The only form of knowledge about ethnicity that officially and permanently attaches to individuals in Kenya is the register of citizens kept by the National Registration Bureau, which issues ID cards. In this chapter, I briefly trace the history of the ID card in colonial labour control practices (not civil registration), but focus on the deeply ambiguous role of ethnicity in registration over recent years. I show how there is a disconnect between the lack of a place for ethnicity in law or regulation surrounding IDs, yet its continued presence in practice. I then examine several cases of minority ethnic community leaders engaged in what I call ‘code seeking’, where they successfully lobbied for recognition as ‘tribes of Kenya’ as a path to securing ID cards – de facto proof of citizenship for people otherwise stateless. However, I also show that other people, in this example, the Galje’el people, a sub-clan of Somalis, have not been and likely will not be successful with this strategy. This chapter draws our attention to the benefits of both classification and vagueness, while remaining vigilant about their risks.

Complex post-traumatic stress disorder (CPTSD) describes the mind’s response to severe and sustained environmental adversity, particularly in the early years of life when we are learning about, and developing adaptive responses to, our environment. There is now consistent evidence that our own early experiences predict outcomes in our children, and probably also our children’s children, and that part of this transmission is mediated by our mental well-being as parents during the perinatal period. In most of the published literature in this field, and in the perinatal mental health literature, the mental health problems studied across the generations have not included CPTSD or the symptom profiles associated with the diagnosis. This is mainly because even recently published studies were designed before the inclusion of CPTSD in the WHO International Classification of Diseases (ICD-11) or are using the USA’s Diagnostic and Statistical Manual (DSM-5), and/or simply because the researchers involved were not aware of the profound significance of CPTSD to intergenerational and perinatal mental health. We will briefly review what makes CPTSD so important, particularly in the perinatal period, for women and their families, for clinicians and researchers, and indeed for decision-makers across society.

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.

José Victorino Lastarria (1817–1888) was a Chilean writer and politician closely aligned with the opposition to the three consecutive governments of Joaquín Prieto (1831–1841), Manuel Bulnes (1841–1851), and Manuel Montt (1851–1861). A liberal, he later turned into a Comtean positivist. As a politician, he served in the Congress and later in government as diplomat and cabinet member. A prolific writer, he was the author of Recuerdos literarios (1878), one of the best, if biased, accounts of Chilean intellectual life in the nineteenth century. He was also the author of La América (1865), Lecciones de política positiva (1874), and numerous contributions to the press. The fragment included here is the introduction to his work on the legacies of the colonial past, which he presented at the first anniversary of the inauguration of the University of Chile (1844). He promoted a view of history that saw in the past the guidance for shaping the present and future, meaning specifically the demolition of colonial institutions and ideas. This presentation became a part of a significant debate on the writing of history in the 1840s.

Francisco Bilbao (1823–1865) was a Chilean writer and political activist educated at the Instituto Nacional. He rose to notoriety when he published an essay, the “Sociabilidad chilena” (1844), condemning the role of both the Catholic Church and the legacies of colonialism in Chile. He was brought to trial for violating the laws regulating press freedoms. As a result, he left Chile for Europe, where he established contact with Edgar Quinet and Hugues-Félicité Robert de Lamennais and witnessed the European revolutions of 1848. Returning to Chile, he founded the Society of Equality in 1850 and participated in the uprising of April 1851, which led to his exile in Peru, Europe, and Argentina, where he died. His principal works, in addition to “Sociabilidad,” are La América en peligro (1862), and El evangelio americano (1864). The essay included here is representative of his views regarding the radical contradiction between Catholicism and republicanism, which was in turn an expression of his views on the struggle between despotism and freedom.

Contrary to what modern observers might have you believe, tax dodging during the 1950s and 1960s was more about tax cuts than tax increases. Faced with a high tax rate it did not support, but, for political reasons, it could not lower, Congress did the next best thing. It riddled the tax laws with “leaks, loopholes, exemptions, and preferences,” while looking the other way at much of the widespread “income-tax chiseling” in American society and only occasionally passing watered down legislation targeting the more high-profile tax dodging schemes.1 In effect, it cut the tax rates implicitly, rather than explicitly, which amounted to a tax cut of the worst kind. It was not transparent, it was not evenly distributed among the taxpayers or even targeted to achieve any policy objective in some cases, and, because it was too unpredictable for taxpayers to rely upon for planning purposes, it was inefficient.

In this chapter we return to the topic of Chapter 3, (small) sieves, which we now treat, at least initially, in some generality. However our objective is to give nothing more than an introduction and some applications to what has become a vast and complex subject. Readers who wish to see the many aspects of the subject in more detail are advised to consult the standard reference on the subject, Friedlander & Iwaniec (2010).

Minkowski’s convex body theorem provides a fundamental bridge between algebraic number theory and Euclidean geometry. The theorem asserts that any sufficiently large, centrally symmetric, convex set in n-dimensional real space must contain a nonzero lattice point.

Classical applications include representing integers as sums of two or four squares, where the existence of lattice points in two- and four-dimensional balls yields concrete arithmetic solutions.

The theorem’s deeper significance appears in the context of number fields, which are finite extensions of the rational numbers. These fields can be embedded into Euclidean space, turning arithmetic problems into geometric ones. The ring of integers and its ideals can be regarded as lattices, making Minkowski’s theorem directly applicable.

This geometric viewpoint leads to central results in algebraic number theory, such as the finiteness of the class group and explicit bounds on its size.

Likewise, the group of units can be regarded as a lattice in Euclidean space via the logarithmic embedding, and Minkowski’s theorem is instrumental in determining its structure (Dirichlet’s unit theorem).

These ideas come together in the analytic class number formula, which relates the size of the class group to the residue of the Dedekind zeta function at s = 1 and to the regulator.

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.

Talking with women in the pre-conception or perinatal periods about psychotropic medication is an essential, sometimes difficult, part of the work of the perinatal psychiatrist. Understanding the current evidence base; knowing how and when to acknowledge the uncertainty inherent in current knowledge and how that translates to the individual woman; balancing risks of medication with risks of not treating and benefits of treating; sharing decision-making while not putting all of the responsibility on the woman; communicating with the woman, her partner, other professionals, services and agencies; and knowing when and how to seek further help or advice, are all essential components of good practice when prescribing in pregnancy and breastfeeding.

The expansion of the Roman Empire into the Mediterranean in the early second century BCE represented a gradual diminution of the independence and autonomy of the Greek cities. At the same time, processes internal to the poleis were moving them in a more elitist direction, as the “big benefactors,” ultrawealthy men who bestowed ever-greater favors on their cities, moved toward monopolizing participation in civic magistracies. The council and other political bodies became off-limits to citizens who were not among the euergetistic elite. Still, democratic institutions and ideas of the previous period persisted, especially in the popular assembly. Christianity, the centralization of administrative power in the Roman Empire under Constantine, and various crises combined to deprive the cities of the last vestiges of dēmokratia in the fourth century CE, when popular assemblies largely disappear from the poleis.

This chapter is devoted to continued fractions, a classical yet often overlooked subject. Continued fractions provide an efficient representation of real numbers, expressing them as sequences of integers through iterative fraction expansions. Every real number has a continued fraction expansion – finite for rationals and infinite for irrationals – with convergents that yield optimal rational approximations. This property makes continued fractions invaluable in Diophantine approximation, where one seeks to approximate irrational numbers by rationals with minimal denominators.

The theory of continued fractions is deeply connected to Möbius transformations, which describe their recursive structure. The action of the group of 2 × 2 invertible integer matrices on quadratic irrationals via Möbius transformations leads to periodicity in their expansions – a characteristic property of quadratic irrationals. This periodicity is intimately related to solutions of Pell’s equation and to units in quadratic rings.

Among the applications of continued fractions are the characterization of integers that are sums of two squares and the solution of some quadratic Diophantine equations (Cornacchia’s algorithm). Beyond number theory, continued fractions have practical applications such as Wiener’s attack on RSA, which exploits the continued fraction expansion of ratios related to the private exponent to break poorly chosen keys.