The most combinatorially interesting maximal subgroups of M24 are the stabilizers of an octad, a duum, a sextet and a trio. In this chapter we investigate the way in which the stabilizer of one of these objects acts on the others. This involves some basic but fascinating character theory; the approach given here is intended to be self-contained. For each of the four types of object we draw a graph in which each member is joined to members of the shortest orbit of its stabilizer. Thus in the octad graph we join two octads if they are disjoint; we join two dua if they cut one another 8.4/4.8; we join two sextets if the tetrads of one cut the tetrads of the other (22.04)6; and we join two trios if they have an octad in common. A diagram of each of these four graphs is included as is the way in which these graphs decompose under the action of one of the other stabilizers. Each of these graphs is, of course, preserved by M24.

This chapter introduces state-space descriptions for computational graphs (structures) representing discrete-time LTI systems. They are not only useful in theoretical analysis, but can also be used to derive alternative structures for a transfer function starting from a known structure. The chapter considers systems with possibly multiple inputs and outputs (MIMO systems); systems with a single input and a single output (SISO systems) are special cases. General expressions for the transfer matrix and impulse response matrix are derived in terms of state-space descriptions. The concept of structure minimality is discussed, and related to properties called reachability and observability. It is seen that state-space descriptions give a different perspective on system poles, in terms of the eigenvalues of the state transition matrix. The chapter also revisits IIR digital allpass filters and derives several equivalent structures for them using so-called similarity transformations on state-space descriptions. Specifically, a number of lattice structures are presented for allpass filters. As a practical example of impact, if such a structure is used to implement the second-order allpass filter in a notch filter, then the notch frequency and notch quality can be independently controlled by two separate multipliers.

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.

This chapter reviews vectors and matrices, and basic properties like shape, orthogonality, determinant, eigenvalues, and trace. It also reviews operations like multiplication and transpose. These operations are used throughout the book and are pervasive in the literature. In short, arranging data into vectors and matrices allows one to apply powerful data analysis techniques over a wide spectrum of applications. Throughout, this chapter (and book) illustrates how the ideas are implemented in practice in Julia.

Gives a short review of the linear dynamics of mechanical system, starting with a single degree of system, continuing with multibody systems, and ending with a continuous beam. Both frequency domain solutions and time domain methods are discussed.

Review of mathematical and statistical concepts includes some foundational materials such as probability densities, Monte Carlo methods and Bayes’ rule are covered. We provide concept reviews that provide additional learning to the previous chapters. We aim to generate first an intuitive understanding of statistical concepts, then, if the student is interested, dive deeper into the mathemetical derivations. For example, principal component analysis can be taught by deriving the equations and making the link with eigenvalue decomposition of the covariance matrix. Instead, we start from simple two- and three-dimensional datasets and appeal to the student’s insight into the geometrical aspect: the study of an ellipse, and how we can transform it to a circle. This geometric aspect is explained without equations, but instead with plots and figures that appeal to intuition starting from geometry. In general, it is our experience that students in the geosciences retain much more practical knowledge when presented with material starting from case studies and intuitive reasoning.

To analyse a discrete walk we need to compute the eigenvalues and eigenvectors of unitary matrix. The matrices that arise in practice are products of two reflections. We develop machinery that takes advantage of this structure to complete the specrtal information we need.

We provide a short introduction to the area of Lieb–Thirring inequalities and their applications. We also explain the structure of the book and summarize some of our notation and conventions.

We provide a brief, but self-contained, introduction to the theory of self-adjoint operators. In a first section we give the relevant definitions, including that of the spectrum of a self-adjoint operator, and we discuss the proof of the spectral theorem. In a second section, we discuss the connection between lower semibounded, self-adjoint operators and lower semibounded, closed quadratic forms, and we derive the variational characterization of eigenvalues in the form of Glazman’s lemma and of the Courant–Fischer–Weyl min-max principle. Furthermore, we discuss continuity properties of Riesz means and present in abstract form the Birman–Schwinger principle.

Describes the fractional p-Laplacian, gives fundamental information about its eigenvalues and establishes inequalities of Faber–Krahn type.

We extend the mathematical description of quantum mechanics by using operators to represent physical observables. The only possible results of measurements are the eigenvalues of operators. The eigenvectors of the operator are the basis states corresponding to each possible eigenvalue. We find the eigenvalues and eigenvectors by diagonalizing the matrix representing the operator, which allows us to predict the results of measurements. We characterize quantum mechanical measurements of an observable A by the expectation value and the uncertainty. We quantify the disturbance that measurement inflicts on quantum systems through the quantum mechanical uncertainty principle. We also introduce the projection postulate, which states how the quantum state vector is changed after a measurement.

Chapter 7 describes the second-order reliability method (SORM), which employs a second-order approximation of the limit-state surface fitted at the design point in the standard normal space. Three distinct SORM approximations are presented. The classical SORM fits the second-order approximating surface to the principal curvatures of the limit-state surface at the design point. This approach requires computing the Hessian (second-derivative matrix) of the limit-state function at the design point and its eigenvalues as the principal curvatures. The second approach computes the principal curvatures iteratively in the process of finding the design point. This approach requires only first-order derivatives of the limit-state function but repeated solutions of the optimization problem for finding the design point. One advantage is that the principal curvatures are found in decreasing order of magnitude and, hence, the computations can be stopped when the curvature found is sufficiently small. The third approach fits the approximating second-order surface to fitting points in the neighborhood of the design point. This approach also avoids computing the Hessian. Furthermore, it corrects for situations where the curvature is zero but the surface is curved, e.g., when the design point is an inflection point of the surface. Results from the three methods are compared numerically.