Root systems are beautiful geometric objects in Euclidean space, which crop up in many parts of mathematics, including Lie algebras, singularity theory, mathematical physics and graph theory.

In this chapter, we will use graph theory to discuss the famous ADE classification of root systems in which all roots have the same length. This will then be used to determine the graphs whose adjacency matrix has least eigenvalue −2 or greater in Chapter 9, where we will see a connection between the Shrikhande graph and exceptional root systems.

We work in the Euclidean space V = Rd, with the standard inner product.

Given a non-zero vector u ∈ V, there is a unique hyperplane Hu through the origin which is perpendicular to u.We define the reflection ru in this hyperplane to be the linear map which fixes every vector in Hu and maps u to −u.

Language education policies outside the Anglosphere that prioritise English as an additional language (EAL) in response to its ubiquity as a global lingua franca direct attention to the work of English language teacher educators (ELTEs). This dimension of their work is complicated by issues around standard and local varieties, and even more so in former colonial states where structural-historical circumstances have resulted in class ideologies that position English proficiencies as pivotal to social status and opportunities for education and employment. In this chapter we draw on interviews with eight ELTEs to explore such a situation in Sri Lanka using the perspectives of linguistic shame and emotional labour. Participants have to manage their experiences of linguistic shame in the workplace which can be complicated by contradictory responses to the ideologies that perpetuate linguistic shame attached to the use of local Englishes. Based on the insights offered by the experiences of participants, it is evident that the dimension of linguistic shame in the emotional labour of English language teacher education in post-colonial settings needs recognition and further investigation.

The institutional terrain in which chaplains operated shifted in moral terms. Until the twenty-first century, the Army was overwhelmingly white, male, heterosexual, Christian and resistant to change. The RAChD did little to challenge that status quo or to combat the racism, sexism and homophobia that arose from it. If, with the same complaisance, it went on to adhere to an externally driven equality and diversity agenda, it had a lot more to say about the ethics of war. Significantly, its Journal aired debates over nuclear weapons and issues of war and peace with surprising freedom. Evidently, killing still raised strong ethical concerns in a professional culture informed by Christian morality and by the just war tradition, with its problematics exacerbated by a growing tangle of legal expectations and constraints. Despite imputations to the contrary, and especially in the complex and controversial counter-insurgency campaigns of the later twentieth century, there is scant evidence to justify the suspicion that unit chaplains, as military pastors, were simply predisposed to connive in the misconduct of soldiers and some clear indications that their influence was one of restraint.

In this chapter, we, as two transnational language teacher educators (LTEs), critically reflect on our evolving identities as LTEs, by explicitly focusing on our emotions and agency. Methodologically, we use collaborative autoethnography (CAE) to blend dialogic co-interviews, autobiographical writing, and self-reflection, situating our identities at the intersection of personal, political, and professional experiences. Our stories include these takeaway points: (1) border crossing and in-betweenness are integral aspects of our identities, (2) teacher educators, specifically those with transnational backgrounds, require more support and opportunities for reflective practice to reconcile their multifaceted identities and responsibilities, (3) navigating our transnational identities involve significant emotion labor and emotion work, (4) our sociopolitically-situated agency is intertwined with the emotional and reflective identity work, and (5) our personal and social identities, vis-a-vis culture, language, gender, nationality, religion amongst others, are inseparable from ongoing professional identity work.

This chapter reports on the self-inquiry of a language teacher educator who explored her emotions as she integrated an innovative intervention during a practicum course. More specifically, she examined the emotions she experienced as she implemented a pedagogical intervention based on positive psychology intended to build peace in the practicum, as well as how such emotions were enacted. Framed within self study in teacher education practices (S-STEP), data were collected by means of reflection journals throughout a practicum term and were subjected to thematic analysis. The findings showed that the language teacher educator experienced emotional dissonance when she realized the outcomes of the innovative intervention were not the ones she had expected, and emotional harmony when the outcomes of the intervention aligned with her beliefs. Moreover, the findings revealed the language teacher educator experienced emotional contagion, which led her to increase positivity and better regulate her emotions. The chapter highlights the value of self-reflection on one’s emotions as a means to inform language teacher educators’ practices and better understand their identities.

International law and policy addressing the management of chemicals and wastes have undergone enormous change between the Trail Smelter arbitration in 1941 and the 2022 decision of the UN Environment Assembly to negotiate a global treaty on plastics in line with a circular economy approach. The link between the two may not be immediately evident. This chapter attempts to shed light on it by reviewing and analysing the developments that took place in the intervening 81 years but, more specifically, between the historic Stockholm Conference in 1972 and the UN International Meeting Stockholm+50 in 2022. Following an overview of the relevant developments, it discusses the different ‘lenses’ through which chemicals and waste management have been viewed over the past five decades, which has influenced policy approaches. It concludes with an outlook towards the future, proposing ways of building a comprehensive international regime for the management of chemicals and wastes, still elusive in 2022.

This chapter enacts a practice of “critical commonplacing” to assemble a new global archive of Romanticism, taking as its examples twentieth- and twenty-first-century remediations from Buenos Aires, New York, and Tokyo. Commonplacing a new Romantic archive finds a model in the world of collecting, which valorizes marginalia, marks, scratches, cut-and-pastes – capturing flashes of ephemera over static texts and images. From Japanese depictions of Mary Shelley’s creature as bakemono, to Julio Cortázar’s biography on John Keats during the Latin American Boom, to Audre Lorde and Diane Di Prima’s schooldays clique “The Branded,” this chapter expands the archive of Romanticism beyond 1780–1830, across different languages and media. Turning away from the anthology and canon, this approach replaces static texts with the dynamic media of seemingly fleeting forms, often ephemeral and ghostly dispersed. Each example showcases the experimental quality of commonplacing, aligned with progressive youth culture, learning, and play.

Chapter 4 further considers how the city informs young women’s means for realising their much hoped-for futures by focusing on how they navigate the social infrastructure that underpins its daily life. Paying particular attention to young women’s friendships with other young women, the chapter details this group’s fears of ‘fake friends’ and the anxieties they have towards those close to them having the potential to cause them (and their futures) harm. As the ethnography shows, mobile phone communication has afforded young women new styles of communication that allow them to overcome the fears of social intimacy, helping them to stay connected with others while maintaining social distance. Enabling young women to remain visible in urban life from the confines of their homes, and to engage in conversation without revealing personal information, mobile phones provide young women with an alternative social life, re-ordering their experiences of the city while enabling them to remain embedded within the social relationships that sustain it.

In this chapter we examine other areas in which the Shrikhande graph has a role to play, including Seidel switching and equiangular line sets (which give rise to our final construction of the Shrikhande graph), design theory, Hadamard matrices and distance-regular graphs.

We begin with a short section indicating a few directions in which the study of the Shrikhande graph has been taken. Most of the detail is omitted, and we refer to the cited papers.

Western philosophy has neglected the body for much of its history, even though it is the body that enables us to have a hold on the world and interact with other people. Beauvoir wrote extensively not about ‘the body’ in general but about the diverse bodies that humans have, and the various phases and states that a single body undergoes. Her presentation of female biology is rather controversial and undoubtedly dated, but her insistence that a body’s characteristics are only good or bad in relation to a particular physical and social environment is still crucial. The ‘facts’ that she borrowed from the science of her time, as well as her views, are placed in dialogue with those of current authors and scientists. The exploration of ageing and sick bodies that she carried out both in general philosophical terms and in her poignant memoirs of Sartre’s last decade and her mother’s last days has lost none of its significance.

In this chapter we consider strongly regular graphs, an important class of graphs for many applications, which include both the Shrikhande graph and the graphs associated with Latin squares, all of which play a part in our story.

This theorem was proved by Erdős, Rényi and Sós in the 1960s. It was an early success for the methods of algebraic graph theory.

We assume that friendship is an irreflexive and symmetric relation on a set of individuals: that is, nobody is his or her own friend, and if A is B’s friend then B is A’s friend. (It may be doubtful if these assumptions are valid in the age of social media – but we are doing mathematics, not sociology.) In other words, the situation is described by a graph, in which the vertices are the individuals and two vertices are joined if they are friends.

9.1 INTRODUCTION [LO1]

Stresses given by relatively simple equations in the strength of materials for structures or machine members are based on the assumed continuity of the elastic medium. However, the presence of discontinuity destroys the assumed regularity of stress distribution in a member and a sudden increase in stresses occurs in the neighborhood of the discontinuity. In developing machines, it is impossible to avoid abrupt changes in cross-sections, holes, notches, shoulders, etc. Abrupt changes in cross-section also occur at the roots of gear teeth and threads of bolts. Some examples are shown in Figure 9.1.

Any such discontinuity acts as a stress raiser. Ideally, discontinuity in materials such as non-metallic inclusions in metals, casting defects, residual stresses from welding may also act as stress raisers. In this chapter, however, we shall consider only the geometric discontinuity that arises from design considerations of structures or machine parts.

Many theoretical methods and experimental techniques have been developed to determine stress concentrations in different structural and mechanical systems. In order to understand the concept, we shall begin with a plate with a centrally located hole. The plate is subjected to uniformly distributed tensile loading at the ends, as shown in Figure 9.2.