Ramanujan s Lost Notebook

Author: George E. Andrews
Publisher: Springer Science & Business Media
ISBN: 9781461440819
Release Date: 2013-06-04
Genre: Mathematics

​​​​In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge, to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated, "Ramanujan's lost notebook." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony. This volume is the fourth of five volumes that the authors plan to write on Ramanujan’s lost notebook.​ In contrast to the first three books on Ramanujan's Lost Notebook, the fourth book does not focus on q-series. Most of the entries examined in this volume fall under the purviews of number theory and classical analysis. Several incomplete manuscripts of Ramanujan published by Narosa with the lost notebook are discussed. Three of the partial manuscripts are on diophantine approximation, and others are in classical Fourier analysis and prime number theory. Most of the entries in number theory fall under the umbrella of classical analytic number theory. Perhaps the most intriguing entries are connected with the classical, unsolved circle and divisor problems. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited." - MathSciNet Review from the first volume: "Andrews and Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society​

Ramanujan s Lost Notebook

Author: George E. Andrews
Publisher: Springer Science & Business Media
ISBN: 9781461440819
Release Date: 2013-06-04
Genre: Mathematics

​​​​In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge, to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated, "Ramanujan's lost notebook." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony. This volume is the fourth of five volumes that the authors plan to write on Ramanujan’s lost notebook.​ In contrast to the first three books on Ramanujan's Lost Notebook, the fourth book does not focus on q-series. Most of the entries examined in this volume fall under the purviews of number theory and classical analysis. Several incomplete manuscripts of Ramanujan published by Narosa with the lost notebook are discussed. Three of the partial manuscripts are on diophantine approximation, and others are in classical Fourier analysis and prime number theory. Most of the entries in number theory fall under the umbrella of classical analytic number theory. Perhaps the most intriguing entries are connected with the classical, unsolved circle and divisor problems. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited." - MathSciNet Review from the first volume: "Andrews and Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society​

Ramanujan s Notebooks

Author: Bruce C. Berndt
Publisher: Springer Science & Business Media
ISBN: 9781461210887
Release Date: 2012-12-06
Genre: Mathematics

Srinivasa Ramanujan is, arguably, the greatest mathematician that India has produced. His story is quite unusual: although he had no formal education inmathematics, he taught himself, and managed to produce many important new results. With the support of the English number theorist G. H. Hardy, Ramanujan received a scholarship to go to England and study mathematics. He died very young, at the age of 32, leaving behind three notebooks containing almost 3000 theorems, virtually all without proof. G. H. Hardy and others strongly urged that notebooks be edited and published, and the result is this series of books. This volume dealswith Chapters 1-9 of Book II; each theorem is either proved, or a reference to a proof is given.

Collected Papers of Srinivasa Ramanujan

Author: Srinivasa Ramanujan Aiyangar
Publisher: American Mathematical Soc.
ISBN: 9780821820766
Release Date: 1927
Genre: Mathematics

The influence of Ramanujan on number theory is without parallel in mathematics. His papers, problems, and letters have spawned a remarkable number of later results by many different mathematicians. Here, his 37 published papers, most of his first two and last letters to Hardy, the famous 58 problems submitted to the Journal of the Indian Mathematical Society, and the commentary of the original editors (Hardy, Seshu Aiyar and Wilson) are reprinted again, after having been unavailable for some time. In this printing of Ramanujan's collected papers, Bruce Berndt provides an annotated guide to Ramanujan's work and to the mathematics it inspired over the last three-quarters of a century. The historical development of ideas is traced in the commentary and by citations to the copious references. The editor has done the mathematical world a tremendous service that few others would be qualified to do.

Number Theory in the Spirit of Ramanujan

Author: Bruce C. Berndt
Publisher: American Mathematical Soc.
ISBN: 9780821841785
Release Date: 2006
Genre: Mathematics

Ramanujan is recognized as one of the great number theorists of the twentieth century. Here now is the first book to provide an introduction to his work in number theory. Most of Ramanujan's work in number theory arose out of $q$-series and theta functions. This book provides an introduction to these two important subjects and to some of the topics in number theory that are inextricably intertwined with them, including the theory of partitions, sums of squares and triangular numbers, and the Ramanujan tau function. The majority of the results discussed here are originally due to Ramanujan or were rediscovered by him. Ramanujan did not leave us proofs of the thousands of theorems he recorded in his notebooks, and so it cannot be claimed that many of the proofs given in this book are those found by Ramanujan. However, they are all in the spirit of his mathematics.The subjects examined in this book have a rich history dating back to Euler and Jacobi, and they continue to be focal points of contemporary mathematical research. Therefore, at the end of each of the seven chapters, Berndt discusses the results established in the chapter and places them in both historical and contemporary contexts. The book is suitable for advanced undergraduates and beginning graduate students interested in number theory.

The Lost Notebook and Other Unpublished Papers

Author: Srinivasa Ramanujan Aiyangar
Publisher: Springer
ISBN: 038718726X
Release Date: 1988
Genre: Fonctions thêta

The so-called Lost Notebook of S.R. Ramanujan was brought to light in 1976 as part of the Watson bequest, by G.E. Andrews with whose introduction this collection of unpublished manuscripts opens. A major portion of the Lost Notebook - really just 90 unpaginated sheets of work on q-series and other topics - is reproduced here in facsimile. Letters from Ramanujan to Hardy as well as various other sheets of seemingly related notes are then included, on topics including coefficients in the 1/q3 and 1/q2 problems and the mock theta functions. The next 180 pages consist of unpublished manuscripts of Ramanujan, including 28 pages from the 'Loose Papers` held in the Trinity College Library. Finally a number of interesting letters that were exchanged between Ramanujan, Littlewood, Hardy and Watson, with a bearing on Ramanujan's work are collected together here with other extracts and fragments.

Ramanujan

Author: Srinivasa Ramanujan Aiyangar
Publisher: American Mathematical Soc.
ISBN: 9780821804704
Release Date: 1995
Genre: Mathematics

The letters that Ramanujan wrote to G. H. Hardy on January 16 and February 27, 1913, are two of the most famous letters in the history of mathematics. These and other letters introduced Ramanujan and his remarkable theorems to the world and stimulated much research, especially in the 1920s and 1930s. This book brings together many letters to, from, and about Ramanujan. The letters came from the National Archives in Delhi, the Archives in the State of Tamil Nadu, and a variety of other sources. Helping to orient the reader is the extensive commentary, both mathematical and cultural, by Berndt and Rankin; in particular, they discuss in detail the history, up to the present day, of each mathematical result in the letters. Containing many letters that have never been published before, this book will appeal to those interested in Ramanujan's mathematics as well as those wanting to learn more about the personal side of his life. Ramanujan: Letters and Commentary was selected for the CHOICE list of Outstanding Academic Books for 1996.

Surveys in Number Theory

Author: Krishnaswami Alladi
Publisher: Springer Science & Business Media
ISBN: 9780387785103
Release Date: 2009-03-02
Genre: Mathematics

Number theory has a wealth of long-standing problems, the study of which over the years has led to major developments in many areas of mathematics. This volume consists of seven significant chapters on number theory and related topics. Written by distinguished mathematicians, key topics focus on multipartitions, congruences and identities (G. Andrews), the formulas of Koshliakov and Guinand in Ramanujan's Lost Notebook (B. C. Berndt, Y. Lee, and J. Sohn), alternating sign matrices and the Weyl character formulas (D. M. Bressoud), theta functions in complex analysis (H. M. Farkas), representation functions in additive number theory (M. B. Nathanson), and mock theta functions, ranks, and Maass forms (K. Ono), and elliptic functions (M. Waldschmidt).

Ramanujan

Author: Godfrey Harold Hardy
Publisher: Taylor & Francis US
ISBN: 0821820230
Release Date: 1999-01-01
Genre: Mathematics

Ramanujan occupies a unique place in analytic number theory. His formulas, identities, and calculations are still amazing three-quarters of a century after his death. Many of his discoveries seem to have appeared as if from the ether. His mentor and primary collaborator was the famous G. H. Hardy. Here, Hardy collects twelve of his own lectures on topics stemming from Ramanujan's life and work. The topics include partitions, hypergeometric series, Ramanujan's $\tau$-function and round numbers. Hardy was the first to recognize the brilliance of Ramanujan's ideas. As one of the great mathematicians of the time, it is fascinating to read Hardy's accounts of their importance and influence. The book concludes with a chapter by chapter overview written by Bruce C. Berndt. In this overview, Berndt gives references to current literature, developments since Hardy's original lectures, and background information on Ramanujan's research, including his unpublished papers.

Computers and Mathematics

Author: Erich Kaltofen
Publisher: Springer Science & Business Media
ISBN: 9781461396475
Release Date: 2012-12-06
Genre: Computers

Advances in computer technology have had a tremendous impact on mathematics in the last two decades. In June of 1989, an international conference was held at MIT, bringing together mathematicians and computer scientists, to survey the work that has been done in computational mathematics, to report recent results in this field, and to discuss research directions as well as educational issues. This book presents a fascinating collection of contributions on topics ranging from computational algebra, and parallel computing, to mathematics education. Mathematicians interested in the computational aspects of their discipline as well as computer scientists interested in mathematical applications will enjoy the integrative view provided by this book.

My Search for Ramanujan

Author: Ken Ono
Publisher: Springer
ISBN: 9783319255682
Release Date: 2016-04-20
Genre: Mathematics

"The son of a prominent Japanese mathematician who came to the United States after World War II, Ken Ono was raised on a diet of high expectations and little praise. Rebelling against his pressure-cooker of a life, Ken determined to drop out of high school to follow his own path. To obtain his father’s approval, he invoked the biography of the famous Indian mathematical prodigy Srinivasa Ramanujan, whom his father revered, who had twice flunked out of college because of his single-minded devotion to mathematics. Ono describes his rocky path through college and graduate school, interweaving Ramanujan’s story with his own and telling how at key moments, he was inspired by Ramanujan and guided by mentors who encouraged him to pursue his interest in exploring Ramanujan’s mathematical legacy. Picking up where others left off, beginning with the great English mathematician G.H. Hardy, who brought Ramanujan to Cambridge in 1914, Ono has devoted his mathematical career to understanding how in his short life, Ramanujan was able to discover so many deep mathematical truths, which Ramanujan believed had been sent to him as visions from a Hindu goddess. And it was Ramanujan who was ultimately the source of reconciliation between Ono and his parents. Ono’s search for Ramanujan ranges over three continents and crosses paths with mathematicians whose lives span the globe and the entire twentieth century and beyond. Along the way, Ken made many fascinating discoveries. The most important and surprising one of all was his own humanity."

The Legacy of Alladi Ramakrishnan in the Mathematical Sciences

Author: Krishnaswami Alladi
Publisher: Springer Science & Business Media
ISBN: 1441962638
Release Date: 2010-08-26
Genre: Mathematics

In the spirit of Alladi Ramakrishnan’s profound interest and contributions to three fields of science — Mathematics, Statistics, and Physics — this volume contains invited surveys and research articles from prominent members of these communities who also knew Ramakrishnan personally and greatly respected his influence in these areas of science. Historical photos, telegrams, and biographical narratives of Alladi Ramakrishnan’s illustrious career of special interest are included as well.

Development of Elliptic Functions According to Ramanujan

Author: K. Venkatachaliengar
Publisher: World Scientific
ISBN: 9789814366458
Release Date: 2012
Genre: Mathematics

This unique book provides an innovative and efficient approach to elliptic functions, based on the ideas of the great Indian mathematician Srinivasa Ramanujan. The original 1988 monograph of K Venkatachaliengar has been completely revised. Many details, omitted from the original version, have been included, and the book has been made comprehensive by notes at the end of each chapter. The book is for graduate students and researchers in Number Theory and Classical Analysis, as well for scholars and aficionados of Ramanujan's work. It can be read by anyone with some undergraduate knowledge of real and complex analysis.

The Lives of Animals

Author: J. M. Coetzee
Publisher: Princeton University Press
ISBN: 9781400883523
Release Date: 2016-09-20
Genre: Fiction

The idea of human cruelty to animals so consumes novelist Elizabeth Costello in her later years that she can no longer look another person in the eye: humans, especially meat-eating ones, seem to her to be conspirators in a crime of stupefying magnitude taking place on farms and in slaughterhouses, factories, and laboratories across the world. Costello's son, a physics professor, admires her literary achievements, but dreads his mother’s lecturing on animal rights at the college where he teaches. His colleagues resist her argument that human reason is overrated and that the inability to reason does not diminish the value of life; his wife denounces his mother’s vegetarianism as a form of moral superiority. At the dinner that follows her first lecture, the guests confront Costello with a range of sympathetic and skeptical reactions to issues of animal rights, touching on broad philosophical, anthropological, and religious perspectives. Painfully for her son, Elizabeth Costello seems offensive and flaky, but—dare he admit it?—strangely on target. In this landmark book, Nobel Prize–winning writer J. M. Coetzee uses fiction to present a powerfully moving discussion of animal rights in all their complexity. He draws us into Elizabeth Costello’s own sense of mortality, her compassion for animals, and her alienation from humans, even from her own family. In his fable, presented as a Tanner Lecture sponsored by the University Center for Human Values at Princeton University, Coetzee immerses us in a drama reflecting the real-life situation at hand: a writer delivering a lecture on an emotionally charged issue at a prestigious university. Literature, philosophy, performance, and deep human conviction—Coetzee brings all these elements into play. As in the story of Elizabeth Costello, the Tanner Lecture is followed by responses treating the reader to a variety of perspectives, delivered by leading thinkers in different fields. Coetzee’s text is accompanied by an introduction by political philosopher Amy Gutmann and responsive essays by religion scholar Wendy Doniger, primatologist Barbara Smuts, literary theorist Marjorie Garber, and moral philosopher Peter Singer, author of Animal Liberation. Together the lecture-fable and the essays explore the palpable social consequences of uncompromising moral conflict and confrontation.

Himalayan Tribal Tales

Author: Stuart H. Blackburn
Publisher: BRILL
ISBN: 9789004171336
Release Date: 2008
Genre: Social Science

This study of an oral tradition in northeast India is the first of its kind in this part of the eastern Himalayas. A comparative analysis reveals parallel stories in an area stretching from central Arunachal Pradesh into upland Southeast Asia and southwest China. The subject of the volume, the Apatanis, are a small population of Tibeto-Burman speakers who live in a narrow valley halfway between Tibet and Assam. Their origin myths, migration legends, oral histories, trickster tales and ritual chants, as well as performance contexts and genre system, reveal key cultural ideas and social practices, shifts in tribal identity and the reinvention of religion.