In the early part of the 20th century, Hermann Minkowski developed a novel geometric approach to several questions in number theory. This approach developed into a field called the geometry of numbers and it had an influence on fields outside number theory as well, particularly functional analysis and the study of Banach spaces, and more recently on cryptography and discrete optimization. This chapter covers those aspects of the geometry of numbers that are most relevant for the second part of the book on optimization. Topics include the basic theory of lattices (including Minkowski’s convex body theorem), packing and covering radii, shortest and closest lattice vector problems (SVPs and CVPs), Dirichlet-Voronoi cells, Khinchine’s flatness theorem, and maximal lattice-free convex sets. Several topics like lattice basis reduction and SVP/CVP algorithms are presented without making a rationality assumption as is common in other expositions. This presents a slightly more general perspective on these topics that contains the rational setting as a special case.

Adolf Hitler sought to transform Berlin as capital of the Nazi Third Reich. Berlin’s topography of power changed dramatically, including the construction of grandiose new buildings and the imposition of ideological visions on both housing and public spaces. Nazi plans for ‘Germania’ were never fully achieved; instead, the impact of Nazi rule was destructive. The overwhelming majority of Berlin Jews who did not flee abroad in time were ultimately deported to ghettos, slave labour, and extermination camps, as were Roma and Sinti; only a tiny minority survived. During the war, Berlin’s already militarised architecture was massively augmented by the building of flak towers, bunkers and air-raid shelters. By the time of defeat in 1945, the city lay in ruins, devastated by the effects of bombing and having been forced to continue fighting to the last. Surviving Berliners eked out a living among ruins, searching for food and fuel, and fearing rape or robbery by Soviet troops.

How was the British imperial state transformed from an ancien régime in 1819 to a liberal one that depended upon a new science of government?

Ganciclovir is related to aciclovir but is more active against CMV. It is also more toxic. It causes profound myelosuppression when given with zidovudine; the two should not be given together, particularly during initial ganciclovir therapy.

By the 1960s, quite different societies were developing in the two halves of Berlin, one under capitalist auspices and the other under communism. With West German Chancellor Willy Brandt’s Ostpolitik in the early 1970s, a series of treaties resulted in recognition of the GDR and easing of travel restrictions from west to east. Under GDR leader Erich Honecker, the modernity of ‘actually existing socialism’ was expressed in an energetic housing development programme, significant public buildings, and some leeway for cultural experimentation; yet the secret police or Stasi continued to expand, and the physical apparatus of the Wall was further refined. Divergent identities between East and West were evident not only in differing political ideologies, social profiles, and degrees of freedom, consumerism, and material well-being, but also in patterns of historical memorialisation. In the 1980s the economic and environmental decline of the GDR became ever more visible. Growing unrest culminated in challenges to communist rule and the opening of the Wall on 9 November 1989, and led to German reunification in October 1990.

The revival of the heliocentric model by Copernicus in the sixteenth century led to speculation about planets orbiting other stars. In a heliocentric model, stars must show annual parallax as the Earth moves around the Sun.

With the foundation of Imperial Germany in 1871, Berlin became capital of an enlarged and increasingly significant Empire, or Reich. Unification precipitated an economic boom, soon followed by a crash; but industry continued to expand rapidly, with an exponential growth of the population through immigrants seeking work in the city. In the half century following unification the population quadrupled, from around one million in 1871 to nearly four million in the expanded metropolis of Greater Berlin in 1920. New forms of transportation altered the dynamics of the city, while adequate housing and public health became matters of growing concern. In an era of competing nation states, Imperial Germany too began to acquire overseas colonies, including in southwest Africa and eastern Africa (today’s Namibia and Tanzania) as well as elsewhere in Africa and the Pacific. But defeat in the First World War shattered the dreams of the newly rich, the imperialists and colonisers, those who trumpeted racial superiority and dreamed of world mastery.

The argument here is that German industry and finance were preprogrammed to participate in the murder of the Jews by decisions made before the war that could not be reversed. Big business thus collaborated fully in the process, becoming “bagmen” and “fences” for stolen Jewish property and providers of goods and services to death camps.