The independence number of a hypergraph H is the maximum size of an independent set of vertices of H – a set of vertices containing no edges of H. The problem of determining the maximum size of an independent set in an r-uniform hypergraph with a prescribed number of vertices and edges is equivalent to the notorious Turán problem for hypergraphs. In this chapter, we consider the fundamental problem of lower bounds on the independence number of H when H is an n-vertex r-uniform linear hypergraph of average degree d, with an illustration of applications to the Erdős–Heilbronn triangle problem and to lower bounds on extremal numbers for bipartite graphs. In particular, we give a novel proof of the Ajtai–Komlós–Pintz–Spencer– Szemerédi Theorem, giving large independent sets in uncrowded hypergraphs.

The proliferation of artificial intelligence (AI) technologies has introduced a profound paradox: while AI promises efficiency, automation and insight, its growing energy consumption, computational demands and infrastructural dependencies increasingly strain ecological and technological limits. This chapter addresses the behavioural and systemic dynamics that shape the sustainability of AI itself – not only in terms of ethics or outcomes, but in terms of its continued viability in a resource-constrained world. Drawing from systems theory, behavioural modelling and computational sustainability, we examine how AI behaves as a system-of-systems, how those behaviours accumulate energy and material costs and what design principles might render AI more compatible with long-term environmental and economic thresholds. By analysing patterns such as runaway model scaling, overfitting to hardware architectures and unsustainable data pipelines, the chapter outlines a Behavioural Data Science approach to understanding and modifying AI’s systemic impact. Through case studies and modelling strategies, it argues that AI’s future must be governed not only by performance metrics, but by its capacity to sustain itself – ecologically, computationally and institutionally – within the planetary boundaries.

An r-sunflower is a collection of sets such that there is a set C (possibly the empty set) called the core, for which every two of the sets have intersection exactly C. Sunflowers are one of the most important structures in extremal set theory since they are the only configuration that one is guaranteed to contain in any sufficiently large collection of k-elements sets. The simple structure of a sunflower enables one to prove many theorems in combinatorics and beyond. This chapter considers the basic theory of sunflowers, culminating in recent breakthroughs.

This chapter explores the use of natural language processing and large databases of digitised text to make inferences about the psychology of authors, including their emotional state, cognitive abilities and beliefs. By analysing longitudinal data, researchers can investigate how behaviour, psychology and culture change over time. The chapter addresses various research questions, including how language evolves, how mental health and rationality change over time and the relationship between mental suffering and artistic output. The chapter aims to provide readers with an understanding of the methodologies, workflows and resources used to conduct this type of research, as well as potential pitfalls and ways to avoid them. It also highlights exceptional work in this field. The chapter emphasises that natural language processing and behavioural insights are allowing researchers to answer questions about human psychology and culture at an unprecedented scale.

This chapter in introduces the basic elements of Ramsey Theory, including the Erdős–Szekeres Theorem, Erdős’ lower bound on Ramsey numbers via random graphs, and the recent approach to Ramsey theory via pseudorandom graphs. We cover the topics of graph Ramsey theory, including Shannon capacity, Ramsey numbers for bounded-degree graphs, multicolor Ramsey numbers, canonical Ramsey theory, and the Erdős–Hajnal conjecture.

After presenting motivating examples, cyclotomic extensions, the relation between the conductor of Galois representations and the level of the corresponding automorphic forms, and the Grothendieck–Ogg–Shafarevich formula, we sketch the fact that fundamental properties of the main objects of the book, lower and upper ramification groups, show sharp contrasts. The structure of the book together with the content of each chapter are explained. Required backgrounds are also listed.

We present historical background, contemporary status, and potential future development of the psychology of religion (PoR) in Romania, beginning with its origins: the formative people, places, and various intellectual schools of thought. Writing about the current state of the topic, we reflect on influential factors that are either facilitating or inhibiting the study of PoR, including publication options, topical emphases, practitioners, orientations, methodologies, and professional organizations.

We offer opinions concerning future development topics that are emerging as important in the immediate future and/or are perennially important in order to stimulate creative and useful research including Western theoretical relevance, the extent to which Western PoR theories may or may not contain reasonable expectations and concepts for this region, contextual nuances, Indigenous theoretical concerns, collaborative research opportunities, and common faux pas – reflections on what people unfamiliar with this region commonly and incorrectly assume about conducting PoR work in this context.

I present historical background, contemporary status, and potential future development of the psychology of religion (PoR) in India (Eastern traditions), beginning with its origins: the formative people, places, and various intellectual schools of thought. Writing about the current state of the topic, I reflect on influential factors that are either facilitating or inhibiting the study of PoR, including publication options, topical emphases, practitioners, orientations, methodologies, and professional organizations.

I offer opinions concerning future development topics that are emerging as important in the immediate future and/or are perennially important in order to stimulate creative and useful research including Western theoretical relevance, the extent to which Western PoR theories may or may not contain reasonable expectations and concepts for this region, contextual nuances, Indigenous theoretical concerns, collaborative research opportunities, and common faux pas – reflections on what people unfamiliar with this region commonly and incorrectly assume about conducting PoR work in this context.

(1) the first introduction is a brief survey of the Song dynasty, the sources to study this period, and the structure, themes, and oraganization of the volume

This chapter narrows in scope to focus on a single individual who hauled the suppression of ‘piracy’ beyond the frontier of British imperial expansion and stretched the term to its discursive limits by mid-century. In establishing his dynasty, James Brooke became a paragon and pariah of empire, consciously trying to embody its noblest virtues from the rivers of northwest Borneo. However, the rajah’s relationship to England was always estranged—he sought the empire’s backing, its sea power, and its status; but it was his estrangement that made Brooke a vexed icon of imperial culture. His campaigns against Sarawak’s neighbouring river-systems earned him both fame and infamy by the early 1850s. In seeking to treat pirates ‘like gentlemen’—subjecting them to violence rather than the rule of law—Brooke’s legacy sits uncomfortably within Britain’s imperial annals. Having observed the discourse of piracy’s spread across Indian, Arabian, and Malay maritime spaces, it was his confrontation with the so-called ‘humanity mongers’ that swelled the wave of antipiracy to its mid-century height, forcing it finally to break and recede into British waters.

After the Jurchen conquest of northern China in 1127, Lin’an became the capital of the Southern Song. This period marked a pivotal transition in urban history: commercialization expanded beyond city walls, water management became essential for urban operations, and religion became more integrated into urban life. Businesses relocated from the old capital, while new resources from South China enriched city life. The exchange of goods and movement across the city blended suburban commerce and natural landscapes with city life. Commercial development and the public space offered by West Lake encouraged intense interactions in markets, teahouses, and gardens. Using capital journals, fiction, biji, and gazetteer maps, this chapter explores Lin’an’s environment, layout, commercial development, water management, and secular and religious life. The city’s blend of natural beauty and rapid commercial development marked a significant transition in Chinese urban history. The Southern Song’s influence elevated Hangzhou from a regional hub to a national political, religious, and aesthetic center, with impacts resonating across East Asia and beyond.

While ownership of private goods enables an actor to exclude others’ ownership, public goods are available to others. Because of this, there is a social dilemma associated with public goods: why would a person contribute to a public good if they can use it even if they don’t contribute? The traditional response is that some cost must be added to not paying so that the public good such as community parks could be created and maintained. This response was a clear outcome of early economic theories. However, empirical anomalies emerged that did not support these earlier theories. Cooperation among theorists enabled the development of new theories of how social characteristics of group members could intervene and solve some social dilemmas.

This chapter explores how online data and mobile phone technologies have transformed our ability to study collective human behaviour. By analysing vast digital traces – from web searches, Wikipedia usage and social media posts to geolocation and sensor data collected via smartphones – researchers can now observe, model and predict individual and societal behaviours at unprecedented scales and temporal resolutions. This chapter highlights key methodological developments and applications in domains such as public health, economics, finance, mental well-being and urban mobility. Through detailed examples, we illustrate how digital behavioural data reveal patterns of attention, intention and sentiment, offering new insights into how people respond to cultural events, policy shifts and environmental changes. They also address the ethical and privacy challenges posed by the collection and use of granular digital data, especially the limitations of traditional anonymisation techniques. Ultimately, the chapter highlights both the promise and the responsibility that come with leveraging online and mobile data for understanding and shaping collective behaviour in the digital age.

The ABCs – attention, behaviour and consequences – provide a framework for understanding approaches to behavioural influence. They help identify what initiated a behaviour, why a certain action was taken, why some actions are not completed and how to motivate the desired actions through consequences. The ABCs are a lens to organise existing frameworks and identify what to look for in new approaches. There are various strategies and techniques that fall under the lens of ABC, such as the EAST framework, Thaler and Sunstein’s nudge approach and boosting. These approaches focus on making behaviour easy, capturing attention and reducing effortful processing. Although human behaviour is complex, triggering behaviour, helping it to happen and making the ending worthwhile are a good rule of thumb. Data science can help identify influential factors, but they are more likely to be successful when informed by strong behavioural intuitions. It is important to test these strategies and techniques through experiments and A/B testing. While none of these strategies will work in every situation, most are supported by the weight of evidence, including meta-analyses that combine research from numerous studies.

An important application of ramification groups is to the conductor of Galois representations. We define the Swan conductor using a rescaling function. A refinement called the refined Swan conductor is defined in the quotient of the multiplicative group divided by the units congruent to 1 modulo the maximal ideal. The integrality of the conductor is a consequence of the Hasse–Arf theorem and proved by establishing the induction formula.