We now address additive questions involving prime numbers, particularly the problem of expressing an integer as a sum of k primes:

Martina Barros Borgoño (1850–1944) was a pioneering Chilean who advanced women’s rights in significant ways, not least in terms of advocating for suffrage, but also in terms of a critical approach to nineteenth-century thinking on gender distinctions. Born in a socially distinguished milieu, Martina Barros encountered John Stuart Mill’s The Subjection of Women (1869) at a young age and embarked on the translation and commentary of the source by 1872. Barros’ prologue celebrated some aspects of Mill’s work, but bemoaned others. She acknowledged the role of her husband, the liberal physician Augusto Orrego Luco, in the writing of the prologue, but affirmed that the ideas were her own in her autobiography. Both had previously read Mill’s On Liberty (1859), and shared his condemnation of the tyranny of customs, challenging gender traditions and participating actively in publications and intellectual circles. Her views, however, were far from radical. As she stated in her autobiography, “my aim in promoting the independence and culture of women was not to make them rivals of men, but rather their dignified companions.”

Eating disorders can have a profound impact on women during the pre-conception, antenatal and postnatal periods, and this has implications for their care and treatment. This chapter describes the rate, course and risk factors for eating disorders within the context of the perinatal period. It covers what is known from current research and clinical evidence about the effect of the most common eating disorders on pre-conception health, pregnancy and birth outcomes. Drawing on existing clinical guidance and research evidence, it provides an overview of the guidance and recommendations for the assessment, management and treatment of eating disorders from pre-conception through to the postnatal period.

Chapter 1 examines three distinct organic farming movements that arose in Germany in the 1920s as pioneering examples of environmental ideals in practice. Though disparate in their origins and political commitments, all three organic tendencies found considerable common ground with Nazism, in some cases well before the Nazis came to power. Tracing their divergent fates under Hitler’s regime after 1933, the chapter offers a dense historical reconstruction of early environmental ambitions that were sometimes thwarted and sometimes fulfilled through active cooperation with Nazi agencies. A core aspect of the analysis centers on the implicit and explicit influence of racial ideologies within the emerging organic milieu and the opportunities and challenges this opened up under the conditions of Nazi rule. The chapter is built around a comprehensive range of archival sources, many of which have never been examined before, and provides the fullest portrait yet in German or English of the inception of organic farming currents in the context of proto-environmental politics.

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.

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.

Millions of women and girls worldwide experience violence. Violence against women and girls takes many forms, including physical, emotional and sexual violence and abuse, which is associated with a range of adverse impacts on women, their families and society as a whole. Health professionals supporting women during the perinatal period should assess the risks posed by exposure to previous or current violence and how this may affect them during pregnancy. As an important risk factor in a woman’s mental health presentation, psychiatrists working with pregnant and postpartum women should consider the presence of violence in their formulation; it can increase the risk of anxiety, depression and post-traumatic stress disorder (PTSD). Domestic violence and abuse increase the risk of domestic homicide and may play a role in many perinatal suicides. Sensitive assessment and effective management of women exposed to violence can improve engagement with mental health services and response to treatment.

In 1962, John F. Kennedy proposed withholding for taxes on dividends and interest to close the large gap between dividends and interest paid and reported. Despite the familiarity with wage withholding, the proposal encountered an enormous wave of public opposition, generating one of the most significant letter-writing campaign ever mounted. Congress relented and stripped the dividend and interest withholding provision from the bill in favor of new information reporting requirements. Why did dividend and interest withholding generate such a populist revolt? In part, the populism on this issue was manufactured by the business community. Banks and corporations mobilized their depositors and investors to contact their congressmen to protest the proposal. This is only part of the story, however. The industry-led campaign struck a chord with taxpayers who had become disaffected by the special tax preferences and shelters enjoyed by high bracket taxpayers. They viewed omitting dividends and interest as their form of self-help, while others were indignant that Congress would attack tax evasion by going after them before solving high-end tax evasion first.

This concluding chapter puts land at the heart of the “China model,” linking legal, fiscal, financial, and political features of the system to explain the roots of China’s contemporary economic challenges, including the real estate crisis, land-backed debt, and abortive property tax initiative. It also extends the theory beyond the Chinese case in three ways. First, it revisits the paradigmatic case of post–Glorious Revolution England in light of China’s experience, suggesting that, in the context of technological change, property rights over land were less secure and governance less democratic in the early eighteenth century than presented in some of the development literature. Second, it examines the relationship between the ease or difficulty of using law to reassign land rights and promotion of transformative economic growth in the case of contemporary India. These comparisons point to the significance of regime type—authoritarian vs. democratic. Regime type shapes the ease with which the state can reassign land rights and how the state manages the conflict that results from the redefinition of property rights. Third, the chapter examines the redefinition of property rights over personal data as a driver of growth in the new information economy as well as a new source of conflict.

This chapter theorises ethnicity as a mode of thought and identification around which ways of being, acting and relating are organised. It is one among many possible anchors for identification, solidarity and difference, though it is the most prominent in Kenya. I discuss how this became so, describing identity and community before colonialism, and offering a history of how ethnicity organised social life under and after colonial rule, especially around elections. I provide a sketch of varied ethnic identifications in Kenya, demonstrating immense variety, not all of which obviously fit an ethnic framework, and many of which entail politics quite different from the ‘big 5’ which dominate studies of elections. Finally, I situate the case of Kenya in a comparative context, highlighting key features of how ethnic classification has operated in Kenya, including reification, colonial penetration, nationhood, demography, and differences between direct and diffuse effects of identification. This section shows that both ethnicity and its classification can be conducive to pluralism and solidarity in Kenya, but perhaps not in other contexts.

Our next aim is to extend the results of Chapter 2 and introduce a notion of weak solution to gradient flows in metric measure spaces in a fairly general setting. Our main assumption is that the functional only depends on the differential of a function. In particular, this setting covers the case when the functional only depends on the function through its minimal p-weak upper gradient. In this entire chapter, we assume that p > 1 and that we work with a convex and lower semicontinuous functional defined on L2, which is given by a composition of the differential and a non-negative, continuous, convex, and coercive functional defined on the cotangent space. We first present the general framework under the minimal structural assumptions described above. Then, we apply the newly developed techniques to study a specific functional with inhomogeneous growth, which is the sum of two Cheeger energies for different exponents.