Network science is a broadly interdisciplinary field, pulling from computer science, mathematics, statistics, and more. The data scientist working with networks thus needs a broad base of knowledge, as network data calls for—and is analyzed with—many computational and mathematical tools. One needs good working knowledge in programming, including data structures and algorithms to effectively analyze networks. In addition to graph theory, probability theory is the foundation for any statistical modeling and data analysis. Linear algebra provides another foundation for network analysis and modeling because matrices are often the most natural way to represent graphs. Although this book assumes that readers are familiar with the basics of these topics, here we review the computational and mathematical concepts and notation that will be used throughout the book. You can use this chapter as a starting point for catching up on the basics, or as reference while delving into the book.

With this chapter starts the process of renormalization, how to assign in a consistent way a finite value to all these diverging integrals. The procedure itself is fully rigorous. It is the fact that the result of this procedure brings us information about the physical world which is largely an act of faith (which is however supported by experimental data to an astonishing degree of precision). We perform preliminary work, giving necessary conditions for rational functions to be integrable by comparing the degrees of the numerator and denominator, conditions which turn out to be sufficient for the rational functions of interest in computing the values of Feynman’s diagrams, a result known as Weinberg’s power counting theorem.

In the first few sections we examine how we might define Hamiltonians which make physical sense, and we observe that the interaction picture, on which the entire approach is built, is fraught with mathematical inconsistencies. Nonetheless we proceed using it to compute the S-matrix in some of the simplest possible models. This is the heart of the theory. In a very progressive fashion we introduce the main tools, Wick’s theorem and the Feynman propagator, a very special tempered distribution. We then introduce Feynman’s diagrams. Each diagram encodes a term of a complicated calculation, and we give an algorithm to compute the value of such a diagram by a complicated integral. We pay great attention to clarify the nature and the role of the so-called symmetry factors. We then receive the bad news. As soon as the diagrams contain loops the integral giving its value has an irresistible tendency to diverge, a consequence of having attempted an ill-defined multiplication of distributions. We then show how to get a sensible physical prediction out of these infinite integrals, first in the relatively easy case of diagrams with one loop, and then in the much deeper case of diagrams with two loops, which involves a remarkable “cancellation of infinities”. We also introduce the physicist’s counter-term method to produce such cancellations.

When speech sounds are produced, articulatory movements for one sound overlap with those of the surrounding sounds, generating articulatory and acoustic signals that at any point in time are informative about two or more sounds, not just one. This process of intermingling of information about several speech sounds in the articulatory and acoustic signals is called coarticulation. This chapter synthesises theories and experimental findings of the last century on the nature of coarticulation, and shows how our modern understanding of this complex process is deeply rooted in theories that have evolved over decades due to novel experimental findings as well as critique from competing theories. After discussion of our current understanding of coarticulation, some suggestions for initiating students into the surprising effects of coarticulation are introduced.

Objective: Delve into programming logic and flow, VBA syntax, and debugging tools. Become familiar with code structure, communication with spreadsheets, dynamic data storage, conditional statements and loops, calling worksheet functions, and creating user-defined ones.