A central notion in extremal set theory is the shadow of a family H of k-element subsets of an n-element set. It consists of all the k − 1-element subsets that are contained in some set of H. A matching of H is a collection of pairwise disjoint sets in H, and the matching number of H is the largest size of a matching in H. In this chapter, we discuss the basic problems surrounding shadows and matchings in hypergraphs.

This chapter considers the conflicting views on whether the Declaration of Independence is law. It sets forth the arguments on either side, ultimately concluding that neither side is completely persuasive. It argues that the Declaration was a legal document of continuing legal significance, even if it is not directly controlling law in the sense of a modern statute. It then sets forth the numerous ways in which American courts have relied on the Declaration, in cases ranging from citizenship to states’ admission to the union to the interpretation of provisions in the federal constitution.

This chapter examines the regulatory framework governing annual and consolidated accounts within EU law. It explores the harmonization efforts aimed at ensuring transparency, comparability and reliability of financial reporting across Member States. The chapter analyses the requirements for preparing, auditing and publishing financial statements, highlighting the role of consolidated accounts in presenting a comprehensive view of corporate groups. It also addresses challenges in implementation, compliance and enforcement, as well as the impact of recent reforms on corporate accountability and investor protection. The discussion underscores the significance of financial reporting in fostering trust in the EU’s internal market.

Logic simplifies and clarifies ordinary language and thereby improves communication by reducing or eliminating errors of interpretation. While English and other languages offer many ways of saying things, logic uses simple structures whose meanings are always clear because they are stated using only a few well-known forms. This chapter considers sentential logic in which every sentence may be either true or false but not both or of unknown truth value. Such logic simplifies communication from the uncertainty that is common in ordinary language. Negation, conjunction, disjunction and implication (or conditionals) are illustrated by truth tables. All are valuable in theory development and enable the prediction of outcomes from premises.

‘The price of victory’ was how Jean Monnet, pioneer of European integration, described Britain’s self-exclusion from the foundational years of European integration. As a late entrant (1973), it was obliged to accept many arrangements that did not suit its interests. Margaret Thatcher and her successors tried to negotiate opt-outs to protect what they deemed Britain’s vital interests as the European Community developed into the European Union (EU), until David Cameron tried to resolve the issue once and for all through renegotiation and an in-or-out referendum. Cameron was outwitted by Boris Johnson – a brilliant political showman – but he in turn had no strategy for his surprise Brexit victory. The chapter explores the struggles of successive Tory premiers – especially May, Johnson and Sunak – to hold their party together and get Brexit done, without isolating Northern Ireland from the rest of the UK. After the Tories lost the 2024 election, the new Labour government was left on the sidelines of the EU, trying to sort out piecemeal opt-ins – an ironic inversion of Britain’s posture when it was inside the EU.

Ferromagnetism and the Curie temperature were explained by Weiss in terms of a huge internal ‘molecular field’ proportional to the magnetization. The theory is applicable both to localized and delocalized electrons. No such magnetic field really exists, but it is a useful way of approximating the effect of the interatomic Coulomb interaction in quantum mechanics, which Heisenberg described by the Hamiltonian describing the localized spins on two adjacent atoms. When J > 0, ferromagnetic exchange leads to ferromagnetic order in three dimensions. Spin waves are the low-energy excitations of the exchange-coupled magnetic lattice. In the delocalized electron picture, a ferromagnet has spontaneously spin-split energy bands. The density of ↑ and ↓ states is calculated using spin-dependent density functional theory. Important physical phenomena associated with ferromagnetism are then discussed, including magnetic anisotropy, magnetoresistance, magnetoelastic and magneto-optic effects.

We present historical background, contemporary status, and potential future development of the psychology of religion (PoR) in Argentina, 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.

We present historical background, contemporary status, and potential future development of the psychology of religion (PoR) in Nigeria, 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.

We define filtration by upper ramification subgroups. Using the geometric construction in the previous chapter, we construct a filtration on the fiber functor. The proof of the fact that this filtration defines the subgroups is based on the going down theorem. We prove that the first steps of the filtration are in fact the inertia group and the wild inertia group, using the computation in the monogenic case. We also prove semi-continuity. The proof relies on the stable reduction theorem and a general version of the reduced fiber theorem.

Sterling depreciated by around 10 per cent on a trade-weighted basis immediately after the vote, raising prices faced by consumers by an estimated 2.9 per cent in the two years after, with a consequent effect on reducing real living standards. For firms, the referendum and its aftermath of political instability increased uncertainty and raised barriers to investment. We focus mostly on the longer-term consequences. We start by outlining, briefly, the key economic claims and forecasts made at the time of the referendum. These matter not just for historical purposes but because ‘consensus’ estimates and assumptions made by key policymakers, notably at the Office for Budget Responsibility, have changed little since then. We then present some of the basic data on UK economic performance over the period and draw comparisons with similar countries. We then consider in more detail what has happened to trade and business investment, the key routes through which any economic impacts are likely to have been transmitted. We briefly review claims that big regulatory changes might have had positive effects.

We present historical background, contemporary status, and potential future development of the psychology of religion (PoR) in Turkey, 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.

In his book Race of Nation: A Conflict of Divided Loyalties (1925), Italian–American lawyer and diplomat Gino Speranza (1887–1935) focuses on immigrants as a threat to American civilization. While Speranza claims that he holds “no brief for the superiority of Anglo-Saxon civilization,” he claims the US is an Anglo-Saxon country with an Anglo-Saxon civilization and “that people of Anglo-Saxon stock are the best adapted to running its government and shaping its special national life.” Speranza coins the term ‘nation-spiritual’ to distance himself from a very particular material, racialized, and conformist American identity that he details and promotes in his book; the term is, effectively, an ineffable shibboleth that can’t be specified because it is only available to the exclusive class of ‘true Americans’ (like Speranza) who, apparently, know it when they see it, or sense it, or feel it, and who arrogate to themselves the role of ‘decider’ of who ‘counts’ as a true American, and what the ‘essential principles and ideals of American life’ are.

The essays in this volume were commissioned to address how British literature developed in the first decade of the nineteenth century, as seen from our current vantage, one-quarter of the way through the twenty-first. They examine the literary forms and cultural formations that emerged during this paradoxical era of aftermath and new beginnings. As with the other volumes in the Nineteenth-Century Literature in Transition series, a decadal ‘cut’ into literary history lends each of the chapters its frame. Our collective goal has been to re-examine a period within the Romantic period, and within the larger context of the nineteenth century. This introduction offers a brief summary of the fourteen chapters that comprise the volume and draws connections among them.