This chapter explores the behavior of random walks on graphs, framed within the broader context of Markov chains. It introduces finite-state Markov chains, explaining key concepts such as transition matrices, the Markov property, and the computation of stationary distributions. The chapter then discusses the long-term behavior of Markov chains, including the convergence to equilibrium under conditions of irreducibility and aperiodicity. The chapter delves into the application of random walks on graphs, particularly in the context of PageRank, a method for identifying central nodes in a network. It also discusses Markov chain Monte Carlo (MCMC) methods, specifically the Metropolis–Hastings algorithm and Gibbs sampling, which are used to generate samples from complex probability distributions. The chapter concludes by illustrating the application of Gibbs sampling to generate images of handwritten digits using a restricted Boltzmann machine.

Human rights instruments in jurisdictions throughout the world assert the right to equality and non-discrimination. These principles lie at the heart of human rights. This chapter initially considers the meanings of equality and non-discrimination. It then examines equality and non-discrimination in the context of Article 14 of the ECHR. This includes an examination of direct and indirect discrimination and intersectionality. Examples from the Strasbourg jurisprudence illustrate the applicability of Article 14 ECHR to specific protected categories including race/ethnicity, religion, sex, gender, sexual orientation, gender identity, and sex characteristics. Article 14 ECHR and its influence in domestic UK law is analysed. The final section considers the UK Equality Act 2010 (EqA)and the complexities raised by competing equality claims. There are case studies concerning sexual orientation and religious belief, employment law, education and a discussion of the concept of ‘woman’ and ‘sex’ in the EqA as referring to a biological woman and biological sex.

The goal of this chapter is to introduce the concepts of Justice, Equity, Diversity, and Inclusion. It begins with a discussion of the American Dream, the notion that the US is a land of opportunity, equality, and the rule of law, and that anyone can be successful if they work hard. This notion of the American Dream is challenged by evidence and discussed from the perspective of Naima Coster, a Black Latina award-winning author. The chapter reviews some characteristics of minoritized communities in the US, how they experience discrimination and barriers to achieve the dream, and some popular myths about racial/gender progress. The chapter defines Justice, Equity, Diversity, and Inclusion, issues at the heart of demands to make the dream true. Justice, Equity, Diversity, and Inclusion represent a growing field of research, a set of initiatives, and social movements to rethink power asymmetries, how resources and opportunities are distributed, and why social hierarchies are maintained. The chapter includes a Food for Thought section on college student debt and the cycle of poverty. It ends with a discussion of Naima Coster, tokenism, and the need for structural change.

Chapter 2 explores the fundamental concept of least squares, covering its geometric, algebraic, and numerical aspects. The chapter begins with a review of vector spaces and matrix inverses, then introduces the geometry of least squares through orthogonal projections. It presents the QR decomposition and Householder transformations as efficient methods for solving least-squares problems. The chapter concludes with an application to regression analysis, demonstrating how to fit linear and polynomial models to data. Key topics include the normal equations, orthogonal decomposition, and the Gram–Schmidt algorithm. The chapter also addresses the issue of overfitting in polynomial regression, highlighting the importance of model selection in data analysis. The chapter includes practical Python implementations and numerical examples to reinforce the theoretical concepts.

The goal of this chapter is to introduce anti-Asian racism and the COVID-19 pandemic. It begins with the story of Kristina Arevalo, an Asian American scholar pursuing a PhD in developmental psychology in New York City who experienced racial discrimination during the COVID-19 pandemic. This is consistent with the history of racial scapegoating in the US, a strategy of blaming Asian Americans for health and economic crises, as well as wars, that relies on fear and hate to justify violence against them. This chapter is informed by Asian critical race theory (AsianCrit), discusses who are Asian Americans, Native Hawaiians, and Pacific Islanders, some popular myths about them, and their place in a racial position model. It examines how anti-Asian racism is supported by the model minority myth, the Yellow Peril racist trope, the perpetual foreigner stereotype, and Orientalism. The chapter includes a Food for Thought section on the Delano grape strike and linked fate. It ends with a discussion of Kristina Arevalo and rethinking who is considered a “Real American.”

This chapter introduces the foundational mathematical concepts behind neural networks, backpropagation, and stochastic gradient descent (SGD). It begins by generalizing the Chain Rule and providing a brief overview of automatic differentiation, which is essential for efficiently computing derivatives in machine learning models. The chapter then explains backpropagation within the context of multilayer neural networks, specifically focusing on multilayer perceptrons (MLPs). It covers the implementation of SGD, highlighting its advantages in optimizing large datasets. Practical examples using the PyTorch library are provided, including the classification of images from the Fashion-MNIST dataset. The chapter provides a solid foundation in the mathematical tools and techniques that underpin modern AI.

The goal of this chapter is to introduce xenophobia against Latinx immigrants. It begins with the story of Carlos Gregorio Hernández Vásquez, a boy from an Indigenous Maya village in Guatemala, an exceptional student who migrated to the US with his sister, was separated from her, and died in custody of US Border Patrol in 2019. The story of family separation of Carlos illustrates anti-immigration policies and xenophobia against asylum seekers, refugees, and undocumented immigrants in the US. Informed by Latinx critical race theory (LatCrit), this chapter discusses who are Latinx people, some popular myths about them, and how they are treated in the US under laws such as Arizona SB 1070 that criminalized undocumented immigrants. It examines the differences between internal and international migration, voluntary and involuntary migration, and emigration and immigration. The chapter includes a Food for Thought section on 9/11, the War on Terror, and Islamophobia. It ends with a discussion of Carlos Hernández and the need for immigration reform.

Critical Security Studies (CSS) is a diverse and multidisciplinary field that approaches traditional security studies through a critical lens and examines the ways in which security discourses and practices reify and reinforce existing power relations and contribute to the marginalization, oppression, and precarity of various groups of people. CSS scholars ask whose security we center when we talk “Security,” and whose security we neglect or sacrifice, what issues are present/absent, who is afforded agency, and who appear only as voiceless victims. They examine the ways in which security and power are intertwined so that evoking security can generate power, enable various kinds of interventions, perpetuate relations of domination and subjugation, and reproduce social hierarchies. Many CSS scholars adopt an interpretivist methodology and normative approach to scientific knowledge; they are interested in analysis not just for the sake of it but for bringing about change to the status quo.

While the UK may not have a single, codified constitution or Bill of Rights instrument, it nevertheless has a long history of rights protection under the common law and through various legislative enactments including, most notably, the Human Rights Act 1998 which gives effect in domestic law to the core rights enshrined in the ECHR. In this chapter, we examine how rights are protected (including their enforcement) in domestic law in the UK, paying particular attention to the principle of legality and the powers conferred on the courts under the Human Rights Act 1998. Common law rights continue to develop and evolve alongside the Human Rights Act and they still act as a vehicle to protect rights and fundamental values.

To link the economic sphere of international relations to the security sphere of international politics in this chapter, we treat economics as a function of politics and security. While controversial in some circles, this need not be so. Economists, historians, and political scientists have distinct answers to questions concerning the economy. That they differ in scope, interest, and focus should be viewed as alternatives for assessing the empirical world, not mutually exclusive representations of it. This is fundamental to the interdisciplinary approach of International Security. It should be no surprise that the vastness and complexity of the global economic system intersect with realms outside the purview of economics. Security is an arena in which the politics of economic decision-making are felt most intensely.

This chapter considers the nature of human rights and its critiques. The language of human rights has become the common lexicon of social justice and the critical standard for assessing political institutions. Yet, whilst human rights law occupies a central place in our moral and legal discourse, there are many aspects of the human rights project which are contested. This chapter introduces the key debates as to the origins of human rights rights and the justifications advanced for the existence of human rights. This includes examining institutional critiques which focus on how rights should be enforced; as well as ideological critiques, which argue that the gaps in what is protected and who is protected mean that human rights law is itself part of the problem.

Freedom to protest is important for similar reasons to freedom of expression, it is, after all, a core form of political expression. Yet while protest is important, protests can also be disruptive, annoying, offensive, harmful and violent. In a rights-based system we thus have to consider where lines should be drawn between competing rights and interests and by whom. This chapter examines how the domestic and ECtHR case law reflects these tensions, with some cases leaning towards a more deferential approach and others establishing a more robust role for the courts. This chapter primarily examines Article 11 the right to freedom of association and assembly, but it also highlights the ways in which many of the other Convention rights safeguard the freedom to protest. In respect of domestic law, this chapter observes that the post-HRA period has also seen the enactment of far wider legislative restrictions than existed previously. There are also concerns about the growing use of surveillance, police brutality and increased restrictions on access to land.