Recursion is a fundamental concept in computer science. A recursive algorithm is one that defines a solution to a problem in terms of itself. That is, recursive techniques solve large problems by building up solutions of smaller instances of the same problem. This turns out to be a powerful technique, because many advanced algorithmic problems and data structures are fundamentally self-similar.

An overview of digital communications techniques is given. The notions of source, transmitter, channel, receiver, and sink are explained. Examples of digital communication schemes and respective applications are given. The main quantities and performance measures are introduced and summarized. The fundamental trade-off between both power efficiency and bandwidth efficiency is characterized.

Bjerknes pointed out for the first time that the evolution of the atmosphere is governed by a complete set of seven equations with seven unknowns. If we know both the equations and the initial conditions with sufficient accuracy, we can predict the weather. In this chapter, we first introduce the continuous equations that govern the atmosphere. From these equations, we then discuss several fundamental wave oscillations existing in the atmosphere and their filtering approximations. For comparison, we then introduce the primitive equations for the oceans and discuss the Kelvin and Equatorially trapped waves, a special kind of waves that can appear in both the atmosphere and the ocean.

Pinterest is a social media platform that allows users to assemble images or other media into customized lists, then share those lists with others. Pinterest calls these lists “pinboards” and the items added to each board “pins,” analogous to real-world physical bulletin boards. Like other social media systems, Pinterest wants to recommend new content to its users to keep them engaged with the service. In 2018, Pinterest introduced a system called Pixie as a component of their overall recommendation infrastructure (Eksombatchai et al., 2018). It uses a graph model to represent the connections among items, then explores that graph in a randomized way to generate recommendations. In this chapter, we’ll build our own system based on the graph algorithms used by Pixie.

We live in a networked world. Professional networks, social networks, neural networks – we’re all familiar with the idea that connections matter. This chapter introduces graphs, our last major topic. Graphs are the primary tool for modeling connections or relationships among a set of items; binary trees, for example, are a special type of graph. Graph models illustrate the power of abstraction: They capture the underlying structure of a network, independent of what the elements actually represent. Therefore, graph algorithms are flexible – they’re not tied to one particular application or problem domain.

Subgrid-scale processes refer to the processes that are vital for describing atmospheric motion but cannot be explicitly resolved due to insufficient model resolution. Although these processes occur at small scales, they depend on and, in turn, affect the larger-scale fields and processes that are explicitly resolved by a numerical model. Due to this two-way interaction, neglecting those subgrid-scale processes will degrade the quality of the weather forecast. To reproduce this two-way interaction, the subgrid-scale processes are “parameterized” by formulating their effects in terms of the resolved fields. Using the prognostic equation for water vapor as an example, we illustrate the general principle of parameterization. We then outline the crucial processes parameterized in today’s numerical weather prediction models. To facilitate the understanding about how parameterizations are implemented in a weather model, a simplified general circulation model with simple parameterizations, SPEEDY, is introduced to the readers.