This chapter is primarily a preparation for the study of the total variation flow. This is due to the fact that in order for the 1-Cheeger energy to be lower semicontinuous, it needs to be defined on the space of functions of bounded variation and not on the Sobolev space with exponent equal to one. Thus, we need to extend parts of the linear differential structure to the BV case, and to this end, we will require that we can approximate BV functions with Lipschitz functions in a suitable way. Our strategy will be to present a metric version of the Anzellotti pairings between a vector field with integrable divergence and a BV function due to Anzellotti and prove a generalised integration by parts formula. To be precise, we will prove the existence of two such pairings, separately for the case of the whole space and for a bounded domain with a sufficiently regular boundary. They are defined slightly differently, in particular, in the second case, the Gauss-Green formula takes into account the boundary effects.

The increasing intensity of industrialisation is taking its toll on nonhuman animals, leading to what is now described as the ‘the sixth mass extinction’ cause by Homo Sapiens. Since European settlement in 1788, the Australian continent has recorded the highest rate of mammal extinction in the world, currently a ten percent loss since pre-colonial times, with hundreds of other endangered species at high risk of disappearing. The Australian plays and performances considered in this chapter, which include Extinction by Hannie Rayson, Piece for Person and Ghetto Blaster and Green Screen by Nicola Gunn and Whale by Fleur Kilpatrick, explore what it means for humans to practice an ethical obligation of care for individual species, their place within particular habitats and ecosystems as well as the entanglement of human-nonhuman well-being and survival.

This chapter draws from work and social-cognitive psychology, which is concerned with understanding people at work, and specifically the social and individual cognitive dimensions to these serious misconducts. It offers a distinct perspective on sexual harm, by focusing on three interconnected elements: the individual, the specific types of behaviours, and the environment. Understanding these distinct elements and how they combine, alongside insight into different inhibitors is critical not only understanding why these events occur, but also why they persist. This lens highlights the role of power and its abuse by elites, and why others may be reluctant to challenge and raise their concerns. It also reveals why a professional, undertaking morally praiseworthy activities may paradoxically be more at risk from the distorting cognitive processes of moral licensing. Psychology offers new perspectives into these phenomena and more critically into upstream preventative responses, to show why remediative sanctions may not be so simple here.

Suicide remains a leading cause of maternal death in the UK and in other high-income countries. While there are clear risks in severe mental disorder, those who die have experienced a range of mental illnesses and often come from more deprived communities. The Confidential Enquiries highlight the distinctive patterns of occurrence and progression in perinatal mental illness and the need for improvements in clinical evaluation of risk, effective risk management and the availability of high-quality perinatal mental healthcare as important factors in helping reduce progression to suicide.

This book comes in two parts; the first, consisting of §§1–7, offers an informal axiomatic introduction to the basics of set theory, including a thorough discussion of the axiom of choice and some of its equivalents. The second part, consisting of §§8–14, is written at a somewhat more advanced level, and treats selected topics in transfinite algebra; that is, algebraic themes where the axiom of choice, in one form or another, is useful or even indispensable.

Signers create depictions in modeling spaces ahead of them. Within these spaces, the hand depicts an entity’s location/movement. Signers create these depictions with depicting verbs composed of a verb root and an entity morpheme. These lexical forms lack features that specify their articulatory locations. For some depicting verbs, the hand’s orientation is also unspecified. It is up to the signer to articulate these verbs at locations that suit the needs of the depiction. Signers also create depictions in modeling spaces by gesturally placing and moving meaningful handshapes within modeling spaces. As a result, it can be difficult to distinguish between a depicting verb and a physically similar gestural movement of a meaningful handshape. Identifying a depicting verb will depend on the ability to identify its lexical form and its lexically encoded meaning. In contrast, a fully gestural articulation has no lexical form and does not express a lexical meaning.

The Riemann zeta function is one of the most profound and extensively studied objects in analytic number theory, with deep connections to the distribution of prime numbers. Initially defined for values of s with real part greater than 1, it can be extended to a meromorphic function on the entire complex plane, except for a simple pole at s = 1.

A central feature of the zeta function is its functional equation, which relates its values at s and 1-s, revealing a deep symmetry. This equation is often written in a symmetric form involving the Gamma function, a fundamental object in complex analysis that generalizes the factorial. The Gamma function plays a key role in the functional equation and links the behavior of the zeta function to complex analytic properties.

Another crucial element in the theory is the Jacobi theta function, whose transformation properties are essential in deriving the functional equation of the zeta function.

Analysing fascism in India has been rather unnecessarily polarized, both by Marxist approaches overemphasizing economic causality and by non-Marxist approaches overemphasizing ideology, politics, organizational aspects, and social psychology.1 This difference is important in the historical condition and context in which an analysis of the current regime in India is being made. Whereas Antonio Gramsci defined fascism on an international scale as ‘an attempt to resolve problems of production and exchange with machine-guns and pistol shots’, in India, the rise of an authoritarian regime with fascist tendencies is certainly not a result of the nation being caught up in an international war. It could be more significant to examine the social reality that lends consent to the authoritarian model of politics and governance and how the forms in which it surfaces exhibit fascist tendencies (Gramsci 1984).2 The fascist regimes during the Second World War were different from the post-war ones, specifically with reference to the experience of developing nations like India. In this context, the distinction between fascist movements and fascist regimes is important, and there seems to have been a right-wing extremist movement pushing for the rise of a regime in India (Dimitrov 1984; Reich 1980; Koves and Mazumdar 2005). If Narendra Modi's regime cannot be characterized as a fascist regime, it certainly has been an authoritarian one with fascist tendencies, and what needs to be explained is how such a regime manages to manufacture popular consent.

Syntactic reconstruction poses a unique set of challenges to comparative philologists, and this has led some authors to go so far as to claim it is impossible. This chapter begins by evaluating these challenges and how troubling they are for the enterprise of syntactic reconstruction. With this baseline established, the author turns to the specific attempts that have been made at reconstructing syntax, in particular with reference to Proto-Indo-European. Although some aspects of syntax were treated as early as the Neo-Grammarians, the earliest concerted efforts to treat Proto-Indo-European syntax on its own terms date to the latter half of the twentieth century. There have been several different approaches to syntactic reconstruction since then, which fall broadly into four categories: Typological reconstruction; Pattern-based approaches; Construction Grammar; and Minimalist reconstruction. This chapter argues that, while it is not the only viable methodology, Minimalist Reconstruction provides the most suitable means for the task of reconstructing relative clause syntax in Proto-Indo-European.

This chapter presents the basic commutative algebra tools for algebraic number theory. It begins with the Chinese Remainder Theorem for arbitrary commutative rings, which decomposes ring elements modulo the intersection of pairwise coprime ideals – illustrating a fundamental local – global principle. This principle is further refined through localization, a technique that isolates the behavior of rings at prime ideals that do not intersect a specified multiplicative subset.

A central class of rings in this study is Noetherian rings, characterized by the ascending chain condition on ideals, which ensures desirable finiteness properties. The Hilbert Basis Theorem guarantees that polynomial rings over Noetherian rings remain Noetherian, extending these finiteness conditions to broader algebraic contexts.

Further structural insights arise from studying fractional ideals and the Picard group, which classify invertible ideals and play a crucial role in the study of rings of Krull dimension 1, particularly Dedekind domains.

The chapter concludes with finite-dimensional commutative algebras over fields, a particular case of rings of Krull dimension 0. Under semisimple and separable conditions, these algebras decompose into direct products of fields, permitting a complete classification.

This book offers a selection of key texts mostly written by leading figures in the history of Spanish American political thought during the first century of independence. Political thinkers in the region had to grapple with rather unique and extraordinary circumstances after three centuries of Spanish colonial rule. The emergence of a significant number of new independent polities that adopted representative institutions in an era when absolutism still prevailed in Western Europe, their general adoption of republicanism (except for Mexico during the brief rule of Agustín de Iturbide in 1822–1823, and Maximilian in 1864–1867), and their complex demographic composition, all posed serious challenges for the formation and consolidation of national states in Spanish America. In dialogue with the major currents of thought in Western Europe and North America, Spanish American thinkers often reflected upon these and other related problems while being politically engaged, either in government or in opposition.

In the 1950s, as one columnist recently pointed out, “the wealthiest people in the U.S. were not corporate executives … Rather, they were entertainers.”2 In 1958, for example, when Arthur B. Homer, the president of Bethlehem Steel, was making $623,336, Frank Sinatra made nearly $4 million.3 Given this reality, when entertainers got together to talk amongst themselves in mid-century America, “the U.S. tax laws [were] deeply involved.”4 It should not be surprising, therefore, that when it came to tax dodging during this period, entertainers were on the cutting edge.

Medicine is a profession built on the pillars of compassion and healing. Paradoxically, the medical community is plagued by a pervasive culture of bullying, harassment, and abuse. Women in medicine face particular challenges, often experiencing gendered forms of harassment that further marginalize them. The fear of retaliation, stigma, and career repercussions deters many from reporting such incidents, perpetuating a culture of silence.

This toxic environment not only harms individuals but also compromises patient care. Early exposure to such behaviors during medical training can have lasting negative effects on professional and personal identity and well-being. To address this pressing issue, it is imperative to foster a supportive and inclusive culture within medical institutions, where individuals feel empowered to speak up and seek help without fear. When organising care, providers need to be aware of the complexity of treating doctors who may themselves be therapists. The complexities of the relationships between doctors and their doctor patients need to be considered, especially when stigma and shame influence care.