From this chapter on, we study graded quotients of upper ramification subgroups. In this chapter we prove that the graded quotients are abelian and annihilated by p. The proof is by reduction to the classical case where the extension is totally ramified. The geometric fact behind this property is that the closed fiber of the covering of stable integral models is étale. Other tools used in the proof of the reduction are the first homology group of the cotangent complex and the Tor group of the module of relative differential forms.

Providing a cohesive reference for advanced undergraduates, graduate students, and even experienced researchers, this text contains both introductory and advanced material in extremal graph theory, hypergraph theory, and Ramsey theory. Along the way, this book includes many modern proof techniques in the field, such as the probabilistic method and algebraic methods. Several recent breakthroughs are presented with complete proofs, for example, recent results on the sunflower problem, and off-diagonal and geometric Ramsey theory. It is perhaps unique in containing material on both hypergraph regularity and containers.

Featuring an extensive list of exercises, this book serves as a valuable teaching resource for a variety of courses in extremal combinatorics. Each of the two parts can form the basis of separate courses, and the majority of sections are designed to match the length of a single lecture.

The surge in archaeological activity since the 1950s has led to the discovery of a large number of Song burials in northern China, adorned with pictorial scenes and architectural motifs. This chapter investigates the decoration of these tombs, shifting our focus from the elaborate tombs themselves to the individuals who constructed them. By emphasizing the significance of burial practices during the Song dynasty, it becomes clear that tombs not only continued to reflect principles of filial piety and ancestral reverence but also became a critical force in shaping cultural values and social order. These tombs can be interpreted as a form of self-expression by the wealthy class, emerging from the interplay between ritual regulations and regional social customs, while simultaneously serving as focal points for family bonds, ancestor veneration, and belief systems.

I present historical background, contemporary status, and potential future development of the psychology of religion (PoR) in Egypt, beginning with its origins: the formative people, places, and various intellectual schools of thought. Writing about the current state of the topic, I reflect on influential factors that are either facilitating or inhibiting the study of PoR, including publication options, topical emphases, practitioners, orientations, methodologies, and professional organizations.

I offer opinions concerning future development topics that are emerging as important in the immediate future and/or are perennially important in order to stimulate creative and useful research including Western theoretical relevance, the extent to which Western PoR theories may or may not contain reasonable expectations and concepts for this region, contextual nuances, Indigenous theoretical concerns, collaborative research opportunities, and common faux pas – reflections on what people unfamiliar with this region commonly and incorrectly assume about conducting PoR work in this context.

Szemerédi’s graph regularity lemma has proved to be one of the most important tools in extremal graph theory with many applications in combinatorics and beyond. In order to extend these applications to the hypergraph setting, we need a corresponding regularity lemma for hypergraphs. In this chapter, we consider such generalizations of Szemerédi’s regularity lemma.

We construct an injection from the dual of the graded quotients of upper ramification groups to a twisted homology group of the cotangent complex. We call the injection the characteristic form. The construction is given in the case where the characteristic is p and the index is an integer. In the geometric case, the construction is based on the group structure on the boundary of the dilatation obtained from the groupoid structure of the self products.

This chapter examines transatlantic news in the first newspaper of seventeenth-century France, the famous Gazette of Théophraste Renaudot, which appeared in Paris every week between 1631 and 1653. Renaudot has often been portrayed in French scholarship as a mere mouthpiece of Cardinal Richelieu, but his coverage of events in the Atlantic world shows that he was in fact a very accomplished news writer who gave a more balanced account of events than contemporaries and historians have given him credit for. Those who wanted to follow events in the South Atlantic were often better off reading Renaudot’s Gazette than the French-language editions of the Amsterdam newspapers, or the embryonic periodicals from Lisbon. As news of the climax in the war in Brazil reached Paris in the late 1640s, the Gazette reported events from as many as four different perspectives – giving equal weight to narratives of the Portuguese and Dutch generals, as well as those of the native American and (formerly enslaved) African commanders who had participated in the final battle.

The first decade of the nineteenth century was a crucial period in the development of Haiti and Black identity in the Caribbean. This chapter focuses on the first half of the decade: L’Ouverture’s reign during the abolition of slavery, his capture and exile, and Dessalines’s declaration of Haitian independence. I trace evolving attitudes towards architectural destruction and renovation, focusing on three stages: uncertainty about violence; creolization, mimicry, and renovation; and acceptance of the obliteration of the architectural archive as an anti-colonial act. Drawing on texts by Romantic authors including John Thelwall, William Wordsworth, Marcus Rainsford, and Leonora Sansay, I recover how Romantic ideology and revolutionary practice were not merely transmitted across the Atlantic then passively accepted in the Caribbean. Haitians actively adapted and complicated Romantic and revolutionary ideology, using the common tropes of the period to reimagine their place in the world and assert their rejection of objectification and colonialism

For abelian extensions, the cohomological filtration on Galois groups is defined by using the cup-product with values in the Brauer groups. In the classical case where the residue field is finite, the filtration is defined as the image of the filtration on the multiplicative group by the reciprocity mapping. The graded quotients are captured by the refined Swan conductor. The proof of its existence in general is postponed to the last chapter. The cohomological filtration and the refined Swan conductor are closely linked to the differential forms with log pole of the residue field.

Walk into most large organisations and you’ll notice two distinct knowledge cultures – one fluent in human insight, the other fluent in data. This is a persistent divide – sometimes historical, sometimes epistemological – between two influential groups: behavioural scientists and data scientists. In the past, this divide was more visible, even physical. During the early data analytics era of the 1990s and early 2000s, data scientists were often ‘tucked away’ in back offices or lower floors – an arrangement at one point satirised by the British comedy The IT Crowd, where technical experts were literally hidden in the basement. Meanwhile, behavioural scientists – those working on organisational culture, consumer insights and human-centred design – tended to sit closer to the executive suite, advising leadership on strategy and the ‘why’ behind behaviour.

Recently large language models (LLMs), such as ChatGPT, have received unrivalled attention not only in AI research but also in the greater public. The massive coverage in the mainstream media makes it utterly hard for an end user to get a decent overview of systems and an understanding of what these systems are actually capable of, plus how they are best and safely used. In the field of LLMs, where news and updates about the newest features are published on almost a daily basis, users, or prospective users, are overwhelmed by new possibilities on how to retrieve information. What is the best approach to tackle the challenging situation where users are inundated with various options of different LLMs, and which performs best for various use cases? This chapter presents the outcomes of a project initiated by AI bachelor students that aim to better understand different LLMs from an end-user perspective. We have thoroughly tested the capabilities of these systems in various group sessions and present our results in this chapter. As well as try to offer some guidance on how to learn about these systems by testing them in a structured way, we reflect on to which extent different LLMs met our expectations and compare them to determine which LLM to use in different scenarios. We believe this chapter can serve as a decent introduction to LLMs for researchers of all disciplines, as well as practitioners interested in exploring the potential of various systems.

Our modern sense of what the poetry of the 1800s is and means has been heavily determined by the standards proposed in William Wordsworth and Samuel Taylor Coleridge’s ‘Preface’ to Lyrical Ballads. However, during the 1800s, the ‘Preface’ was a controversial, oppositional proposition. Rather than being seen as the ‘breath and finer spirit of all knowledge’, poetry was more commonly perceived to be a social form of networked writing that was tightly aligned with performances of politeness, privilege, affiliation, and discernment. This essay explores three 1800s publications that sought to regulate poetry as a social world: the Poetical Register (commenced in 1801), the Edinburgh Review (commenced in 1802), and Lord Byron’s English Bards and Scotch Reviewers (1809). These works serve as exemplars of the contradictions inherent in attempts to delimit poetry during a decade in which its forms and manifestations were becoming increasingly popular, diverse, and contested.