Showing papers in "The Mathematical Gazette in 2016"

The incomplete gamma functions

Abstract: Recall the integral definition of the gamma function: for a > 0. By splitting this integral at a point x ⩾ 0, we obtain the two incomplete gamma functions:(1)(2)Γ(a, x)is sometimes called the complementary incomplete gamma function. These functions were first investigated by Prym in 1877, and Γ(a, x) has also been called Prym's function. Not many books give these functions much space. Massive compilations of results about them can be seen stated without proof in [1, chapter 9] and [2, chapter 8]. Here we offer a small selection of these results, with proofs and some discussion of context. We hope to convince some readers that the functions are interesting enough to merit attention in their own right.

Defining trigonometric functions via complex sequences

Abstract: In the literature we find several different ways of introducing elementary functions. For the exponential function, we mention the following ways of characterising the exponential function:(a) (b) , also for complex values of x;(c) x → exp (x) is the unique solution to the initial value problem [4](d) x → exp (x) is the inverse of (e)x → exp (x) is the unique continuous function satisfying thefunctional equation f (x + y) = f (x) f (y) and f(0) = 1 [6]; the corresponding definition is done for the logarithm in [7];(f) Define dr for rational r, and then use a continuity/density argument [8].All of them have their advantages and disadvantages. We like (a) and (c), mostly because they have natural interpretations, (a) in the setting of compound interest and (c) being a simple model of many processes in physics and other sciences, but also because they are related to methods and ideas that are (usually) introduced rather early to the students.

Analysis and geometry of Markov diffusion operators by Dominique Bakry, Ivan Gentil & Michel Ledoux, pp. 553, £99.00 (hard), ISBN 978-3-319-00226-2, Springer Verlag (2014).

Abstract: The text offers a step-by-step learning with motivation, help and guidance; lucidity takes priority over elegant prose. The format is unusual in that each chapter is divided into two parts: Exposition and Expansion. In the exposition part, definitions, examples, simple lemmas and exercises to illustrate the new concepts are given, and there are comments and occasional warnings highlighting the most common source of beginners' mistakes in elementary topology. The propositions and theorems are then gently brought in, and the reader is encouraged to attempt the proofs by following the guides. Solutions to the more taxing exercises and proofs of the propositions and theorems are then given in the expansion part at the end of each chapter. The exposition is leisurely and yet, in only 144 pages, the excellent book covers most of the topics that a well-read graduate student is expected to know in basic general topology. 10.1017/mag.2016.88 PETER SHIU 353 Fulwood Road, Sheffield S10 3BQ e-mail: p.shiu@yahoo.co.uk

What is Bayesian statistics

Abstract: Bayesian statistics is included in few elementary statistics courses, and many mathematicians have heard of it, perhaps through collateral readings from popular literature or [1], selected as an Editor's Choice in the New York Times Book Review. ‘Bayesian statistics’ provides for a way to incorporate prior beliefs, experience, or information into the analysis of data. Bayesian thinking is natural, and that is an advantage. For example, on a summer morning, if we see dark rain clouds up in the sky, we leave home for work with an umbrella because prior experience tells us that doing so is beneficial. In general, the idea is simple; schematically, it looks like this:(prior belief) + (data: new information) ⇒ (posterior belief).Thus, we begin with a prior belief that we allow to be modified or informed by new data to produce a posterior belief, which then becomes our new prior, and this process is never-ending. We are always willing to update our beliefs according to new information.

On the classification of convex quadrilaterals

Abstract: We live in a golden age regarding the opportunities to explore Euclidean geometry. The access to dynamic geometry computer programs for everyone has made it very easy to study complex configurations in a way that has never been possible before. This is especially apparent in the study of triangle geometry, where the flow of new problems, properties, and papers is probably the highest it has ever been in the history of mathematics. Even though it has increased a bit in recent years, the interest in quadrilateral geometry is significantly lower. Why are triangles so much more popular than quadrilaterals? In fact, we think it would be more logical if the situation were reversed, since there are so many classes of quadrilaterals to explore. This is the primary reason we think that quadrilaterals are a lot more interesting to study than triangles.

The dice and numbers game

Abstract: In maths lessons in secondary school, on certain occasions my teacher entertained our class with the following game. Each player starts with an empty 2 × 5 grid, as shown in Figure 1(a). In each round of the game, a die is rolled, and each player has to choose one of the empty boxes in their grid and fill it with the number rolled. For example, after 3 rounds, a player's grid may look like the one in Figure 1(b). After 10 rounds everyone's grid is filled, forming two 5-digit numbers. A player's score is the sum of her/his two 5-digit numbers. Players with the largest sum win the game. We shall call this the ‘dice-and-numbers game’. It is easy to think of other variants. Any child who is comfortable with column addition and understands how place value affects the size of numbers will develop a reasonable strategy for the game.

Multisection of series

Abstract: In a recent paper [1], the authors gave a combinatorial interpretation to sums of equally spaced binomial coefficients. Others have been interested in finding such sums, known as multisection of series. For example, Gould [2] derived interesting formulas but much of his work involved complicated manipulations of series. When the combinatorial approach can be implemented, it is neat and efficient. In this paper, we present another approach for finding equally spaced sums. We consider both infinite sums and partial finite sums based on generating functions and extracting coefficients. While generating functions were first introduced by Abraham de Moivre at the end of seventeen century, its systematic use in combinatorial analysis was inspired by Leonhard Euler. Generating functions got a new birth in the twentieth century as a part of symbolic methods. As a central mathematical tool in discrete mathematics, generating functions are an essential part of the curriculum in the analysis of algorithms [3, 4]. They provide a bridge between discrete and continuous mathematics, as illustrated by the fact that the generating functions presented here appear as solutions to corresponding differential equations.

Remembering spherical trigonometry

Abstract: Although high school textbooks from early in the 20th century show that spherical trigonometry was still widely taught then, today very few mathematicians have any familiarity with the subject. The first thing to understand is that all six parts of a spherical triangle are really angles — see Figure 1.This shows a spherical triangle ABC on a sphere centred at O. The typical side is a = BC is a great circle arc from to that lies in the plane OBC; its length is the angle subtended at O. Similarly, the typical angle between the two sides AB and AC is the angle between the planes OAB and OAC.

A simple proof of Euler's continued fraction of e^{1/M}

Abstract: A continued fraction is an expression of the formand we will denote it by the notation [f0, (g0, f1), (g1, f2), (g2, f3), … ]. If the numerators gi are all equal to 1 then we will use a shorter notation [f0, f1, f2, f3, … ]. A simple continued fraction is a continued fraction with all the gi coefficients equal to 1 and with all the fi coefficients positive integers except perhaps f0.The finite continued fraction [f0, (g0, f1), (g1, f2),…, (gk–1, fk)] is called the k th convergent of the infinite continued fraction [f0, (g0, f1), (g1, f2),…]. We defineif this limit exists and in this case we say that the infinite continued fraction converges.

Properties of Pythagorean quadrilaterals

Abstract: There are many named quadrilaterals. In our hierarchical classification in [1, Figure 10] we included 18, and at least 10 more have been named, but the properties of the latter have only scarcely (or not at all) been studied. However, only a few of all these quadrilaterals are defined in terms of properties of the sides alone. Two well-known classes are the rhombi and the kites, defined to be quadrilaterals with four equal sides or two pairs of adjacent equal sides respectively. The orthodiagonal quadrilaterals are defined to have perpendicular diagonals, but an equivalent defining condition is quadrilaterals where the consecutive sides a, b, c, d satisfy a 2 + c 2 = b 2 + d 2 . Then it is possible to prove that the diagonals are perpendicular and that no other quadrilaterals have perpendicular diagonals (see [2, pp. 13-14]). In the same way tangential quadrilaterals can be defined to be convex quadrilaterals where a + c = b + d . Starting from this equation, it is possible to prove that these and only these quadrilaterals have an incircle (since this equation is a characterisation of tangential quadrilaterals, see [3, pp. 65-67]).

Euler, the clothoid and

Abstract: One of the many definite integrals that Euler was the first to evaluate was(1)He did this, almost as an afterthought, at the end of his short, seven-page paper catalogued as E675 in [1] and with the matter-of-fact title, On the values of integrals from x = 0 to x = ∞. It is a beautiful Euler miniature which neatly illustrates the unexpected twists and turns in the history of mathematics. For Euler's derivation of (1) emerges as the by-product of a solution to a problem in differential geometry concerning the clothoid curve which he had first encountered nearly forty years earlier in his paper E65, [1]. As highlighted in the recent Gazette article [2], E675 is notable for Euler's use of a complex number substitution to evaluate a real-variable integral. He used this technique in about a dozen of the papers written in the last decade of his life. The rationale for this manoeuvre caused much debate among later mathematicians such as Laplace and Poisson and the technique was only put on a secure footing by the work of Cauchy from 1814 onwards on the foundations of complex function theory, [3, Chapter 1]. Euler's justification was essentially pragmatic (in agreement with numerical evidence) and by what Dunham in [4, p. 68] characterises as his informal credo, ‘Follow the formulas, and they will lead to the truth.’ Smithies, [3, p. 187], contextualises Euler's approach by noting that, at that time, ‘a function was usually thought of as being defined by an analytic expression; by the principle of the generality of analysis, which was widely and often tacitly accepted, such an expression was expected to be valid for all values, real or complex, of the independent variable’. In this article, we examine E675 closely. We have tweaked notation and condensed the working in places to reflect modern usage. At the end, we outline what is, with hindsight, needed to make Euler's arguments watertight: it is worth noting that all of his conclusions survive intact and that the intermediate functions of one and two variables that he introduces in E675 remain the key ingredients for much subsequent work on these integrals.

The most elementary proof that

Abstract: Here we present a simplification of one of the standard proofs that . We then look at extensions of the new approach, and add comments on the nature of the simplification (which relates to Step 1 below) and finally on the literature. As in most proofs of the result, we shall actually prove that ; because this is equivalent to the result sought.

100.33 A surprising relationship between Pell-Lucas, Pell, square and centred square numbers

Abstract: for and . The Pell numbers are defined recursively by , and for all . The Pell-Lucas numbers are defined recursively by , and for all . The purpose of this Note is to show an unexpected relationship between Pell-Lucas numbers, Pell numbers and − the square numbers that are centred square numbers. n ≥ 0 k ≥ 3 P0 = 0 P1 = 1 Pn + 2 = 2Pn + 1 + Pn n ≥ 0 Q0 = 2 Q1 = 2 Qn + 2 = 2Qn + 1 + Qn n ≥ 0 u (n; 4)

100.31 Heron-like formulas for quadrilaterals

Abstract: References 1. R. Honsberger, Episodes in nineteenth and twentieth century Euclidean geometry, Mathematical Association of America (1995) p. 19. 2. F. Smarandache, Proposed problems of mathematics Vol. II, Kishinev University Press, Kishinev (1997), Problem 62, pp. 42-43. 3. W. Koepf and M. Brede, Relations between some characteristic lengths in a triangle, The International Journal for Technology in Mathematics Education 12 (4) (2005) pp. 149-154. 4. T. A. Sarasvati Amma, Geometry in ancient and medieval India, Motilal Banarsidass, Delhi, India (1979) p. 127.

Constructing the 15th pentagon that tiles the plane

Abstract: z = 2 (F6nF6n+ 1 + F6n+ 1F6n+ 2 − 1) = 2 (F2 6n+ 2 − F 6n − 1). 4. The case where form an isosceles triangle with corresponds to or . Standard theory of the Pell equation shows that all solutions of are given by the recurrence relation , , with , always odd numbers. So all solutions to are generated by , , . The first few are shown in Table 2: note how the successive convergents of the continued fraction expansion of occur in the entries for . x, y, z x = y 2T (x) = T (z) 2 (2x + 1) − (2z + 1) = 1 a2 − 2b = −1 a1 = b1 = 1 an + 1 = 3an + 4bn bn + 1 = 2an + 3bn an bn 2T (x) = T (z) x1 = z1 = 0 xn + 1 = 3xn + 2zn + 2 zn + 1 = 4xn + 3zn + 3 2 a, b, c, d

100.28 Mixing angle trisection with Pythagorean triples

Abstract: The problem is of course reminiscent of Pythagorean triples, triples of integers that satisfy the Pythagorean theorem , and hence are lengths of the sides of a right triangle. Euclid in Elements X.29 gives a solution without a hint of how it was obtained (trial and error by Pythagoreans seems likely), or why it gives all the triples. A constructive method first appears in Diophantus's Arithmetica (c. 250 AD) when solving the famous problem II.8, ‘partition a given square into two squares’, the one whose margins were too narrow in Fermat's copy of the book, see [1] for a nice commentary. a, b, c c2 = a2 + b

100.29 Some odd series for π

Abstract: probably representing the quadrature of the circle. In the square on the left Leibniz quotes the Roman poet Virgil: God loves odd numbers, and he illustrates this in the circle on the right. Note that the series (1) converges very slowly, something that Leibniz also knew. It appears that Leibniz's series is but one of an infinite number of similar series for , using only odd numbers, the next two being π / 4 π 4 = ∑ ∞

UK pension changes in 2015: some mathematical considerations

Abstract: Since April 2015 there has been no legal requirement in the UK to purchase an annuity with pension savings [1] while for those who reach state pension age on or after 6th April 2016 the UK Government changed the state pension deferral arrangements [2]. The latter refers to an arrangement whereby a pensioner can receive an enhanced state pension by deferring its uptake for an arbitrary number of years. These two changes raise certain questions for prospective pensioners which are worthy of some mathematical consideration.An annuity is a guaranteed income for life in exchange for a certain sum of money: the pension pot. An alternative to the annuity since April 2015 is a ‘draw down scheme’ in which the pension pot can be used almost like an ordinary bank account and money periodically withdrawn. These two choices arise from ‘defined contribution’ pension arrangements. By contrast ‘defined benefit’ work-based (company) pensions allow no such choice and are not considered further here apart from the special case of the UK state pension. With an annuity a further option to consider, and one which predates the 2015 changes, is whether to take payments that are fixed or index-linked to inflation. Only the UK state pension offers a late retirement enhanced pension if its uptake is deferred.

100.32 Linear quadrisection and the Kiepert hyperbola

Abstract: In this Note we will locate two significant points lying on different branches of the hyperbola and also encounter three examples of linear quadrisection. However, we opt just to outline some of the routine coordinate geometry. In our Cartesian framework, the vertices , , , centroid and orthocentre lie on the curve . Thus , with for rectangularity. The origin is the Kiepert centre and and two of the parameters may be taken as positive. A (a, a) B (b, b) C (c, c) G (g, g) H (h, h) xy = 1 3g = a + b + c g = a + b + c abch = −1 O (0, 0) g, h a, b, c We will need the following result. Lemma: We have (b + c) (c + a) (a + b) = (3g − a) (3g − b) (3g − c) = 27g − 9(a + b + c)g + 3(bc + ca + ab)g − abc

Picking genuine zeros of cubics in the Tschirnhaus method

Abstract: In 1683, the German mathematician Ehrenfried Walther von Tschirnhaus introduced a polynomial transformation which, he claimed, would eliminate all intermediate terms in a polynomial equation of any degree, thereby reducing it to a binomial form from which roots can easily be extracted [1]. As mathematicians at that time were struggling to solve quintic equations in radicals with no sign of any success, the Tschirnhaus transformation gave them some hope, and in 1786, Bring was able to reduce the general quintic to the form x 5 + ax + b = 0, even though he didn't succeed in his primary mission of solving it. It seems Bring's work got lost in the archives of University of Lund. Unaware of Bring's work, Jerrard (1859) also arrived at the same form of the quintic using a quartic Tschirnhaus transformation [2]. From the works of Abel (1826) and Galois (1832) it is now clear that the general polynomial equation of degree higher than four cannot be solved in radicals.

100.09 Extremal distance ratios

Abstract: Thus and are on the same Apollonius circle by the angle bisector theorem. By angle chasing, , hence is the centre of the above Apollonius circle . As is tangent to , maximises under the constraint for the following reason: being a continuous function of with value at inside , an Apollonius circle given by a ratio greater than on lies around inside and has thus no intersection with . Q T XA / XB = QA / QB TQ′ = QQ′ Q′ Q XA / XB X ∈ XA / XB X ∞ X = B XA / XB B

Mathematics and art, a cultural history by Lynn Gamwell, $49.50, pp 556, ISBN 978-0-691-16528-8, Princeton University Press (2015).

Abstract: a very tiresome process. Suppose, for example, that you wish to insert the equation into a document. Call up the Insert Equation facility which produces a menu and a small window in which to work. You then select a format for the squared expression, which is two boxes . Next you enter a 2 into the index box and a bracket into the other box. Now you type into the bracket remembering that to create you have to go through the ‘two boxes’ process. Having produced the left-hand side of the equation, you do something similar for the rest. This is a very frustrating way of proceeding as you are being forced to work ‘from the outside in’ and beyond this there are issues about trying to maintain a consistent font style and size throughout a piece of mathematical text. (1 + x2) = 5x − 1

A simple approach to solving cubic equations

Abstract: Finding the roots of cubic equations has been the focus of research by many mathematicians. Omar Khayyam, the 11th century Iranian mathematician, astronomer, philosopher and poet, discovered a geometrical method for solving cubic equations by intersecting conic sections [1]. In more recent times, various methods have been presented to find the roots of cubic equations. Some methods require complex number calculations, a number of techniques use graphical methods to find the roots [e.g. 2, 3] and some other techniques use trigonometric functions [e.g. 4]. The method presented in this paper does not use graphical techniques as in [2] and [3], does not involve complex number calculations, and does not require using trigonometric functions. By using this fairly simple method, the roots of cubic equations can be found in a short time without using complicated formulas.

100.25 Normals to the Fermat line for a scalene triangle

Abstract: 3. O. Giering, Affine and projective generalization of Wallace lines, Journal for Geometry and Graphics 1 (1997), No. 2, pp. 119-133. 4. P. Pech, On the Simson-Wallace theorem and its generalizations, Journal for Geometry and Graphics, 9 (2005), No. 2, pp. 141-153. 5. T. O. Dao, Advanced plane geometry, message 1781, September 26, 2014, available at https://groups.yahoo.com/neo/groups/AdvancedPlaneGeometry/ conversations/messages/1781 6. A. Bogomolny, A generalization of the Simson line, available at http://www.cut-the-knot.org/m/Geometry/GeneralizationSimson.shtml 10.1017/mag.2016.77 NGUYEN LE PHUOC 501 CT7G, Duong Noi, Le Van Luong Stree, Ha Noi, Viet Nam e-mail: Nguyenlephuoc2006@gmail.com NGUYEN CHUONG CHI Rowna 4 str., 05-075 Warsaw, Poland e-mail: Nguyenchuongchi@yahoo.com

100.19 A quick evaluation of

Abstract: 2. H. Iwasawa, Gaussian integral puzzle, Mathematical Intelligencer 31 (2009) pp. 38-41. 3. P. M. Lee, The probability integral, http://www.york.ac.uk/depts/maths/histstat/normal_history.pdf 4. P. J. Nahin, Inside interesting integrals, Springer (2015) pp. 120-123. 5. E. Artin, The gamma function (English translation), Holt, Rinehart and Winston (1964). 6. W. W. R. Ball, Mathematical recreations and essays (10th edn.), Macmillan and Co. (1922) p. 348. 7. R. P. Feynman, What is science? The Physics Teacher 7 (1969) pp. 313-320. 10.1017/mag.2016.71 HSUAN-CHI CHEN Anderson School of Management, University of New Mexico, Albuquerque, NM 87131, USA e-mail: chenh@unm.edu

100.18 Newton's semicircle and the probability integral

Abstract: is conceptually inferior to vector algebra. One is left with the impression that somehow (2) intrinsically evades direct and short proofs. However, that need not be the case. I hope to have shown in this Note that viewing the problem as a planar one does lead to a short and fairly straightforward derivation of (2), provided that the validity of the planar proposition is first demonstrated. Since that proposition is significant in itself, it should be established anyway if planar vector algebra is properly developed. Drawing attention to the need for a more expressive planar vector algebra is a goal of this Note, apart from proving (2). It is particularly important to stress that by putting to use and , a wealth of properties, such as orientation, relative location, and related phenomena receive due consideration that is impossible when restricted to only linear operations and the scalar product. Making use of and brings to the calculations in planar geometry many benefits, here only glimpsed at. ∧ ⊥

100.04 Converses of a ‘curious’ theorem of Cesàro on conditionally convergent series

Abstract: References 1. T. P. Jameson, The probability integral by volume of revolution, Math. Gaz. 78 (1994) pp. 339-340. 2. C. P. Nicholas, Another look at the probability integral, Amer. Math. Monthly 64 (1957) pp. 739-741. 3. C. P. Nicholas and R. C. Yates, The probability integral, Amer. Math. Monthly 57 (1950) pp. 412-413. 4. R. M. Young, On evaluating the probability integral: a simple onevariable proof, Math. Gaz. 89 (2005) pp. 252-254.

Some properties of Kiepert lines of a triangle

Abstract: This article describes an investigation into Kiepert lines, and leads to some surprising and little-known relationships between the Fermat, Napoleon and Vecten points of a triangle.If we draw similar isosceles triangles A'BC, B'CA and C'AB outwards on the sides of a given scalene triangle ABC as in Figure 1, Kiepert's theorem tells us that the lines A'A, B'B and C'C meet in a single point - a Kiepert point [1, Chapter 11]. Since its position depends on the common base angles θ of the isosceles triangles, I label it K(θ), taking θ as the parameter of this point.