Chapter 13 covers LOAC that permits or restricts the targeting of enemy objects: “military objectives.” Lawful military objectives are defined. Economic targets and war-sustaining targets, contentious issues, are described, with the US minority position noted. World War II’s area bombing by all parties to the conflict is shown to be unlawful, unproductive and, today, unlawful. The widely used targeting criteria, lawful by virtue of nature, location, purpose or use, is expanded upon, with examples of each qualifier. That is followed by relating the six-step targeting process employed by US units in combat. Collateral damage and its avoidance efforts are discussed at length, with emphasis on Chapter 7’s core principles of distinction and proportionality. Unique targeting issues such as “dangerous forces” and dual-use targets are covered, with examples. Precautions in the attack do not sound particularly significant but they are essential LOAC constants. Two recent trial records are included in the Cases and Materials section to illustrate real-world legal consequences of disregarding the illegalities of targeting human beings, even ones’ enemies.

Chapter 5 explains the varieties of armed conflicts. Conflict status determines the law that applies to that conflict, including prohibitions and duties applicable to those involved. Conflict status is the first question a student should answer when looking at any armed conflict. This chapter details the factors that determine conflict status. Common Article 2 conflicts (the designation referring to the Article which is common to all four of the 1949 Geneva Conventions) are international armed conflicts. Those conflicts are discussed, as are common Article 3 conflicts, those between a nation and an armed opposition group – rebels or terrorists. Given that states beyond those directly involved may intervene, interfere, assist or impede one side or the other, this is a challenging but important chapter. It gives the student the information necessary to recognize any conflict’s status. Also discussed are recent developments in conflict classification, for example, gray zone conflicts and hybrid warfare, as well as the LOAC of cross-border terrorist attacks and the lawful ability of attacked state forces to pursue and engage those terrorists across a neighboring state’s border.

Very few problems in quantum mechanics can be solved exactly. For example, in the case of the helium atom, including the inter-electron electrostatic repulsion term in the Hamiltonian changes the problem into one which cannot be solved analytically. Perturbation theory provides a method for finding approximate energy eigenvalues and eigenstates for a system whose Hamiltonian is of the form.

In this chapter we describe the motivation and requirements for converting analog signals to digital time series so that they can be analyzed using digital computers. The process of digitizing analog geophysical signals, for example continuous voltages from seismometers, must ensure that all information about signal amplitudes (the dynamic range) and range of frequencies (the frequency bandwidth) is retained. Important elements of analog to digital conversion include the number of samples to be taken (the sampling rate) and how to represent samples as numbers in a computer. The chapter also presents the standard statistics used to describe time series and introduces the logarithmic decibel scale as a convenient way to compare the relative magnitudes of signals.

Chapter 19 covers a critical 1980 Convention involving battlefield weapons and practices. Some weapons have long been banned: bayonets with serrated edges, explosive bullets, and poison, for example. The 1980 Convention addresses five weapon types through five protocols. Looking to Chapter 7’s core principle of distinction, Protocol I bans nondetectable fragments: glass bullets that cannot be detected by x-rays in casualty treatment. Protocol II limits (but does not ban) antipersonnel land mines and booby-traps. Both are banned in areas frequented by civilians but continue to be allowed elsewhere. Examples of their limited uses are provided. The “Ottawa Convention,” banning antipersonnel mines entirely, is also covered. Protocol III limits use of incendiary weapons, banning their use on targets where civilians might be present. Protocol IV bans blinding laser weapons. Nonblinding lasers are unaffected. Protocol V requires ratifying states to retrieve unexploded ordnance, land mines, cluster munitions, dud artillery rounds, at the conflict’s conclusion. All good intentions that are difficult to put into practice, aided by language that invites evasion. But better than nothing.

The Discrete Fourier Transform (DFT) is one of the most important tools in geophysical data processing and in many other fields. The DFT may be understood from a number of viewpoints, but here we emphasize that it is a Fourier series representing a uniformly sampled time series as a sum of sampled complex sinusoids. We refer to the time series as being in the time domain while the set of its complex-valued sinusoidal coefficients computed using the DFT is in the frequency domain. The inverse DFT (IDFT) computes time series values by adding together Fourier frequency sinusoids, each scaled by a frequency domain sinusoidal coefficient. We develop the DFT by converting the ordinary Fourier series to complex form, transitioning to sampled time series, and finally explaining standard normalization and the usual (and often baffling) frequency and time ordering conventions. The DFT came into widespread use only in the 1960s after the development of Fast Fourier Transform (FFT) algorithms. The speed of FFT algorithms has led to many important applications. Those presented here include the interpolation and computation of analytic signals for real-valued time series. In later chapters we show the important roles of the DFT in linear filtering and spectral analysis.

Chapter 10 looks at command responsibility, the other side of the “obedience to orders” coin. A soldier who knowingly obeys an unlawful order is guilty of the resulting crimes but the superior who gave the order is equally culpable. Like obedience to orders, command responsibility is an ancient concomitant of command authority. The modern establishment of command responsibility arguably started with the case of Japanese General Yamashita, at the end of World War II. His military commission ruled that, although he issued no unlawful order, he knew or should have known of the crimes being committed by his subordinates. He was hanged for his inaction. Nazi generals were hanged or imprisoned for orders they issued or passed on. The commander’s knowledge, actual or constructive, is required for conviction. A commander’s seven routes to trial – the ways in which a combatant leader may be exposed to legal liability for their subordinates’ actions – are explained and illustrated by modern-day cases. The chapter also recites instances of officers themselves disobeying orders they view as unlawful. Today, 1977 Additional Protocol spells out a commander’s responsibility for unlawful acts.

Chapter 9 examines the most-frequently raised defense to war crime charges, obedience to orders. The defense is rarely successful but has been raised as long ago as 1474. Examples are culled from Chapter 3’s Leipzig Trials that followed World War I. A history of the defense in US and UK military law provides background and context. The defense of superior orders was valid in the UK and the US until late in World War II, when the UK and US decided to preclude Nazi war criminals from its use. The case of a noted World War II US submarine commander who ordered machine-gunning of survivors of a ship he had sunk, is recounted. The post–World War II Nuremberg Military Tribunal is thought to have finally ended the defense but it only briefly slowed its use. It was Calley’s My Lai defense in 1971 and it continues around the world today – although constrained. What constitutes an unlawful order is parsed through actual examples. Readers are advised what to do, should they be given an unlawful order. Use of the defense in foreign jurisdictions is demonstrated. Again, excerpts from modern-day trials are in the Cases and Materials, at the chapter’s end.

In this chapter we are going to set up the formalism to describe observables in quantum mechanics. This is an essential part of the formulation of the theory, as it deals with the description of the outcome of experiments. Beyond any theoretical sophistication, a physical theory is first and foremost a description of natural phenomena; therefore it requires a very precise framework that allows the observer to relate the outcome of experiments to theoretical predictions. As we will see, this is particularly true for quantum phenomena. The necessary formalism is very different from the intuitive one used in classical mechanics. A subtle point is that the state of the system is not an observable by itself. As seen in the , the state of the system is specified by a complex vector.

Chapter 2 describes how this text will develop, moving from LOAC basics to somewhat more difficult legal situations and their resolution, to complex LOAC concepts and their often less-than-clear resolutions. A thumbnail sketch of the “father” of LOAC, Francis Lieber, is given, along with his LOAC foundational Code of 1863, including a basic element of that Code, the “combatant’s privilege,” which allows lawful combatants to kill and wound, destroy, and damage, without penalty of law – but only lawful combatants. The first Geneva Convention (1864) is described, along with brief accounts of one or two other basic LOAC treaties. Finally, the actual “law” of the law of armed conflict is described: the 1899 and 1907 Hague Peace Conferences, both of which continue into today to be international laws for which violators are being convicted. Finally, brief extracts from actual trials that illustrate elements of the chapter are provided to demonstrate to the student that the reading represents real-world events with modern application.