Chapter 1 relates a brief history of the law of war and the emergence of rules in conflicts. Those rules turned to royal mandates which, in turn, became laws that are today’s law of armed conflict, or LOAC. The chapter relates why modern states should obey the law of armed conflict – often referred to as “international humanitarian law.” The chapter details the five bases of LOAC: domestic laws, customary laws, treaties, judicial decisions, and publicists (i.e., academics). Common LOAC terms are explained to the student new to the subject. Finally, human rights, often seen as contrary or antagonistic to LOAC, particularly by the United States, is shown to now be complementary to LOAC, rather than contrary to it. This is essentially due to a still-evolving US position on the subject that is more willing to accept human rights objectives on the battlefield.

In order to fully appreciate the Discrete Fourier Transform (DFT), a mainstay of geophysical data processing, which will be developed in the , we review here the Fourier series of continuous functions which represents analog signals as sums of sinusoids. In addition, the chapter provides a concise development of complex numbers, complex sinusoids, and complex number multiplication, all of which are essential to understanding the basic elements of the DFT, such as the distinction between positive and negative frequencies.

Linear time domain filters are useful in data processing and as models of physical systems. This chapter reviews examples of digital filter design methods, with applications to data processing, as approximations to physical systems (a seismometer is used as an example) and as a description of physical processes such as echoes and reverberations, gravity anomalies, and ground motion amplification in an earthquake. The focus here is on time domain filters. However, many of these problems may also be addressed in the frequency domain using the Discrete Fourier Transform.

For observables like position and momentum, in quantum mechanics the quantum states in general do not give them an absolute existence. Their value in a particular system is generally only known once the measurement is made. Nevertheless, certain correlations can be present in a system. For a system that is made up of two or more parts that can be measured separately, such as at distinctly different spatial positions, the measurement of one part of the system may immediately imply what the measurement at another part of the system will be. This is a feature that can emerge in a quantum system which is entangled.

Quantum mechanics emerged as a natural extension of classical mechanics. As physics probed into the microscopic realm, it could be argued it would be almost impossible not to discover quantum mechanics. The spectra of atoms, the blackbody spectra, the photoelectric effect and the behaviour of particles through an array of slits had characteristically non-classical features. These phenomena were waiting their time for a theory to explain them. That does not diminish from the huge scientific insights of the founders of the subject. In physics, the great accomplishments come more often than not from insight rather than foresight. Knowing what will be the right physics 50 years into the future is a game of speculation. Recognising what is the important physics in the present and being able to explain it is the work of scientific insight. Thus, whereas we might say Democritus had great foresight millennia ago to envision the discrete nature of particles, it was Albert Einstein, Max Planck, Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Paul Dirac, Max Born and Wolfgang Pauli who had the insights to develop quantum mechanics. And since their foundational work, our understanding of the physical world grew dramatically like never before.

Chapter 11 is a brief chapter on seldom encountered legal issues: ruses and perfidy. Acts that invite an enemy’s confidence that he is entitled to protection under the rules of LOAC, with an intent to betray that confidence, is the crime of perfidy. It has been a codified war crime since 1907, though seldom prosecuted. False flags of truce, informing an opponent that the war is over so you can come on out, are perfidy, as is fighting in the enemy’s uniform. Feigning being wounded, however, is not perfidy, because it does not invite an enemy’s confidence. Examples in recent years are related: in Columbia against the FARC, in the Falklands against the British. Ruses, on the other hand, are lawful: deceit employed in the interest of military operations for the purpose of misleading the enemy. They do not invite the confidence of the enemy with respect to the protection of LOAC. Dummy artillery pieces, inflatable “tanks,” mock operations by nonexistent troops, all lawful.

Chapter 15 involves much of what has gone before; targeting both objects and human beings, core principles, individual status, and more. Artificial intelligence is described as applied to autonomous weapons, then as applied to LOAC’s core principles – difficult values for autonomous weapons to meet. To whom does criminal liability attach, should such weapons go awry? Designers? Builders? Users? These remain difficult LOAC issues that this chapter examines. Drones and their military use are discussed, including the American CIA’s use. Since CIA personnel are civilians, their involvement in targeting in armed conflict is unlawful, an issue discussed in this chapter. Targeted killing and its lawfulness are examined at length, as well as their relationship to assassination, an illegal act in US law. Targeted killing’s weak link, who decides which individuals should be killed, is also discussed. In the Cases and Materials section, the wrongful shooting down of an Iranian civilian airliner in 1988, that killed 290, is examined – a case study of autonomous weapons gone bad.

Building on what we have discussed in the previous two chapters, we now turn to the problem of dealing with the addition of two angular momenta. For example, we might wish to consider an electron which has both an intrinsic spin and some orbital angular momentum, as in a real hydrogen atom. Or we might have a system of two electrons and wish to know what possible values the total spin of the system can take.

Quantum mechanics describes the behaviour of matter and light at the atomic scale, where physical systems behave very differently from what we experience in everyday life – the laws of physics of the quantum world are different from the ones we have learned in classical mechanics. Despite this ‘unusual’ behaviour, the principles of scientific inquiry remain unchanged: the only way we can access natural phenomena is through experiment; therefore our task in these first chapters is to develop the tools that allow us to compute predictions for the outcome of experiments starting from the postulates of the theory. The new theory can then be tested by comparing theoretical predictions to experimental results. Even in the quantum world, computing and testing remain the workhorses of physics.

Using the commutation relations for the components of the angular momentum, we have found that the allowed eigenvalues for ${\widehat{J}}^2$ are ${ħ}^{2}j(j+1)$, where $j=0,\frac{1}{2},1\frac{3}{2},\dots$. For each value of 𝑗, the eigenvalues of $J_z$ are $ħm$, with $m=-j,-j+1,\dots ,j-1,j$.

Chapter 12 explores rules of engagement. ROE are neither LOAC nor mentioned in the Geneva Conventions, but they are the important means by which commanders control use of deadly force by their subordinates. First appearing in the 1950’s US-North Korean conflict, they initially made little sense to warfighters. Their formulation is explained step-by-step in this chapter. ROE never limit the exercise of self-defense and they are not tactical in nature – never instructing combatants in how a mission should be executed. Instead, they restrict the use of force in certain circumstances, putting some targets off-limits, restricting the use of force in some locations. Junior officers seldom see the full ROE and troops never do; instead they are given greatly distilled versions, often on pocket-size cards. Before a US infantryman may fire in Afghanistan, for example, his ROE require that he observe “hostile intent,” or actually be the target of a “hostile act,” terms explained in this chapter. Other targets must be positively identified. It is understandable that infantrymen dislike ROE, but they are essential to the commander’s operational plans and helpful in the observance of LOAC.

This chapter surveys least squares and related methods for designing inverse, prediction, and interpolation filters. The name Wiener filter is associated with these methods. Correlations and autocorrelations play a central role in forming normal equations for filter coefficients. The underlying theory assumes that true correlations are available, but in practice correlations and autocorrelations are estimated from data. We will find the surprising result that a least squares inverse to a linear filter with impulse response $l_t$ can be found from its autocorrelation, without knowledge of the impulse response itself. The least squares inverse filter is closely related to a filter for predicting future values of a time series. Similar ideas may be used to design interpolation filters and to develop linear filter models for time series. This chapter also reviews the correlation filters used in geophysical systems such as radar, sonar, and exploration seismology. We also discuss how correlation filtering is the essential element enabling navigation with the Global Positioning System (GPS).

A filter may be a physical system or computational algorithm with an input and output. If the filter is linear, then the relationship between input and output is the same regardless of the amplitude of the input. Throughout this chapter and the entire book we consider only time-invariant linear filters, that is, linear filters whose properties do not change with time. As a consequence, linear systems and filters obey a superposition principle, so that when two inputs are added together the output is the sum of the separate outputs that would result from separate inputs. Another consequence is that a single-frequency sinusoidal input produces a sinusoidal output at exactly the same frequency and no other. As a result, linear systems and filters are preferred models for physical processes and for data processing because they allow analysis and implementation in both the frequency and time domains. The DFT presented in theis the main tool for frequency domain analysis and implementation. This chapter develops important elements used in time domain implementation: digital filter equations and discrete convolution; the transfer function; and the impulse response. These concepts are extended to the properties of a cascade of linear filters (the successive application of several different filters to a time series) and are used to define the concept of an inverse filter. Example applications of linear filters in data processing, as models of physical processes, and in methods for finding practical inverse filters appear inand .