Showing papers in "The Mathematical Gazette in 2003"

Statistical inference (2nd edn), by Paul H. Garthwaite, Ian T. Jolliffe and Byron Jones. Pp.328. £40 (hbk). 2002. ISBN 0 19 857226 3 (Oxford University Press).

Abstract: However, despite its claims, this book is an excellent introduction to mathematical modelling, with a wealth of interesting case studies and lovely mathematics that will appeal to many of its target audience final year undergraduates, Masters and PhD students working in 'practical' mathematical modelling. The source of most of the case studies comes from the Study Groups with Industry and Mathematics that originated in 1968 at Oxford, organized by Alan Taylor and Leslie Fox, but is now a global activity. The book describes the type of problem, how they are approached and funded (typically a surprisingly cheap £10 000 to £15 000). The first chapter covers the mathematical preliminaries for the case studies which are all based on continuous media, and so all that is needed is diffusion and flow and deformation of solids and fluids. The next four chapters are based on elliptic partial differential equations and relate to simulation of a type of compressor, determination of the viscosity of a carbon paste used in smelting, simulation of an instrument for determining density, and modelling a form of nondestructive testing of pressure vessels. There follow six chapters devoted to a range of applications of the diffusion equation and various non-linear effects. These cover the cooking of a single cereal grain, waves of epidemics in animal populations, catalytic converters, problems with blast furnaces, the cooling of hot glass, and crystal growth. Chapter 11 looks at the load responses of filled viscoelastic materials. The last two chapters, based on hyperbolic equations, model the noise produced by a bird scare gun and the paper tension variations in a printing press.

A course in combinatorics (2nd edn), by J. H. van Lint and R. M. Wilson. Pp. 602. £24.95. 2001. ISBN 0 521 00601 5 (Cambridge University Press).

Abstract: axioms for geometry. Plane projective geometry is developed axiomatically and there is a competent discussion both of finite projective planes and of various models of projective geometry (although Figure 8.15, the punctured sphere model, is incorrect). A chapter on non-Euclidean geometry examines variants of the fifth postulate; hyperbolic geometry is described in some depth, whilst elliptic geometry gets rather short shrift (and there is no mention of spherical geometry at all). After a short digression into equivalence relations and some linear algebra, affine and projective coordinate systems are introduced and the classic results of projective geometry are presented alongside general properties of conies and algebraic curves. Another digression into elementary Galois Theory is followed by a demonstration of the impossibility of the 'classical problems'. The final two chapters are very short: the one on fractals does little more than discuss Hausdorff dimension and that on catastrophe theory outlines the mathematics behind the cusp catastrophe.

87.65 Can growth be faster than exponential, and just how slow is the logarithm?

Abstract: underestimating although the revised 'smooth' model 6 did show an improvement in every case. Model 5 using one of Dixon's constructions was very variable in its predictions but gave the best result in three cases. The final column in the table comparing nab with the measured volume showed the ratio was very close to one sixth. As a final exercise we photographed an actual egg and used the preceding work to find the algebraic equations of the curves which best approximated to its outline. The superimposition of the mathematical curve onto the real egg in Figure 8 shows the 'goodness of fit' of model 4. Hens are clearly trying their best to produce perfect ellipsoids.

CRC Concise encyclopedia of mathematics (2nd edn), by Eric W. Weisstein. Pp. 3242. £66.99 (hbk). 2002. ISBN 1 58488 347 2 (Chapman and Hall). Also available as a CD-ROM (ISBN 084919463) at the same price.

Abstract: CRC Concise encyclopedia of mathematics (2nd edn), by Eric W. Weisstein. Pp. 3242. £66.99 (hbk). 2002. ISBN 1 58488 347 2 (Chapman and Hall). Also available as a CD-ROM (ISBN 084919463) at the same price. When it was published in 1998, the first edition of this encyclopaedia received overwhelming praise for its massive scope as well as its accessibility and ease of use. For this new edition, only four years later, the author has updated most of the original entries and added 1000 extra pages of new entries. The result is a substantial tome weighing in at several kilos and printed on wafer-thin paper, but sturdily bound so that it lies open on the desk at any page without any apparent strain. The book is also available as a CD-ROM, and much of the text seems to be available free on the internet on the site 'Eric Weisstein's World of Mathematics'. Despite these options, I am quite happy with the hard copy, which affords endless opportunities for random browsing as well an immediate reference point without the need to turn on the PC and wait for the operating system to behave itself.

Diagrams for difficult problems in probability

Abstract: We have developed a novel diagrammatic approach for understanding and teaching probability theory — Probability Space diagrams [1]. Our studies of learning and instruction with Probability Space (PS) diagrams have demonstrated that they can significantly enhance students' conceptual understanding. This article illustrates the utility of PS diagrams by applying them to the explanation of some difficult concepts and notoriously counterintuitive problems in probability. We first outline the nature of the system.

87.18 A curious identity revisited

Abstract: 2 ^ V 5 X 3 A 6 X 6 A 9 X 9 A l 2 x 12^ ••• ill (1*1 x 2l(5_xJZ x 5 .1(9*11 x 7 W l 3 x l 5 x 2~| n V2x4 x 2 A 6 x 8 x 4 A l 0 x l 2 x 6 A l 4 x l 6 x 8 ) • • • £H (2x7 x S U l l x l ? x 9 V l 9 x 2 3 x 131(27x31 x XZ") ?r ~ \4^t x 4 / U 2 x l 6 x 8A2oS24 x 12A28x32 x 1 6 A " j i / L (2>Oi2 x 8 l ( ' 3 x 1 6 x U V 19x22 x M l 4^4jr " x 6 3/^9x12 * 6 / U 5 x l 8 x 9 A21x24 x 111--ssm(r?r/s) (s-r 2s-r i + r U l i r 4s r 2s + A ( 5 s r 6s-r 3.s+r\ 2 r / s r a <• s 2s s A 3s 4s 2s A 5s 6s 3s / •• • V5 (1x3x5 x 3 V 7x9x11 „ 5 \ ( 13x 15x 17 „ 71(19x21 x23 . . 91 yfyt ~ V2x4x6 x 2 ^ 8 x 1 0 x 1 2 x 4^14x16x18 x 6^20x22x24 x » ) • • • 2 (1x3x5 ., 3x5W 7x9x11 „ 7x9W13xl5x l7 ~ l l x l 3 \ ^ n 12x4x6 x 2 x 4 A 8 x l 0 x l 2 A 6 x 8 A l 4 x l 6 x l 8 * 10x12,*

87.76 Two more proofs of Lester’s theorem

Abstract: 87.76 Two more proofs of Lester's theorem In this note we use areal coordinates ([1]) to prove the recent theorem for a scalene triangle that the circumcentre O, the nine-point centre N, and the two Fermat points P and Q are concyclic ([2]). The first proof makes use of the point of intersection M of the Euler line OM and the Fermat line PQ, and invokes the power lemma for concyclic points; the second proof arises from the surdic conjugacy of the areal coordinates for P and Q. We define the actual areal coordinates of a general point W with respect to the triangle of reference ABC by the triple [x, y, z], where AWBC _ AWCA _ AWAB AABC' y ~ AABC' Z ~ AABC' so that x + y + z = 1. Relative areal coordinates (Ax, ky, Az) (A # 0) are sometimes preferred, with square/round brackets for actual/relative areal coordinates, respectively. The alternative expression given in [1] for the distance WW' between the points W and W with actual areal coordinates [x, y, z] and [x', y', z'] is

87.10 On the ratio of the inradius to the circumradius of a triangle

Abstract: 87.10 On the ratio of the inradius to the circumradius of a triangle The formula r = cos A + cosB + cosC 1 (1) R for the ratio of the inradius r to the circumradius R of a triangle ABC deserves to be better known. It was first derived by Carnot [1, p. 169], as a consequence of the theorem in [2] which bears his name: 'The sum of the distances of the circumcentre of a triangle from the three sides is equal to the circumradius increased by the radius,' in other words if hA = R cosA, etc, then hA + hB + hc = R + r. (2) Equation (1) is related also to a result obtained by Hajja [3], but in his proof the three angles A, B and C are not treated symmetrically. It seems worthwhile, therefore, to supply a proof of equation (1) in which A, B and C are treated symmetrically, and which is shorter and simpler than the usual proof of equation (2) given for example in [2], and also by Carnot himself [1, p. 168]. The following proof uses essentially areas and projections. Let P be any point in the plane, and let £, ij and £ denote the directed perpendicular distances from P to the three sides a, b and c of the triangle ABC. (|, T), t, are thus trilinear coordinates of P.) Then, by summing the areas of the three triangles BPC, CPA and APB we have

The wonder of Horner’s method

Abstract: Apart from false position and double false position, another numerical method for calculating roots of equations was known to the Ancient Chinese. Chia Hsien in the eleventh century is reputed to have given an algorithm for calculating roots as well as describing Pascal’s triangle. The algorithm was mentioned again by the twelfth century scholar Liu I. The Chinese used the method to solve quadratics and cubics as early as 100 BC, but it was not until 1247 that Ch’in Kiu-Shao from South China published its extension to higher order polynomials in his work, Mathematics in nine chapters. A year later in the book, Sea-mirror of circle measurements, Li Yeh, who was from north China, took root-finding for granted. The fact that these quite independent writers published similar work suggests that finding the zeros of polynomials was well known by the middle of the thirteenth century. It was left to one of China’s greatest mathematicians, Chu Shi-kie’ (ca. 1280–1303), to give this algorithm its name fan fa, which means the method of the Celestial Element or sometimes the Celestial Unknown [1,2]. Translations and spellings of these older Chinese words do not always give the same result. This algorithm eventually became known as the Ruffini-Horner Method or more simply Horner’s method.

87.40 An elementary single-variable proof of

Abstract: and would then talk them through the way this leads to the power rules (the reverse process to that shown here). This was not, in any sense, a substitute for a formal proof, but rather a pre-calculus taster. With the advent of the Internet and Java it has now become possible to supplement the original worksheet (e-mail the author for a free copy) with dynamic interactive investigations, which can be seen at www.powermaths.org.uk . SIDNEY SCHUMAN 25 Wellmeadow Road, London SE13 6SY e-mail: sidney@jusiruma.co.uk

87.09 Fermat row-eliminated matrices and some binomial determinants *

Abstract: Given Y, we shall construct an X in P(n) so that q> (X) = Y. First, we must obviously have x^ = yy for i, j e {l, ... , n 1}, and this alone ensures that

87.69 On a precursor to Stirling’s formula

Abstract: is well known to most students of calculus, and can be obtained in a variety of ways. In [1], it is used to show that .. VST l hm — = , n-»~ n e which is sometimes referred to as a 'baby Stirling's formula'. Along similar lines, below we use a refinement of (1) to show very simply that en\\ lim —•= = L n -»~ n \\Jn exists (and satisfies 2 < L < e). From here, Stirling's formula lim ^ = \\l7jt is just a short (and familiar) step away, via Wallis' product [37

A polynomial that is a statistical prism

Abstract: The roots of a polynomial can be represented as points in the complex plane The time value of money (TVM) equation commonly used in finance is a polynomial Concepts from financial mathematics can be obtained from the pattern of the roots of the TVM equation The concepts are given in terms of distances between the roots and other salient points in the plane This note shows that this particular polynomial, and the technique, can be applied more generally When a series of data is fed into the coefficients of the polynomial, the mean and standard deviation of the data are seen in the complex plane as combinations of distances between the roots and other salient points The results are aesthetically pleasing as well as mathematically interesting

87.80 On a metric-related function for the triangle

Abstract: satisfying the tangent property implies the concurrence of the angle bisectors, say at the point O. So consider a hinged quadrilateral ABCD with points {X, Y, Z, W) satisfying the tangent condition. Then the angle bisectors meet at O. We can then drop perpendiculars from O to meet all of the sides internally (since each base angle in triangles of the form OAB are acute). Call these points {X\\ Y', Z', W'}; they clearly satisfy the tangent condition and are points of contact of the inscribed circle centred at O. What can we now say about {X, Y, Z, W}l Draw a circle, E, centred at O passing through X. Then a reflection on l\\ will carry Z onto itself and X into Y. Thus Y must also lie on £. Similarly Z and W are on Z. So each set of four points with the tangent property is concyclic, and all such quadruples lie on a circle centred at O, the meeting place of the angle bisectors. So we can complete the previous diagram by drawing the missing circles:

87.21 Curious numbers revisited

Abstract: lnrB + 1 = 2\"[lnr! / ( « ) ] where 1 1 1 1 f(n) = — lnl + — ln2 + ln3 + ... + — Inn 2 2 2 2\" from which the conclusions can be drawn: (i) If In T\\ = f (n), i.e. T\\ = exp [f (n)] then the term Tn+i = 1 and the convergence will be rapid thereafter, (ii) IfTi < exp [/(n)] the term T„ +1 < 1 and rapid convergence follows. For n large,/(n) = 0.507833... and e x p [ / » ] = 1.661687... so that convergence will result if 7̂ < 1.6616 87....

87.04 When is n a member of a Pythagorean triple

Abstract: 87.04 When is n a member of a Pythagorean triple? Pythagoras' theorem is perhaps the best-known result in the whole of mathematics, yet many things remain unknown (or perhaps just unstudied) about the consequences of this 'simple' theorem. In this article we investigate which numbers can be part of triples such as {3, 4, 5} and {5, 12, 13}—right-angled triangles with integer sides. Perhaps the most interesting results concern instances where a number is a member of a triple in which none of the members have a common factor. In this case any odd number > 1 and any multiple > 4 of 4 is the shortest side of some such triangle while an odd number is the middle side if, and only if, it can be written as xy (x and v co-prime) where v < x < (l + \\ll)y. Similarly, an even number is the middle side if, and only if, it can be written as 2xy where x and y satisfy the inequality above, are co-prime and are of opposite parity. On the other hand, a number can be the hypotenuse if, and only if, all of its prime factors are of the form 4k + 1.

87.23 Triangles with a 60° angle and sides of integer length

Abstract: References 1. D. F. Lawden, On the solution of linear difference equations, Math. Gaz. XXXVI (1952) pp. 193-196. 2. L. Moser, King paths on a chessboard, Math. Gaz. XXXIX (1955) p. 54. 3. R. Grassl, T. Mingus, Equivalence classes and a familiar combinatorial identity, Math. Gaz. 82 (1998) pp. 98-100. 4. S. Simons, A curious identity, Math. Gaz. 85 (2001) pp. 296-298. MICHEL BATAILLE 12 Rue Sainte-Catherine, 76000 Rouen, France e-mail: michelbataille @ wanadoo.fr

87.39 Tinkering with the calculus power rules

Abstract: 87.39 Tinkering with the calculus power rules The graph of the power function y = kx\", x > 0, n > 0 yields an intriguing piece of geometry in relation to the calculus power rules. Given the simplest form of the differential and integral power rules (omitting any coefficients and constants), it can be shown that the power V is equal to AIB, where A and B are significant dimensions (Figure 1) or areas (Figure 2).

Multi-objective optimization using evolutionary algorithms, by Kalyanmoy Deb, Pp.487. £60. 2001. ISBN 0 471 87339 X (Wiley).

Abstract: Multi-objective optimization using evolutionary algorithms, by Kalyanmoy Deb, Pp.487. £60. 2001. ISBN 0 471 87339 X (Wiley). Classical optimization is about finding maxima and minima of a function of n variables, the objective function. Often there are constraints, equations or inequalities in the variables. There is a large body of good theory and algorithms for many types of problem of practical interest. Multi-objective optimization is a much harder subject: it is about optimizing several objectives at the same time. Of course, this is usually impossible. For example, when a consumer wants the best quality at the lowest price, there is no optimal solution: the two objectives conflict. That conflict is the essence of multi-objective optimization. Deb's book begins with an introduction to the theory of multi-objective optimization problems (MOPs), but most of the text deals with research on the solution of MOPs using evolutionary computation (aka genetic algorithms). There is a final chapter on applications in engineering (such as designing a structural component to maximize strength and minimize cost and weight).

A unique property of right-angled triangles

Abstract: The familiar fact that we may cut any triangle into three polygonal pieces which can be rearranged to form a rectangle is one of the most elementary problems of polygonal dissection. This fact is shown in two ways in Figure 1 where both sides of the rectangle are predetermined after setting up the triangle: one side of the triangle and half of the altitude to this side, or one altitude and half of the side at right angles to it.

87.02 Sums of consecutive positive integers

Abstract: Notes 87.01 Sums of squares with digit restrictions In this note we consider the number of ways of representing an integer as a sum of squares of integers but with an added restriction: the numbers to be squared may only use certain digits in their base 10 representations. For example, if we restrict our choice of digits to those in {0, 1, 2}, we are restricted to the numbers in {0, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101,...}. In number theory we often pose questions about the number of representations of a number as a sum of squares. For example, we might ask what is the smallest number that can be expressed as a sum of three squares in 2 different ways. If we allow repetitions, and allow the square of zero to be counted, the answer is 9, since 9 = 0 + 0 + 3 = l + 2 + 2. We can ask the same sorts of questions of our digit-restricted numbers. Clearly, this opens up a new field of questions—here is just one of them: what is the smallest number that can be expressed as the sum of three squares of digit-restricted numbers in two different ways, when the digits are restricted to digit sets of size three? For example, if the digits are restricted to the set {0, 1, 2} the smallest number with the required property is 545 = l + 12 + 20 = 2 + 10 + 21. If the digits are instead restricted to the set {1, 2, 4} then the smallest number is 2401 = 12 + 24 + 41 = 14 + 42 + 21(= 49). These solutions may be tested by a computer search, but is there a general analytic method for problems such as these?

Some properties of repetends

Abstract: We wish to discuss some aspects of repetends, the repeating sequence of digits in the expansion of a fraction (for illuminating introductions to the subject see [1, 2]). For the most part we restrict consideration here to fractions with a prime denominator. But we do consider the general condition for the length of repetends and examine some special cases when the base of the number system is varied. An illustration of the use of other bases than 10 is given. Then we consider the multiplication of repetends and show a connection with group theory, giving an old result by a new twist.

87.71 Evaluation of a class of improper integrals of the first kind

Abstract: r^*-? a) o x 2 is related to the names of Laplace and Dirichlet. In fact, it was first obtained using a residue method by Euler in 1781 (see any standard textbook on complex functions for undergraduates, for example [1, pp. 226-227] and [2, pp. 168-170]). An elegant calculation of formula (1), depending on the partial fraction decomposition i = } + X / l i ( l ) ' ( i ^ + T+7s) a n ^ due to the noted geometrician N. I. Lobatschevski, is provided in [3, pp. 436-437] and [4, pp. 382-384]. Another polished proof of identity (1) is given by G. Klambauer in [4, pp. 381-382] based on the result in Problem 13 in [4, pp. 331-332]: if/ is Riemann integrable on the interval [a, b], then lim iaf(x) sin (px)dx = 0.

The story of Blanche Descartes

Abstract: Recently a poem, written by Blanche Descartes in the 1930s, was republished in the Gazette, with the additional attribution ‘communicated via Cedric A. B. Smith’. Cedric had reported ‘it was written after Blanche heard that Hector Petard was about to be married. She composed the verse as a wedding present. Hector is still active mathematically, though I suspect that he may be in his mid nineties’. We felt that readers might be interested to know something about the mysterious Blanche Descartes, and the origins of the poem. It makes quite a story. At the risk of boring you, we will start at the beginning.

87.67 Sums involving square-free integers

Abstract: References 1. J. Stewart, Calculus: concepts and contexts, Brooks/Cole Publishing Company, USA (1998). 2. D. MacHale, My favourite polynomial, Math. Gaz. 75 (June 1991) pp. 157-165. 3. R. Vein and P. Dale, Determinants and their applications in mathematical physics, Springer-Verlag (1999). H. AZAD and A. LARADJI Department of Mathematical Sciences, King Fahd University, Dhahran 31261, Saudi Arabia