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.

The epilogue examines the persistence of the term ‘achievements’ in Egyptian governmental media today, which is indicative of the concept’s resilience. This persistence raises an important question around the social and historical reasons undergirding the continuity of achievement praxis. Why are cultural and media institutions reproducing the achievement state in Egypt? The answer would seem to be that the current bureaucratic apparatus inherited, via institutional means, certain ways of thinking and working established after the 1952 revolution. This simple answer belies my ethnographic experience, because contemporary bureaucrats – with few exceptions – have a very faint sense of the history of the bureaucratic apparatus prior to their own entry into the workforce. A more likely answer, I suggest, is that the institutional context within which bureaucrats work did not change in some identifiable ways since 1952. The continuity of achievement praxis is tied to the institutional environment in which it thrives, rather than a conscious will among state officials transmitted across generations.

Theatrical presentation encompasses diverging perspectives on water ecologies, ecological divisions and extremes of wet and dry in tropical and desert climates. While twentieth-century drama points to how water sources in Australia have been divided up to restrict access through land proprietorship, polarising attitudes and racial injustice, innovative twenty-first century performance emphasises the interconnectedness of water flows, seepage and below ground storage. An appreciation of water flow is particularly evident in First Nations performance, which includes the influential work of Bangarra Dance Theatre. Performance explores values and practices that resist the way water is polluted and detrimentally reconfigured in binary divisions to restrict access and divert flows and highlights the need for water availability for all species in a climate change era. Australian theatrical performance points to emotional feelings and values that protect and preserve water and its river flows even as human impact on the climate means its patterns are no longer predictable.

This chapter takes up a series of popular folktales about radical inequality (Aesop, Gevia ben Psisa/ben Qosem) to argue that provincial communities become increasingly interested in reframing quotidian interactions as legal interactions. Legal dialogue came to be imagined as a register of discourse capable of controlling powerful people. Interestingly, however, all of these stories feature a protagonist somehow marked as physically deformed or otherwise grotesque: Aesop was the ugliest slave imaginable, Geviah a hunchback. The very bizarreness of these characters offers a standing challenge to normative understandings of power: in each case, it is the most degraded members of society who manage to wield legal logos to control their superiors, society’s notional elites.

Building on Chapter 1, Chapter 2 presents an in-depth case study of the fourth major organic farming movement in interwar Germany, biodynamic agriculture. With its Demeter and Weleda brands, this was the organic current that was most successful in garnering consistent support from Nazi patrons, and it remains the most high-profile form of organics in Germany today. Early biodynamic proponents particularly emphasized the ecological dimension of their work, framing their approach as the way to “heal the earth” from environmental harm. In part because of pre-existing ties to several groups of Nazi activists, the biodynamic movement flourished for much of the Third Reich until falling prey to intra-party disputes in 1941. Leading biodynamic figures worked closely with life reform officials within the Nazi apparatus, illustrating the active connections linking multiple strands of environmental advocacy across far-flung sectors of the regime. Through a detailed critical account based on previously unused archival sources, the chapter carefully delineates the reasons for the rise and eventual fall of the Reich League for Biodynamic Agriculture, concluding with a differentiated assessment of the space available for grassroots environmental initiatives in Nazi Germany.

This chapter analyses the left periphery of PIE with specific reference to the interaction between pragmatic fronting (topicalisation and focalisation) and clitic placement (Wackernagel’s law). This constitutes a mapping out of the CP layer in PIE, which forms a crucial part of relative clause structure, and lays the groundwork for analysing the precise syntactic behaviour of the relative pronoun, *REL.

Early Christianity, and the Pauline letters specifically, was concerned with questions of legality and its transcendence. In a process analogous to that described in the previous chapter, the flesh of Christ holds out the possibility of transcending the law itself, and remaking oneself and one’s community in the wake of the historical disruptions of the first century BC/AD. Communities throughout the empire, intrigued by the possibility of taking on new and different laws or, by contrast, freeing themselves from all laws, saw in the possibility of Christianity the opportunity for transformation. For Paul, real law is not concerned about materiality, but transcendence. There are important and underexplored commonalities between Paul’s interpretive moves and those of the sophists of the following chapter. The key body, however, is not the body of the orator, but the perfect body of Christ, which deserves to be imitated.

Juana Manso (1819–1875) has been described as “the first woman to be appointed to an official government position and arguably the most radical feminist in nineteenth century Argentina.” Born in Argentina, she left the country with her family in 1840, after her father was exiled by the Rosas regime. When she returned to Buenos Aires in 1853, she was already an experienced educator, author, and editor. In 1859, Domingo Faustino Sarmiento, then head of the Elementary Schools Department, appointed her as director of the Escuela de Ambos Sexos, the first co-educational school in Buenos Aires, and editor of the educational journal Anales de la Educación Común. As Argentine president, Sarmiento appointed her to the Board of Public Instruction in 1871. Manso not only advocated for women’s education but also for popular education more widely – her selected passages in our volume, originally published in the Album de Señoritas, serve to illustrate her most passionate interests in these causes.