Many applications require solving a system of linear equations 𝑨𝒙 = 𝒚 for 𝒙 given 𝑨 and 𝒚. In practice, often there is no exact solution for 𝒙, so one seeks an approximate solution. This chapter focuses on least-squares formulations of this type of problem. It briefly reviews the 𝑨𝒙 = 𝒚 case and then motivates the more general 𝑨𝒙 ≈ 𝒚 cases. It then focuses on the over-determined case where 𝑨 is tall, emphasizing the insights offered by the SVD of 𝑨. It introduces the pseudoinverse, which is especially important for the under-determined case where 𝑨 is wide. It describes alternative approaches for the under-determined case such as Tikhonov regularization. It introduces frames, a generalization of unitary matrices. It uses the SVD analysis of this chapter to describe projection onto a subspace, completing the subspace-based classification ideas introduced in the previous chapter, and also introduces a least-squares approach to binary classifier design. It introduces recursive least-squares methods that are important for streaming data.

This chapter provides an introduction to psychodynamic theory as applied to settings outwith the specialist psychotherapy clinic, paving the way for the chapters that follow in Part 4. An individual’s internal world affects how they relate to others. Others may be unconsciously invited into playing old roles that are familiar to the individual (such as rejecting, not listening, criticising), even though these roles bring difficulty and distress to both sides. This chapter explores how these powerful but sometimes ‘invisible’ interpersonal dynamics may play out between service users and staff in settings where the human relationship is at the fore (such as schools, social service agencies, and hospitals). We also discuss splitting within a clinical team and other system dynamics. In circumstances where services and professionals can sustain a good-enough therapeutic environment in the face of unconscious invitations to repeat a problematic relationship, trust may develop between service user and service and many people are able to discover new ways of forming relationships. This depends partly on the capacities and current state of the person using a service, but also, crucially, on the capacity of the professionals and services to observe and be reflective about both sides of the relationship.

This chapter explores the complex area of working with patients who experience relational difficulties and who may function predominantly at a borderline level of psychological organization. These patients are influenced by early traumatic experiences, which can shape the therapeutic encounter. They often don’t have the kind of early experience that enables them to develop the capacity to recognise feelings and to know that they are not dangerous, that they are bearable, and will pass. Acts of self-harm are frequently a response to manage unbearable feelings. These and the experience of suicidal thoughts can be understood as a wish to get rid of these feelings. The nature of self-harm and what it evokes in the clinician are discussed. Individuals with these difficulties have often experienced a lack of a consistent and containing other and can enter crisis in response to experiences of rejection or threats of abandonment. This is important both during therapy but particularly when ending the therapy. If we understand what underpins the relational difficulties that these patients have, we can take them into account in the therapeutic work. Some adaptations of technique when working with patients with borderline level difficulties are considered.

This chapter describes the theoretical foundations of the study. The study is located at the interface of two scientific areas that had not had much contact before: Conversation Analysis and World Englishes. In the first two sections of the chapter, central theoretical and methodological tenets in both fields are introduced. The last section addresses epistemological differences between the traditions and provides a rationale for why and how Conversation Analysis and World Englishes can still be reconciled in a fruitful way.

This chapter focuses on grammatical resources for construing logical relations between clauses. The chapter begins by clarifying the distinction between a clause simplex and a clause complex and the distinction between a clause complex and a verbal group complex. Subsequently the basic oppositions between parataxis and hypotaxis on the one hand and between projection and expansion on the other are introduced and relevant resources are then presented, section by section, in more detail. A sample analysis of a longer clause complex rounds off the discussion.

This chapter argues that Ishiguro’s novels frame ethical issues through questions of agency. Hannah Arendt’s ideas about agency and action provide a way to understand this in detail: for Arendt, speech and action reveal ‘who’ the speaker is, and shows their involvement with the ‘web of human relationships’ and the ramifications of their actions; the representation of action is inextricable from style and form. Using these ideas, the chapter demonstrates that there are significant changes over Ishiguro’s work: the first three novels concern reflections on past actions; the second three explore different conditions of agency in both content and in style; the two most recent novels deal with the impact and risks of actions and reactions. This also illuminates two recognizable literary devices used by Ishiguro: the way his characters ‘project’ themselves onto others, and what he calls the ‘dream grammar’ in relation to some aspects of his prose and plotting.

Basic higher-dimensional Euclidean theory

Basic one-dimensional theory

This chapter compares Heidegger’s transcendental approach to social ontology with that found in Husserl. I argue that Husserl and Heidegger are united by the idea that ’the world’ or ’transcendence’ constitutes the most basic form of intersubjectivity, but that their different understandings of the concept of the world lead to divergent conceptions of both subjectivity and intersubjectivity. In short, Husserl takes the world to involve irreducible references to others since perceptual objects can only appear as real or as transcendent if we assume that they possess an inexhaustive number of unperceived aspects that are, in principle, available to other (transcendental) subjects. Heidegger, on the contrary, rejects both Husserl’s interest in objectivity and his notion of the transcendental subject. Instead, he claims that Dasein’s relation to the world must be understood in terms of practical and affective engagement within a field of possibilities, that is, in terms of existential projections. Accordingly, the most basic form of intersubjectivity is found in the transcendental necessity that the same field of entities can be subjected to a multitude of existential projections.

This chapter presents the most influential linguistic approaches to presupposition. Going beyond the traditional analyses of the problem of presupposition projection, it also considers recent developments in linguistics that link the analysis of presuppositions to general processes of cognition and reasoning, such as attention, probabilistic reasoning, theory of mind, information structure, attitudes and perspectival structure. I discuss some outstanding questions: whether presuppositions form one coherent group or should be thought of as different types of phenomena, why we have presuppositions at all, and why we see the presuppositions that we see (aka the triggering problem). Overall, the chapter stresses the need to consider the intricacies of the interaction of presuppositions with the broader discourse context.

We extend the mathematical description of quantum mechanics by using operators to represent physical observables. The only possible results of measurements are the eigenvalues of operators. The eigenvectors of the operator are the basis states corresponding to each possible eigenvalue. We find the eigenvalues and eigenvectors by diagonalizing the matrix representing the operator, which allows us to predict the results of measurements. We characterize quantum mechanical measurements of an observable A by the expectation value and the uncertainty. We quantify the disturbance that measurement inflicts on quantum systems through the quantum mechanical uncertainty principle. We also introduce the projection postulate, which states how the quantum state vector is changed after a measurement.

This chapter is dedicated to Hegel’s student Ludwig Feuerbach. It begins by giving an overview of Feuerbach’s life and writings. The main focus of the chapter is Feuerbach’s most famous work, The Essence of Christianity. Feuerbach tries to argue that it is a mistake to think of God as an objective, transcendent entity that is fundamentally different from human beings, as is traditionally done in theology. Instead, God is simply the essence of what is human projected onto an external entity. For this reason he refers to his undertaking not as theology or philosophy of religion but as anthropology; that is, a study of the human. It is shown that Feuerbach takes up the key Hegelian concepts of recognition and alienation. We take God to be something different and other, but in fact he is a reflection of our self-consciousness. Humans are alienated from their own positive qualities, which they have denied to themselves in order to project them onto God. Humans are thus not separated from something else or other but rather from themselves or their own nature. Feuerbach’s plea is that we restore our energy and efforts to ourselves by, for the first time, dedicating them to ourselves.

Chapter 2 is methodological, offering a primer on multimodal network analysis. It proceeds by quickly reviewing 1-mode network analysis, paying special attention to summarizing several measures of network centrality and how they relate to power. Often, relational data that are 2-mode or multimodal are “projected” into one of the node sets. Ties are then defined by their shared relations to the second-mode nodes so that 1-mode measures of centrality and algorithms for community detection can be employed. We discuss the loss of information on structure and agency that projection entails and argue that, in many cases, projection is neither helpful nor necessary. We then proceed to detailed discussions of methods for 2-mode and 3-mode network analysis, from first principles of matrix algebra to centrality measures and core-periphery analysis; faction analysis and community detection; as well as structural/regular equivalence and blockmodeling. We conclude with a brief introduction to recent advances in statistical network modelling that facilitate inferences about multimodal networks.

In Chapter 1, we introduced the supply chain planning matrix and its different planning functions, discussed how scheduling fits within this matrix, and mentioned that integration across functions can lead to better solutions. Chemical production scheduling interacts directly with two functions: (1) production planning, and (2) process automation and control (though the latter are not typically defined as functions of the SC matrix). Integration with automation and control were discussed in Chapter 14. In the present chapter, we discuss the integration of production planning and scheduling. We start, in Section 15.1, with some preliminary concepts and motivation for the need to integrate planning with scheduling. In Section 15.2, we present a formulation for an introductory planning-scheduling problem. We continue, in Section 15.3, with an approach for more complex problems, both in single- and multiunit environments. Finally, in Section 15.4, we overview a general but also algorithmically more advanced approach that is applicable to any production environment. For simplicity, in Sections 15.2 and 15.3, we do not consider special processing features, such as complex storage policies and utility constraints. The method in Section 15.4 can in principle be applied to any facility with any processing feature.