{"title":"Freeman Dyson FRS (1923–2020)","authors":"J. Dainton","doi":"10.1098/rsta.2020.0139","DOIUrl":"https://doi.org/10.1098/rsta.2020.0139","url":null,"abstract":"The editorial staff of Philosophical Transactions of the Royal Society A are saddened to hear of the death of one of its most distinguished members, Freeman Dyson, on 28 February 2020. Prof. Dyson served as a member of the Phil Trans A Editorial Board during the years 2004–2009. Prof. Dyson’s remarkable career of more than 70 years played a huge part in the golden age of science and technology which followed World War 2 (WW2). Throughout, Dyson’s towering intellect led to achievements which ranged over many diverse topics in mathematics and theoretical physics. He also turned his attention to issues concerned with the politics of science and technology, philosophy, education and even the place of religion in life, often bringing together strands from each to influence important aspects of social issues of the moment.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91305742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Coupling in complex systems as information transfer across time scales","authors":"M. Paluš","doi":"10.1098/rsta.2019.0094","DOIUrl":"https://doi.org/10.1098/rsta.2019.0094","url":null,"abstract":"Complex systems such as the human brain or the Earth's climate consist of many subsystems interacting in intricate, nonlinear ways. Moreover, variability of such systems extends over broad ranges of spatial and temporal scales and dynamical phenomena on different scales also influence each other. In order to explain how to detect cross-scale causal interactions, we review information-theoretic formulation of the Granger causality in combination with computational statistics (surrogate data method) and demonstrate how this method can be used to infer driver-response relations from amplitudes and phases of coupled nonlinear dynamical systems. Considering complex systems evolving on multiple time scales, the reviewed methodology starts with a wavelet decomposition of a multi-scale signal into quasi-oscillatory modes of a limited bandwidth, described using their instantaneous phases and amplitudes. Then statistical associations, in particular, causality relations between phases or between phases and amplitudes on different time scales are tested using the conditional mutual information. As an application, we present the analysis of cross-scale interactions and information transfer in the dynamics of the El Niño Southern Oscillation. This article is part of the theme issue ‘Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"100 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79499098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Ramanujan's legacy: the work of the SASTRA prize winners†","authors":"K. Alladi","doi":"10.1098/rsta.2018.0438","DOIUrl":"https://doi.org/10.1098/rsta.2018.0438","url":null,"abstract":"The SASTRA Ramanujan Prize is an annual $10 000 prize given to mathematicians not exceeding the age of 32 for revolutionary contributions to areas influenced by Srinivasa Ramanujan. The prize has been unusually successful in recognizing highly gifted mathematicians at an early stage of their careers who have gone on to shape the development of mathematics. We describe the fundamental contributions of the winners and the impact they have had on current research. Several aspects of the work of the awardees either stem from or have been strongly influenced by Ramanujan's ideas. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"344 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75942090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Selmer groups in Iwasawa theory and congruences","authors":"R. Sujatha","doi":"10.1098/rsta.2018.0442","DOIUrl":"https://doi.org/10.1098/rsta.2018.0442","url":null,"abstract":"This article outlines the behaviour of Iwasawa μ-invariants for Selmer groups of elliptic curves when the residual representations are equivalent. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"57 2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83411361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Srinivasa Ramanujan and signal-processing problems","authors":"P. Vaidyanathan, S. Tenneti","doi":"10.1098/rsta.2018.0446","DOIUrl":"https://doi.org/10.1098/rsta.2018.0446","url":null,"abstract":"The Ramanujan sum cq(n) has been used by mathematicians to derive many important infinite series expansions for arithmetic-functions in number theory. Interestingly, this sum has many properties which are attractive from the point of view of digital signal processing. One of these is that cq(n) is periodic with period q, and another is that it is always integer-valued in spite of the presence of complex roots of unity in the definition. Engineers and physicists have in the past used the Ramanujan-sum to extract periodicity information from signals. In recent years, this idea has been developed further by introducing the concept of Ramanujan-subspaces. Based on this, Ramanujan dictionaries and filter banks have been developed, which are very useful to identify integer-valued periods in possibly complex-valued signals. This paper gives an overview of these developments from the view point of signal processing. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86486119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Asymptotics and Ramanujan's mock theta functions: then and now","authors":"A. Folsom","doi":"10.1098/rsta.2018.0448","DOIUrl":"https://doi.org/10.1098/rsta.2018.0448","url":null,"abstract":"This article is in commemoration of Ramanujan's election as Fellow of The Royal Society 100 years ago, as celebrated at the October 2018 scientific meeting at the Royal Society in London. Ramanujan's last letter to Hardy, written shortly after his election, surrounds his mock theta functions. While these functions have been of great importance and interest in the decades following Ramanujan's death in 1920, it was unclear how exactly they fit into the theory of modular forms—Dyson called this ‘a challenge for the future’ at another centenary conference in Illinois in 1987, honouring the 100th anniversary of Ramanujan's birth. In the early 2000s, Zwegers finally recognized that Ramanujan had discovered glimpses of special families of non-holomorphic modular forms, which we now know to be Bruinier and Funke's harmonic Maass forms from 2004, the holomorphic parts of which are called mock modular forms. As of a few years ago, a fundamental question from Ramanujan's last letter remained, on a certain asymptotic relationship between mock theta functions and ordinary modular forms. The author, with Ono and Rhoades, revisited Ramanujan's asymptotic claim, and established a connection between mock theta functions and quantum modular forms, which were not defined until 90 years later in 2010 by Zagier. Here, we bring together past and present, and study the relationships between mock modular forms and quantum modular forms, with Ramanujan's mock theta functions as motivation. In particular, we highlight recent work of Bringmann–Rolen, Choi–Lim–Rhoades and Griffin–Ono–Rolen in our discussion. This article is largely expository, but not exclusively: we also establish a new interpretation of Ramanujan's radial asymptotic limits in the subject of topology. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83001737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Three-manifold quantum invariants and mock theta functions","authors":"Miranda C. N. Cheng, Francesca Ferrari, G. Sgroi","doi":"10.1098/rsta.2018.0439","DOIUrl":"https://doi.org/10.1098/rsta.2018.0439","url":null,"abstract":"Mock modular forms have found applications in numerous branches of mathematical sciences since they were first introduced by Ramanujan nearly a century ago. In this proceeding, we highlight a new area where mock modular forms start to play an important role, namely the study of three-manifold invariants. For a certain class of Seifert three-manifolds, we describe a conjecture on the mock modular properties of a recently proposed quantum invariant. As an illustration, we include concrete computations for a specific three-manifold, the Brieskorn sphere Σ(2, 3, 7). This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87750232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"How Ramanujan may have discovered the mock theta functions","authors":"G. Andrews","doi":"10.1098/rsta.2018.0436","DOIUrl":"https://doi.org/10.1098/rsta.2018.0436","url":null,"abstract":"Mock theta functions appeared out of the blue in Ramanujan's last letter to Hardy. What would lead Ramanujan to consider the possibility of such functions in the first place? This paper seeks to provide a plausible answer to this question. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"121 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82185976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Ramanujan conjecture and its applications","authors":"Wen-Ching Winnie Li","doi":"10.1098/rsta.2018.0441","DOIUrl":"https://doi.org/10.1098/rsta.2018.0441","url":null,"abstract":"In this paper, we review the Ramanujan conjecture in classical and modern settings and explain its various applications in computer science, including the explicit constructions of the spectrally extremal combinatorial objects, called Ramanujan graphs and Ramanujan complexes, points uniformly distributed on spheres, and Golden-Gate Sets in quantum computing. The connection between Ramanujan graphs/complexes and their zeta functions satisfying the Riemann hypothesis is also discussed. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80234423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Srinivasa Ramanujan: in celebration of the centenary of his election as FRS","authors":"K. Ono","doi":"10.1098/rsta.2019.0386","DOIUrl":"https://doi.org/10.1098/rsta.2019.0386","url":null,"abstract":"Srinivasa Ramanujan, the so-called Man Who Knew Infinity, was one of the most influential, as well as most enigmatic, mathematicians in the recent history of mathematics. With a letter written to G. H. Hardy in 1913, the impoverished Hindu college dropout, self-taught in mathematics, reaching for worlds beyond the shores of India, introduced himself to the history of science. He had spent his youth sitting on cool stone floors in the neighbourhood temple, surrounded by Hindu deities, his mind wandering the world of mathematics. After absorbing the mysterious equations in the letter, Hardy invited Ramanujan to study in England, an extraordinary offer for an Indian under colonial rule. Together they innovated vast tracts of mathematics, before Ramanujan returned to India in fragile health. Tragically, he died at 32 from a misdiagnosed illness, leaving behind three enigmatic notebooks. Ramanujan’s notebooks and research papers have continued to inspire developments in modern mathematics and physics. His formulae and observations now play central roles in fields extending well beyond the realm of pure mathematics. For these reasons, we felt the need to honour the legacy of this great man. To celebrate the centenary of Srinivasa Ramanujan’s election as a Fellow of the Royal Society,1 we organized a public discussion meeting at which leading scientists spoke about Ramanujan’s legacy to mathematics and science. This meeting was held on 15–16 October 2018 at Carlton House. Fifteen distinguished scientists spoke about Ramanujan’s mathematics and his extraordinary legacy across Computer Science, Electrical Engineering, Mathematics and Physics. They were:","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87107054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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