Solving quadratic Diophantine equations amounts to finding the values taken by quadratic forms, a problem that can be fruitfully approached by finding the equivalents of a given form under change of variables. This approach was initiated by Lagrange and developed to a high level by Gauss. However, the way Gauss did it involved an apparently difficult operation called composition of forms, clarified only later by the concept of Abelian group.

Diophantine equations are polynomial equations for which integer (or sometimes rational) solutions are sought. The oldest examples date from ancient Greek times, and Diophantus in particular solved many such equations. His methods and the questions they raised inspired much of modern number theory, beginning with the work of Fermat and Euler. Euler, and later Gauss, introduced algebraic integers to solve Diophantine equations, implicitly or explicitly using "unique prime factorization" to do so.

The previous chapters established the centrality and importance of Kant’s theory of magnitudes. Before investigating Kant’s views more deeply, Chapter 6 provides critical background concerning the theory of magnitude, which is rooted in Euclid’s Elements. It describes the organization and deductive structure of the Elements to explain tensions between Euclid’s account of number and continuous spatial magnitude that persisted into the eighteenth century. It then explains the theory of proportions, in particular, the crucial definition of sameness of ratio. It also describes the mathematical nature of the continuous spatial magnitudes implicitly defined by that theory, including their homogeneity. Euclid also covers number and arithmetic and he regards numbers as collections of units. Much of the theory of proportions covers numbers as well as continuous spatial magnitudes, although Euclid gives numbers a separate treatment. This commonality inspired the search for a universal mathematics that covered continuous magnitudes more generally as well as number. The chapter traces the history and development of these themes in the Euclidean mathematical tradition, and describes its influence on three of Kant’s immediate predecessors: Wolff, Kästner, and Euler. It explains why these philosophers and mathematicians regarded mathematics as a science of magnitudes and their measurement.

A vehicle in an airstream sets up a pressure field on its surface, resulting in forces acting on it. Thus, the aerodynamic design task becomes: determine the shape that produces a surface pressure distribution yielding optimal flight performance. Based on the principles of flow physics, computational fluid dynamics (CFD) maps out how an aircraft's shape affects the flow patterns around it. Combined with mathematical techniques for shape optimization, CFD offers a powerful tool for sophisticated aerodynamic design. The goal is to achieve those vital features stemming from the concept of a "healthy flow," namely that these specific flow patterns and associated surface pressures are efficient means of generating aerodynamic lift with acceptable drag and are capable of persisting in a steady and stable form over ranges of Mach numbers, Reynolds numbers, angles of incidence, and sideslip embracing the flight envelope of the aircraft. In the parlance of multidisciplinary design and optimization, this chapter talks about the level of fidelity of the models and solutions. L0 methods are based on empiricisms and statistics. L1–L3 are physics-based models. The governing equations in L1 are linear potential flow, in L2 are inviscid compressible flow, and in L3 are nonlinear viscous turbulent flow.

As we saw in Chapter 1, Newton’s laws are valid only for observers at rest in an inertial frame of reference. But to an observer in a non-inertial frame, like an accelerating car or a rotating carnival ride, the same object will generally move in accelerated curved paths even when no forces act upon it. How then can we do mechanics from the vantage point of actual, non-inertial frames? In many tabletop situations, the effects of the non-inertial perspective are small and can be neglected. Yet even in these situations we often still need to quantify how small these effects are. Furthermore, learning how to study dynamics from the non-inertial vantage point turns out to be critical in understanding many other interesting phenomena, including the directions of large-scale ocean currents, the formation of weather patterns -- including hurricanes and tornados, life inside rotating space colonies or accelerating spacecraft, and rendezvousing with orbiting space stations. There is an infinity of ways a frame might accelerate relative to an inertial frame. Two stand out as particularly interesting and useful: linearly uniformly accelerating frames, and rotating frames.

This contribution covers the topic of my talk at the 2016-17 Warwick-EPSRC Symposium: 'PDEs and their applications'. As such it contains some already classical material and some new observations. The main purpose is to compare several avatars of the Kato criterion for the convergence of a Navier-Stokes solution, to a regular solution of the Euler equations, with numerical or physical issues like the presence (or absence) of anomalous energy dissipation, the Kolmogorov 1/3 law or the Onsager C^{0,1/3} conjecture. Comparison with results obtained after September 2016 and an extended list of references have also been added.

We give a survey of recent results on weak-strong uniqueness for compressible and incompressible Euler and Navier-Stokes equations, and also make some new observations. The importance of the weak-strong uniqueness principle stems, on the one hand, from the instances of nonuniqueness for the Euler equations exhibited in the past years; and on the other hand from the question of convergence of singular limits, for which weak-strong uniqueness represents an elegant tool.