The chapter gives the historic background in bounded arithmetic and describes how it lead to the development of the presented theory. It lists prerequisites and some notation and terminology to be used.

1.1 Introduction

The first-order equations–linear and non-linear–and the second-order linear equations, with constant or variable coefficients, are considered in this chapter. For solving the first-order equations, familiar methods such as the method of separation of variables or a method that can be reduced to this are used. Also discussed are the exact differential equations or those equations that can be reduced to this form using a suitable integrating factor (IF). We also emphasize the peculiarities that may arise in an initial value problem (IVP) when sufficient conditions imposed in the Cauchy–Peano existence theorem or in the method of Picard's iterations are not satisfied. Many exercises deal with the maximal interval of existence of a solution to an IVP.

Only second-order linear equations are considered here. The non-linear equations or, more generally, the two-dimensional systems of first-order equations are treated in Chapter 5 on qualitative analysis. The treatment of equations with constant coefficients is straightforward. The equations with variable coefficients are more difficult to deal with, and, in general, it is not possible to obtain the solution in explicit form. However, the structure of solutions to the homogeneous and inhomogeneous equations is well-understood.

A general first-order ordinary differential equation (ODE) takes the form f(t, x(t), x′(t)) = 0, where f is a given function and x = x(t) is the unknown function to be determined. A general theory for the above equation is rather difficult.

The history of FD approximations goes back even further than that of calculus. The classical definition of a derivative is an example of a very simple FD formula. Although many basic FD properties follow quite immediately from Taylor expansions, numerous additional perspectives have proven very helpful both for deriving a wide range of FD formulas, and for understanding their different features (such as their accuracy near boundaries vs. in domain interiors). This chapter focuses on FD approximations on equispaced grids in one dimension, generalizing quite directly to Cartesian grids in higher dimensions. Further generalizations to mesh-free node layouts in multiple dimensions are discussed in Chapter 5.

Chapter 1 presents a brief overview of the book and the basics on inpainting, visual perception and Gestalt laws, together with a presentation of the Fitzwilliam Museum dataset of illuminated manuscripts, selected to represent different types of damage and consequent restoration challenges, which will be used throughout the book.

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.

This chapter explores the pivotal role of modeling as a conduit between diverse data representations and applications in real, complex systems. The emphasis is on portraying modeling in terms of multivariate probabilities, laying the foundation for the probabilistic data-driven modeling framework.

The K-stability of Fano varieties has been a major area of research over the last decade, ever since the Yau–Tian–Donaldson conjecture was resolved. This is one of the first books to give a comprehensive algebraic treatment of this emerging field. It introduces all the notions of K-stability that have been used over the development of the subject, proves their equivalence, and discusses newly developed theory, including several new proofs for existing theorems. Aiming to be as self-contained as possible, the text begins with a chapter covering essential background knowledge and includes exercises throughout to test understanding. Written by an author at the forefront of developments in the area, it will be a source of inspiration for graduate students and researchers who work in algebraic geometry.

The geometric motivation for the theory is combinatorial data associated with matrices, vector arrangements, hyperplane arrangements, and subspaces of real vector spaces. Interpretations of this data are given in terms of linear algebra, discrete geometry, and the Plucker embedding of the Grassmannian. Elementary proofs of cryptomorphisms for realizable oriented matroids are provided. The chapter finishes with an application of Gale Diagrams.

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.