In Introduction we mostly discuss nearest neighbour Markov chains which represent one of the two classes of Markov chains whose either invariant measure in the case of positive recurrence or Green function in the case of transience is available in closed form. Closed form makes possible direct analysis of such Markov chains: classification, tail asymptotics of the invariant probabilities or Green function. This discussion sheds some light on what we may expect for general Markov chains. Another class is provided by diffusion processes which are also discussed in Introduction.

In this chapter, we introduce some background knowledge.

Minkowskian geometry provides a mathematical model of spacetime that resolves a number of perplexing issues that had arisen in physics by the dawn of the twentieth century. The model leads to surprising predictions for physics, which have been confirmed experimentally. In the chapter we review several well-known features of Minkowski spacetime, including Lorentz transformations, time dilation and Lorentz contraction, as well as its conformal compactification.

This chapter introduces the notation used in the book and discusses the mixed integer programming (MIP) computational framework in which heuristics are developed, used, and evaluated. The chapter starts by formally definining MIP and presenting the basic complete algorithms to solve it. Then, the more important building block concepts at the core of primal heuristics are presented, as well as the way in which they are incorporated in the MIP framework and their impact.

This chapter discusses 1-polygraphs, which are simply directed graphs, thought of here as abstract rewriting systems: they consist of vertices, which represent the objects of interest, and arrows, which indicate that one object can be rewritten into another. After formally introducing those, it will be shown that they provide a notion of presentation for sets, by generators and relations. Of course presentations of sets are of little interest in themselves, but merely used here as a gentle introduction to some of the main concepts discussed in this work: in particular, the notion of Tietze transformations is introduced, which generates the equivalence between two presentations of the same set. In this context, an important question consists in deciding when two objects are equivalent, i.e., represent the same element of the presented set. In order to address it, the theory of abstract rewriting systems is developed.

In this chapter the main topic of the book is put in context.

The subject of elliptic curves is one of the jewels of nineteenth-century mathematics, originated by Abel, Gauss, Jacobi, and Legendre. This book, reissued with a new Foreword, presents an introductory account of the subject in the style of the original discoverers, with references to and comments about more modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic.

After an informal preparatory chapter, the book follows an historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions. This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves.

Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics, with many exercises with hints scattered throughout the text.

This chapter formally defines a financial market and associated constructs, and lays the foundations for arbitrage pricing and dynamic replication (or hedging) through trading strategies.

We motivate the book based on categorical formulations of recursion and induction. We also discuss the background that readers should have and preview many of the topics in the book.

This chapter reviews the use of enriched diagrams and enriched Mackey functors in equivariant homotopy theory and the theory of stable model categories. The results outlined here are not used directly in the main body of the work, but they provide an important motivating context. The goal of this chapter is, therefore, to outline some of the main ideas and provide numerous references to the literature for further treatment.

Accurate predictions with quantifiable uncertainty are essential to many practical turbulent flows in engineering, geophysics, and astrophysics typically comprising extreme geometrical complexity and broad ranges of length and timescales. Dominating effects of the flow instabilities can be captured with coarse-graining (CG) modeling based on the primary conservation equations and effectively codesigned physics and algorithms. The collaborative computational and laboratory experiments unavoidably involve inherently intrusive coarse-grained observations – intimately linked to their subgrid scale and supergrid (initial and boundary conditions) specifics. We discuss turbulence fundamentals and predictability aspects and introduce the CG modified equation analysis. Modeling and predictability issues for underresolved flow and mixing driven by underresolved velocity fields and underresolved initial and boundary conditions are revisited in this context. CG simulations modeling prototypical shock-tube experiments are used to exemplify relevant actual issues, challenges, and strategies.

This chapter covers the basics from real analysis to linear algebra and the theory of computation that is foundational for the rest of the book. A careful discussion of different models of computation is taken up, which discusses several issues that are often ignored in other presentations of optimization theory and algorithms.

The text proper of the book begins with Littlewood’s three principles. The first – ‘any measurable set is nearly a finite union of intervals’ – is essentially regularity of Lebesgue measure. The second – ‘any measurable function is nearly continuous’ – is Lusin’s Theorem. The third – ‘any convergent sequence of measurable functions is nearly uniformly continuous’ – is Egorov’s Theorem. Then what will be needed from general topology is summarised, with references, going as far as para-compactness. Modes of convergence – in measure (in probability), almost everywhere (almost sure), etc. – are discussed. The Borel hierarchy – the result of applying, to (say) the open sets, the sigma and delta operations (union and intersection) alternately – is developed, as far as the Souslin operation. Analytic sets – much used in the book – are briefly treated here.

This section gives a general overview of abelian model structures and their homotopy categories. It is also meant to be a survey of the most fundamental examples of such homotopy categories. These include the chain homotopy category of a ring, the derived category of a ring, and the stable module category of a quasi-Frobenius (or Iwanaga–Gorenstein) ring.

I will describe the earlier efforts in both hydrodynamic instability and turbulence mixing research and provide a broad non-mathematical overview of the significance of turbulence mixing on scientific and engineering applications. I will briefly explain several varied applications in which hydrodynamic instability plays a critical role, namely, inertial confinement fusion (ICF), supernova explosions, solar prominences, paintings, and combustions and detonations, among others, to provide the reader with an idea of what will be discussed later in the book.

We present basic ideas used throughout the book: sets, lists, relations, equivalence relations, and equivalence classes.