Herman's self-stabilisation algorithm, introduced 25 years ago, is a well-studied synchronous randomised protocol for enabling a ring of $N$ processes collectively holding any odd number of tokens to reach a stable state in which a single token remains. Determining the worst-case expected time to stabilisation is the central outstanding open problem about this protocol. It is known that there is a constant $h$ such that any initial configuration has expected stabilisation time at most $h N^2$. Ten years ago, McIver and Morgan established a lower bound of $4/27 \approx 0.148$ for $h$, achieved with three equally-spaced tokens, and conjectured this to be the optimal value of $h$. A series of papers over the last decade gradually reduced the upper bound on $h$, with the present record (achieved in 2014) standing at approximately $0.156$. In this paper, we prove McIver and Morgan's conjecture and establish that $h = 4/27$ is indeed optimal.

Cross-disciplinary collaborations have become an increasingly important part of science. They are seen as a key factor for finding solutions to pressing societal challenges on a global scale including green technologies, sustainable food production and drug development. This has also been realized by regulators and policy-makers, as it is reflected in the 80 billion Euro "Horizon 2020" EU Framework Programme for Research and Innovation. This programme puts special emphasis at breaking down barriers between fields to create a path breaking environment for knowledge, research and innovation. However, igniting and successfully maintaining cross-disciplinary collaborations can be a delicate task. In this article we focus on the specific challenges associated with cross-disciplinary research in particular from the perspective of the theoretician. As research fellows of the 2020 Science project (http://www.2020science.net) and collaboration partners, we bring broad experience of developing interdisciplinary collaborations [2–12]. We intend this guide for early career computational researchers as well as more senior scientists who are entering a cross disciplinary setting for the first time. We describe the key benefits, as well as some possible pitfalls, arising from collaborations between scientists with backgrounds in very different fields. This paper has inter alia been cited by Times Higher education: http://www.timeshighereducation.co.uk/news/people/the-secrets-to-successful-interdisciplinary-work/2020267.article .

In order to attempt to understand the complexity inherent in nature, mathematical, statistical and computational techniques are increasingly being employed in the life sciences. In particular, the use and development of software tools is becoming vital for investigating scientific hypotheses, and a wide range of scientists are finding software development playing a more central role in their day-to-day research. In fields such as biology and ecology, there has been a noticeable trend towards the use of quantitative methods for both making sense of ever-increasing amounts of data [1] and building or selecting models [2]. As Research Fellows of the “2020 Science” project (http://www.2020science.net), funded jointly by the EPSRC (Engineering and Physical Sciences Research Council) and Microsoft Research, we have firsthand experience of the challenges associated with carrying out multidisciplinary computation-based science [3]–[5]. In this paper we offer a jargon-free guide to best practice when developing and using software for scientific research. While many guides to software development exist, they are often aimed at computer scientists [6] or concentrate on large open-source projects [7]; the present guide is aimed specifically at the vast majority of scientific researchers: those without formal training in computer science. We present our ten simple rules with the aim of enabling scientists to be more effective in undertaking research and therefore maximise the impact of this research within the scientific community. While these rules are described individually, collectively they form a single vision for how to approach the practical side of computational science. Our rules are presented in roughly the chronological order in which they should be undertaken, beginning with things that, as a computational scientist, you should do before you even think about writing any code. For each rule, guides on getting started, links to relevant tutorials, and further reading are provided in the supplementary material (Text S1).