A Euclidean domain is an integral domain with a function enabling a division algorithm similar to that of the ring of integers, while a principal ideal domain (PID) is a domain where every ideal is principal – generated by a single element. Every Euclidean domain is a PID, and every PID is a unique factorization domain (UFD), ensuring unique factorization into irreducibles. This chapter develops the theory of finitely generated modules over PIDs. The central result is their decomposition into direct sums of cyclic submodules. The uniqueness of invariant factors, together with the module’s rank, provides complete classification up to isomorphism. The cyclic decomposition relates closely to the Smith normal form of matrices over PIDs, a canonical form revealing the cokernel structure of linear transformations between finite-rank free modules. We apply the primary decomposition of torsion modules over a PID to the classification of finitely generated abelian groups and to the classification of linear operators over algebraically closed fields, yielding the classical Jordan canonical form.

Our best unconditional bounds for ψ(x, χ), such as Theorem 11.16, are not very good owing to our rather limited knowledge of the zero-free region of L(s, χ). It is possible to extend the range for the modulus q of χ at the expense of the error term or by including zeros close to 1. See Corollary 11.20 or the chapter on Vinogradov’s mean value theorem in the forthcoming Volume III. If we assume GRH, then we have a much better estimate (cf. Theorem 13.7). However, in some situations, a good bound for an average of |ψ (x, χ) | is all that is required, and such bounds can be obtained unconditionally by combining our methods of Chapter 17 with the large sieve.

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.

Fear-inducing tragedy encompasses the deadly consequences of contaminating the atmosphere, including through nuclear radiation. Theatrical works juxtapose the threat of extinction with the benefits of energy obtained from nuclear, coal and oil power sources as they indicate that the earth’s atmosphere can no longer be regarded as an exploitable resource. The degradation of breathable air is life-threatening for human and nonhuman species alike. The depiction of human-induced contamination from mid twentieth-century theatre encompasses life-threatening atmospheric nuclear bomb testing in the Pacific Ocean region and on mainland Australia. First Nations theatrical performance about nuclear-damaged ecological systems broadens the emotional commons of climate change theatre in which characters confront atmospheric contamination and the threat of annihilation.

Based on extensive in-depth interviews and primary documents, Chapter 5 presents a case study of a middle-performing county in central China to illustrate the core arguments of the book. It shows how the local state reassigns rural land rights in an effort to drive urbanization and industrialization. It provides data on how local officials respond to performance targets by using land to attract investment in both real estate and industry. The chapter also provides data on the county’s debt-fueled infrastructural development employing land-backed LGFVs and presents data on economic activity and fiscal revenue. The chapter illustrates the nature of popular dissatisfaction over the reassignment of land rights and how local officials use law to deflect discontent and conflict away from the state itself, while exacerbating conflict between villagers and village collectives and among neighbors. It shows how villagers, unable to effectively challenge the state, seek to exclude others from sharing in land-taking compensation funds by using contested ideas about household registration (hukou), marriage, village charters, and land contracts. Villagers manifest legal consciousness, including awareness of legal aid, and engage in legal mobilization, including negotiation, mediation, petition, litigation, and protest.

What is a legal culture, and how do we understand and describe it? Historians have done a good job, over the past century, of describing legal institutions. They have been less successful at understanding legal cultures. Yet the eastern Roman Empire is suffused with attempts to articulate understandings of state power and capacity in the language of law. The current "institutional" approach does very little to explain why law was meaningful to subjects of empire: it merely attempts to explain "how it worked," hypothesizing that decent functioning incentivizes the use of the system. This is problematic: it relies anachronistically on a positivist understanding of law. Instead, law is shown to be implicated in multiple acts of community self-definition, in public rituals, and in popular consciousness. This raises the questions: why did legality play such an important role in the provincial imagination? And with what effects on the state itself?

The pre-birth planning meeting is an essential aspect of the care of any pregnant woman who has a current or previous severe or complex mental illness. It brings together the pregnant woman, her partner and/or other family members and all the professionals involved. The meeting ensures that relevant information is shared so that everyone has a good understanding of the concerns, risks and strengths. At the meeting a perinatal mental health care plan is devised collaboratively. This outlines the woman’s care for the remainder of her pregnancy, her maternity admission for the birth of her baby and for the early postnatal period. It also includes a crisis plan. This process helps all the professionals to work in partnership with the woman and her family and to ensure she and her family have the best possible care and outcomes.

No figure did more to promote the myth of blood and soil than Richard Darré, the Nazi minister of agriculture. Darré’s environmental legacy is disputed; current scholarship recognizes his racial obsessions while typically denying any substantive ecological orientation. This chapter takes a different perspective, arguing that Darré’s views on race and on care for the soil were inseparable ideologically as well as in practice. Moreover, his environmental allegiances shifted significantly over time, reflecting a series of fierce debates about the politics of nature that involved competing Nazi factions. Peering below the level of famous Nazis, the core of the chapter presents a detailed appraisal of often neglected mid-level officials in interaction with conservationists, organic adherents, and life reform activists, with particular attention to the increasingly influential biodynamic representatives on Darré’s staff. The resulting profile of blood and soil principles in action casts Nazi environmental efforts in a new light: never fully at the center of power but never entirely marginalized, ecological politics played out in tandem with conflicting economic priorities and overarching racial objectives.

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.

Ulpian’s successors followed his lead in imagining a world without legal politics. To articulate their vision, they constructed a law of government: a body of law devoted to the administration of cities, and to criminal punishment. It focused on questions of public order and administration, and sought to eliminate the scope of, if not the need for, collective participation. It was concerned to limit the jurisdiction of governors, who might become enmeshed in local political systems. Within this system, jurists reserved the capacity for affective judgment for emperors alone. This is the vision of law that would be taken up over the long course of Late Antiquity: only the emperor would be permitted affect and discretion; all others were construed as responsible to the law itself. Together, jurists and the emperors created a vision of law that was radically opposed to the society upon which it was enacted.

There is a strong evidence base for the management of perinatal mental illness and all healthcare professionals should be aware of that; women can be offered hope of recovery with treatment.

NHS England and Scotland have published pathways of care for perinatal mental health care, but these do not deal with illness below the threshold of specialist care. Each area needs to develop effective local pathways of care that cover the full spectrum of disease and include all services.

The purpose of this chapter is to describe the building blocks that contribute to the rest of the pathway for women with perinatal mental illness. Although the chapter mainly covers the situation in England, it also gives information about the devolved nations, Scotland, Wales and Northern Ireland.

What follows covers principles of teamwork and stepped care.

Chapter 4 focuses on the themes of sacred kingship and prophecy in New Spain’s support for the Bourbon monarchy. It examines how preachers and intellectuals drew upon the long tradition of messianic prophecy to frame Felipe V as a providential ruler destined to inaugurate a new, redemptive age aligned with the prophecies of the thirteenth-century mystic Joachim de Fiore. Orators in New Spain tapped into both European and American prophetic traditions to suggest that Bourbon rule marked the arrival of a long-awaited third era of Christian history. Using Old Testament genealogies, especially the lineage of David and Jesse, preachers positioned Felipe V within a sacred genealogy that linked the Spanish Crown to biblical kingship. This chapter argues that these narratives were more than rhetorical flourish—they constituted a powerful ideological tool, allowing contemporaries to reimagine the imperial transition as divinely ordained and embedded in eschatological time.

Substance misuse is defined as the harmful or hazardous use of psychoactive substances, including alcohol and nicotine. Substance misuse in the perinatal period may also increase the risk of adverse maternal and child sequelae. These include reduced engagement with antenatal care and obstetric and neonatal complications such as low birth weight and prematurity. Substance misuse has also been implicated in maternal deaths in the UK; 23% of those who died between 2019 and 2021 were smokers and 14% were using other substances. Clearly, studying longer-term outcomes in offspring is challenging, with small sample sizes and unmeasured confounding factors characteristic of many of the studies in this area. Despite this there is some evidence from prospective, longitudinal birth cohorts that maternal substance misuse is associated with a range of emotional and behavioural difficulties in exposed children and even in a recent US cohort with future substance misuse at age 30.

In this chapter we discuss how psychiatrists and other healthcare professionals can support families affected by substance misuse, from the pre-conception period, through pregnancy and in the postpartum.

Optimising women’s mental health at the time of conception, as a means of improving pregnancy outcomes, is of increasing interest. Women with pre-existing or new onset mental illness in the perinatal period, like those with pre-existing or new onset physical health conditions, are considered as high-risk pregnancies. Strategies to mitigate pre-conception risk factors are emerging from the evidence linking pre-conception health to pregnancy and birth outcomes. Yet data on the prevalence and effectiveness of psychiatric preconception health and care remain scant and inconclusive. The remits of pre-conception advice extend beyond the dilemma of prescribing psychotropic medication in childbearing women. Pre-conception counselling can inform women of the physiological and emotional changes occurring in pregnancy, explore expectations about parenthood and evaluate how the woman’s own experience of being parented may affect her parenting style. Equipping women and their partners with unbiased information through specialist advice will empower them to make an informed decision about their reproductive choices.

The aim of this chapter is to provide a best practice framework to guide pre-conception mental health advice to women with a mental illness. It will not detail the evidence on the association between the exposure of psychotropic medication and adverse outcomes.

In this chapter, we study the p-Laplacian evolution equation for p > 1, that is, the gradient flow of the p-Cheeger energy. Our goal is to use the Hilbertian theory of gradient flows to provide a notion of weak solution to the gradient flow associated to the p-Cheeger energy and prove their existence and uniqueness for initial data in L2. We assume that the metric space is complete and separable and that the associated measure is finite on bounded sets (we keep this assumption throughout the whole chapter). Then, Ambrosio, Gigli, and Savaré proved that the p-Cheeger energy is convex and lower semicontinuous; since it’s the domain of definition is dense in L2 as it contains Lipschitz functions with bounded support, by the Brezis–Komura theorem, for every initial data in L2, there exists a unique strong solution to the abstract Cauchy problem. Our goal is to introduce a notion of weak solutions, improving upon the above definition; we will express the subdifferential of the p-Cheeger energy in terms of the differential structure due to Gigli presented in Chapter 1.