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The aim of these lecture notes is to provide a self-contained exposition of several fascinating formulas discovered by Srinivasa Ramanujan. Two central results in these notes are: (1) the evaluation of the RogersOCoRamanujan continued fraction OCo a result that convinced G H Hardy that Ramanujan was a OC mathematician of the highest classOCO, and (2) what G. H. Hardy called Ramanujan''s OC Most Beautiful IdentityOCO. This book covers a range of related results, such as several proofs of the famous RogersOCoRamanujan identities and a detailed account of Ramanujan''s congruences. It also covers a range of techniques in q-series."
The aim of these lecture notes is to provide a self-contained exposition of several fascinating formulas discovered by Srinivasa Ramanujan. Two central results in these notes are: (1) the evaluation of the Rogers–Ramanujan continued fraction — a result that convinced G H Hardy that Ramanujan was a “mathematician of the highest class”, and (2) what G. H. Hardy called Ramanujan's “Most Beautiful Identity”. This book covers a range of related results, such as several proofs of the famous Rogers–Ramanujan identities and a detailed account of Ramanujan's congruences. It also covers a range of techniques in q-series. Contents:Jacob's Triple Product IdentityThe Rogers-Ramanujan IdentitiesThe rogers-Ramanujan continued FractionFrom the “Most Beautiful Identity” to Ramanujan's Congruences Readership: Graduate students and researchers in number theory. Keywords:RogersâRamanujan Continued Fraction;RogersâRamanujan Identities;Ramanujan's âMost Beautiful Identityâ;Ramanujan CongruencesReviews: “A great strength of this book is that almost every major result is proven in several different ways, illustrating the breadth of approaches to q-series … This book is well written with results that are well motivated and clearly explained. It is an excellent and statisfying introduction to q-series.” Mathematical Reviews
$q$-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra by George E. Andrews Pdf
Integrates developments and related applications in $q$-series with a historical development of the field. This book develops important analytic topics (Bailey chains, integrals, and constant terms) and applications to additive number theory.
An Introduction to q-analysis by Warren P. Johnson Pdf
Starting from simple generalizations of factorials and binomial coefficients, this book gives a friendly and accessible introduction to q q-analysis, a subject consisting primarily of identities between certain kinds of series and products. Many applications of these identities to combinatorics and number theory are developed in detail. There are numerous exercises to help students appreciate the beauty and power of the ideas, and the history of the subject is kept consistently in view. The book has few prerequisites beyond calculus. It is well suited to a capstone course, or for self-study in combinatorics or classical analysis. Ph.D. students and research mathematicians will also find it useful as a reference.
Reading Frege's Grundgesetze by Richard G. Heck Jr. Pdf
Gottlob Frege's Grundgesetze der Arithmetik, or Basic Laws of Arithmetic, was intended to be his magnum opus, the book in which he would finally establish his logicist philosophy of arithmetic. But because of the disaster of Russell's Paradox, which undermined Frege's proofs, the more mathematical parts of the book have rarely been read. Richard G. Heck, Jr., aims to change that, and establish it as a neglected masterpiece that must be placed at the center of Frege's philosophy. Part I of Reading Frege's Grundgesetze develops an interpretation of the philosophy of logic that informs Grundgesetze, paying especially close attention to the difficult sections of Frege's book in which he discusses his notorious 'Basic Law V' and attempts to secure its status as a law of logic. Part II examines the mathematical basis of Frege's logicism, explaining and exploring Frege's formal arguments. Heck argues that Frege himself knew that his proofs could be reconstructed so as to avoid Russell's Paradox, and presents Frege's arguments in a way that makes them available to a wide audience. He shows, by example, that careful attention to the structure of Frege's arguments, to what he proved, to how he proved it, and even to what he tried to prove but could not, has much to teach us about Frege's philosophy.
The aim of this lecture notes is to provide a self-contained exposition of several fascinating formulas discovered by Srinivasa Ramanujan. Two central results in these notes are: (1) the evaluation of the Rogers–Ramanujan continued fraction — a result that convinced G H Hardy that Ramanujan was a “mathematician of the highest class”, and (2) what G H Hardy called Ramanujan's “Most Beautiful Identity”. This book covers a range of related results, such as several proofs of the famous Rogers–Ramanujan identities and a detailed account of Ramanujan's congruences. It also covers a range of techniques in q-series.
The Philosophy of Mathematics Today by Matthias Schirn Pdf
Representing the state of the art in the field of the philosophy of mathematics, this collection of 20 essays deals with fundamental issues, ranging from the nature of mathematical knowledge to sets and natural 'number'.
The unpredictable cosmic entity known only as Q has plagued Captain Jean-Luc Picard and the crew of the Starship Enterprise™ since their very first voyage together. But little was known of Q's mysterious past or of the unearthly realm from which he hails. Until now. A brilliant scientist may have found a way to breach the energy barrier surrounding the Milky Way galaxy, and the Enterprise is going to put it to the test. The last thing Captain Picard needs is a surprise visit from Q, but the omnipotent trickster has more in mind than his usual pranks. Kidnapping Picard, he takes the captain back through time to the moment the Q Continuum faced its greatest threat. Now Picard must learn Q's secrets -- or all of reality may perish!
Fundamentals of Group Theory provides a comprehensive account of the basic theory of groups. Both classic and unique topics in the field are covered, such as an historical look at how Galois viewed groups, a discussion of commutator and Sylow subgroups, and a presentation of Birkhoff’s theorem. Written in a clear and accessible style, the work presents a solid introduction for students wishing to learn more about this widely applicable subject area. This book will be suitable for graduate courses in group theory and abstract algebra, and will also have appeal to advanced undergraduates. In addition it will serve as a valuable resource for those pursuing independent study. Group Theory is a timely and fundamental addition to literature in the study of groups.
Essays on Frege's Basic Laws of Arithmetic by Philip A. Ebert,Marcus Rossberg Pdf
This volume is the first collective study of a foundational text in modern philosophy and logic, Gottlob Frege's Basic Laws of Arithmetic which appeared in two volumes in 1893 and 1903. Twenty-two Frege scholars discuss a wide range of philosophical and logical topics arising from Basic Lawsof Arithmetic, and demonstrate the technical and philosophical richness of the work. Their original contributions make vivid the importance of this magnum opus not just for Frege scholars but for the study of the history of logic, mathematics, and philosophy.
George E. Andrews 80 Years of Combinatory Analysis by Krishnaswami Alladi,Bruce C. Berndt,Peter Paule,James A. Sellers,Ae Ja Yee Pdf
This book presents a printed testimony for the fact that George Andrews, one of the world’s leading experts in partitions and q-series for the last several decades, has passed the milestone age of 80. To honor George Andrews on this occasion, the conference “Combinatory Analysis 2018” was organized at the Pennsylvania State University from June 21 to 24, 2018. This volume comprises the original articles from the Special Issue “Combinatory Analysis 2018 – In Honor of George Andrews’ 80th Birthday” resulting from the conference and published in Annals of Combinatorics. In addition to the 37 articles of the Andrews 80 Special Issue, the book includes two new papers. These research contributions explore new grounds and present new achievements, research trends, and problems in the area. The volume is complemented by three special personal contributions: “The Worlds of George Andrews, a daughter’s take” by Amy Alznauer, “My association and collaboration with George Andrews” by Krishna Alladi, and “Ramanujan, his Lost Notebook, its importance” by Bruce Berndt. Another aspect which gives this Andrews volume a truly unique character is the “Photos” collection. In addition to pictures taken at “Combinatory Analysis 2018”, the editors selected a variety of photos, many of them not available elsewhere: “Andrews in Austria”, “Andrews in China”, “Andrews in Florida”, “Andrews in Illinois”, and “Andrews in India”. This volume will be of interest to researchers, PhD students, and interested practitioners working in the area of Combinatory Analysis, q-Series, and related fields.
From Foundations to Philosophy of Mathematics by Joan Roselló Pdf
From Foundations to Philosophy of Mathematics provides an historical introduction to the most exciting period in the foundations of mathematics, starting with the discovery of the paradoxes of logic and set theory at the beginning of the twentieth century and continuing with the great foundational debate that took place in the 1920s. As a result of the efforts of several mathematicians and philosophers during this period to ground mathematics and to clarify its nature from a certain philosophical standpoint, the four main schools in the philosophy of mathematics that have largely dominated the twentieth century arose, namely, logicism, intuitionism, formalism and predicativism. It was due precisely to the insufficiencies of the first three foundational programs and the objections raised against them, that interest in Platonism was renewed in the 1940s, mainly by Gödel. Not only does this book pay special attention to the foundational programs of these philosophies of mathematics, but also to some technical accomplishments that were developed in close connection with them and have largely shaped our understanding of the nature of mathematics, such as Russell’s type theory, Zermelo’s set theory and Gödel’s incompleteness theorems. Finally, it also examines some current research programs that have been pursued in the last decades and have tried, at least to some extent, to show the feasibility of the foundational programs developed in the schools mentioned above. This is the case of neologicism, constructivism, and predicativist and finitist reductionism, this last one developed closely with the research program of reverse mathematics.
These proceedings cover all the presentations from this year's programme which was jam-packed with novel and application-driven presentations that will keep you informed about the TPE industry of today. You will be fully up-to-date on the all there is to know, gaining first hand insight on the latest TPE materials for the consumer electronic market, healthcare applications, right through to the key material requirements for automotive manufacturers.