Asymptotics and Ramanujan's mock theta functions: then and now.

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'.

Making Fair Choices on the Path to Universal Health Coverage: Applying Principles to Difficult Cases.

Abstract-Progress toward universal health coverage (UHC) requires making difficult trade-offs. In this journal, Dr. Margaret Chan, the World Health Organization (WHO) Director-General, has endorsed the principles for making such decisions put forward by the WHO Consultative Group on Equity and UHC. These principles include maximizing population health, priority for the worse off, and shielding people from health-related financial risks. But how should one apply these principles in particular cases, and how should one adjudicate between them when their demands conflict? This article by some members of the Consultative Group and a diverse group of health policy professionals addresses these questions. It considers three stylized versions of actual policy dilemmas. Each of these cases pertains to one of the three key dimensions of progress toward UHC: which services to cover first, which populations to prioritize for coverage, and how to move from out-of-pocket expenditures to prepayment with pooling of funds. Our cases are simplified to highlight common trade-offs. Though we make specific recommendations, our primary aim is to demonstrate both the form and substance of the reasoning involved in striking a fair balance between competing interests on the road to UHC.

Three Case Studies in Making Fair Choices on the Path to Universal Health Coverage.

The goal of achieving Universal Health Coverage (UHC) can generally be realized only in stages. Moreover, resource, capacity, and political constraints mean governments often face difficult trade-offs on the path to UHC. In a 2014 report, Making fair choices on the path to UHC, the WHO Consultative Group on Equity and Universal Health Coverage articulated principles for making such trade-offs in an equitable manner. We present three case studies which illustrate how these principles can guide practical decision-making. These case studies show how progressive realization of the right to health can be effectively guided by priority-setting principles, including generating the greatest total health gain, priority for those who are worse off in a number of dimensions (including health, access to health services, and social and economic status), and financial risk protection. They also demonstrate the value of a fair and accountable process of priority setting.

U.S.-based recruitment of foreign-educated nurses: implications of an emerging industry.

OBJECTIVES: Despite an increase in the number of foreign-educated nurses (FENs) working in U.S. hospitals and nursing homes, very little is known about the industry that brought them here. Our objectives were to learn more about the size and scope of the international nurse recruitment industry, its business models, and the range of countries where companies actively recruit. Based on reports from focus groups of FENs in New York City, we also sought to identify some of the problems that have occurred in the areas of contracting and clinical orientation. METHODS: We used a combination of qualitative methods and secondary data sources, which included U.S.-based international nurse recruitment company Web sites, interviews with 20 executives from international nurse recruitment companies, two focus groups with FENs in New York City, and letters sent to the Philippine Nurses Association of America by FENs seeking legal advice. RESULTS: Through a July 2007 Internet search, we found that at least 273 U.S. companies were actively recruiting FENs. While most such companies focused on the Philippines and India, about 20 companies were active in Africa. (A second search revealed that, as of January, at least 211 U.S. companies were actively recruiting FENs abroad.) Within the industry there is growing use of the staffing-agency model, which typically requires nurses to sign 18-to-36-month contracts and imposes high breach-of-contract fees. The focus group discussions with FENs in New York City revealed inadequate orientation programs and several types of labor abuses. CONCLUSIONS: Concerns about recruitment practices, which were expressed by many industry executives and FENs, reveal the need for accountability within the industry.

The spt-function of Andrews.

Recently, Andrews introduced the function s(n) = spt(n) which counts the number of smallest parts among the integer partitions of n. We show that its generating function satisfies an identity analogous to Ramanujan's mock theta identities. As a consequence, we are able to completely determine the parity of s(n). Using another type of identity, one based on Hecke operators, we obtain a complete multiplicative theory for s(n) modulo 3. These congruences confirm unpublished conjectures of Garvan and Sellers. Our methods generalize to all integral moduli.

Five years later: poor women's health care coverage after welfare reform.

The 1996 welfare reform law aims to increase poor women's participation in the work force and encourage their financial independence. Because women's ability to obtain and retain employment is affected by their health status, welfare reform's success is fundamentally tied to poor women's access to health care and to health insurance. Despite this, the rate of uninsurance among poor women with children has grown by half in recent years, leaving 37% of poor mothers uninsured in 2000. Coverage through employer-sponsored insurance has increased only slightly, and Medicaid participation has dropped. Although many factors contributed to this, welfare policies and procedures and low Medicaid eligibility levels had unintended yet significant negative effects on women's health care coverage. The sharp decline in poor women's health care coverage is likely to be one of several health-related issues that Congress will consider as it debates the reauthorization of the welfare law in 2002. Both public and private efforts will be necessary to improve coverage for poor women with children. Much progress has been made during the past 5 years in covering poor and near-poor children, but their parents have been left behind. The same efforts that proved successful for children, including broadening eligibility for coverage and simplifying the application process, can be used to improve the health and well-being of parents and to strengthen their ability to care for and support their families.