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This introductory chapter gives an overall view of the realm of density functional theory based electronic structure calculations and their impact on complex materials research. There are already many excellent methods and packages that use different strategies and which focus on different aspects of materials or specific properties. The unique role of the OLCAO method is emphasized.

The use of computers in mathematics raises a number of questions of a generally epistemological character. This chapter surveys the ways in which computers are used in mathematics, clarifies some of the questions that arise, and assesses the philosophical methods that may be brought to bear. In particular, it explores the sense in which computational methods provide ‘evidence’ for mathematical assertions, as well as mathematical ‘understanding’.

The goal of decision support is to develop methods that assist decision makers. In this chapter computational methods are brought to bear on a multi-dimensional choice problem with the two-part challenge of efficiently determining the decision-maker’s preferences and then finding the best choice. Computer-aided decision support is growing rapidly. This chapter introduces some technical methods of broad applicability in this area, including specification of utility functions with uncertainty, Markov decision processes, and robust optimization. The author exploits the notion of minimax regret, where the goal is to find an option that minimizes the maximum regret relative to all possible manifestations of an uncertain utility function. This approach is robust to incomplete information and facilitates the important process of preference elicitation from the client. There are opportunities for using such approaches to investigate human decision making based on incomplete information; the impact of cognitive costs, biases, and heuristics; and choices made by groups.

This chapter examines the role of computational methods in determining brain mechanisms. Computation is a key basis for modeling complex phenomena such as cognition. Computation’s language is the best available source for providing information regarding information encoding, its storage, modification, and use by natural and synthetic systems. Computation also involves the investigation of natural and artificial information-gathering techniques. Computing principles are categorized into a number of sections, including computation, communication, and coordination. These sections or categories form the basis of computation’s language. Tools and concepts available within these categories emerge as a strong and suitable medium for expressing natural information process theories. Computational language has also played a key role in understanding perceptions.

This chapter discusses the differences between digital humanities and speculative computing. It explains that the early character of digital humanities was formed by concessions to the exigencies of computational disciplines and humanists played by the rules of computer science and its formal logic. Speculative computing, on the other hand, is a set of principles through which to push back on the cultural authority by which computational methods instrumentalize their effects across many disciplines.

When determining the neural correlates of decision making, a major difficulty is the non-stationarity of the brain's response. The evaluation of the same action should vary over time depending on the subject's choice and reward history. The speed at which such subjective evaluations changes should differ among subjects, or even in the same subject in different sessions. A recently emerging paradigm for coping with this difficulty is to estimate the time course of the internal variables of a mathematical model of decision making from each subject's sequence of stimuli, actions, and obtained rewards in each experimental session, and use that as a marker for detecting the neurons or brain regions implicated. This chapter reviews mathematical models describing the adaptive process of decision making, computational methods for estimating the model variables from observed data, and examples of applications of such model-based analysis to behavioural tests, neural recording, and functional brain imaging.

Computational methods derived from evolutionary biology are increasingly being applied to the study of cultural evolution. This is particularly the case in studies of language evolution, where phylogenetic methods have recently been used to test hypotheses about divergence dates, rates of lexical change, borrowing, and putative language universals. This chapter outlines three new and related questions that could be productively tackled with computational phylogenetic methods: What drives language diversification? What drives differences in the rate of linguistic change (disparity)? Can we identify cultural and linguistic homelands?

This chapter studies the evolution of perturbations in the nonlinear regime. It focuses for the most part on analytic models that shed light on the physical processes involved. The advent of computer technology has made numerical studies of nonlinear evolution almost routine, and many of today's theoretical calculations follow this path. The analytic approaches described here inform these calculations, but the numerical simulations help to sharpen this chapter's conclusions and predictions. The chapter discusses this synergy and describes “semianalytic” models that can be written analytically but whose ultimate justification lies in their good agreement with numerical simulations. It also describes the fundamental aspects of computational methods.

The most interesting theoretical problems in Earth system science cannot be solved by analytical methods; their solutions cannot be expressed as algebraic expressions; and so numerical solutions are needed. In this chapter I shall introduce a method of numerical solution that can be applied to a wide range of simulations and yet is easy to use. In later chapters I shall elaborate and apply this method to a variety of situations. All numerical solutions of differential equations involve some degree of approximation. Derivatives—properly defined in terms of infinitesimally small increments—are approximated by finite differences: dy/dx is approximated by dely/delx, whose accuracy increases as the finite difference, delx, is reduced. A large value of delx may even cause numerical instability, yielding a numerical solution that is altogether different from the true solution of the original system of differential equations. But a small value of delx generally implies that many steps must be taken to evolve the solution from a starting value of x to a finishing value. A numerical solution, therefore, requires a trade-off between computational speed and accuracy. We seek an efficient and stable method of calculation that provides accuracy appropriate to our knowledge of the physical system being simulated. The problem described in this chapter can be easily solved analytically, and the analytical solution serves as a check on the accuracy of the numerical solutions. As a simple example of a global geochemical simulation, consider the exchange of carbon dioxide between the ocean and the atmosphere. The atmosphere contains 5.6 × 1016 moles of carbon dioxide (cf. Walker, 1977), a quantity that I assume to be in equilibrium with the ocean. In this illustration I assume that the oceanic reservoir is very large and therefore does not change with time. According to Broecker and Peng (1982, p. 680) the annual exchange of carbon dioxide between ocean and atmosphere is 6.5 × 1015 moles. The rate of transfer from atmosphere to ocean is proportional to the amount in the atmosphere; the flow from ocean to atmosphere is constant. Figure 2-1 summarizes this system. The amount of carbon dioxide in the atmosphere is proportional to the partial pressure.

This chapter introduces the reader to developing and state-of-the-art tools for measuring musical expressivity. It summarizes recent quantitative measurement and analysis techniques of three domains of musical expressiveness: body motion, musical sound, and listeners’ continuous response to musical sound. It outlines computational methods to quantitatively assess expressive aspects of the body movements of the performing musicians, to extract expressive information from the musical sound itself, and to examine the perception of expressiveness through self-report continuous response methods. The chapter highlights the critical philosophical implications of making measurements of expressiveness, specifically in distinguishing between the measurement of physical aspects, such as motion and musical characteristics, and experiential aspects—the actual sensation of expressiveness experienced by the perceiver. The key philosophical point that unifies the chapter is that each of these aspects is understood through the world of ideas, definitions, and, most importantly, instruments for measuring physical and psychometric signals.

Bever calls grammar “the epicenter of all language behavior” but observes that the relation between grammar and models of linguistic behavior remains unclear. This chapter reviews some major advances in our understanding of common features of diverse grammar formalisms. Stabler argues that mild context-sensitive grammars can both define the sentences of human languages (weak adequacy) and also provide the structures of those languages (strong adequacy). He also points out that different formalisms (context-free grammars (CFG), tree-adjoining grammars (TAG), combinatory categorial grammars (CCG), set-local multicomponent grammars (MCTAG), abstract categorial grammars (ACG_2,4), multiple context-free grammars (MCFG), Minimalist grammars (MG), and context-sensitive grammars (CSG)) are weakly equivalent in the sense that they define exactly the same sets of sentences. These grammars are expressive enough to define the discontinuous dependencies of human languages. Furthermore, computational methods provide tools for describing rather abstract similarities of structures and languages.

This chapter suggests that discussion of new media should be considered in its relation to aesthetics. This would bring digital art into dialogue with other artistic practices that are part of a contemporary landscape of imaginative and creative work and would put the fate of aesthetics in an era of new media under consideration. This chapter also criticizes the notion of the so-called hybrid aesthetics and generative aesthetics as problematic and proposes a speculative aesthetics that is grounded in the language of computational method.

This chapter introduces the necessary concepts of probability theory for discrete random events, which is then used in Chapter 4. The problems at the end of the chapter emphasize computational methods, which are extremely helpful in developing intuition about the properties of random variables.

This chapter explains what financial crashes are and why, how, and when they occur. More specifically, it examines the mechanisms underlying crashes; whether we can forecast crashes; whether we could control or at least have some influence on crashes; whether crashes indicate a fundamental instability in the world financial structure; and what could be changed to modify or suppress these instabilities. The chapter considers the stock market crash of October 1987 and the main explanations for its occurrence, as well as other financial crashes in history, including the tulip mania, the South Sea bubble, and the great crash of October 1929. It also discusses extreme events in complex systems and concludes with an analysis of a new set of computational methods that can be used in the prediction of large-scale financial crashes.