Every network has a corresponding matrix representation. This is powerful. We can leverage tools from linear algebra within network science, and doing so brings great insights. The branch of graph theory concerned with such connections is called spectral graph theory. This chapter will introduce some of its central principles as we explore tools and techniques that use matrices and spectral analysis to work with network data. Many matrices appear in different cases when studying networks, including the modularity matrix, nonbacktracking matrix, and the precision matrix. But one matrix stands out—the graph Laplacian. Not only does it capture dynamical processes unfolding over a networks structure, its spectral properties have deep connections to that structure. We show many relationships between the Laplacians eigendecomposition and network problems, such as graph bisection and optimal partitioning tasks. Combining the dynamical information and the connections with partitioning also motivates spectral clustering, a powerful and successful way to find groups of data in general. This kind of technique is now at the heart of machine learning, which well explore soon.

This chapter discusses how British modernists engaged with Europe. It takes account of the wider landscape of modernists active on British soil: the impact of art circulation (including key moments like the vogue for Ivan Meštrović, British performances of the Ballets Russes, and F. T. Marinetti’s visit to England), continental European travel and residence, translation, the rise of disciplines like anthropology, attitudes like ‘Byzantinism’ and ‘primitivism’, and the redrafting of European nation-state boundaries. For key figures such as W. B. Yeats, Wyndham Lewis, and Virginia Woolf, one of the driving forces of British modernism was the question of what Europe is and what it might become if its geographical, cultural, and social boundaries were redefined. British modernists’ interest in Europe was intimately connected to their obsession with borders and boundaries – of spaces, political possibilities, customs, conditions, and humans. Reimagining others and themselves as insiders, outsiders, or in-outsiders, the writers and artists of the period adopted a range of identities (from regional identities to British to European and/or cosmopolitan) and developed corresponding artistic practices.

Ensuring the future of France – its children – meant fighting on multiple dimensions. One set of enemies included infectious diseases, especially tuberculosis and the influenza pandemic; the other set comprises illnesses and infant mortality attendant to poverty and malnutrition. Thousands of volunteers from the United States fought these battles with treatment and prevention strategies. They toured the Franco-American colonies, organized large antiepidemic campaigns, and produced leaflets providing practical advice on managing the care of babies and children during wartime. With the help of the Children’s Bureau of the American Red Cross, the American Commission for the Prevention of Tuberculosis in France, and the Rockefeller Foundation’s International Health Division, thousands of leaflets were distributed to the Franco-American colonies of the CFAPCF, fatherless children supported through the FCFS, as well as to schools and mothers across France. With the spread of tuberculosis in 1917 and the 1918 influenza pandemic, American medical experts realized that a sanitary ironclad was needed to block the spread of contagious diseases to the United States: to protect France was to protect the United States.

In this survey we aim to give a comprehensive overview of results using sublinear expanders. The term sublinear expanders refers to a variety of definitions of expanders, which typically are defined to be graphs G such that every not-too-small and not-too-large set of vertices U has neighbourhood of size at least α|U|, where α is a function of n and |U|. This is in contrast with linear expanders, where α is typically a constant. We will briefly describe proof ideas of some of the results mentioned here, as well as related open problems.

This chapter focuses on information knowledge (also known as declarative or ‘what’ knowledge) and looks at how digital tools can be used to learn information. It considers how extended reality, gamification and simulations are used for learning and teaching, and explores why and how digital technologies are used in inquiry-based learning, in particular, challenge-based learning.

After the Gothic wars, significant areas of the Italian peninsula were taken by the newly arrived Lombards. Chapter 5 discusses how convincing Lombard leaders, vacillating between Nicene and Christian Christianity, to embrace one of the available versions of the Christian faith became the goal of the representatives of various sides and how the Lombard religious ambiguity created a special environment in which different doctrines could coexist and compete.