Showing papers in "The Mathematical Gazette in 2004"
TL;DR: Marker's model-theoretic proof of the Mordell-Lang conjecture on algebraic varieties and function fields is a good illustration of this as discussed by the authors, as is Marker's proof of Morley's theorem.
Abstract: celebrated result there is Morley's theorem, which is proved in the sixth chapter. David Marker's explicit goal is to integrate these two approaches in a way which permits each to enlighten the other and to develop model-theoretic methods which allow proofs of sophisticated theorems in modern algebra, and the closing result of his book, Hrushovski's proof of the Mordell-Lang conjecture on algebraic varieties and function fields, is a good illustration of this.
256 citations
TL;DR: In this article, the authors admit that there are no definitive answers, considering, inter alia, the following questions: how convinced are we that the trends in climate change over the past thirty years are an indication of global warming rather than just random fluctuations? how much belief can there be in miracles? is the movement of share prices better explained by chaos theory than by statistics?
Abstract: distribution, queuing theory, random walks, and so on. On many topical issues he is prepared to admit that there are no definitive answers, considering, inter alia, the following questions: how convinced are we that the trends in climate change over the past thirty years are an indication of global warming rather than just random fluctuations? how much belief can there be in miracles? is the movement of share prices better explained by chaos theory than by statistics? He also emphasizes that issues such as psychology and economic efficiency sometimes have as much of a bearing on eventual decisions as purely statistical considerations.
219 citations
TL;DR: Schatzman as discussed by the authors has a poetic turn of phrase, describing her subject as'more like a heavy matron than a gracious ballerina' but firmly grounds this comprehensive introduction in reality.
Abstract: Numerical analysis: a mathematical introduction, by Michelle Schatzman, translated by John Taylor. Pp. 496. £49.95 (hb), £24.95 (pb). 2002. ISBN 0 19 850279 6 (hb), 0 19 850852 2 (pb) (Oxford University Press). Schatzman has a poetic turn of phrase, describing her subject as 'more like a heavy matron than a gracious ballerina' but firmly grounds this comprehensive introduction in reality. In the opening chapter she is especially good at reorienting the reader to understand the effects of machine representations of numbers on calculations. She sets exercises in sections like 'Even the obvious problems are rotten' and 'Even the easy problems are hard' that bring out the fundamental issues. The second chapter continues the theme of re-education, with its 'self-guided visit in the garden of approximations of the continuous by the discrete' including approximation of natural logarithms, construction of the exponential and even matrix exponentials. Understanding that numerical analysis is hard for students, who lack the necessary maturity, Schatzman includes a third introductory chapter covering linear algebra.
56 citations
TL;DR: In this article, the authors present various different methods and approaches, and the power in the appropriate use of language and notation in mathematics, and most of the text and most the examples are presented in a concrete manner, but there is also a lengthy and very readable appendix offering the abstract setting.
Abstract: various different methods and approaches, and the power in the appropriate use of language and notation in mathematics. Much of the text and most the examples are presented in a concrete manner, but there is also a lengthy and very readable appendix offering the abstract setting. The writing style is clear, and may be described as exceedingly enthusiastic, with much of the text peppered with bold and capital letters. Unfortunately there are also some careless misprints which should have been removed during proof-reading. The book should complement linear algebra texts which place more emphasis on vector spaces and linear transformations. P. SHIU
37 citations
TL;DR: 88.27 More on spreads and non-arithmetic means Nick Lord points out that if all numbers in a set become proportionately farther from their fixed arithmetic mean, their geometric mean decreases, but in general if the numbers are arbitrarily altered to increase their standard deviation, their geometry mean may increase or decrease.
Abstract: 88.27 More on spreads and non-arithmetic means Nick Lord [1] points out that (1) if all numbers in a set become proportionately farther from their fixed arithmetic mean (of course increasing their standard deviation), their geometric mean decreases, but (2) in general if the numbers are arbitrarily altered to increase their standard deviation, again with fixed arithmetic mean, their geometric mean may increase or decrease.
35 citations
TL;DR: In this paper, Jost fleshes out Dirichlet's principle by splitting the question of existence of (weak) solutions lying in an appropriate Sobolev space from the trickier question of regularity with a gradually increasing ratchet of difficulty.
Abstract: needed to make Dirichlet's principle watertight: that the solution of Laplace's equation with given boundary values 'ought' to be the one which minimises j |grad u (x)\\ dx. Jost fleshes out the 'ought' by splitting the question of existence of (weak) solutions lying in an appropriate Sobolev space from the trickier question of regularity with a gradually increasing ratchet of difficulty as he traverses the Ltheory, strong solutions (with a glimpse of Calderon-Zygmund territory), C-theory (and Schauder-type arguments) and the ideas of de Giorgi/Nash/Moser. Throughout, Jost achieves an impeccable blend of motivation, orientation and analysis and he persuasively makes his case (p. 230) that, '... in the theory of PDEs, the situation becomes much clearer when a more abstract approach is developed.' Beautifully written and superbly well-organised, I strongly recommend this book to anyone seeking a stylish, balanced, up-to-date survey of this central area of mathematics.
28 citations
TL;DR: Gian Carlo Rota as discussed by the authors was born in Italy in 1932, but moved to the United States, where he studied at Princeton and Yale and obtained his PhD under Jacob T. Schwartz in 1956 and remained at MIT until his death in 1999.
Abstract: Gian-Carlo Rota was born in Italy in 1932, but moved to the United States, where he studied at Princeton and Yale. He earned his PhD under Jacob T. Schwartz in 1956 then moved to Massachusetts Institute of Technology, where he largely remained until his death in 1999. Renowned as a bon viveur, Rota juggled his mathematical career with consultancy work for the US government. This mixture of reprinted papers and articles about Rota tell the story of his mathematical journey.
13 citations
TL;DR: Frederickson as discussed by the authors introduced the notion of hinged dissections, whereby the dissected pieces of polygonal shapes are hinged together, permitting them to swing into new positions to form various other polygonally shapes.
Abstract: Hinged dissections: swinging and twisting, by Greg N. Frederickson. Pp. 287. £35 (hbk). 2002. ISBN 0 521 81192 9 (Cambridge University Press). This book is a companion volume to the author's introductory text on dissections [1], the two books combining to constitute a comprehensive coverage of the topic. The present work is confined to the notion of hinged dissections, whereby the dissected pieces of polygonal shapes are hinged together, permitting them to swing into new positions to form various other polygonal shapes. For example, the hinged dissection of the triangle, below, can be swung into various positions, eventually forming a square, as shown.
12 citations
TL;DR: In this paper, a new physical invariance for magic squares is reported for the moment of inertia of these squares which depends only on the order of the magic square, which is called the second moment.
Abstract: Magic squares are characterised by having the sum of the elements of all rows, columns, and main diagonals having the same sum. A new physical invariance for magic squares is reported for the moment of inertia of these squares which depends only on the order of the square. The numbers in the magic square are replaced by corresponding multiples of a unit mass placed on a square unit lattice. 1 Introduction Classical magic squares of the whole numbers, 1:::N, have the same line sum (magic constant) for each row, column, and main diagonals [1]: CN = N 2 (N + 1) (1) This line sum invariance depends only on the order, N , of the magic square. Here we report a second invariance which was found by considering the analogue of the physical moment of inertia of a square array of masses which are taken to be proportional to the numbers in the magic square. In the mathematical context this would be called the second moment. The smallest and earliest is the ancient Chinese 3 3 Lo-shu magic square of the sequence 1::9, with a magic constant of 15, as shown below: 4 9 2 3 5 7 8 1 6 (2) De
ne the moment of inertia, IN , of a magic square of order N about an axis perpendicular to its centre by summing mr for each cell, where m is the
10 citations
TL;DR: In this paper, the authors show that if X * Z = Z, then the problem is solvable in Z and if X = Z*, then X = W, Y = ± 60, 6 e Z, and if Z * 0, then, 3F + Z > 0, so, X > 0.
Abstract: AX = 3Y + Z. (4) If Z = 0, the solutions are easily described as X = W, Y = ±60, 6 e Z. If Z * 0, then, 3F + Z > 0, so, by (4), X > 0. Now, 12X 3Z = (3F). If we had X * Z, then, by the proposition, (1) would be solvable in Z. Therefore, X = Z*, and hence, by (4), Y = Z. This reduces here to X = Z and Y = ±Z. Finally, all solutions of (4) are given by: (X, Y, Z) = (36>, ±66>, 0) or (<9, ±9, d), 9 € Z.
10 citations
TL;DR: The importance of teaching students at all levels how to use computers as a problem-solving tool is well-recognised as discussed by the authors, and using numerical approximations to discover unknown formulas can be a powerful example.
Abstract: The importance of teaching our students at all levels how to use computers as a problem-solving tool is well-recognised. Using numerical approximations to discover unknown formulas can be a powerful example. It is usually pointed out to the students that computer exploration alone is not sufficient and should be followed up whenever possible with a proof or at least some additional analysis. The importance of this is illustrated in the example below, in which there is compelling numerical evidence for an incorrect conclusion.
TL;DR: The last chapter of "Chung's book" is a highly authoritative account of random walks and Markov chains which includes a discussion of communication classes, recurrence and transience and steady state probabilities as mentioned in this paper.
Abstract: the Poisson and normal and gives proofs of the central limit theorem and the (weak) law of large numbers. Regrettably, since this book was first written, students' knowledge of and ability with elementary analysis has declined and it may be that the treatment herein of these topics is no longer appropriate for first years. The last chapter of 'Chung's book' is a highly authoritative account of random walks and Markov chains which includes a discussion of communication classes, recurrence and transience and steady state probabilities. Elementary accounts of branching processes and birth and death processes appear as examples.
TL;DR: In this article, the authors considered the case of substituting x = 1 in Gregory's arctangent series and showed that it is possible to speed up the convergence of the series.
Abstract: 88.38 Some observations on the method of arctangents for the calculation of n In 1671, the Scottish mathematician James Gregory discovered the arctangent series _, ^ (,-lfx\"* , tan x = Y —• , -1 < x < 1. n = o 2n + 1 When x = 1, this becomes Leibniz's series of 1674. Beckman [1] has the comment 'Yet it is unthinkable that the discoverer of arctangent series, a man who had worked on the transcendence of n, should have overlooked the obvious case of substituting x = 1 in his series. More likely he did not consider it important because its convergence was too slow to be of practical use for numerical calculations. If this was the reason why he did not specifically mention it, he was of course quite right.' Before 1983, it was usual to use Gregory's series (with various modifications to accelerate its convergence). Let us consider the angle a whose tangent is 1/5: 1 tana = .
TL;DR: In this article, the authors consider the problem of numbers with n discs and show that if the number of different sets of discs labelled with positive integers can be represented by a binomial coefficient, then (binary numbers) all the consecutive numbers from B to B + 2* can be generated.
Abstract: numbers with n discs needs nothing deeper than an understanding of how binary numbers work: it is intriguing the extent to which the context of the problem seems to camouflage this connection in the classroom! Suppose then that we have n discs marked with positive integers a,,b, where a, > bx so that dt = a, bt > 0, with labels chosen in such a way that di < d2 <... < dn. Our target is the 2\" consecutive numbers starting with B = b\\ + b2 + ... + bn. These totals are realised by adding to B any of the 2\" subsets of {dy, d2, ... , dn}. In order to produce B + 1, we must have d\\ = 1 and, arguing inductively, if dt = 2' ~' for 1 < i < k, then (binary numbers!) all totals from B to B + 2* 1 are catered for so that dk + i = 2* to create the next total, B + 2 . And, clearly, if dt = 2' ~ 1 for 1 < i < n, then (binary notation!), all of the consecutive totals from B to B + 2\" 1 can be generated. Finally, the number of different sets of discs labelled with positive integers equals the number of choices of positive integers b\\, b2, ... , bn with b\\ + b2+ ... +bn = B. This is the same as the number of choices of non-negative integers 0 < b\\, b2, ... , bn with bi + b2+ ... +bn = B-n, which in turn is the number of ways of placing n 1 'separators' into a row of B n ones, i.e. the binomial coefficient
TL;DR: A Latin square of order 3 and order 4 is a diagonal Latin square as discussed by the authors, where the elements in its left diagonal, au, a22, •••, am, are distinct.
Abstract: are Latin squares of order 3 and order 4 respectively. A Latin square is termed left semi-diagonal if the elements in its left diagonal, au, a22, ••• , am, are distinct. Similarly if the elements in its right diagonal are distinct then the Latin square is termed right semi-diagonal. If a Latin square is both left semi-diagonal and right semi-diagonal then such a square is termed a diagonal Latin square. L\\ above is a left semi-diagonal Latin square and LQ. is a diagonal Latin square. The number of Latin squares of order n = 1, 2, 3, 4, 5, 6 are 1, 2, 12, 576,161280, 812851200. The two Latin squares of order 2 are given by
TL;DR: In this article, the use of projective transformations, circular transformations, Cremona transformations, the Kleinian viewpoint and Lie theory is discussed under five chapter headings: higher spaces and higher space elements • Geometrical transformations.
Abstract: • Higher spaces and higher space elements • Geometrical transformations. The last of these examines the use of projective transformations, circular transformations, Cremona transformations, the Kleinian viewpoint and Lie theory. Covering all of this in a mere fifty pages means that the treatment is somewhat condensed, but the historical thread emerges with clarity. The final part of the book deals with differential geometry under five chapter headings:
TL;DR: In this paper, it was shown that of the five Platonic solids, the regular tetrahedron, cube and octahedron have analogues in all higher dimensions, called regular polytopes.
Abstract: There are infinitely many regular polygons, but we find, on extending the idea of polygons to three dimensions, that there are only five regular polyhedra, the Platonic solids. What happens then if we try to extend this idea beyond three dimensions? It turns out that, of the five Platonic solids, just the regular tetrahedron, cube and regular octahedron have analogues in all higher dimensions, the so-called regular polytopes. Brief descriptions of these mathematical objects are to be found in [1], for example.
TL;DR: It is shown that S(0) is easily seen to have the value 1, so that the behaviour of S(x) for 0 < x < 1 is unclear, and the problem " 1 becomes more difficult since ]£ — is divergent as n —» oo, as is the integral term in (2).
Abstract: l + I + I + ... + I = ?„« (3) 2 3* n is the nth partial sum of the Riemann zeta function £ (x) which is convergent as n —» oo, as is the integral term in (2). When JE is an even integer, simple analytic results are known. For example, when x = 2, £ (2) = n16 and the corresponding integral is j — = 1. Hence 5(2) = jt/6 1 = 0.6449.... Likewise, when * = 4, £(4) = ^/90 and the corresponding integral has the value 1/3. Hence S(4) = 0.7489.... As x -» oo, it is easy to see that S (x) -> 1. We note that S(0) is easily seen to have the value 1, so that the behaviour of S(x) for 0 < x < 1 is unclear. In this range, the problem \" 1 becomes more difficult since ]£ — is divergent as n ) « , as is the
TL;DR: In this paper, Stedall makes a case for the existence of transcendental numbers in the quadrature of the circle, which Wallis attacked in his Arithmetica infinitorum.
Abstract: The enigmatic John Pell published virtually nothing in his lifetime and could hardly be persuaded to allow his name to appear in print, but he was a skilled and methodical algebraist and much of the work Wallis presents in his treatise can be attributed to him. Ironically, his name is remembered nowadays only for an equation which he did nothing to solve! Wallis himself is famous for his infinite product for 4/ n, and Dr Stedall paints a fascinating picture of the methods he employed to achieve it. It arose from a much more significant problem, that of the quadrature of the circle, which Wallis attacked in his Arithmetica infinitorum. Geometrical methods for finding areas, volumes and arc lengths were well established; the originality of Wallis was that he attempted to algebraicise the methods. Essentially his approach was to interpolate values corresponding to fractional indices into a table which related to whole number powers, and he displayed great ingenuity (as well as confidence in his own intuitions) in doing so. His interpretation of his product shows remarkable prescience, and Dr Stedall makes a case for him postulating the existence of transcendental numbers.
TL;DR: Teaching mathematics using information and communications technology, by Adrian Oldknow and Ron Taylor as mentioned in this paper, is the second edition of a book first published in 2000, and the focus is the teaching of mathematics using the wide range of computer-based resources both hardware and software which is now becoming widely available in schools.
Abstract: Teaching mathematics using information and communications technology, by Adrian Oldknow and Ron Taylor. Pp. 262. £19.99. 2003. ISBN 0826470599 (Continuum). This is the second edition of a book first published in 2000. The focus is the teaching of mathematics using the wide range of computer-based resources both hardware and software which is now becoming widely available in schools, partly due to government funding and encouragement. It is not, as the authors explain in the newly-written introduction, about ICT for its own sake, or the separate subject of that name in the national curriculum. The key, therefore, is the integration of ICT into mathematics teaching of 11 16 year olds, and throughout it is emphasized that there is little point in using 'new technology' unless it enhances and develops the quality of teaching and learning. This is a point I will return to later.
TL;DR: Schatzman as discussed by the authors has a poetic turn of phrase, describing her subject as'more like a heavy matron than a gracious ballerina' but firmly grounds this comprehensive introduction in reality.
Abstract: Numerical analysis: a mathematical introduction, by Michelle Schatzman, translated by John Taylor. Pp. 496. £49.95 (hb), £24.95 (pb). 2002. ISBN 0 19 850279 6 (hb), 0 19 850852 2 (pb) (Oxford University Press). Schatzman has a poetic turn of phrase, describing her subject as 'more like a heavy matron than a gracious ballerina' but firmly grounds this comprehensive introduction in reality. In the opening chapter she is especially good at reorienting the reader to understand the effects of machine representations of numbers on calculations. She sets exercises in sections like 'Even the obvious problems are rotten' and 'Even the easy problems are hard' that bring out the fundamental issues. The second chapter continues the theme of re-education, with its 'self-guided visit in the garden of approximations of the continuous by the discrete' including approximation of natural logarithms, construction of the exponential and even matrix exponentials. Understanding that numerical analysis is hard for students, who lack the necessary maturity, Schatzman includes a third introductory chapter covering linear algebra.
TL;DR: The two oldest representations for the number π are infinite product expansions, due to Vieta in 1592 and Wallis's product dating from 1655 as discussed by the authors, which are the two oldest product expansions for π.
Abstract: The two oldest representations for the number π are infinite product expansions. The first, is due to Vieta in 1592. The second is Wallis's product dating from 1655:
TL;DR: In this article, it was shown that the table cannot be turned in such a way that at least two persons are sitting in front of their nameplates, even when n is even.
Abstract: of a is in front of 2a for every a. By construction every a is sitting in front of his nameplate after a rotation of the people over a persons in the chosen direction. We will always rotate the people instead of the table for the sake of elegance. We prove by contradiction that it is possible when n is even. Suppose that the table cannot be turned in such a way that at least two persons are sitting in front of their nameplate. After every rotation there is thus exactly one person sitting correctly. Consider the permutation n defined by n(i) =j if, and only if, the nameplate of i is in front of j . Identify the rotations with the integers modulo n by calling a rotation over a persons in the chosen direction simply a. Then we have
TL;DR: In this paper, the structural aspects of the Fibonacci numbers are linked with trigonometric and hyperbolic functions, but no evidence that the link has been systematically developed.
Abstract: This article started life as an investigation into certain aspects of the Fibonacci numbers, ‘morphed’ seamlessly into the structure of some infinite matrices and finally resolved into a general set of results that link structural aspects of Fibonacci numbers with trigonometric and hyperbolic functions. It is a surprising fact, but while I can find evidence that the link between these areas has been noted in the past, I can find no evidence that the link has been systematically developed.
TL;DR: In the 1930s several mathematicians, principally Alonzo Church, Stephen Kleene, and Alan Turing, began investigating the notion of effective calculability, which is a process of defining a function by specifying each of its values in terms of previously defined values.
Abstract: In the 1930s several mathematicians, principally Alonzo Church (1903-1995), Stephen Kleene (1909-1994), Emil Post (1897-1954) and Alan Turing (1912-1954), began investigating the notion of effective calculability . (A function from natural numbers to natural numbers is effectively calculable if there is some finite rule or mechanism which will calculate the value of the function for any natural number.) Central to this activity was the notion of recursiveness. Loosely, recursion is a process of defining a function by specifying each of its values in terms of previously defined values.
TL;DR: A Watt quadrilateral is defined in this paper as a quadrailateral with a pair of opposite sides of equal length, and the Watt linkage was devised by James Watt about 1784 to constrain the steam engine piston rod, connected at E, the midpoint of CD.
Abstract: We define a Watt quadrilateral to be a quadrilateral with a pair of opposite sides of equal length. See Figure 1. 2. Linkages The Watt linkage (Figure 2) has equal-length cranks AD and BC, A and B fixed, and coupler bar CD. It was devised by James Watt about 1784 to constrain the steam-engine piston rod, connected at E, the midpoint of CD, to approximate straight-line motion over a limited range.
TL;DR: In this paper, Finbar Holland et al. discuss the Tournament of the towns 1984-1989, Australian International Centre for Mathematics Enrichment (1992), which was held in Australia from 1984 to 1989.
Abstract: References 1. Steve Abbott, Rank and file: vision and visualisations, Math. Gaz. 85 (November 2001) pp. 386-402. 2. P. J. Taylor (ed.), Tournament of the towns 1984-1989, Australian International Centre for Mathematics Enrichment (1992). 3. G. Hardy, J. E. Littlewood and G. Polya, Some simple inequalities satisfied by convex functions, Messenger of Math. 58 (1929) pp. 145152. FINBARR HOLLAND Department of Mathematics, University College Cork, Cork, Ireland e-mail: fholland@ucc.ie
TL;DR: In this paper, it was shown that a singular matrix can be made nonsingular by very minor tweaking, such as changing just one entry, for example, will make a matrix of rank n − 1 into a full rank (nonsingular) matrix.
Abstract: Just how many matrices have inverses? In elementary linear algebra courses, many of the matrices encountered are singular, but perhaps the reason is that such matrices provide rich and interesting examples. How many of them occur naturally? Many beginning students observe that a singular matrix can be made nonsingular by very minor tweaking – changing just one entry, for example, will make a matrix of rank n – 1 into a full rank (nonsingular) matrix. In fact, changing one entry just a tiny bit will do it. Looking at the question from that point of view, with a little experimentation students begin to discover that singular matrices are quite rare. Entire rows (or columns) have to be rigged exactly right in order to get one, while minor changes in individual entries undo all the work and give us another nonsingular matrix. If we were to reach into a hat full of all the numbers – each equally likely to be chosen – and draw enough to fill in a square matrix randomly, we would certainly expect to get a nonsingular one.