**Author**: George E. Andrews,Bruce C. Berndt

**Publisher:**Springer Science & Business Media

**ISBN:**1461440815

**Category:**Mathematics

**Page:**439

**View:**781

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# Search Results for: ramanujan-s-lost-notebook-part-i-pt-1

**Author**: George E. Andrews,Bruce C. Berndt

**Publisher:** Springer Science & Business Media

**ISBN:** 1461440815

**Category:** Mathematics

**Page:** 439

**View:** 781

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

**Author**: Bruce C. Berndt

**Publisher:** Springer Science & Business Media

**ISBN:** 9780387949413

**Category:** Mathematics

**Page:** 624

**View:** 1414

The fifth and final volume to establish the results claimed by the great Indian mathematician Srinivasa Ramanujan in his "Notebooks" first published in 1957. Although each of the five volumes contains many deep results, the average depth in this volume is possibly greater than in the first four. There are several results on continued fractions - a subject that Ramanujan loved very much. It is the authors wish that this and previous volumes will serve as springboards for further investigations by mathematicians intrigued by Ramanujans remarkable ideas.

**Author**: Srinivasa Ramanujan Aiyangar,Godfrey Harold Hardy,P. Veṅkatesvara Seshu Aiyar,Bertram Martin Wilson

**Publisher:** American Mathematical Soc.

**ISBN:** 0821820761

**Category:** Mathematics

**Page:** 426

**View:** 1655

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.
*Letters and Commentary*

**Author**: Srinivasa Ramanujan Aiyangar,Bruce C. Berndt,Robert Alexander Rankin

**Publisher:** American Mathematical Soc.

**ISBN:** 0821804707

**Category:** Mathematics

**Page:** 347

**View:** 6566

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.

**Author**: Bruce C. Berndt

**Publisher:** American Mathematical Soc.

**ISBN:** 0821841785

**Category:** Mathematics

**Page:** 187

**View:** 4884

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.
*A Life of the Genius Ramanujan*

**Author**: Robert Kanigel

**Publisher:** Simon and Schuster

**ISBN:** 1476763496

**Category:** Biography & Autobiography

**Page:** 464

**View:** 7823

A biography of the Indian mathematician Srinivasa Ramanujan. The book gives a detailed account of his upbringing in India, his mathematical achievements, and his mathematical collaboration with English mathematician G. H. Hardy. The book also reviews the life of Hardy and the academic culture of Cambridge University during the early twentieth century.

**Author**: M. Ram Murty,V. Kumar Murty

**Publisher:** Springer Science & Business Media

**ISBN:** 8132207696

**Category:** Mathematics

**Page:** 188

**View:** 6548

Srinivasa Ramanujan was a mathematician brilliant beyond comparison who inspired many great mathematicians. There is extensive literature available on the work of Ramanujan. But what is missing in the literature is an analysis that would place his mathematics in context and interpret it in terms of modern developments. The 12 lectures by Hardy, delivered in 1936, served this purpose at the time they were given. This book presents Ramanujan’s essential mathematical contributions and gives an informal account of some of the major developments that emanated from his work in the 20th and 21st centuries. It contends that his work still has an impact on many different fields of mathematical research. This book examines some of these themes in the landscape of 21st-century mathematics. These essays, based on the lectures given by the authors focus on a subset of Ramanujan’s significant papers and show how these papers shaped the course of modern mathematics.

**Author**: George E. Andrews,Bruce C. Berndt

**Publisher:** Springer Science & Business Media

**ISBN:** 1461440815

**Category:** Mathematics

**Page:** 439

**View:** 7221

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
*The Lives of Animals [Princeton Classics]*

**Author**: J. M. Coetzee

**Publisher:** Princeton University Press

**ISBN:** 1400883520

**Category:** Fiction

**Page:** 144

**View:** 9964

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.

**Author**: Krishnaswami Alladi,John R. Klauder,Calyampudi R. Rao

**Publisher:** Springer Science & Business Media

**ISBN:** 9781441962638

**Category:** Mathematics

**Page:** 575

**View:** 4704

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.
*How I Learned to Count*

**Author**: Ken Ono,Amir D. Aczel

**Publisher:** Springer

**ISBN:** 3319255681

**Category:** Mathematics

**Page:** 238

**View:** 4831

"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."
*Twelve Lectures on Subjects Suggested by His Life and Work*

**Author**: Godfrey Harold Hardy

**Publisher:** Taylor & Francis US

**ISBN:** 9780821820230

**Category:** Mathematics

**Page:** 254

**View:** 5509

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.
*5th International Workshop, DAS 2002, Princeton, NJ, USA, August 19-21, 2002. Proceedings*

**Author**: Daniel Lopresti,Jianying Hu,Ramanusan Kashi

**Publisher:** Springer Science & Business Media

**ISBN:** 3540440682

**Category:** Computers

**Page:** 574

**View:** 9569

**Author**: Ravindra B. Bapat,Steve J. Kirkland,K. Manjunatha Prasad,Simo Puntanen

**Publisher:** Springer Science & Business Media

**ISBN:** 8132210530

**Category:** Mathematics

**Page:** 277

**View:** 2723

This book consists of eighteen articles in the area of `Combinatorial Matrix Theory' and `Generalized Inverses of Matrices'. Original research and expository articles presented in this publication are written by leading Mathematicians and Statisticians working in these areas. The articles contained herein are on the following general topics: `matrices in graph theory', `generalized inverses of matrices', `matrix methods in statistics' and `magic squares'. In the area of matrices and graphs, speci_c topics addressed in this volume include energy of graphs, q-analog, immanants of matrices and graph realization of product of adjacency matrices. Topics in the book from `Matrix Methods in Statistics' are, for example, the analysis of BLUE via eigenvalues of covariance matrix, copulas, error orthogonal model, and orthogonal projectors in the linear regression models. Moore-Penrose inverse of perturbed operators, reverse order law in the case of inde_nite inner product space, approximation numbers, condition numbers, idempotent matrices, semiring of nonnegative matrices, regular matrices over incline and partial order of matrices are the topics addressed under the area of theory of generalized inverses. In addition to the above traditional topics and a report on CMTGIM 2012 as an appendix, we have an article on old magic squares from India.

**Author**: David H. Bailey,Jonathan Borwein,Neil Calkin,Russell Luke,Roland Girgensohn,Victor Moll

**Publisher:** CRC Press

**ISBN:** 1439864330

**Category:** Mathematics

**Page:** 337

**View:** 9447

With the continued advance of computing power and accessibility, the view that "real mathematicians don't compute" no longer has any traction for a newer generation of mathematicians. The goal in this book is to present a coherent variety of accessible examples of modern mathematics where intelligent computing plays a significant role and in so doing to highlight some of the key algorithms and to teach some of the key experimental approaches.

**Author**: Christof Teuscher

**Publisher:** Springer Science & Business Media

**ISBN:** 3662056429

**Category:** Computers

**Page:** 542

**View:** 3948

Written by a distinguished cast of contributors, Alan Turing: Life and Legacy of a Great Thinker is the definitive collection of essays in commemoration of the 90th birthday of Alan Turing. This fascinating text covers the rich facets of his life, thoughts, and legacy, but also sheds some light on the future of computing science with a chapter contributed by visionary Ray Kurzweil, winner of the 1999 National Medal of Technology. Further, important contributions come from the philosopher Daniel Dennett, the Turing biographer Andrew Hodges, and from the distinguished logician Martin Davis, who provides a first critical essay on an emerging and controversial field termed "hypercomputation".

**Author**: Shaun Cooper

**Publisher:** Springer

**ISBN:** 3319561723

**Category:** Mathematics

**Page:** 687

**View:** 3092

Theta functions were studied extensively by Ramanujan. This book provides a systematic development of Ramanujan’s results and extends them to a general theory. The author’s treatment of the subject is comprehensive, providing a detailed study of theta functions and modular forms for levels up to 12. Aimed at advanced undergraduates, graduate students, and researchers, the organization, user-friendly presentation, and rich source of examples, lends this book to serve as a useful reference, a pedagogical tool, and a stimulus for further research. Topics, especially those discussed in the second half of the book, have been the subject of much recent research; many of which are appearing in book form for the first time. Further results are summarized in the numerous exercises at the end of each chapter.
*Selected Writings*

**Author**: Jonathan M. Borwein

**Publisher:** PSIpress

**ISBN:** 193563805X

**Category:** Experimental mathematics

**Page:** 297

**View:** 1954

A quiet revolution in mathematical computing and scientific visualization took place in the latter half of the 20th century. These developments have dramatically enhanced modes of mathematical insight and opportunities for "exploratory" computational experimentation. This volume collects the experimental and computational contributions of Jonathan and Peter Borwein over the past quarter century.

**Author**: Jane Piirto

**Publisher:** Springer Science & Business Media

**ISBN:** 9460914632

**Category:** Education

**Page:** 195

**View:** 4206

VERY practical, on target for schools today—good balance of theory with anecdotal connections.” “At first I was worried about the time involved. I discovered when given 5 minutes . . . the time is a continuation to their work in progress. Realizing that creativity does not have to consume large chunks of time is more meaningful than tokens.” “I like the tone of the writing. It feels like there is a conversation going on.” “I like the stories of famous people and how their creativity influenced and changed their lives.” CREATIVITY FOR 21ST CENTURY SKILLS describes what many creative people really do when they create. It focuses on the practical applications of a theoretical approach to creativity training the author has developed. Many suggestions for enhancing creativity focus on ideas that are over 60 years old. This new approach may be helpful for those seeking to develop 21st Century Skills of creativity. Five core attitudes (Naiveté, Risk-taking, Self-Discipline, Tolerance for Ambiguity, and Group Trust), Seven I’s (Inspiration, Intuition, Improvisation, Imagination, Imagery, Incubation, and Insight), and several General Practices—the use of ritual, meditation, solitude, exercise, silence, and a creative attitude to the process of life, with corresponding activities, are described, discussed, and illustrated. A discussion of how to be creative within an educational institution is also included. JANE PIIRTO is Trustees’ Distinguished Professor at Ashland University. Her doctorate is in educational leadership. She has worked with students pre-K to doctoral level as a teacher, administrator, and professor. She has published 11 books, both literary and scholarly, and many scholarly articles in peer-reviewed journals and anthologies, as well as several poetry and creative nonfiction chapbooks. She has won Individual Artist Fellowships from the Ohio Arts Council in both poetry and fiction and is one of the few American writers listed as both a poet and a writer in the Directory of American Poets and Writers. She is a recipient of the Mensa Lifetime Achievement Award, of an honorary Doctor of Humane Letters, was named an Ohio Magazine educator of distinction. In 2010 she was named Distinguished Scholar by the National Association for Gifted Children.

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