This is a collection of many of the open problems discussed in the book.

There are many applications of the low-rank signal-plus-noise model 𝒀 = 𝑿 + 𝒁 where 𝑿 is a low-rank matrix and 𝒁 is noise, such as denoising and dimensionality reduction. We are interested in the properties of the latent matrix 𝑿, such as its singular value decomposition (SVD), but all we are given is the noisy matrix 𝒀. It is important to understand how the SVD components of 𝒀 relate to those of 𝑿 in the presence of a random noise matrix 𝒁. The field of random matrix theory (RMT) provides insights into those relationships, and this chapter summarizes some key results from RMT that help explain how the noise in 𝒁 perturbs the SVD components, by analyzing limits as matrix dimensions increase. The perturbations considered include roundoff error, additive Gaussian noise, outliers, and missing data. This is the only chapter that requires familiarity with the distributions of continuous random variables, and it provides many pointers to the literature on this modern topic, along with several demos that illustrate remarkable agreement between the asymptotic predictions and the empirical performance even for modest matrix sizes.

This chapter starts out a short, two-chapter section on very basic mathematics of quantum mechanics, appropriate for those who have taken undergraduate science or engineering courses. The method of “unit analysis” is used as a way of getting at when quantum mechanics will play a role in the behavior of things.

In Chapter 8, I deal with the threat that the present account strips arithmetical knowledge of all the important characteristics traditionally associated with it: apriority, objectivity, necessity and universality. I argue that apriority can be saved in the strong sense of arithmetical knowledge being contextually a priori in the context set by our cognitive and physical capacities. Objectivity can be saved in the sense of maximal inter-subjectivity, while necessity can be saved in the sense of arithmetical theorems being true in all possible worlds where cognitive agents with proto-arithmetical abilities have developed. Finally, universality of arithmetical truths is saved through arithmetic being universally applicable and shared by all members of cultures that develop arithmetic based on proto-arithmetical abilities.

Human rights have an important dual function: they are claims based on particular values or principles and often also legal rights that entail entitlements and freedoms. Philosophical and political conceptions of human rights are broader than international human rights law, which is essentially a normative term referring to rights validated in recognised sources. While the two spheres are closely intertwined, they do not necessarily share a causal or automatic relationship, i.e. that every claim must transform into a legally recognised right. Nor is the relationship always harmonious. A legally recognised right may be defined too narrowly and may therefore exclude certain categories: for example, age may not explicitly fall within the purview of the right to non-discrimination, or conversely a recognised right may be wider than thin theories of human rights based on a limited number of core rights. Theories of human rights abound, including substantive (based on moral values or foundational postulates), formal (constructive, pragmatic, discourse), subaltern (human rights as distinctive practices born out of struggle) and post-modern (empathy for the other) approaches, as well as political theories, such as liberal or socialist notions of human rights. It is in particular the purported universality of human rights, i.e. their applicability to everyone, everywhere and anytime, that has given rise to enduring debates. Those often, somewhat misleadingly, labelled ‘cultural relativists’ have raised important challenges regarding the supposed origins, validity, scope of application and politics of human rights.

Chapter 1 introduces the importance of metaphor to an understanding of time and, in that context, introduces two gaps in our understanding of spatial metaphors for time that the current volume aims to fill.

The conclusion ties together some of the arguments that have recurred throughout this book. Productions of Shakespeare often cast a spotlight on core debates within a conflict, but do not have fixed wartime identities and are, instead, malleable and responsive as a result of their multi-layered networks of production and reception, which is the core methodological framework proposed in this book. It examines the recent use of Hamlet during the Russian invasion of Ukraine in 2022 to reflect on the future of ‘wartime Shakespeare’ and the need for further studies that emphasize the transnational mobilization of Shakespeare that reflects the increasing ‘place-less-ness’ of modern conflict.

Chapter 8 concludes that the ICC’s existence has not resulted in a short-term reduction in atrocities. It is not a shortcut to Utopia, but its long-term preventive impact is too early to assess. Through its expressive function, the Court is having normative impact. In terms of “systemic effect,” the Court may have the most impact where it is the least needed, or its impact may be hindered by “parallelism.” At the same time, there is potential in the internalization of the Rome Statute by domestic legal systems over time. In two peace negotiations, a measure of punishment for perpetrators was included, although this did not necessarily mean imprisonment. However, its normative impact is undermined by its lack of societal impact, including a lack of impact on victims. In this respect the ICC represents “law” rather than “justice.” Victims’ rights are recognized in the Rome Statute, but these have not yielded concrete remedies. The Court also suffers from negative perceptions among affected populations. To move from law to justice, the Court should seek to maximize its impact, through a better understanding of the local context and through a focus on fewer situations. However, justice has no “universal formula” and may require approaches beyond the ICC.

In creating the WTO and GATT before it, the world’s trading nations did more than enter into a contract. They also created a compact that is imbued with a series of values. No understanding of the WTO would be complete without understanding this qualitative aspect, often explicit, but sometimes only to be derived from the operation of the WTO and its agreements.

Kenneth Wilson introduced the renormalization-group (RG) approach in 1971. This approach gave new life to the study of the Ising model. The implications of this breakthrough were immediately recognized by researchers in the field, and Wilson and the RG technique were awarded the Nobel Prize in Physics soon thereafter. One of the distinguishing features of RG methods is that they explicitly include the effects of fluctuations. In addition, the RG approach gives a natural understanding of the universality that is seen in critical phenomena in general, and in critical exponents in particular. In many respects, the RG approach gives a deeper understanding not only of the Ising model itself, but of all aspects of critical phenomena. The original version of the renormalization-group method was implemented in momentum space – which is a bit like studying a system with Fourier transforms. It is beyond the scope of this presentation. Following that, various investigators extended the approach to position space, which is more intuitive in many ways and is certainly much easier to visualize. We present the basics of position-space renormalization group methods in this chapter. We will also explain the origin of the terms “renormalization” and “group” in the RG part of the name.

In this chapter, we explore Ising systems that consist of just one or a few spins. We define a Hamiltonian for each system and then carry out straightforward summations over all the spin states to obtain the partition function. No phase transitions occur in these systems – in fact, an infinite system is needed to produce the singularities that characterize phase transitions. Even so, our study of finite systems yields a number of results and insights that are important to the study of infinite systems.

Few models in theoretical physics have been studied for as long, or in as much detail, as the Ising model. It’s the simplest model to display a nontrivial phase transition, and as such it plays a unique role in theoretical physics. In addition, the Ising model can be applied to a wide range of physical systems, from magnets and binary liquid mixtures, to adsorbed monolayers and superfluids, to name just a few. In this chapter, we present some of the background material that sets the stage for a detailed study of the Ising model in the chapters to come.

Far from resulting from a rational design, the Human Rights Council is in fact a perpetual work in progress. In terms of policymaking practices, the level of open-endedness and the role of trial and error in the institutionalization of this UN body is particularly striking. The making of the HRC is testimony to the prevalence of bricolage in global public policymaking: even its anchoring practice, the Universal Periodic Review (UPR), is subject to unending revisions. In terms of value debates, this chapter shows how the consensus against the politicization of human rights, which originally led to the demise of the COMMISSION ON HUMAN RIGHTS, actually hides deep cleavages over universality, equal treatment, and dialog. A lot of these debates pit the North against the South, although the lines demarcating these two camps are often blurred. Today’s HRC forms an amazing bricolage of governance practices and universal values, one characterized by constant adaptation and constructive ambiguity.

Chapter 6 characterizes the development of Kant’s views of the sublime in Observations on the Feeling of the Beautiful and Sublime and other pre-Critical writings and materials. The description of the sublime in the third Critique is shaped by Kant’s moral turn and his interest in a principle of natural purposiveness. The chapter shows how the early Kant synthesizes the ideas of Edmund Burke on the one hand and Alexander Baumgarten and Moses Mendelssohn on the other. It reveals how Kant shifts from a psychological–anthropological account of the sublime to a non-empirical, transcendental one.

The closing remark offers three points for further reflection. First, Kant’s Critique of the Power of Judgment is full of inner tensions. Some of them can be traced back to his working through various sources and to his drawing from different aspects of his earlier views. Second, Kant’s early aesthetics may have more to offer than scholars and commentators realize, and some of his early ideas are worth examining more closely. In addition, his mature or Critical account continues to inform current discussions in the philosophy of art and art criticism. Third, not all of Kant’s ideas, whether early or late, can be readily applied to aesthetic theory and philosophy of art today.

This chapter introduces the concepts of convergence and fragmentation in international human rights law (IHRL) by providing and discussing their definitions and meanings. Building on the conclusion of the 2006 ILC Report on Fragmentation in international law and drawing on existing literature on the matter, the chapter assesses the extent of normative fragmentation and proposes a new definition of judicial fragmentation in IHRL. Moreover, this chapter also engages in a discussion of how convergence and fragmentation relate to the concepts of universality and relativism, key for any comparative discussion on human rights.