This chapter introduces gradient descent and Newton's method, two of the most popular techniques in optimization. They are simple, effective, and versatile in a wide range of scenarios. For example, stochastic gradient descent is the backbone of machine learning training. Gradient-based approaches use the first derivative of the objective function to iteratively search for local minima where the gradient is zero. Newton's method, and its related quasi-Newton methods, use the second derivative of the objective to converge in fewer, more expensive iterations. We will develop several first-order and second-order algorithms and analyze their convergence rate, stability, computational complexity, and scaling to higher-dimensional problems. In all iterative methods, the resulting solution is only locally optimal, unless the problem is convex. We stress the importance of the initial guess for nonconvex problems, as this determines which local minima the algorithm converges to and how fast it converges. Here we mostly consider unconstrained problems, although gradient-based approaches are also useful for constrained problems.

The International Convention on the Elimination of All Forms of Racial Discrimination is the oldest UN human rights treaty, and for over forty years, the Committee overseeing its implementation, CERD, has had the power to decide individual communications. Despite this long history, a settled evidentiary framework has not materialised yet. The Committee rarely discussed evidence, and when it did, the results could differ markedly: In Dawas and Shava v. Denmark (2012), a case on mob violence, the Committee did not directly engage with the evidence, which led to a resurfacing of evidentiary questions during the follow-up phase, when they could no longer be addressed. Far preferable is the approach adopted in Zapescu v. Moldova (2021), dealing with discriminatory employment practices, where the Committee discussed the standard of proof for procedural violations and the necessary evidence. More elaborations of this kind are needed for a clear evidentiary pathway to emerge.

Although the arts are often sidelined as electives in US education, discussed as secondary to more "important" disciplines, and even stereotyped as mere entertainment and leisure, this book applies interdisciplinary perspectives to emphasize and demonstrate the multifaceted ways in which the arts are indelible: foundational to who we are, to what we know, to how we know, and to how we interact with the world around us. This chapter describes what it means for something to matter, takes up foundational questions about the nature of art, and introduces the central argument of the book. The chapter draws on philosophical arguments and scientific research to argue that the arts cultivate a specific set of skills that enable us to better connect with the world: they sharpen our capacity for observation, foster our emotional intelligence, and encourage and enable our open-mindedness and flexible thinking. Moreover, they provide an avenue for self-reflection, expression, and belief revision. Ultimately, this chapter suggests that these cognitive and affective processes that are engaged through art not only enhance our relationship with art itself but also contribute to a richer life, helping us to navigate challenges and to find greater meaning in our everyday experiences.

In his chapter Steven Hyman describes several paths forward for a more mechanistic understanding of psychiatric disorder, articulating the promise of such an approach as well as illustrating how it encompasses a variety of projects.

Chapter 9 focuses on using the CERIC method to compare two research articles. This chapter first motivates this practice as an essential activity in placing research in context and understanding the broader scope of a research area. It then discusses how one finds similar papers for comparison, emphasizing references, citations, and “similar paper” searches to select suitable articles for comparison. This chapter then delves into using the CERIC elements to conduct a detailed comparative analysis across texts, highlighting case studies to demonstrate this approach. This chapter describes methods of visually comparing CERIC elements to aid in interpreting commonalities and differences. It guides readers through turning the original CERIC reviews and comparisons into a coherent summary illustrated with practical examples.

The author of the following chapter, Denny Borsboom, has played a leading role in the rise of the network model in psychopathology. This model has developed into a viable alternative to the older conventional common-factor model. These two models provide quite different explanations for why psychiatric disorders tend to cluster together in disorders. Let us illustrate this using two commonly co-occurring symptoms in individuals with major depression: insomnia and tiredness. The common-factor model assumes that the two symptoms co-occur because they are both caused by a latent construct: the psychobiological state of depression. By contrast, the network model implies a more common (and perhaps naïve) presentation, assuming a direct causal link between the two symptoms, especially in the direction of “I sleep poorly on night x” followed by “during day x + 1 I am tired.”

This chapter explores the complexities of Black/White biracial (BWB) identity through a Critical Race Theory (CRT) framework, highlighting the intersections of white supremacy, anti-Blackness, and colorism in shaping identity formation and belonging. Drawing on personal narrative and existing research, it challenges traditional monoracial frameworks by framing biracial identity as fluid, liminal, and intersectional rather than bifurcated. Key dynamics include racial invalidation, colorism, multiracial microaggressions, and racial homelessness, all of which undermine mental health, belonging, and self-esteem. The chapter emphasizes how BWB individuals must navigate polarized family and community expectations, societal stereotypes, and shifting notions of authenticity, often experiencing both privilege and oppression simultaneously. Implications for social work practice include the need for culturally responsive and critically conscious approaches that affirm biracial identities, organizational policies that dismantle monoracial assumptions, and educational curricula that embed multiracial perspectives into clinical and social work training. By integrating biracial voices and perspectives, the profession advances a critical race praxis that challenges systemic inequities while fostering healing, resilience, and equity for biracial populations.

The Real Pain of Punishment explores the true pains of incarceration using insights from empirical sciences and people with lived prison experiences. The book highlights the concept of “belonging” as an unprecedented lens for critically interrogating the legitimacy of incarceration across penal theory, sentencing practice, and human rights frameworks. The chapters chart pathways for bridging the gap between the normative idea of punishment and the stark realities of prison life. The final chapter, written with scholars currently and formerly incarcerated in a New York State facility, reflects on how embracing belonging within penal approaches can inform responses to harm grounded in humanization, proximity, empowerment, and collaboration. With this chapter and more, the book advances a call for deeper epistemic dialogue within legal discourse on crime, punishment, and justice. This title is part of the Flip it Open Programme and may also be available Open Access. Check our website Cambridge Core for details.

This introductory chapter illustrates why evidence in the individual communications procedure of the United Nations human rights treaty bodies (UNTB) is an issue requiring reflection and clarification. The chapter firstly contextualises this central topic of this book by broadly introducing the UNTBs’ mandates, composition and ways of working, as well as some general features of their individual communications procedures. Indications are given of how this legal, institutional and procedural setting interacts with the handling of evidence by the UNTBs, as well as some of the key questions it raises. The chapter further outlines some of the particular research challenges encountered in tackling the questions at the heart of this book, and how they have been addressed. It then goes on to introduce the four-part structure of the book and its ten chapters, including the final chapter, containing recommendations. Finally, this introduction discusses cross-cutting themes which emerge from the contributions.

Campbell’s contribution investigates the possible function of scientific laws in psychopathology, specifically the idea that the standard notion of scientific law could function in the way it has traditionally done in scientific explanation. An important way of thinking about this, as outlined by Campbell, is to introduce scientific laws as general axioms that act as premises in a deductive argument through a covering law. Properties of an individual case then feature as the explanandum. Variants on this approach generally emulate Aristotle’s well-known modus ponens argument scheme that deduces the mortality of Socrates from the premises that all men are mortal and that Socrates is a man. Although many different theories of explanation have been produced over the past century, I think it is fair to say that most working scientists have something like this model in mind when they think about explanations of individual cases.

The chapter analyzes the conflict between nascent bright green environmentalism and well-established fossil fuel-based developmentalisms, set against a backdrop of global population growth and resource concerns in the 1970s. It addresses the questions of how and why fossil fuel-based developmentalism contributed to the prolonged marginalization of bright green environmentalist ideas about zero-carbon offshore energy generation. Intellectually, architect Kikutake Kiyonori and engineer John P. Craven faced the spatial constraints inherent in island contexts, in this case Hawai’i and Japan. They focused on employing technology to decouple socio-economic development from ecosystem service and resource overexploitation to enable continuous industrial growth. Applying an oceanic-vertical perspective to island ecosystems and their littoral marine regions gives insights in some of the origins of zero-carbon energy technologies like offshore wind, floating solar photovoltaics, and floating nuclear plants. US and Japanese governmental support for already feasible offshore oil and gas exploitation and terrestrial energy generation nevertheless marginalized these ideas for decades.