Mid-century corporate executives received most of their compensation from salaries and cash bonuses, making them highly vulnerable to the top marginal income tax rates. Because executives were also able to negotiate custom pay packages, they adopted policies to dodge those rates. Most importantly, executives were influential in spreading and legitimizing tax dodging not only within their own companies, where they could affect the nature and structure of their own pay packages and those of their employees, but they could influence compensation, benefit and perquisite, and reimbursement policies at thousands more companies in part by lobbying for legislative and regulatory action that officially sanctioned these new policies. Although executive tax dodging was not the only reason corporations partnered with the government in creating the plans that form the backbone of our employer-sponsored retirement and health insurance systems, as well as the stock-based compensation that helped to drive the wealth and pay gap between executives and workers, it was a powerful force in a system that has endured for decades.

The large sieve takes various forms, as a mean square upper bound for a trigonometric polynomial at well-spaced points, as a mean square upper bound for the distribution of a set of integers into arithmetic progressions, and as a mean square upper bound for character sums. We take the trigonometric form to be fundamental and derive the other versions from it.

The distribution of prime numbers, one of the central questions in number theory, is elegantly described by the Prime Number Theorem, which states that the number of primes up to a given value x grows asymptotically like x/log(x). We present a complete proof of this theorem following the approach of Newman and Zagier, using only basic tools from complex analysis.

A deeper refinement is provided by the von Mangoldt exact formula, which expresses the Chebyshev weighted prime-counting function in terms of the zeros of the Riemann zeta function. This formula makes explicit the connection between prime numbers and the non-trivial zeros of the zeta function, showing how each zero contributes to the oscillations in the distribution of primes and to deviations from smooth, regular growth.

The Riemann explicit formula offers an even more direct relationship between the number of primes up to x and the zeros of the zeta function, capturing the deviations from the Prime Number Theorem as a sum over these zeros. Together, these formulas deepen our understanding of prime distribution and highlight the profound connection between analytic and arithmetic structures.

We outline the derivation of both exact formulas, omitting some of the most technical steps.

Chapter 1 introduces the War of the Spanish Succession as it unfolded in both Europe and Spanish America, emphasizing its profound impact on the social and political fabric of New Spain. It argues that persistent fears of foreign invasion and economic instability generated heightened anxieties across racial, class, and regional lines. The chapter examines how viceregal authorities mobilized mixed-race militias, co-opted Creole elites, and engaged municipal governments to defend the viceroyalty. These efforts were not only reactive but also instrumental in reinforcing loyalty to the Bourbon cause. At the same time, the war imposed economic burdens on broad sectors of society, even as it created opportunities for advancement among Creole families with access to patronage networks. Through sermons, official correspondence, and public rituals, the chapter highlights how political uncertainty intensified the stakes of dynastic legitimacy and fueled a sense of urgency surrounding the future of the empire.

This chapter examines the key principles of applying mental health and capacity legislation in the perinatal period. The four nations of the United Kingdom have different legislative frameworks. England and Wales are governed by the same legislation – the Mental Health Act 1983 (MHA) and the Mental Capacity Act 2005 (MCA), although with some minor variations. Scotland has an entirely different framework – the Mental Health (Care and Treatment) Act 2003 and the Adults with Incapacity Act 2000. Northern Ireland is in a (slow) transition from having mental health legislation (the Mental Health (Northern Ireland) Order 1986) sitting alongside, in effect, no formal framework for thinking about capacity, to ‘fused’ legislation (the Mental Capacity Act (Northern Ireland) 2016) with no stand-alone mental health legislation. This chapter focuses on the position in England and Wales, primarily because it has the largest body of case law to help understand how to think through the dilemmas covered; for those in other parts of the United Kingdom grappling with those dilemmas, the most useful resource is the BMA’s Ethics Toolkit which has specific sections for each of the nations.

For many people the journey to becoming a parent is not a straightforward one. The following chapter examines how experiences with infertility, assisted conception and perinatal loss can impact on an individual’s mental health, relationships with others and future children. The chapter is divided into two sections: the first examining infertility and assisted conception, and the second focusing on perinatal loss. Each section will briefly define key concepts, definitions and prevalence before describing the psychosocial impact of these difficulties and treatment options.

Although this chapter is divided into two sections, it should be noted that women who experience infertility and utilise artificial reproductive techniques (ART) are also highly likely to have experienced some form of perinatal loss. The experience of infertility alone is its own form of loss and individuals who experience it may have a grief response and share similar psychological difficulties to those who experience pregnancy loss or baby loss. The aim of this chapter is to increase awareness around the often complex and lengthy journey to parenthood, and to support clinicians in understanding the experiences of those they come across at various stages of this journey and how they can best support them.

This chapter begins with the study of Dedekind domains: integrally closed Noetherian integral domains of Krull dimension one. Dedekind domains are the natural framework for studying rings of integers in number fields.

A key feature of Dedekind domains is that every fractional ideal is invertible, and the set of fractional ideals forms a group under multiplication – in fact, a free abelian group with basis given by the set of prime ideals. This recovers unique factorization, but for ideals rather than elements.

The theory of modules over Dedekind domains is particularly elegant: finitely generated torsion-free modules decompose into direct sums of ideals and are classified up to isomorphism by their rank and Steinitz invariant. Finitely generated torsion modules are classified by their invariant factors, similarly to the case of PIDs.

In arithmetic applications, ramification in field extensions describes how prime ideals factor when extended to larger rings. The explicit factorization of a prime ideal is governed by the factorization of the minimal polynomial of a primitive element modulo that prime.

José María Samper (1828-1888) was born in the provincial town of Honda, Colombia. A prolific author, he combined his writing activities with politics, journalism, and teaching. In his earlier career, Samper became prominent within a young generation of radical liberals who advocated universal and direct suffrage, the separation of church and state, and the abolition of the standing army. In 1855, In what was perhaps one of the first usages of the expression “Latin American,” Samper proposed the formation of a federation of Spanish American republics to defend their emerging “democracies” and their interests from the ambitions of the European monarchies, the Brazilian Empire, and US expansionism (Reflexiones sobre la federación colombiana, 1855). “Colombia” was the name Samper gave to his proposed union. Later in his life, however, Samper tempered the radicalism of his youth and moved closer to the Conservative Party. Yet in the convention that adopted the 1886 centralist, “conservative” constitution, Samper’s interventions revealed that he continued defending key liberal principles. Samper’s selected passages in our volume come from his earlier liberal phase.

Perinatal anxiety is characterised by intense symptoms of anxiety and fear and includes affective, cognitive and behavioural components. Anxiety disorders include generalised anxiety disorder, panic disorder, specific phobias and social anxiety disorder. Obsessive compulsive disorder and post-traumatic stress disorder are also included here to remain in line with the majority of published evidence, and because of agreement that anxiety is a central characteristic of these disorders. Some level of antenatal anxiety is common but when it becomes extreme or persistent it can be associated with maternal, fetal and infant morbidity, as well as emotional and cognitive problems in children and teenagers and risk of the child developing an anxiety disorder.

This chapter provides an overview of the different anxiety and associated disorders. It focuses on issues to consider in perinatal anxiety, namely the consideration of symptoms versus disorders, normal versus pathological anxiety and pregnancy-specific versus general anxiety. The subsequent sections of the chapter focus on each of the anxiety and associated disorders in turn: their epidemiology, aetiology, presentation, course and outcomes including for the infant and family; and, finally, their management.

Andrés Bello (1781–1865) was born in Caracas, Venezuela. Early in his career he occupied various administrative positions in the government of the Captaincy General. In 1810, the newly inaugurated Junta of Caracas appointed him secretary of the commission sent to Great Britain to request support for the new government. He remained in London for nineteen years. He lived in Santiago for the remaining thirty-six years of his life, serving as government official, senator, and rector of University of Chile. He was in addition a poet, a literary critic, and a jurist. His most important works include Principios de derecho de jentes (1832), Gramática de la lengua castellana (1847), and the Civil Code of the Republic of Chile (1855). A towering intellectual figure in Spanish America, he has only recently been translated into English. The current selection concerns a specific development in Ecuador but involves a central principle of international relations where Bello took a strong position against foreign intervention, especially from emerging states that sought the political and military assistance of stronger ones. The polemical tone of the article is typical of the press debates in Chile in the 1840s.

Of humble origins and mixed race, Bernardo Monteagudo (1789–1825) was born in Tucumán, in the River Plate (Argentina today). He graduated in law from the University of Chuquisaca and soon became involved in the wars of independence against Spain, first in Upper Peru and then in Buenos Aires, where he stood out for his radical republicanism, which he originally displayed as a polemicist in newspapers. Embroiled in the internal conflicts of the revolutionary movement in the River Plate, he was forced into exile in 1815. In 1817, he joined the army of José de San Martín, which drove the Spaniards out of Lima in 1821, when Peruvian independence was declared. Monteagudo became San Martín’s right hand and was practically in charge of governing Lima while San Martín continued the fight against the royalist forces elsewhere. Monteagudo’s authoritarian rule provoked wide resentment and, following two days of riots, he was expelled from the city in 1822. Written a year later as a defence of his actions in government, his Memoir offered a systematic examination of the conditions that, in his view, made democracy unworkable in Peru.

This chapter examines the enumeration of ethnic populations in the census, where ‘the tribe question’ has been included since 1948. I trace its evolution – from its origins as self-evidently important with a self-evident list of groups – through numerous changes up to 2019. The powerful social imaginary of ‘42+ tribes’ comes from the 1969 census, despite the numerous changes since then. I show how changes in classifications over time, as well as the way they have been used and narrated by the state, reveal the multiple political purposes of classifying and counting ethnicity. In the colonial period, this centred on ethnic population distribution to support indirect rule via ethnicity, as well as tax collection and labour control. In the postcolonial period, ethnic demographic posturing for electoral purposes or ‘the tyranny of numbers’ became a major driver of interest in ‘the tribe question’. However, since 2009, the census has also been a site of recognition for minorities and of the painting of a portrait of a nation defined by its diversity. In this chapter, I also show how the quintessentially unambiguous nature of ethnic census codes has been rendered ambiguous in useful ways.