4.1. Introduction.- 4.2. Harmonic Functions and the Dirichlet Problem.- 4.3. The One-Dimensional Heat Equation.- 4.4. The Formulas of Feynman and Kac.- 4.5. Solutions to selected problems.- 4.6. Notes.- 5 Stochastic Differential Equations.- 5.1. Introduction.- 5.2. Strong Solutions.- 5.3. Weak Solutions.- 5.4. The Martingale Problem of Stroock and Varadhan.- 5.5. A Study of the One-Dimensional Case.- 5.6. Linear Equations.- 5.7. Connections with Partial Differential Equations.- 5.8. Applications to Economics.- 5.9. Solutions to Selected Problems.- 5.10. Notes.- 6 P. Lévy’s Theory of Brownian Local Time.-6.1. Introduction.- 6.2. Alternate Representations of Brownian Local Time.- 6.3. Two Independent Reflected Brownian Motions.- 6.4. Elastic Brownian Motion.- 6.5. An Application: Transition Probabilities of Brownian Motion with Two-Valued Drift.- 6.6. Solutions to Selected Problems.- 6.7. Notes.

This book is designed as a text for graduate courses in stochastic processes. It is written for readers familiar with measure-theoretic probability and discrete-time processes who wish to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and these in turn permit a presentation of recent advances in financial economics (option pricing and consumption/investment optimization). This book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The text is complemented by a large number of problems and exercises.

A perennial best-seller, now in its fourth printingBrownian motion is currently a hot topic in mathematicsKaratzas is one of the leaders in the field of stochastics and finance

9780387976556

/Mathematics and Computing/Probability Theory/Mathematics//Classical Mechanics/Classical and Continuum Physics/Physics and Astronomy/Physical Sciences/////

10.1007/978-1-4612-0949-2

978-3-030-41067-4

Matthew F. Dixon, Illinois Institute of Technology, Chicago, IL, USA; Igor Halperin, New York University, Brooklyn, NY, USA; Paul Bilokon, Imperial College London, London, UK

Chapter 1. Introduction.- Chapter 2. Probabilistic Modeling.- Chapter 3. Bayesian Regression & Gaussian Processes.- Chapter 4. Feed Forward Neural Networks.- Chapter 5. Interpretability.- Chapter 6. Sequence Modeling.- Chapter 7. Probabilistic Sequence Modeling.- Chapter 8. Advanced Neural Networks.- Chapter 9. Introduction to Reinforcement learning.- Chapter 10. Applications of Reinforcement Learning.- Chapter 11. Inverse Reinforcement Learning and Imitation Learning.- Chapter 12. Frontiers of Machine Learning and Finance.

This book introduces machine learning methods in finance. It presents a unified treatment of machine learning and various statistical and computational disciplines in quantitative finance, such as financial econometrics and discrete time stochastic control, with an emphasis on how theory and hypothesis tests inform the choice of algorithm for financial data modeling and decision making. With the trend towards increasing computational resources and larger datasets, machine learning has grown into an important skillset for the finance industry. This book is written for advanced graduate students and academics in financial econometrics, mathematical finance and applied statistics, in addition to quants and data scientists in the field of quantitative finance.

Machine Learning in Finance: From Theory to Practice is divided into three parts, each part covering theory and applications. The first presents supervised learning for cross-sectional data from both a Bayesian and frequentist perspective. The more advanced material places a firm emphasis on neural networks, including deep learning, as well as Gaussian processes, with examples in investment management and derivative modeling. The second part presents supervised learning for time series data, arguably the most common data type used in finance with examples in trading, stochastic volatility and fixed income modeling. Finally, the third part presents reinforcement learning and its applications in trading, investment and wealth management. Python code examples are provided to support the readers' understanding of the methodologies and applications. The book also includes more than 80 mathematical and programming exercises, with worked solutions available to instructors. As a bridge to research in this emergent field, the final chapter presents the frontiers of machine learning in finance from a researcher's perspective, highlighting how many well-known concepts in statistical physics are likely to emerge as important methodologies for machine learning in finance.

This book introduces machine learning methods in finance. It presents a unified treatment of machine learning and various statistical and computational disciplines in quantitative finance, such as financial econometrics and discrete time stochastic control, with an emphasis on how theory and hypothesis tests inform the choice of algorithm for financial data modeling and decision making. With the trend towards increasing computational resources and larger datasets, machine learning has grown into an important skillset for the finance industry. This book is written for advanced graduate students and academics in financial econometrics, mathematical finance and applied statistics, in addition to quants and data scientists in the field of quantitative finance.

Machine Learning in Finance: From Theory to Practice is divided into three parts, each part covering theory and applications. The first presents supervised learning for cross-sectional data from both a Bayesian and frequentist perspective. The more advanced material places a firm emphasis on neural networks, including deep learning, as well as Gaussian processes, with examples in investment management and derivative modeling. The second part presents supervised learning for time series data, arguably the most common data type used in finance with examples in trading, stochastic volatility and fixed income modeling. Finally, the third part presents reinforcement learning and its applications in trading, investment and wealth management. Python code examples are provided to support the readers' understanding of the methodologies and applications. The book also includes more than 80 mathematical and programming exercises, with worked solutions available to instructors. As a bridge to research in this emergent field, the final chapter presents the frontiers of machine learning in finance from a researcher's perspective, highlighting how many well-known concepts in statistical physics are likely to emerge as important methodologies for machine learning in finance.

<p>Introduces fundamental concepts in machine learning for canonical modeling and decision frameworks in finance</p><p>Presents a unified treatment of machine learning, financial econometrics and discrete time stochastic control problems in finance</p><p>Chapters include examples, exercises and Python codes to reinforce theoretical concepts and demonstrate the application of machine learning to algorithmic trading, investment management, wealth management and risk management</p><p>Request lecturer material: sn.pub/lecturer-material</p>

Paul Bilokon, Ph.D., is CEO and Founder of Thalesians Ltd. Paul has made contributions to mathematical logic, domain theory, and stochastic filtering theory, and, with Abbas Edalat, has published a prestigious LICS paper. He is a member of the British Computer Society, the Institution of Engineering and the European Complex Systems Society.

Matthew Dixon, FRM, Ph.D., is an Assistant Professor of Applied Math at the Illinois Institute of Technology and an Affiliate of the Stuart School of Business. He has published over 20 peer reviewed publications on machine learning and quant finance and has been cited in Bloomberg Markets and the Financial Times as an AI in fintech expert. He is Deputy Editor of the Journal of Machine Learning in Finance, Associate Editor of the AIMS Journal on Dynamics and Games, and is a member of the Advisory Board of the CFA Quantitative Investing Group.

Igor Halperin, Ph.D., is a Research Professor in Financial Engineering at NYU, and an AI Research associate at Fidelity Investments. Igor has published more than 50 scientific articles in machine learning, quantitative finance and theoretic physics. Prior to joining the financial industry, he held postdoctoral positions in theoretical physics at the Technion and the University of British Columbia.

9783030410674

Statistics in Business, Management, Economics, Finance, Insurance

/Statistics in Business, Management, Economics, Finance, Insurance/Applied Statistics/Statistics/Mathematics and Computing/

/Statistics in Business, Management, Economics, Finance, Insurance/Applied Statistics/Statistics/Mathematics and Computing//Applications of Mathematics/Mathematics and Computing/Mathematics//Statistics/Mathematics and Computing////

1: A Special Case of Fermat’s Conjecture.- 2: Number Fields and Number Rings.- 3: Prime Decomposition in Number Rings.- 4: Galois Theory Applied to Prime Decomposition.- 5: The Ideal Class Group and the Unit Group.- 6: The Distribution of Ideals in a Number Ring.- 7: The Dedekind Zeta Function and the Class Number Formula.- 8: The Distribution of Primes and an Introduction to Class Field Theory.- Appendix A: Commutative Rings and Ideals.- Appendix B: Galois Theory for Subfields of C.- Appendix C: Finite Fields and Rings.- Appendix D: Two Pages of Primes.- Further Reading.- Index of Theorems.- List of Symbols.

Requiring no more than a basic knowledge of abstract algebra, this textbook presents the basics of algebraic number theory in a straightforward, 'down-to-earth' manner. It thus avoids local methods, for example, and presents proofs in a way that highlights key arguments. There are several hundred exercises, providing a wealth of both computational and theoretical practice, as well as appendices summarizing the necessary background in algebra.Now in a newly typeset edition including a foreword by Barry Mazur, this highly regarded textbook will continue to provide lecturers and their students with an invaluable resource and a compelling gateway to a beautiful subject. From the reviews:“A thoroughly delightful introduction to algebraic number theory” – Ezra Brown in the Mathematical Reviews“An excellent basis for an introductory graduate course in algebraic number theory” – Harold Edwards in the Bulletin ofthe American Mathematical Society

Requiring no more than a basic knowledge of abstract algebra, this textbook presents the basics of algebraic number theory in a straightforward, 'down-to-earth' manner. It thus avoids local methods, for example, and presents proofs in a way that highlights key arguments. There are several hundred exercises, providing a wealth of both computational and theoretical practice, as well as appendices summarizing the necessary background in algebra.

Now in a newly typeset edition including a foreword by Barry Mazur, this highly regarded textbook will continue to provide lecturers and their students with an invaluable resource and a compelling gateway to a beautiful subject.



From the reviews:

“A thoroughly delightful introduction to algebraic number theory” – Ezra Brown in the Mathematical Reviews

“An excellent basis for an introductory graduate course in algebraic number theory” – Harold Edwards in the Bulletin of the American Mathematical Society

Contains over 300 exercisesAssumes only basic abstract algebraCovers topics leading up to class field theory

Daniel A. Marcus received his PhD from Harvard University in 1972. He was a J. Willard Gibbs Instructor at Yale University from 1972 to 1974 and Professor of Mathematics at California State Polytechnic University, Pomona, from 1979 to 2004. He published research papers in the areas of graph theory, number theory and combinatorics. The present book grew out of a lecture course given by the author at Yale University.

9783319902326

/Number Theory/Mathematics and Computing/Mathematics//Algebra/Mathematics and Computing/Mathematics/////

978-0-387-97993-9

John H. Conway; Richard Guy, University of Calgary Dept. Mathematics & Statistics, Calgary, AB, Canada

1 The Romance of Numbers.- 2 Figures from Figures: Doing Arithmetic and Algebra by Geometry.- 3 What Comes Next?.- 4 Famous Families of Numbers.- 5 The Primacy of Primes.- 6 Further Fruitfulness of Fractions.- 7 Geometric Problems and Algebraic Numbers.- 8 Imagining Imaginary Numbers.- 9 Some Transcendental Numbers.- 10 Infinite and Infinitesimal Numbers.

Journey through the world of numbers with the foremost authorities and writers in the field.
John Horton Conway and Richard K. Guy are two of the most accomplished, creative, and engaging number theorists any mathematically minded reader could hope to encounter. In this book, Conway and Guy lead the reader on an imaginative, often astonishing tour of the landscape of numbers.
The Book of Numbers is just that - an engagingly written, heavily illustrated introduction to the fascinating, sometimes surprising properties of numbers and number patterns. The book opens up a world of topics, theories, and applications, exploring intriguing aspects of real numbers, systems, arrays and sequences, and much more. Readers will be able to use figures to figure out figures, rub elbows with famous families of numbers, prove the primacy of primes, fathom the fruitfulness of fractions, imagine imaginary numbers, investigate the infinite and infinitesimal and more.

9780387979939

10.1007/978-1-4612-4072-3

Chapter 1: Vectors and Geometry.- Chapter 2: Linear systems and Subspaces.- Chapter 3: Unraveling Matrices.- Appendix A: Mathematical Preliminaries.- Appendix B: Additional Proofs.- Appendix C: Selected Exercises Solutions.

This textbook emphasizes the interplay between algebra and geometry to motivate the study of linear algebra. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. By focusing on this interface, the author offers a conceptual appreciation of the mathematics that is at the heart of further theory and applications. Those continuing to a second course in linear algebra will appreciate the companion volume Advanced Linear and Matrix Algebra. Starting with an introduction to vectors, matrices, and linear transformations, the book focuses on building a geometric intuition of what these tools represent. Linear systems offer a powerful application of the ideas seen so far, and lead onto the introduction of subspaces, linear independence, bases, and rank. Investigation then focuses on the algebraic properties of matrices that illuminate the geometry of the linear transformations that they represent. Determinants, eigenvalues, and eigenvectors all benefit from this geometric viewpoint. Throughout, “Extra Topic” sections augment the core content with a wide range of ideas and applications, from linear programming, to power iteration and linear recurrence relations. Exercises of all levels accompany each section, including many designed to be tackled using computer software.Introduction to Linear and Matrix Algebra is ideal for an introductory proof-based linear algebra course. The engaging color presentation and frequent marginal notes showcase the author’s visual approach. Students are assumed to have completed one or two university-level mathematics courses, though calculus is not an explicit requirement. Instructors will appreciate the ample opportunities to choose topics that align with the needs of each classroom, and the online homework sets that are available through WeBWorK.

This textbook emphasizes the interplay between algebra and geometry to motivate the study of linear algebra. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. By focusing on this interface, the author offers a conceptual appreciation of the mathematics that is at the heart of further theory and applications. Those continuing to a second course in linear algebra will appreciate the companion volume Advanced Linear and Matrix Algebra.

Starting with an introduction to vectors, matrices, and linear transformations, the book focuses on building a geometric intuition of what these tools represent. Linear systems offer a powerful application of the ideas seen so far, and lead onto the introduction of subspaces, linear independence, bases, and rank. Investigation then focuses on the algebraic properties of matrices that illuminate the geometry of the linear transformations that they represent. Determinants, eigenvalues, and eigenvectors all benefit from this geometric viewpoint. Throughout, “Extra Topic” sections augment the core content with a wide range of ideas and applications, from linear programming, to power iteration and linear recurrence relations. Exercises of all levels accompany each section, including many designed to be tackled using computer software.

Introduction to Linear and Matrix Algebra is ideal for an introductory proof-based linear algebra course. The engaging color presentation and frequent marginal notes showcase the author’s visual approach. Students are assumed to have completed one or two university-level mathematics courses, though calculus is not an explicit requirement. Instructors will appreciate the ample opportunities to choose topics that align with the needs of each classroom, and the online homework sets that are available through WeBWorK.

<p>Motivates the study of linear algebra by exploring the interplay between algebra and geometry</p><p>Engages readers with a visual approach that uses color to enhance both content and learning</p><p>Features a wide selection of theoretical and applied topics to complement the core material</p><p>Incorporates exercises of all levels, including many designed for computer software</p><p>Offers corresponding online homework sets through WeBWorK</p><p>Includes supplementary material: sn.pub/extras</p>

Nathaniel Johnston is an Associate Professor of Mathematics at Mount Allison University in New Brunswick, Canada. His research makes use of linear algebra, matrix analysis, and convex optimization to tackle questions related to the theory of quantum entanglement. His companion volume, Advanced Linear and Matrix Algebra, is also published by Springer.

9783030528102

/Linear Algebra/Algebra/Mathematics and Computing/Mathematics/

/Linear Algebra/Algebra/Mathematics and Computing/Mathematics//////

978-3-030-52814-0

Chapter 1: Vector Spaces.- Chapter 2: Matrix Decompositions.- Chapter 3: Tensors and Multilinearity.- Appendix A: Mathematical Preliminaries.- Appendix B: Additional Proofs.- Appendix C: Selected Exercise Solutions.

This textbook emphasizes the interplay between algebra and geometry to motivate the study of advanced linear algebra techniques. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. Building on a first course in linear algebra, this book offers readers a deeper understanding of abstract structures, matrix decompositions, multilinearity, and tensors. Concepts draw on concrete examples throughout, offering accessible pathways to advanced techniques.Beginning with a study of vector spaces that includes coordinates, isomorphisms, orthogonality, and projections, the book goes on to focus on matrix decompositions. Numerous decompositions are explored, including the Shur, spectral, singular value, and Jordan decompositions. In each case, the author ties the new technique back to familiar ones, to create a coherent set of tools. Tensors and multilinearity complete the book, with a study of the Kronecker product, multilinear transformations, and tensor products. Throughout, “Extra Topic” sections augment the core content with a wide range of ideas and applications, from the QR and Cholesky decompositions, to matrix-valued linear maps and semidefinite programming. Exercises of all levels accompany each section.
Advanced Linear and Matrix Algebra offers students of mathematics, data analysis, and beyond the essential tools and concepts needed for further study. The engaging color presentation and frequent marginal notes showcase the author’s visual approach. A first course in proof-based linear algebra is assumed. An ideal preparation can be found in the author’s companion volume, Introduction to Linear and Matrix Algebra.

This textbook emphasizes the interplay between algebra and geometry to motivate the study of advanced linear algebra techniques. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. Building on a first course in linear algebra, this book offers readers a deeper understanding of abstract structures, matrix decompositions, multilinearity, and tensors. Concepts draw on concrete examples throughout, offering accessible pathways to advanced techniques.

Beginning with a study of vector spaces that includes coordinates, isomorphisms, orthogonality, and projections, the book goes on to focus on matrix decompositions. Numerous decompositions are explored, including the Shur, spectral, singular value, and Jordan decompositions. In each case, the author ties the new technique back to familiar ones, to create a coherent set of tools. Tensors and multilinearity complete the book, with a study of the Kronecker product, multilinear transformations, and tensor products. Throughout, “Extra Topic” sections augment the core content with a wide range of ideas and applications, from the QR and Cholesky decompositions, to matrix-valued linear maps and semidefinite programming. Exercises of all levels accompany each section.


Advanced Linear and Matrix Algebra offers students of mathematics, data analysis, and beyond the essential tools and concepts needed for further study. The engaging color presentation and frequent marginal notes showcase the author’s visual approach. A first course in proof-based linear algebra is assumed. An ideal preparation can be found in the author’s companion volume, Introduction to Linear and Matrix Algebra.

<p>Motivates a deeper understanding of the abstract structures needed to tackle questions in mathematics, data analysis, and quantum information theory</p><p>Engages readers with a visual approach that uses color to enhance both content and learning</p><p>Features a wide selection of theoretical and applied topics to complement the core material</p><p>Incorporates exercises of all levels</p><p>Includes supplementary material: sn.pub/extras</p>

Nathaniel Johnston is an Associate Professor of Mathematics at Mount Allison University in New Brunswick, Canada. His research makes use of linear algebra, matrix analysis, and convex optimization to tackle questions related to the theory of quantum entanglement. His companion volume, Introduction to Linear and Matrix Algebra, is also published by Springer.

9783030528140

/Linear Algebra/Algebra/Mathematics and Computing/Mathematics/

/Linear Algebra/Algebra/Mathematics and Computing/Mathematics//////

978-3-030-47294-8

1 Apéritif.- 2 Foundations.- 3 The p-adic Numbers.- 4 Exploring ℚp.- 5 Elementary Analysis in ℚp.- 6 Vector Spaces and Field Extensions.- 7 Analysis in ℂp.- 8 Fun With Your New Head.- A Sage and GP: A (Very) Quick Introduction.- B Hints and Comments on the Problems.- C A Brief Glance at the Literature.- Bibliography.- Index.

There are numbers of all kinds: rational, real, complex, p-adic, and more. The p-adic numbers are not as well known as the others, but they play a fundamental role in number theory and in other parts of mathematics, capturing information related to a chosen prime number p. They also allow us to use methods from calculus and analysis to obtain results in algebra and number theory.This book is an elementary introduction to p-adic numbers. Most other books on the subject are written for more advanced students; this book provides an entryway to the subject for students with an undergraduate mathematics education. Readers who want to have an idea of and appreciation for the subject will probably find what they need in this book. Readers on the way to becoming experts can begin here before moving on to more advanced texts.This third edition has been thoroughly revised to correct mistakes, make the exposition clearer, and call attention to significant aspects that are usually reserved for advanced books. The most important addition is the integration of mathematical software for computations with p-adic numbers and functions. A final chapter includes a selection of problems for further exploration.
From the reviews of the first and second editions:'Perhaps the most suitable text for beginners' - The Mathematical Gazette'This text perfectly fulfills what it proposes' - Mathematical Reviews'An extraordinarily nice manner to introduce the uninitiated to the subject' - Mededelingen van het wiskundig genootschap'If I had to recommend one book on the subject to a student – or even to a fully grown mathematician ...– it would still be this book' - MAA Reviews

There are numbers of all kinds: rational, real, complex, p-adic, and more. The p-adic numbers are not as well known as the others, but they play a fundamental role in number theory and in other parts of mathematics, capturing information related to a chosen prime number p. They also allow us to use methods from calculus and analysis to obtain results in algebra and number theory.This book is an elementary introduction to p-adic numbers. Most other books on the subject are written for more advanced students; this book provides an entryway to the subject for students with an undergraduate mathematics education. Readers who want to have an idea of and appreciation for the subject will probably find what they need in this book. Readers on the way to becoming experts can begin here before moving on to more advanced texts.This third edition has been thoroughly revised to correct mistakes, make the exposition clearer, and call attention to significant aspects that are usually reserved for advanced books. The most important addition is the integration of mathematical software for computations with p-adic numbers and functions. A final chapter includes a selection of problems for further exploration.
From the reviews of the first and second editions:'Perhaps the most suitable text for beginners' - The Mathematical Gazette'This text perfectly fulfills what it proposes' - Mathematical Reviews'An extraordinarily nice manner to introduce the uninitiated to the subject' - Mededelingen van het wiskundig genootschap'If I had to recommend one book on the subject to a student – or even to a fully grown mathematician ...– it would still be this book' - MAA Reviews

Written for undergraduate students and beginnersMore than 300 exercises, most with hints or solutionsTeaches the use of open source mathematical software in number theory

Fernando Q. Gouvêa is a number theorist and historian of mathematics. In number theory he has been interested in the connection between p-adic modular forms and deformations of Galois representations. As a historian his main focus is on the early history of algebraic number theory, but he has also written on other historical topics. He enjoys books and has written many book reviews for a wide range of publications. For many years he served as editor of MAA Focus and of MAA Reviews. His current project is a forthcoming book on the history of p-adic numbers and p-adic analysis in the first decades of the twentieth century. Gouvêa’s other books include Arithmetic of p-adic Modular Forms, A Guide to Groups, Rings, and Fields, and (with William P. Berlinghoff) Math through the Ages: A Gentle History for Teachers and Others.

9783030472948

/Number Theory/Mathematics and Computing/Mathematics//Field Theory and Polynomials/Algebra/Mathematics and Computing/Mathematics//Analysis/Mathematics and Computing/Mathematics////

978-3-031-25408-6

Gal Gross, University of Toronto, Toronto, ON, Canada; Eckhard Meinrenken, University of Toronto, Toronto, ON, Canada

1. Introduction.- 2. Manifolds.- 3. Smooth maps.- 4. Submanifolds.- 5. Tangent spaces.- 6. Vector fields.- 7. Differential forms.- 8. Integration.- 9. Vector bundles.- Notions from set theory.- Notions from algebra.- Topological properties of manifolds.- Hints and answers to in-text questions.- References.- List of Symbols.- Index.

<div>This textbook serves as an introduction to modern differential geometry at a level accessible to advanced undergraduate and master's students. It places special emphasis on motivation and understanding, while developing a solid intuition for the more abstract concepts. In contrast to graduate level references, the text relies on a minimal set of prerequisites: a solid grounding in linear algebra and multivariable calculus, and ideally a course on ordinary differential equations. Manifolds are introduced intrinsically in terms of coordinate patches glued by transition functions. The theory is presented as a natural continuation of multivariable calculus; the role of point-set topology is kept to a minimum.

Questions sprinkled throughout the text engage students in active learning, and encourage classroom participation. Answers to these questions are provided at the end of the book, thus making it ideal for independent study. Material is further reinforced with homework problems ranging from straightforward to challenging. The book contains more material than can be covered in a single semester, and detailed suggestions for instructors are provided in the Preface.
</div>

This textbook serves as an introduction to modern differential geometry at a level accessible to advanced undergraduate and master's students. It places special emphasis on motivation and understanding, while developing a solid intuition for the more abstract concepts. In contrast to graduate level references, the text relies on a minimal set of prerequisites: a solid grounding in linear algebra and multivariable calculus, and ideally a course on ordinary differential equations. Manifolds are introduced intrinsically in terms of coordinate patches glued by transition functions. The theory is presented as a natural continuation of multivariable calculus; the role of point-set topology is kept to a minimum.

Questions sprinkled throughout the text engage students in active learning, and encourage classroom participation. Answers to these questions are provided at the end of the book, thus making it ideal for independent study. Material is further reinforced with homework problems ranging from straightforward to challenging. The book contains more material than can be covered in a single semester, and detailed suggestions for instructors are provided in the Preface.

For undergraduates! Required background material is typically covered in the first 2 or 3 years of universityThe role of point set topology is kept to a minimumTheory of manifolds appears as a natural continuation of multivariable calculus

Gal Gross is a Ph.D. student in mathematics at the University of Toronto, working in combinatorics and algebra with a special interest in additive combinatorics. Gross' other mathematical interests include differential geometry, set theory and foundational questions.

Eckhard Meinrenken is a professor of mathematics at the University of Toronto, working in the fields of differential geometry and mathematical physics. His contributions include a proof of the Guillemin-Sternberg conjecture in symplectic geometry and the development, with Alekseev and Malkin, of the theory of group-valued momentum maps. In 2002 he was an invited speaker at the ICM in Beijing, and in 2008 he was elected Fellow of the Royal Society of Canada. Meinrenken's book Clifford Algebras and Lie Theory was published (c) 2013 in Springer's Ergebnisse series

9783031254086

/Manifolds and Cell Complexes/Topology/Mathematics and Computing/Mathematics/

/Manifolds and Cell Complexes/Topology/Mathematics and Computing/Mathematics//Differential Geometry/Geometry/Mathematics and Computing/Mathematics/////

978-3-540-43871-7

Jean Jacod, Université Paris VI - Pierre et Marie Curie, Paris, Cedex 05, France; Philip Protter, Columbia University, Protter, NY, USA

1 Introduction.- 2 Axioms of Probability.- 3 Conditional Probability and Independence.- 4 Probabilities on a Finite or Countable Space.- 5 Random Variables on a Countable Space.- 6 Construction of a Probability Measure.- 7 Construction of a Probability Measure on R.- 8 Random Variables.- 9 Integration with Respect to a Probability Measure.- 10 Independent Random Variables.- 11 Probability Distributions on R.- 12 Probability Distributions on Rn.- 13 Characteristic Functions.- 14 Properties of Characteristic Functions.- 15 Sums of Independent Random Variables.- 16 Gaussian Random Variables (The Normal and the Multivariate Normal Distributions).- 17 Convergence of Random Variables.- 18 Weak Convergence.- 19 Weak Convergence and Characteristic Functions.- 20 The Laws of Large Numbers.- 21 The Central Limit Theorem.- 22 L2 and Hilbert Spaces.- 23 Conditional Expectation.- 24 Martingales.- 25 Supermartingales and Submartingales.- 26 Martingale Inequalities.- 27 Martingale Convergence Theorems.- 28 The Radon-Nikodym Theorem.- References.

We have made small changes throughout the book, including the exercises, and we have tried to correct if not all, then at least most of the typos. We wish to thank the many colleagues and students who have commented c- structively on the book since its publication two years ago, and in particular Professors Valentin Petrov, Esko Valkeila, Volker Priebe, and Frank Knight. Jean Jacod, Paris Philip Protter, Ithaca March, 2002 Preface to the Second Printing of the Second Edition We have bene?ted greatly from the long list of typos and small suggestions sent to us by Professor Luis Tenorio. These corrections have improved the book in subtle yet important ways, and the authors are most grateful to him. Jean Jacod, Paris Philip Protter, Ithaca January, 2004 Preface to the First Edition We present here a one semester course on Probability Theory. We also treat measure theory and Lebesgue integration, concentrating on those aspects which are especially germane to the study of Probability Theory. The book is intended to ?ll a current need: there are mathematically sophisticated s- dents and researchers (especially in Engineering, Economics, and Statistics) who need a proper grounding in Probability in order to pursue their primary interests. Many Probability texts available today are celebrations of Pr- ability Theory, containing treatments of fascinating topics to be sure, but nevertheless they make it di?cult to construct a lean one semester course that covers (what we believe) are the essential topics.

In the words of one reviewer: "Normally graduate students need two books, one on measure theory and one on probability theory.This book contains (most of) the essentials of both fields and students can go on directly to Oksendal's book on SDE theory.I would personally recommend it to my students.In my eyes this book is, in this respect, a small gold mine."Includes supplementary material: sn.pub/extras

9783540438717

/Mathematics and Computing/Probability Theory/Mathematics//Mathematics in Business, Economics and Finance/Applications of Mathematics/Mathematics and Computing/Mathematics/////

978-0-387-98650-0

Robin Hartshorne, Department of Mathematics University of California at Berkeley, Berkeley, CA, USA

1. Euclid’s Geometry.- 2. Hilbert’s Axioms.- 3. Geometry over Fields.- 4. Segment Arithmetic.- 5. Area.- 6. Construction Problems and Field Extensions.- 7. Non-Euclidean Geometry.- 8. Polyhedra.- Appendix: Brief Euclid.- Notes.- References.- List of Axioms.- Index of Euclid’s Propositions.

In recent years, I have been teaching a junior-senior-level course on the classi­ cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa­ rately. The remainder of the book is an exploration of questions that arise natu­ rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks.

9780387986500

10.1007/978-0-387-22676-7

978-0-387-98592-3

Serge Lang, Yale University Dept. Mathematics, New Haven, CT, USA

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.

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.

<p>Written by a well-known and somewhat controversial Springer author whose writing style has achieved a considerable following</p><p>Unique approach should appeal to those who want to obtain new results in this field</p><p>Goes beyond the basic material to present special topics of current interest</p><p>Additional topics covered in the second part give readers a great deal of flexibility in structuring their learning of the subject</p><p>Extensively revised edition, includes new examples and exercises, and hundreds of minor improvements throughout</p>

9780387985923

978-0-387-90053-7

I. Basic Concepts.- 1. Definitions and first examples.- 2. Ideals and homomorphisms.- 3. Solvable and nilpotent Lie algebras.- II. Semisimple Lie Algebras.- 4. Theorems of Lie and Cartan.- 5. Killing form.- 6. Complete reducibility of representations.- 7. Representations of sl (2, F).- 8. Root space decomposition.- III. Root Systems.- 9. Axiomatics.- 10. Simple roots and Weyl group.- 11. Classification.- 12. Construction of root systems and automorphisms.- 13. Abstract theory of weights.- IV. Isomorphism and Conjugacy Theorems.- 14. Isomorphism theorem.- 15. Cartan subalgebras.- 16. Conjugacy theorems.- V. Existence Theorem.- 17. Universal enveloping algebras.- 18. The simple algebras.- VI. Representation Theory.- 20. Weights and maximal vectors.- 21. Finite dimensional modules.- 22. Multiplicity formula.- 23. Characters.- 24. Formulas of Weyl, Kostant, and Steinberg.- VII. Chevalley Algebras and Groups.- 25. Chevalley basis of L.- 26. Kostant’s Theorem.- 27. Admissible lattices.- References.- Afterword (1994).- Index of Terminology.- Index of Symbols.

This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. A good knowledge of linear algebra (including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector spaces) is presupposed, as well as some acquaintance with the methods of abstract algebra. The first four chapters might well be read by a bright undergraduate; however, the remaining three chapters are admittedly a little more demanding. Besides being useful in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results. Since Jacobson's book appeared a decade ago, improvements have been made even in the classical parts of the theory. I have tried to incor­ porate some of them here and to provide easier access to the subject for non-specialists. For the specialist, the following features should be noted: (I) The Jordan-Chevalley decomposition of linear transformations is emphasized, with 'toral' subalgebras replacing the more traditional Cartan subalgebras in the semisimple case. (2) The conjugacy theorem for Cartan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry.

9780387900537

10.1007/978-1-4612-6398-2

978-0-387-97245-9

John B Conway, The George Washington Univ Dept of Mathematics, Fairfax, VA

,978-1-4757-3830-8,978-1-4757-3829-2,978-0-387-96042-5,978-1-4757-3828-5

I Hilbert Spaces.- II Operators on Hilbert Space.- III Banach Spaces.- IV Locally Convex Spaces.- V Weak Topologies.- VI Linear Operators on a Banach Space.- VII Banach Algebras and Spectral Theory for Operators on a Banach Space.- VIII C*-Algebras.- IX Normal Operators on Hilbert Space.- X Unbounded Operators.- XI Fredholm Theory.- Appendix A Preliminaries.- §1. Linear Algebra.- §2. Topology.- List of Symbols.

Functional analysis has become a sufficiently large area of mathematics that it is possible to find two research mathematicians, both of whom call themselves functional analysts, who have great difficulty understanding the work of the other. The common thread is the existence of a linear space with a topology or two (or more). Here the paths diverge in the choice of how that topology is defined and in whether to study the geometry of the linear space, or the linear operators on the space, or both. In this book I have tried to follow the common thread rather than any special topic. I have included some topics that a few years ago might have been thought of as specialized but which impress me as interesting and basic. Near the end of this work I gave into my natural temptation and included some operator theory that, though basic for operator theory, might be considered specialized by some functional analysts.

9780387972459

10.1007/978-1-4757-4383-8

978-0-387-92299-7

Peter D. Hoff, University of Washington Department of Statistics, Seattle, WA, USA

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.

This book provides a compact self-contained introduction to the theory and application of Bayesian statistical methods. The book is accessible to readers having a basic familiarity with probability, yet allows more advanced readers to quickly grasp the principles underlying Bayesian theory and methods. The examples and computer code allow the reader to understand and implement basic Bayesian data analyses using standard statistical models and to extend the standard models to specialized data analysis situations. The book begins with fundamental notions such as probability, exchangeability and Bayes' rule, and ends with modern topics such as variable selection in regression, generalized linear mixed effects models, and semiparametric copula estimation. Numerous examples from the social, biological and physical sciences show how to implement these methodologies in practice.
Monte Carlo summaries of posterior distributions play an important role in Bayesian data analysis. The open-source R statistical computing environment provides sufficient functionality to make Monte Carlo estimation very easy for a large number of statistical models and example R-code is provided throughout the text. Much of the example code can be run ``as is'' in R, and essentially all of it can be run after downloading the relevant datasets from the companion website for this book.
Peter Hoff is an Associate Professor of Statistics and Biostatistics at the University of Washington. He has developed a variety of Bayesian methods for multivariate data, including covariance and copula estimation, cluster analysis, mixture modeling and social network analysis. He is on the editorial board of the Annals of Applied Statistics.

Provides a nice introduction to Bayesian statistics with sufficient grounding in the Bayesian framework without being distracted by more esoteric pointsThe material is well-organized, weaving applications, background material and computation discussions throughout the bookR examples also facilitate how the approaches workIncludes supplementary material: sn.pub/extras

9780387922997

/Mathematics and Computing/Probability Theory/Mathematics//Operations Research, Management Science /Optimization/Mathematics and Computing/Mathematics//Statistical Theory and Methods/Statistics/Mathematics and Computing//Sociological Methods/Humanities and Social Sciences/Society/Sociology//Probability and Statistics in Computer Science/Mathematics of Computing/Computer Science/Mathematics and Computing//Econometrics/Quantitative Economics/Economics/Humanities and Social Sciences/

10.1007/978-0-387-92407-6

Introduction: Prerequisites and Preliminaries.- 1. Logic.- 2. Sets and Classes.- 3. Functions.- 4. Relations and Partitions.- 5. Products.- 6. The Integers.- 7. The Axiom of Choice, Order and Zorn’s Lemma.- 8. Cardinal Numbers.- I: Groups.- 1. Semigroups, Monoids and Groups.- 2. Homomorphisms and Subgroups.- 3. Cyclic Groups.- 4. Cosets and Counting.- 5. Normality, Quotient Groups, and Homomorphisms.- 6. Symmetric, Alternating, and Dihedral Groups.- 7. Categories: Products, Coproducts, and Free Objects.- 8. Direct Products and Direct Sums.- 9. Free Groups, Free Products, Generators & Relations.- II: The Structure of Groups.- 1. Free Abelian Groups.- 2. Finitely Generated Abelian Groups.- 3. The Kruli-Schmidt Theorem.- 4. The Action of a Group on a Set.- 5. The Sylow Theorems.- 6. Classification of Finite Groups.- 7. Nilpotent and Solvable Groups.- 8. Normal and Subnormal Series.- III: Rings.- 1. Rings and Homomorphisms.- 2. Ideals.- 3. Factorization in Commutative Rings.- 4. Rings of Quotients and Localization.- 5. Rings of Polynomials and Formal Power Series.- 6. Factorization in Polynomial Rings.- IV: Modules.- 1. Modules, Homomorphisms and Exact Sequences.- 2. Free Modules and Vector Spaces.- 3. Projective and Injective Modules.- 4. Horn and Duality.- 5. Tensor Products.- 6. Modules over a Principal Ideal Domain.- 7. Algebras.- V: Fields and Galois Theory.- 1. Field Extensions.- 2. The Fundamental Theorem.- 3. Splitting Fields, Algebraic Closure and Normality.- 4. The Galois Group of a Polynomial.- 5. Finite Fields.- 6. Separability.- 7. Cyclic Extensions.- 8. Cyclotomic Extensions.- 9. Radical Extensions.- VI: The Structure of Fields.- 1. Transcendence Bases.- 2. Linear Disjointness and Separability.- VII: Linear Algebra.- 1. Matrices and Maps.- 2. Rank andEquivalence.- 3. Determinants.- 4. Decomposition of a Single Linear Transformation and Similarity.- 5. The Characteristic Polynomial, Eigenvectors and Eigenvalues.- VIII: Commutative Rings and Modules.- 1. Chain Conditions.- 2. Prime and Primary Ideals.- 3. Primary Decomposition.- 4. Noetherian Rings and Modules.- 5. Ring Extensions.- 6. Dedekind Domains.- 7. The Hilbert Nullstellensatz.- IX: The Structure of Rings.- 1. Simple and Primitive Rings.- 2. The Jacobson Radical.- 3. Semisimple Rings.- 4. The Prime Radical; Prime and Semiprime Rings.- 5. Algebras.- 6. Division Algebras.- X: Categories.- 1. Functors and Natural Transformations.- 2. Adjoint Functors.- 3. Morphisms.- List of Symbols.

Algebra fulfills a definite need to provide a self-contained, one volume, graduate level algebra text that is readable by the average graduate student and flexible enough to accomodate a wide variety of instructors and course contents. The guiding philosophical principle throughout the text is that the material should be presented in the maximum usable generality consistent with good pedagogy. Therefore it is essentially self-contained, stresses clarity rather than brevity and contains an unusually large number of illustrative exercises. The book covers major areas of modern algebra, which is a necessity for most mathematics students in sufficient breadth and depth.

9780387905181

978-1-84882-890-2

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.

Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions. It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates. 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.com<br/> Praise for the first edition: 'The text is nicely illustrated, the definitions are well-motivated and the proofs are particularly well-written and student-friendly…this book would make an excellent text for an undergraduate course, but could also well be used for a reading course, or simply read for pleasure.' Australian Mathematical Society Gazette 'Excellent figures supplement a good account, sprinkled with illustrative examples.' Times Higher Education Supplement

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

A revised and expanded second edition which presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject.Includes supplementary material: sn.pub/extrasRequest lecturer material: sn.pub/lecturer-material

Andrew Pressley is Professor of Mathematics at King’s College London, UK.

9781848828902

/Differential Geometry/Geometry/Mathematics and Computing/Mathematics/

/Differential Geometry/Geometry/Mathematics and Computing/Mathematics//////

10.1007/978-1-84882-891-9

Kristopher Tapp, St. Joseph's University Department of Mathematics, Philadelphia, PA, USA

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.

<div>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.</div><div>
</div>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.<div>
</div><div>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.</div><div>
</div>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.

<div>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.</div><div>
</div><div>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.</div><div>
</div><div>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.</div><div>
</div><div>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. </div>

Can be used as a textbook in elementary and more advanced courses in differential geometryFocuses on applications of differential geometry, lending simplicity to more difficult and abstract conceptsFeatures full-color text and inserts to distinguish fundamental definitions and theoremsIncludes supplementary material: sn.pub/extras

<div>Kristopher Tapp is Professor of Mathematics at Saint Joseph's University. He has been awarded two National Science Foundation research grants to support research in differential geometry, and several teaching awards. He is the author of Symmetry: A Mathematical Exploration (Springer, 2012) and over twenty research papers featured in top journals.
</div>

9783319397986

/Differential Geometry/Geometry/Mathematics and Computing/Mathematics/

/Differential Geometry/Geometry/Mathematics and Computing/Mathematics//////

Jay L. Devore, California Polytechnic State University, San Luis Obispo, CA, USA; Kenneth N. Berk, Illinois State University, Normal, IL, USA; Matthew A. Carlton, California Polytechnic State University, San Luis Obispo, CA, USA

,978-1-4614-0390-6,978-1-4614-0392-0,978-1-4614-0391-3,978-1-4939-4221-3

Preface.- 1 Overview and Descriptive Statistics.- 2 Probability.- 3 Discrete Random Variables and Probability Distributions.- 4 Continuous Random Variables and Probability Distributions.- 5 Joint Probability Distributions and Their Applications.- 6 Statistics and Sampling Distributions.- 7 Point Estimation.- 8 Statistical Intervals Based on a Single Sample.- 9 Tests of Hypotheses Based on a Single Sample.- 10 Inferences Based on Two Samples.- 11 The Analysis of Variance.- 12 Regression and Correlation.- 13 Chi-Squared Tests.- 14 Chi-Squared Tests.- 15 Introduction to Bayesian Estimation.- Appendix Tables.- Index.

This 3rd edition of Modern Mathematical Statistics with Applications tries to strike a balance between mathematical foundations and statistical practice. The book provides a clear and current exposition of statistical concepts and methodology, including many examples and exercises based on real data gleaned from publicly available sources. Here is a small but representative selection of scenarios for our examples and exercises based on information in recent articles:Use of the “Big Mac index” by the publication The Economist as a humorous way to compare product costs across nationsVisualizing how the concentration of lead levels in cartridges varies for each of five brands of e-cigarettesDescribing the distribution of grip size among surgeons and how it impacts their ability to use a particular brand of surgical staplerEstimating the true average odometer reading of used Porsche Boxsters listed for sale on www.cars.comComparing head acceleration after impact when wearing a football helmet with acceleration without a helmetInvestigating the relationship between body mass index and foot load while runningThe main focus of the book is on presenting and illustrating methods of inferential statistics used by investigators in a wide variety of disciplines, from actuarial science all the way to zoology. It begins with a chapter on descriptive statistics that immediately exposes the reader to the analysis of real data. The next six chapters develop the probability material that facilitates the transition from simply describing data to drawing formal conclusions based on inferential methodology. Point estimation, the use of statistical intervals, and hypothesis testing are the topics of the first three inferential chapters. The remainder of the book explores the use of these methods in a variety of more complex settings.This edition includes many new examples and exercises as well as an introduction to the simulation of events and probability distributions. There are more than 1300 exercises in the book, ranging from very straightforward to reasonably challenging. Many sections have been rewritten with the goal of streamlining and providing a more accessible exposition. Output from the most common statistical software packages is included wherever appropriate (a feature absent from virtually all other mathematical statistics textbooks). The authors hope that their enthusiasm for the theory and applicability of statistics to real world problems will encourage students to pursue more training in the discipline.

This 3rd edition of Modern Mathematical Statistics with Applications tries to strike a balance between mathematical foundations and statistical practice. The book provides a clear and current exposition of statistical concepts and methodology, including many examples and exercises based on real data gleaned from publicly available sources. Here is a small but representative selection of scenarios for our examples and exercises based on information in recent articles: Use of the “Big Mac index” by the publication The Economist as a humorous way to compare product costs across nationsVisualizing how the concentration of lead levels in cartridges varies for each of five brands of e-cigarettesDescribing the distribution of grip size among surgeons and how it impacts their ability to use a particular brand of surgical staplerEstimating the true average odometer reading of used Porsche Boxsters listed for sale on www.cars.comComparing head acceleration after impact when wearing a football helmet with acceleration without a helmetInvestigating the relationship between body mass index and foot load while running The main focus of the book is on presenting and illustrating methods of inferential statistics used by investigators in a wide variety of disciplines, from actuarial science all the way to zoology. It begins with a chapter on descriptive statistics that immediately exposes the reader to the analysis of real data. The next six chapters develop the probability material that facilitates the transition from simply describing data to drawing formal conclusions based on inferential methodology. Point estimation, the use of statistical intervals, and hypothesis testing are the topics of the first three inferential chapters. The remainder of the book explores the use of these methods in a variety of more complex settings.

This edition includes manynew examples and exercises as well as an introduction to the simulation of events and probability distributions. There are more than 1300 exercises in the book, ranging from very straightforward to reasonably challenging. Many sections have been rewritten with the goal of streamlining and providing a more accessible exposition. Output from the most common statistical software packages is included wherever appropriate (a feature absent from virtually all other mathematical statistics textbooks). The authors hope that their enthusiasm for the theory and applicability of statistics to real world problems will encourage students to pursue more training in the discipline.

Features an extensive range of real-world and relevant applications to connect students to the concepts and theory, making the volume useful for quantitative courses in a wide variety of majors (business, mathematics, statistics, social sciences, sciences, and engineering, among others)Includes updates on the latest methods in statistical practice, as well as the latest in statistical software packages, in this new editionIncludes sample syllabi for one- and two-term courses in mathematical statistics, which serve as guides for instructors in smoothly adjusting to a new textIncludes supplementary material: sn.pub/extrasRequest lecturer material: sn.pub/lecturer-material

Jay L. Devore received a B.S. in Engineering Science from the University of California, Berkeley, and a Ph.D. in Statistics from Stanford University. He previously taught at the University of Florida and Oberlin College, and has had visiting positions at Stanford, Harvard, the University of Washington, New York University, and Columbia. He has been at California Polytechnic State University, San Luis Obispo, since 1977, where he was chair of the Department of Statistics for seven years and recently achieved the exalted status of Professor Emeritus.Jay has previously authored or coauthored five other books, including Probability and Statistics for Engineering and the Sciences, which won a McGuffey Longevity Award from the Text and Academic Authors Association for demonstrated excellence over time. He is a Fellow of the American Statistical Association, has been an associate editor for both the Journal of the American Statistical Association and The American Statistician, and received the Distinguished Teaching Award from Cal Poly in 1991. His recreational interests include reading, playing tennis, traveling, and cooking and eating good food.Kenneth N. Berk has a B.S. in Physics from Carnegie Tech (now Carnegie Mellon) and a Ph.D. in Mathematics from the University of Minnesota. He is Professor Emeritus of Mathematics at Illinois State University and a Fellow of the American Statistical As­sociation. He founded the Software Reviews section of The American Statistician and edited it for six years. He served as secretary/treasurer, program chair, and chair of the Statistical Computing Section of the American Statistical Association, and he twice co-chaired the Interface Symposium, the main annual meeting in statistical computing. His published work includes papers on time series, statistical computing, regression analysis, and statistical graphics, as well as the book Data Analysis with Microsoft Excel (with Patrick Carey).Matthew A. Carlton is Professor of Statistics at California Polytechnic State University, San Luis Obispo, where he joined the faculty in 1999. He received a B.A. in Mathematics from the University of California, Berkeley and a Ph.D. in Mathematics from the University of California, Los Angeles, with an emphasis on pure and applied probability; his thesis research involved applications of the Poisson-Dirichlet random process. Matt has published papers in the Journal of Applied Probability, Human Biology, Journal of Statistics Education, and The American Statistician. He was also the lead content adviser for the “Statistically Speaking” video series, designed for community college statistics courses, and he has published a variety of educational materials for high school statistics teachers. Matt was responsible for developing both the applied probability course and the probability and random processes course at Cal Poly, which in turn inspired him to get involved in writing this text. His professional research focus involves applications of probability to genetics and engineering. Personal interests include travel, good wine, and college sports.

9783030551551

/Statistical Theory and Methods/Statistics/Mathematics and Computing/

/Statistical Theory and Methods/Statistics/Mathematics and Computing//Statistics in Business, Management, Economics, Finance, Insurance/Applied Statistics/Statistics/Mathematics and Computing////

978-1-4471-6394-7

The Laplace Transform.- Further Properties of the Laplace Transform.- Convolution and the Solution of Ordinary Differential Equations.- Fourier Series.- Partial Differential Equations.- Fourier Transforms.- Wavelets and Signal Processing.- Complex Variables and Laplace Transforms.

Laplace transforms continue to be a very important tool for the engineer, physicist and applied mathematician. They are also now useful to financial, economic and biological modellers as these disciplines become more quantitative. Any problem that has underlying linearity and with solution based on initial values can be expressed as an appropriate differential equation and hence be solved using Laplace transforms. In this book, there is a strong emphasis on application with the necessary mathematical grounding. There are plenty of worked examples with all solutions provided. This enlarged new edition includes generalised Fourier series and a completely new chapter on wavelets. Only knowledge of elementary trigonometry and calculus are required as prerequisites. An Introduction to Laplace Transforms and Fourier Series will be useful for second and third year undergraduate students in engineering, physics or mathematics, as well as for graduates in any disciplinesuch as financial mathematics, econometrics and biological modelling requiring techniques for solving initial value problems.

In this book, there is a strong emphasis on application with the necessary mathematical grounding. There are plenty of worked examples with all solutions provided. This enlarged new edition includes generalised Fourier series and a completely new chapter on wavelets.
Only knowledge of elementary trigonometry and calculus are required as prerequisites. An Introduction to Laplace Transforms and Fourier Series will be useful for second and third year undergraduate students in engineering, physics or mathematics, as well as for graduates in any discipline such as financial mathematics, econometrics and biological modelling requiring techniques for solving initial value problems.

Provides an easy-to-read account of fourier series, wavelets and laplace transformsContains many examplesProvides solutions to all exercisesIncludes supplementary material: sn.pub/extras

Phil Dyke has over 40 years experience teaching at UK Universities, and for the past 6 years has based a course on the subject of this book. He has also used Laplace transforms and Fourier methods in his research. He has been a professor of applied mathematics at Plymouth University for over 20 years.

9781447163947

/Integral Transforms and Operational Calculus/Analysis/Mathematics and Computing/Mathematics/

/Integral Transforms and Operational Calculus/Analysis/Mathematics and Computing/Mathematics//Fourier Analysis/Analysis/Mathematics and Computing/Mathematics//Functions of a Complex Variable/Analysis/Mathematics and Computing/Mathematics//Mathematical and Computational Engineering Applications/Technology and Engineering//Mathematical Methods in Physics/Theoretical, Mathematical and Computational Physics/Physics and Astronomy/Physical Sciences/

10.1007/978-1-4471-6395-4

978-3-658-43030-6

Ulrich Görtz, Universität Duisburg-Essen, Essen, Germany; Torsten Wedhorn, TU Darmstadt, Darmstadt, Germany

Introduction.- 17 Differentials.- 18 Étale and smooth morphisms.- 19 Local complete intersections.- 20 The étale topology.- 21 Cohomology of sheaves of modules.- 22 Cohomology of quasi-coherent modules.- 23 Cohomology of projective and proper schemes.- 24 Theorem on formal functions.- 25 Duality.- 26 Curves.- 27 Abelian schemes.- F Homological algebra.- G Commutative algebra II.

<div>This book completes the comprehensive introduction to modern algebraic geometry which was started with the introductory volume Algebraic Geometry I: Schemes.
</div><div>
</div>It begins by discussing in detail the notions of smooth, unramified and étale morphisms including the étale fundamental group. The main part is dedicated to the cohomology of quasi-coherent sheaves. The treatment is based on the formalism of derived categories which allows an efficient and conceptual treatment of the theory, which is of crucial importance in all areas of algebraic geometry. After the foundations are set up, several more advanced topics are studied, such as numerical intersection theory, an abstract version of the Theorem of Grothendieck-Riemann-Roch, the Theorem on Formal Functions, Grothendieck's algebraization results and a very general version of Grothendieck duality. The book concludes with chapters on curves and on abelian schemes, which serve todevelop the basics of the theory of these two important classes of schemes on an advanced level, and at the same time to illustrate the power of the techniques introduced previously.<div>
</div><div>The text contains many exercises that allow the reader to check their comprehension of the text, present further examples or give an outlook on further results.</div><div>
</div><div>Contents</div><div>
</div><div>Differentials - Étale and smooth morphisms - Local complete intersections - The étale topology - Cohomology of sheaves of modules - Cohomology of quasi-coherent sheaves - Cohomology of projective and proper schemes - Theorem on formal functions - Duality - Curves - Abelian schemes - Appendix: Homological Algebra - Appendix: Commutative Algebra</div><div>
</div><div>About the Authors</div><div>
</div>Prof. Dr. Ulrich Görtz, Department of Mathematics, University of Duisburg-Essen
<div>Prof. Dr. TorstenWedhorn, Department of Mathematics, Technical University of Darmstadt</div>

This book completes the comprehensive introduction to modern algebraic geometry which was started with the introductory volume Algebraic Geometry I: Schemes.<div>
</div>It begins by discussing in detail the notions of smooth, unramified and étale morphisms including the étale fundamental group. The main part is dedicated to the cohomology of quasi-coherent sheaves. The treatment is based on the formalism of derived categories which allows an efficient and conceptual treatment of the theory, which is of crucial importance in all areas of algebraic geometry. After the foundations are set up, several more advanced topics are studied, such as numerical intersection theory, an abstract version of the Theorem of Grothendieck-Riemann-Roch, the Theorem on Formal Functions, Grothendieck's algebraization results and a very general version of Grothendieck duality. The book concludes with chapters on curves and on abelian schemes, which serve to develop the basics of the theory of these two important classes of schemes on an advanced level, and at the same time to illustrate the power of the techniques introduced previously.<div>
</div><div>The text contains many exercises that allow the reader to check their comprehension of the text, present further examples or give an outlook on further results.</div>

A systematic approach combined with explicit motivation of theoryContaining lots of concrete examplesYour companion into the field of modern algebraic geometry

Prof. Dr. Ulrich Görtz, Department of Mathematics, University of Duisburg-EssenProf. Dr. Torsten Wedhorn, Department of Mathematics, Technical University of Darmstadt

9783658430306

/Algebraic Geometry/Algebra/Mathematics and Computing/Mathematics/

/Algebraic Geometry/Algebra/Mathematics and Computing/Mathematics/////

978-0-387-97710-2

Saunders MacLane, Heidelberg, Germany; Ieke Moerdijk, Universiteit Nijmegen, Nijmegen, Netherlands

Prologue.- Categorial Preliminaries.- I. Categories of Functors.- 1. The Categories at Issue.- 2. Pullbacks.- 3. Characteristic Functions of Subobjects.- 4. Typical Subobject Classifiers.- 5. Colimits.- 6. Exponentials.- 7. Propositional Calculus.- 8. Heyting Algebras.- 9. Quantifiers as Adjoints.- Exercises.- II. Sheaves of Sets.- 1. Sheaves.- 2. Sieves and Sheaves.- 3. Sheaves and Manifolds.- 4. Bundles.- 5. Sheaves and Cross-Sections.- 6. Sheaves as Étale Spaces.- 7. Sheaves with Algebraic Structure.- 8. Sheaves are Typical.- 9. Inverse Image Sheaf.- Exercises.- III. Grothendieck Topologies and Sheaves.- 1. Generalized Neighborhoods.- 2. Grothendieck Topologies.- 3. The Zariski Site.- 4. Sheaves on a Site.- 5. The Associated Sheaf Functor.- 6. First Properties of the Category of Sheaves.- 7. Subobject Classifiers for Sites.- 8. Subsheaves.- 9. Continuous Group Actions.- Exercises.- IV. First Properties of Elementary Topoi.- 1. Definition of a Topos.- 2. The Construction of Exponentials.- 3. Direct Image.- 4. Monads and Beck’s Theorem.- 5. The Construction of Colimits.- 6. Factorization and Images.- 7. The Slice Category as a Topos.- 8. Lattice and Heyting Algebra Objects in a Topos.- 9. The Beck-Chevalley Condition.- 10. Injective Objects.- Exercises.- V. Basic Constructions of Topoi.- 1. Lawvere-Tierney Topologies.- 2. Sheaves.- 3. The Associated Sheaf Functor.- 4. Lawvere-Tierney Subsumes Grothendieck.- 5. Internal Versus External.- 6. Group Actions.- 7. Category Actions.- 8. The Topos of Coalgebras.- 9. The Filter-Quotient Construction.- Exercises.- VI. Topoi and Logic.- 1. The Topos of Sets.- 2. The Cohen Topos.- 3. The Preservation of Cardinal Inequalities.- 4. The Axiom of Choice.- 5. The Mitchell-Bénabou Language.- 6. Kripke-Joyal Semantics.- 7. Sheaf Semantics.- 8. Real Numbers in a Topos.- 9. Brouwer’s Theorem: All Functions are Continuous.- 10. Topos-Theoretic and Set-Theoretic Foundations.- Exercises.- VII. Geometric Morphisms.- 1. Geometric Morphismsand Basic Examples.- 2. Tensor Products.- 3. Group Actions.- 4. Embeddings and Surjections.- 5. Points.- 6. Filtering Functors.- 7. Morphisms into Grothendieck Topoi.- 8. Filtering Functors into a Topos.- 9. Geometric Morphisms as Filtering Functors.- 10. Morphisms Between Sites.- Exercises.- VIII. Classifying Topoi.- 1. Classifying Spaces in Topology.- 2. Torsors.- 3. Classifying Topoi.- 4. The Object Classifier.- 5. The Classifying Topos for Rings.- 6. The Zariski Topos Classifies Local Rings.- 7. Simplicial Sets.- 8. Simplicial Sets Classify Linear Orders.- Exercises.- IX. Localic Topoi.- 1. Locales.- 2. Points and Sober Spaces.- 3. Spaces from Locales.- 4. Embeddings and Surjections of Locales.- 5. Localic Topoi.- 6. Open Geometric Morphisms.- 7. Open Maps of Locales.- 8. Open Maps and Sites.- 9. The Diaconescu Cover and Barr’s Theorem.- 10. The Stone Space of a Complete Boolean Algebra.- 11. Deligne’s Theorem.- Exercises.- X. Geometric Logic and Classifying Topoi.- 1. First-OrderTheories.- 2. Models in Topoi.- 3. Geometric Theories.- 4. Categories of Definable Objects.- 5. Syntactic Sites.- 6. The Classifying Topos of a Geometric Theory.- 7. Universal Models.- Exercises.- Appendix: Sites for Topoi.- Epilogue.- Index of Notation.

We dedicate this book to the memory of J. Frank Adams. His clear insights have inspired many mathematicians, including both of us. In January 1989, when the first draft of our book had been completed, we heard the sad news of his untimely death. This has cast a shadow on our subsequent work. Our views of topos theory, as presented here, have been shaped by continued study, by conferences, and by many personal contacts with friends and colleagues-including especially O. Bruno, P. Freyd, J.M.E. Hyland, P.T. Johnstone, A. Joyal, A. Kock, F.W. Lawvere, G.E. Reyes, R Solovay, R Swan, RW. Thomason, M. Tierney, and G.C. Wraith. Our presentation combines ideas and results from these people and from many others, but we have not endeavored to specify the various original sources. Moreover, a number of people have assisted in our work by pro­ viding helpful comments on portions of the manuscript. In this respect, we extend our hearty thanks in particular to P. Corazza, K. Edwards, J. Greenlees, G. Janelidze, G. Lewis, and S. Schanuel.

9780387977102

/Geometry/Mathematics and Computing/Mathematics//K-Theory/Algebra/Mathematics and Computing/Mathematics//Mathematical Logic and Foundations/Mathematics and Computing/Mathematics////

10.1007/978-1-4612-0927-0

978-3-030-46039-6

Jean Gallier, University of Pennsylvania, Philadelphia, PA, USA; Jocelyn Quaintance, University of Pennsylvania, Philadelphia, PA, USA

1. The Matrix Exponential; Some Matrix Lie Groups.- 2. Adjoint Representations and the Derivative of exp.- 3. Introduction to Manifolds and Lie Groups.- 4. Groups and Group Actions.- 5. The Lorentz Groups ⊛.- 6. The Structure of O(p,q) and SO(p, q).- 7. Manifolds, Tangent Spaces, Cotangent Spaces.- 8. Construction of Manifolds From Gluing Data ⊛.- 9. Vector Fields, Integral Curves, Flows.- 10. Partitions of Unity, Covering Maps ⊛.- 11. Basic Analysis: Review of Series and Derivatives.- 12. A Review of Point Set Topology.-13. Riemannian Metrics, Riemannian Manifolds.- 14. Connections on Manifolds.- 15. Geodesics on Riemannian Manifolds.- 16. Curvature in Riemannian Manifolds.- 17. Isometries, Submersions, Killing Vector Fields.- 18. Lie Groups, Lie Algebra, Exponential Map.- 19. The Derivative of exp and Dynkin's Formula ⊛.- 20. Metrics, Connections, and Curvature of Lie Groups.- 21. The Log-Euclidean Framework.- 22. Manifolds Arising from Group Actions.

This textbook offers an introduction to differential geometry designed for readers interested in modern geometry processing. Working from basic undergraduate prerequisites, the authors develop manifold theory and Lie groups from scratch; fundamental topics in Riemannian geometry follow, culminating in the theory that underpins manifold optimization techniques. Students and professionals working in computer vision, robotics, and machine learning will appreciate this pathway into the mathematical concepts behind many modern applications.

Starting with the matrix exponential, the text begins with an introduction to Lie groups and group actions. Manifolds, tangent spaces, and cotangent spaces follow; a chapter on the construction of manifolds from gluing data is particularly relevant to the reconstruction of surfaces from 3D meshes. Vector fields and basic point-set topology bridge into the second part of the book, which focuses on Riemannian geometry.

Chapters on Riemannian manifolds encompass Riemannian metrics, geodesics, and curvature. Topics that follow include submersions, curvature on Lie groups, and the Log-Euclidean framework. The final chapter highlights naturally reductive homogeneous manifolds and symmetric spaces, revealing the machinery needed to generalize important optimization techniques to Riemannian manifolds. Exercises are included throughout, along with optional sections that delve into more theoretical topics.

Differential Geometry and Lie Groups: A Computational Perspective offers a uniquely accessible perspective on differential geometry for those interested in the theory behind modern computing applications. Equally suited to classroom use or independent study, the text will appeal to students and professionals alike; only a background in calculus and linear algebra is assumed. Readers looking to continue on to more advanced topics will appreciate the authors’ companion volume Differential Geometry andLie Groups: A Second Course.

This textbook offers an introduction to differential geometry designed for readers interested in modern geometry processing. Working from basic undergraduate prerequisites, the authors develop manifold theory and Lie groups from scratch; fundamental topics in Riemannian geometry follow, culminating in the theory that underpins manifold optimization techniques. Students and professionals working in computer vision, robotics, and machine learning will appreciate this pathway into the mathematical concepts behind many modern applications.

Starting with the matrix exponential, the text begins with an introduction to Lie groups and group actions. Manifolds, tangent spaces, and cotangent spaces follow; a chapter on the construction of manifolds from gluing data is particularly relevant to the reconstruction of surfaces from 3D meshes. Vector fields and basic point-set topology bridge into the second part of the book, which focuses on Riemannian geometry.

Chapters on Riemannian manifolds encompass Riemannian metrics, geodesics, and curvature. Topics that follow include submersions, curvature on Lie groups, and the Log-Euclidean framework. The final chapter highlights naturally reductive homogeneous manifolds and symmetric spaces, revealing the machinery needed to generalize important optimization techniques to Riemannian manifolds. Exercises are included throughout, along with optional sections that delve into more theoretical topics.

Differential Geometry and Lie Groups: A Computational Perspective offers a uniquely accessible perspective on differential geometry for those interested in the theory behind modern computing applications. Equally suited to classroom use or independent study, the text will appeal to students and professionals alike; only a background in calculus and linear algebra is assumed. Readers looking to continue on to more advanced topics will appreciate the authors’ companion volume Differential Geometry and Lie Groups: A Second Course.

Illuminates the mathematical theory behind modern geometry processingOffers a uniquely accessible entry-point that is suitable for students and professionals alikeBuilds the mathematical theory behind modern applications in medical imaging, computer vision, robotics, and machine learningIncludes exercises throughout that are suitable for class use or independent study

Jean Gallier is Professor of Computer and Information Science at the University of Pennsylvania, Philadelphia. His research interests include geometry and its applications, geometric modeling, and differential geometry. He is also a member of the University of Pennsylvania’s Department of Mathematics, and its Center for Human Modelling and Simulation.

Jocelyn Quaintance is postdoctoral researcher at the University of Pennsylvania who has contributed to the fields of combinatorial identities and power product expansions. Her recent mathematical books investigate the interplay between mathematics and computer science. Covering areas as diverse as differential geometry, linear algebra, optimization theory, and Fourier analysis, her writing illuminates the mathematics behind topics relevant to engineering, computer vision, and robotics.

9783030460396

/Differential Geometry/Geometry/Mathematics and Computing/Mathematics/

/Differential Geometry/Geometry/Mathematics and Computing/Mathematics//Topological Groups and Lie Groups/Algebra/Mathematics and Computing/Mathematics//Computational Mathematics and Numerical Analysis/Mathematics and Computing/Mathematics////

10.1007/978-3-030-46040-2

978-3-030-25442-1

1. Preliminaries via Gambles.- 2. Options.- 3. Modeling.- 4. Some Probability.- 5. The Black–Scholes Equation.- 6. Solutions of Black–Scholes.- 7. Partial information: the Greeks.- 8. Sketching and the American Options.- 9. Embellishments.

This textbook invites the reader to develop a holistic grounding in mathematical finance, where concepts and intuition play as important a role as powerful mathematical tools. Financial interactions are characterized by a vast amount of data and uncertainty; navigating the inherent dangers and hidden opportunities requires a keen understanding of what techniques to apply and when. By exploring the conceptual foundations of options pricing, the author equips readers to choose their tools with a critical eye and adapt to emerging challenges.

Introducing the basics of gambles through realistic scenarios, the text goes on to build the core financial techniques of Puts, Calls, hedging, and arbitrage. Chapters on modeling and probability lead into the centerpiece: the Black–Scholes equation. Omitting the mechanics of solving Black–Scholes itself, the presentation instead focuses on an in-depth analysis of its derivation and solutions. Advanced topics that follow include the Greeks, American options, and embellishments. Throughout, the author presents topics in an engaging conversational style. “Intuition breaks” frequently prompt students to set aside mathematical details and think critically about the relevance of tools in context.

Mathematics of Finance is ideal for undergraduates from a variety of backgrounds, including mathematics, economics, statistics, data science, and computer science. Students should have experience with the standard calculus sequence, as well as a familiarity with differential equations and probability. No financial expertise is assumed of student or instructor; in fact, the text’s deep connection to mathematical ideas makes it suitable for a math capstone course. A complete set of the author’s lecture videos is available on YouTube, providing a comprehensive supplementary resource for a course or independent study.

This textbook invites the reader to develop a holistic grounding in mathematical finance, where concepts and intuition play as important a role as powerful mathematical tools. Financial interactions are characterized by a vast amount of data and uncertainty; navigating the inherent dangers and hidden opportunities requires a keen understanding of what techniques to apply and when. By exploring the conceptual foundations of options pricing, the author equips readers to choose their tools with a critical eye and adapt to emerging challenges.

Introducing the basics of gambles through realistic scenarios, the text goes on to build the core financial techniques of Puts, Calls, hedging, and arbitrage. Chapters on modeling and probability lead into the centerpiece: the Black–Scholes equation. Omitting the mechanics of solving Black–Scholes itself, the presentation instead focuses on an in-depth analysis of its derivation and solutions. Advanced topics that follow include the Greeks, American options, and embellishments. Throughout, the author presents topics in an engaging conversational style. “Intuition breaks” frequently prompt students to set aside mathematical details and think critically about the relevance of tools in context.


Mathematics of Finance is ideal for undergraduates from a variety of backgrounds, including mathematics, economics, statistics, data science, and computer science. Students should have experience with the standard calculus sequence, as well as a familiarity with differential equations and probability. No financial expertise is assumed of student or instructor; in fact, the text’s deep connection to mathematical ideas makes it suitable for a math capstone course. A complete set of the author’s lecture videos is available on YouTube, providing a comprehensive supplementary resource for a course or independent study.

Promotes critical thinking skills to develop intuition about financial optionsHighlights the mathematical concepts fundamental to finance by offering an intuitive approachOffers instructors potentially new to the area a valuable resource for teaching a mathematical finance courseSimplifies complex mathematical concepts, such as the derivation of the Black–Scholes equation and its solutions, by emphasizing the concepts behind a formulaIncludes supplementary material: sn.pub/extras

Donald G. Saari is Emeritus Professor of Mathematics and Economics at the University of California, Irvine. His contributions to voting theory, economics, and celestial mechanics are widely celebrated, and his achievements include election to the U.S. National Academy of Sciences and the Chauvenet Prize for mathematical exposition. His books Geometry of Voting and Basic Geometry of Voting are celebrated for their seminal contributions to mathematical voting theory.

9783030254421

/Mathematics in Business, Economics and Finance/Applications of Mathematics/Mathematics and Computing/Mathematics/

/Mathematics in Business, Economics and Finance/Applications of Mathematics/Mathematics and Computing/Mathematics//Game Theory/Quantitative Economics/Economics/Humanities and Social Sciences//Economics/Humanities and Social Sciences/Financial Economics//Economics/Humanities and Social Sciences/Macroeconomics and Monetary Economics///

10.1007/978-3-030-25443-8

Răzvan Gelca, Texas Tech University, Lubbock, TX, USA; Titu Andreescu, University of Texas at Dallas, Richardson, TX, USA

Preface to the Second Edition.- Preface to the First Edition.- A Study Guide.- 1. Methods of Proof.- 2. Algebra.- 3. Real Analysis.- 4. Geometry and Trigonometry.- 5. Number Theory.- 6. Combinatorics and Probability.- Solutions.- Index of Notation.- Index.

This book takes the reader on a journey through the world of college mathematics, focusing on some of the most important concepts and results in the theories of polynomials, linear algebra, real analysis, differential equations, coordinate geometry, trigonometry, elementary number theory, combinatorics, and probability. Preliminary material provides an overview of common methods of proof: argument by contradiction, mathematical induction, pigeonhole principle, ordered sets, and invariants. Each chapter systematically presents a single subject within which problems are clustered in each section according to the specific topic. The exposition is driven by nearly 1300 problems and examples chosen from numerous sources from around the world; many original contributions come from the authors. The source, author, and historical background are cited whenever possible. Complete solutions to all problems are given at the end of the book. This second edition includes new sections on quadratic polynomials, curves in the plane, quadratic fields, combinatorics of numbers, and graph theory, and added problems or theoretical expansion of sections on polynomials, matrices, abstract algebra, limits of sequences and functions, derivatives and their applications, Stokes' theorem, analytical geometry, combinatorial geometry, and counting strategies. Using the W.L. Putnam Mathematical Competition for undergraduates as an inspiring symbol to build an appropriate math background for graduate studies in pure or applied mathematics, the reader is eased into transitioning from problem-solving at the high school level to the university and beyond, that is, to mathematical research. This work may be used as a study guide for the Putnam exam, as a text for many different problem-solving courses, and as a source of problems for standard courses in undergraduate mathematics. Putnam and Beyond is organized for independent study by undergraduate and graduate students, as well as teachers and researchers in the physical sciences who wish to expand their mathematical horizons. Reviews of the first edition: The reviewer recommends this book to all students curious about the force of mathematics, especially those who are bored at school and ready for a challenge. Teachers would find this book to be a welcome resource, as will contest organizers. —Teodora-Liliana Radulescu, Zentralblatt MATH, Vol. 1122 (24), 2007   …This extraordinary book can be read for fun. However, it can also serve as a textbook for preparation for the Putnam … for an advanced problem-solving course, or even as an overview of undergraduate mathematics. … it could certainly serve as a great review for senior-level students.    — Donald L. Vestal, MathDL, December, 2007

This book takes the reader on a journey through the world of college mathematics, focusing on some of the most important concepts and results in the theories of polynomials, linear algebra, real analysis, differential equations, coordinate geometry, trigonometry, elementary number theory, combinatorics, and probability. Preliminary material provides an overview of common methods of proof: argument by contradiction, mathematical induction, pigeonhole principle, ordered sets, and invariants. Each chapter systematically presents a single subject within which problems are clustered in each section according to the specific topic. The exposition is driven by nearly 1300 problems and examples chosen from numerous sources from around the world; many original contributions come from the authors. The source, author, and historical background are cited whenever possible. Complete solutions to all problems are given at the end of the book. This second edition includes new sections on quadratic polynomials, curves in the plane, quadratic fields, combinatorics of numbers, and graph theory, and added problems or theoretical expansion of sections on polynomials, matrices, abstract algebra, limits of sequences and functions, derivatives and their applications, Stokes' theorem, analytical geometry, combinatorial geometry, and counting strategies. Using the W.L. Putnam Mathematical Competition for undergraduates as an inspiring symbol to build an appropriate math background for graduate studies in pure or applied mathematics, the reader is eased into transitioning from problem-solving at the high school level to the university and beyond, that is, to mathematical research. This work may be used as a study guide for the Putnam exam, as a text for many different problem-solving courses, and as a source of problems for standard courses in undergraduate mathematics. Putnam and Beyond is organized for independent study by undergraduate and graduate students, as well as teachers and researchers in the physical sciences who wish to expand their mathematical horizons.

<p>Second edition includes new sections on quadratic polynomials, functions in real analysis, curves in the plane, quadratic fields, combinatorics of numbers, and graph theory</p><p>Second edition includes added problems and theoretical expansion of several sections</p><p>Undergraduate textbook with an emphasis on problem-solving; driven by over 1300 problems</p><p>Structured topically to assist undergraduates in gaining proficiency across a broad spectrum of subjects: algebra, real analysis, geometry and trigonometry, number theory, combinatorics and probabilities</p><p>Fills a gap in the market for problem-based texts that specifically target the Putnam exams and undergraduate mathematics majors</p><p>Includes supplementary material: sn.pub/extras</p>

Răzvan Gelca, Texas Tech University, works in Chern-Simons theory, a   field of mathematics that blends low dimensional topology, mathematical physics, geometry, and the theory of group representations. He is also involved in mathematics competitions such as the mathematical Olympiads and the W.L. Putnam Mathematical Competition. He is co-author of 2 published books (with Titu Andreescu), namely “Mathematical Olympiad Challenges” and the first edition of “Putnam and Beyond.” In 2015 Gelca and Andreescu will also publish a monograph on Pell’s Equations.<br/>Titu Andreescu, University of Texas-Dallas,  is highly involved with mathematics contests and olympiads. He was the Director of AMC (as appointed by the Mathematical Association of America), Director of MOP, Head Coach of the USA IMO Team and Chairman of the USAMO. He has also authored a large number of books on the topic of problem solving and olympiad-style mathematics including the first edition of “Putnam and Beyond” (with Razvan Gelca), “Mathematical Olympiad Treasures” and “Mathematical Olympiad Challenges” (with Razvan Gelca). Additional Springer publications include “Mathematical Bridges”, “Complex Numbers from A to …Z”, “Number Theory” and a new monograph on Pell’s Equations to be published in 2015.<br/>

9783319589862

/Algebra/Mathematics and Computing/Mathematics//Analysis/Mathematics and Computing/Mathematics//Geometry/Mathematics and Computing/Mathematics//Number Theory/Mathematics and Computing/Mathematics//Discrete Mathematics/Mathematics and Computing/Mathematics//

978-0-387-90328-6

,978-0-387-90062-9,978-0-387-90061-2,978-1-4615-9973-9,978-1-4615-9972-2

I. The Complex Number System.- §1. The real numbers.- §2. The field of complex numbers.- §3. The complex plane.- §4. Polar representation and roots of complex numbers.- §5. Lines and half planes in the complex plane.- §6. The extended plane and its spherical representation.- II. Metric Spaces and the Topology of ?.- §1. Definition and examples of metric spaces.- §2. Connectedness.- §3. Sequences and completeness.- §4. Compactness.- §5. Continuity.- §6. Uniform convergence.- III. Elementary Properties and Examples of Analytic Functions.- §1. Power series.- §2. Analytic functions.- §3. Analytic functions as mapping, Möbius transformations.- IV. Complex Integration.- §1. Riemann-Stieltjes integrals.- §2. Power series representation of analytic functions.- §3. Zeros of an analytic function.- §4. The index of a closed curve.- §5. Cauchy’s Theorem and Integral Formula.- §6. The homotopic version of Cauchy’s Theorem and simple connectivity.- §7. Counting zeros; the Open Mapping Theorem.- §8. Goursat’s Theorem.- V. Singularities.- §1. Classification of singularities.- §2. Residues.- §3. The Argument Principle.- VI. The Maximum Modulus Theorem.- §1. The Maximum Principle.- §2. Schwarz’s Lemma.- §3. Convex functions and Hadamard’s Three Circles Theorem.- §4. Phragm>én-Lindel>üf Theorem.- VII. Compactness and Convergence in ihe Space of Analytic Functions.- §1. The space of continuous functions C(G, ?).- §2. Spaccs of analytic functions.- §3. Spaccs of meromorphic functions.- §4. The Riemann Mapping Theorem.- §5. Weierstrass Factorization Theorem.- §6. Factorization of the sine function.- $7. The gamma function.- §8. The Riemann zeta function.- VIII. Runge’s Theorem.- §1. Runge’s Theorem.- §2. Simple connectedness.- §3.Mittag-Leffler’s Theorem.- IX. Analytic Continuation and Riemann Surfaces.- §1. Schwarz Reflection Principle.- $2. Analytic Continuation Along A Path.- §3. Monodromy Theorem.- §4. Topological Spaces and Neighborhood Systems.- $5. The Sheaf of Germs of Analytic Functions on an Open Set.- $6. Analytic Manifolds.- §7. Covering spaccs.- X. Harmonic Functions.- §1. Basic Properties of harmonic functions.- §2. Harmonic functions on a disk.- §3. Subharmonic and superharmonic functions.- §4. The Dirichlet Problem.- §5. Green’s Functions.- XI. Entire Functions.- §1. Jensen’s Formula.- §2. The genus and order of an entire function.- §3. Hadamard Factorization Theorem.- XII. The Range of an Analytic Function.- §1. Bloch’s Theorem.- §2. The Little Picard Theorem.- §3. Schottky’s Theorem.- §4. The Great Picard Theorem.- Appendix A: Calculus for Complex Valued Functions on an Interval.- Appendix B: Suggestions for Further Study and Bibliographical Notes.- References.- List of Symbols.

This book is intended as a textbook for a first course in the theory of functions of one complex variable for students who are mathematically mature enough to understand and execute E - 8 arguments. The actual pre­ requisites for reading this book are quite minimal; not much more than a stiff course in basic calculus and a few facts about partial derivatives. The topics from advanced calculus that are used (e.g., Leibniz's rule for differ­ entiating under the integral sign) are proved in detail. Complex Variables is a subject which has something for all mathematicians. In addition to having applications to other parts of analysis, it can rightly claim to be an ancestor of many areas of mathematics (e.g., homotopy theory, manifolds). This view of Complex Analysis as 'An Introduction to Mathe­ matics' has influenced the writing and selection of subject matter for this book. The other guiding principle followed is that all definitions, theorems, etc.

9780387903286

/Calculus of Variations and Optimization/Optimization/Mathematics and Computing/Mathematics/

/Calculus of Variations and Optimization/Optimization/Mathematics and Computing/Mathematics//Analysis/Mathematics and Computing/Mathematics/////

Introduction.- Basic Principles of Survival Analysis.- Nonparametric Survival Curve Estimation.- Nonparametric Comparison of Survival Distributions.- Regression Analysis Using the Proportional Hazards Model.- Model Selection and Interpretation.- Model Diagnostics.- Time Dependent Covariates.- Multiple Survival Outcomes and Competing Risks.- Parametric Models.- Sample Size Determination for Survival Studies.- Additional Topics.- References.- Appendix A.- Index.- R Package Index.

Applied Survival Analysis Using R covers the main principles of survival analysis, gives examples of how it is applied, and teaches how to put those principles to use to analyze data using R as a vehicle. Survival data, where the primary outcome is time to a specific event, arise in many areas of biomedical research, including clinical trials, epidemiological studies, and studies of animals. Many survival methods are extensions of techniques used in linear regression and categorical data, while other aspects of this field are unique to survival data. This text employs numerous actual examples to illustrate survival curve estimation, comparison of survivals of different groups, proper accounting for censoring and truncation, model variable selection, and residual analysis.Because explaining survival analysis requires more advanced mathematics than many other statistical topics, this book is organized with basic concepts and most frequently used procedures covered in earlier chapters, with more advanced topics near the end and in the appendices. A background in basic linear regression and categorical data analysis, as well as a basic knowledge of calculus and the R system, will help the reader to fully appreciate the information presented. Examples are simple and straightforward while still illustrating key points, shedding light on the application of survival analysis in a way that is useful for graduate students, researchers, and practitioners in biostatistics.Clearly illustrates concepts of survival analysis principles and analyzes actual survival data using R, in addition to including an appendix with a basic introduction to ROrganized via basic concepts and most frequently used procedures, with advanced topics toward the end of the book and in appendicesIncludes multiple original data sets that have not appeared in other textbooksDirk F. Moore is Associate Professor of Biostatisticsat the Rutgers School of Public Health and the Rutgers Cancer Institute of New Jersey. He received a Ph.D. in biostatistics from the University of Washington in Seattle and, prior to joining Rutgers, was a faculty member in the Statistics Department at Temple University. He has published numerous papers on the theory and application of survival analysis and other biostatistics methods to clinical trials and epidemiology studies.

<div>Applied Survival Analysis Using R covers the main principles of survival analysis, gives examples of how it is applied, and teaches how to put those principles to use to analyze data using R as a vehicle. Survival data, where the primary outcome is time to a specific event, arise in many areas of biomedical research, including clinical trials, epidemiological studies, and studies of animals. Many survival methods are extensions of techniques used in linear regression and categorical data, while other aspects of this field are unique to survival data. This text employs numerous actual examples to illustrate survival curve estimation, comparison of survivals of different groups, proper accounting for censoring and truncation, model variable selection, and residual analysis.</div><div>
</div><div>Because explaining survival analysis requires more advanced mathematics than many other statistical topics, this book is organized with basic concepts and most frequently used procedures covered in earlier chapters, with more advanced topics near the end and in the appendices. A background in basic linear regression and categorical data analysis, as well as a basic knowledge of calculus and the R system, will help the reader to fully appreciate the information presented. Examples are simple and straightforward while still illustrating key points, shedding light on the application of survival analysis in a way that is useful for graduate students, researchers, and practitioners in biostatistics.</div><div>
</div>

Clearly illustrates concepts of survival analysis principles and analyzes actual survival data using R, in addition to including an appendix with a basic introduction to ROrganized via basic concepts and most frequently used procedures, with advanced topics toward the end of the book and in appendicesIncludes multiple original data sets that have not appeared in other textbooksIncludes supplementary material: sn.pub/extras

Dirk F. Moore is Associate Professor of Biostatistics at the Rutgers School of Public Health and the Rutgers Cancer Institute of New Jersey. He received a Ph.D. in biostatistics from the University of Washington in Seattle and, prior to joining Rutgers, was a faculty member in the Statistics Department at Temple University. He has published numerous papers on the theory and application of survival analysis and other biostatistics methods to clinical trials and epidemiology studies.

9783319312439

/Biostatistics/Applied Statistics/Statistics/Mathematics and Computing/

/Biostatistics/Applied Statistics/Statistics/Mathematics and Computing//Statistical Theory and Methods/Statistics/Mathematics and Computing//Epidemiology/Biomedical Research/Life Sciences/Health Sciences////

978-0-387-23229-4

Fred Diamond, King's College London Dept. Mathematics, London, UK; Jerry Shurman, Reed College Dept. Mathematics, Portland, OR, USA

Modular Forms, Elliptic Curves, and Modular Curves.- Modular Curves as Riemann Surfaces.- Dimension Formulas.- Eisenstein Series.- Hecke Operators.- Jacobians and Abelian Varieties.- Modular Curves as Algebraic Curves.- The Eichler-Shimura Relation and L-functions.- Galois Representations.

<div>This book introduces the theory of modular forms with an eye toward the Modularity Theorem:</div>All rational elliptic curves arise from modular forms. <div>The topics covered include</div><div>
</div><div>• elliptic curves as complex tori and as algebraic curves,</div><div>
</div><div>• modular curves as Riemann surfaces and as algebraic curves,</div><div>
</div><div>• Hecke operators and Atkin–Lehner theory,</div><div>
</div><div>• Hecke eigenforms and their arithmetic properties,</div><div>
</div><div>• the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms,</div><div>
</div><div>• elliptic and modular curves modulo p and the Eichler–Shimura Relation,</div><div>
</div>• the Galois representations associated to elliptic curves and to Hecke eigenforms.<div>
</div><div>As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory.</div><div>A First Course in Modular Forms is written for beginning graduate students and advanced undergraduates. It does not require background in algebraic number theory or algebraic geometry, and it contains exercises throughout.</div><div>Fred Diamond received his Ph.D from Princeton University in 1988 under the direction of Andrew Wiles and now teaches at King's College London. Jerry Shurman received his Ph.D from Princeton University in 1988 under the direction of Goro Shimura and now teaches at Reed College.</div><div> </div>

<div>This book introduces the theory of modular forms with an eye toward the Modularity Theorem:</div>All rational elliptic curves arise from modular forms. <div>The topics covered include</div><div>
</div><div>• elliptic curves as complex tori and as algebraic curves,</div><div>
</div><div>• modular curves as Riemann surfaces and as algebraic curves, </div><div>
</div><div>• Hecke operators and Atkin–Lehner theory, </div><div>
</div><div>• Hecke eigenforms and their arithmetic properties, </div><div>
</div><div>• the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms, </div><div>
</div><div>• elliptic and modular curves modulo p and the Eichler–Shimura Relation, </div><div>
</div><div>• the Galois representations associated to elliptic curves and to Hecke eigenforms.</div><div>
</div><div>
</div><div>As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory.</div><div>A First Course in Modular Forms is written for beginning graduate students and advanced undergraduates. It does not require background in algebraic number theory or algebraic geometry, and it contains exercises throughout.</div><div>Fred Diamond received his Ph.D from Princeton University in 1988 under the direction of Andrew Wiles and now teaches at King's College London. Jerry Shurman received his Ph.D from Princeton University in 1988 under the direction of Goro Shimura and now teaches at Reed College.</div><div> </div>

Covers many topics not covered in other texts on elliptic curvesIncludes supplementary material: sn.pub/extras

9780387232294

/Number Theory/Mathematics and Computing/Mathematics//Algebraic Geometry/Algebra/Mathematics and Computing/Mathematics/////

10.1007/978-0-387-27226-9

978-0-387-98931-0

Carl M. Bender, Washington University Dept. Physics, St. Louis, MO, USA; Steven A. Orszag, Yale University Dept. Mathematics, New Haven, CT, USA

I Fundamentals.- 1 Ordinary Differential Equations.- 2 Difference Equations.- II Local Analysis.- 3 Approximate Solution of Linear Differential Equations.- 4 Approximate Solution of Nonlinear Differential Equations.- 5 Approximate Solution of Difference Equations.- 6 Asymptotic Expansion of Integrals.- III Perturbation Methods.- 7 Perturbation Series.- 8 Summation of Series.- IV Global Analysis.- 9 Boundary Layer Theory.- 10 WKB Theory.- 11 Multiple-Scale Analysis.

The triumphant vindication of bold theories-are these not the pride and justification of our life's work? -Sherlock Holmes, The Valley of Fear Sir Arthur Conan Doyle The main purpose of our book is to present and explain mathematical methods for obtaining approximate analytical solutions to differential and difference equations that cannot be solved exactly. Our objective is to help young and also establiShed scientists and engineers to build the skills necessary to analyze equations that they encounter in their work. Our presentation is aimed at developing the insights and techniques that are most useful for attacking new problems. We do not emphasize special methods and tricks which work only for the classical transcendental functions; we do not dwell on equations whose exact solutions are known. The mathematical methods discussed in this book are known collectively as­ asymptotic and perturbative analysis. These are the most useful and powerful methods for finding approximate solutions to equations, but they are difficult to justify rigorously. Thus, we concentrate on the most fruitful aspect of applied analysis; namely, obtaining the answer. We stress care but not rigor. To explain our approach, we compare our goals with those of a freshman calculus course. A beginning calculus course is considered successful if the students have learned how to solve problems using calculus.

9780387989310

/Analysis/Mathematics and Computing/Mathematics//Mathematical and Computational Engineering Applications/Technology and Engineering//Mathematical Methods in Physics/Theoretical, Mathematical and Computational Physics/Physics and Astronomy/Physical Sciences//Theoretical, Mathematical and Computational Physics/Physics and Astronomy/Physical Sciences//

10.1007/978-1-4757-3069-2

978-0-387-95584-1

László Lovász, Microsoft Research, Redmond, WA, USA; József Pelikán, Eotvos Lorand University Dept. of Algebra and Number Theory, Budapest, Hungary; Katalin Vesztergombi, Seattle, WA, USA

Let’s Count!.- Combinatorial Tools.- Binomial Coefficients and Pascal’s Triangle.- Fibonacci Numbers.- Combinatorial Probability.- Integers, Divisors, and Primes.- Graphs.- Trees.- Finding the Optimum.- Matchings in Graphs.- Combinatorics in Geometry.- Euler’s Formula.- Coloring Maps and Graphs.- Finite Geometries, Codes, Latin Squares, and Other Pretty Creatures.- A Glimpse of Complexity and Cryptography.- Answers to Exercises.

Discrete mathematics is quickly becoming one of the most important areas of mathematical research, with applications to cryptography, linear programming, coding theory and the theory of computing. This book is aimed at undergraduate mathematics and computer science students interested in developing a feeling for what mathematics is all about, where mathematics can be helpful, and what kinds of questions mathematicians work on. The authors discuss a number of selected results and methods of discrete mathematics, mostly from the areas of combinatorics and graph theory, with a little number theory, probability, and combinatorial geometry. Wherever possible, the authors use proofs and problem solving to help students understand the solutions to problems. In addition, there are numerous examples, figures and exercises spread throughout the book.
László Lovász is a Senior Researcher in the Theory Group at Microsoft Corporation. He is a recipient of the 1999 Wolf Prize and theGödel Prize for the top paper in Computer Science. József Pelikán is Professor of Mathematics in the Department of Algebra and Number Theory at Eötvös Loránd University, Hungary. In 2002, he was elected Chairman of the Advisory Board of the International Mathematical Olympiad. Katalin Vesztergombi is Senior Lecturer in the Department of Mathematics at the University of Washington.

Discrete mathematics is quickly becoming one of the most important areas of mathematical research, with applications to cryptography, linear programming, coding theory and the theory of computing. This book is aimed at undergraduate mathematics and computer science students interested in developing a feeling for what mathematics is all about, where mathematics can be helpful, and what kinds of questions mathematicians work on. The authors discuss a number of selected results and methods of discrete mathematics, mostly from the areas of combinatorics and graph theory, with a little number theory, probability, and combinatorial geometry. Wherever possible, the authors use proofs and problem solving to help students understand the solutions to problems. In addition, there are numerous examples, figures and exercises spread throughout the book. Laszlo Lovasz is a Senior Researcher in the Theory Group at Microsoft Corporation. He is a recipient of the 1999 Wolf Prize andthe Godel Prize for the top paper in Computer Science. Jozsef Pelikan is Professor of Mathematics in the Department of Algebra and Number Theory at Eotvos Lorand University, Hungary. In 2002, he was elected Chairman of the Advisory Board of the International Mathematical Olympiad. Katalin Vesztergombi is Senior Lecturer in the Department of Mathematics at the University of Washington.

László Lovász is a Senior Researcher in the Theory Group at Microsoft Corporation. He is a recipient of the 1999 Wolf Prize and the Gödel Prize for the top paper in Computer Science. József Pelikán is Professor of Mathematics in the Department of Algebra and Number Theory at Eötvös Loránd University, Hungary. In 2002, he was elected Chairman of the Advisory Board of the International Mathematical Olympiad. Katalin Vesztergombi is Senior Lecturer in the Department of Mathematics at the University of Washington.

9780387955841

/Discrete Mathematics/Mathematics and Computing/Mathematics/

/Discrete Mathematics/Mathematics and Computing/Mathematics//Number Theory/Mathematics and Computing/Mathematics/////

978-3-642-20211-7

Michel Ledoux, Université Toulouse III Lab. Statist. et Probabilités, Toulouse CX, France; Michel Talagrand, Université Paris VI Dépt. Mathématiques, Paris CX 05, France

Notation.- 0. Isoperimetric Background and Generalities.- 1. Isoperimetric Inequalities and the Concentration of Measure Phenomenon.- 2. Generalities on Banach Space Valued Random Variables and Random Processes.- I. Banach Space Valued Random Variables and Their Strong Limiting Properties.- 3. Gaussian Random Variables.- 4. Rademacher Averages.- 5. Stable Random Variables.- 6 Sums of Independent Random Variables.- 7. The Strong Law of Large Numbers.- 8. The Law of the Iterated Logarithm.- II. Tightness of Vector Valued Random Variables and Regularity of Random Processes.- 9. Type and Cotype of Banach Spaces.- 10. The Central Limit Theorem.- 11. Regularity of Random Processes.- 12. Regularity of Gaussian and Stable Processes.- 13. Stationary Processes and Random Fourier Series.- 14. Empirical Process Methods in Probability in Banach Spaces.- 15. Applications to Banach Space Theory.- References.

Isoperimetric, measure concentration and random process techniques appear at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (integrability and limit theorems for vector valued random variables, boundedness and continuity of random processes) and of some of their links to Geometry of Banach spaces (via the type and cotype properties). Its purpose is to present some of the main aspects of this theory, from the foundations to the most important achievements. The main features of the investigation are the systematic use of isoperimetry and concentration of measure and abstract random process techniques (entropy and majorizing measures). Examples of these probabilistic tools and ideas to classical Banach space theory are further developed.

Isoperimetric, measure concentration and random process techniques appear at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (integrability and limit theorems for vector valued random variables, boundedness and continuity of random processes) and of some of their links to Geometry of Banach spaces (via the type and cotype properties). Its purpose is to present some of the main aspects of this theory, from the foundations to the most important achievements. The main features of the investigation are the systematic use of isoperimetry and concentration of measure and abstract random process techniques (entropy and majorizing measures). Examples of these probabilistic tools and ideas to classical Banach space theory are further developed.

<p>A very comprehensive book which develops a wide variety of the methods existing in this field</p><p>An event for mathematicians working or interested in probability in Banach spaces</p><p>a presentation of the main aspects of the theory of probability in Banach spaces</p>

Michel Ledoux held first a research position with CNRS, and since 1991 is Professor at the University of Toulouse. He is moreover, since 2010, a senior member of the Institut Universitaire de France, having been also a junior member from 1997 to 2002. He has held associate editor appointments for various journals, including the Annals of Probability and Probability Theory and Related Fields (current). His research interests centre on probability, random matrices, logarithmic Sobolev inequalities, probability in Banach spaces.Michel Talagrand has held a research position with the CNRS since 1974. His thesis was directed by Gustave Choquet and his interests revolve around the theory of stochastic processes and probability in Banach spaces, as well as the mathematical theory of spin glasses.  He was invited to deliver a lecture at the International Congress of Mathematicians in 1990, and to deliver a plenary lecture at the same congress in 1998. He received the Loeve Prize (1995) and the Fermat Prize (1997) for his work in probability theory. He was elected to the Paris Academy of Sciences in 2004.

9783642202117

/Mathematics and Computing/Probability Theory/Mathematics//Real Functions/Analysis/Mathematics and Computing/Mathematics//Systems Theory, Control /Optimization/Mathematics and Computing/Mathematics//Calculus of Variations and Optimization/Optimization/Mathematics and Computing/Mathematics///

10.1007/978-3-642-20212-4

978-3-662-48790-7

V. A. Zorich, Moscow State University Dept. Mathematics & Mechanics, Moscow, Russia

Original Russian edition (6th edition) published by MCCME, Moscow, Russia, 2012

,978-3-642-00249-6,978-3-540-87451-5,978-3-540-40386-9,978-3-540-87452-2

1 Some General Mathematical Concepts and Notation.- 2 The Real Numbers.- 3 Limits.- 4 Continuous Functions.- 5 Differential Calculus.- 6 Integration.- 7 Functions of Several Variables.- 8 Differential Calculus in Several Variables.- Some Problems from the Midterm Examinations.- Examination Topics.- Appendices.- References.- Subject Index.- Name Index.

VLADIMIR A. ZORICH is professor of mathematics at Moscow State University. His areas of specialization are analysis, conformal geometry, quasiconformal mappings, and mathematical aspects of thermodynamics. He solved the problem of global homeomorphism for space quasiconformal mappings. He holds a patent in the technology of mechanical engineering, and he is also known by his book Mathematical Analysis of Problems in the Natural Sciences .
This second English edition of a very popular two-volume work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds; asymptotic methods; Fourier, Laplace, and Legendre transforms; elliptic functions; and distributions. Especially notable in this course are the clearly expressed orientation toward the natural sciences and the informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of instructive exercises, problems, and fresh applications to areas seldom touched on in textbooks on real analysis. The main difference between the second and first English editions is the addition of a series of appendices to each volume. There are six of them in the first volume and five in the second. The subjects of these appendices are diverse. They are meant to be useful to both students (in mathematics and physics) and teachers, who may be motivated by different goals. Some of the appendices are surveys, both prospective and retrospective. The final survey establishes important conceptual connections between analysis and other parts of mathematics. The first volume constitutes a complete course in one-variable calculus along with the multivariable differential calculus elucidated in an up-to-date, clear manner, with a pleasant geometric and natural sciences flavor.
“...Complete logical rigor of discussion...is combined with simplicity and completeness as well as with the development of the habit to work with real problems from natural sciences. ” From a review by A.N. Kolmogorov of the first Russian edition of this course“...We see here not only a mathematical pattern, but also the way it works in the solution of nontrivial questions outside mathematics. ...The course is unusually rich in ideas and shows clearly the power of the ideas and methods of modern mathematics in the study of particular problems....In my opinion, this course is the best of the existing modern courses of analysis.” From a review by V.I.Arnold

This second edition of a very popular two-volume work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds; asymptotic methods; Fourier, Laplace, and Legendre transforms; elliptic functions; and distributions. Especially notable in this course are the clearly expressed orientation toward the natural sciences and the informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of instructive exercises, problems, and fresh applications to areas seldom touched on in textbooks on real analysis. The main difference between the second and first editions is the addition of a series of appendices to each volume. There are six of them in the first volume and five in the second. The subjects of these appendices are diverse. They are meant to be useful to both students (in mathematics and physics) and teachers, who may be motivated by different goals. Some of the appendices are surveys, both prospective and retrospective. The final survey establishes important conceptual connections between analysis and other parts of mathematics. The first volume constitutes a complete course in one-variable calculus along with the multivariable differential calculus elucidated in an up-to-date, clear manner, with a pleasant geometric and natural sciences flavor.

Thoroughness of coverage, from elementary to very advancedClarity of expositionOriginality and variety of exercises and examplesComplete logical rigor of discussionVarious new appendicesUseful not only to mathematicians, but also to physicists and engineersIncludes supplementary material: sn.pub/extras

VLADIMIR A. ZORICH is professor of mathematics at Moscow State University. His areas of specialization are analysis, conformal geometry, quasiconformal mappings, and mathematical aspects of thermodynamics. He solved the problem of global homeomorphism for space quasiconformal mappings. He holds a patent in the technology of mechanical engineering, and he is also known by his book “Mathematical Analysis of Problems in the Natural Sciences”.

9783662487907

/Analysis/Mathematics and Computing/Mathematics//Theoretical, Mathematical and Computational Physics/Physics and Astronomy/Physical Sciences/////

10.1007/978-3-662-48792-1

978-1-0716-0297-3

Israel M. Gelfand, Deceased, New Brunswick, NJ; Tatiana Alekseyevskaya (Gelfand), Rutgers, The State University of New Jersey, Highland Park, NJ, USA

Points and Lines: A Look at Projective Geometry.- Parallel Lines: A Look at Affine Geometry.- Area: A Look at Symplectic Geometry.- Circles: A Look at Euclidean Geometry.

<div><div>This text is the fifth and final in the series of educational books written by Israel Gelfand with his colleagues for high school students. These books cover the basics of mathematics in a clear and simple format – the style Gelfand was known for internationally. Gelfand prepared these materials so as to be suitable for independent studies, thus allowing students to learn and practice the material at their own pace without a class.</div><div>
</div><div>Geometry takes a different approach to presenting basic geometry for high-school students and others new to the subject. Rather than following the traditional axiomatic method that emphasizes formulae and logical deduction, it focuses on geometric constructions. Illustrations and problems are abundant throughout, and readers are encouraged to draw figures and “move” them in the plane, allowing them to develop and enhance their geometrical vision, imagination, and creativity. Chapters are structured so that only certain operations and the instruments to perform these operations are available for drawing objects and figures on the plane. This structure corresponds to presenting, sequentially, projective, affine, symplectic, and Euclidean geometries, all the while ensuring students have the necessary tools to follow along. </div><div>
</div><div>Geometry is suitable for a large audience, which includes not only high school geometry students, but also teachers and anyone else interested in improving their geometrical vision and intuition, skills useful in many professions. Similarly, experienced mathematicians can appreciate the book’s unique way of presenting plane geometry in a simple form while adhering to its depth and rigor. </div><div>
</div><div>“Gelfand was a great mathematician and also a great teacher. The book provides an atypical view of geometry. Gelfand gets to the intuitive core of geometry, to the phenomena of shapes and how they move in the plane, leading us to a better understanding of what coordinate geometry and axiomatic geometry seek to describe.” </div><div>
</div><div>Mark Saul, PhD, Executive Director, Julia Robinson Mathematics Festival</div><div>
</div><div>“The subject matter is presented as intuitive, interesting and fun. No previous knowledge of the subject is required. Starting from the simplest concepts and by inculcating in the reader the use of visualization skills, [and] after reading the explanations and working through the examples, you will be able to confidently tackle the interesting problems posed. I highly recommend the book to any person interested in this fascinating branch of mathematics.” </div><div>
</div><div>Ricardo Gorrin, a student of the Extended Gelfand Correspondence Program in Mathematics (EGCPM)</div><div>
</div></div>

<div>This text is the fifth and final in the series of educational books written by Israel Gelfand with his colleagues for high school students. These books cover the basics of mathematics in a clear and simple format – the style Gelfand was known for internationally. Gelfand prepared these materials so as to be suitable for independent studies, thus allowing students to learn and practice the material at their own pace without a class.</div><div>
</div><div>Geometry takes a different approach to presenting basic geometry for high-school students and others new to the subject. Rather than following the traditional axiomatic method that emphasizes formulae and logical deduction, it focuses on geometric constructions. Illustrations and problems are abundant throughout, and readers are encouraged to draw figures and “move” them in the plane, allowing them to develop and enhance their geometrical vision, imagination, and creativity. Chapters are structured so that only certain operations and the instruments to perform these operations are available for drawing objects and figures on the plane. This structure corresponds to presenting, sequentially, projective, affine, symplectic, and Euclidean geometries, all the while ensuring students have the necessary tools to follow along. </div><div>
</div><div>Geometry is suitable for a large audience, which includes not only high school geometry students, but also teachers and anyone else interested in improving their geometrical vision and intuition, skills useful in many professions. Similarly, experienced mathematicians can appreciate the book’s unique way of presenting plane geometry in a simple form while adhering to its depth and rigor. </div><div>
</div><div>“Gelfand was a great mathematician and also a great teacher. The book provides an atypical view of geometry. Gelfand gets to the intuitive core of geometry, to the phenomena of shapes and how they move in the plane, leading us to a better understanding of what coordinate geometry and axiomatic geometry seek to describe.” </div><div>
</div><div>- Mark Saul, PhD, Executive Director, Julia Robinson Mathematics Festival</div><div>
</div><div>“The subject matter is presented as intuitive, interesting and fun. No previous knowledge of the subject is required. Starting from the simplest concepts and by inculcating in the reader the use of visualization skills, [and] after reading the explanations and working through the examples, you will be able to confidently tackle the interesting problems posed. I highly recommend the book to any person interested in this fascinating branch of mathematics.” </div><div>
</div><div>- Ricardo Gorrin, a student of the Extended Gelfand Correspondence Program in Mathematics (EGCPM)</div><div>
</div>

<p>The final book in Israel Gelfand's internationally renowned correspondence course</p><p>Focuses on geometric constructions and includes over 400 figures in order to develop students' geometrical intuition</p><p>Contains a large number of exercises with answers, as well as problems suggested for further study</p>

<div>Israel Gelfand (1913-2009) is often considered one of the greatest mathematicians of the Twentieth Century. He published dozens of books and over 400 articles in a variety of mathematical fields, including group theory, representation theory, and functional analysis. Gelfand was known internationally as an outstanding and passionate teacher, as well as for his famous seminars in mathematics and biology, which were attended by the most prominent specialists in the field. He had a remarkable ability to adapt his presentation of difficult concepts so they would be easily understood by his audience, whether that was children or experienced professors. </div><div>
</div><div>In 1964, he created the Correspondence School in Mathematics (ZMSH) in Moscow, and later on, the Gelfand Correspondence Program in Mathematics (GCPM) at Rutgers University, both of which made mathematics available to a broad range of students. His goal was to pass on to students his belief that mathematics is simple, beautiful, and a part of human culture which anyone can learn and enjoy, just like literature, poetry, art, and music.</div><div>
</div><div>Tatiana Alekseyevskaya (Gelfand) graduated from the Department of Cybernetics and Applied Mathematics at Kiev State University in Ukraine. She then received her PhD in Mathematics in Moscow for her research on systems of quasi-linear equations and related geometrical constructions describing isotachophoresis, a process used in biological studies of protein molecules. She has extensive experience teaching mathematics to undergraduate students in both Russia and the United States, and worked closely with Israel Gelfand at Rutgers University, preparing assignments to be used in the GCPM.</div>

9781071602973

/Geometry/Mathematics and Computing/Mathematics//Mathematical Logic and Foundations/Mathematics and Computing/Mathematics/////

10.1007/978-1-0716-0299-7

Michael Barot, Universidad Nacional Autónoma de México, Mexico City, Mexico

Matrix Problems.- Representations of Quivers.- Algebras.- Module Categories.- Elements of Homological Algebra.- The Auslander-Reiten Theory.- Knitting.- Combinatorial Invariants.- Indecomposables and Dimensions.

This book gives a general introduction to the theory of representations of algebras. It starts with examples of classification problems of matrices under linear transformations and explains the three common setups: representation of quivers, modules over algebras and additive functors over certain categories. The main part is devoted to (i) module categories, presenting the unicity of the decomposition into indecomposable modules, the Auslander–Reiten theory and the technique of knitting; (ii) the use of combinatorial tools such as dimension vectors and integral quadratic forms; and (iii) deeper theorems such as Gabriel‘s Theorem, the trichotomy and the Theorem of Kac – all accompanied by further examples.
Each section includes exercises to facilitate understanding. By keeping the proofs as basic and comprehensible as possible and introducing the three languages at the beginning, this book is suitable for readers from the advanced undergraduate level onwards and enables them to consult related, specific research articles.

This book gives a general introduction to the theory of representations of algebras. It starts with examples of classification problems of matrices under linear transformations, explaining the three common setups: representation of quivers, modules over algebras and additive functors over certain categories. The main part is devoted to (i) module categories, presenting the unicity of the decomposition into indecomposable modules, the Auslander–Reiten theory and the technique of knitting; (ii) the use of combinatorial tools such as dimension vectors and integral quadratic forms; and (iii) deeper theorems such as Gabriel‘s Theorem, the trichotomy and the Theorem of Kac – all accompanied by further examples.
Each section includes exercises to facilitate understanding. By keeping the proofs as basic and comprehensible as possible and introducing the three languages at the beginning, this book is suitable for readers from the advanced undergraduate level onwards and enables them to consult related, specific research articles.

<p>A down-to-earth approach to the subject</p><p>Introduces all established descriptions within the field</p><p>Provides detailed and comprehensible proofs for all statements</p><p>Contains numerous exercises within the chapters</p><p>Includes supplementary material: sn.pub/extras</p>

Michael Barot was a researcher at the Instituto de Matemáticas of the Universidad Nacional Autónoma de México.

9783319114743

/Associative Rings and Algebras/Algebra/Mathematics and Computing/Mathematics/

/Associative Rings and Algebras/Algebra/Mathematics and Computing/Mathematics//Category Theory, Homological Algebra/Algebra/Mathematics and Computing/Mathematics//General Algebraic Systems/Algebra/Mathematics and Computing/Mathematics///

978-0-387-96201-6

Serge Lang, Yale University Dept. Mathematics, New Haven, CT, USA

Originally published by Addison-Wesley Publishing Company, 1978, 1973, 1968, 1964

One Review of Basic Material.- I Numbers and Functions.- II Graphs and Curves.- Two Differentiation and Elementary Functions.- III The Derivative.- IV Sine and Cosine.- V The Mean Value Theorem.- VI Sketching Curves.- VII Inverse Functions.- VIII Exponents and Logarithms.- Three Integration.- IX Integration.- X Properties of the Integral.- XI Techniques of Integration.- XII Applications of Integration.- Four Taylor’s Formula and Series.- XIII Taylor's Formula.- XIV Series.- Five Functions of Several Variables.- XV Vectors.- XVI Differentiation of Vectors.- XVII Functions of Several Variables.- XVIII The Chain Rule and the Gradient.- Answer.

The purpose of a first course in calculus is to teach the student the basic notions of derivative and integral, and the basic techniques and applica­ tions which accompany them. The very talented students, with an ob­ vious aptitude for mathematics, will rapidly require a course in functions of one real variable, more or less as it is understood by professional is not primarily addressed to them (although mathematicians. This book I hope they will be able to acquire from it a good introduction at an early age). I have not written this course in the style I would use for an advanced monograph, on sophisticated topics. One writes an advanced monograph for oneself, because one wants to give permanent form to one's vision of some beautiful part of mathematics, not otherwise ac­ cessible, somewhat in the manner of a composer setting down his sym­ phony in musical notation. This book is written for the students to give them an immediate, and pleasant, access to the subject. I hope that I have struck a proper com­ promise, between dwelling too much on special details and not giving enough technical exercises, necessary to acquire the desired familiarity with the subject. In any case, certain routine habits of sophisticated mathematicians are unsuitable for a first course. Rigor. This does not mean that so-called rigor has to be abandoned.

9780387962016

/Real Functions/Analysis/Mathematics and Computing/Mathematics/

/Real Functions/Analysis/Mathematics and Computing/Mathematics/////

978-3-030-94945-7

- 1. Metric Spaces. - 2. Basic Theory of Metric Spaces. - 3. Completeness of the Classical Spaces. - 4. Compact Spaces. - 5. Separable Spaces. - 6. Properties of Complete Spaces. - 7. Connected Spaces. - Afterword.

This textbook presents the theory of Metric Spaces necessary for studying analysis beyond one real variable. Rich in examples, exercises and motivation, it provides a careful and clear exposition at a pace appropriate to the material.<div>
</div><div>The book covers the main topics of metric space theory that the student of analysis is likely to need. Starting with an overview defining the principal examples of metric spaces in analysis (chapter 1), it turns to the basic theory (chapter 2) covering open and closed sets, convergence, completeness and continuity (including a treatment of continuous linear mappings). There is also a brief dive into general topology, showing how metric spaces fit into a wider theory. The following chapter is devoted to proving the completeness of the classical spaces. The text then embarks on a study of spaces with important special properties. Compact spaces, separable spaces, complete spaces and connected spaces each have a chapter devoted to them. A particular feature of the book is the occasional excursion into analysis. Examples include the Mazur–Ulam theorem, Picard’s theorem on existence of solutions to ordinary differential equations, and space filling curves.</div><div>
</div><div>This text will be useful to all undergraduate students of mathematics, especially those who require metric space concepts for topics such as multivariate analysis, differential equations, complex analysis, functional analysis, and topology. It includes a large number of exercises, varying from routine to challenging. The prerequisites are a first course in real analysis of one real variable, an acquaintance with set theory, and some experience with rigorous proofs.</div>

This textbook presents the theory of Metric Spaces necessary for studying analysis beyond one real variable. Rich in examples, exercises and motivation, it provides a careful and clear exposition at a pace appropriate to the material.<div>
</div><div>The book covers the main topics of metric space theory that the student of analysis is likely to need. Starting with an overview defining the principal examples of metric spaces in analysis (chapter 1), it turns to the basic theory (chapter 2) covering open and closed sets, convergence, completeness and continuity (including a treatment of continuous linear mappings). There is also a brief dive into general topology, showing how metric spaces fit into a wider theory. The following chapter is devoted to proving the completeness of the classical spaces. The text then embarks on a study of spaces with important special properties. Compact spaces, separable spaces, complete spaces and connected spaces each have a chapter devoted to them. A particular feature of the book is the occasional excursion into analysis. Examples include the Mazur–Ulam theorem, Picard’s theorem on existence of solutions to ordinary differential equations, and space filling curves.</div><div>
</div><div>This text will be useful to all undergraduate students of mathematics, especially those who require metric space concepts for topics such as multivariate analysis, differential equations, complex analysis, functional analysis, and topology. It includes a large number of exercises, varying from routine to challenging. The prerequisites are a first course in real analysis of one real variable, an acquaintance with set theory, and some experience with rigorous proofs. </div>

Provides a lucid and clear exposition which includes additional motivation and explanation for delicate pointsPresents metric spaces as a tool for advanced analysis, topology and related subjectsIncludes many exercises with hints

Robert Magnus was born in the UK and studied mathematics at the Universities of Cambridge and Sussex. He has taught nearly all subjects associated with analysis and has published papers in the areas of bifurcation theory, catastrophe theory, analytic operator functions and nonlinear partial differential equations. Since 1977 he has lived and worked in Iceland.

9783030949457

/Analysis/Mathematics and Computing/Mathematics//Topology/Mathematics and Computing/Mathematics//Geometry/Mathematics and Computing/Mathematics////

10.1007/978-3-030-94946-4

978-3-319-77648-4

Part I Preliminaries.- 1 Relations and Functions.- 2 The Integers and Modular Arithmetic.- Part II Groups.- 3 Introduction to Groups.- 4 Factor Groups and Homomorphisms.- 5 Direct Products and the Classification of Finite Abelian Groups.- 6 Symmetric and Alternating Groups.- 7 The Sylow Theorems.- Part III Rings.- 8 Introduction to Rings.- 9 Ideals, Factor Rings and Homomorphisms.- 10 Special Types of Domains.- Part IV Fields and Polynomials.- 11 Irreducible Polynomials.- 12 Vector Spaces and Field Extensions.- Part V Applications.- 13 Public Key Cryptography.- 14 Straightedge and Compass Constructions.- A The Complex Numbers.- B Matrix Algebra.- Solutions.- Index.

This carefully written textbook offers a thorough introduction to abstract algebra, covering the fundamentals of groups, rings and fields. The first two chapters present preliminary topics such as properties of the integers and equivalence relations. The author then explores the first major algebraic structure, the group, progressing as far as the Sylow theorems and the classification of finite abelian groups. An introduction to ring theory follows, leading to a discussion of fields and polynomials that includes sections on splitting fields and the construction of finite fields. The final part contains applications to public key cryptography as well as classical straightedge and compass constructions.Explaining key topics at a gentle pace, this book is aimed at undergraduate students. It assumes no prior knowledge of the subject and contains over 500 exercises, half of which have detailed solutions provided.

This carefully written textbook offers a thorough introduction to abstract algebra, covering the fundamentals of groups, rings and fields.

The first two chapters present preliminary topics such as properties of the integers and equivalence relations. The author then explores the first major algebraic structure, the group, progressing as far as the Sylow theorems and the classification of finite abelian groups. An introduction to ring theory follows, leading to a discussion of fields and polynomials that includes sections on splitting fields and the construction of finite fields. The final part contains applications to public key cryptography as well as classical straightedge and compass constructions.

Explaining key topics at a gentle pace, this book is aimed at undergraduate students. It assumes no prior knowledge of the subject and contains over 500 exercises, half of which have detailed solutions provided.

Provides a gentle, yet thorough, introduction to abstract algebraIncludes careful proofs of theorems and numerous worked examplesWritten in an informal, readable style

Gregory T. Lee is a professor at Lakehead University specializing in group rings, a branch of abstract algebra. He has published numerous papers on the subject, as well as a monograph with Springer.

9783319776484

/Group Theory and Generalizations/Algebra/Mathematics and Computing/Mathematics/

/Group Theory and Generalizations/Algebra/Mathematics and Computing/Mathematics//Associative Rings and Algebras/Algebra/Mathematics and Computing/Mathematics//Field Theory and Polynomials/Algebra/Mathematics and Computing/Mathematics////

978-3-540-41160-4

Originally published as volume 224 in the series: Grundlehren der mathematischen Wissenschaften

,978-3-642-96381-0,978-3-642-96380-3,978-3-540-08007-7,978-3-642-96379-7

1. Introduction.- I. Linear Equations.- 2. Laplace’s Equation.- 3. The Classical Maximum Principle.- 4. Poisson’s Equation and the Newtonian Potential.- 5. Banach and Hubert Spaces.- 6. Classical Solutions; the Schauder Approach.- 7. Sobolev Spaces.- 8. Generalized Solutions and Regularity.- 9. Strong Solutions.- II. Quasilinear Equations.- 10. Maximum and Comparison Principles.- 11. Topological Fixed Point Theorems and Their Application.- 12. Equations in Two Variables.- 13. Hölder Estimates for the Gradient.- 14. Boundary Gradient Estimates.- 15. Global and Interior Gradient Bounds.- 16. Equations of Mean Curvature Type.- 17. Fully Nonlinear Equations.- Epilogue.- Notation Index.

From the reviews:
'This is a book of interest to any having to work with differential equations, either as a reference or as a book to learn from. The authors have taken trouble to make the treatment self-contained. It (is) suitable required reading for a PhD student. Although the material has been developed from lectures at Stanford, it has developed into an almost systematic coverage that is much longer than could be covered in a year's lectures'. Newsletter, New Zealand Mathematical Society, 1985
' ... as should be clear from the previous discussion, this book is a bibliographical monument to the theory of both theoretical and applied PDEs that has not acquired any flaws due to its age. On the contrary, it remains a crucial and essential tool for the active research in the field. In a few words, in my modest opinion, “. . . this book contains the essential background that a researcher in elliptic PDEs should possess the day s/he gets a permanent academic position. . . .” SIAM Newsletter

Biography of David Gilbarg
David Gilbarg was born in New York in 1918, and was educated there through udergraduate school. He received his Ph.D. degree at Indiana University in 1941. His work in fluid dynamics during the war years motivated much of his later research on flows with free boundaries. He was on the Mathematics faculty at Indiana University from 1946 to 1957 and at Stanford University from 1957 on. His principal interests and contributions have been in mathematical fluid dynamics and the theory of elliptic partial differential equations.
Biography of Neil S. Trudinger
Neil S. Trudinger was born in Ballarat, Australia in 1942. After schooling and undergraduate education in Australia, he completed his PhD at Stanford University, USA in 1966. He has been a Professor of Mathematics at the Australian National University, Canberra since 1973. His research contributions, while largely focussed on non-linear elliptic partial differential equations, have also spread into geometry, functional analysis and computational mathematics. Among honours received are Fellowships of the Australian Academy of Science and of the Royal Society of London.

9783540411604

/Differential Equations/Analysis/Mathematics and Computing/Mathematics/

/Differential Equations/Analysis/Mathematics and Computing/Mathematics//////

10.1007/978-3-642-61798-0

Stanley J. Miklavcic, University of South Australia (Mawson Lakes Campus), Adelaide, SA, Australia

1. Preliminary Ideas.- 2. Introduction to Differentiation.- 3. Applications of the Differential Calculus.- 4. Introduction to Integration.- 5. Vector Calculus.- Glossary of Symbols.- Bibliography.- Index.

This textbook focuses on one of the most valuable skills in multivariable and vector calculus: visualization. With over one hundred carefully drawn color images, students who have long struggled picturing, for example, level sets or vector fields will find these abstract concepts rendered with clarity and ingenuity. This illustrative approach to the material covered in standard multivariable and vector calculus textbooks will serve as a much-needed and highly useful companion.<div>
</div><div>Emphasizing portability, this book is an ideal complement to other references in the area. It begins by exploring preliminary ideas such as vector algebra, sets, and coordinate systems, before moving into the core areas of multivariable differentiation and integration, and vector calculus. Sections on the chain rule for second derivatives, implicit functions, PDEs, and the method of least squares offer additional depth; ample illustrations are woven throughout. Mastery Checks engage students in material on the spot, while longer exercise sets at the end of each chapter reinforce techniques.
An Illustrative Guide to Multivariable and Vector Calculus will appeal to multivariable and vector calculus students and instructors around the world who seek an accessible, visual approach to this subject. Higher-level students, called upon to apply these concepts across science and engineering, will also find this a valuable and concise resource.</div>

This textbook focuses on one of the most valuable skills in multivariable and vector calculus: visualization. With over one hundred carefully drawn color images, students who have long struggled picturing, for example, level sets or vector fields will find these abstract concepts rendered with clarity and ingenuity. This illustrative approach to the material covered in standard multivariable and vector calculus textbooks will serve as a much-needed and highly useful companion.

Emphasizing portability, this book is an ideal complement to other references in the area. It begins by exploring preliminary ideas such as vector algebra, sets, and coordinate systems, before moving into the core areas of multivariable differentiation and integration, and vector calculus. Sections on the chain rule for second derivatives, implicit functions, PDEs, and the method of least squares offer additional depth; ample illustrations are woven throughout. Mastery Checks engage students in material on the spot, while longer exercise sets at the end of each chapter reinforce techniques.


An Illustrative Guide to Multivariable and Vector Calculus will appeal to multivariable and vector calculus students and instructors around the world who seek an accessible, visual approach to this subject. Higher-level students, called upon to apply these concepts across science and engineering, will also find this a valuable and concise resource.

<p>Offers an in-depth visual approach to multivariable and vector calculus</p><p>Complements existing textbooks on the subject by being concise and portable</p><p>Includes over one hundred carefully drawn figures that illustrate the material with clarity and ingenuity</p><p>Includes supplementary material: sn.pub/extras</p>

Stanley J. Miklavcic is a Professor of Mathematics at the University of South Australia. He was awarded a BSc Hons in Applied Mathematics and the University Medal by the University of New South Wales and holds a PhD from the Australian National University. His research interests include the application of mathematics and modelling in biology, physics and chemistry. A one-time recipient of a Queen Elizabeth Research Fellowship, Stanley has held academic positions in both Sweden and Australia and has published over 150 papers. Stanley is a Fellow of the Australian Mathematical Society and Member of both the Australasian Colloid and Interface Society and the Australian Society of Plant Scientists.

9783030334581

/Integral Transforms and Operational Calculus/Analysis/Mathematics and Computing/Mathematics/

/Integral Transforms and Operational Calculus/Analysis/Mathematics and Computing/Mathematics//////

978-0-387-97926-7

I General Topology.- II Differentiable Manifolds.- III Fundamental Group.- IV Homology Theory.- V Cohomology.- VI Products and Duality.- VII Homotopy Theory.- Appendices.- App. A. The Additivity Axiom.- App. B. Background in Set Theory.- App. C. Critical Values.- App. D. Direct Limits.- App. E. Euclidean Neighborhood Retracts.- Index of Symbols.

The golden age of mathematics-that was not the age of Euclid, it is ours. C. J. KEYSER This time of writing is the hundredth anniversary of the publication (1892) of Poincare's first note on topology, which arguably marks the beginning of the subject of algebraic, or 'combinatorial,' topology. There was earlier scattered work by Euler, Listing (who coined the word 'topology'), Mobius and his band, Riemann, Klein, and Betti. Indeed, even as early as 1679, Leibniz indicated the desirability of creating a geometry of the topological type. The establishment of topology (or 'analysis situs' as it was often called at the time) as a coherent theory, however, belongs to Poincare. Curiously, the beginning of general topology, also called 'point set topology,' dates fourteen years later when Frechet published the first abstract treatment of the subject in 1906. Since the beginning of time, or at least the era of Archimedes, smooth manifolds (curves, surfaces, mechanical configurations, the universe) have been a central focus in mathematics. They have always been at the core of interest in topology. After the seminal work of Milnor, Smale, and many others, in the last half of this century, the topological aspects of smooth manifolds, as distinct from the differential geometric aspects, became a subject in its own right.

9780387979267

/Topology/Mathematics and Computing/Mathematics//Geometry/Mathematics and Computing/Mathematics/////

10.1007/978-1-4757-6848-0

978-3-030-40182-5

Introduction.-Warming Up.- Integration Theory for Probability.- Probability and Expectation.- Convergence of random sequences.- Markov Chains.- Martingale Sequences.- Ergodic Sequences.- Generalities on Stochastic Processes.- Poisson Processes.- Continuous-Time Markov Chains.- Renewal Theory in Continuous Time.- Brownian Motion.- Wide-sense Stationary Stochastic Processes.- An Introduction to Itô’s Calculus.- Appenndix: Number Theory and Linear Algebra.- Analysis.- Hilbert Spaces.- Z-Transforms.- Proof of Paul Lévy’s Criterion.- Direct Riemann Integrability.- Bibliography.- Index. <div><div>
</div></div>

The ultimate objective of this book is to present a panoramic view of the main stochastic processes which have an impact on applications, with complete proofs and exercises. Random processes play a central role in the applied sciences, including operations research, insurance, finance, biology, physics, computer and communications networks, and signal processing.

In order to help the reader to reach a level of technical autonomy sufficient to understand the presented models, this book includes a reasonable dose of probability theory. On the other hand, the study of stochastic processes gives an opportunity to apply the main theoretical results of probability theory beyond classroom examples and in a non-trivial manner that makes this discipline look more attractive to the applications-oriented student.

One can distinguish three parts of this book. The first four chapters are about probability theory, Chapters 5 to 8 concern random sequences, or discrete-time stochastic processes, and the rest of the book focuses on stochastic processes and point processes. There is sufficient modularity for the instructor or the self-teaching reader to design a course or a study program adapted to her/his specific needs. This book is in a large measure self-contained.

The ultimate objective of this book is to present a panoramic view of the main stochastic processes which have an impact on applications, with complete proofs and exercises. Random processes play a central role in the applied sciences, including operations research, insurance, finance, biology, physics, computer and communications networks, and signal processing.In order to help the reader to reach a level of technical autonomy sufficient to understand the presented models, this book includes a reasonable dose of probability theory. On the other hand, the study of stochastic processes gives an opportunity to apply the main theoretical results of probability theory beyond classroom examples and in a non-trivial manner that makes this discipline look more attractive to the applications-oriented student. One can distinguish three parts of this book. The first four chapters are about probability theory, Chapters 5 to 8 concern random sequences, or discrete-time stochastic processes, and the rest of the book focuses on stochastic processes and point processes. There is sufficient modularity for the instructor or the self-teaching reader to design a course or a study program adapted to her/his specific needs. This book is in a large measure self-contained.

Mathematically rigorous but written in a convivial styleTreats the general theory as well as special models of proven interest in applicationsSelf-contained with exercises and a helpful appendix on analysis

Pierre Brémaud graduated from the École Polytechnique and obtained his Doctorate in Mathematics from the University of Paris VI and his PhD from the department of Electrical Engineering and Computer Science at the University of California, Berkeley. He is a major contributor to the theory of stochastic processes and their applications, and has authored or co-authored several reference books and textbooks on the subject.

9783030401825

/Mathematics and Computing/Probability Theory/Mathematics//Statistical Theory and Methods/Statistics/Mathematics and Computing/////

Nicolas Privault, Nanyang Technological University, Singapore, Singapore

Probability Background.- Gambling Problems.- Random Walks.- Discrete-Time Markov Chains.- First Step Analysis.- Classification of States.- Long-Run Behavior of Markov Chains.- Branching Processes.- Continuous-Time Markov Chains.- Discrete-Time Martingales.- Spatial Poisson Processes.- Reliability Theory.

This book provides an undergraduate-level introduction to discrete and continuous-time Markov chains and their applications, with a particular focus on the first step analysis technique and its applications to average hitting times and ruin probabilities. It also discusses classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes. It first examines in detail two important examples (gambling processes and random walks) before presenting the general theory itself in the subsequent chapters. It also provides an introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times, together with a chapter on spatial Poisson processes. The concepts presented are illustrated by examples, 138 exercises and 9 problems with their solutions.

This book provides an undergraduate-level introduction to discrete and continuous-time Markov chains and their applications, with a particular focus on the first step analysis technique and its applications to average hitting times and ruin probabilities. It also discusses classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes. It first examines in detail two important examples (gambling processes and random walks) before presenting the general theory itself in the subsequent chapters. It also provides an introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times, together with a chapter on spatial Poisson processes. The concepts presented are illustrated by examples, 138 exercises and 9 problems with their solutions.

Easily accessible to both mathematics and non-mathematics majors who are taking an introductory course on Stochastic ProcessesFilled with numerous exercises to test students' understanding of key conceptsA gentle introduction to help students ease into later chapters, also suitable for self-studyAccompanied with computer simulation codes in R and PythonRequest lecturer material: sn.pub/lecturer-material

The author is an associate professor from the Nanyang Technological University (NTU) and is well-established in the field of stochastic processes and a highly respected probabilist. He has authored the book, Stochastic Analysis in Discrete and Continuous Settings: With Normal Martingales, Lecture Notes in Mathematics, Springer, 2009 and was a co-editor for the book, Stochastic Analysis with Financial Applications, Progress in Probability, Vol. 65, Springer Basel, 2011. Aside from these two Springer titles, he has authored several others. He is currently teaching the course M27004-Probability Theory and Stochastic Processes at NTU. The manuscript has been developed over the years from his courses on Stochastic Processes.

9789811306587

Statistics in Engineering, Physics, Computer Science, Chemistry and Earth Sciences

/Mathematics and Computing/Probability Theory/Mathematics//Statistical Theory and Methods/Statistics/Mathematics and Computing//Statistics in Engineering, Physics, Computer Science, Chemistry and Earth Sciences/Applied Statistics/Statistics/Mathematics and Computing////

Michele Conforti, University of Padova, Padova, Italy; Gérard Cornuéjols, Carnegie Mellon University, Pittsburgh, PA, USA; Giacomo Zambelli, London School of Economics & Political Science, London, UK

Preface.- 1 Getting Started.- 2 Integer Programming Models.- 3 Linear Inequalities and Polyhedra.- 4 Perfect Formulations.- 5 Split and Gomory Inequalities.- 6 Intersection Cuts and Corner Polyhedra.- 7 Valid Inequalities for Structured Integer Programs.- 8 Reformulations and Relaxations.- 9 Enumeration.- 10 Semidefinite Bounds.- Bibliography.- Index.

This book is an elegant and rigorous presentation of integer programming, exposing the subject’s mathematical depth and broad applicability. Special attention is given to the theory behind the algorithms used in state-of-the-art solvers. An abundance of concrete examples and exercises of both theoretical and real-world interest explore the wide range of applications and ramifications of the theory. Each chapter is accompanied by an expertly informed guide to the literature and special topics, rounding out the reader’s understanding and serving as a gateway to deeper study.Key topics include:formulationspolyhedral theorycutting planesdecompositionenumerationsemidefinite relaxationsWritten by renowned experts in integer programming and combinatorial optimization, Integer Programming is destined to become an essential text in the field.

This book is an elegant and rigorous presentation of integer programming, exposing the subject’s mathematical depth and broad applicability. Special attention is given to the theory behind the algorithms used in state-of-the-art solvers. An abundance of concrete examples and exercises of both theoretical and real-world interest explore the wide range of applications and ramifications of the theory. Each chapter is accompanied by an expertly informed guide to the literature and special topics, rounding out the reader’s understanding and serving as a gateway to deeper study.Key topics include:formulationspolyhedral theorycutting planesdecompositionenumerationsemidefinite relaxationsWritten by renowned experts in integer programming and combinatorial optimization, Integer Programming is destined to become an essential text in the field.

Gives a concise yet in-depth treatment of the theory and practice of integer programmingProvides ample motivation and concrete illustration of theoretical resultsGuides the reader to many special topics in the literatureIncludes supplementary material: sn.pub/extras

Michelangelo Conforti is Professor of Mathematics at the University of Padova. Together with G. Cornuéjols and M. R. Rao, he received the 2000 Fulkerson Prize in discrete mathematics.Gérard Cornuéjols is IBM University Professor of Operations Research at Carnegie Mellon University. His research has been recognized by numerous honors, among them the Fulkerson Prize, the Frederick W. Lanchester Prize, the Dantzig Prize, and the John von Neumann Theory Prize.Giacomo Zambelli is Associate Professor (Reader) of Management Science at the London School of Economics and Political Sciences.All three authors are leading experts in the fields of integer programming, graph theory, and combinatorial optimization.

9783319110073

/Operations Research, Management Science /Optimization/Mathematics and Computing/Mathematics/

/Operations Research, Management Science /Optimization/Mathematics and Computing/Mathematics//Convex and Discrete Geometry/Geometry/Mathematics and Computing/Mathematics//Algorithms/Computational Mathematics and Numerical Analysis/Mathematics and Computing/Mathematics///

978-0-387-00451-8

1 Foundations.- 2 Generating Random Numbers and Random Variables.- 3 Generating Sample Paths.- 4 Variance Reduction Techniques.- 5 Quasi-Monte Carlo.- 6 Discretization Methods.- 7 Estimating Sensitivities.- 8 Pricing American Options.- 9 Applications in Risk Management.- A Appendix: Convergence and Confidence Intervals.- A.1 Convergence Concepts.- A.2 Central Limit Theorem and Confidence Intervals.- B Appendix: Results from Stochastic Calculus.- B.1 Itô’s Formula.- B.2 Stochastic Differential Equations.- B.3 Martingales.- B.4 Change of Measure.- C Appendix: The Term Structure of Interest Rates.- C.1 Term Structure Terminology.- C.2 Interest Rate Derivatives.- References.

Monte Carlo simulation has become an essential tool in the pricing of derivative securities and in risk management. These applications have, in turn, stimulated research into new Monte Carlo methods and renewed interest in some older techniques. This book develops the use of Monte Carlo methods in finance and it also uses simulation as a vehicle for presenting models and ideas from financial engineering. It divides roughly into three parts. The first part develops the fundamentals of Monte Carlo methods, the foundations of derivatives pricing, and the implementation of several of the most important models used in financial engineering. The next part describes techniques for improving simulation accuracy and efficiency. The final third of the book addresses special topics: estimating price sensitivities, valuing American options, and measuring market risk and credit risk in financial portfolios. The most important prerequisite is familiarity with the mathematical tools used to specify and analyze continuous-time models in finance, in particular the key ideas of stochastic calculus. Prior exposure to the basic principles of option pricing is useful but not essential. The book is aimed at graduate students in financial engineering, researchers in Monte Carlo simulation, and practitioners implementing models in industry.

Monte Carlo simulation has become an essential tool in the pricing of derivative securities and in risk management. These applications have, in turn, stimulated research into new Monte Carlo methods and renewed interest in some older techniques. This book develops the use of Monte Carlo methods in finance and it also uses simulation as a vehicle for presenting models and ideas from financial engineering. It divides roughly into three parts. The first part develops the fundamentals of Monte Carlo methods, the foundations of derivatives pricing, and the implementation of several of the most important models used in financial engineering. The next part describes techniques for improving simulation accuracy and efficiency. The final third of the book addresses special topics: estimating price sensitivities, valuing American options, and measuring market risk and credit risk in financial portfolios. The most important prerequisite is familiarity with the mathematical tools used to specify and analyze continuous-time models in finance, in particular the key ideas of stochastic calculus. Prior exposure to the basic principles of option pricing is useful but not essential. The book is aimed at graduate students in financial engineering, researchers in Monte Carlo simulation, and practitioners implementing models in industry. Mathematical Reviews, 2004: '... this book is very comprehensive, up-to-date and useful tool for those who are interested in implementing Monte Carlo methods in a financial context.'

9780387004518

/Applications of Mathematics/Mathematics and Computing/Mathematics/

/Applications of Mathematics/Mathematics and Computing/Mathematics//Public Economics/Economics/Humanities and Social Sciences//Mathematics and Computing/Probability Theory/Mathematics//Mathematics in Business, Economics and Finance/Applications of Mathematics/Mathematics and Computing/Mathematics//Statistical Theory and Methods/Statistics/Mathematics and Computing//Quantitative Economics/Economics/Humanities and Social Sciences/

1. Introduction.- 2. Background: 2.1 Data representation.- 2.2 Topological abstractions.- 2.3 Algorithms and applications.- 3. Abstraction: 3.1 Efficient topological simplification of scalar fields.- 3.2 Efficient Reeb graph computation for volumetric meshes.- 4. Interaction: 4.1 Topological simplification of isosurfaces.- 4.2 Interactive editing of topological abstractions.- 5. Analysis: 5.1 Exploration of turbulent combustion simulations.- 5.2 Quantitative analysis of molecular interactions.- 6. Perspectives: 6.1 Emerging constraints.- 6.2 Emerging data types.- 7. Conclusion.

Combining theoretical and practical aspects of topology, this book delivers a comprehensive and self-contained introduction to topological methods for the analysis and visualization of scientific data.Theoretical concepts are presented in a thorough but intuitive manner, with many high-quality color illustrations. Key algorithms for the computation and simplification of topological data representations are described in details, and their application is carefully illustrated in a chapter dedicated to concrete use cases.With its fine balance between theory and practice, 'Topological Data Analysis for Scientific Visualization' constitutes an appealing introduction to the increasingly important topic of topological data analysis, for lecturers, students and researchers.

Combining theoretical and practical aspects of topology, this book provides a comprehensive and self-contained introduction to topological methods for the analysis and visualization of scientific data.Theoretical concepts are presented in a painstaking but intuitive manner, with numerous high-quality color illustrations. Key algorithms for the computation and simplification of topological data representations are described in detail, and their application is carefully demonstrated in a chapter dedicated to concrete use cases.With its fine balance between theory and practice, 'Topological Data Analysis for Scientific Visualization' constitutes an appealing introduction to the increasingly important topic of topological data analysis for lecturers, students and researchers.

84 color images, including 20 high quality illustrations in the background section, nicely introducing in images the key concepts of topological data analysisSpecial chapter on application examples, illustrating concrete use cases of topological data analysis pipelines in combustion and chemistrySpecial chapter on the perspectives of topological data analysis regarding the upcoming generation of super-computers (exascale computing)

Julien Tierny received the Ph.D. degree in Computer Science from Lille 1 University in 2008 and the Habilitation degree (HDR) from Sorbonne Universités UPMC in 2016. He is currently a CNRS permanent research scientist, affiliated with Sorbonne Universities (LIP6, UPMC Paris 6, France) since September 2014 and with Telecom ParisTech from 2010 to 2014. Prior to his CNRS tenure, he held a Fulbright fellowship (U.S. Department of State) and was a post-doctoral research associate at the Scientific Computing and Imaging Institute at the University of Utah. His research expertise includes topological data analysis for scientific visualization. Dr. Julien Tierny received several awards for his research, including best paper awards (IEEE VIS 2017, IEEE VIS 2016, IEEE SciVis Contest 2016, EGPGV 2013). He is the lead developer of the Topology ToolKit (TTK), an open source library for topological data analysis.

9783319715063

/Data and Information Visualization/Statistics and Computing/Statistics/Mathematics and Computing/Computational Mathematics and Numerical Analysis/Mathematics/

/Data and Information Visualization/Statistics and Computing/Statistics/Mathematics and Computing/Computational Mathematics and Numerical Analysis/Mathematics//Topology/Mathematics and Computing/Mathematics//Computer Imaging, Vision, Pattern Recognition and Graphics/Computer Science/Mathematics and Computing//Algorithms/Computational Mathematics and Numerical Analysis/Mathematics and Computing/Mathematics//Discrete Mathematics in Computer Science/Mathematics of Computing/Computer Science/Mathematics and Computing//Mathematical Software/Computational Mathematics and Numerical Analysis/Mathematics and Computing/Mathematics/

978-3-319-91040-6

Peter J. Olver, University of Minnesota, Minneapolis, MN, USA; Chehrzad Shakiban, University of St. Thomas, St. Paul, MN, USA

Preface.- 1. Linear Algebraic Systems.- 2. Vector Spaces and Bases.- 3. Inner Products and Norms.- 4. Minimization and Least Squares Approximation.- 5. Orthogonality.- 6. Equilibrium.- 7. Linearity.- 8. Eigenvalues.- 9. Linear Dynamical Systems.- 10. Iteration of Linear Systems.- 11. Boundary Value Problems in One Dimension.- References.- Index.

This textbook develops the essential tools of linear algebra, with the goal of imparting technique alongside contextual understanding. Applications go hand-in-hand with theory, each reinforcing and explaining the other. This approach encourages students to develop not only the technical proficiency needed to go on to further study, but an appreciation for when, why, and how the tools of linear algebra can be used across modern applied mathematics.Providing an extensive treatment of essential topics such as Gaussian elimination, inner products and norms, and eigenvalues and singular values, this text can be used for an in-depth first course, or an application-driven second course in linear algebra. In this second edition, applications have been updated and expanded to include numerical methods, dynamical systems, data analysis, and signal processing, while the pedagogical flow of the core material has been improved. Throughout, the text emphasizes the conceptual connections between each application and the underlying linear algebraic techniques, thereby enabling students not only to learn how to apply the mathematical tools in routine contexts, but also to understand what is required to adapt to unusual or emerging problems.No previous knowledge of linear algebra is needed to approach this text, with single-variable calculus as the only formal prerequisite. However, the reader will need to draw upon some mathematical maturity to engage in the increasing abstraction inherent to the subject. Once equipped with the main tools and concepts from this book, students will be prepared for further study in differential equations, numerical analysis, data science and statistics, and a broad range of applications. The first author’s text, Introduction to Partial Differential Equations, is an ideal companion volume, forming a natural extension of the linear mathematical methods developed here.

This textbook develops the essential tools of linear algebra, with the goal of imparting technique alongside contextual understanding. Applications go hand-in-hand with theory, each reinforcing and explaining the other. This approach encourages students to develop not only the technical proficiency needed to go on to further study, but an appreciation for when, why, and how the tools of linear algebra can be used across modern applied mathematics.Providing an extensive treatment of essential topics such as Gaussian elimination, inner products and norms, and eigenvalues and singular values, this text can be used for an in-depth first course, or an application-driven second course in linear algebra. In this second edition, applications have been updated and expanded to include numerical methods, dynamical systems, data analysis, and signal processing, while the pedagogical flow of the core material has been improved. Throughout, the text emphasizes the conceptual connections between each application and the underlying linear algebraic techniques, thereby enabling students not only to learn how to apply the mathematical tools in routine contexts, but also to understand what is required to adapt to unusual or emerging problems.No previous knowledge of linear algebra is needed to approach this text, with single-variable calculus as the only formal prerequisite. However, the reader will need to draw upon some mathematical maturity to engage in the increasing abstraction inherent to the subject. Once equipped with the main tools and concepts from this book, students will be prepared for further study in differential equations, numerical analysis, data science and statistics, and a broad range of applications. The first author’s text, Introduction to Partial Differential Equations, is an ideal companion volume, forming a natural extension of the linear mathematical methods developed here.

Visit here for updated student and instructor solutions https://www-users.cse.umn.edu/~olver/ala.htmlDevelops a strong conceptual grounding for applying linear algebra in numerous modern applicationsWeaves the theory of linear algebra with applications across engineering, science, computing, data analysis, and beyondProvides an engaging and full color preparation for future study in applied differential equations

Peter Olver is Professor of Mathematics at University of Minnesota, Twin Cities. His research centers around Lie groups, differential equations, and various areas of applied mathematics. His previous books include Introduction to Partial Differential Equations (Springer, UTM, 2014), and Applications of Lie Groups to Differential Equations (Springer, GTM 107, 1993).

Chehrzad Shakiban is Professor of Mathematics at University of St. Thomas, St. Paul. Her interests include calculus of variations, computer vision, and innovative learning experiences that illustrate connections between mathematics and the arts.

9783319910406

/Linear Algebra/Algebra/Mathematics and Computing/Mathematics/

/Linear Algebra/Algebra/Mathematics and Computing/Mathematics//Mathematical Physics/Theoretical, Mathematical and Computational Physics/Physics and Astronomy/Physical Sciences/Applications of Mathematics/Mathematics and Computing/Mathematics/////

Peter Petersen, University of California, Los Angeles, Los Angeles, CA, USA

,978-0-387-51000-2,978-1-4419-2123-9,978-0-387-29246-5,978-0-387-29403-2

Preface.- 1. Riemannian Metrics.-2. Derivatives.- 3. Curvature.- 4. Examples.- 5. Geodesics and Distance.- 6. Sectional Curvature Comparison I.- 7. Ricci Curvature Comparison.- 8. Killing Fields.- 9. The Bochner Technique.- 10. Symmetric Spaces and Holonomy.- 11. Convergence.- 12. Sectional Curvature Comparison II.- Bibliography.- Index.

Intended for a one year course, this text serves as a single source, introducing readers to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few Works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory. The book will appeal to a readership that have a basic knowledge of standard manifold theory, including tensors, forms, and Lie groups. Important revisions to the third edition include:a substantial addition of unique and enriching exercises scattered throughout the text;
inclusion of an increased number of coordinate calculations of connection and curvature;
addition of general formulas for curvature on Lie Groups and submersions;
integration of variational calculus into the text allowing for an early treatment of the Sphere theorem using a proof by Berger;
incorporation of several recent results about manifolds with positive curvature;
presentation of a new simplifying approach to the Bochner technique for tensors with application to bound topological quantities with general lower curvature bounds.
From reviews of the first edition:'The book can be highly recommended to all mathematicians who want to get a more profound idea about the most interesting achievements in Riemannian geometry. It is one of the few comprehensive sources of this type.'―Bernd Wegner, ZbMATH

Intended for a one year course, this text serves as a single source, introducing readers to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few Works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory. The book will appeal to a readership that have a basic knowledge of standard manifold theory, including tensors, forms, and Lie groups.Important revisions to the third edition include:a substantial addition of unique and enriching exercises scattered throughout the text;
inclusion of an increased number of coordinate calculations of connection and curvature;
addition of general formulas for curvature on Lie Groups and submersions;
integration of variational calculus into the text allowing for an early treatment of the Sphere theorem using a proof by Berger;
incorporation of several recent results about manifolds with positive curvature;
presentation of a new simplifying approach to the Bochner technique for tensors with application to bound topological quantities with general lower curvature bounds.
From reviews of the first edition:'The book can be highly recommended to all mathematicians who want to get a more profound idea about the most interesting achievements in Riemannian geometry. It is one of the few comprehensive sources of this type.'―Bernd Wegner, ZbMATH

<p>Includes a substantial addition of unique and enriching exercises</p><p>Exists as one of the few Works to combine both the geometric parts of Riemannian geometry and analytic aspects of the theory</p><p>Presents a new approach to the Bochner technique for tensors that considerably simplifies the material</p><p>Includes supplementary material: sn.pub/extras</p>

Peter Petersen is a Professor of Mathematics at UCLA. His current research is on various aspects of Riemannian geometry. Professor Petersen has authored two important textbooks for Springer: Riemannian Geometry in the GTM series and Linear Algebra in the UTM series.

9783319266527

/Differential Geometry/Geometry/Mathematics and Computing/Mathematics/

/Differential Geometry/Geometry/Mathematics and Computing/Mathematics//////

978-0-387-90190-9

Jean-Pierre Serre, College de France Paris Chaire d'Algebre et Geometrie, Paris CX 05, France

Title of the original French edition: Representations lineaires des groupes finis

I Representations and Characters.- 1 Generalities on linear representations.- 2 Character theory.- 3 Subgroups, products, induced representations.- 4 Compact groups.- 5 Examples.- Bibliography: Part I.- II Representations in Characteristic Zero.- 6 The group algebra.- 7 Induced representations; Mackey’s criterion.- 8 Examples of induced representations.- 9 Artin’s theorem.- 10 A theorem of Brauer.- 11 Applications of Brauer’s theorem.- 12 Rationality questions.- 13 Rationality questions: examples.- Bibliography: Part II.- III Introduction to Brauer Theory.- 14 The groups RK(G), Rk
(G), and Pk(G).- 15 The cde triangle.- 16 Theorems.- 17 Proofs.- 18 Modular characters.- 19 Application to Artin representations.- Index of notation.- Index of terminology.

This book consists of three parts, rather different in level and purpose: The first part was originally written for quantum chemists. It describes the correspondence, due to Frobenius, between linear representations and charac­ ters. This is a fundamental result, of constant use in mathematics as well as in quantum chemistry or physics. I have tried to give proofs as elementary as possible, using only the definition of a group and the rudiments of linear algebra. The examples (Chapter 5) have been chosen from those useful to chemists. The second part is a course given in 1966 to second-year students of I'Ecoie Normale. It completes the first on the following points: (a) degrees of representations and integrality properties of characters (Chapter 6); (b) induced representations, theorems of Artin and Brauer, and applications (Chapters 7-11); (c) rationality questions (Chapters 12 and 13). The methods used are those of linear algebra (in a wider sense than in the first part): group algebras, modules, noncommutative tensor products, semisimple algebras. The third part is an introduction to Brauer theory: passage from characteristic 0 to characteristic p (and conversely). I have freely used the language of abelian categories (projective modules, Grothendieck groups), which is well suited to this sort of question. The principal results are: (a) The fact that the decomposition homomorphism is surjective: all irreducible representations in characteristic p can be lifted 'virtually' (i.e., in a suitable Grothendieck group) to characteristic O.

9780387901909

/Group Theory and Generalizations/Algebra/Mathematics and Computing/Mathematics/

/Group Theory and Generalizations/Algebra/Mathematics and Computing/Mathematics//////

10.1007/978-1-4684-9458-7

978-3-030-62340-1

Probability review.- Convergence and sampling.- Linear algebra review.- Distances and nearest neighbors.- Linear Regression.- Gradient descent.- Dimensionality reduction.- Clustering.- Classification.- Graph structured data.- Big data and sketching.

This textbook, suitable for an early undergraduate up to a graduate course, provides an overview of many basic principles and techniques needed for modern data analysis. In particular, this book was designed and written as preparation for students planning to take rigorous Machine Learning and Data Mining courses. It introduces key conceptual tools necessary for data analysis, including concentration of measure and PAC bounds, cross validation, gradient descent, and principal component analysis. It also surveys basic techniques in supervised (regression and classification) and unsupervised learning (dimensionality reduction and clustering) through an accessible, simplified presentation. Students are recommended to have some background in calculus, probability, and linear algebra. Some familiarity with programming and algorithms is useful to understand advanced topics on computational techniques.

This textbook, suitable for an early undergraduate up to a graduate course, provides an overview of many basic principles and techniques needed for modern data analysis. In particular, this book was designed and written as preparation for students planning to take rigorous Machine Learning and Data Mining courses. It introduces key conceptual tools necessary for data analysis, including concentration of measure and PAC bounds, cross validation, gradient descent, and principal component analysis. It also surveys basic techniques in supervised (regression and classification) and unsupervised learning (dimensionality reduction and clustering) through an accessible, simplified presentation. Students are recommended to have some background in calculus, probability, and linear algebra. Some familiarity with programming and algorithms is useful to understand advanced topics on computational techniques.

Provides accessible, simplified introduction to core mathematical language and conceptsIntegrates examples of key concepts through geometric illustrations and Python codingAddresses topics in locality sensitive hashing, graph-structured data, and big data processing as well as basic linear algebraIncludes perspectives on ethics in data

Jeff M. Phillips is an Associate Professor in the School of Computing within the University of Utah. He directs the Utah Center for Data Science as well as the Data Science curriculum within the School of Computing. His research is on algorithms for big data analytics, a domain with spans machine learning, computational geometry, data mining, algorithms, and databases, and his work regularly appears in top venues in each of these fields. He focuses on a geometric interpretation of problems, striving for simple, geometric, and intuitive techniques with provable guarantees and solve important challenges in data science. His research is supported by numerous NSF awards including an NSF Career Award.

9783030623401