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34754 results in Cambridge Textbooks

Page 1629 of 1738

  • Foreword

  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp xi-xiv

  • Summary

    Inductive logic is unlike deductive or symbolic logic. In deductive reasoning, when you have true premises and a valid argument, the conclusion must be true too. Valid deductive arguments do not take risks.

    Inductive logic takes risks. You can have true premises, a good argument, but a false conclusion. Inductive logic uses probability to analyse that kind of risky argument.

    Good News

    Inductive reasoning is a guide in life. People make risky decisions all the time. It plays a much larger part in everyday affairs than deductive reasoning.

    Bad News

    People are very bad when reasoning about risks. We make a lot of mistakes when we use probabilities.

    This book starts with a list of seven Odd Questions. They look pretty simple. But most people get some of the answers wrong. The last group of nine-year-olds I tested did better than a group of professors. Try the Odd Questions. Each one is discussed later in the book.

    Practical Aims

    This book can help you understand, use, and act on probabilities, risks, and statistics. We live our lives taking chances, acting when we don't know enough. Every day we experience a lot of uncertainties. This book is about the kinds of actions you can take when you are uncertain what to do. It is about the inferences you can draw when your evidence leaves you unsure what is true.

  • 4 - Elementary Probability Ideas

  • from HOW TO CALCULATE PROBABILITIES
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 37-46

  • Summary

    This chapter explains the usual notation for talking about probability, and then reminds you how to add and multiply with probabilities.

    WHAT HAS A PROBABILITY?

    Suppose you want to take out car insurance. The insurance company will want to know your age, sex, driving experience, make of car, and so forth. They do so because they have a question in mind:

    What is the probability that you will have an automobile accident next year?

    That asks about a proposition (statement, assertion, conjecture, etc.):

    “You will have an automobile accident next year.”

    The company wants to know: What is the probability that this proposition is true?

    The insurers could ask the same question in a different way:

    What is the probability of your having an automobile accident next year?

    This asks about an event (something of a certain sort happening). Will there be “an automobile accident next year, in which you are driving one of the cars involved”?

    The company wants to know: What is the probability of this event occurring?

    Obviously these are two different ways of asking the same question.

    PROPOSITIONS AND EVENTS

    Logicians are interested in arguments from premises to conclusions. Premises and conclusions are propositions. So inductive logic textbooks usually talk about the probability of propositions.

    Most statisticians and most textbooks of probability talk about the probability of events.

    So there are two languages of probability, propositions and events.

  • 3 - The Gambler's Fallacy

  • from HOW TO CALCULATE PROBABILITIES
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 23-36

  • Summary

    Most of the main ideas about probability come up right at the beginning. Two major ones are independence and randomness. Even more important for clear thinking is the notion of a probability model.

    ROULETTE

    A gambler is betting on what he thinks is a fair roulette wheel. The wheel is divided into 38 segments, of which:

  • ▪ 18 segments are black.

  • ▪ 18 segments are red.

  • ▪ 2 segments are green, and marked with zeroes.

    • If you bet $10 on red, and the wheel stops at red, you win $20. Likewise if you bet $10 on black and it stops at black, you win $20. Otherwise you lose. The house always wins when the wheel stops at zero.

    Now imagine that there has been a long run–a dozen spins–in which the wheel stopped at black. The gambler decides to bet on red, because he thinks:

    The wheel must come up red soon.

    This wheel is fair, so it stops on red as often as it stops on black.

    Since it has not stopped on red recently, it must stop there soon. I'll bet on red.

    The argument is a risky one. The conclusion is, “The wheel must stop on red in the next few spins.” The argument leads to a risky decision. The gambler decides to bet on red. There you have it, an argument and a decision. Do you agree with the gambler?

  • 13 - Personal Probabilities

  • from PROBABILITY AS A MEASURE OF BELIEF
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 151-162

  • Summary

    How personal degrees of belief can be represented numerically by using imaginary gambles.

    Chapters 1–10 were often deliberately ambiguous about different kinds of probability. That was because the basic ideas usually applied, across the board, to most kinds of probability.

    Now we develop ideas that matter a lot for belief-type probabilities. They do not matter so much from the frequency point of view.

    THE PROGRAM

    There are three distinct steps in the argument, and each deserves a separate chapter.

  • ▪ This chapter shows how you might use numbers to represent your degrees of belief.

  • ▪ Chapter 14 shows why these numbers should satisfy the basic rules of probability. (And hence they should obey Bayes' Rule.)

  • ▪ Chapter 15 shows how to use Bayes' Rule to revise or update personal probabilities in the light of new evidence. This is the fundamental motivation for the group of chapters, 13–15.

    • In these chapters we are concerned with a person's degrees of belief. We are talking about personal probabilities. But this approach can be used for other versions of belief-type probability, such as the logical perspective of Keynes and Carnap.

    Because Bayes' Rule is so fundamental, this approach is often called Bayesian. “Belief dogmatists” are often simply called Bayesians because the use of Bayes' rule as a model of learning from experience plays such a large part in their philosophy. But notice that there are many varieties of Bayesian thinking. This perspective ranges from the personal to the logical.

  • 9 - Maximizing Expected Value

  • from HOW TO COMBINE PROBABILITIES AND UTILITIES
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 98-113

  • Summary

    How do you choose among possible acts? The most common decision rule is an obvious one. Perform the action with the highest expected value. There are, however, a few more paradoxes connected with this simple rule.

    RISKY DECISIONS

    Logic analyzes reasons and arguments. We can give reasons for our beliefs. We can also give reasons for our actions and our decisions. What is the best thing to do under the circumstances? Inductive logic analyzes risky arguments. It also helps with decision theory, the theory of making risky decisions.

    Should I go out in a thunderstorm to fetch a book, even though I am scared of lightning? I go out in a thunderstorm because I believe I left a book outside. I believe it will get wet and be ruined. I also believe I will not be struck by lightning. But I also go outside because I want the book, among other things. Of course, my beliefs are not certainties–I am pretty confident I left the book there. I am pretty sure it will get wet if it is there. I know it is not probable that I will be hit by lightning.

    Decisions depend on two kinds of thing:

  • ▪ What we believe.

  • ▪ What we want.

    • Sometimes we can represent our degrees of belief or confidence by probabilities. Sometimes we can represent what we want by dollar values, or at least by judgments of value, which we call utilities.

  • A - Mathematical methods for the solution of Fick's Second Law

  • from Appendices
  • David S. Wilkinson, McMaster University, Ontario
  • Book:
    Mass Transport in Solids and Fluids
    Published online:
    28 May 2018
    Print publication:
    02 November 2000, pp 236-241

  • Summary

    Separation of variables method

    We will assume that the solution for solute concentration C(y,t) can be separated into two functions, one depending only on position, the other on time, i.e.

    If we substitute this into Fick's Second Law then

    We then separate the y- and t-dependent terms,

    For the first equality in this expression to be valid, each side must be constant. We set this constant to be (—λ2). We now solve each side of the equation separately. The time-dependent equation

    can be integrated to yield

    where ⊖0 is a constant of integration. The spatial equation

    is integrated twice, giving

    where A′ and B′ are integration constants. Combining these expressions gives

    where A = A′⊖0 and B′ = B′⊖0 . In general, there are a multitude of solutions to Fick's Second Law, all of which have this form, but with different values of A, B and λ. Therefore, the most general solution is given by the sum of these

    This solution is generally valid for diffusion in one dimension. Similar expressions can be derived for cylindrical or spherical symmetry.

    In order to proceed further we need to define the initial and boundary conditions appropriate to the problems at hand. We will consider only one example, namely diffusion out of a plate with an initially uniform concentration, and the concentration fixed at the surface of the plate.

  • 4 - Applications to problems in materials engineering

  • from Part B - Solid-state diffusion in dilute alloys
  • David S. Wilkinson, McMaster University, Ontario
  • Book:
    Mass Transport in Solids and Fluids
    Published online:
    28 May 2018
    Print publication:
    02 November 2000, pp 96-98

  • Summary

    ‘Cheshire Puss’, she began, rather timidly…

    ‘Would you tell me please, which way I ought to go from here?’

    ‘That depends a good deal on where you want to get to,’ said the cat.

    ‘I don't much care where –’ said Alice.

    ‘Then it doesn't much matter which way you go,’ said the cat.

    Lewis Carroll, Alice Through the Looking Glass

    We now turn our attention to the application of Fick's Laws for the solution of problems of interest to materials engineers. In this brief chapter the groundwork is laid through a discussion of the kinds of boundary conditions that can be applied.

    Introduction

    In the next few chapters we will build on our understanding of solutions to Fick's Laws, by learning how to apply them to a large variety of problems involving diffusion. This process involves a number of well-defined steps. These steps are required in order to turn a general statement of a problem into a framework which can be solved mathematically. The procedure we will adopt is generally applicable to all problems involving the solution of partial differential equations such as Fick's Laws. As we discussed in the last chapter, these equations fall under the general category of what are called ‘boundary-value problems’. That is, we need to solve a differential equation (such as Fick's Second Law), subject to certain bounds.

  • D - Solving problems by developing conceptual models

  • from Appendices
  • David S. Wilkinson, McMaster University, Ontario
  • Book:
    Mass Transport in Solids and Fluids
    Published online:
    28 May 2018
    Print publication:
    02 November 2000, pp 258-266

  • Summary

    Introduction

    Along with learning about specific aspects of matter transport and its relationship to materials engineering, a course in this subject area is an excellent place to learn about the role of modelling in materials science and engineering. Models are used to develop a better understanding of the processes we are trying to control in order to produce better, more reliable and consistent materials. We usually start with a problem (which is often ill-defined, at least initially) and some data. A model is an attempt to draw general conclusions about the behaviour of the system of interest in a quantitative way which allows us to make predictions about the behaviour of the system. Models invariably require certain assumptions and approximations to be made. Thus the model will be limited both in its precision and in its scope or range of applicability. However, a model can be a powerful tool if used carefully, and all materials engineers should have expertise in how to model materials behaviour.

    Consider the problem of predicting weather behaviour. A set of observations of many days might lead to the hypothesis that the weather on any given day in a particular location is correlated to the weather on the previous day. Thus if you predict that the weather tomorrow is going to be the same as the weather today you will be correct say 50% of the time.

  • Preface

  • David S. Wilkinson, McMaster University, Ontario
  • Book:
    Mass Transport in Solids and Fluids
    Published online:
    28 May 2018
    Print publication:
    02 November 2000, pp xiii-xiv

  • Summary

    The field of matter transport is central to understanding the processing and subsequent behaviour of materials. While thermodynamics tells us the state in which a material system would like to exist, the kinetics of matter transport tell us how fast it can get there. The materials which we use in service are often not at equilibrium. This is particularly so in solid materials. Here kinetics are sufficiently slow that materials are almost never in complete thermodynamic equilibrium but attain a stable state which is kinetically limited. However, kinetics are equally important in fluids since the transport of matter in a liquid or gas often limits the rate at which many processes for materials fabrication can proceed. The aim of this book is to give students a solid grounding in the principles of matter transport and their application to the solution of engineering problems.

    It is this emphasis on engineering application that distinguishes this text from many other books in this field. In the field of solid-state diffusion, for example, much has been written about the mechanisms of diffusion at the atomic scale. This is a fascinating topic and of great interest to materials scientists. However, only an elementary understanding is required in order to treat many practical diffusion problems. Thus, the emphasis throughout this book is on developing methods for solving problems involving the transport of matter in materials.

  • 10 - General driving force for diffusion

  • from Part D - Alternative driving forces for diffusion
  • David S. Wilkinson, McMaster University, Ontario
  • Book:
    Mass Transport in Solids and Fluids
    Published online:
    28 May 2018
    Print publication:
    02 November 2000, pp 223-235

  • Summary

    Applied science is a conjuror, whose bottomless hat yields

    impartially the softest of Angora rabbits and the most

    petrifying of Medusas.

    Aldous Huxley, Tomorrow and Tomorrow and Tomorrow

    Up to now we have assumed that only concentration gradients provide a driving force for diffusion. We now generalize this, by including all forms of free-energy gradient. This will lead us fairly easily to a general theory of diffusion that includes Fick's First Law as a special case. We will apply our new understanding to problems such as diffusion due to an electrical field or a mechanical stress. We will finish the chapter by seeing how this method can help to understand diffusion in multicomponent systems.

    Theory

    Throughout this book we have considered that diffusional processes are solely driven by concentration gradients. This is based on the concept that a material strives towards equilibrium, and that equilibrium is defined as a state of uniform concentration. In materials science, however, we recognize that this is a rather naive definition of equilibrium. To be more precise, we should state that a body is at equilibrium when the free energy is uniform throughout. Using this definition, we can broaden considerably the range of situations in which diffusion occurs. Thus, we expect diffusion not only in response to concentration gradients, but also due to electrical and thermal fields, and mechanical

  • 5 - Applications involving gas-solid reactions

  • from Part B - Solid-state diffusion in dilute alloys
  • David S. Wilkinson, McMaster University, Ontario
  • Book:
    Mass Transport in Solids and Fluids
    Published online:
    28 May 2018
    Print publication:
    02 November 2000, pp 99-113

  • Summary

    Though this be madness, yet there is method in't.

    William Shakespeare, Hamlet

    In this chapter we consider problems involving diffusion from a gas phase into the solid. The boundary conditions at the gas/solid interface are determined, assuming that local equilibrium exists, from a knowledge of thermodynamics. We will also consider an example in which the rate of reaction at the interface is the controlling step.

    Introduction

    We start by considering the interaction between gases and solid surfaces. This is a useful starting point for several reasons. First, this area includes a number of important technological problems in materials engineering. Second, these problems generally involve only a single diffusing species. This greatly simplifies matters. Examples of gas species which diffuse readily into solids include carbon and nitrogen in steel, hydrogen in many metals and ceramics, and water (H+ or H3O+ ions) into glass. One common feature of these gases is that they involve rather small ions, which diffuse interstitially into crystalline solids. They therefore exhibit fairly high diffusion coefficients.

    There is a related set of problems, involving diffusion from liquids into solids, which can be handled in the same way. For example, glasses can be strengthened by allowing K+ or Ag+ to diffuse into the surface, through ion exchange from a liquid. Similarly, ions can be leached out of a glass when in contact with a liquid.

  • 3 - Transient diffusion problems

  • from Part B - Solid-state diffusion in dilute alloys
  • David S. Wilkinson, McMaster University, Ontario
  • Book:
    Mass Transport in Solids and Fluids
    Published online:
    28 May 2018
    Print publication:
    02 November 2000, pp 47-95

  • Summary

    It was a saying of the ancients that ‘truth lies in a well’;

    and to carry on the metaphor, we may justly add, that

    logic supplies us with steps whereby we may go down to

    reach the water.

    Isaac Watts

    Fick's First Law is still valid even when steady-state conditions do not exist. However, it is not very convenient to use it in this form for solving transient diffusion problems, for which the concentration at any position depends on both time and position. It is therefore more convenient to derive a second version of this equation, Fick's Second Law, which contains an explicit dependence on time. In this chapter we will first derive the equation; a second-order partial differential equation. We will then learn how solutions to this equation can be obtained for a large variety of situations depending on the problem geometry, the initial and boundary conditions and the time-frame over which we need a valid solution.

    Fick's Second Law

    We consider the situation illustrated in Fig. 3.1. At two positions along the y-axis at y1 and y2 , a distance Δy apart, the concentration of the solute is C1 and C2 respectively. We can use Fick's First Law to determine the amount of solute per unit time which enters this element at y1 and which leaves it at y2 . Because matter must be conserved, the difference between these must be equal to the rate at which solute accumulates within the element.

  • 9 - Advanced topics

  • from Part C - Mass transport in concentrated alloys and fluids
  • David S. Wilkinson, McMaster University, Ontario
  • Book:
    Mass Transport in Solids and Fluids
    Published online:
    28 May 2018
    Print publication:
    02 November 2000, pp 202-220

  • Summary

    For who would lose,

    Though full of pain, this intellectual being,

    Those thoughts that wander through eternity,

    To perish, swallowed up and lost

    In the wide womb of uncreated night,

    devoid of sense and motion?

    John Milton, Paradise Lost

    In this chapter we introduce several advanced topics. The first deals with different approaches that can be taken to develop overall mass balances within a vessel. This can be used to simplify the analysis of a complex process, as we will see. We then turn to the problem of multi-phase resistances. Up to now we have been able to make assumptions about which of two phases in contact dominates the mass transfer kinetics across the interface between them. However, there are situations in which both phases may contribute to the overall mass transfer, at least over a limited range of conditions. We will develop an approach for dealing with this situation. Finally, we will consider a specific case involving topochemical reactions in porous solids.

    Overall mass balance

    We have used mass balances extensively throughout this book. Most often we have used them to develop an expression for the motion of an interface separating two reacting phases. In fluids, however, this process can become quite complex. Consider, for example, what happens when a gas stream is passed through or over a bed containing some reactive species. The reaction occurs slowly as the gas stream passes.

Page 1629 of 1738