Introduced by Erdős in 1950, a covering system of the integers is a finite collection of infinite arithmetic progressions whose union is the set of all integers. Many beautiful questions and conjectures about covering systems have been posed over the past several decades, but until the last decade little was known about their properties. Most famously, the so-called minimum modulus problem of Erdős was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most a fixed constant. The ideas of Hough were simplified and extended in 2018 by Balister, Bollobas, Morris, Sahasrabudhe and Tiba, to give solutions (or progress towards solutions) to a number of related questions. We give a summary of this and other progress that has been made since.

The chapter focuses on the influence of French cuisine in Britain. The innovations of French courtly cuisine were frequently mocked in Britain which had its own tradition of sound and economical country house cooking. The Industrial Revolution brought a loss of cooking skills among the urban poor. The affluent benefitted from the flight of French chefs after the French Revolution, leading to the culinary pretensions of the (equally mocked) Regency period. French chefs set up French restaurants and cooking schools, popularising French cuisine, thus influencing the tastes of the middle classes and stimulating a range of gastronomical writings. Examples from Antony Trollope’s Vanity Fair, Virgina Woolf’s To the Lighthouse, Joseph Conrad’s The Secret Agent, Mike Leigh’s Life is Sweet, Peter Greenaway’s The Cook, the Thief, the Wife and Her Lover and Elizabeth David’s postwar books serve to gauge the extent of French infuence and help define what makes British food, British.

Network science is a broadly interdisciplinary field, pulling from computer science, mathematics, statistics, and more. The data scientist working with networks thus needs a broad base of knowledge, as network data calls for—and is analyzed with—many computational and mathematical tools. One needs good working knowledge in programming, including data structures and algorithms to effectively analyze networks. In addition to graph theory, probability theory is the foundation for any statistical modeling and data analysis. Linear algebra provides another foundation for network analysis and modeling because matrices are often the most natural way to represent graphs. Although this book assumes that readers are familiar with the basics of these topics, here we review the computational and mathematical concepts and notation that will be used throughout the book. You can use this chapter as a starting point for catching up on the basics, or as reference while delving into the book.

This chapter investigates Gwen Harwood’s subversion of gendered presumptions of authorship and style. After discussing her skilful redress of a male-dominant literary culture through hoax poetry, it considers how Harwood mobilised male personae to critique the cultural valuing of science and reason, explore sexual immorality, and address women’s experience of domesticity. It discusses how Harwood celebrated motherhood but was also one of the earliest writers to articulate its associated realities of exhaustion, loss of self, and feelings of despair and rage. The chapter argues that Harwood lays important groundwork for second-wave feminism while representing the ambiguities of care and connection. The chapter also engages with Harwood’s later exploration of death and the dynamic between sex and spirituality.

As we have seen, network data are necessarily imperfect. Missing and spurious nodes and edges can create uncertainty in what the observed data tell us about the original network. In this chapter, we dive deeper into tools that allow us to quantify such effects and probe more deeply into the nature of an unseen network from our observations. The fundamental challenge of measurement error in network data is capturing the error-producing mechanism accurately and then inferring the unseen network from the (imperfectly) observed data. Computational approaches can give us clues and insights, as can mathematical models. Mathematical models can also build up methods of statistical inference, whether in estimating parameters describing a model of the network or estimating the networks structure itself. But such methods quickly become intractable without taking on some possibly severe assumptions, such as edge independence. Yet, even without addressing the full problem of network inference, in this chapter, we show valuable ways to explore features of the unseen network, such as its size, using the available data.

This concise and rigorous textbook introduces students to the subject of continuum thermodynamics, providing a complete treatment of the subject with practical applications to material modelling.

• Presents mathematical prerequisites and the foundations of continuum mechanics, taking the student step-by-step through the subject to allow full understanding of the theory.

• Introduces more advanced topics such as theories for the investigation of material models, showing how they relate to real-world practical applications.

• Numerous examples and illustrations, alongside end-of-chapter problems with helpful hints, help describe complex concepts and mathematical derivations.

Sexual offending is astonishingly prevalent and is associated with significant mental health and psychosocial consequences for victims. It is also the basis of a range of severe criminal justice sanctions, which in various jurisdictions can include indeterminate sentences as well as post-sentence restrictions. Of all forms of offending, it is sexual offending which is most stigmatised and despised, and this may lead to vigilantism and persecution of offenders or those perceived to be offenders [1].

This chapter provides a general overview of the market and institutional environments existing in emerging economies. Despite the divergences existing across jurisdictions, this chapter shows that most emerging economies share some common features, including the existence of an institutional environment that generally comprises inefficient judicial systems, high levels of corruption, low levels of protection of property rights, and a weak rule of law. Other features commonly found in emerging economies include the existence of underdeveloped financial systems and the prevalence of micro- and small enterprises and large controlled firms. This chapter, and therefore the understanding of the market and institutional environments existing in emerging economies, will provide the basis for the understanding of the insolvency framework for emerging economies suggested in this book.

The opening lines of Ajax are spoken by the goddess Athena, who addresses her favourite, Odysseus, as an adherent of Harm Enemies: he is tracking down an enemy as usual, in a manner worthy of his traditionally tricky persona. She is the dearest of gods to him, and they enjoy a solidarity inherited from the Odyssey. He places himself in her hands, as he has always done in the past. But despite the bond between them, a conflict of values emerges. When Odysseus is reluctant to view the mad Ajax Athena scolds him as a coward. She implies that any kind of fear or ’reluctance’ constitutes cowardice.

In 426, five years after the outbreak of its great war with Athens, Sparta founded a colony near Trachis, a short distance from Thermopylai – Herakleia Trachinia. By 394 decolonization had begun there, and by 370 nothing Spartan was left of it. The story of this colony is one of mismanagement, arbitrary rule, ethnic tension, and shifts in the control and composition of the citizenry. The foundation was marked by memories of Sparta’s Dorianism and Herakleid heritage. The myth of Herakles functioned in this colonization both through its geographical localization and through its content: Herakles spent his last days at Trachis as the guest of its king and died on the pyre on Mt Oita. In the more political versions of the myth Herakles becomes the original founder of Trachis, the conqueror and destroyer of the abhorrent brigands who infested the land. In slightly later versions which probably belong to the decolonization era, these same native ’brigands’ become Herakles’ friendly companions, the co-founders of Herakleia Trachinia.