This chapter involves the input--output model. Here, there are several goods under production, and some of each is needed to meet the production of the others, and there is also an external demand for each good. The model involves a matrix known as the technology matrix and a related matrix know as the Leontief matrix. It is shown how to solve such problems and it is explained that, in general (under very reasonable conditions), there always will be a solution. It is also shown how to approximate the solution using powers of the technology matrix.

Previous chapters presented linear models for responses of marine systems in regular, harmonic waves and various probabilistic properties of random processes, e.g. ocean waves. This chapter combines the two topics - a system’s deterministic response in the frequency domain and the statistics of that system’s random response when excited by a random, irregular sea. Several models for ocean wave spectra are presented and input/output relations for linear systems subject to stochastic excitation developed. The ocean wave environment is described by a single-sided wave spectrum based on various empirical formulae: P-M spectrum (single parameter, wind speed or significant wave height for the North Atlantic); ISSC spectrum (two parameter, significant crossing period and wave height); JONSWAP spectrum (six parameter, fetch limited, typical of the North Sea); and the Ochi six parameter spectrum (combined wind and swell). Short crested seas are defined and their effects discussed. The output spectrum of a linear system subject to stochastic input is derived and its Gaussian PDF given. By invoking a narrow banded assumption, PDF’s of the output follow the Rayleigh most probable extremes.

This chapter discusses the necessary components of second language acquisition, that is, input, interaction, and output. While all language learning theories support the importance of input (written and spoken), they diverge in ways which input is connected to second language acquisition. The chapter then examines some of the second language acquisition theories, such as the interaction hypothesis, the noticing hypothesis, the cognitive-interactionist approach, the output hypothesis, and sociocultural theory, all of which explains how interaction leads to second language acquisition. The chapter moves onto specific pedagogical frameworks that support interaction, including communicative language teaching and task-based language teaching. Finally, the chapter delves into the role that output plays by discussing its functions for second language acquisition.

This section lays the foundation for the analysis of random marine dynamics. A platform’s dynamics, which result from excitation due to irregular waves, can generally by expressed in a Fourier series - a consequence of linearity and the principal of linear superposition. Fourier representation, either through Fourier series or Fourier transforms, allows for frequency or time domain analysis, both of which are developed in this chapter. The frequency domain representation implies a harmonic solution in time. Consequently, the system of second order ordinary differential equations with constant coefficients become a set of simultaneous linear algebraic equations whose solutions are the complex motion amplitudes. This system of equations represents the response to harmonic forcing and does not include transient behavior associated with initial conditions. A time domain representation of floating bodies requires a means to include system memory effects. These memory effects are modeled by convolution integrals in the equations of motion where the kernel function in the convolution integral is related to the Fourier cosine transform of the damping coefficient of the floating body.

This chapter studies the case of a small efficient firm in a perfectly competitive market. Breakeven and startup points are defined. Relationships between marginal cost, average cost and average variable cost at breakeven and startup points are investigated, and it is shown how to derive the supply set of such firms.

A distinguishing factor of marine dynamics is the presence of the air-water interface. In order to determine the dynamic fluid forces acting on floating bodies - the wave exciting forces and the radiation forces (i.e. added mass and damping) - in addition to the hydrostatic forces, a lower order model of water waves based on the velocity potential and a linearized form of Bernoulli’s equation is given. The air-water interface is defined by two boundary conditions: kinematic and dynamic boundary conditions. Examining limits of the free surface boundary conditions allows a limiting process in the estimation of fluid added mass without having to solve a free surface boundary value problem. A low order model of plane progressive waves is simply a harmonic function in the lateral plane multiplied by an exponentially decaying function in the vertical coordinate. Application of the linear free surface conditions yields the important dispersion relation - a relation between the temporal wave frequency and the spatial wave frequency.

The presentation is necessarily brief and references for a more comprehensive development are listed.

This chapter discusses a language component that has not received sufficient research attention, that is, pragmatics. The chapter answers socially oriented questions such as How can I make this request politely? and What’s the best way to address this person? The chapter explains how pragmatics knowledge helps learners be aware of the requirements of the larger social context surrounding the language, and understand which grammatical and lexical forms are appropriate and helpful in different social contexts. Different pragmatic knowledge is discussed, that is, sociopragmatics (i.e., knowledge about the context) and pragmalinguistics (i.e., knowledge about specific linguistic forms). In addition, the chapter discusses speech acts, such as requests and compliments, which are used to achieve goals in our daily life by using language appropriately and effectively. The chapter then explains how pragmatics is embedded in our society by discussing illocutionary force and intercultural communicative competence. Finally, the chapter explores different ways of teaching pragmatics.

This final chapter revisits all the issues discussed in the textbook in light of their applicability to the real classroom, that is, the relationship between research and practice. It argues for the importance of considering the usefulness of research if and when a researcher hopes to influence classroom teaching via their research. The chapter begins with a description as to how research in general is connected to our daily lives (e.g., medicine, engineering, education). It then overviews the research examining the impact of research on educational practices including second language teaching. Practical and epistemological obstacles to bridging the research–practice gap from practitioners’ and researchers’ sides are discussed. The chapter proposes a variety of actions that practitioners and researchers can take in order to foster a bidirectional, constructive, productive, equitable, and mutually beneficial relationship. The chapter ends with a series of activities designed to ensure the learning outcomes from the textbook.

The chapter starts by discussing how we can determine the long-term qualitative behaviour of the solutions to second-order recurrence equations. In particular, in some cases, it can be seen that the solution is oscillatory. In the context of the multiplier-accelerator model, this corresponds to what are known as business cycles. The chapter concludes with an analysis of a dynamic macroeconomic model that is more realistic than the multiplier-accelerator one.

This chapter discusses different ways that grammar has been viewed, and answers questions such as: Is there always just one grammatical form that’s correct? and How do linguists and lay people think about grammar? The chapter reviews different ways of understanding and investigating grammar learning such as pedagogical grammar, systemic functional linguistics, contrastive analysis, CALF (complexity, accuracy, lexical complexity, and fluency), language-related episodes, and languaging. The chapter then explores several issues that are specific to grammar learning, such as rule-learning and developmental stages. In particular, it discusses processability theory, teachability theory, and communicative competence. The chapter continues by examining possible answers to the question What’s the best way to teach grammar? The chapter includes the traditional, explicit approach to grammar instruction, and continues with other approaches that are more communicative. Different approaches include proactive and reactive grammar teaching, isolated and integrated form-focused instruction, focused and unfocused tasks, and concept-based instruction.

The chapter starts by formulating the standard problem in the theory of the firm: namely, to minimise combined capital and labour costs while producing a certain output. A general formulation of a constrained optimisation problem (with one constraint) is given and it is explained how to solve such problems by the method of Lagrange multipliers. This leads to a method to calculate the cost function of firms given their production functions and capital and labour unit costs. This enables us to derive the supply sets for efficient small firms with Cobb--Douglas production functions.

This chapter provides an introduction to mathematical modelling in economics through the study of supply and demand sets, equilibrium and the effect of the imposition of an excise tax.

This chapter examines second language learning and teaching from a skills perspective, that is, reading, writing, listening, and speaking. The chapter first explains the differences among the four skills (e.g., receptive vs. productive; the necessity of formal instruction). Then, the chapter dissects each skill. As for reading and writing, the chapter discusses the necessity of formal instruction, and how the two skills are developed cognitively. The chapter also distinguishes writing-to-learn-language and learning-to-write. In terms of listening, the chapter explains different ways of processing incoming information (i.e., parsing). In relation to speaking, the chapter discusses differences between monologues or dialogues and the influence that time pressure has on second language production. By drawing on the previous chapters focusing on specific language components, the chapter ends with pedagogical recommendations for teaching the academic skills (e.g., how to target and sequence different skills).