This chapter begins by exploring the concept of legitimacy, which the CCP regime seeks to achieve in part through its project of legal construction. It employs official data and primary documents to present multiple aspects of access to justice nominally afforded by the legal system: training of a cadre of legal professionals, provision of institutions for dispute resolution—including mediation, petition, and litigation, establishment of state-sponsored legal aid, and implementation of an official campaign to imbue Chinese citizens with legal consciousness. It concludes with an assessment of China’s model of legal development, reviewing arguments about law and order, order maintenance, pure legality, normative and prerogative aspects of the dual state, and legal dualism. The illiberal system of law is a powerful tool in the hands of the party-state.

The UK has led the development of specialist perinatal mental health services since the latter half of the last century. However, new innovations in service design and delivery continue to present challenges for commissioners and providers. This chapter reviews the evolution of specialist care across the UK and in Ireland and identifies how services may need to respond to changing demand.

Postpartum psychosis is a condition of great clinical and public health importance. Severe episodes of mental illness in the perinatal period can result in significant distress, may disrupt the developing relationship between mother and child, and have long-term implications for the well-being of the woman, her baby, family and wider society.

In this chapter we will discuss what we know about this condition and its relationship to bipolar disorder, how it might best be defined, what we still need to find out, and consider how it should be managed.

If f is monotonic, then we can estimate S by using the Prime Number Theorem and integration by parts. If f is multiplicative, then we can gain information concerning S by studying the properties of the associated Dirichlet series Σ f(n)n–S.

In their book, The Triumph of Injustice: How the Rich Dodge Taxes and How to Make Them Pay, economists Emmanuel Saez and Gabriel Zucman lamented the retreat of the United States’ tax system from its heyday between the 1930s and 1970s when it was, in their words, “perhaps the most progressive in world history.”1 As seen in Figure I.1, the top rate shot up from 25 percent after World War I to more than 60 percent in the early 1930s and settled at the astronomically high rate of 91 percent for over a decade between 1951 and 1963. That turned out to be the high point for the top marginal rate. Over the next several decades the rate dropped to 70 percent, then 50 percent, and finally a low of 28 percent in the late 1980s. Although it has crept up since then, the top rate has never approached anything close to what it was at mid-century, remaining below 40 percent for the last four decades.

Mariano Otero (1817–1850) was born in Guadalajara, Mexico. A lawyer by training, Otero made early choices in favor of liberalism, advocating a return to the federalism embodied in the Constitution of 1824, although with some alterations, which included the separation of the clergy from politics. He was also concerned about the political instability of the post-independence period. He participated in the liberal revolt of 1846, was part of the Constituent Assembly of 1847, and was one of the members of Congress who opposed the Treaty of Guadalupe Hidalgo on the grounds that the conflict was an illegitimate war of conquest. He became Minister of Foreign Relations in the government of José Joaquín de Herrera in 1848. His premature death was due to the cholera epidemic of 1850. In this selection, which he wrote during the Mexican–American War, he tried to explain the causes that led to defeat, tracing them back to the colonial and independence periods, but eloquently rejecting any justifications based on race.

Personality disorder (PD) is a complex condition, which has been the subject of both debate and research. However, maternal PD has only been the focus of research in the last two decades. In this chapter I discuss that research in the context of what is known generally about PD, as a disorder with a predictable presentation of signs and symptoms; an aetiology, and indications for effective treatment. There has been more research on maternal PD in the last 15 years, which shows that mothers with PD may struggle to care for their children, especially in the postnatal period, and their mental health may also deteriorate during pregnancy. In this chapter I describe the issues described above and discuss how clinicians approach the management of mothers with PD. I place special emphasis on the impact of maternal PD on mother-child relationships and attachment, and the implications for child health.

Selected passages from “Nuestros indios” (1904) by Manuel Gonzalez Prada (1844–1918).

During the first three years of life our brain develops exponentially, and synaptic connections are formed faster than at any other period of our lives. The organisation of its processes is directly affected by early life experiences. It is therefore vitally important that families, healthcare providers and early intervention professionals understand early brain infant development so that they can fully support infants on to achieving their full potential. This chapter provides an insight into the typical development of core occupations and sensory systems by age, in the first year of life, highlighting the essential role that parents, carers and health professionals have at each stage. Sections on sensory deprivation and carers’ mental health issues are included.

This chapter begins with elementary results concerning Euclidean division and the Euclidean algorithm. We show that the algorithm’s complexity, measured in division steps, has logarithmic growth.

Gently introducing the reader to analytic methods in number theory, we present a proof of the divergence of the Euler series that sums the reciprocals of prime numbers. This not only provides an elegant analytic proof of the infinitude of primes but also offers insight into their distribution.

We then turn to classical arithmetic functions and derive recursive formulas for the partition function, culminating in a complete proof of Euler’s pentagonal number theorem.

This chapter also includes the first of three proofs in this book of Gauss’s theorema fundamentale, the law of quadratic reciprocity, which reveals a remarkable symmetry in the solvability of quadratic congruences modulo two distinct odd primes.

These theoretical foundations underpin practical applications, including the Miller–Rabin primality test – a probabilistic method for identifying prime numbers – and RSA encryption, a cornerstone of modern cryptography that relies on the computational difficulty of factoring large integers.

Florentino González (1805–1875) was one of the “founders” of classical liberalism in nineteenth-century Colombia. His early life was marked by the experience of independence since his family was forced to move from their home by the loyalists when he was still a child. He completed his studies in jurisprudence in Bogota in 1825. As Gran Colombia tore apart, González participated in the plot to assassinate Bolívar in 1828, and subsequently suffered prison and exile. He was back in Bogota shortly after Bolívar’s death and became actively involved in politics and journalism for the next two decades, when he held a succession of important posts, including elected member of Congress and State Secretary of Finance. In 1840, he published Elementos de ciencia administrativa, a two-volume treatise about public administration, a subject he then taught at the university in Bogota. He authored a significant number of essays, some of them in the newspapers he edited. Appointed to a diplomatic mission that took him to Lima and Santiago de Chile, he resigned it in 1861 and remained in exile until the end of his life, first in Chile and later in Argentina.

We now study the total variation flow on bounded domains in metric measure spaces. In Section 6.1, we consider the Neumann problem; using the techniques developed in Chapters 4 and 5, we give a definition of weak solution to the Neumann problem for initial data in L2 based on the Gigli differential structure adapted to a bounded domain and prove their existence and uniqueness. We also introduce the notion of entropy solution for initial data in L1. In Section 6.2, we consider the Dirichlet problem for initial data in L2 and boundary data in L1. We prove lower semicontinuity of the associated functional, give a definition of weak solution, and prove their existence and uniqueness.