A Beautiful Math John Nash, Game Theory, and the Modern Quest for a Code of Nature Tom Siegfried

National Academies Press (September 21, 2006)

Summary: Millions have seen the movie and thousands have read the book but few have fully appreciated the mathematics developed by John Nash's beautiful mind. Today Nash's beautiful math has become a universal language for research in the social sciences and has infiltrated the realms of evolutionary biology, neuroscience, and even quantum physics. John Nash won the 1994 Nobel Prize in economics for pioneering research published in the 1950s on a new branch of mathematics known as game theory. At the time of Nash's early work, game theory was briefly popular among some mathematicians and Cold War analysts. But it remained obscure until the 1970s when evolutionary biologists began applying it to their work. In the 1980s economists began to embrace game theory. Since then it has found an ever expanding repertoire of applications among a wide range of scientific disciplines. Today neuroscientists peer into game players' brains, anthropologists play games with people from primitive cultures, biologists use games to explain the evolution of human language, and mathematicians exploit games to better understand social networks. A common thread connecting much of this research is its relevance to the ancient quest for a science of human social behavior, or a Code of Nature, in the spirit of the fictional science of psychohistory described in the famous Foundation novels by the late Isaac Asimov. In A Beautiful Math, acclaimed science writer Tom Siegfried describes how game theory links the life sciences, social sciences, and physical sciences in a way that may bring Asimov's dream closer to reality. Language: English ISBN: 9780309101929 Category: History of Mathematics

A Beautiful Mind A Biography of John Forbes Nash, Jr., Winner of the Nobel Prize in Economics, 1994 Sylvia Nasar

Simon & Schuster (1998)

Summary: In this dramatic and moving biography, Sylvia Nasar re-creates the life of a mathematical genius whose brilliant career was cut short by schizophrenia and who, after three decades of devastating mental illness, miraculously recovered and was honored with a Nobel Prize. "A Beautiful Mind" traces the meteoric rise of John Forbes Nash, Jr., from his lonely childhood in West Virginia to his student years at Princeton, where he encountered Albert Einstein, John von Neumann, and a host of other mathematical luminaries. At twenty-one, the handsome, ambitious, eccentric graduate student invented what would become the most influential theory of rational human behavior in modern social science. Nash's contribution to game theory would ultimately revolutionize the field of economics.As a young professor at MIT, still in his twenties, Nash dazzled the mathematical world by solving a series of deep problems deemed "impossible" by other mathematicians. As unconventional in his private life as in his mathematics, Nash fathered a child with a woman he did not marry. At the height of the McCarthy era, he was expelled as a security risk from the supersecret RAND Corporation -- the Cold War think tank where he was a consultant. At thirty, Nash was poised to take his dreamed-of place in the pantheon of history's greatest mathematicians. His associates included the most renowned mathematicians and economists of the era: Norbert Wiener, John Milnor, Alexandre Grothendieck, Kenneth Arrow, Robert Solow, and Paul Samuelson. He married an exotic and beautiful MIT physics student, Alicia Larde. They had a son. Then Nash suffered a catastrophic mental breakdown.Nasar details Nash's harrowing descent into insanity -- his bizarre delusions that he was the Prince of Peace; his resignation from MIT, flight to Europe, and attempt to renounce his American citizenship; his repeated hospitalizations, from the storied McLean, where he came to know the poet Robert Lowell, to the crowded wards of a state hospital; his "enforced interludes of rationality" during which he was able to return briefly to mathematical research. Nash and his wife were divorced in 1963, but Alicia Nash continued to care for him and for their mathematically gifted son, who was diagnosed with schizophrenia as a teenager. Saved from homelessness by his loyal ex-wife and protected by a handful of mathematical friends, Nash lived quietly in Princeton for many years, a dreamy, ghostlike figure who scrawled numerological messages on blackboards, all but forgotten by the outside world.His early achievements, however, fired the imagination of a new generation of scholars. At age sixty-six, twin miracles -- a spontaneous remission of his illness and the sudden decision of the Nobel Prize committee to honor his contributions to game theory -- restored the world to him. Nasar recounts the bitter behind-the-scenes battle in Stockholm over whether to grant the ultimate honor in science to a man thought to be "mad". She describes Nash's current ambition to pursue new mathematical breakthroughs and his efforts to be a loving father to his adult sons.Based on hundreds of interviews with Nash's family, friends, and colleagues and scores of letters and documents, "A Beautiful Mind" is a heartbreaking but inspiring story about the most remarkable mathematician of our time and his triumph over a tragic illness. Language: English ISBN: 9780684819068 Category: History of Mathematics

A Course in Combinatorics J. H. van Lint, R. M. Wilson

Cambridge University Press (November 22, 2001)

Summary: This is the second edition of a popular book on combinatorics, a subject dealing with ways of arranging and distributing objects, and which involves ideas from geometry, algebra and analysis. The breadth of the theory is matched by that of its applications, which include topics as diverse as codes, circuit design and algorithm complexity. It has thus become essential for workers in many scientific fields to have some familiarity with the subject. The authors have tried to be as comprehensive as possible, dealing in a unified manner with, for example, graph theory, extremal problems, designs, colorings and codes. The depth and breadth of the coverage make the book a unique guide to the whole of the subject. The book is ideal for courses on combinatorical mathematics at the advanced undergraduate or beginning graduate level. Working mathematicians and scientists will also find it a valuable introduction and reference. Language: English ISBN: 9780521006019 Category: Mathematics

A Course in Game Theory Martin J. Osborne, Ariel Rubinstein

MIT Press (July 12, 1994)

Summary: A Course in Game Theory presents the main ideas of game theory at a level suitable for graduate students and advanced undergraduates, emphasizing the theory's foundations and interpretations of its basic concepts. The authors provide precise definitions and full proofs of results, sacrificing generalities and limiting the scope of the material in order to do so. The text is organized in four parts: strategic games, extensive games with perfect information, extensive games with imperfect information, and coalitional games. It includes over 100 exercises. Language: English ISBN: 9780262650403 Category: Mathematics

A First Course in Differential Equations with Modeling Applications Dennis G. Zill

Cengage Learning (March 15, 2012)

Summary: A FIRST COURSE IN DIFFERENTIAL EQUATIONS WITH MODELING APPLICATIONS, 10th Edition strikes a balance between the analytical, qualitative, and quantitative approaches to the study of differential equations. This proven and accessible text speaks to beginning engineering and math students through a wealth of pedagogical aids, including an abundance of examples, explanations, Remarks boxes, definitions, and group projects. Written in a straightforward, readable, and helpful style, this book provides a thorough treatment of boundary-value problems and partial differential equations.Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version. Language: English ISBN: 9781285401102 Category: Mathematics

A History of Mathematics Carl B. Boyer

Wiley (March 20, 1991)

Summary: "Boyer and Merzbach distill thousands of years of mathematics into this fascinating chronicle. From the Greeks to Godel, the mathematics is brilliant; the cast of characters is distinguished; the ebb and flow of ideas is everywhere evident. And, while tracing the development of European mathematics, the authors do not overlook the contributions of Chinese, Indian, and Arabic civilizations. Without doubt, this is--and will long remain--a classic one-volume history of mathematics and mathematicians who create it." --William Dunham Author, Journey Through Genius, The Great Theorems of Mathematics "When we read a book like A History of Mathematics, we get the picture of a mounting structure, ever taller and broader and more beautiful and magnificent--and with a foundation, moreover, that is as untainted and as functional now as it was when Thales worked out the first geometrical theorems nearly 26 centuries ago." --From the Foreword by Isaac Asimov "One of the most useful and comprehensive general introductions to the subject." --J. W. Dauben The City University of New York "Both readable and scholarly, this book can serve as a fine introduction to the topic and also a reference book." --J. David Bolter University of North Carolina Author of Turing's Man Revised to make it more accessible to a general audience, A History of Mathematics paints a vivid picture of humankind's relationship with numbers. Updated and expanded, it now offers broadened coverage of twentieth century advances in probability and computers, and updated references to further reading. A feature that will be of interest to every reader is an appendix containing an extensive chronological table of mathematical and general historical developments. Language: English ISBN: 9780471543978 Category: History of Mathematics

Abstract Algebra David S. Dummit, Richard M. Foote

John Wiley & Sons (2004)

Summary: Widely acclaimed algebra text. This book is designed to give the reader insight into the power and beauty that accrues from a rich interplay between different areas of mathematics. The book carefully develops the theory of different algebraic structures, beginning from basic definitions to some in-depth results, using numerous examples and exercises to aid the reader's understanding. In this way, readers gain an appreciation for how mathematical structures and their interplay lead to powerful results and insights in a number of different settings. * The emphasis throughout has been to motivate the introduction and development of important algebraic concepts using as many examples as possible. Language: English ISBN: 9780471433347 Category: Mathematics

Algebra Michael Artin

Prentice Hall India Pvt., Limited (2011)

Language: English ISBN: 9788120343290 Category: Mathematics

Algebraic Topology Allen Hatcher

Cambridge University Press (2002)

Summary: In most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises. The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally. The author emphasizes the geometric aspects of the subject, which helps students gain intuition. A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers. Language: English ISBN: 9780521795401 Category: Mathematics

An Historical Introduction To The Philosophy Of Mathematics a reader Russell Marcus, Mark (Mark V.) McEvoy

Bloomsbury Academic, an imprint of Bloomsbury Publsihing Plc (2016)

ISBN: 9781472525673 Category: History of Mathematics

An Imaginary Tale the story of [the square root of minus one] Paul J. Nahin

Princeton University Press (1998)

Summary: Today complex numbers have such widespread practical use--from electrical engineering to aeronautics--that few people would expect the story behind their derivation to be filled with adventure and enigma. In An Imaginary Tale, Paul Nahin tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them. In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number. The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for i. In the first century, the mathematician-engineer Heron of Alexandria encountered I in a separate project, but fudged the arithmetic; medieval mathematicians stumbled upon the concept while grappling with the meaning of negative numbers, but dismissed their square roots as nonsense. By the time of Descartes, a theoretical use for these elusive square roots--now called "imaginary numbers"--was suspected, but efforts to solve them led to intense, bitter debates. The notorious i finally won acceptance and was put to use in complex analysis and theoretical physics in Napoleonic times. Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts and mathematical discussions, including the application of complex numbers and functions to important problems, such as Kepler's laws of planetary motion and ac electrical circuits. This book can be read as an engaging history, almost a biography, of one of the most evasive and pervasive "numbers" in all of mathematics.Some images inside the book are unavailable due to digital copyright restrictions. Language: English ISBN: 9780691027951 Category: History of Mathematics

An Introduction to Linear Algebra T.A. Whitelaw

Blackie & Son (1983)

Summary: This popular textbook was thoughtfully and specifically tailored to introducing undergraduate students to linear algebra. The second edition has been carefully revised to improve upon its already successful format and approach. In particular, the author added a chapter on quadratic forms, making this one of the most comprehensive introductory texts on linear algebra. Language: English ISBN: 9780751401592 Category: Mathematics

Applied Combinatorics, Second Edition Fred Roberts, Barry Tesman

CRC Press (June 3, 2009)

Summary: Now with solutions to selected problems, Applied Combinatorics, Second Edition presents the tools of combinatorics from an applied point of view. This bestselling textbook offers numerous references to the literature of combinatorics and its applications that enable readers to delve more deeply into the topics. After introducing fundamental counting rules and the tools of graph theory and relations, the authors focus on three basic problems of combinatorics: counting, existence, and optimization problems. They discuss advanced tools for dealing with the counting problem, including generating functions, recurrences, inclusion/exclusion, and Pólya theory. The text then covers combinatorial design, coding theory, and special problems in graph theory. It also illustrates the basic ideas of combinatorial optimization through a study of graphs and networks. Language: English ISBN: 9781420099836 Category: Mathematics

Approche élémentaire de l'étude des fonctions arithmétiques J. M. de Koninck, Armel Mercier

Presses Université Laval (1982)

Summary: Introduction à la théorie analytique des nombres. Les auteurs se sont limités à trois thèmes principaux: la répartition des nombres premiers dans la suite des nombres naturels (chap. 1 à 3), les fonctions arithmétiques (chap. 4 à 7) et les méthodes du crible moderne (chap. 8). "La présentation de ces thèmes se limite à l'utilisation de l'analyse réelle." Language: French ISBN: 9782763769479 Category: Mathematics

Calculus Tom M. Apostol

John Wiley & Sons (1967)

ISBN: 9780471000051 Category: Mathematics

Chaos Making a New Science James Gleick

Penguin (1988)

Summary: The author describes how scientists studying the growth of complexity in nature are discovering order and pattern in chaos. He explains concepts such as nonlinearity, the Butterfly Effect, universal constants, fractals, and strange attractors, and examines the work of scientists such as Mitchell J. Feigenbaum, Edward Lorenz, and Benoit Mandelbrot. Language: English ISBN: 9780140092509 Category: History of Mathematics

Differential Geometry of Curves and Surfaces Manfredo Perdigão do Carmo

Prentice-Hall (1976)

Summary: This volume covers local as well as global differential geometry of curves and surfaces. Language: English ISBN: 9780132125895 Category: Mathematics

Differential Topology Victor Guillemin, Alan Pollack

American Mathematical Soc. (2010)

Summary: Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. In the years since its first publication, Guillemin and Pollack's book has become a standard text on the subject. It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. The text is mostly self-contained, requiring only undergraduate analysis and linear algebra. By relying on a unifying idea--transversality--the authors are able to avoid the use of big machinery or ad hoc techniques to establish the main results. In this way, they present intelligent treatments of important theorems, such as the Lefschetz fixed-point theorem, the Poincaré-Hopf index theorem, and Stokes theorem. The book has a wealth of exercises of various types. Some are routine explorations of the main material. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem. An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. The book is suitable for either an introductory graduate course or an advanced undergraduate course. Language: English ISBN: 9780821851937 Category: Mathematics

Dr. Euler's Fabulous Formula Cures Many Mathematical Ills Paul J. Nahin

Princeton University Press (2006)

Summary: I used to think math was no fun 'Cause I couldn't see how it was done Now Euler's my hero For I now see why zero Equals e[pi] i+1 --Paul Nahin, electrical engineer In the mid-eighteenth century, Swiss-born mathematician Leonhard Euler developed a formula so innovative and complex that it continues to inspire research, discussion, and even the occasional limerick. Dr. Euler's Fabulous Formula shares the fascinating story of this groundbreaking formula--long regarded as the gold standard for mathematical beauty--and shows why it still lies at the heart of complex number theory. This book is the sequel to Paul Nahin's An Imaginary Tale: The Story of I [the square root of -1], which chronicled the events leading up to the discovery of one of mathematics' most elusive numbers, the square root of minus one. Unlike the earlier book, which devoted a significant amount of space to the historical development of complex numbers, Dr. Euler begins with discussions of many sophisticated applications of complex numbers in pure and applied mathematics, and to electronic technology. The topics covered span a huge range, from a never-before-told tale of an encounter between the famous mathematician G. H. Hardy and the physicist Arthur Schuster, to a discussion of the theoretical basis for single-sideband AM radio, to the design of chase-and-escape problems. The book is accessible to any reader with the equivalent of the first two years of college mathematics (calculus and differential equations), and it promises to inspire new applications for years to come. Or as Nahin writes in the book's preface: To mathematicians ten thousand years hence, "Euler's formula will still be beautiful and stunning and untarnished by time." Language: English ISBN: 9780691118222 Category: History of Mathematics

E=MC2 A Biography of the Worlds most Famous Equation A biography of the world's most famous equation David Bodanis

n/a (2000)

ISBN: 9780965006934 Category: History of Mathematics

Elements of Real Analysis Herbert S. Gaskill, P. P. Narayanaswami

Prentice Hall (1998)

Summary: Comprehensive in coverage, this book explores the principles of logic, the axioms for the real numbers, limits of sequences, limits of functions, differentiation and integration, infinite series, convergence, and uniform convergence for sequences of real-valued functions. Concepts are presented slowly and include the details of calculations as well as substantial explanations as to how and why one proceeds in the given manner. Uses the words WHY? and HOW? throughout; inviting readers to become active participants and to supply a missing argument or a simple calculation. Contains more than 1000 individual exercises. Stresses and reviews elementary algebra and symbol manipulation as essential tools for success at the kind of computations required in dealing with limiting processes. Language: English ISBN: 9780138970673 Category: Mathematics

Equations of eternity Speculations on consciousness, meaning, and the mathematical rules that orchestrate the cosmos David J. Darling

Hyperion (1993)

ISBN: 9781562828752 Category: History of Mathematics

Fermat's Enigma The Epic Quest to Solve the World's Greatest Mathematical Problem Simon Singh

Paw Prints (April 8, 2009)

Summary: A national best-seller traces the quest of scientists around the world to solve the theorem devised by seventeenth- century French mathematician Pierre de Fermat, a story of extraordinary human emotion, ambition, and desperation. Reprint. Language: English ISBN: 9781442006898 Category: History of Mathematics

Foundations of p-adic Teichmüller Theory Shinichi Mochizuki

American Mathematical Soc. (January 6, 2014)

Summary: This book lays the foundation for a theory of uniformization of p-adic hyperbolic curves and their moduli. On one hand, this theory generalizes the Fuchsian and Bers uniformizations of complex hyperbolic curves and their moduli to nonarchimedian places. That is why in this book, the theory is referred to as p-adic Teichmüller theory, for short. On the other hand, the theory may be regarded as a fairly precise hyperbolic analog of the Serre-Tate theory of ordinary abelian varieties and their moduli. The theory of uniformization of p-adic hyperbolic curves and their moduli was initiated in a previous work by Mochizuki. And in some sense, this book is a continuation and generalization of that work. This book aims to bridge the gap between the approach presented and the classical uniformization of a hyperbolic Riemann surface that is studied in undergraduate complex analysis. Features: Presents a systematic treatment of the moduli space of curves from the point of view of p-adic Galois representations.Treats the analog of Serre-Tate theory for hyperbolic curves.Develops a p-adic analog of Fuchsian and Bers uniformization theories.Gives a systematic treatment of a "nonabelian example" of p-adic Hodge theory. Titles in this series are co-published with International Press of Boston, Inc., Cambridge, MA. Language: English ISBN: 9781470412265 Category: Mathematics

Fractals The Patterns of Chaos : a New Aesthetic of Art, Science, and Nature John Briggs

Simon and Schuster (1992)

Summary: Fractals are unique patterns left behind by the unpredictable movements -- the chaos -- of the world at work. The branching patterns of trees, the veins in a hand, water twisting out of a running tap -- all of these are fractals. Learn to recognize them and you will never again see things in quite the same way.Fractals permeate our lives, appearing in places as tiny as the surface of a virus and as majestic as the Grand Canyon. From ancient tribal peoples to modern painters to the animators of Star Wars, artists have been captivated by fractals and have utilized them in their work. Computer buffs are wild about fractals as well, for they can be generated on ordinary home computers.In Fractals: The Patterns of Chaos, science writer John Briggs uses over 170 illustrations to clearly explain the significance -- and more importantly, the beauty -- of fractals. He describes how fractals were discovered, how they are formed, and the unique properties different fractals share. Fractals is a breathtaking guided tour of a brand new aesthetic of art, science, and nature. It will revolutionize the way you see the world and your place within it.* Contains a special bibliography listing fractal generating software for desktop computers Language: English ISBN: 9780671742171 Category: History of Mathematics God Created The Integers Stephen Hawking

Running Press (October 4, 2005)

Summary: Bestselling author and physicist Stephen Hawking explores the "masterpieces" of mathematics, 25 landmarks spanning 2,500 years and representing the work of 15 mathematicians, including Augustin Cauchy, Bernard Riemann, and Alan Turing. This extensive anthology allows readers to peer into the mind of genius by providing them with excerpts from the original mathematical proofs and results. It also helps them understand the progression of mathematical thought, and the very foundations of our present-day technologies. Each chapter begins with a biography of the featured mathematician, clearly explaining the significance of the result, followed by the full proof of the work, reproduced from the original publication. Language: English ISBN: 9780762419227 Category: History of Mathematics

Groupes et Algebres de Lie Algèbres de Lie libres ; groupes de Lie. Chap. 2,3 Nicolas Bourbaki

Hermann (1972)

Language: French Category: Mathematics

Groupes et algèbres de Lie Chapitres 7 et 8 N. Bourbaki

Springer Berlin Heidelberg (September 15, 2006)

Summary: Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce troisième volume du Livre sur les Groupes et algèbres de Lie, neuvième Livre du traité, poursuit l’étude des algèbres de Lie et leurs représentations. Il comprend les chapitres: 7. Sous-algèbres de Cartan, éléments réguliers; 8. Algèbres de Lie semi-simples déployées. Ce volume contient également un appendice sur la topologie de Zariski. Ce volume est une réimpression de l’édition de 1975. Language: French ISBN: 9783540339397 Category: Mathematics

Height Pairings on Elliptic Curves Andrew John Plater

Date Published: June 20, 1991 Category: Mathematics Notes: A dissertation submitted for the degree of Doctor of Philosophy at the University of Cambridge

How to Solve it A New Aspect of Mathematical Method George Pólya

Princeton University Press (1957)

Language: English Category: Mathematics

Introduction To Commutative Algebra M.F. Atiyah And I.G. Macdonald

Sarat Book House (2007)

Language: English ISBN: 9788187169871 Category: Mathematics

Introduction to Probability and Statistics Henry L. Alder, Edward B. Roessler

Date Published: 1968 Language: English Category: Mathematics

Introduction to the Arithmetic Theory of Automorphic Functions Goro Shimura

Princeton University Press (August 21, 1971)

Summary: The theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost ubiquitous in number theory. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their number-theoretical aspects. After two chapters geared toward elementary levels, there follows a detailed treatment of the theory of Hecke operators, which associate zeta functions to modular forms. At a more advanced level, complex multiplication of elliptic curves and abelian varieties is discussed. The main question is the construction of abelian extensions of certain algebraic number fields, which is traditionally called "Hilbert's twelfth problem." Another advanced topic is the determination of the zeta function of an algebraic curve uniformized by modular functions, which supplies an indispensable background for the recent proof of Fermat's last theorem by Wiles. Language: English ISBN: 9780691080925 Category: Mathematics

Journey through genius the great theorems of mathematics William Dunham

Penguin Books (August 1, 1991)

Summary: A rare combination of the historical, biographical, and mathematicalgenius, this book is a fascinating introduction to a neglected field of human creativity. Dunham places mathematical theorem, along with masterpieces of art, music, and literature and gives them the attention they deserve. Language: English ISBN: 9780140147391 Category: History of Mathematics

Lectures on the Topology of 3-manifolds An Introduction to the Casson Invariant Nikolai Saveliev

De Gruyter (2012)

Summary: Progress in low-dimensional topology has been very quick in the last three decades, leading to the solutions of many difficult problems. Among the earlier highlights of this period was Casson's λ-invariant that was instrumental in proving the vanishing of the Rohlin invariant of homotopy 3-spheres. The proof of the three-dimensional Poincar� conjecture has rendered this application moot but hardly made Casson's contribution less relevant: in fact, a lot of modern day topology, including a multitude of Floer homology theories, can be traced back to his λ-invariant. The principal goal of this book, now in its second revised edition, remains providing an introduction to the low-dimensional topology and Casson's theory; it also reaches out, when appropriate, to more recent research topics. The book covers some classical material, such as Heegaard splittings, Dehn surgery, and invariants of knots and links. It then proceeds through the Kirby calculus and Rohlin's theorem to Casson's invariant and its applications, and concludes with a brief overview of recent developments. The book will be accessible to graduate students in mathematics and theoretical physics familiar with some elementary algebraic and differential topology, including the fundamental group, basic homology theory, transversality, and Poincar� duality on manifolds. Language: English ISBN: 9783110250350 Category: Mathematics

Mathematical Understanding of Nature Essays on Amazing Physical Phenomena and Their Understanding by Mathematicians V. I. Arnold

American Mathematical Society (September 4, 2014)

Summary: This collection of 39 short stories gives the reader a unique opportunity to take a look at the scientific philosophy of Vladimir Arnold, one of the most original contemporary researchers. Topics of the stories included range from astronomy, to mirages, to motion of glaciers, to geometry of mirrors and beyond. In each case Arnold's explanation is both deep and simple, which makes the book interesting and accessible to an extremely broad readership. Original illustrations hand drawn by the author help the reader to further understand and appreciate Arnold's view on the relationship between mathematics and science. Language: English ISBN: 9781470417017 Category: History of Mathematics

Mathematics Queen and Servant of Science E. T. Bell

Mathematical Association of America (September 5, 1996)

Summary: An absorbing account of pure and applied mathematics from the geometry of Euclid to that of Riemann, and its application in Einstein's theory of relativity. The twenty chapters cover such topics as: algebra, number theory, logic, probability, infinite sets and the foundations of mathematics, rings, matrices, transformations, groups, geometry, and topology. Mathematics was republished in 1987 with corrections and an added foreword by Martin Gardner. Language: English ISBN: 9780883854471 Category: History of Mathematics

Measure and Integral An Introduction to Real Analysis Richard Wheeden, Richard L. Wheeden, Antoni Zygmund

CRC Press (November 1, 1977)

Summary: This volume develops the classical theory of the Lebesgue integral and some of its applications. The integral is initially presented in the context of n-dimensional Euclidean space, following a thorough study of the concepts of outer measure and measure. A more general treatment of the integral, based on an axiomatic approach, is later given.Closely related topics in real variables, such as functions of bounded variation, the Riemann-Stieltjes integral, Fubini's theorem, L(p)) classes, and various results about differentiation are examined in detail. Several applications of the theory to a specific branch of analysis--harmonic analysis--are also provided. Among these applications are basic facts about convolution operators and Fourier series, including results for the conjugate function and the Hardy-Littlewood maximal function.Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis for student interested in mathematics, statistics, or probability. Requiring only a basic familiarity with advanced calculus, this volume is an excellent textbook for advanced undergraduate or first-year graduate student in these areas. Language: English ISBN: 9780824764999 Category: Mathematics

Non-compact Groups in Particle Physics Proceedings A. O. Barut, L. C. Biedenharn, N. Makunda, et. al.

W. A. Benjamin (1966)

Editor: Yutze Chow Language: English Category: Mathematics

Principles of mathematical analysis Walter Rudin

ISBN: 9781259064784 Category: Mathematics

Real analysis Modern techniques and their applications G. B. Folland

[World Book Publishing Company]

ISBN: 9787506282758 Category: Mathematics

Real Analysis N. L. Carothers

Cambridge University Press (August 15, 2000)

Summary: This is a course in real analysis directed at advanced undergraduates and beginning graduate students in mathematics and related fields. Presupposing only a modest background in real analysis or advanced calculus, the book offers something to specialists and non-specialists alike, including historical commentary, carefully chosen references, and plenty of exercises. Language: English ISBN: 9780521497565 Category: Mathematics

Real and Complex Analysis Walter Rudin

Tata McGraw-Hill (1987)

Language: English ISBN: 9780070619876 Category: Mathematics

Rings with Minimum Condition Emil Artin, Cecil Nesbitt, Robert Thrall

University of Michigan Press (October 31, 2016)

Summary: Dealing with algebraic ring theory, this book was written for mathematicians who are familiar with groups, rings, fields, and their properties. Concepts such as vector spaces, matrix representations, simple rings, and semisimple rings are considered. Language: English ISBN: 9780472750092 Category: Mathematics

Sacred Geometry Philosophy and Practice Robert Lawlor

Thames and Hudson (1994)

Language: English Category: History of Mathematics

Symmetry and the Monster : One of the greatest quests of mathematics One of the greatest quests of mathematics Mark Ronan

Oxford University Press, UK (May 18, 2006)

Summary: The hunt for the 'Monster' of symmetry is one of the great mathematical quests, alongside Fermat's Last Theorem, the Riemann Hypothesis, and Poincar--eacute--; Conjecture. The Monster is a giant snowflake in 196,884 dimensions - the largest exception to our neat classifications of symmetry, with a beautiful structure which may turn out to unlock our understanding of symmetry, string theory, and the very fabric of our universe. The story of its discovery became the biggest joint mathematical project of all time - involving determination, luck, and some very extraordinary characters. - ;Mathematics is being driven forward by the quest to solve a small number of major problems - generating excitement in the mathematical world and beyond. Four famous challenges have been Fermat's Last Theorem, the Riemann Hypothesis, Poincar--eacute--;'s Conjecture, and, now, the quest for the 'Monster' of Symmetry. It is this latter that forms the topic of this book. Although its roots go back much further, the quest to understand symmetry really begins with the tragic young genius Evariste Galois, who died at the age of 20 in a duel. He used symmetry to understand algebraic equations, and he discovered that there were building blocks or 'atoms of symmetry'. Most fit into a table, rather like the periodic table of elements, but there are 26 exceptions. The biggest of these was dubbed 'the Monster' - a giant snowflake in 196,884 dimensions. At first the Monster was only dimly seen. Did it really exist, or was it a mirage? Many mathematicians became involved. The Monster became clearer, and it was no longer monstrous but a beautiful form that pointed out deep connections between symmetry, string theory, and the very fabric and form of the universe. The story of the discovery involves some extraordinary characters, and Mark Ronan brings these people to life, and recreates in accessible language the growing excitement of what became the biggest joint project ever in the field of mathematics - the hunt for the Monster. - ;...includes entertaining glimpses of the personalities involved ...but best of all gives an admirable amount of detail... - TLS;a fascinating book that will appeal to anyone with an appetite for exploration and discovery, and which is accessble to all. - Language: English ISBN: 9780192807229 Category: History of Mathematics

The Art of Problem Solving The Basics Sandor Lehoczky, Richard Rusczyk

Mu Alpha Theta, National High School Mathematics Club (1995)

Language: English ISBN: 9781885875006 Category: Mathematics

The Art of Problem Solving And Beyond, Text Sandor Lehoczky, Richard Rusczyk

Mu Alpha Theta, National High School Mathematics Club (April 1, 1994)

Language: English ISBN: 9781885875037 Category: Mathematics

The Artist and the Mathematician Amir D. Aczel

Basic Books (April 29, 2009)

Summary: Nicolas Bourbaki, whose mathematical publications began to appear in the late 1930s and continued to be published through most of the twentieth century, was a direct product as well as a major force behind an important revolution that took place in the early decades of the twentieth century that completely changed Western culture. Pure mathematics, the area of Bourbaki's work, seems on the surface to be an abstract field of human study with no direct connection with the real world. In reality, however, it is closely intertwined with the general culture that surrounds it. Major developments in mathematics have often followed important trends in popular culture; developments in mathematics have acted as harbingers of change in the surrounding human culture. The seeds of change, the beginnings of the revolution that swept the Western world in the early decades of the twentieth century — both in mathematics and in other areas — were sown late in the previous century. This is the story both of Bourbaki and the world that created him in that time. It is the story of an elaborate intellectual joke — because Bourbaki, one of the foremost mathematicians of his day — never existed. Language: English ISBN: 9780786732883 Category: History of Mathematics

The Banach-Tarski Paradox Stan Wagon

Cambridge University Press (September 24, 1993)

Summary: This volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, and logic. It unifies the results of contemporary research on the paradox and presents several new results including some unusual paradoxes in hyperbolic space. It also provides up to date proofs and discusses many unsolved problems. Language: English ISBN: 9780521457040 Category: Mathematics

The Code Book: The Secret History of Codes and Code- breaking The Secret History of Codes and Code-breaking Simon Singh

HarperCollins Publishers (June 23, 2010)

Summary: The Science of Secrecy from Ancient Egypt to Quantum Cryptography From the best-selling author of Fermat’s Last Theorem, The Code Book is a history of man’s urge to uncover the secrets of codes, from Egyptian puzzles to modern day computer encryptions. As in Fermat’s Last Theorem, Simon Singh brings life to an anstonishing story of puzzles, codes, languages and riddles that reveals man’s continual pursuit to disguise and uncover, and to work out the secret languages of others. Codes have influenced events throughout history, both in the stories of those who make them and those who break them. The betrayal of Mary Queen of Scots and the cracking of the enigma code that helped the Allies in World War II are major episodes in a continuing history of cryptography. In addition to stories of intrigue and warfare, Simon Singh also investigates other codes, the unravelling of genes and the rediscovery of ancient languages and most tantalisingly, the Beale ciphers, an unbroken code that could hold the key to a $20 million treasure. Note that it has not been possible to include the same picture content that appeared in the original print version. Language: English ISBN: 9780007378302 Category: History of Mathematics

The Golden Ratio The Story of Phi, the World's Most Astonishing Number Mario Livio

Broadway Books (2003)

Summary: Throughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, Mario Livio tells the tale of a number at the heart of that mystery: phi, or 1.6180339887...This curious mathematical relationship, widely known as "The Golden Ratio," was discovered by Euclid more than two thousand years ago because of its crucial role in the construction of the pentagram, to which magical properties had been attributed. Since then it has shown a propensity to appear in the most astonishing variety of places, from mollusk shells, sunflower florets, and rose petals to the shape of the galaxy. Psychological studies have investigated whether the Golden Ratio is the most aesthetically pleasing proportion extant, and it has been asserted that the creators of the Pyramids and the Parthenon employed it. It is believed to feature in works of art from Leonardo da Vinci's Mona Lisa to Salvador Dali's The Sacrament of the Last Supper, and poets and composers have used it in their works. It has even been found to be connected to the behavior of the stock market! The Golden Ratio is a captivating journey through art and architecture, botany and biology, physics and mathematics. It tells the human story of numerous phi-fixated individuals, including the followers of Pythagoras who believed that this proportion revealed the hand of God; astronomer Johannes Kepler, who saw phi as the greatest treasure of geometry; such Renaissance thinkers as mathematician Leonardo Fibonacci of Pisa; and such masters of the modern world as Goethe, Cezanne, Bartok, and physicist Roger Penrose. Wherever his quest for the meaning of phi takes him, Mario Livio reveals the world as a place where order, beauty, and eternal mystery will always coexist. From the Hardcover edition. Language: English ISBN: 9780767908160 Category: History of Mathematics

The Knot Book An Elementary Introduction to the Mathematical Theory of Knots Colin Conrad Adams

American Mathematical Soc. (2004)

Summary: Knots are familiar objects. We use them to moor our boats, to wrap our packages, to tie our shoes. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. ""The Knot Book"" is an introduction to this rich theory, starting with our familiar understanding of knots and a bit of college algebra and finishing with exciting topics of current research. ""The Knot Book"" is also about the excitement of doing mathematics. Colin Adams engages the reader with fascinating examples, superb figures, and thought-provoking ideas. He also presents the remarkable applications of knot theory to modern chemistry, biology, and physics. This is a compelling book that will comfortably escort you into the marvelous world of knot theory. Whether you are a mathematics student, someone working in a related field, or an amateur mathematician, you will find much of interest in ""The Knot Book"".Colin Adams received the Mathematical Association of America (MAA) Award for Distinguished Teaching and has been an MAA Polya Lecturer and a Sigma Xi Distinguished Lecturer. Other key books of interest available from the ""AMS"" are ""Knots and Links"" and ""The Shoelace Book: A Mathematical Guide to the Best (and Worst) Ways to Lace your Shoes"". Language: English ISBN: 9780821836781 Category: Mathematics

The Lady Or the Tiger? And Other Logic Puzzles Raymond M. Smullyan

Alfred A. Knopf (1982) Summary: "Another scintillating collection of brilliant problems and paradoxes by the most entertaining logician and set theorist who ever lived." — Martin Gardner "Smullyan is not your run-of-the-mill puzzlemeister; he polishes up old chestnuts, spins variations on a theme, and peoples his logical world with a delightful cast of characters." — Science 82 "I believe Ray Smullyan to be the Lewis Carroll of our times. His little books of logic puzzles will be remembered long after most of us are forgotten." — Peter Denning, Chairman of the Computer Science Department, Naval Postgraduate School "You may experience small frissons of delight as you follow Smullyan into the dizzying heights of Gödel's proof and the very nature of proof, truth, and logic in mathematics." — Kirkus ReviewsDiscover scintillating new perspectives on the principles of mathematical logic with this puzzle treasury. Inspired by the classic tale of a prisoner's choice between two doors, these whimsically themed challenges allow readers to base their decisions on logic rather than luck. Nineteen chapters advance from relatively simple puzzles and meta-puzzles to highly complex paradoxes involving probability, time, and change. The author, a well-known philosopher and magician as well as a celebrated mathematician and logician, was acclaimed by The New York Times as "a master at translating difficult ideas into stories and puzzles that require no formal background, only patience and a passion to learn." Language: English ISBN: 394514661 Category: History of Mathematics

The Life of Pythagoras With His Symbols and Golden Verses. Together with the Life of Hierocles, and His Commentaries Upon the Verses. Collected Out of the Choicest Manuscripts, and Tr. Into French, with Annotations André Dacier

J. Tonson (1707)

Language: English Category: History of Mathematics

The Man Who Loved Only Numbers The Story of Paul Erdos and the Search for Mathematical Truth Paul Hoffman

Hyperion Books (May 12, 1999)

Summary: Based on a National Magazine Award-winning article, this masterful biography of Hungarian-born Paul Erdos is both a vivid portrait of an eccentric genius and a layman's guide to some of this century's most startling mathematical discoveries. Language: English ISBN: 9780786884063 Category: History of Mathematics

The Mathematical Tourist New and Updated Snapshots of Modern Mathematics Ivars Peterson

Macmillan (April 15, 1998)

Summary: In the first edition of The Mathematical Tourist, renowned science journalist Ivars Peterson took readers on an unforgettable tour through the sometimes bizarre, but always fascinating, landscape of modern mathematics. Now the journey continues in a new, updated edition that includes all the latest information on mathematical proofs, fractals, prime numbers, and chaos, as well as new material on* the relationship between mathematical knots and DNA* how computers based on quantum logic can significantly speed up the factoring of large composite numbers* the relationship between four-dimensional geometry and physical theories of the nature of matter* the application of cellular automata models to social questions and the peregrinations of virtual ants* a novel mathematical model of quasicrystals based on decagon-shaped tilesBlazing a trail through rows of austere symbols and dense lines of formulae, Peterson explores the central ideas behind the work of professional mathematicians-- how and where their pieces of the mathematical puzzle fit in, the sources of their ideas, their fountains of inspiration, and the images that carry them from one discovery to another. Language: English ISBN: 9780805071597 Category: History of Mathematics

The Mathematics Bible The Definitive Guide to the Last 4,000 Years of Theories Colin Beveridge

Firefly Books (2016)

Summary: Mathematics has a history filled with brilliant minds and world-changing discoveries. It just needs to be made accessible. And that's exactly what The Mathematics Bible does. It describes the history and development of mathematics in easily understood language. It introduces the most important players, societies and cultures, like the Ancient Egyptians and Pythagoreans, and key figures such as Galileo, Dodgson, Babbage and Lovelace. It brings the ancient science and art of mathematics into the contemporary world of the 21st century. Accessible, well-informed and fully illustrated, this is a book that shows perfectly just how varied and fascinating mathematics can be. These definitive guides to their subjects together have sold over three million copies worldwide. Their success is undoubtedly owed to the comprehensiveness and quality of content, for an excellent price, and the smaller size is nonetheless filled with 400 illustrations. Language: English ISBN: 9781770857933 Category: History of Mathematics

The Mathematics of Life Ian Stewart

Basic Books (June 7, 2011)

Summary: A wonderful and engaging introduction to the role of mathematics in life sciences, from cellular organization to the behavior and evolution of entire organisms Language: English ISBN: 9780465022380 Category: History of Mathematics

The most beautiful mathematical formulas Lionel Salem, Frédéric Testard, Coralie Salem

J. Wiley (1992)

Summary: A lighthearted tour through 49 of the most interesting and useful mathematical formulas ever derived Now in paperback, this whimsical book reacquaints the reader with the pleasure of playing with numbers. Both entertaining and practical, it reaches a level of sophistication consistently high enough to make intelligent people think, but never aims so high that it is difficult to follow. Accompanying the formulas are over 70 amusing cartoons and diverting stories that point up how everyday events can lead to fundamental mathematical insights. * Features 49 short, inviting chapters, written in simple, clear language * Proves that math is fun, functional and accessible--not an arcane subject reserved for specialists LIONEL SALEM (Paris, France) is an internationally renowned theoretical chemist at the University of Paris, and has an appointment to the world famous French National Research Center. FREDERIC TESTARD (Nice, France) is a mathematician at the University of Nice. CORALIE SALEM (Paris, France) is an artist/illustrator who provided the book's graphics. Language: English ISBN: 9780471552765 Category: History of Mathematics

The Unimaginable Mathematics of Borges' Library of Babel William Goldbloom Bloch

Oxford University Press, USA (August 25, 2008)

Summary: "The Library of Babel" is arguably Jorge Luis Borges' best known story--memorialized along with Borges on an Argentine postage stamp. Now, in The Unimaginable Mathematics of Borges' Library of Babel, William Goldbloom Bloch takes readers on a fascinating tour of the mathematical ideas hidden within one of the classic works of modern literature.Written in the vein of Douglas R. Hofstadter's Pulitzer Prize-winning Godel, Escher, Bach, this original and imaginative book sheds light on one of Borges' most complex, richly layered works. Bloch begins each chapter with a mathematical idea--combinatorics, topology, geometry, information theory--followed by examples and illustrations that put flesh on the theoretical bones. In this way, he provides many fascinating insights into Borges' Library. He explains, for instance, a straightforward way to calculate how many books are in the Library--an easily notated but literally unimaginable number--and also shows that, if each book were the size of a grain of sand, the entire universe could only hold a fraction of the books in the Library. Indeed, if each book were the size of a proton, our universe would still not be big enough to hold anywhere near all the books.Given Borges' well-known affection for mathematics, this exploration of the story through the eyes of a humanistic mathematician makes a unique and important contribution to the body of Borgesian criticism. Bloch not only illuminates one of the great short stories of modern literature but also exposes the reader--including those more inclined to the literary world--to many intriguing and entrancing mathematical ideas. Language: English ISBN: 9780195334579 Category: History of Mathematics

Topology James R. Munkres

Prentice-Hall of India (2004)

ISBN: 9788120320468 Category: Mathematics

Vector Bundles and K-Theory Allen Hatcher

Unpublished

Category: Mathematics

Weil Conjectures, Perverse Sheaves and l’adic Fourier Transform Reinhardt Kiehl, Rainer Weissauer

Springer Science & Business Media (August 14, 2001)

Summary: In this book the authors describe the important generalization of the original Weil conjectures, as given by P. Deligne in his fundamental paper "La conjecture de Weil II". The authors follow the important and beautiful methods of Laumon and Brylinski which lead to a simplification of Deligne's theory. Deligne's work is closely related to the sheaf theoretic theory of perverse sheaves. In this framework Deligne's results on global weights and his notion of purity of complexes obtain a satisfactory and final form. Therefore the authors include the complete theory of middle perverse sheaves. In this part, the l-adic Fourier transform is introduced as a technique providing natural and simple proofs. To round things off, there are three chapters with significant applications of these theories. Language: English ISBN: 9783540414575 Category: Mathematics

Zero The Biography of a Dangerous Idea Charles Seife

Viking (2000)

Summary: "Zero follows the number zero from its birth as an Eastern philosophical concept to its struggle for acceptance in Europe and its apotheosis as the mystery of a black hole. Here are the legendary thinkers who battled over the meaning of this mysterious number - scholars and mystics, cosmologists and clergymen whose clashes over zero shook the foundations of philosophy, science, mathematics, and religion." "Charles Seife's account takes us from Aristotle to superstring theory by way of Pythagoras, Descartes, the Kabbalists, and Einstein. It is a concise tour of a universe of ideas bound up in the simple notion of nothing."--BOOK JACKET.Title Summary field provided by Blackwell North America, Inc. All Rights Reserved Language: English ISBN: 9780670884575 Category: History of Mathematics

Hyperfunctions on Hypo-analytic Manifolds Paulo D. Cordaro, Francois Treves

Princeton University Press (1994)

Summary: In the first two chapters of this book, the reader will find a complete and systematic exposition of the theory of hyperfunctions on totally real submanifolds of multidimensional complex space, in particular of hyperfunction theory in real space. The book provides precise definitions of the hypo-analytic wave-front set and of the Fourier-Bros-Iagolnitzer transform of a hyperfunction. These are used to prove a very general version of the famed Theorem of the Edge of the Wedge. The last two chapters define the hyperfunction solutions on a general (smooth) hypo-analytic manifold, of which particular examples are the real analytic manifolds and the embedded CR manifolds. The main results here are the invariance of the spaces of hyperfunction solutions and the transversal smoothness of every hyperfunction solution. From this follows the uniqueness of solutions in the Cauchy problem with initial data on a maximally real submanifold, and the fact that the support of any solution is the union of orbits of the structure. Series: Annals of Mathematics Studies Volume: 136 Language: English ISBN: 9780691029924 Category: Mathematics

Etale Cohomology Of Rigid Analytic Varieties And Adic Spaces Roland Huber

Vieweg+teubner Verlag (2013)

Series: Aspects of Mathematics ISBN: 9783663099925 Category: Mathematics

Spectral Methods in Linear Transport Theory Operator Theory: Advances and Applications H. G. Kaper, Cornelis Gerrit Lekkerkerker, Johann Hejtmanek

Birkhäuser Verlag (1982)

Series: Birkhäuser Volume: 5 Language: English ISBN: 9783764313722 Category: Mathematics

A course on set theory Ernest Schimmerling

Cambridge University Press (2011)

Series: Cambridge studies in advanced mathematics Language: English ISBN: 9781107400481 Category: Mathematics

Finite Group Theory M. Aschbacher

Cambridge University Press (June 26, 2000)

Summary: During the last 40 years the theory of finite groups has developed dramatically. The finite simple groups have been classified and are becoming better understood. Tools exist to reduce many questions about arbitrary finite groups to similar questions about simple groups. Since the classification there have been numerous applications of this theory in other branches of mathematics. Finite Group Theory develops the foundations of the theory of finite groups. It can serve as a text for a course on finite groups for students already exposed to a first course in algebra. It could supply the background necessary to begin reading journal articles in the field. For specialists it also provides a reference on the foundations of the subject. This second edition has been considerably improved with a completely rewritten Chapter 15 considering the 2-Signalizer Functor Theorem, and the addition of an appendix containing solutions to exercises. Series: Cambridge studies in advanced mathematics Volume: 10 Language: English ISBN: 9780521786751 Category: Mathematics

An Introduction to Homological Algebra Charles A. Weibel

Cambridge University Press (October 27, 1995)

Summary: The landscape of homological algebra has evolved over the past half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras is also described. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras illustrate these topics. Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras. The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors. Series: Cambridge studies in advanced mathematics Volume: 38 Language: English ISBN: 9780521559874 Category: Mathematics

Hodge Theory and Complex Algebraic Geometry I: Claire Voisin

Cambridge University Press (December 20, 2007)

Summary: This is a modern introduction to Kaehlerian geometry and Hodge structure. Coverage begins with variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory (with the latter being treated in a more theoretical way than is usual in geometry). The book culminates with the Hodge decomposition theorem. In between, the author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The second part of the book investigates the meaning of these results in several directions. Series: Cambridge studies in advanced mathematics Volume: 76 Language: English ISBN: 9780521718011 Category: Mathematics

Central Simple Algebras and Galois Cohomology Philippe Gille, Tamás Szamuely

Cambridge University Press (August 10, 2006)

Summary: This book is the first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields. Starting from the basics, it reaches such advanced results as the Merkurjev-Suslin theorem. This theorem is both the culmination of work initiated by Brauer, Noether, Hasse and Albert and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, but no homological algebra, the book covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi-Brauer varieties, residue maps and, finally, Milnor K-theory and K-cohomology. The last chapter rounds off the theory by presenting the results in positive characteristic, including the theorem of Bloch-Gabber-Kato. The book is suitable as a textbook for graduate students and as a reference for researchers working in algebra, algebraic geometry or K-theory. Series: Cambridge studies in advanced mathematics Volume: 101 Language: English ISBN: 9780521861038 Category: Mathematics

An Introduction to Lie Groups and Lie Algebras Alexander Kirillov

Cambridge University Press (July 31, 2008)

Summary: This classic graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semisimple Lie algebras. Lie theory, in its own right, has become regarded as a classical branch of mathematics. Written in an informal style, this is a contemporary introduction to the subject which emphasizes the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of New York at Stony Brook, the book includes numerous exercises and worked examples and is ideal for graduate courses on Lie groups and Lie algebras. Series: Cambridge studies in advanced mathematics Volume: 113 Language: English ISBN: 9780521889698 Category: Mathematics

Introduction to Model Spaces and their Operators Stephan Ramon Garcia, Javad Mashreghi, William T. Ross

Cambridge University Press (May 17, 2016)

Summary: The study of model spaces, the closed invariant subspaces of the backward shift operator, is a vast area of research with connections to complex analysis, operator theory and functional analysis. This self-contained text is the ideal introduction for newcomers to the field. It sets out the basic ideas and quickly takes the reader through the history of the subject before ending up at the frontier of mathematical analysis. Open questions point to potential areas of future research, offering plenty of inspiration to graduate students wishing to advance further. Series: Cambridge studies in advanced mathematics Volume: 148 Language: English ISBN: 9781107108745 Category: Mathematics

Optimal Control and Geometry: Integrable Systems Velimir Jurdjevic

Cambridge University Press (July 4, 2016)

Summary: The synthesis of symplectic geometry, the calculus of variations and control theory offered in this book provides a crucial foundation for the understanding of many problems in applied mathematics. Focusing on the theory of integrable systems, this book introduces a class of optimal control problems on Lie groups, whose Hamiltonians, obtained through the Maximum Principle of optimality, shed new light on the theory of integrable systems. These Hamiltonians provide an original and unified account of the existing theory of integrable systems. The book particularly explains much of the mystery surrounding the Kepler problem, the Jacobi problem and the Kovalevskaya Top. It also reveals the ubiquitous presence of elastic curves in integrable systems up to the soliton solutions of the non-linear Schroedinger's equation. Containing a useful blend of theory and applications, this is an indispensable guide for graduates and researchers in many fields, from mathematical physics to space control. Series: Cambridge studies in advanced mathematics Volume: 154 Language: English ISBN: 9781107113886 Category: Mathematics

The Three-Dimensional Navier-Stokes Equations Classical Theory James C. Robinson, José L. Rodrigo, Witold Sadowski

Cambridge University Press (September 7, 2016)

Summary: A rigorous but accessible introduction to the mathematical theory of the three-dimensional Navier-Stokes equations, this book provides self-contained proofs of some of the most significant results in the area, many of which can only be found in research papers. Highlights include the existence of global-in-time Leray-Hopf weak solutions and the local existence of strong solutions; the conditional local regularity results of Serrin and others; and the partial regularity results of Caffarelli, Kohn, and Nirenberg. Appendices provide background material and proofs of some 'standard results' that are hard to find in the literature. A substantial number of exercises are included, with full solutions given at the end of the book. As the only introductory text on the topic to treat all of the mainstream results in detail, this book is an ideal text for a graduate course of one or two semesters. It is also a useful resource for anyone working in mathematical fluid dynamics. Series: Cambridge studies in advanced mathematics Volume: 157 Language: English ISBN: 9781107019669 Category: Mathematics

Lectures on K3 Surfaces Daniel Huybrechts

Cambridge University Press (September 26, 2016)

Summary: K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi-Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin-Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers. Series: Cambridge studies in advanced mathematics Volume: 158 Language: English ISBN: 9781107153042 Category: Mathematics

Auxiliary Polynomials in Number Theory David Masser

Cambridge University Press (July 21, 2016)

Summary: This unified account of various aspects of a powerful classical method, easy to understand in its simplest forms, is illustrated by applications in several areas of number theory. As well as including diophantine approximation and transcendence, which were mainly responsible for its invention, the author places the method in a broader context by exploring its application in other areas, such as exponential sums and counting problems in both finite fields and the field of rationals. Throughout the book, the method is explained in a 'molecular' fashion, where key ideas are introduced independently. Each application is the most elementary significant example of its kind and appears with detailed references to subsequent developments, making it accessible to advanced undergraduates as well as postgraduate students in number theory or related areas. It provides over 700 exercises both guiding and challenging, while the broad array of applications should interest professionals in fields from number theory to algebraic geometry. Series: Cambridge Tracts in Mathematics Volume: 207 Language: English ISBN: 9781107061576 Category: Mathematics

Non-homogeneous Random Walks Lyapunov Function Methods for Near-Critical Stochastic Systems Mikhail Menshikov, Serguei Popov, Andrew Wade

Cambridge University Press (December 22, 2016)

Summary: Stochastic systems provide powerful abstract models for a variety of important real-life applications: for example, power supply, traffic flow, data transmission. They (and the real systems they model) are often subject to phase transitions, behaving in one way when a parameter is below a certain critical value, then switching behaviour as soon as that critical value is reached. In a real system, we do not necessarily have control over all the parameter values, so it is important to know how to find critical points and to understand system behaviour near these points. This book is a modern presentation of the 'semimartingale' or 'Lyapunov function' method applied to near-critical stochastic systems, exemplified by non-homogeneous random walks. Applications treat near-critical stochastic systems and range across modern probability theory from stochastic billiards models to interacting particle systems. Spatially non-homogeneous random walks are explored in depth, as they provide prototypical near-critical systems. Series: Cambridge Tracts in Mathematics Volume: 209 Language: English ISBN: 9781107026698 Category: Mathematics

Recent Trends in Orthogonal Polynomials and Approximation Theory International Workshop in Honor of Guillermo López Lagomasino's 60th Birthday, September 8-12, 2008, Universidad Carlos III de Madrid, Leganés, Spain Guillermo Lopez Lagomasino

American Mathematical Soc. (2010)

Summary: This volume contains invited lectures and selected contributions from the International Workshop on Orthogonal Polynomials and Approximation Theory, held at Universidad Carlos III de Madrid on September 8-12, 2008, and which honored Guillermo Lopez Lagomasino on his 60th birthday.This book presents the state of the art in the theory of Orthogonal Polynomials and Rational Approximation with a special emphasis on their applications in random matrices, integrable systems, and numerical quadrature. New results and methods are presented in the papers as well as a careful choice of open problems, which can foster interest in research in these mathematical areas. This volume also includes a brief account of the scientific contributions by Guillermo Lopez Lagomasino. Series: Contemporary Mathematics Volume: 507 Language: English ISBN: 9780821848036 Category: Mathematics

Mathematical Aspects of Quantization Center for Mathematics at Notre Dame, Summer School and Conference, May 31-June 10, 2011, Notre Dame University, Notre Dame, Indiana Sam Evens

American Mathematical Soc. (2012)

Summary: This book is a collection of expository articles from the Center of Mathematics at Notre Dame's 2011 program on quantization. Included are lecture notes from a summer school on quantization on topics such as the Cherednik algebra, geometric quantization, detailed proofs of Willwacher's results on the Kontsevich graph complex, and group- valued moment maps. This book also includes expository articles on quantization and automorphic forms, renormalization, Berezin-Toeplitz quantization in the complex setting, and the commutation of quantization with reduction, as well as an original article on derived Poisson brackets. The primary goal of this volume is to make topics in quantization more accessible to graduate students and researchers. Series: Contemporary Mathematics Volume: 583 Language: English ISBN: 9780821875735 Category: Mathematics

100 Great Problems of Elementary Mathematics Their History and Solution Heinrich Dörrie

Courier Corporation (1965)

Summary: Problems that beset Archimedes, Newton, Euler, Cauchy, Gauss, Monge and other greats, ready to challenge today's would-be problem solvers. Among them: How is a sundial constructed? How can you calculate the logarithm of a given number without the use of logarithm table? No advanced math is required. Includes 100 problems with proofs. Series: Dover Language: English ISBN: 9780486613482 Category: Mathematics

An Investigation of the Laws of Thought On which are Founded the Mathematical Theories of Logic and Probabilities George Boole

Dover Publications (1982)

Summary: A timeless introduction to the field and a landmark in symbolic logic, showing that classical logic can be treated algebraically. Series: Dover Language: Spanish ISBN: 9780486600284 Category: Mathematics

Counterexamples in Analysis Bernard R. Gelbaum, John M. H. Olmsted

Courier Corporation (June 4, 2003)

Summary: These counterexamples, arranged according to difficulty or sophistication, deal mostly with the part of analysis known as "real variables," starting at the level of calculus. The first half of the book concerns functions of a real variable; topics include the real number system, functions and limits, differentiation, Riemann integration, sequences, infinite series, uniform convergence, and sets and measure on the real axis. The second half, encompassing higher dimensions, examines functions of two variables, plane sets, area, metric and topological spaces, and function spaces. This volume contains much that will prove suitable for students who have not yet completed a first course in calculus, and ample material of interest to more advanced students of analysis as well as graduate students. 12 figures. Bibliography. Index. Errata. Series: Dover Language: English ISBN: 9780486428758 Category: Mathematics

Counterexamples in Topology Lynn Arthur Steen, J. Arthur Seebach

Courier Corporation (1995)

Summary: Over 140 examples, preceded by a succinct exposition of general topology and basic terminology. Each example treated as a whole. Over 25 Venn diagrams and charts summarize properties of the examples, while discussions of general methods of construction and change give readers insight into constructing counterexamples. Extensive collection of problems and exercises, correlated with examples. Bibliography. 1978 edition. Series: Dover Language: English ISBN: 9780486687353 Category: Mathematics

Elements of Abstract Algebra Allan Clark

Courier Corporation (1984)

Summary: This concise, readable, college-level text treats basic abstract algebra in remarkable depth and detail. An antidote to the usual surveys of structure, the book presents group theory, Galois theory, and classical ideal theory in a framework emphasizing proof of important theorems. Chapter I (Set Theory) covers the basics of sets. Chapter II (Group Theory) is a rigorous introduction to groups. It contains all the results needed for Galois theory as well as the Sylow theorems, the Jordan-Holder theorem, and a complete treatment of the simplicity of alternating groups. Chapter III (Field Theory) reviews linear algebra and introduces fields as a prelude to Galois theory. In addition there is a full discussion of the constructibility of regular polygons. Chapter IV (Galois Theory) gives a thorough treatment of this classical topic, including a detailed presentation of the solvability of equations in radicals that actually includes solutions of equations of degree 3 and 4 ― a feature omitted from all texts of the last 40 years. Chapter V (Ring Theory) contains basic information about rings and unique factorization to set the stage for classical ideal theory. Chapter VI (Classical Ideal Theory) ends with an elementary proof of the Fundamental Theorem of Algebraic Number Theory for the special case of Galois extensions of the rational field, a result which brings together all the major themes of the book. The writing is clear and careful throughout, and includes many historical notes. Mathematical proof is emphasized. The text comprises 198 articles ranging in length from a paragraph to a page or two, pitched at a level that encourages careful reading. Most articles are accompanied by exercises, varying in level from the simple to the difficult. Series: Dover Language: English ISBN: 9780486647258 Category: Mathematics

Excursions in Number Theory Charles Stanley Ogilvy, John Timothy Anderson

Courier Corporation (1988)

Summary: Challenging, accessible mathematical adventures involving prime numbers, number patterns, irrationals and iterations, calculating prodigies, and more. No special training is needed, just high school mathematics and an inquisitive mind. "A splendidly written, well selected and presented collection. I recommend the book unreservedly to all readers." — Martin Gardner. Series: Dover Language: English ISBN: 9780486257785 Category: Mathematics

Fundamentals of Number Theory William Judson LeVeque

Courier Corporation (1996)

Summary: This excellent textbook introduces the basics of number theory, incorporating the language of abstract algebra. A knowledge of such algebraic concepts as group, ring, field, and domain is not assumed, however; all terms are defined and examples are given — making the book self-contained in this respect. The author begins with an introductory chapter on number theory and its early history. Subsequent chapters deal with unique factorization and the GCD, quadratic residues, number-theoretic functions and the distribution of primes, sums of squares, quadratic equations and quadratic fields, diophantine approximation, and more. Included are discussions of topics not always found in introductory texts: factorization and primality of large integers, p-adic numbers, algebraic number fields, Brun's theorem on twin primes, and the transcendence of e, to mention a few. Readers will find a substantial number of well-chosen problems, along with many notes and bibliographical references selected for readability and relevance. Five helpful appendixes — containing such study aids as a factor table, computer-plotted graphs, a table of indices, the Greek alphabet, and a list of symbols — and a bibliography round out this well-written text, which is directed toward undergraduate majors and beginning graduate students in mathematics. No post-calculus prerequisite is assumed. 1977 edition. Series: Dover Language: English ISBN: 9780486689067 Category: Mathematics

Geometry A Comprehensive Course Daniel Pedoe

Courier Corporation (1970)

Summary: "A lucid and masterly survey." — Mathematics Gazette Professor Pedoe is widely known as a fine teacher and a fine geometer. His abilities in both areas are clearly evident in this self-contained, well-written, and lucid introduction to the scope and methods of elementary geometry. It covers the geometry usually included in undergraduate courses in mathematics, except for the theory of convex sets. Based on a course given by the author for several years at the University of Minnesota, the main purpose of the book is to increase geometrical, and therefore mathematical, understanding and to help students enjoy geometry. Among the topics discussed: the use of vectors and their products in work on Desargues' and Pappus' theorem and the nine-point circle; circles and coaxal systems; the representation of circles by points in three dimensions; mappings of the Euclidean plane, similitudes, isometries, mappings of the inversive plane, and Moebius transformations; projective geometry of the plane, space, and n dimensions; the projective generation of conics and quadrics; Moebius tetrahedra; the tetrahedral complex; the twisted cubic curve; the cubic surface; oriented circles; and introduction to algebraic geometry. In addition, three appendices deal with Euclidean definitions, postulates, and propositions; the Grassmann-Pluecker coordinates of lines in S3, and the group of circular transformations. Among the outstanding features of this book are its many worked examples and over 500 exercises to test geometrical understanding. Series: Dover Language: English ISBN: 9780486658124 Category: Mathematics

Introduction to Combinatorial Analysis John Riordan

Courier Corporation (December 13, 2002)

Summary: This introduction to combinatorial analysis defines the subject as "the number of ways there are of doing some well- defined operation." Chapter 1 surveys that part of the theory of permutations and combinations that finds a place in books on elementary algebra, which leads to the extended treatment of generation functions in Chapter 2, where an important result is the introduction of a set of multivariable polynomials.Chapter 3 contains an extended treatment of the principle of inclusion and exclusion which is indispensable to the enumeration of permutations with restricted position given in Chapters 7 and 8. Chapter 4 examines the enumeration of permutations in cyclic representation and Chapter 5 surveys the theory of distributions. Chapter 6 considers partitions, compositions, and the enumeration of trees and linear graphs.Each chapter includes a lengthy problem section, intended to develop the text and to aid the reader. These problems assume a certain amount of mathematical maturity. Equations, theorems, sections, examples, and problems are numbered consecutively in each chapter and are referred to by these numbers in other chapters. Series: Dover Language: English ISBN: 9780486425368 Category: Mathematics

Introduction to Graph Theory Richard J. Trudeau

Courier Corporation (1993)

Summary: A stimulating excursion into pure mathematics aimed at "the mathematically traumatized," but great fun for mathematical hobbyists and serious mathematicians as well. Requiring only high school algebra as mathematical background, the book leads the reader from simple graphs through planar graphs, Euler's formula, Platonic graphs, coloring, the genus of a graph, Euler walks, Hamilton walks, and a discussion of The Seven Bridges of Konigsberg. Exercises are included at the end of each chapter. "The topics are so well motivated, the exposition so lucid and delightful, that the book's appeal should be virtually universal . . . Every library should have several copies" — Choice. 1976 edition. Series: Dover Language: English ISBN: 9780486678702 Category: Mathematics

Introduction to Topology Bert Mendelson

Courier Corporation (1990)

Summary: Highly regarded for its exceptional clarity, imaginative and instructive exercises, and fine writing style, this concise book offers an ideal introduction to the fundamentals of topology. Originally conceived as a text for a one-semester course, it is directed to undergraduate students whose studies of calculus sequence have included definitions and proofs of theorems. The book's principal aim is to provide a simple, thorough survey of elementary topics in the study of collections of objects, or sets, that possess a mathematical structure.The author begins with an informal discussion of set theory in Chapter 1, reserving coverage of countability for Chapter 5, where it appears in the context of compactness. In the second chapter Professor Mendelson discusses metric spaces, paying particular attention to various distance functions which may be defined on Euclidean "n"-space and which lead to the ordinary topology.Chapter 3 takes up the concept of topological space, presenting it as a generalization of the concept of a metric space. Chapters 4 and 5 are devoted to a discussion of the two most important topological properties: connectedness and compactness. Throughout the text, Dr. Mendelson, a former Professor of Mathematics at Smith College, has included many challenging and stimulating exercises to help students develop a solid grasp of the material presented.Unabridged Dover (1990) republication of the edition published by Allyn and Bacon, Inc., Boston 1975. Series: Dover Language: English ISBN: 9780486663524 Category: Mathematics

Introductory Real Analysis A. N. Kolmogorov, S. V. Fomin

Courier Corporation (June 1, 1975)

Summary: Self-contained and comprehensive, this elementary introduction to real and functional analysis is readily accessible to those with background in advanced calculus. It covers basic concepts and introductory principles in set theory, metric spaces, topological and linear spaces, linear functionals and linear operators, and much more. 350 problems. 1970 edition. Series: Dover Language: English ISBN: 9780486612263 Category: Mathematics

Lie Algebras Nathan Jacobson

Courier Corporation (1979)

Summary: Definitive treatment covers split semi-simple Lie algebras, universal enveloping algebras, classification of irreducible modules, automorphisms, simple Lie algebras over an arbitrary field, and more. Classic handbook for researchers and students; useable in graduate courses or for self-study. Reader should have basic knowledge of Galois theory and the Wedderburn structure theory of associative algebras. Series: Dover Language: English ISBN: 9780486638324 Category: Mathematics

Linear Algebra and Matrix Theory Robert R. Stoll

Courier Corporation (October 17, 2012)

Summary: One of the best available works on matrix theory in the context of modern algebra, this text bridges the gap between ordinary undergraduate studies and completely abstract mathematics. The first five chapters treat topics important to economics, psychology, statistics, physics, and mathematics. Final chapters apply chiefly to students of engineering, physics, and advanced mathematics. Series: Dover Language: English ISBN: 9780486623184 Category: Mathematics

Linear groups with an exposition of the Galois field theory Leonard Eugene Dickson

B. G. Teubner (1901)

Series: Dover Language: English Category: Mathematics

Mathematical Recreations of Lewis Carroll Pillow Problems and a Tangled Tale Lewis Carroll

Courier Corporation (June 1, 1958)

Summary: Virtually unobtainable for many years, these two books by Lewis Carroll (C. L. Dodgson) have now been reprinted in their entirety for the pleasure of modern enthusiasts of mathematical puzzles. Written by the 19th-century mathematician who gave us Alice in Wonderland and Through the Looking Glass, they contain an unusual combination of wit and mathematical intricacy that will test your mathematical ingenuity and provide hours of stimulating entertainment.Pillow-Problems is one of the rarest of all Lewis Carroll's works. It contains 72 mathematical posers ranging from those that can be solved by arithmetic, simple algebra, or plane geometry, to those that require more advanced algebra, trigonometry, algebraical geometry, differential calculus, and transcendental probabilities. Both numerical answers and fully worked out solutions are given, each in a separate section so that you can test your methods of problem-solving even after you have looked up the answer to a problem.In A Tangled Tale, Carroll embodies some of his most perplexing mathematical puzzles in the ten knots or chapters of a delightful story that has all the charm and wit of his better-known works. The Tale was originally printed as a monthly magazine serial, and many readers sent in solutions to the problems that were posed in it. In the long Appendix to The Tale, which contains the answers and solutions to the problems, Carroll uses the answers sent in by readers as the basis for illuminating and entertaining discussions of the many wrong ways in which the problems can be attacked, as well as the right ways. Series: Dover Language: English ISBN: 9780486204932 Category: Mathematics

Symmetry Discovered Concepts and Applications in Nature and Science Joe Rosen

Courier Corporation (1975)

Summary: Symmetry provides an insight into the way nature works and is often used by scientists and technologists to help solve problems. Symmetry has numerous other applications as well — with more being discovered all the time in science, the arts and other fields of human endeavor. This classic work provides an excellent introduction to the basic concepts and terminology (including, optionally, group theory), as well as lucid discussions of geometric symmetry, other symmetries and appropriate symmetry, symmetry in nature, uses of symmetry in science and much more. Readers wishing to pursue specific topics will find many references that reflect the author's wide reading in the subject and his own obvious enthusiasm. For this edition, Dr. Rosen has provided a new preface, solutions to the problems, and an addendum to the bibliography. Series: Dover Language: English ISBN: 9780486294339 Category: Mathematics

Tables of Functions with Formulae and Curves Eugen Jahnke, Fritz Emde

New York (1945)

Series: Dover Language: English Category: Mathematics

The Divine Proportion A Study in Mathematical Beauty H. E. Huntley

Courier Corporation (1970)

Summary: Using simple mathematical formulas, most as basic as Pythagoras's theorem and requiring only a very limited knowledge of mathematics, Professor Huntley explores the fascinating relationship between geometry and aesthetics. Poetry, patterns like Pascal's triangle, philosophy, psychology, music, and dozens of simple mathematical figures are enlisted to show that the "divine proportion" or "golden ratio" is a feature of geometry and analysis which awakes answering echoes in the human psyche. When we judge a work of art aesthetically satisfying, according to his formulation, we are making it conform to a pattern whose outline is laid down in simple geometrical figures; and it is the analysis of these figures which forms the core of Professor Huntley's book. For the philosopher, scientist, poet, art historian, music listener, artist, as well as the general reader who wants to understand more about the fascinating properties of numbers, this is a beautifully written, exciting account of the search for a naturally manifested aesthetic that has occupied man since he first asked the question "why?" "This is a delightful book to read. . . . It wanders here and there through some of the most attractive byways of simple mathematics, returning always to the oddities and pleasures of the golden section. This is a browser's book ? a happy, untidy traveling or bedside book for those who know how to enjoy the charm of numbers and shapes." ? Dr. J. Bronowski, The Salk Institute. Series: Dover Language: English ISBN: 9780486222547 Category: Mathematics

The Moscow Puzzles 359 Mathematical Recreations Boris A. Kordemsky

Courier Corporation (April 10, 1992)

Summary: This is, quite simply, the best and most popular puzzle book ever published in the Soviet Union. Since its first appearance in 1956 there have been eight editions as well as translations from the original Russian into Ukrainian, Estonian, Lettish, and Lithuanian. Almost a million copies of the Russian version alone have been sold.Part of the reason for the book's success is its marvelously varied assortment of brainteasers ranging from simple "catch" riddles to difficult problems (none, however, requiring advanced mathematics). Many of the puzzles will be new to Western readers, while some familiar problems have been clothed in new forms. Often the puzzles are presented in the form of charming stories that provide non-Russian readers with valuable insights into contemporary Russian life and customs. In addition, Martin Gardner, former editor of the Mathematical Games Department, Scientific American, has clarified and simplified the book to make it as easy as possible for an English-reading public to understand and enjoy. He has been careful, moreover, to retain nearly all the freshness, warmth, and humor of the original.Lavishly illustrated with over 400 clear diagrams and amusing sketches, this inexpensive edition of the first English translation will offer weeks or even months of stimulating entertainment. It belongs in the library of every puzzlist or lover of recreational mathematics. Series: Dover Language: English ISBN: 9780486270784 Category: Mathematics

Partial Differential Equations Lawrence C. Evans American Mathematical Soc. (2010)

Summary: This is the second edition of the now definitive text on partial differential equations (PDE). It offers a comprehensive survey of modern techniques in the theoretical study of PDE with particular emphasis on nonlinear equations. Its wide scope and clear exposition make it a great text for a graduate course in PDE. For this edition, the author has made numerous changes, including a new chapter on nonlinear wave equations, more than 80 new exercises, several new sections, a significantly expanded bibliography. About the First Edition: I have used this book for both regular PDE and topics courses. It has a wonderful combination of insight and technical detail...Evans' book is evidence of his mastering of the field and the clarity of presentation (Luis Caffarelli, University of Texas) It is fun to teach from Evans' book. It explains many of the essential ideas and techniques of partial differential equations ...Every graduate student in analysis should read it. (David Jerison, MIT) I use Partial Differential Equations to prepare my students for their Topic exam, which is a requirement before starting working on their dissertation. The book provides an excellent account of PDE's ...I am very happy with the preparation it provides my students. (Carlos Kenig, University of Chicago) Evans' book has already attained the status of a classic. It is a clear choice for students just learning the subject, as well as for experts who wish to broaden their knowledge ...An outstanding reference for many aspects of the field. (Rafe Mazzeo, Stanford University. Series: Graduate Studies in Mathematics Volume: 19 Language: English ISBN: 9780821849743 Category: Mathematics

Function Theory of One Complex Variable Robert Everist Greene, Steven George Krantz

American Mathematical Society (2002)

Summary: Complex analysis is one of the most beautiful subjects that we learn as graduate students. Part of the joy comes from being able to arrive quickly at some real theorems. The fundamental techniques of complex variables are also used to solve real problems in neighbouring subjects, such as number theory or PDEs. Series: Graduate Studies in Mathematics Volume: 40 Language: English ISBN: 9780821829059 Category: Mathematics

Algebra Chapter 0 Paolo Aluffi

American Mathematical Soc. (2009)

Summary: Algebra: Chapter 0 is a self-contained introduction to the main topics of algebra, suitable for a first sequence on the subject at the beginning graduate or upper undergraduate level. The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics. A second feature consists of an emphasis on homological algebra: basic notions on complexes are presented as soon as modules have been introduced, and an extensive last chapter on homological algebra can form the basis for a follow-up introductory course on the subject. Approximately 1,000 exercises both provide adequate practice to consolidate the understanding of the main body of the text and offer the opportunity to explore many other topics, including applications to number theory and algebraic geometry. This will allow instructors to adapt the textbook to their specific choice of topics and provide the independent reader with a richer exposure to algebra. Many exercises include substantial hints, and navigation of the topics is facilitated by an extensive index and by hundreds of cross-references. Series: Graduate Studies in Mathematics Volume: 104 Language: English ISBN: 9780821847817 Category: Mathematics

Categories for the Working Mathematician Saunders Mac Lane

Springer Science & Business Media (April 17, 2013)

Summary: 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. Series: Graduate Texts in Mathematics Volume: 5 Language: English ISBN: 9781475747218 Category: Mathematics

A Course in Arithmetic J-P. Serre

Springer New York (November 7, 1996)

Summary: This book is divided into two parts. The first one is purely algebraic. Its objective is the classification of quadratic forms over the field of rational numbers (Hasse-Minkowski theorem). It is achieved in Chapter IV. The first three chapters contain some preliminaries: quadratic reciprocity law, p-adic fields, Hilbert symbols. Chapter V applies the preceding results to integral quadratic forms of discriminant ± I. These forms occur in various questions: modular functions, differential topology, finite groups. The second part (Chapters VI and VII) uses "analytic" methods (holomor phic functions). Chapter VI gives the proof of the "theorem on arithmetic progressions" due to Dirichlet; this theorem is used at a critical point in the first part (Chapter Ill, no. 2.2). Chapter VII deals with modular forms, and in particular, with theta functions. Some of the quadratic forms of Chapter V reappear here. The two parts correspond to lectures given in 1962 and 1964 to second year students at the Ecole Normale Superieure. A redaction of these lectures in the form of duplicated notes, was made by J.-J. Sansuc (Chapters I-IV) and J.-P. Ramis and G. Ruget (Chapters VI-VII). They were very useful to me; I extend here my gratitude to their authors. Series: Graduate Texts in Mathematics Volume: 7 Language: English ISBN: 9780387900407 Category: Mathematics

General Topology John L. Kelley

Springer Science & Business Media (June 27, 1975)

Summary: Aimed at graduate math students, this classic work is a systematic exposition of general topology and is intended to be a reference and a text. As a reference, it offers a reasonably complete coverage of the area, resulting in a more extended treatment than normally given in a course. As a text, the exposition in the earlier chapters proceeds at a pedestrian pace. A preliminary chapter covers those topics requisite to the main body of work. Series: Graduate Texts in Mathematics Volume: 27 Language: English ISBN: 9780387901251 Category: Mathematics

Algebraic Geometry Robin Hartshorne

Springer Science & Business Media (December 19, 1977)

Summary: Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. After receiving his Ph.D. from Princeton in 1963, Hartshorne became a Junior Fellow at Harvard, then taught there for several years. In 1972 he moved to California where he is now Professor at the University of California at Berkeley. He is the author of "Residues and Duality" (1966), "Foundations of Projective Geometry (1968), "Ample Subvarieties of Algebraic Varieties" (1970), and numerous research titles. His current research interest is the geometry of projective varieties and vector bundles. He has been a visiting professor at the College de France and at Kyoto University, where he gave lectures in French and in Japanese, respectively. Professor Hartshorne is married to Edie Churchill, educator and psychotherapist, and has two sons. He has travelled widely, speaks several foreign languages, and is an experienced mountain climber. He is also an accomplished amateur musician: he has played the flute for many years, and during his last visit to Kyoto he began studying the shakuhachi. Series: Graduate Texts in Mathematics Volume: 52 Language: English ISBN: 9780387902449 Category: Mathematics

Local Fields Jean-Pierre Serre

Springer New York (July 27, 1995)

Summary: The goal of this book is to present local class field theory from the cohomo logical point of view, following the method inaugurated by Hochschild and developed by Artin-Tate. This theory is about extensions-primarily abelian-of "local" (i.e., complete for a discrete valuation) fields with finite residue field. For example, such fields are obtained by completing an algebraic number field; that is one of the aspects of "localisation". The chapters are grouped in "parts". There are three preliminary parts: the first two on the general theory of local fields, the third on group coho mology. Local class field theory, strictly speaking, does not appear until the fourth part. Here is a more precise outline of the contents of these four parts: The first contains basic definitions and results on discrete valuation rings, Dedekind domains (which are their "globalisation") and the completion process. The prerequisite for this part is a knowledge of elementary notions of algebra and topology, which may be found for instance in Bourbaki. The second part is concerned with ramification phenomena (different, discriminant, ramification groups, Artin representation). Just as in the first part, no assumptions are made here about the residue fields. It is in this setting that the "norm" map is studied; I have expressed the results in terms of "additive polynomials" and of "multiplicative polynomials", since using the language of algebraic geometry would have led me too far astray. Series: Graduate Texts in Mathematics Volume: 67 Language: English ISBN: 9780387904245 Category: Mathematics

Algebra Thomas W. Hungerford

Springer Science & Business Media (February 14, 2003)

Summary: 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. Series: Graduate Texts in Mathematics Volume: 73 Language: English ISBN: 9780387905181 Category: Mathematics

Introduction to Elliptic Curves and Modular Forms Neal Koblitz

Springer Science & Business Media (April 29, 1993)

Summary: This textbook covers the basic properties of elliptic curves and modular forms, with emphasis on certain connections with number theory. The ancient "congruent number problem" is the central motivating example for most of the book. My purpose is to make the subject accessible to those who find it hard to read more advanced or more algebraically oriented treatments. At the same time I want to introduce topics which are at the forefront of current research. Down-to- earth examples are given in the text and exercises, with the aim of making the material readable and interesting to mathematicians in fields far removed from the subject of the book. With numerous exercises (and answers) included, the textbook is also intended for graduate students who have completed the standard first-year courses in real and complex analysis and algebra. Such students would learn applications of techniques from those courses. thereby solidifying their under standing of some basic tools used throughout mathematics. Graduate stu dents wanting to work in number theory or algebraic geometry would get a motivational, example-oriented introduction. In addition, advanced under graduates could use the book for independent study projects, senior theses, and seminar work. Series: Graduate Texts in Mathematics Volume: 97 Language: English ISBN: 9780387979663 Category: Mathematics

Representation Theory A First Course William Fulton, Joe Harris

Springer Science & Business Media (1991)

Summary: 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. Series: Graduate Texts in Mathematics Volume: 129 Language: English ISBN: 9780387974958 Category: Mathematics

Tensor Geometry The Geometric Viewpoint and its Uses Christopher T. J. Dodson, Timothy Poston

Springer Science & Business Media (November 23, 2009)

Summary: We have been very encouraged by the reactions of students and teachers using our book over the past ten years and so this is a complete retype in TEX, with corrections of known errors and the addition of a supplementary bibliography. Thanks are due to the Springer staff in Heidelberg for their enthusiastic sup port and to the typist, Armin Kollner for the excellence of the final result. Once again, it has been achieved with the authors in yet two other countries. November 1990 Kit Dodson Toronto, Canada Tim Poston Pohang, Korea Contents Introduction ...... XI O. Fundamental Not(at)ions ...... 1 1. Sets ...... 1 2. Functions ...... 6 3. Physical Background ...... 13 I. Real Vector Spaces ...... 18 1. Spaces ...... 18 Subspace geometry, components 2. Maps...... 24 Linearity, singularity, matrices 3. Operators ...... 31 Projections, eigenvalues, determinant, trace II. Affine Spaces ...... 43 1. Spaces ...... 43 Tangent vectors, parallelism, coordinates 2. Combinations of Points ...... 49 Midpoints, convexity 3. Maps...... 53 Linear parts, translations, components III. Dual Spaces ...... 57 1. Contours, Co- and Contravariance, Dual Basis ...... 57 IV. Metric Vector Spaces ...... 64 1. Metrics ...... 64 Basic geometry and examples, Lorentz geometry 2. Maps...... 76 Isometries, orthogonal projections and complements, adjoints 3. Coordinates ...... 83 Orthonormal bases Contents VIII 4. Diagonalising Symmetric Operators 92 Principal directions, isotropy V. Tensors and Multilinear Forms 98 1. Multilinear Forms ...... 98 Tensor Products, Degree, Contraction, Raising Indices VE Topological Vector Spaces ...... 114 1. Continuity ...... 114 Metrics, topologies, homeomorphisms 2. Limits ...... 125 Convergence and continuity 3. The Usual Topology ...... Series: Graduate Texts in Mathematics Volume: 130 Language: English ISBN: 9783540520184 Category: Mathematics

Algebraic Geometry A First Course Joe Harris

Springer Science & Business Media (September 17, 1992)

Summary: This book is intended to introduce students to algebraic geometry; to give them a sense of the basic objects considered, the questions asked about them, and the sort of answers one can expect to obtain. It thus emplasizes the classical roots of the subject. For readers interested in simply seeing what the subject is about, this avoids the more technical details better treated with the most recent methods. For readers interested in pursuing the subject further, this book will provide a basis for understanding the developments of the last half century, which have put the subject on a radically new footing. Based on lectures given at Brown and Harvard Universities, this book retains the informal style of the lectures and stresses examples throughout; the theory is developed as needed. The first part is concerned with introducing basic varieties and constructions; it describes, for example, affine and projective varieties, regular and rational maps, and particular classes of varieties such as determinantal varieties and algebraic groups. The second part discusses attributes of varieties, including dimension, smoothness, tangent spaces and cones, degree, and parameter and moduli spaces. Series: Graduate Texts in Mathematics Volume: 133 Language: English ISBN: 9780387977164 Category: Mathematics

Topology and Geometry Glen E. Bredon

Springer Science & Business Media (June 24, 1993)

Summary: 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. Series: Graduate Texts in Mathematics Volume: 139 Language: English ISBN: 9780387979267 Category: Mathematics

Commutative Algebra with a View Toward Algebraic Geometry David Eisenbud

Springer Science & Business Media (December 1, 2013)

Summary: Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algebraic geometry. To help beginners, the essential ideals from algebraic geometry are treated from scratch. Appendices on homological algebra, multilinear algebra and several other useful topics help to make the book relatively self- contained. Novel results and presentations are scattered throughout the text. Series: Graduate Texts in Mathematics Volume: 150 Language: English ISBN: 9781461253501 Category: Mathematics

Advanced Topics in the Arithmetic of Elliptic Curves Joseph H. Silverman

Springer New York (September 24, 1999)

Summary: In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted." I included a brief introduction to ten additional topics as an appendix to the first volume, with the tacit understanding that eventually there might be a second volume containing the details. You are now holding that second volume. it turned out that even those ten topics would not fit Unfortunately, into a single book, so I was forced to make some choices. The following material is covered in this book: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-Neron classification of special fibers, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions. Series: Graduate Texts in Mathematics Volume: 151 Language: English ISBN: 9780387943282 Category: Mathematics

An Introduction to Knot Theory W.B.Raymond Lickorish

Springer Science & Business Media (October 3, 1997)

Summary: This account is an introduction to mathematical knot theory, the theory of knots and links of simple closed curves in three-dimensional space. Knots can be studied at many levels and from many points of view. They can be admired as artifacts of the decorative arts and crafts, or viewed as accessible intimations of a geometrical sophistication that may never be attained. The study of knots can be given some motivation in terms of applications in molecular biology or by reference to paral lels in equilibrium statistical mechanics or quantum field theory. Here, however, knot theory is considered as part of geometric topology. Motivation for such a topological study of knots is meant to come from a curiosity to know how the ge ometry of three-dimensional space can be explored by knotting phenomena using precise mathematics. The aim will be to find invariants that distinguish knots, to investigate geometric properties of knots and to see something of the way they interact with more adventurous three-dimensional topology. The book is based on an expanded version of notes for a course for recent graduates in mathematics given at the University of Cambridge; it is intended for others with a similar level of mathematical understanding. In particular, a knowledge of the very basic ideas of the fundamental group and of a simple homology theory is assumed; it is, after all, more important to know about those topics than about the intricacies of knot theory. Series: Graduate Texts in Mathematics Volume: 175 Language: English ISBN: 9780387982540 Category: Mathematics

The Geometry of Schemes David Eisenbud, Joe Harris

Springer (January 25, 2000)

Summary: The theory of schemes is the foundation for algebraic geometry proposed and elaborated by Alexander Grothendieck and his co-workers. It has allowed major progress in classical areas of algebraic geometry such as invariant theory and the moduli of curves. It integrates algebraic number theory with algebraic geometry, fulfilling the dreams of earlier generations of number theorists. This integration has led to proofs of some of the major conjectures in number theory (Deligne's proof of the Weil Conjectures, Faltings' proof of the Mordell Conjecture). This book is intended to bridge the chasm between a first course in classical algebraic geometry and a technical treatise on schemes. It focuses on examples, and strives to show "what is going on" behind the definitions. There are many exercises to test and extend the reader's understanding. The prerequisites are modest: a little commutative algebra and an acquaintance with algebraic varieties, roughly at the level of a one-semester course. The book aims to show schemes in relation to other geometric ideas, such as the theory of manifolds. Some familiarity with these ideas is helpful, though not required. Series: Graduate Texts in Mathematics Volume: 197 Language: English ISBN: 9780387986371 Category: Mathematics

Algebra Serge Lang

Springer Science & Business Media (June 21, 2005)

Summary: "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." NOTICES OF THE AMS "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." MATHEMATICAL REVIEWS 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. Series: Graduate Texts in Mathematics Volume: 211 Language: English ISBN: 9780387953854 Category: Mathematics

The Arithmetic of Hyperbolic 3-Manifolds Colin Maclachlan, Alan W. Reid

Springer Science & Business Media (2003)

Summary: For the past 25 years, the Geometrization Program of Thurston has been a driving force for research in 3-manifold topology. This has inspired a surge of activity investigating hyperbolic 3-manifolds (and Kleinian groups), as these manifolds form the largest and least well-understood class of compact 3-manifolds. Familiar and new tools from diverse areas of mathematics have been utilized in these investigations, from topology, geometry, analysis, group theory, and from the point of view of this book, algebra and number theory. This book is aimed at readers already familiar with the basics of hyperbolic 3-manifolds or Kleinian groups, and it is intended to introduce them to the interesting connections with number theory and the tools that will be required to pursue them. While there are a number of texts which cover the topological, geometric and analytical aspects of hyperbolic 3-manifolds, this book is unique in that it deals exclusively with the arithmetic aspects, which are not covered in other texts. Colin Maclachlan is a Reader in the Department of Mathematical Sciences at the University of Aberdeen in Scotland where he has served since 1968. He is a former President of the Edinburgh Mathematical Society. Alan Reid is a Professor in the Department of Mathematics at The University of Texas at Austin. He is a former Royal Society University Research Fellow, Alfred P. Sloan Fellow and winner of the Sir Edmund Whittaker Prize from The Edinburgh Mathematical Society. Both authors have published extensively in the general area of discrete groups, hyperbolic manifolds and low-dimensional topology. Series: Graduate Texts in Mathematics Volume: 219 Language: English ISBN: 9780387983868 Category: Mathematics

An Introduction to Markov Processes Daniel W. Stroock

Springer Science & Business Media (March 30, 2005) Summary: To some extent, it would be accurate to summarize the contents of this book as an intolerably protracted description of what happens when either one raises a transition probability matrix P (i. e. , all entries (P)»j are n- negative and each row of P sums to 1) to higher and higher powers or one exponentiates R(P — I), where R is a diagonal matrix with non- negative entries. Indeed, when it comes right down to it, that is all that is done in this book. However, I, and others of my ilk, would take offense at such a dismissive characterization of the theory of Markov chains and processes with values in a countable state space, and a primary goal of mine in writing this book was to convince its readers that our offense would be warranted. The reason why I, and others of my persuasion, refuse to consider the theory here as no more than a subset of matrix theory is that to do so is to ignore the pervasive role that probability plays throughout. Namely, probability theory provides a model which both motivates and provides a context for what we are doing with these matrices. To wit, even the term "transition probability matrix" lends meaning to an otherwise rather peculiar set of hypotheses to make about a matrix. Series: Graduate Texts in Mathematics Volume: 230 Language: English ISBN: 9783540234517 Category: Mathematics

The Arithmetic of Dynamical Systems J.H. Silverman

Springer Science & Business Media (June 6, 2007)

Summary: This book is designed to provide a path for the reader into an amalgamation oftwo venerable areas ofmathematics, Dynamical Systems and Number Theory. Many of the motivating theorems and conjectures in the new subject of Arithmetic Dynamics may be viewed as the transposition ofclassical results in the theory ofDiophantine equations to the setting of discrete dynamical systems, especially to the iteration theory ofmaps on the projective line and other algebraic varieties. Although there is no precise dictionary connecting the two areas, the reader will gain a flavor of the correspondence from the following associations: Diophantine Equations Dynamical Systems rational and integral rational and integral points on varieties points in orbits torsion points on periodic and preperiodic abelian varieties points ofrational maps There are a variety of topics covered in this volume, but inevitably the choice reflects the author's tastes and interests. Many related areas that also fall under the heading ofarithmetic or algebraic dynamics have been omitted in order to keep the book to a manageable length. A brief list of some of these omitted topics may be found in the introduction. Online Resources The reader will find additonal material, references and errata at http://www. math. brown. ectu/-jhs/ADSHome. html Acknowledgments The author has consulted a great many sources in writing this book. Every attempt has been made to give proper attribution for all but the most standard results. Series: Graduate Texts in Mathematics Volume: 241 Language: English ISBN: 9780387699035 Category: Mathematics

Journal of Algebra Special Issue Celebrating the 70th Birthday of Bernd Fischer M. Aschbacher, G. Röhrle, F. Timmesfeld, et. al.

Elsevier (June 15, 2006)

Series: Journal of Algebra Volume: 300 Editor: Michel Broué Category: Mathematics Notes: Number 2

Journal of Algebra Special Issue in Honor of Susan Montgomery N. Andruskiewitsch, R. Guralnick, D. Radford, et. al.

Elsevier (December 1, 2010)

Series: Journal of Algebra Volume: 324 Editor: Michel Broué Category: Mathematics Notes: Issue 11

Cohomologie étale Seminaire de Geometrie Algebrique du Bois-Marie SGA 4 1/2 (Lecture Notes in Mathematics) Pierre Deligne

Springer-Verlag (March 21, 1977)

Series: Lecture Notes in Mathematics Language: French ISBN: 9783540080664 Category: Mathematics

Arithmetic Noncommutative Geometry Matilde Marcolli, Jurij Ivanovič Manin

American Mathematical Soc. (2005)

Summary: Arithmetic noncommutative geometry denotes the use of ideas and tools from the field of noncommutative geometry, to address questions and reinterpret in a new perspective results and constructions from number theory and arithmetic algebraic geometry. This general philosophy is applied to the geometry and arithmetic of modular curves and to the fibers at archimedean places of arithmetic surfaces and varieties. The main reason why noncommutative geometry can be expected to say something about topics of arithmetic interest lies in the fact that it provides the right framework in which the tools of geometry continue to make sense on spaces that are very singular and apparently very far from the world of algebraic varieties. This provides a way of refining the boundary structure of certain classes of spaces that arise in the context of arithmetic geometry, such as moduli spaces (of which modular curves are the simplest case) or arithmetic varieties (completed by suitable ``fibers at infinity''), by adding boundaries that are invisible to algebraic geometry, such as degenerations of elliptic curves to noncommutative tori. The text of the book is organized around series of invited lectures delivered by the author at various universities, and the results presented are based on work of the author in collaboration with Alain Connes, Katia Consani, Yuri Manin, and Niranjan Ramachandran. Series: Lecture Notes in Mathematics Volume: 36 Language: English ISBN: 9780821838334 Category: Mathematics

Local Newforms for GSp(4) Brooks Roberts, Ralf Schmidt

Springer Science & Business Media (August 20, 2007)

Summary: Local Newforms for GSp(4) describes a theory of new- and oldforms for representations of GSp(4) over a non- archimedean local field. This theory considers vectors fixed by the paramodular groups, and singles out certain vectors that encode canonical information, such as L-factors and epsilon-factors, through their Hecke and Atkin-Lehner eigenvalues. While there are analogies to the GL(2) case, this theory is novel and unanticipated by the existing framework of conjectures. An appendix includes extensive tables about the results and the representation theory of GSp(4). Series: Lecture Notes in Mathematics Volume: 1,918 Language: English ISBN: 9783540733232 Category: Mathematics

The Laplacian on a Riemannian Manifold An Introduction to Analysis on Manifolds Steven Rosenberg

Cambridge University Press (January 9, 1997)

Summary: This text on analysis on Riemannian manifolds is a thorough introduction to topics covered in advanced research monographs on Atiyah-Singer index theory. The main theme is the study of heat flow associated to the Laplacians on differential forms. This provides a unified treatment of Hodge theory and the supersymmetric proof of the Chern-Gauss- Bonnet theorem. In particular, there is a careful treatment of the heat kernel for the Laplacian on functions. The author develops the Atiyah-Singer index theorem and its applications (without complete proofs) via the heat equation method. Rosenberg also treats zeta functions for Laplacians and analytic torsion, and lays out the recently uncovered relation between index theory and analytic torsion. The text is aimed at students who have had a first course in differentiable manifolds, and the author develops the Riemannian geometry used from the beginning. There are over 100 exercises with hints. Series: London Math. Soc. Student Texts Volume: 31 Language: English ISBN: 9780521468312 Category: Mathematics

Memoirs of the American Mathematical Society Fritz Gesztesy

American Mathematical Society (1950)

Series: Memoirs of the AMS Volume: 563 Language: English ISBN: 9780821804063 Category: Mathematics

Elliptic Partial Differential Operators and Symplectic Algebra William Norrie Everitt, L. Markus (Lawrence)

American Mathematical Soc. (2003)

Summary: This investigation introduces a new description and classification for the set of all self-adjoint operators (not just those defined by differential boundary conditions) which are generated by a linear elliptic partial differential expression $A(\mathbf{x},D)=\sum_{0\,\leq\,\lefts\right\,\leq\,2m}a_{s} (\mathbf{x})D^{s}\;\text{for all}\;\mathbf{x}\in\Omega$ in a region $\Omega$, with compact closure $\overline{\Omega}$ and $C^{\infty}$-smooth boundary $\partial\Omega$, in Euclidean space $\mathbb{E}^{r}$ $(r\geq2).$ The order $2m\geq2$ and the spatial dimension $r\geq2$ are arbitrary. We assume that the coefficients $a_{s}\in C^{\infty}(\overline {\Omega})$ are complex-valued, except real for the highest order terms (where $\lefts\right =2m$) which satisfy the uniform ellipticity condition in $\overline{\Omega}$.In addition, $A(\cdot,D)$ is Lagrange symmetric so that the corresponding linear operator $A$, on its classical domain $D(A):=C_{0}^{\infty}(\Omega)\subset L_{2}(\Omega)$, is symmetric; for example the familiar Laplacian $\Delta$ and the higher order polyharmonic operators $\Delta^{m}$. Through the methods of complex symplectic algebra, which the authors have previously developed for ordinary differential operators, the Stone-von Neumann theory of symmetric linear operators in Hilbert space is reformulated and adapted to the determination of all self-adjoint extensions of $A$ on $D(A)$, by means of an abstract generalization of the Glazman-Krein-Naimark (GKN) Theorem.In particular the authors construct a natural bijective correspondence between the set $\{T\}$ of all such self-adjoint operators on domains $D(T)\supset D(A)$, and the set $\{\mathsf{L}\}$ of all complete Lagrangian subspaces of the boundary complex symplectic space $\mathsf{S}=D(T_{1}\,/\,D(T_{0})$, where $T_{0}$ on $D(T_{0})$ and $T_{1}$ on $D(T_{1})$ are the minimal and maximal operators, respectively, determined by $A$ on $D(A)\subset L_{2}(\Omega)$. In the case of the elliptic partial differential operator $A$, we verify $D(T_{0})=\overset{\text{o}}{W}{}^{2m}(\Omega)$ and provide a novel definition and structural analysis for $D(T_{1})=\overset{A}{W}{}^{2m}(\Omega)$, which extends the GKN-theory from ordinary differential operators to a certain class of elliptic partial differential operators.Thus the boundary complex symplectic space $\mathsf{S}=\overset{A} {W} {}^{2m}(\Omega)\,/\,\overset{\text{o}}{W}{}^{2m}(\Omega)$ effects a classification of all self-adjoint extensions of $A$ on $D(A)$, including those operators that are not specified by differential boundary conditions, but instead by global (i. e. non-local) generalized boundary conditions. The scope of the theory is illustrated by several familiar, and other quite unusual, self-adjoint operators described in special examples. An Appendix is attached to present the basic definitions and concepts of differential topology and functional analysis on differentiable manifolds. In this Appendix care is taken to list and explain all special mathematical terms and symbols - in particular, the notations for Sobolev Hilbert spaces and the appropriate trace theorems. An Acknowledgment and subject Index complete this memoir. Series: Memoirs of the AMS Volume: 770 Language: English ISBN: 9780821832356 Category: Mathematics

Galois Cohomology Jean-Pierre Serre

Springer Science & Business Media (October 23, 2001)

Summary: This volume is an English translation of "Cohomologie Galoisienne" . The original edition (Springer LN5, 1964) was based on the notes, written with the help of Michel Raynaud, of a course I gave at the College de France in 1962-1963. In the present edition there are numerous additions and one suppression: Verdier's text on the duality of profinite groups. The most important addition is the photographic reproduction of R. Steinberg's "Regular elements of semisimple algebraic groups", Publ. Math. LH.E.S., 1965. I am very grateful to him, and to LH.E.S., for having authorized this reproduction. Other additions include: - A proof of the Golod-Shafarevich inequality (Chap. I, App. 2). - The "resume de cours" of my 1991-1992 lectures at the College de France on Galois cohomology of k(T) (Chap. II, App.). - The "resume de cours" of my 1990-1991 lectures at the College de France on Galois cohomology of semisimple groups, and its relation with abelian cohomology, especially in dimension 3 (Chap. III, App. 2). The bibliography has been extended, open questions have been updated (as far as possible) and several exercises have been added. In order to facilitate references, the numbering of propositions, lemmas and theorems has been kept as in the original 1964 text. Jean- Pierre Serre Harvard, Fall 1996 Table of Contents Foreword ...... V Chapter I. Cohomology of profinite groups §1. Profinite groups ...... 3 ...... Series: Monographs in Mathematics Language: English ISBN: 9783540421924 Category: Mathematics

Generic Polynomials Constructive Aspects of the Inverse Galois Problem Christian U. Jensen, Arne Ledet, Noriko Yui

Cambridge University Press (December 9, 2002)

Summary: This book describes a constructive approach to the Inverse Galois problem. The main theme is an exposition of a family of "generic" polynomials for certain finite groups, which give all Galois extensions having the required group as their Galois group. The existence of such generic polynomials is discussed, and where they do exist, a detailed treatment of their construction is given. The book also introduces the notion of "generic dimension" to address the problem of the smallest number of parameters required by a generic polynomial. Series: MSRI Volume: 45 Language: English ISBN: 9780521819985 Category: Mathematics

Topology of Stratified Spaces Greg Friedman

Cambridge University Press (March 28, 2011)

Summary: Appearance of singularities is pervasive in many problems in topology, differential geometry, and algebraic geometry. This book concerns the study of singular spaces using techniques from a variety of areas of geometry and topology and the interactions among them. Expository chapters by well-known experts cover intersection homology, L2 cohomology and differential operators, topology of algebraic varieties, signatures and characteristic classes, mixed Hodge theory, and elliptic genera of singular complex and real algebraic varieties. The book concludes with a list of open problems. Series: MSRI Volume: 58 Language: English ISBN: 9780521191678 Category: Mathematics

Olga Taussky-Todd in memoriam Michael Aschbacher, Don Blasius, Dinakar Ramakrishnan

International Press (1997)

Series: Pacific Journal of Mathematics Language: English ISBN: 9781571460516 Category: Mathematics

Lectures on the spectrum of L [Gamma] Floyd L. Williams

Longman Scientific & Technical (1991)

Series: Pitman Research Notes in Mathematics Volume: 242 Language: English ISBN: 9780582068636 Category: Mathematics

Theorems and Problems in Functional Analysis A. A. Kirillov, A. D. Gvishiani

Springer Science & Business Media (December 6, 2012)

Summary: Even the simplest mathematical abstraction of the phenomena of reality the real line-can be regarded from different points of view by different mathematical disciplines. For example, the algebraic approach to the study of the real line involves describing its properties as a set to whose elements we can apply" operations," and obtaining an algebraic model of it on the basis of these properties, without regard for the topological properties. On the other hand, we can focus on the topology of the real line and construct a formal model of it by singling out its" continuity" as a basis for the model. Analysis regards the line, and the functions on it, in the unity of the whole system of their algebraic and topological properties, with the fundamental deductions about them obtained by using the interplay between the algebraic and topological structures. The same picture is observed at higher stages of abstraction. Algebra studies linear spaces, groups, rings, modules, and so on. Topology studies structures of a different kind on arbitrary sets, structures that give mathe matical meaning to the concepts of a limit, continuity, a neighborhood, and so on. Functional analysis takes up topological linear spaces, topological groups, normed rings, modules of representations of topological groups in topological linear spaces, and so on. Thus, the basic object of study in functional analysis consists of objects equipped with compatible algebraic and topological structures. Series: Problem Books in Mathematics Language: English ISBN: 9781461381532 Category: Mathematics

Publications Mathématiques No. 81 D. W. Masser and G. Wüstholz, J.-L. Waldspurger, Michel Talagrand, Wenzhi Luo and Peter Sarnak

Institut Des Hautes Études Scientifiques (1995)

Series: Publications Mathématiques de I’HÉS Volume: 81 Category: Mathematics

Stochastic calculus for finance Steven E. Shreve

Springer (2004)

Series: Springer Finance ISBN: 9781441923110 Category: Mathematics

Global Analysis Shiing-shen Chern, S. Smale, American Mathematical Society

AMS (1970)

Series: Symposium in Pure Mathematics, Berkeley, July 1968, Proceedings Volume: 14 Language: English Category: Mathematics

Global Analysis, II Shiing-shen Chern, S. Smale, American Mathematical Society

AMS (1970)

Series: Symposium in Pure Mathematics, Berkeley, July 1968, Proceedings Volume: 15 Language: English Category: Mathematics

Seiberg-Witten Gauge Theory Matilde Marcolli

Hindustan Book Agency (1999)

Series: Texts and Readings in Mathematics Volume: 17 Language: English ISBN: 9788185931227 Category: Mathematics

Mathematics abstracts in a room with many mirrors Peter John Hilton

Springer. (1997)

Series: Undergraduate Texts in Mathematics Language: English ISBN: 9783923923335 Category: Mathematics

Rational Points on Elliptic Curves Joseph H. Silverman, John T. Tate

Springer Science & Business Media (November 18, 1994)

Summary: In 1961 the second author deliv1lred a series of lectures at Haverford Col lege on the subject of "Rational Points on Cubic Curves. " These lectures, intended for junior and senior mathematics majors, were recorded, tran scribed, and printed in mimeograph form. Since that time they have been widely distributed as photocopies of ever decreasing legibility, and por tions have appeared in various textbooks (Husemoller [1], Chahal [1]), but they have never appeared in their entirety. In view of the recent inter est in the theory of elliptic curves for subjects ranging from cryptogra phy (Lenstra [1], Koblitz [2]) to physics (Luck-Moussa-Waldschmidt [1]), as well as the tremendous purely mathematical activity in this area, it seems a propitious time to publish an expanded version of those original notes suitable for presentation to an advanced undergraduate audience. We have attempted to maintain much of the informality of the orig inal Haverford lectures. Our main goal in doing this has been to write a textbook in a technically difficult field which is "readable" by the average undergraduate mathematics major. We hope we have succeeded in this goal. The most obvious drawback to such an approach is that we have not been entirely rigorous in all of our proofs. In particular, much of the foundational material on elliptic curves presented in Chapter I is meant to explain and convince, rather than to rigorously prove. Series: Undergraduate Texts in Mathematics Language: English ISBN: 9780387978253 Category: Mathematics

Number Fields Daniel A. Marcus

Springer New York (May 11, 1995)

Summary: Requiring no more than a basic knowledge of abstract algebra, this text presents the mathematics of number fields in a straightforward, "down-to-earth" manner. It thus avoids local methods, for example, and presents proofs in a way that highlights the important parts of the arguments. Readers are assumed to be able to fill in the details, which in many places are left as exercises. Series: Universitext Language: English ISBN: 9780387902791 Category: Mathematics