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36762 results in Cambridge Textbooks

Page 1807 of 1839

  • Chapter E - Exercises

  • from APPENDICES
  • David Williams, Statistical Laboratory, University of Cambridge
  • Book:
    Probability with Martingales
    Published online:
    05 June 2012
    Print publication:
    14 February 1991, pp 224-242

  • Summary

    Starred exercises are more tricky. The first number in an exercise gives a rough indication of which chapter it depends on. ‘G’ stands for ‘a bit of gumption is all that's necessary’. A number of exercises may also be found in the main text. Some are repeated here. We begin with an

    Antidote to measure-theoretic material – just for fun, though the point that probability is more than mere measure theory needs hammering home.

    EG.1. Two points are chosen at random on a line AB, each point being chosen according to the uniform distribution on AB, and the choices being made independently of each other. The line AB may now be regarded as divided into three parts. What is the probability that they may be made into a triangle?

    EG.2. Planet X is a ball with centre O. Three spaceships A, B and C land at random on its surface, their positions being independent and each uniformly distributed on the surface. Spaceships A and B can communicate directly by radio if ∡AOB < 90°. Show that the probability that they can keep in touch (with, for example, A communicating with B via C if necessary) is (π + 2)/(4π).

    EG.3. Let G be the free group with two generators a and b. Start at time 0 with the unit element 1, the empty word.

  • 1 - Introduction

  • L. P. Hughston, King's College London, K. P. Tod, University of Oxford
  • Book:
    An Introduction to General Relativity
    Published online:
    28 May 2018
    Print publication:
    25 January 1991, pp 1-9

  • Summary

    Space, time, and gravitation

    GENERAL RELATIVITY is Einstein's theory of gravitation. It is not only a theory of gravity: it is a theory of the structure of space and time, and hence a theory of the dynamics of the universe in its entirety. The theory is a vast edifice of pure geometry, indisputably elegant, and of great mathematical interest.

    When general relativity emerged in its definitive form in November 1915, and became more widely known the following year with the publication of Einstein's famous exposé Die Grundlage der allgemeinen Relativitätstheorie in Annalen der Physik, the notions it propounded constituted a unique, revolutionary contribution to the progress of science. The story of its rapid, dramatic confirmation by the bending-of-light measurements associated with the eclipse of 1919 is thrilling part of the scientific history. The theory was quickly accepted as physically correct—but at the same time acquired a reputation for formidable mathematical complexity. So much so that it is said that when an American newspaper reporter asked Sir Arthur Eddington (the celebrated astronomer who had led the successful solar eclipse expedition) whether it was true that only three people in the world really understood general relativity, Eddington swiftly replied, “Ah, yes—but who's the third?”

Page 1807 of 1839