In this chapter we review basic knowledge of discrete probability theory. We also take a closer look at conditional expectations and filtrations in the discrete case, as these are important tools in financial mathematics. In particular, we will try to illustrate the ideas behind these two concepts and why they lead to the corresponding definitions. A reader having a good understanding of those concepts can skip this chapter.

This chapter considers the main forms of investment treaties, exploring some leading examples such as the Energy Charter Treaty, the United States–Mexico–Canada Agreement and the Comprehensive Trans-Pacific Partnership. It also discusses the role of some global organizations in facilitating international investment, notably the World Trade Organization and the World Bank.

This chapter develops the Navier–Stokes equations using a Lagrangian description. In doing so, the concept of a stress tensor and its role in the overall force balance on a fluid element is discussed. In addition, the various terms in the stress tensor as well as the individual force terms in the Navier–Stokes equations are investigated. The chapter ends with a discussion on the incompressible Navier–Stokes equations.

This chapter serves as an introduction to the concept of conservation and how conservation principles are used in fluid mechanics. The conservation principle is then applied to mass and an equation known as the continuity equation is developed. Various mathematical operations such as the dot product, the divergence, and the divergence theorem are introduced along the way. The continuity equation is discussed and the idea of an incompressible flow is introduced. Some examples using mass conservation are also given.

In this chapter, a concept known as scaling is introduced. Scaling (also known as nondimensionalization) is essentially a form of dimensional analysis. Dimensional analysis is a general term used to describe a means of analyzing a system based off the units of the problem (e.g. kilogram for mass, kelvin for temperature, meter for length, coulomb for electric change, etc.). The concepts of this chapter, while not entirely about the fluid equations per se, is arguably the most useful in understanding the various concepts of fluid mechanics. In addition, the concepts discussed within this chapter can be extended to other areas of physics, particularly areas that are heavily reliant on differential equations (which is most of physics and engineering).

In addition to the continuity equation, there is another very important equation that is often employed alongside the Navier–Stokes equations: the energy equation. The energy equation is required to fully describe compressible flows. This chapter guides the student through the development of the energy equation, which can be an intimidating equation. A discussion on diffusion and its interplay with advection is also included, leading to the idea of a boundary layer. The chapter ends with the addition of the energy equation in shear-driven and pressure-driven flows.

This chapter considers the role of essential security as a justification for breaching obligations under investment treaties. It notes the link between the protection of national security and the doctrine of necessity under customary international law. The chapter looks at some of the key national security screening instruments of various countries, notably the US FIRRMA.

This chapter introduces the concept of foreign direct investment, tracing its history and economic justifications. The chapter goes on to explain the sources of international investment law, the most significant of which are treaties.