Convex polytopes, or simply polytopes, are geometric objects in some space $\R^{d}$; in fact, they are bounded intersections of finitely many closed halfspaces in $\R^{d}$.The space $\R^{d}$ can be regarded as a linear space or an affine space, and its linear or affine subspaces can be described by linear or affine equations. We introduce the basic concepts and results from linear algebra that allow the description and analysis of these subspaces. A polytope can alternatively be described as the convex hull of a finite set of points in $\R^{d}$, and so it is a convex set. Convex sets are therefore introduced, as well as their topological properties, with emphasis on relative notions as these are based on a more natural setting, the affine hull of the set. We then review the separation and support of convex sets by hyperplanes. A convex set is formed by fitting together other polytopes of smaller dimensions, its faces; Section 1.7 discusses them.Finally, the chapter studies convex cones and lineality spaces of convex sets; these sets are closely connected to the structure of unbounded convex sets.

Wagner and money is a cantus firmus of his biography. Notoriously broke, he is often regarded as a ‘pump genius’. He always demanded financial generosity from anyone who wanted to call themselves his friend. His pre-March criticism of capitalism has its origins in his completely underdeveloped economic mind. In King Ludwig II of Bavaria, he gained his most powerful and significant patron from 1864 onwards. But contrary to the widespread prejudice, it was by no means excessive sums that the king spent on Wagner. Moreover, in times without copyright and regulated royalty payments, artists were always dependent on patrons and gainful employment. Under today’s legal conditions, Wagner would have been a millionaire.

This chapter introduces the notion of identifying primary and secondary stressors as a way of drawing attention to the many sources of stress that arise either as a consequence of disastrous events of long or short duration, or because they frame the context in which those disastrous events occur. The events that we include in this approach are broad in nature; they include emergencies of all kinds, major incidents, outbreaks of high consequence infectious diseases (HCIDs), terrorist attacks, and conflicts. We begin by outlining the context in which the circumstances that cause stress operate. Then we define primary and secondary stressors. This approach helps us to understand sources of stress in all kinds of adverse and disastrous events. We conclude that the sources of stress that we have researched in these differing emergencies differ, but also have much in common.

The proportion of the population living into old age has been increasing worldwide. For the first time in history, there are more older people than children under 5 years of age. The task for public health is to understand the relationships between ageing, health and the environment (physical, social and economic) in which people live, to promote healthy ageing and prevent the disability and subsequent dependency that is often associated with growing old.

This chapter examines the factors that lead to ageing populations and explores the health, social and economic consequences of the change in the population structure. It then goes on to outline strategies that can lead to healthy ageing and other public health actions that could help to manage the challenges posed – and the opportunities afforded – by the relative and absolute increase in the number of older people.

This chapter focuses on early literacy acquisition as embedded in specific social contexts such as the child's home environment. During the last three decades, the role of parental support, or family support when those other than parents are involved in promoting children's literacy at home was broadly examined within a monolingual context. Concerning a bilingual context, numerous studies have shown that literacy acquisition in minority languages in a home or community setting might serve as a springboard for literacy acquisition in the second language. However, less is known on parental support of biliteracy among language-minority children. Thus, together with home support in the monolingual context, this chapter addresses research evidence on literacy practices in different languages provided by family members to children growing up in bilingual or multilingual homes. A brief overview of prominent theoretical frameworks is provided that outlines parental support and literacy development among children. In addition, drawing mainly on examples from North America, Israel, and other high-income-country contexts, research on parental support in a monolingual context is explored.

This chapter offers an empirically informed approach that utilizes the Internet and social media to provide support and resources for health concerns. The topic of self-injury will be used as a case illustration throughout. To this end, initiatives aimed at supporting and offering outreach for people with lived experience of self-injury as well as key stakeholders who can play key support roles will be presented. By the end of the chapter, readers will be able to reflect on how these various considerations can be drawn upon when planning their own outreach efforts aimed at providing support and informational resources for a range of health concerns via the Internet.

This chapter considers the range of possible education support roles in the mathematics classroom that a teacher may work with. This chapter presents effective ways of working collaboratively with education support workers and explores positive planning of learning experiences which considers the affordances of various education support workers’ roles.

More and more donor-conceived people are interested in, and in contact with, same-donor offspring. Moreover, donor-conceived people can find large numbers of same-donor offspring, enabled by more openness regarding donor conception, social media, donor registers and (online) DNA testing. The impact of meeting large numbers of same-donor offspring has not yet been explored. At Fiom, a Dutch organisation providing independent support in the search for biographical information, we conducted a qualitative study interviewing 19 donor-conceived people who participated in same-donor offspring group meetings. Data showed that while relationships between same-donor offspring are commonly experienced as more beneficial than connecting with the donor, they also come with challenges such as how to integrate these new relations into their lives, how to manage group dynamics and a continuously growing network. Data revealed a need for counselling tailored to the context of multiple same-donor offspring contact. Lastly, challenges for regulators are discussed.

Social mentalizing informs the theory and practice of mentalization-based treatment for adolescents (MBT-A). Adolescence is, among other things, a time for establishing a self-identity and learning about how to interact effectively with a peer group. A focus on balancing mentalizing in peer and family interactions is crucial, with special attention to hypermentalizing and the alien self. Involvement of families and schools in treatment is necessary. MBT-A includes individual, family, and group therapy, and its overall aim is to develop the patient’s independence. Crucial aspects of achieving this goal include building up relational stability and supporting the patient’s sense of agency and autonomy within their relational networks.

Why do ordinary citizens believe in an autocratic ideology? Why do people follow their autocratic leader? How can autocrats win the hearts and minds of their people, securing their support? This chapter seeks to absorb the manifold responses that have been given to these questions in the last decades. It theorizes the role of ordinary citizens in autocratic regimes, illuminating the legitimacy claims of autocratic rulers on the one hand and the popular beliefs in these legitimacy claims on the other hand. Drawing on a rich repository of previous research, it highlights the two strategies of autocracies to deal with their people. Autocracies might either overwhelm their citizens through mass-level political indoctrination or systematically underwhelm them, keeping them satisfied and apolitical. While the former relies on ideational legitimation, the latter refers to socioeconomic performance and a managerial, technocratic style of rule. Both strategies share the goal of thwarting the risk of public protest.

Chapter 1 sets out the growth of the Province from 1221 to 1348, the different patterns of development in different parts of the British Isles, and what explains them. The first part of the chapter examines who supported the new foundations, how the English and Scottish kings in particular aided them, but also the role played by nobles and townsfolk. The second part of the chapter considers what lay behind this support, by looking at the friars’ life and ministry in relation to their supporters’ needs, how they met them directly, and how their religious life and training within the cloister enabled them to do this.

In Chapter 4, we consider both women’s and men’s accounts of the birth of their first child. In our focus on birthing experiences, we consider both women’s and men’s experiences separately and explore similarities within couples. For women, we reflect on orientations to ideas of ‘natural’ birth, and for men we focus on accounts of emotional responses to the birth of a child, and their experiences of cutting the umbilical cord (or not). For both women and men, we focus on issues with receiving support while in hospital, experiences of traumatic birth, and men’s involvement in the birth of their child. Overall, this chapter highlights similarities between women and men within some of the couples, as well as marked differences for some couples. The chapter explores how in some instances birthing is naturalised, while in other instances it is seen as highly mediated.

In Chapter 8, we consider how the arrival of a first child impacted on or shaped the couple relationship. Specifically, we contrast the assumption that having a child will bring a couple closer together with the reality of the many challenges that a new child can bring with them. To do this, we look closely at three of the couples we interviewed, as broadly indicative of the range of the experiences of the couples. For some, the couple experienced a strong supportive relationship across the transition to new parenthood. For others, the arrival of a baby exacerbated existing challenges, and for some new challenges arose. Often these challenges were specifically gendered, related to women’s expectations about men’s involvement, and men’s shifting views about how they would be involved. While these challenges often resolved themselves with time and after increased communication in the couple, they presented the couple relationship with an often unexpected weight that had to be grappled with.

This chapter addresses how senior managers (executives, directors) and immediate managers (those to whom employees report directly) can protect, if not enhance, employees’ mental health. We begin by defining the broad concept of employee “mental health.” We then review scientific literature addressing senior and immediate managers’ potential roles in supporting employee mental health. Although more has been published on immediate managers, much of that work has failed to provide practically useful insights, because of vague conceptualizations, poorly developed measures, or insufficient integration across related topics of study. To help fill that gap, we propose a comprehensive behavioral taxonomy of mental health–supportive supervision. This taxonomy integrates evidence-based insights on the types of behavior that immediate managers should avoid, those they should display, and helpful actions advocated by mental health first aid training programs. Lastly, we list several pressing avenues for future research.