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49487 results in Computer Science

Page 2396 of 2475

  • 20 - Where to from here?

  • Peter J. Cameron
  • Book:
    Combinatorics
    Published online:
    28 May 2018
    Print publication:
    06 October 1994, pp 325-338

  • Summary

    This kind of rather highflown speculation is an essential part of my job. Without some capacity for it I could not have qualified as a Mobile, and I received formal training in it on Hain, where they dignify it with the title of Farfetching.

    Ursula K. LeGuin, The Left Hand of Darkness (1969)

    This final chapter has two purposes. A few topics not considered earlier are discussed briefly; usually there is a central problem which has served as a focus for research. Then there is a list of assorted problems in other areas, and some recommended reading for further investigation of some of the main subdivisions of combinatorics.

    Computational complexity

    This topic belongs to theoretical computer science; but many of the problems of greatest importance are of a combinatorial nature. In the first half of this century, it was realised that some well-posed problems cannot be solved by any mechanical procedure. Subsequently, interest turned to those which may be solvable in principle, but for which a solution may be difficult in practice, because of the length of time or amount of resources required. To discuss this, we want a measure of how hard, computationally, it is to solve a problem. The main difficulty here lies in defining the terms!

    Problems .

    Problems we may want to solve are of many kinds: anything from factorising a large number to solving a system of differential equations to predict tomorrow's weather. In practice, we usually have one specific problem to solve; but, in order to do mathematics, we must consider a class of problems.

  • Preface

  • Peter J. Cameron
  • Book:
    Combinatorics
    Published online:
    28 May 2018
    Print publication:
    06 October 1994, pp ix-x

  • Summary

    Ive got to work the E qwations and the low cations Ive got to comb the nations of it.

    Russell Hoban, Riddley Walker (1980)

    We have not begun to understand the relationship between combinatorics and conceptual mathematics.

    J. Dieudonné, A Panorama of Pure Mathematics (1982)

    If anything at all can be deduced from the two quotations at the top of this page, perhaps it is this: Combinatorics is an essential part of the human spirit; but it is a difficult subject for the abstract, axiomatising Bourbaki school of mathematics to comprehend. Nevertheless, the advent of computers and electronic communications have made it a more important subject than ever.

    This is a textbook on combinatorics. It's based on my experience of more than twenty years of research and, more specifically, on teaching a course at Queen Mary and Westfield College, University of London, since 1986. The book presupposes some mathematical knowledge. The first part (Chapters 2–11) could be studied by a second-year British undergraduate; but I hope that more advanced students will find something interesting here too (especially in the Projects, which may be skipped without much loss by beginners). The second half (Chapters 12–20) is in a more condensed style, more suited to postgraduate students.

    I am grateful to many colleagues, friends and students for all kinds of contributions, some of which are acknowledged in the text; and to Neill Cameron, for the illustration on p. 128.

    I have not provided a table of dependencies between chapters. Everything is connected; but combinatorics is, by nature, broad rather than deep. The more important connections are indicated at the start of the chapters.

Page 2396 of 2475