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Coral reef ecosystems have great importance for the countries of the Wider Caribbean Region in terms of both use and non-use values and services. Several of the contributors to this symposium attest to their importance for fisheries and biodiversity (see Ehrhardt et al. in Chapter 11; Appeldoorn in Chapter 10; Appeldoorn et al. in Chapter 12; Horrocks et al. in Chapter 9). Coral reef ecosystems support livelihoods (see McConney and Salas in Chapter 7) and provide critical ecosystem services (Schuhmann et al. in Chapter 8) including for tourism, although this aspect of their value is not developed in detail in Chapter 8. Caribbean coral reef ecosystems have been degraded by many human impacts of both marine and land-based origin (see Sweeney and Corbin in Chapter 4; Gil and Wells in Chapter 5; Yáñez-Arancibia et al. in Chapter 17). They are among the most complex and biologically diverse marine ecosystems, and will require a holistic ecosystem- based approach for their conservation and sustainable use.
This synthesis chapter presents the outputs of a group process aimed at developing a vision and way ahead for ecosystem-based management (EBM) for coral reef ecosystems in the Wider Caribbean, using the methods described earlier (Fanning et al. in Chapter 1). The chapter first describes a vision for coral reef EBM and reports on the priorities assigned to the identified vision elements. It then discusses how the vision might be achieved by taking into account assisting factors (those that facilitate achievement) and resisting factors (those that inhibit achievement). The chapter concludes with guidance on the strategic direction needed to implement the vision, identifying specific actions to be undertaken for each of the vision elements.
The Vision
The occupational breakdown of members of the Coral Reef Ecosystems Working Group reflected the diversity of affiliations present at the EBM Symposium and included governmental, intergovernmental, academic, non-governmental and private sector (fishers and fishing industry and consulting) representatives. With guidance provided by the facilitator, this diverse group of participants was asked to first address the question of “What do you see in place in 10 years’ time when EBM/EAF has become a reality in the Caribbean?”
To report a case series of elective removal of bone-anchored hearing aid implants, and reasons for removal.
Design:
Retrospective review of a prospectively collected database.
Setting:
Two tertiary referral centres in the Manchester area: Manchester Royal Infirmary and Salford Royal University Hospital.
Participants:
A series of 499 adults and children who had undergone a total of 602 implant insertions (1984–2008).
Main outcome measures:
Implant removal rates, and reasons.
Results:
Twenty-seven of the 602 implants (4.5 per cent) required removal. Of these, 12 were due to pain (2.0 per cent), seven to persistent infection (1.2 per cent), three to failure of osseointegration (0.5 per cent), three to trauma (0.5 per cent) and two to other reasons (0.4 per cent).
Conclusion:
Chronic implant site pain represents the main reason why implants are removed electively, and affects 2 per cent of all implants. This complication has important medico-legal implications and should be discussed when obtaining informed consent for implantation.
To develop a virtual-reality subtotal tonsillectomy simulation for surgical training.
Materials and Methods:
Computer models of a male patient's head and throat, and the surgical instrument, were created. These models were combined with custom-built simulation software. Recently developed tissue simulation technology that exploits recent developments in programmable graphics processing units was used to model tonsillar tissue in a way that allows surgical interaction whilst providing accurate tactile feedback. Current real-time rendering techniques were used to provide realistic visuals. Iterative refinements were made to the simulation, and in particular the tissue simulation, in consultation with relevantly experienced surgeons.
Results:
We have used newly developed tissue simulation technology to developed a novel virtual-reality subtotal tonsillectomy simulation for surgical training, the first of its kind.
Conclusion:
Early feedback suggests that this simulator can help surgeons to rapidly acquire subtotal tonsillectomy surgical skills in a risk-free and realistic virtual environment.
It is normal practice when starting the mathematical investigation of a physical problem to assign algebraic symbols to the quantity or quantities whose values are sought, either numerically or as explicit algebraic expressions. For the sake of definiteness, in this chapter, our discussion will be in terms of a single quantity, which we will denote by x most of the time. The extension to two or more quantities is straightforward in principle, but usually entails much longer calculations, or a significant increase in complexity when graphical methods are used.
Once the sought-for quantity x has been identified and named, subsequent steps in the analysis involve applying a combination of known laws, consistency conditions and (possibly) given constraints to derive one or more equations satisfied by x. These equations may take many forms, ranging from a simple polynomial equation to, say, a partial differential equation with several boundary conditions. Some of the more complicated possibilities are treated in the later chapters of this book, but for the present we will be concerned with techniques for the solution of relatively straightforward algebraic equations.
When algebraic equations are to be solved, it is nearly always useful to be able to make plots showing how the functions, fi(x), involved in the problem change as their argument x is varied; here i is simply a label that identifies which particular function is being considered.
This tutorial-style textbook develops the basic mathematical tools needed by first and second year undergraduates to solve problems in the physical sciences. Students gain hands-on experience through hundreds of worked examples, self-test questions and homework problems. Each chapter includes a summary of the main results, definitions and formulae. Over 270 worked examples show how to put the tools into practice. Around 170 self-test questions in the footnotes and 300 end-of-section exercises give students an instant check of their understanding. More than 450 end-of-chapter problems allow students to put what they have just learned into practice. Hints and outline answers to the odd-numbered problems are given at the end of each chapter. Complete solutions to these problems can be found in the accompanying Student Solutions Manual. Fully-worked solutions to all problems, password-protected for instructors, are available at www.cambridge.org/foundation.
In Chapter 6 we discussed how complicated functions f(x) may be expressed as power series. Although they were not presented as such, the power series could all be viewed as linear superpositions of the monomial basic set of functions, namely 1, x, x2, x3, … xn, … Natural though this set may seem, they are in many ways far from ideal: for example they possess no mutual orthogonality properties, a characteristic that is generally of great value when it comes to determining, for any particular function, the multiplying constant for each basic function in the sum. Moreover, this particular set of basic functions can only be used to represent continuous functions.
In the case of original functions f(t) that are periodic, some improvement on this situation can be made by using, as the basic set, sine and cosine functions. For a function with period T, say, the set of sine and cosine functions with arguments 2πnt/T, for all n ≥ 0, form a suitable basic set for expressing f as a series; such a representation is called a Fourier series. One great advantage they possess over the monomial functions is that they are mutually orthogonal when integrated over any continuous period of length T, i.e. the integral from t0 to t0 + T of the product of any sine and any cosine, or of two sines or cosines with different values of n, is equal to zero.
This and the next chapter are concerned with the formalism of probably the most widely used mathematical technique in the physical sciences, namely the calculus. The current chapter deals with the process of differentiation whilst Chapter 4 is concerned with its inverse process, integration. The topics covered are essential for the remainder of the book; once studied, the contents of the two chapters serve as reference material, should that be needed. Readers who have had previous experience of differentiation and integration should ensure full familiarity by looking at the worked examples in the main text and by attempting the problems at the ends of the two chapters.
Also included in this chapter is a section on curve sketching. Most of the mathematics needed as background to this important skill for applied physical scientists was covered in the first two chapters, but delaying our main discussion of it until the end of this chapter allows the location and characterisation of turning points to be included amongst the techniques available.
Differentiation
Differentiation is the process of determining how quickly or slowly a function varies, as the quantity on which it depends, its argument, is changed. More specifically, it is the procedure for obtaining an expression (numerical or algebraic) for the rate of change of the function with respect to its argument.
This chapter introduces space vectors and their manipulation. Firstly we deal with the description and algebra of vectors, then we consider how vectors may be used to describe lines, planes and spheres, and finally we look at the practical use of vectors in finding distances. The calculus of vectors will be developed in a later chapter; this chapter gives only some basic rules.
Scalars and vectors
The simplest kind of physical quantity is one that can be completely specified by its magnitude, a single number, together with the units in which it is measured. Such a quantity is called a scalar, and examples include temperature, time and density.
A vector is a quantity that requires both a magnitude (≥ 0) and a direction in space to specify it completely; we may think of it as an arrow in space. A familiar example is force, which has a magnitude (strength) measured in newtons and a direction of application. The large number of vectors that are used to describe the physical world include velocity, displacement, momentum and electric field. Vectors can also be used to describe quantities such as angular momentum and surface elements (a surface element has a magnitude, defined by its area, and a direction defined by the normal to its tangent plane); in such cases their definitions may seem somewhat arbitrary (though in fact they are standard) and not as physically intuitive as for vectors such as force.
Differential equations are the group of equations that contain derivatives. There are several different types of differential equations, but here we will be considering only the simplest types. As its name suggests, an ordinary differential equation (ODE) contains only ordinary derivatives (no partial derivatives) and describes the relationship between these derivatives of the dependent variable, usually called y, with respect to the independent variable, usually called x. The solution to such an ODE is therefore a function of x and is written y(x). For an ODE to have a closed-form solution, it must be possible to express y(x) in terms of the standard elementary functions such as x2, exp x, In x, sin x, etc. The solutions of some differential equations cannot, however, be written in closed form, but only as an infinite series that carry no special names.
Ordinary differential equations may be separated conveniently into different categories according to their general characteristics. The primary grouping adopted here is by the order of the equation. The order of an ODE is simply the order of the highest derivative it contains. Thus, equations containing dy/dx, but no higher derivatives, are called first order, those containing d2y/dx2 are called second order and so on. In this chapter we consider first-order equations and some of the more straightforward equations of second order.