This chapter explores how international law and its legitimacy could be improved and made more aligned with the demands of justice. It focuses on two types of requirements. First, there are the principles and accompanying procedures on the basis of which actors ask their agency (and their rights) to be recognized by international law and its culture of legitimacy. These principles are consent, justification, accountability, consistency, representation and participation, and non-abuse of power. Second, there are the topics around which this quest for the recognition of agency (and rights) takes place. They are better universality of international law, human rights as a benchmark of the legitimacy of sovereignty, compliance/enforcement/accountability, and human rights supported by public goods. These two kinds of requirements have been at the center of the efforts to make international law more inclusive as well as more legitimate, and they need to be taken more seriously in the future.

The worldwide scope and depth of the present international system and its sense of legitimacy have not been applied in the same way everywhere. There is still much diversity among countries and the courses of action and the policies that they embrace. This explains, in part, the tensions and disagreements concerning the nature and dynamic of this international system as well as the claims of legitimacy in it. The redistribution of power currently underway at the international level, epitomized by the rise of China, could create more stress in the future. Nevertheless, overlooking the scope and depth of the present international order and its culture of legitimacy would be a mistake. The scope and depth of the present international order and its culture of legitimacy are the manifestations and the products of the following elements working together: position of power dominance, means of penetration and integration, values and norms, and secularization and democratization.

This chapter concentrates on the conditions of access to and the nature of membership in the international system as established by international law—specifically, three issues. The first issue is the type of society that is presented as a legitimate collective member of the international order. One of the first steps that international law takes to determine legitimacy at the international level is to identify the criteria necessary for a collective actor to be viewed as a full-fledged legitimate member of the international community. The second issue is that after World War II and the creation of the United Nations (UN), access to international membership in the international order moved toward a form of universality that has been relatively pluralistic. The third issue is that despite this movement toward a pluralistic universality, there are limits to the universality and pluralism of international membership in the international system as defined by international law.

This chapter examines the specificities at work at the international level and their implications for the issue of legitimacy, concentrating on the contemporary context. These specificities have to be understood in relation to the notion and experience of the international community. There are five distinctive features that characterize the international level in connection with the sense of international community. Although the interactions of these characteristics make the international system what it is, this chapter addresses them individually to identify clearly their nature and respective significance for international legitimacy. These features include the ambiguity of the international community; the national bent of international life; cultural diversity and disparity of development; the hegemonic and yet fragmented and contested conception and exercise of power; and the extent to which actors (states and individuals, in particular) have the possibility of ownership—that is, of being represented and of participating at the international level, including in terms of consent.

The chapter is an introduction to basic equilibrium aspects of phase transitions. It starts by reviewing thermodynamics and the thermodynamic description of phase transitions. Next, lattice models, such as the paradigmatic Ising model, are introduced as simple physical models that permit a mechano-statistical study of phase transitions from a more microscopic point of view. It is shown that the Ising model can quite faithfully describe many different systems after suitable interpretation of the lattice variables. Special emphasis is placed on the mean-field concept and the mean-field approximations. The deformable Ising model is then studied as an example that illustrates the interplay of different degrees of freedom. Subsequently, the Landau theory of phase transitions is introduced for continuous and first-order transitions, as well as critical and tricritical behaviour are analysed. Finally, scaling theories and the notion of universality within the framework of the renormalization group are briefly discussed.

This part focuses on the foundational aspects of international human rights law, exploring its theoretical, historical, and philosophical underpinnings. It examines the evolution of human rights ideas, the influence of various philosophical traditions, and the ongoing debates about the nature and universality of human rights. The sections address the epistemological ruptures between philosophy and law, and between law and justice, highlighting the challenges in reconciling these perspectives within a coherent human rights framework. The part also delves into normative pluralism, discussing the coexistence and interaction of multiple legal systems and norms within the global human rights framework. It covers treaty-based structures, customary international law, general principles of law, and the role of judicial decisions and soft law instruments. By critically analyzing these foundational elements, this part aims to provide a deeper understanding of the principles and values that underpin international human rights law and to highlight the complexities and nuances involved in defining and protecting human rights in diverse cultural and legal contexts.

In this chapter, a notion of Tietze transformation for 2-polygraphs is introduced, consisting of elementary operations on 2-polygraphs that preserve the presented category, such that any two finite 2-polygraphs presenting the same category can be transformed into one another by applying such transformations. By using Tietze transformations, the goal is to turn a given presentation of a category into another one that possesses better computational properties. In particular, the Knuth-Bendix completion procedure applies those transformations to turn a presentation into a confluent one. Convergent presentations lead to a solution of the word problem: for those, the equivalence between two words is immediately decided by comparing their normal forms.To tackle the word problem for an arbitrary presentation, a good strategy consists in trying to transform it into a convergent one by using Tietze transformations. From this point of view, a natural question arises as to whether a finite presentation of a category with decidable word problem can always be turned into a convergent one by applying Tietze transformations: this problem is called "universality of convergent presentations".

This chapter highlights the utility of cultural imagination, the ability to see human behaviors not just as the result of their dispositions or immediate situations but also as the result of larger cultural contexts. Our cultural imagination, as researchers, evolves as we are increasingly exposed to ideas from different parts of the world, either through collaboration with other researchers or interacting with individuals outside our immediate cultural context. While cross-cultural research has become simpler with the rise of the Internet, there still remain many challenges. This current chapter delineates concrete steps one can take to conduct an informative cross-cultural study, increasing the diversity of databases for generalizable theories of personality and social behaviors.

Research in “complex physics” or “nonlinear physics” is rapidly expanding across various science disciplines, for example, in mathematics, astrophysics, geophysics, magnetospheric physics, plasma physics, biophysics, and sociophysics. What is common among these science disciplines is the concept of “self-organized criticality systems,” which is presented here in detail for observed astrophysical phenomena, such as solar flares, coronal mass ejections, solar energetic particles, solar wind, stellar flares, magnetospheric events, planetary systems, and galactic and black-hole systems. This book explains fundamental questions: Why do power laws, as hallmarks of self-organized criticality, exist? What power law index is predicted for each astrophysical phenomenon? Which size distributions have universality? What can waiting time distributions tell us about random processes? This book is the first monograph that tests comprehensively astrophysical observations of self-organized criticality systems. The highlight of this book is a paradigm shift from microscopic concepts (such as the traditional cellular automaton algorithms) to macroscopic concepts (formulated in terms of physical scaling laws).

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.