Showing papers in "The Mathematical Gazette in 1989"

Introduction to mathematical control theory (2nd edition), by S. Barnett and R. G. Cameron. Pp 404. £25 (hardback), £12·50 (paperback). 1985. ISBN 0-19-859640-5, 859639-1 (Oxford University Press)

Abstract: The best concise account of the basic mathematical aspects of control has been brought completely up to date while retaining its focus on state-space methods and its emphasis on points of mathematical interest. The authors have written a new chapter on multivariable theory and a new appendix on Kalman filtering, added a large number of new problems, and updated all the references. This book will continue as a fundamental resource for applied mathematicians studying control theory and for control engineers and electrical and mechanical engineers pursuing mathematically oriented studies. From reviews of the first edition: \"Excellent....Strongly recommended.\"--Bulletin of the International Mathematical Association. \"Could hardly be bettered.\"--Times Higher Education Supplement

The history of mathematics: a reader, edited by John Fauvel and Jeremy Gray. Pp 628. £14·25 (paperback), £35·00 (hardback). 1987. ISBN 0-333-42790-4, 0-333-42791-4 (Macmillan Education and Open University)

Abstract: In 1922 Barnes Wallis, who later invented the bouncing bomb immortalized in the movie The Dam Busters, fell in love for the first and last time, aged 35. The object of his affection, Molly Bloxam, was 17 and setting off to study science at University College London. Her father decreed that the two could correspond only if Barnes taught Molly mathematics in his letters. Mathematics with Love presents, for the first time, the result of this curious dictat: a series of witty, tender and totally accessible introductions to calculus, trigonometry and electrostatic induction that remarkably, wooed and won the girl. Deftly narrated by Barnes and Molly's daughter Mary, Mathematics with Love is an evocative tale of a twenties courtship, a surprising insight into the early life of a World War Two hero, and a great way to learn a little mathematics.

A Hilbert space problem book (2nd edition), by P. R. Halmos. Pp 369. £43. 1982. ISBN 0-387-90685-1 (Springer)

Abstract: The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with * include VAT for books; the €(D) includes 7% for Germany, the €(A) includes 10% for Austria. Prices indicated with ** include VAT for electronic products; 19% for Germany, 20% for Austria. All prices exclusive of carriage charges. Prices and other details are subject to change without notice. All errors and omissions excepted. P.R. Halmos A Hilbert Space Problem Book

Chambers science and technology dictionary, edited by P. M. B. Walker. Pp 1008. £30 (hardback), £16·95 (paperback). 1988. ISBN 0-521-96150-3, 96151-1 (Chambers and Cambridge University Press)

Abstract: Chambers science and technology dictionary , Chambers science and technology dictionary , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی

The jealous husbands and The missionaries and cannibals

Abstract: The classical river crossing problem of the jealous husbands involves three couples who have to cross a river using a boat that holds just two people. The jealousy of the husbands requires that no wife can be in the presence of another man without her husband being present. This can be accomplished in 11 crossings (i.e. one-way trips). Tartaglia gave a sketchy solution for four couples but Bachet pointed out that this was erroneous and that four couples could not get across the river. In 1879, De Fontenay pointed out that four or more couples could cross the river if there was an island in the river and gave a solution for n couples in 8n – 8 crossings. Dudeney improved the solution for n = 4 and Ball noted that this gives 6n – 7 crossings for n couples.

Circles, chords and epicycloids

Abstract: In a recent session of the Royal Institution Master Classes in Cambridge, the enthusiastic participants explored a computer graphics program, designed and written by Derek and Alan Ball. The program first draws a circle, and, given an integer N , marks N equally spaced points on it, these being cyclically labelled 1,2,…, N . The operator then inputs some rule, say n ↦ 5 n + 8, and the program draws, for each n , the chord from the point marked n to the point marked 5 n + 8 (modulo N ). Before long, the children took N to be large and flower-like envelopes of the chords started to appear (as illustrated below). Here, we show how the equations of these envelopes can be obtained in a simple way with some general parameter k , and that for positive k they are all epicycloids, that is, curves obtained by rolling a circle around a fixed circle, the simplest one being a cardioid. A similar analysis shows that for negative k they are hypocycloids (see for example E. H. Lockwood’s Book of curves , C.U.P. 1961) except when k = 1 when parallel chords occur.

Geometries and groups, by V. V. Nikulin and I. R. Shafarevich, translated from the Russian by M. Reid. Pp 251. DM 58. 1987. ISBN 0-540-15281-4 (Springer)

Abstract: I. Forming geometrical intuition statement of the main problem.- 1. Formulating the problem.- 2. Spherical geometry.- 3. Geometry on a cylinder.- 3.1. First acquaintance.- 3.2. How to measure distances.- 3.3. The study of geometry on a cylinder.- 4. A world in which right and left are indistinguishable.- 5. A bounded world.- 5.1. Description of the geometry.- 5.2. Lines on the torus.- 5.3. Some applications.- 6. What does it mean to specify a geometry?.- 6.1. The definition of a geometry.- 6.2. Superposing geometries.- II. The theory of 2-dimensional locally Euclidean geometries.- 7. Locally Euclidean geometries and uniformly discontinuous groups of motions of the plane.- 7.1. Definition of equivalence by means of motions.- 7.2. The geometry corresponding to a uniformly discontinuous group.- 8. Classification of all uniformly discontinuous groups of motions of the plane.- 8.1. Motions of the plane.- 8.2. Classification: generalities and groups of Type I and II.- 8.3. Classification: groups of Type III.- 9. A new geometry.- 10. Classification of all 2-dimensional locally Euclidean geometries.- 10.1. Constructions in an arbitrary geometry.- 10.2. Coverings.- 10.3. Construction of the covering.- 10.4. Construction of the group.- 10.5. Conclusion of the proof of Theorem 1.- III. Generalisations and applications.- 11. 3-dimensional locally Euclidean geometries.- 11.1. Motions of 3-space.- 11.2. Uniformly discontinuous groups in 3-space: generalities.- 11.3. Uniformly discontinuous groups in 3-space: classification.- 11.4. Orientability of the geometries.- 12. Crystallographic groups and discrete groups.- 12.1. Symmetry groups.- 12.2. Crystals and crystallographic groups.- 12.3. Crystallographic groups and geometries: discrete groups.- 12.4. A typical example: the geometry of the rectangle.- 12.5. Classification of all locally Cn or Dn geometries.- 12.6. On the proof of Theorems 1 and 2.- 12.7. Crystals and their molecules.- IV. Geometries on the torus, complex numbers and Lobachevsky geometry.- 13. Similarity of geometries.- 13.1. When are two geometries defined by uniformly discontinuous groups the same?.- 13.2. Similarity of geometries.- 14. Geometries on the torus.- 14.1. Geometries on the torus and the modular figure.- 14.2. When do two pairs of vectors generate the same lattice?.- 14.3. Application to number theory.- 15. The algebra of similarities: complex numbers.- 15.1. The geometrical definition of complex numbers.- 15.2. Similarity of lattices and the modular group.- 16. Lobachevsky geometry.- 16.1. 'Motions'.- 16.2. 'Lines'.- 16.3. Distance.- 16.4. Construction of the geometry concluded.- 17. The Lobachevsky plane, the modular group, the modular figure and geometries on the torus.- 17.1. Discreteness of the modular group.- 17.2. The set of all geometries on the torus.- Historical remarks.- List of notation.- Additional Literature.

Mathematics in sport

Abstract: Illustrative examples which draw on everyday experience are particularly useful in the classroom. For this reason, sport can be a good source of examples. We present four illustrations of the ways that mathematics arises in different sports. The ideas should be understandable to those studying A-level mathematics.

A-level mathematics: where next?

Abstract: Since the introduction in 1978 of the idea of a “common-core” for A-level mathematics, there was been a continuing debate over the structure and content of the subject at this level. Much of this debate remains pertinent in 1989: see, for example, [10], [15] and [28]. However, the setting-up of the Higginson Committee to review A-levels in March 1987 had the effect of putting much of the discussion specific to mathematics on the back-boiler. It is well known that the committee’s report of May 1988 recommended a wider subject-spread, roughly speaking on the model of the International Baccalaureate, and that its immediate rejection by the government created considerable furore in educational circles.

Factors in recurrence relations

Abstract: Suppose we are given a sequence of numbers defined by a linear recurrence relation with constant coefficients, for example the Fibonacci numbers which are defined by: