In Chapter 3 we explored transformations where a finite group of Ising spins is summed to produce effective interactions among the remaining spins. In all of these cases a finite sum of Boltzmann factors is sufficient to solve the problem. We turn now to infinite systems, where a straightforward, brute-force summation is not possible. Instead, we develop a number of new techniques that allow us to evaluate an infinite summation in full detail.

As insurance companies hold portfolios of insurance policies that may result in claims, it is a good management practice to assess the exposure of the company to such risks. A risk measure, which summarizes the overall risk exposures of the company, helps the company evaluate if there is sufficient capital to overcome adverse events.

Chapter 11: Many facts about matrices can be revealed (or questions about them answered) by performing a suitable transformation that puts them into a special form. Such a form typically contains many zero entries in strategic locations. In this chapter, we show that every square complex matrix is unitarily similar to an upper triangular matrix. This is a powerful result with a host of important consequences.

Kenneth Wilson introduced the renormalization-group (RG) approach in 1971. This approach gave new life to the study of the Ising model. The implications of this breakthrough were immediately recognized by researchers in the field, and Wilson and the RG technique were awarded the Nobel Prize in Physics soon thereafter. One of the distinguishing features of RG methods is that they explicitly include the effects of fluctuations. In addition, the RG approach gives a natural understanding of the universality that is seen in critical phenomena in general, and in critical exponents in particular. In many respects, the RG approach gives a deeper understanding not only of the Ising model itself, but of all aspects of critical phenomena. The original version of the renormalization-group method was implemented in momentum space – which is a bit like studying a system with Fourier transforms. It is beyond the scope of this presentation. Following that, various investigators extended the approach to position space, which is more intuitive in many ways and is certainly much easier to visualize. We present the basics of position-space renormalization group methods in this chapter. We will also explain the origin of the terms “renormalization” and “group” in the RG part of the name.

Explaining historical change is difficult because it involves analyzing a moving object. Historical explanations address this problem by dividing historical change into moments of discontinuity and periods of continuity. They explain discontinuities by retracing the generative processes that ultimately produced a change. Historical explanations explain continuities by drawing on path-dependent explanations. Such explanations involve specifying an early mover advantage during a historical discontinuity and following up by identifying so-called increasing return mechanisms that compound the causal effects of the early mover advantage over time. This compounding effect serves to epxlain why certain changes, once they are in place, reproduce themselves and hence endure.

In this chapter, we explore Ising systems that consist of just one or a few spins. We define a Hamiltonian for each system and then carry out straightforward summations over all the spin states to obtain the partition function. No phase transitions occur in these systems – in fact, an infinite system is needed to produce the singularities that characterize phase transitions. Even so, our study of finite systems yields a number of results and insights that are important to the study of infinite systems.

Chapter 10: In this chapter, we identify the eigenvalues of a square complex matrix as the zeros of its characteristic polynomial. We show that an n × n complex matrix is diagonalizable (similar to a diagonal matrix) if and only if it has n linearly independent eigenvectors. If A is a diagonalizable matrix and if f is a complex-valued function on the spectrum of A, we discuss a way to define f(A) that has many desirable properties.

CHA’s focus on exploration, description, and theorizing does not mean that questions of causal inference are ignored. CHA values exploration just as much as testing, on the grounds that only an equal weighing of them will translate results into actual answers. Its approach to methodology is more heterodox than that of methodologies that concentrate narrowly on questions of causal inference. This heterodoxy reflects an older, traditional understanding of methodology as a series of research cycles. Each cycle has an exploratory stage that involves exploration, description, conceptualization, and theorizing and a confirmatory stage that involves data collection, data analysis, and replications. In addition to having those stages, research cycles are iterative and constantly update our knowledge. CHA employs those elements of the research cycle in nonlinear fashion and thus engages in a kind methodological bricolage or what is sometimes called abduction. The goal of bricolage is to align the ontological presuppositions of methods with the ontological attributes of questions. Methods thus differ in their appropriateness rather than in their sophistication.

Credibility models were first proposed in the beginning of the twentieth century to update predictions of insurance losses in light of recently available data of insurance claims. The oldest approach is the limited-fluctuation credibility method, also called the classical approach, which proposes to update the loss prediction as a weighted average of the prediction based purely on the recent data and the rate in the insurance manual. Full credibility is achieved if the amount of recent data is sufficient, in which case the updated prediction will be based on the recent data only. If, however, the amount of recent data is insufficient, only partial credibility is attributed to the data, and the updated prediction depends on the manual rate as well.

Chapter 14: Every square complex matrix is unitarily similar to an upper triangular matrix, but which matrices are unitarily similar to a diagonal matrix? The answer is the main result of this chapter: the spectral theorem for normal matrices. Hermitian, skew-Hermitian, unitary, and circulant matrices are unitarily diagonalizable. As a consequence, they have special properties that we investigate in this and following chapters.

Few models in theoretical physics have been studied for as long, or in as much detail, as the Ising model. It’s the simplest model to display a nontrivial phase transition, and as such it plays a unique role in theoretical physics. In addition, the Ising model can be applied to a wide range of physical systems, from magnets and binary liquid mixtures, to adsorbed monolayers and superfluids, to name just a few. In this chapter, we present some of the background material that sets the stage for a detailed study of the Ising model in the chapters to come.