Linear programming is an important class of convex problems with constraints, where the objective function is linear, and the problem is almost entirely determined by the linear inequality constraints. Most real-world problems involve constrained resources, such as time, space, money, materials, and labor. There are real tradeoffs in solving such problems, which we express as inequality constraints. Some of the many applications include: resource allocation, workforce scheduling, factory layout, space usage, production scheduling, inventory management, crop planning and irrigation, portfolio optimization, supply chain logistics, fleet routing, and healthcare resource and staff allocation. These examples are all characterized by a relatively simple objective function that is linearly related (e.g., profit or cost), and constrained resources. There is no closed-form solution to a general linear program. However, there are accurate and efficient algorithms that scale to large problems. In this chapter, we define the mathematical ingredients of a linear program, provide examples to illustrate several applications, and discuss solution techniques and extensions.

This chapter explores constraints and duality, two of the most important topics in optimization. Most real-world problems are constrained by resources, time, material properties, cost, etc. We begin with Lagrange multipliers and the Karush–Kuhn–Tucker (KKT) conditions to enforce constraints, including geometric interpretations. Next, we explore duality, which provides insights into the interactions between constraints and the objective. The dual problem gives tight bounds on the original problem and provides methods for efficient computation. Duality leads to interior point methods for constrained optimization. Next, we introduce the alternating direction method of multipliers (ADMM), a primal-dual method to handle nonsmooth regularizers. Finally, we introduce infinite dimensional optimization and the calculus of variations. The Euler-Lagrange equations, used to derive equations of motion in physics, also provide optimality conditions for infinite-dimensional optimization problems. We investigate several infinite dimensional optimization problems, such as the classic Brachistochrone and Tautochrone problems.

As a preliminary to the proof of the Hasse–Arf theorem in the next chapter, the trace mapping of differential forms with log pole is constructed and studied. The construction is based on a logarithmic variant of that in the book Residues and Duality. In the last section, the general construction is applied to inseparable extensions of fields of positive characteristic.

This appendix gives some examples of presentations of 2-categories by 3-polygraphs. In many examples, the presented 2-categories are in fact monoidal categories and, actually, PROs.

This chapter presents the basic concepts of session types, using the pi calculus as a core concurrent programming language for which a type system is defined. It assumes some familiarity with the pi calculus and the concepts of operational semantics and type systems. References to background reading are included at the end of the chapter.

This chapter introduces some basic mathematical notions that are used throughout the book. Convex sets and functions, optimization problems, feasible solutions, and optimal solutions are first defined. The chapter then covers duality theory, including the definition of the Lagrangian function and the dual function, which are used to derive the duals of linear programs. Weak and strong duality are then defined and related to certain classes of optimization problems. The Karush–Kuhn–Tucker (KKT) conditions are defined, and their relation to the optimal solution of mathematical programs is discussed. KKT conditions are a fundamental concept used extensively in the book in order to understand the properties and economic interpretations of the various economic models encountered. Subgradients are subsequently defined in order to establish the relation between Lagrange multipliers and the sensitivity of an optimization model with respect to changes in the right-hand side parameters of its constraints. These sensitivity results are also used repeatedly in the book, for instance in order to derive locational marginal prices in chapter 5.

This chapter looks at the extensive body of empirical research bearing on the major governance best practices recommended for boards of directors: (1) majority (and super-majority) independent directors; (2) independent board committees for things like audit and compensation oversight; (3) board diversity; (4) separating the CEO and board Chair roles; (5) reducing director commitments outside of the company, often referred to as “director busyness” or “overboarding”; and (6) avoiding interlocking directorships. The chapter finds that these best practices do not produce any real-world corporate outcomes that we care about. The possible reasons for these failures are considered.

Does agency cost theory work in the real world? The various hypotheses drawn from agency cost theory are considered in light of the relevant empirical evidence. Agency cost theory appears to be able to explain almost anything, but it predicts nothing.

This book concludes by reiterating the importance of avoiding grand narratives in research on sustainable development in international law. While each chapter revolves around its unique theme, my adoption of TWAIL helped unite these separate parts to tell a single story on Africa’s intersection with sustainable development’s legal evolution, conceptualisation, and implementation. Even so, this book is more than just writing about sustainable development or Africa as it deeply explores how international law should evolve, going forward. Finally, I end this book by drawing on TWAIL’s hopeful agenda by foreshadowing my future research interests in re-reading the law and politics of ecological crises as everyday occurrences and not as episodic events in international law.

Branching processes, which are the focus of this chapter, arise naturally in the study of stochastic processes on trees and locally tree-like graphs. Similarly to martingales, finding a hidden branching process within a probabilistic model can lead to useful bounds and insights into asymptotic behavior. After a review of the extinction theory of branching processes and of a fruitful random-walk perspective, we give a couple examples of applications in discrete probability. In particular we analyze the height of a binary search tree, a standard data structure in computer science. We also give an introduction to phylogenetics, where a “multitype” variant of the Galton–Watson branching process plays an important role; we use the techniques derived in this chapter to establish a phase transition in the reconstruction of ancestral molecular sequences. We end this chapter with a detailed look into the phase transition of the Erdos–Renyi graph model. The random-walk perspective mentioned above allows one to analyze the “exploration” of a largest connected component, leading to information about the “evolution” of its size as edge density increases.