Chapter 15: Many interesting mathematical ideas evolve from analogies. If we think of matrices as analogs of complex numbers, then the representation z = a + bi suggests the Cartesian decomposition A = H + iK of a square complex matrix, in which Hermitian matrices play the role of real numbers. Hermitian matrices with nonnegative eigenvalues are natural analogs of nonnegative real numbers. They arise in statistics (correlation matrices and the normal equations for least-squares problems), Lagrangian mechanics (the kinetic energy functional), and quantum mechanics (density matrices). They are the subject of this chapter.

Eventful analysis employs the most unfrozen and hence the most exploratory strand of CHA. It employs historical comparisons and explores transformation patterns, that is, patterns of qualitative change. It uses two key tools: historical description and conceptualization. The aim of historical description is to figure out what is going on, to gain a basic understand of a phenomenon before proceeding to explain it. Often this involves de-redescribing a phenomena that has qualitatively changed over time. Historical description, in turn, involves six concrete steps: fact gathering, chronicling, concatenation, periodizing, looking for intercurrence patterns, and rethinking research questions. Conceptualization serves to make historical description more comparativist and to explore broader patterns. The chapter discusses how to replace proper names with broader concepts by defining both the positive and the negative pole of concepts. It lists criteria for assessing the content and temporcal validity of concepts.

Some problems arising from loss modeling may be analytically intractable. Many of these problems, however, can be formulated in a stochastic framework, with a solution that can be estimated empirically. This approach is called Monte Carlo simulation. It involves drawing samples of observations randomly according to the distribution required, in a manner determined by the analytic problem.

In this chapter, we discuss some applications of Monte Carlo methods to the analysis of actuarial and financial data. We first revisit the tests of model misspecification introduced in Chapter 13.

In the chapters so far, we have studied a number of exact methods of calculation for Ising models. These studies culminated in the exact solution for an infinite one-dimensional Ising model, as well as the corresponding solution on a 2 × ∞ lattice. Neither of these systems shows a phase transition, however. In this chapter, we start with Onsager’s exact solution for the two-dimensional lattice, which quite famously does have a phase transition. Next, we explore exact series expansions from low and high temperature, and show how these results can be combined, via the concept of duality, to give the exact location of the phase transition in two dimensions.

Some models assume that the failure-time or loss variables follow a certain family of distributions, specified up to a number of unknown parameters. To compute quantities such as average loss or VaR, the parameters of the distributions have to be estimated. This chapter discusses various methods of estimating the parameters of a failure-time or loss distribution.

Chapter 18: If a Hermitian matrix is perturbed by adding a rank-1 Hermitian matrix, or by bordering to obtain a larger Hermitian matrix, the eigenvalues of the respective matrices are related by interlacing inequalities. We use subspace intersections to study eigenvalue interlacing and the related inequalities of Weyl. We discuss applications of eigenvalue interlacing, including Sylvester's principal-minor criterion for positive definiteness, singular value interlacing between a matrix and a submatrix, and majorization inequalities between the eigenvalues and diagonal entries of a Hermitian matrix. We also prove Sylvester's inertia theorem, and then use the polar decomposition to prove a generalization of the inertia theorem for normal matrices.

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.