Showing papers in "The Mathematical Gazette in 2005"

Set theory: the third millennium edition , by Thomas Jech. Pp. 769. £77. 2003. ISBN 3 540 44085 2 (Springer).

Abstract: You are then told of a rule that if a card has a vowel on one side, it has an even number on the other. Which cards do you need to turn over to make sure they all satisfy this rule? The blurb speaks of 'thorough revisions that improve accessibility'; that accessibility has been improved cannot be disputed, but the thoroughness can. For example, reference is made incorrectly to a theorem 4.1; an exercise requires an understanding of idempotence although the concept is not discussed in the text; and most curiously, a section on how to solve algebraic inequalities is sandwiched in the chapter on set theory, between a description of the notation for real intervals and an explanation of indexed sets. Again, when so much has been done to improve accessibility in other sections, it is disappointing to note that the description of injections and surjections is inferior to that in many other texts; some schematic diagrams would have helped enormously.

Unsolved problems in number theory (3rd edn), by Richard K. Guy. Pp. 437. £54.00. 2004. ISBN 0 387 20860 (Springer-Verlag).

Abstract: references. The index of notations used in the text fills six pages. There are 29 tables plus other lists. There are however no graphs or other illustrations. A plot of each index (or the number of digits) of the Mersenne primes on a logarithmic scale against their dates of discovery would show the significance of GIMPS (Great Internet Mersenne Prime Search) since 1996, which is clear from the table. But what is not easily seen from the table is the increase in the rate of finding these large primes in this period. In 1983, the year when the 29th Mersenne prime was found, I predicted from such a plot, and assuming no changes in the trends for resources or methods, that the 40th Mersenne prime would be found in 2008 ± 3 years [1]. It was found in November 2003 and confirmed soon afterwards, over a year before my window, due to GIMPS. A similar plot for the largest known primes during the last 30 years can be seen at www.utm.edu/research/primes/notes/by_year.html.

Fractals and chaos. The Mandelbrot set and beyond, by Benoit B. Mandelbrot. Pp. 308. £38.50. 2004, ISBN 0 387 20158 0 (Springer-Verlag).

Abstract: Fractals and chaos. The Mandelbrot set and beyond, by Benoit B. Mandelbrot. Pp. 308. £38.50. 2004, ISBN 0 387 20158 0 (Springer-Verlag). Fractals, fractal geometry or chaos theory have been a hot topic in scientific research. It may come as a surprise that much of the theory as we know it was initiated during the last 30 years and by the vision of one man: Benoit Mandelbrot. The story starts in 1975 with Mandelbrot's small booklet Les objets fractals. Forme, hasard et dimension (published by Flammarion, Paris). Already in 1977 it was translated and expanded into Fractals: Form, Chance and Dimension (W. H. Freeman, San Francisco), but the breakthrough came in 1980 with the first picture of the Mandelbrot set in 'Fractal aspects of the iteration of z >-» Xz (1 z) for complex X and z', Ann. New York Acad. Sci. and reprinted on pp. 37—51 in the present volume. It was, however, Mandelbrot's 1982 masterpiece The Fractal Geometry of Nature (W. H. Freeman, New York) that popularized the subject. Mandelbrot's book is a scientific, philosophic and pictorial treatise at the same time and it is one of the rare specimen of serious mathematics books that can be read and re-read at many different levels. The volume under review is 'Selecta C of Mandelbrot's oeuvre. It is a selection of papers which appeared between 1980 and 2003, dealing with (non-)quadratic rational dynamics, iterated (nonlinear) function systems and multifractal measures. Alongside some important and very technical original papers, there is a highly readable (also for the non-specialist) introduction and survey-type original contributions, extracts from his 1982 monograph as well as unpublished material. The last chapter is devoted to a brief historical account of the subject's early heroes: Pierre Fatou and Gaston Julia. Rather than being a juxtaposition of papers, Mandelbrot succeeded in creating a readable selection of material which contains new original contributions. The papers featured in the book are sometimes corrected and annotated; that in this process the original pagination was lost is somewhat unfortunate. The style is what one could call 'truly Mandelbrotian', a mixture of hard science, often with a personal touch, some (sometimes quite lop-sided) personal notes and recollections and always the urge to convey a message.

Stochastic differential equations: An introduction with applications (6th edn), by Bernt Oksendal. Pp. 360. £27. 2004. ISBN 3 540 04758 1 (Springer-Verlag).

Abstract: Finally, Chapter 25 is devoted to Conway's Game of Life. It is fully in keeping with the whimsical nature of Winning ways that the last chapter features a no-person game (one which plays itself once set up) and that what began as a piece of recreational fun turns out to be a universal Turing machine with possible ramifications for Life, The Universe and Everything. The first edition of Winning ways was the definitive work on combinatorial game theory for nearly quarter of a century with a uniquely rich and distinctive fare for mathematicians of all abilities to savour. The tweaking, polishing and updating evident in the second edition is surely destined to preserve this pre-eminence in all likelihood for a similar span of time.

James Stirling’s Methodus Differentialis: An annotated translation of Stirling’s text, by Ian Tweddle. Pp. 295. €129.95, £ 75.00, sFr 210.00, $ 129.00. 2003. ISBN 1 85233 723 0 (Springer).

Abstract: classroom as well as Anthony Ferzola's account of Euler's use of differentials as 'absolute zeros', a non-rigorous intuitive approach which was exonerated in retrospect by Abraham Robinson's non-standard model of arithmetic. It is Euler above all who emerges as a vital and inspirational historical figure whose boldness and imagination can still motivate an intuitive spark in today's students. For these essays alone, this book would be worth its price, but there is much more besides. I can highly recommend it for your library.

Sets, functions, and logic: an introduction to abstract mathematics (3rd edn), by Keith Devlin. Pp. 160. £24.99. 2003. ISBN 1 58488 449 5 (Chapman & Hall/CRC).

Abstract: Partial differential equations with numerical methods, by S. Larsson and V. Thomee. Pp. 259. £38.50. 2003. ISBN 3 540 01772 0 (Springer-Verlag). This book, which is aimed at beginning graduate students of applied mathematics and engineering, provides an up to date synthesis of mathematical analysis, and the corresponding numerical analysis, for elliptic, parabolic and hyperbolic partial differential equations. Each of the three types of equation is allotted three chapters. In one, basic properties such as the existence, uniqueness, stability and regularity of solutions are reviewed; the other two focus on the finite difference and finite element numerical methods, including aspects of their practical implementation. Together, these nine chapters form the meat of the book: the rest of the sandwich reprises physical motivation, boundary and initial value problems for ordinary differential equations and background 'eigen-theory', functional analysis and numerical linear algebra. Historically, finite difference methods were the first numerical methods to be developed: they seek an approximating solution at a mesh of points with derivatives replaced by difference quotients, and are still widely used for hyperbolic equations. Finite element methods, which are more suited for general domains and are now the most popular methods for elliptic and parabolic equations, use approximating solutions that are locally piecewise polynomial.

Elements of mathematics. Functions of a real variable elementary theory , by Nicolas Bourbaki. Pp. 338. £77.00 (hbk). 2004. ISBN 3 540 65340 6 (Springer)

Abstract: The last 100 pages of the book include interesting applications of the prime number theorem, such as counting primes in intervals and the enumeration of numbers with a given number of prime factors. There is a proof of Dirichlet's theorem that the primes are (asymptotically) equally distributed in residue classes r mod k for any given k and taken over those r with gcd (r, k) = 1. This section uses the elements of group theory (characters of abelian groups) and extends the methods of proof of the prime number theorem. Because the prime number theorem is an asymptotic formula, error estimates are rather crucial, and the author gives a nice treatment of these, relating them to the Riemann hypothesis, which has recently found its way into the general consciousness as the Hardest Unsolved Problem in mathematics. A number of topics are mentioned here, including the famous result of Littlewood that n (x) > li (x) for some (very large) x. For all values actually calculated, the reverse inequality holds, and the best estimate for such as x is that it is < 10. Finally there is an account of the 'elementary' proof of the prime number theorem by Selberg and Erd6s, that is, a proof that avoids complex analysis but is, naturally, allowed the techniques of real analysis since the very statement of the theorem is about limits and integrals.

89.90 Iterated sequences of triangles

Abstract: three sides of a triangle. We can rephrase this in terms of the three coins. If one coin has a diameter greater than the sum of the diameters of the other two, then you can place their rims tangentially in two ways. Either (a) all the coins are flat on the table (ru r2, r? all positive); or (b) the two smaller coins can lie on the largest, touching each other and the interior rim of the larger coin (r$ < —r\\ r2 < 0). These two arrangements of course give distinct sets of lengths {a, b, c) for ABC.

89.80 A Heron-type formula for the reciprocal area of a triangle

Abstract: Proof: Let a, b, c be the sides forming the bases of the respective altitudes x, y, z. Then A = jax = \\by = \\cz so a = 2A/jt, b = 2A/y and c = 2A/z. For the semi-perimeter s we have s = j(a + b + c) = A.(^ + y' + z~) and s-a = A . ( J T ' +>>\"' +z~), s-b = A.(x~ -y~ + z~), sc = A.(x + y~ z). Substituting these into Heron's formula, dividing both sides by A, and using h = j(x~ + y~ + z~) in this gives (4).

Proofs from THE BOOK (3rd edn), by Martin Aigler and Günter M Ziegler. Pp. 240. £23. 2004. ISBN 3 540 40460 0 (Springer).

Abstract: Numbers: facts, figures and fiction, by Richard Phillips. Pp. 128. £12.00. 2004. ISBN 095465620 2 (Badsey Publications). This entertaining and accessible book is even more attractive in its second edition: its colourful, glossy presentation is packed with full-colour photographs and diagrams that are a feast to the eyes. As a light-hearted and idiosyncratic reference book for all ages it is full of facts, both mathematical and cultural, tantalising problems and anecdotes. The subject material is the non-negative integers, dealt with individually from 0 to 156, then more selectively to 999, and finally 'a few large numbers'.

An introduction to financial option valuation: Mathematics, stochastics and computation, by Desmond J. Higham. Pp. 273. £50.00. 2004. ISBN 0 521 83884 3 (hbk); £24.99. ISBN 0 521 54757 1 (pbk) (Cambridge University Press).

Abstract: are well chosen and are in the main based on or from original scripts. Ancient tables of chords and sines at the beginning of chapter 5 are taken from Ptolemy's Almagest (according to Toomer). Explanations of their use are given. The mathematicians of the Islamic world extended ancient methods of trigonometry to the use of six functions (sine, cosine, tangent, cotangent, secant and cosecant) their methods are explained. Abu 1-Wafa's proof of the addition theorem for sines is explained. An application of trigonometry is given using Al-BIrunf s measurement of the Earth. Interpolation procedures are explained with an excellent extract from a table attributed to Ibn Yunus. The chapter ends with Al-Kashf s approximation to sine and the usual exercises and bibliography.

89.54 Another decreasing sequence of triangles

Abstract: 89.54 Another decreasing sequence of triangles We first need to explain that the term pedal triangle, which often refers to the triangle formed by the feet of the altitudes of a triangle ABC, may also be used for the triangle formed by the feet of the perpendiculars from a point P inside the triangle ABC. A theorem of Neuberg states that every third member of the sequence of nested pedal triangles derived from an interior point P and a host triangle ABC is similar ([1]). The proof for this ternary similarity follows easily by repeated use of the theorem for equal angles in the same segment of a cyclic quadrilateral. It is also easy to show that the pedal triangle of greatest area is that for the circumcentre O (the medial triangle) so that the area ratio of successive triangles in the Neuberg sequence cannot exceed \ and depends on both the choice of host triangle and point P within. In this note we show that the sequence of triangles obtained by using the lengths of the medians of a triangle as the sides of its successor has binary similarity.

Dynamical systems in population biology , by Xiao-Qiang Zhao. Pp. 276. £65.50. 2003. ISBN 0 387 00308 8 (Springer-Verlag).

Abstract: like a textbook in itself. The authors succeed admirably in their objective of writing so that the pure mathematician can safely skip the applications, while the applied mathematician can skim over the mathematical analysis. Having developed much of the basic mathematical theory in Chapter 1, Chapters 2 4 look at various types of single-species models in great depth, concentrating on reaction-diffusion models, but placing these in the context of other approaches that may be more familiar to practitioners in ecology. On moving, in Chapter 5, to interacting species, the prime concept used is that of permanence, a criterion for a model of interacting biological species to predict coexistence. Very simple illustrative models are used to lead the reader gently into this notion, worked out in great detail, but this soon escalates to clear introductions to the sort of dynamical systems modelling that is required to study more involved models. Further chapters develop such ideas and models to the very latest work, and provide many examples of the mathematical methods that may be extended to other fields.

Mathematics handbook for science and engineering (5th edn), by Lennart Råde and Bertil Westergren. Pp. 562. £38.50 (hbk). 2004. ISBN 3 540 211411 (Springer).

Abstract: May I begin with a short reminiscence which I promise is relevant to this review? Many moons ago I changed half way through my degree from physics to applied mathematics. This was mostly due to my growing excitement at the power of such methods as contour integration, vector calculus, integral transforms and Fourier analysis, and at the marvellous connections between them. At this time I had a friend studying English literature who had an intriguing approach to a difficult text. Using every commentary he could find he would make copious notes on the text. Then he would make notes on his notes, ruthlessly purging anything that was clumsy or redundant. Then, as exams approached, he would make ultra concise notes on his notes on his notes. Just before the exam he would carry out a final condensation, so that all of Beowulf (for example) was summarised in half a dozen words. These words were all he remembered for the exam, but they were seeds, which he could expand as required to answer the Beowulf question.

The arithmetic of infinitesimals. John Wallis 1656. Translated from the Latin and edited by Jacqueline Stedall. Pp. 192. £92.50/$119.00/€119.50. 2004. ISBN 0 387 20709 0 (Springer-Verlag).

Abstract: Quite a lot of the text is unavoidably devoted to providing necessary background in probability, statistics, simulation and numerics. This is nicely done, and as it is all good material that students should see anyway makes the book worth reading even if one is not specially interested in finance. But the main focus is on the core financial material, and I feel this is as well covered as could be done in the space available, and starting from scratch.

89.36 On evaluating the probability integral: a simple one-variable proof

Abstract: The first equality requires Fubini's theorem while the second makes use of the change of variable theorem to convert to polar coordinates. (For another more recondite multivariable argument, see [4].) In this note we offer a simple one-variable proof that should be accessible to every first-year calculus student. When the region in the first quadrant bounded by the coordinate axes and the curve z = e is revolved about the z-axis, we obtain a solid of revolution Q whose volume V is readily obtained by the method of cylindrical shells:

Lie groups, Lie algebras, and representations: An elementary introduction, by Brian Hall. Pp. 351. £50. 2003. ISBN 0 387 401229 (Springer-Verlag).

Abstract: theory of generalised Taylor expansions, the Euler-MacLaurin formula and the Gamma function. The topics of the book are mainly of an advanced undergraduate level, but they are presented in the generality needed for more advanced purposes. For example, functions are allowed to take values in topological vector spaces and asymptotic expansions are treated on a filtered set equipped with a comparison scale. This generality however, obscures some of the more everyday topics of mathematics, like differentiation and integration, and I found the treatment of specific subjects such as convex functions or the Gamma function easier reading.

Rethinking proof with The Geometer’s Sketchpad CD-ROM , by Michael D. De Villiers. Pp. 214. £19.95. 2003. ISBN 1 55953 646 2 (Key Curriculum Press).

Abstract: The rest of the book, in looking at a variety of topics within algebra, draws upon the early chapters in making very specific suggestions as to how our understanding of relevant research can inform classroom practice. One simple but striking example of this is representing the expression 5 r as 5 squares of length /, with suitable diagrams, as a means of avoiding confusion with (5f)Other approaches include the use of number patterns, puzzles, graphs and diagrams, including a rearrangement of the squares to show that x 1 = (x l)(x + 1) which has now become part of my PGCE and CPD repertoire. Almost all of these examples avoid the 'all singing, all dancing' approach and leave the reader feeling empowered, able to make realistic, achievable changes to the way algebra is approached to the benefit of the children in the classroom.

89.11 Relations between Euler's constant, Riemann's zeta function and Bernoulli numbers

Abstract: If, on the other hand, t = s + 1 then k = m + d. Numbers of the form \\lm + m + d (remembering that 1 < d < m) have continued fraction representations of the form [m; 1, c2, ... ] so that Pi Pi , , ,Pi m + c2(m + 1) c2 m, — = m + 1 and— = = m + qx q2 q-i 1 + c2 1 + c2 Since d is a factor of m (and thus gcd (m + m + d, m + l ) = l ) w e also have, in this case, that m + 1 + (m + m + d) d + 1 a\\=m,a2 = m+\\ anda3 = = m H . l + ( m + l ) m + 2 d + 1 c2 Therefore, if a3 is to equal p 3 / q3, we require to be equal to sd + 2 1 + c2 We see that, with this condition, either s = 1 or s = 2.

The little book of bigger primes (2nd edn), by Paulo Ribenboim. Pp. 356. £29.50 (pbk)., 2004. ISBN 0 387 20169 6 (Springer).

Abstract: The little book of bigger primes (2nd edn), by Paulo Ribenboim. Pp. 356. £29.50 (pbk)., 2004. ISBN 0 387 20169 6 (Springer). Ribenboim gives in the introduction questions it is natural to ask when studying a set of numbers, plus a final topic. The six chapters of the book follow this admirable scheme: how many are there? how is one recognised? are there functions defining them? how are they distributed? what special kinds are there? experiments and predictions. The first question is easily answered in the first paragraph. The remainder of the first chapter is an impressive collection of ten or more proofs of the infinitude of the number of primes, from Euclid to the twentieth century. The uncertainty arises from deciding which are variants rather than different proofs. Ribenboim remarks 'There are of course more (but not quite infinitely many)'. The bracketed phrase is at the expense of those who refer to the almost infinite. But how to prove that there are not infinitely may proofs? There are certainly infinitely many variants. Goldbach's proof depends on finding an infinite sequence of natural numbers each pair of which are relatively prime. Goldbach used the Fermat numbers, each of which must be divisible by a prime different from any that divides a smaller number in the sequence. Each infinite subset of the Fermat numbers will suffice, and there are considerably more than not quite infinitely many of these.

89.65 A simple method for finding tangents to polynomial graphs

Abstract: 89.65 A simple method for finding tangents to polynomial graphs It is a standard result that, when the polynomial P(x) is divided by (x a), the remainder is /?, (x) = P(a) + P\" (a) (x a). Perhaps it is overlooked that y = Ri (x) is in fact the equation of the tangent to the polynomial at x = a. Therefore we can find the equation of the tangent to a polynomial without the use of calculus. For example, to find the equation of the tangent at x = 2 to y = x 8x + 23x 25x + 1: x 4x + 3 x 4x + 4 | x 8x + 23x 25x + ~ x 4x + 4x 4x + 19x 25x 4x + 16x 16* 3x 9x + 1 3x 12x + 12

Cyclotomic polynomial factors

Abstract: The n th roots of unity play a key role in abstract algebra, providing a rich link between groups, vectors, regular n-gons, and algebraic factorizations. This richness permits extensive study. A historical example of this interest is the 1938 challenge levelled by the Soviet mathematician N. G. Chebotarëv (see [1] or [2]). His question was ‘Are the coefficients of the irreducible factors in Z[n] of xn − 1 always from the set {−1, 0, 1}?’ Massive tables of data were compiled, but attempts to prove the results for all n failed. Three years later, V. Ivanov [3] proved that all polynomials xn 1, where n < 105, had the property that when fully factored over the integers all coefficients were in the set {−1, 0, 1}. However, one of the factors of x105 − 1 contains two coefficients that are −2. Ivanov further proved for which n such factorisations would occur and which term in the factor would have the anomalous coefficients. A twist that makes this historical episode more intriguing is that Bloom credited Bang with making this discovery in 1895, predating the Chebotarëv challenge by more than four decades.

89.29 An extension of the fundamental theorem on rightangled triangles

Abstract: {3, 4, 5} is perhaps the most famous Pythagorean Triple with interest in such triples dating back many thousands of years to the ancient people of Mesopotamia. In this article, we shall consider such triples, with the restriction that the elements of these triples must not have any common factors they are Primitive Pythagorean Triples (PPTs). In particular, we shall consider the question of how many PPTs a given integer can be a member of. The answer to this simple question is, surprisingly, that a given integer n can play the role of a specified side in either 0 or 2k−1 different PPTs, where k is the number of distinct prime factors of n. Our result is a generalisation of what Fermat grandly called the Fundamental Theorem on right-angled triangles ([2], chapter 5), which states that:

89.35 Another look at the calculus power rules

Abstract: At point P on the graph of y = x\" (x > 0), a rectangle is formed by lines perpendicular to the coordinate axes and a right triangle is formed by a line tangent to the curve. The rectangle is divided by the curve into regions A and B, the triangle has base c. A previous article of mine [1] used two diagrams to support the algebra that showed how, on the one hand, AIB = n is obtained from the integral power rule and, on the other hand, x/c = n is obtained from the differential power rule. These were treated separately with no attempt to show a connection between them. This connection is, of course, a heavily-disguised version of the inverse relationship between the calculus power rules. This relationship is shown easily enough with a standard piece of algebra, but this begs an interesting, albeit naive, question: 'Why is an area rule the inverse of a gradient rule?' It occurred to me that combining these diagrams into a single diagram (see Figure 1) might provide a way of addressing that question, but I needed something other than a static diagram. I needed a dynamic picture, which I obtained by using the Java applet at [2]. The applet seems to suggest that, as P moves along the curve, the tangent line 'projects' the ratio AlB onto the x-axis. In other words, just as region B is always 1 / n th of region A, so the base of the triangle is always 1 / n th of the base of the rectangle. Since these ratios are simply 'boileddown' versions of the calculus power rules, this could be a way of showing graphically the reason why these rules are connected. I wonder, if it is possible to devise a mathematical formulation, based on this 'dynamic' geometry, that answers my naive question simply and convincingly?