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38626 results in Cambridge Textbooks

Page 1507 of 1932

  • 11 - Inequality among and within nations: past, present, future

  • Karl Gunnar Persson, University of Copenhagen, Paul Sharp, University of Southern Denmark
  • Book:
    An Economic History of Europe
    Published online:
    28 May 2018
    Print publication:
    12 March 2015, pp 238-255

  • Summary

    Why is there inequality?

    Inequality refers to unequal access to welfare as manifested in consumption, health, life expectancy and schooling. It is usually allied to inequality of income. However, income is not an end but a means of acquiring a good life, which has a number of attributes apart from consumption. Needless to say, income is an imperfect guide to welfare distribution because some aspects of welfare are only vaguely linked to income. For example, income inequality on a world scale went on increasing until recent decades while inequality in terms of literacy has fallen sharply since 1950. The dramatic fall in child mortality during the last two centuries is also only remotely linked to income, and inequality in terms of life expectancy across nations has fallen. In the advanced welfare states of Europe, a growing number of services such as health, childcare, schooling and access to cultural sites such as theatres is provided at subsidized rates that again weaken the link between income and actual consumption. Despite these reservations, income inequality is an important, although insufficient, guide to welfare distribution.

    The major sources of income are work, acquired or inherited wealth and, from the twentieth century onward, transfers such as pensions. Excluding property income, the income inequality we observe is closely related to skills acquired through formal education and on-the-job training. However, throughout history we have seen that discrimination can distort the relationship between skill and reward. Property income is not necessarily related to one's own efforts in the past but simply to the sheer luck of being born well endowed.

    On a world scale we note that the poor in poor nations are usually much poorer than the poor in rich nations, while the rich in poor nations are almost as rich as the rich in rich nations, although less numerous.

  • 7 - Basic calculus of random processes

  • Bruce Hajek, University of Illinois, Urbana-Champaign
  • Book:
    Random Processes for Engineers
    Published online:
    11 May 2018
    Print publication:
    12 March 2015, pp 206-247

  • Summary

    The calculus of deterministic functions revolves around continuous functions, derivatives, and integrals. These concepts all involve the notion of limits. See the appendix for a review of continuity, differentiation, and integration. In this chapter the same concepts are treated for random processes. We've seen four different senses in which a sequence of random variables can converge: almost surely (a.s.), in probability (p.), in mean square (m.s.), and in distribution (d.). Of these senses, we will use the mean square sense of convergence the most, and make use of the correlation version of the Cauchy criterion for m.s. convergence, and the associated facts that for m.s. convergence, the means of the limits are the limits of the means, and correlations of the limits are the limits of correlations (Proposition 2.11 and Corollaries 2.12 and 2.13). Ergodicity and the Karhunen–Loéve expansions are discussed as applications of integration of random processes.

    Continuity of random processes

    The topic of this section is the definition of continuity of a continuous-time random process, with a focus on continuity defined using m.s. convergence. Chapter 2 covers convergence of sequences. Limits for deterministic functions of a continuous variable can be defined in either of two equivalent ways. Specifically, a function f on ℝ has a limit y at to, written as lims→to f(s) = y, if either of the two equivalent conditions is true:

    (1) (Definition based on ε and δ) Given ε > 0, there exists δ > 0 so |f(s) − y|≤ ε whenever |s − to| ≤ δ.

    (2) (Definition based on sequences) f(sn) → y for any (sn) such that sn → to.

    Let us check that (1) and (2) are equivalent. Suppose (1) is true, and let (sn) be such that sn → to. Let ε > 0 and then let δ be as in condition (1). Since sn → to, it follows that there exists no so that |sn − to| ≤ δ for all n ≥ no.

  • 10 - Martingales

  • Bruce Hajek, University of Illinois, Urbana-Champaign
  • Book:
    Random Processes for Engineers
    Published online:
    11 May 2018
    Print publication:
    12 March 2015, pp 304-324

  • Summary

    This chapter builds on the brief introduction to martingales given in Chapter 4, to give a glimpse of how martingales can be used to obtain bounds and prove convergence in many contexts, such as for estimation and control algorithms in a random environment. On one hand, the notion of a martingale is weak enough to include processes arising in applications involving estimation and control, and on the other hand, the notion is strong enough that important tools for handling sums of independent random variables, such as the law of large numbers, the central limit theorem, and large deviation estimates, extend to martingales.

    Two other topics in this book are closely related to martingales. The first is the use of linear innovations sequences discussed in Chapter 3. As explained in Example 10.7 below, martingale difference sequences arise as innovations sequences when the linearity constraint on predictors, imposed for linear innovations sequences, is dropped. The other topic in this book closely related to martingales is the Foster–Lyapunov theory for Markov processes, discussed in Chapter 6. A central feature of the Foster–Lyapunov theory is the drift of a function of a Markov process: E[V(Xt+1) − V(Xt)|Xt = x]. If this drift were zero then V(Xt) would be a martingale. The assumptions used in the Foster–Lyapunov theory allow for a controlled difference from the martingale assumption. In a sense, martingale theory is what is left when the linearity and Markov assumptions are both dropped.

    The chapter is organized as follows. The definition of a martingale involves conditional expectations, so to give the general definition of a martingale we first revisit the definition of conditional expectation in Section 10.1. The standard definition of martingales, in which σ-algebras are used to represent information, is given in Section 10.2. Section 10.3 explains how the Chernoff bound, central to large deviations theory, readily extends to sequences that are not independent.

  • 3 - Population, economic growth and resource constraints

  • Karl Gunnar Persson, University of Copenhagen, Paul Sharp, University of Southern Denmark
  • Book:
    An Economic History of Europe
    Published online:
    28 May 2018
    Print publication:
    12 March 2015, pp 47-66

  • Summary

    Historical trends in population growth

    Economics is sometimes called the dismal science because many of its pioneers expressed pessimism about the possibility of sustained economic growth in a world of limited resources. We meet this view today in the worries about shortages of raw materials, such as oil, eventually putting an end to economic growth. Late-nineteenth-century economists worried about coal shortages, but the concern today is rather that coal generates too much CO2, which might in the long run harm growth. The first economist to develop a coherent theory of limited resources as a binding constraint for sustained long-term economic growth was Thomas Malthus (1766–1834), whose An Essay on the Principle of Population was first published in 1798. The date of publication is not without interest, since it falls within the first decades of the Industrial Revolution in Britain, which combined unprecedented population growth with increasing (or at least constant) income per head. Malthus argued that given limited land the supply of food would eventually constrain income and population growth. We will soon return to a detailed exposition of the Malthusian view, but we will first review the evidence on long-term population growth in Europe. Reasonably precise population estimates are available only from the sixteenth century onward; before that, populations are estimated from projections based on scarce data and conjectures of the carrying capacity of a given area using the prevailing technology.

    Cultures based on hunter-gatherer technology, which preceded the breakthrough of agriculture and sedentary civilization in the Middle East some 12,000 years ago, are very demanding in terms of land, and this limited world population to an estimated 8–12 million. A hunter-gatherer culture satisfies its food requirements from nature without actually controlling it: consequently an equilibrium between the stock of animals and the stock of men will evolve. If mankind over-exploits the animal stock the reproduction of both will be disturbed.

  • 8 - Derivatives: Tangents

  • from PART II - FUNCTIONS
  • R. P. Burn, University of Exeter
  • Book:
    Numbers and Functions
    Published online:
    11 May 2018
    Print publication:
    19 February 2015, pp 197-217

  • Summary

    Preliminary reading: Bryant ch. 4, Courant and John ch. 2.

    Concurrent reading: Hart, Spivak ch. 9 and ch. 10.

    Further reading: Mason, Tall (1982).

    Definition of derivative

    It is easy enough to say when a straight line is a tangent to a circle or an ellipse. For these curves, a straight line – infinitely extended – meets the curve in 0, 1 or 2 points. Each line with a unique point of intersection is a tangent. However, if we try to use such a test to identify tangents to other curves we are in for a disappointment, and on several counts.

    1 At how many points does the line x = 1 intersect the parabola y = x2? Draw a sketch. Is this line a tangent to the curve?

    2 At how many points does the line y = −2 intersect the cubic curve y = x3 − 3x? Draw a sketch. Is this line a tangent to the curve?

    From qn 2 we learn that whether a line is a tangent to a curve or not is a local question, which must be asked relative to the particular point of intersection, that is, inside a sufficiently small neighbourhood of that point.

    3 At how many points does the line y = mx intersect the cubic curve y = x3? For how many values of m might the line be a tangent to the curve?

    Check that the line y = h2x is the chord joining the two points (0, 0) and (h, h3) on the curve, provided h ≠ 0.

    If h → 0, to what does the slope (or gradient) of the chord tend?

    4 Write down the equation of the line with slope m through the point(a, f(a)).

    Write down the equation of the chord joining the points (a, f(a)) and (a + h, f(a + h)).

  • Preface to the second edition (2015)

  • Joint Association of Classical Teachers' Greek Course
  • Book:
    The Intellectual Revolution
    Published online:
    28 May 2018
    Print publication:
    19 February 2015, pp xi-xi

  • Summary

    Preface to the second edition (2015)

    The second, fully revised and reset edition of Reading Greek (2007) consists of two volumes, one of Text and Vocabulary and one of Grammar and Exercises. The fully revised and reset second editions of The Intellectual Revolution and A World of Heroes contain the same selections of authors and follow the same principles as the first editions in the provision of vocabulary and learning vocabulary, and rely on Greek Vocabulary in the same way, but there are two vital differences:

  • all help is given in the order in which it appears in the text;

  • the help now includes extensive grammatical and syntactical notes, together with comment of the sort one would expect to find in an edition of the works in hand.

    • The revision of The Intellectual Revolution was undertaken by Keith Maclennan (Euripides), to whom the Project is (again) most grateful for his care and expertise, and Dr Jones (Thucydides and Plato). Dr Jones is responsible for the final editing.

    My very best thanks go to Professor Colin Leach for work on the proofs and to Cambridge University Press’s Leigh Mueller and Elizabeth Hanlon for the care they took over the editing of this new edition. Thanks also go to Cecilia Mackay for gathering the pictures and permissions.

Page 1507 of 1932