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.

Data-centric socialities are media-centric socialities. One way to make sense of the novelty of digital media is to ask what scientists can do with them that they could not do before. This chapter does so for digital imaging in astronomy. It explores two arithmetic operations – adding and subtracting digital exposures pixel by pixel – and their surprising practical and organizational consequences. Setting out from a lecture on image processing to undergraduate students, it traces astronomers’ understandings of digital data’s affordances. It argues that the introduction of charge-coupled devices in the 1980s provided solutions to a set of practical problems that astronomers had formulated with increasing clarity since the 1950s. Subsequently, new organizational possibilities for astronomical research emerged. These include mobilizing data beyond local contexts, rendering abstract time as an object of management, sharing data as nonrivalrous goods, assessing others’ work remotely, and building new forms of collaboration – elements of a novel medial middle ground in data-rich science.

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.

Although Thomas Jefferson claimed on his tombstone to be the author of the Declaration of Independence, this claim is significantly overstated. Jefferson drafted the initial document, but it was heavily edited by the Continental Congress. Moreover, Jefferson was not drafting the document in his own voice, and interpretations of the Declaration that rely on Jefferson’s intellectual peculiarities are strained. The voice of the document is best described, not as the voice of the states, or even of the delegates to the Continental Congress, but the voice of the American people.

The dipole moment is the elementary magnetic quantity, and magnetization is its mesoscopic volume average. The primary magnetic field B is related to the auxiliary magnetic field H. Sources of magnetic field are electric currents and magnetized material. The field produced by a given distribution of magnetization can be calculated by integrating the dipole field due to each volume element, or by using the equivalent distributions of electric currents or magnetic charge. Internal, external and demagnetizing fields are distinguished. The internal field may be defined on a mesoscopic or a macroscopic scale. Magnetic forces and energies are related to magnetization and externally applied field, and force densities to derivatives of the thermodynamic free energy.

This chapter turns back to the scope of national powers, and argues that the original meaning of the Commerce Clause limits Congress to regulating the exchange of goods across state lines, and critiques the Supreme Court’s modern, expansive interpretation of the clause. It demonstrates, however, that the national commerce power was understood to be exclusive, lending support to the modern dormant Commerce Clause doctrine that restricts states from interfering with interstate commerce and that many originalists have opposed. It argues that, as originally understood, States retain police powers to regulate the health, safety, morals, and welfare of their populations and can incidentally affect interstate commerce through police regulations, as long as they have a legitimate purpose and do not target interstate commerce. Congress, similarly, could incidentally affect the states’ police powers or private conduct beyond the enumeration of powers so long as it was genuinely exercising its power over interstate commerce, such as when it closed the channels of interstate commerce to goods produced with child labor. This formalist approach reveals a symmetry in the original distribution of power between Congress and the states.

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.

This chapter examines the development of uniform company law within the EU, focusing on supranational company forms such as the Societas Europaea (SE) and the Societas Cooperativa Europaea (SCE). It explores the legal framework governing their formation, structure and operation, highlighting the balance between EU regulation and national company laws. The chapter assesses the goals of legal harmonization, cross-border mobility and enhanced corporate flexibility. It also considers the practical challenges and limited uptake of these forms, along with their potential in promoting European economic integration. Comparative insights and recent legislative developments are discussed to evaluate future prospects for uniform company law.