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.

Chapter 4 brings the student fully into today’s LOAC with details of 1977’s Additional Protocol I and Protocol II. It explains why there was a need to update the 1949 Geneva Conventions and why A.P.s I and II, initially favored by the US, turned out not to be what the US and her allies had anticipated. There were positive changes and additions in both Protocols, as well as regrettable LOAC modifications. Both sides of that coin are examined, with discussion of the lasting results of both the positive and negative changes. It is significant to note that US self-interest did not always prevail, and to explain the basis of that lack of international consensus. Relaxing the requirements for prisoner of war status continues to block US ratification of either Protocol. In the A.P. negotiations the US paid a price for its somewhat ill-advised Vietnam venture but America has, by and large, learned to live with the AP I and II provisions it initially fought. Additional Protocol III (2005), far less significant than the earlier Protocols I and II, is also discussed, if briefly.

There are various calculational methods beyond the perturbation theory of thethat can be applied in specific circumstances to give either exact or approximate results. In this chapter some of the most common methods are explained. We start with the Rayleigh–Ritz variational method that can be used to obtain an upper-bound estimate of the ground-state energy of a quantum-mechanical system. Next we examine multi-electron atoms. In such a case simple application of perturbation theory becomes difficult and more needs to be done.