We start by introducing the key ingredients in macroeconomic modelling: investment, production, income and consumption, and explain the corresponding equilibrium conditions. Modelling these quantities in discrete time, we describe the multiplier-accelerator model, a classic model of macroeconomic dynamics, and an example of a second-order recurrence equation. We then embark on describing how to solve linear constant-coefficient second-order recurrence equations in general. The general solution is the sum of the solution of a corresponding homogeneous equation and a particular solution. There is a general method for determining the solution of the homogeneous equation, involving the solution of a corresponding quadratic equation known as the auxiliary equation.

A reduced order model for marine vehicle dynamics is the simple linear spring-mass-damper system. However, the various terms in the equation of motion differ in detail from their mechanical counterparts. The usual balance between mechanical inertial, damping, and stiffness loads with external forcing is maintained, but now includes additional effects reflecting the presence of the fluid. Individual coefficient matrices correspond to the mass of the platform plus the mass of the water being accelerated; the linear damping coefficient of the system due to viscous effects and the generation of radiating waves due to platform motion; a linear restoring force/moment coefficient due to hydrostatic pressure and/or mooring lines; and an external exciting force/moment due to incident waves, wind, tow lines, etc. Ideal fluid theory is introduced to model the hydrodynamic forces implicit in the marine system’s equations of motion. The purpose is not to give a detailed derivation of basic hydrodynamics, but rather to describe the assumptions necessary to apply the useful ideal, potential theory and understanding when the theory will be successful and, equally important, when it will not.

It is explained what is meant by a function defined implicitly and how the derivative of an implicitly defined function can be determined via partial differentiation. The general concept of the contour of a two-variable function is presented, together with the special case of this when the function is a production function, and the contours are known as isoquants. It is explained how the slopes of contours can be determined. Then, the concept of homogeneous functions and the connected economic interpretation of returns to scale are considered, along with Euler's Theorem and its economic interpretation in terms of marginal product of labour and marginal product of capital.

Basic concepts in finance are introduced and modelled via first-order recurrence equations. In particular, we discuss compound interest, present value and the present value of an annuity.

The concept of consumer surplus is introduced and this motivates the problem of determining the area under the graph of a function. We indicate the connection between this problem and anti-derivatives (or integrals), defining what we mean by a definite integral. We illustrate with some examples after developing a repertoire of standard integrals.

This chapter introduces the important idea of a vector through the example of bundles of goods. The dot product of two vectors is defined and it is shown how a budget constraint can be expressed in terms of dot product. It is explained how, in order to rank bundles according to a particular consumer's preference, we can use a utility function. Indifference curves are defined as the contours of the utility function. Linear and convex combinations and the concept of a convex set are explained. The utility maximisation problem -- to maximise utility subject to a budget constraint -- is explored and the relevance of convexity is emphasised.

Regarding the key components of macroeconomics as continuous (rather than discrete) and considering the corresponding dynamics leads to differential equations. In this chapter, we focus on first-order differential equations and consider two methods (applicable to certain types of equation): separation and the use of integrating factors. We look at economic applications to continuous-time price adjustment and continuous cash flows.

A mathematical discrete-time population model is presented, which leads to a system of two interlinked, or coupled, recurrence equations. We then turn to the general issue of how to solve such systems. One approach is to reduce the two coupled equations to a single second-order equation and solve using the techniques already developed, but there is another more sophisticated way. To this end, we introduce eigenvalues and eigenvectors, show how to find them and explain how they can be used to diagonalise a matrix.

In many applications, dimensionality reduction is important. Uses of dimensionality reduction include visualization, removing noise, and decreasing compute and memory requirements, such as for image compression. This chapter focuses on low-rank approximation of a matrix. There are theoretical models for why big matrices should be approximately low rank. Low-rank approximations are also used to compress large neural network models to reduce computation and storage. The chapter begins with the classic approach to approximating a matrix by a low-rank matrix, using a nonconvex formulation that has a remarkably simple singular value decomposition solution. It then applies this approach to the source localization application via the multidimensional scaling method and to the photometric stereo application. It then turns to convex formulations of low-rank approximation based on proximal operators that involve singular value shrinkage. It discusses methods for choosing the rank of the approximation, and describes the optimal shrinkage method called OptShrink. It discusses related dimensionality reduction methods including (linear) autoencoders and principal component analysis. It applies the methods to learning low-dimensionality subspaces from training data for subspace-based classification problems. Finally, it extends the method to streaming applications with time-varying data. This chapter bridges the classical singular value decomposition tool with modern applications in signal processing and machine learning.