The method of hypergraph containers is a modern incarnation of a classical proof technique concerned with enumerating certain classes of graphs. Perhaps the origins of the method lie in a paper of Kleitman and Winston about enumerating the number of n-vertex graphs containing no 4-cycle. The full generality and power of this approach became evident only in the past decade when very general theorems were proved with wide applicability to a host of questions. These results are concerned with counting the number of independent sets in a hypergraph. Two essentially equivalent approaches were developed by Saxton and Thomason and by Balogh, Morris, and Samotij, and our goal is to state one of these results and give some applications.

This introduction outlines a comprehensive study of the U.S. Constitution’s structure, marking its 237th year as the oldest modern constitution. It aims to equip readers – from law students to everyday Americans – with tools to understand its framework of representation, separation of powers, checks and balances, federalism, and the Bill of Rights, addressing enduring questions like the legitimacy of secession, the scope of national powers, presidential authority in foreign affairs and war, judicial review’s origins, and Congress’s taxing and spending powers. The book challenges myths, such as the existence of a “general welfare clause,” and critiques both originalist and living constitutionalist views, arguing, for instance, that enumeration limited executive, not just national, power, and that functionalism fuels the imperial presidency. Organized into seven parts and eighteen chapters, the book explores the Constitution’s compound republic, the distribution of legislative, executive, and judicial powers, and federal-state dynamics. By engaging recent scholarship, it seeks to clarify the Constitution’s original design while advocating its continued relevance today.

The Epilogue draws out the broader implications for the rise of the Atlantic world in the early modern public sphere, demonstrating, first, that what happened in and across the ocean mattered to people at home, and, second, why managing news became a concern for authorities who gradually lost their information monopoly regarding developments across the Atlantic. It reflects on the historiographical repercussions for the fields of Atlantic history and the history of news – making fresh connections to the transoceanic exchange of people, commodities, values, and ideas.

In this chapter, we deal with extremal problems for uniform hypergraphs that are motivated by Turán’s graph theorem. These extremal problems are substantially more difficult than for graphs, and indeed the extremal problem for hypergraph cliques remains unsolved, even asymptotically as the number of vertices grows. Perhaps the most notorious conjecture is the Turán conjecture from 1941 about the maximum density of a triple system on n points that contains no copy of the tetrahedron. This chapter explores a variety of Turán-type problems for hypergraphs while introducing the important methods in this area.

We present historical background, contemporary status, and potential future development of the psychology of religion (PoR) in United Kingdom, beginning with its origins: the formative people, places, and various intellectual schools of thought. Writing about the current state of the topic, we 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.

We 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.

It has been said that Brexit was a solution in search of a problem. The campaign encouraged the idea that voting for Brexit could satisfactorily shift the practice and discourse of national political priorities; the downside proved to be the confusion over exactly what specific issues it was meant to address. An indeterminate number of loosely focused grievances could not easily be translated into a programme. The protracted parliamentary stand-off in the following years encouraged suspicion about any attempts to adjust or fine-tune precise legal provisions – suspicion of elite interests, of reversion to rule by anonymous experts and of legal process itself. A distinctly close vote in the actual referendum was rapidly mythologised as an overwhelming popular mandate, even though it was unclear what exactly it was a mandate for. The lasting legacy has been to reinforce a ‘theatrical’ approach to politics, in which actual problem-solving and long-term strategising yield to performative or gestural decisions. Both globally and nationally, this is an increasingly disturbing and destabilising trend; those on both sides of the Brexit debate need to acknowledge this as a real issue about the health of a critical and engaged democracy.

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.

Learning from data is the point of making and using them. But how can scientists pass on to potential users what they have learned about their data? This chapter proposes answers to this question by focusing on the medium of data and its social uses. It argues that scientific data do not merely represent information but can be structured and presented to have a pragmatic function oriented to enable users’ understanding. It demonstrates this by describing how the MUWAGS collaboration, discussed in Chapters 6 and 7, designed its catalog – a table of the measured and estimated properties of galaxies –, to guide users to self-correct wrong uses and delimit itself from being held accountable for misuses. Some astronomers argue that catalogs encode their makers’ collective knowledge of their data. This chapter examines this claim ethnographically, suggests elements of a pragmatics of data reuse, and ends with a reflection on socio-computational orders – entanglements of the social and the computational in scientific work with large datasets.

Resonance arises when the energy levels of a quantized system of electronic or nuclear moments are Zeeman-split by a uniform magnetic field. The system absorbs energy from an oscillating magnetic field at sharply defined frequencies, which correspond to transitions between levels. Classically, the resonance occurs when a transverse AC field is applied at the Larmor frequency. Resonance methods are valuable for investigating the structure and magnetic properties of solids, and they are used for sophisticated imaging and other applications. The resonant magnetic moment may be an isolated ionic spin, free radical or an electron associated with a crystal defect in electron paramagnetic resonance (EPR), or the ordered magnetization, in ferromagnetic resonance (FMR). Otherwise it can be a nuclear spin in nuclear magnetic resonance (NMR). Damping is critical for control of magnetization reversal. Resonant spin dynamics are also associated with spin waves, and domain walls. The related techniques of Mossbauer spectroscopy and muon spin resonance provide information on hyperfine interactions in solids. Spin dynamics can be excited with sub-picosecond laser pulses and tracked using magneto-optic Kerr effect.

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.