This chapter starts by distinguishing between the primitive and conservative equations of MHD in 1-D, emphasising that the former deal only with continuous flow, whereas the latter admit flow discontinuities. The first application is to MHD waves including Alfvén, slow, fast, and magneto-acoustical waves. An intuitive analogy is given describing what one might experience in an MHD atmosphere when a “thunder clap” occurs. The MHD Rankine–Hugoniot jump conditions for MHD are introduced and solved (using difference theory) revealing tangential/contact/rotational discontinuities and, most importantly, shock waves including slow, intermediate, and fast shocks. In the context of the not strictly hyperbolic nature of the MHD equations, both the entropy and evolutionary conditions are used to determine the physicality and uniqueness of the shock solution. Finally, discussion of MHD shocks includes the special cases of switch-on/off shocks and Euler shocks.

In this chapter, we exclusively consider vector spaces over the field of reals unless otherwise stated. First, we present a general discussion on bilinear and quadratic forms and their matrix representations. We also show how a symmetric bilinear form may be uniquely represented by a self-adjoint mapping. Then we establish the main spectrum theorem for self-adjoint mappings based on a proof of the existence of an eigenvalue using Calculus. Next we focus on characterizing the positive definiteness of self-adjoint mappings. After these we study the commutativity of self-adjoint mappings. As applications, we show the effectiveness of using self-adjoint mappings in computing the norm of a mapping between different spaces and in the formalism of least squares approximations.

In this chapter, we extend our study of linear algebraic structures to multilinear ones that have broad and profound applications beyond those covered by linear structures. First, we give some remarks on the rich applications of multilinear algebra and consider multilinear forms in a general setting as a starting point that directly generates bilinear forms already studied. Next, we specialize our discussion to consider tensors and their classifications. Then, we elaborate on symmetric and antisymmetric tensors and investigate their properties and characterizations. Finally, we discuss exterior algebras and the Hodge dual correspondence.

In this chapter, we extend our study on real quadratic forms and self-adjoint mappings to the complex situation. We begin by a discussion on the complex version of bilinear forms and the Hermitian structures. We will relate the Hermitian structure of a bilinear form with representing it by a unique self-adjoint mapping. Then we establish the main spectrum theorem for self-adjoint mappings. Next we focus again on the positive definiteness of self-adjoint mappings. We explore the commutativity of self-adjoint mappings and apply it to obtain the main spectrum theorem for normal mappings. We also show how to use self-adjoint mappings to study a mapping between two spaces.

In this chapter, we present two important and related problems in data analysis: the low-rank approximation and principal component analysis (PCA), both based on singular value decomposition. First, we consider the low-rank approximation problem for mappings between two vector spaces. Next we specialize on the low-rank approximation problem for matrices in both induced norm and the Frobenius norm, which are of independent interest for applications. Then we consider PCA. These results are also useful in machine learning. Furthermore, as an extension of the ideas and methods, we present a study of some related matrix nearness problems.

This chapter gives an overview of the different topics covered in the book. As an outlook, we also share our view on potential exciting directions.

Distinguishing between different phases of matter and detecting phase transitions are some of the most central tasks in many-body physics. Traditionally, these tasks are accomplished by searching for a small set of low-dimensional quantities capturing the macroscopic properties of each phase of the system, so-called order parameters. Because of the large state space underlying many-body systems, success generally requires a great deal of human intuition and understanding. In particular, it can be challenging to define an appropriate order parameter if the symmetry breaking pattern is unknown or the phase is of topological nature and thus exhibits nonlocal order. In this chapter, we explore the use of machine learning to automate the task of classifying phases of matter and detecting phase transitions. We discuss the application of various machine learning techniques, ranging from clustering to supervised learning and anomaly detection, to different physical systems, including the prototypical Ising model that features a symmetry-breaking phase transition and the Ising gauge theory which hosts a topological phase of matter.