In this chapter, we describe basic machine learning concepts connected to optimization and generalization. Moreover, we present a probabilistic view on machine learning that enables us to deal with uncertainty in the predictions we make. Finally, we discuss various basic machine learning models such as support vector machines, neural networks, autoencoders, and autoregressive neural networks. Together, these topics form the machine learning preliminaries needed for understanding the contents of the rest of the book.

In a steady-state, axisymmetric atmosphere surrounding a gravitating point mass, three constants of flow along lines of induction (equivalently, streamlines) are identified, collectively referred to as the Weber–Davis constants. The MHD Bernoulli function, the fourth constant along a line of induction, is derived from examining Euler’s equation in a rotating reference frame, and a link is made between the centrifugal terms and the magnetic terms found in an inertial reference frame. From the four constants, two types of magneto-rotational forces arise which, acting in tandem, can accelerate material from an accretion disc to escape velocities provided the line of induction emerges from the disc at an angle less than 60°. Two astrophysical examples are then described. The first is a quantitative account of Weber and Davis’ model for a stellar wind, including the derivation of specific fluid profiles along a poloidal line of induction. The second looks at how the four constants can arise naturally in an axisymmetric, non-steady-state simulation of an astrophysical jet.

In this chapter, we consider vector spaces over a field that is either the real or complex numbers. We shall start from the most general situation of scalar products. We then consider the situations when scalar products are nondegenerate and positive definite, respectively.

In this chapter, we review the growing field of research aiming to represent quantum states with machine learning models, known as neural quantum states. We introduce the key ideas and methods and review results about the capacity of such representations. We discuss in details many applications of neural quantum states, including but not limited to finding the ground state of a quantum system, solving its time evolution equation, quantum tomography, open quantum system dynamics and steady-state solution, and quantum chemistry. Finally, we discuss the challenges to be solved to fully unleash the potential of neural quantum states.

In this chapter, we present an introduction to an important area of contemporary quantum physics: quantum information and quantum entanglement. After a brief introduction regarding why and how linear algebra is so useful in this area, we first consider the concepts of quantum bits and quantum gates in quantum information theory. We next explore some geometric features of quantum bits and quantum gates. Then we study the phenomenon of quantum entanglement. In particular, we shall clarify the notions of untangled and entangled quantum states and establish a necessary and sufficient condition to characterize or divide these two different categories of quantum states. Finally, we present Bell’s theorem which is of central importance for the mathematical foundation of quantum mechanics implicating that quantum mechanics is nonlocal.

In this chapter, we introduce the reader to basic concepts in machine learning. We start by defining the artificial intelligence, machine learning, and deep learning. We give a historical viewpoint on the field, also from the perspective of statistical physics. Then, we give a very basic introduction to different tasks that are amenable for machine learning such as regression or classification and explain various types of learning. We end the chapter by explaining how to read the book and how chapters depend on each other.

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.