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Showing papers in "The Mathematical Gazette in 2001"


Journal ArticleDOI
Kiril Bankov1
TL;DR: Aigner and Ziegler as discussed by the authors presented beautiful proofs of some of the most exciting and historically important mathematical theorems, focusing on human endeavour to make the proofs more and more simple and striking.
Abstract: Proofs from THE BOOK, by Martin Aigner and Giinter M. Ziegler. Pp.199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag). I would like to begin with the authors' words from the Preface of this book: Paul Erd6s liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. ErdSs also said that 'you need not believe in God but, as a mathematician, you should believe in The Book'. This excellent book is dedicated to the memory of Paul ErdSs. Paul himself wrote several pages but he is not listed as a co-author because of his death in the summer of 1996. The idea of the book is to present beautiful proofs of some of the most exciting and historically important mathematical theorems. The book focuses on human endeavour to make the proofs more and more simple and striking.

203 citations






Journal ArticleDOI
TL;DR: The Delaunay triangulation as discussed by the authors is a generalization of the Voronoi diagram of P. This is the subdivision of the plane into regions, each containing one point p of P, where the interior of p's region contains the points of the planes closer to p than to any other point of p. The main difference is that no point of the set P lies inside the circumcircle of any of the triangles.
Abstract: Computational geometry: algorithms and applications (2nd edn.), by M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf. Pp. 367. £20.50. 2000. ISBN 3 540 65620 0 (Springer-Verlag). Suppose we are given spot heights for a piece of terrain at a random and not too sparse collection of points—just think of a surface z = f(x, y) where the values of z are given for a finite set of points (x„ v,) in general position. How can we get a reasonably good idea of what the terrain looks like? Over each (JC„ y,) there is a pole (in the telegraph sense) of height / (x„ y,), but what about other points? One natural idea is to triangulate the base plane, using the set P of points (x„ y,) as vertices, and draw the corresponding triangles in 3-space with vertices at the tops of the poles. But how best to triangulate? It is easy to produce situations where a bad choice of triangles results in an unexpected valley or ridge in the landscape. It turns out that a good choice is a triangulation which maximises the minimum angle of any triangle. That is, long thin triangles are a bad thing. What kind of a triangulation is that? It turns out to be something called a Delaunay triangulation, named, I was enlightened to read, after Boris Nikolaevich Delone and spelt in the French manner to make him sound decidedly French instead of Russian. The Delaunay triangulation can also be characterised by requiring that no point of the set P lies inside the circumcircle of any of the triangles. The Delaunay triangulation is dual to the Voronoi diagram of P. This is the subdivision of the plane into regions, each containing one point p of P, where the interior of p's region contains the points of the plane closer to p than to any other point of P. (And Voronoi even sounds Russian.) Voronoi diagrams are extensively used in shape analysis and pattern recognition.

119 citations


Journal ArticleDOI
TL;DR: In revising his 1988 book An introduction to cryptology, Henk van Tilborg has added new chapters on elliptic curve methods and on authentication codes, and the most significant addition is probably the CD-ROM, which contains an interactive Mathematica based tutorial.
Abstract: Fundamentals of cryptology: a professional reference and interative tutorial, by Henk C. A. van Tilborg. Pp. 505 with a CD-ROM. £110.50. 1999. ISBN 0 7923 8675 2 (Kluwer Academic Publishers). In a fast-changing field like cryptology, a textbook needs regular updating. In revising his 1988 book An introduction to cryptology, Henk van Tilborg has added new chapters on elliptic curve methods and on authentication codes. He has also updated and extended other sections. However, the most significant addition is probably the CD-ROM, which contains an interactive Mathematica based tutorial. The book starts gently with classical cryptosystems (such as the Caesar and Playfair ciphers), shift-register sequences and block ciphers. There is then a short chapter on Shannon theory, including the concepts of entropy and redundancy, followed by a discussion of data-compression techniques and Huffman codes. Attention then shifts to public-key cryptography based on discrete logarithms and RSA systems. Next come chapters on coding systems based on elliptic curves, coding theory, and the knapsack method. The new chapter on authentication is followed by a short discussion of zero-knowledge protocols and longer consideration of secret sharing systems. Distinctive feature of this book include the many references to the interactive CD-ROM and the attention paid to methods of breaking the codes, which in turn motivates the discussion of refinements and alternative methods. There are appendices on number theory, finite fields and relevant famous mathematicians. This book would suit someone who wanted an in-depth and practical understanding of cryptology.

114 citations



Journal ArticleDOI
Bill Cox1
TL;DR: In each chapter there follows a selection of questions and the whole is followed by a list of over 250 references as mentioned in this paper, which is a useful feature which is to be welcomed, as it allows the reader to get a good overview of the methods used.
Abstract: In each chapter there follows a selection of questions and the whole is followed by a list of over 250 references. A useful feature, which is to be welcomed, is a list of symbols used. The book seems to fall between two stools, being halfway between a textbook and a reference book, the text being a mixture of analysis and descriptions of and references to other methods not dealt with in detail. There is very little in the way of worked examples and many of the exercises at the end of the book are in fact supplementary text, rather than examples to be worked out. The book reads well and the printing is excellent. However, there is one serious fault. The hardback edition published in 1987 (not 11987 as page iv suggests!) and has gone through several reprints before the present paperback reprint in 2000, thereby clearly proving its worth. The opportunity could have been taken to revise and provide a third edition and certainly the list of references could have been updated; the latest reference being 1987, thirteen years before the appearance of the paperback edition.

28 citations


Journal ArticleDOI
TL;DR: Mowaffaq Hajja and Ross Honsberger as discussed by the authors presented a vector proof of a theorem of Bang, Amer. Math. No. 108 (2001) pp. 562-564.
Abstract: References 1. George S. Carr, Formulas and theorems in pure mathematics, (2nd edn.), Chelsea (1970). 2. Mowaffaq Hajja, A vector proof of a theorem of Bang, Amer. Math. Monthly 108 (2001) pp. 562-564. 3. Ross Honsberger, Mathematical gems II, Mathematical Association of America, Dolciani Math. Expos. No. 2 (1976). 4. Murray S. Klamkin, Vector proofs in solid geometry, Amer. Math. Monthly 77 (1970) pp. 1051-1065. MOWAFFAQ HAJJA Department of Mathematics, Yarmouk University, Irbid, Jordan

21 citations


Journal ArticleDOI
TL;DR: Two proofs concerning 'Octagon loops' are given in this article, where the authors present a brief description of the 'rules' of the "Octagon Loops' investigation.
Abstract: 85.03 Two proofs concerning 'Octagon loops' I must admit that I had never heard of the mathematical investigation 'Octagon loops'* when I arrived in deepest Ilford to take up my first teaching post. I first encountered it during a departmental meeting at which investigative material for year nine groups was being issued, and it immediately fascinated me. I am sure that not all readers will be familiar with it, so here is a brief description of the 'rules':

Journal ArticleDOI
TL;DR: The technique enables the determination of an estimate of the slope and intercept of a straight line relationship between two quantities or variables x and y.
Abstract: 85.13 Least squares revisited The method of least squares is a popular statistical tool which is used widely, particularly in science. In its simplest form, the technique enables the determination of an estimate of the slope and intercept of a straight line relationship between two quantities or variables x and y. Although a theoretical relationship may exist between x and y of the form y = mx + c, in practice experimental or measurement errors will occur, and the observed or measured values X and Y may not lie exactly on a straight line. Given a set of n observed values (X„ y,-), the standard least squares estimates for m and c are given by

Journal ArticleDOI
TL;DR: There has been a long history in the approximation of ellipses by circular arcs in order to simplify their construction and manipulation as discussed by the authors, which was of use for a wide variety of applications, in fields such as mathematics (generating figures), astronomy (analysing orbits), art (marking out large oval frames for ceiling painting), architecture (building masonry arches, floor plans, etc), and conversion of fonts from a general conic specification to circular arcs.
Abstract: There has been a long history in the approximation of ellipses by circular arcs in order to simplify their construction and manipulation. Such approximation was of use for a wide variety of applications, in fields such as mathematics (generating figures), astronomy (analysing orbits), art (marking out large oval frames for ceiling painting), architecture (building masonry arches, floor plans, etc), and, more recently, the conversion of fonts from a general conic specification to circular arcs. Documented evidence goes as far back as the Italian Renaissance when various schemes were published by the architect Sebastiano Serlio in the sixteenth century. More contentiously, it has been argued that fifteen centuries previously the Romans used such approximations when designing and building their amphitheatres.


Journal ArticleDOI
TL;DR: First published 25 years ago in the London Mathematical Society Monographs series, ONAG, as the book has since been commonly known, was apparently written in a burst of creative energy released essentially all within one week.
Abstract: On numbers and games (2nd edn), by J. H. Conway. Pp. 256. £27.00. 2001. ISBN 1 56881 127 6 (A. K. Peters). First published 25 years ago in the London Mathematical Society Monographs series, ONAG, as the book has since been commonly known, was apparently written in a burst of creative energy released essentially all within one week. The highly original theory being propounded is that a relationship between a new class of transfinite numbers and mathematical games can be set up. Presented in a delightfully whimsical style, the book soon became an exciting phenomenon leading to many research papers and a number of books, including one called Surreal numbers [1] by Donald Knuth.

Journal ArticleDOI
TL;DR: Pbraza et al. as discussed by the authors proposed a model for square-triangular numbers, which is based on elementary number theory, 2nd edn., Addison Wesley (1988).
Abstract: 4. Leonard E. Dickson, Theory of numbers, Volume 3, Chelsea Publishing (1966). 5. Donald Keedwell, Square-triangular numbers, Math. Gaz84 (July 2000) pp. 292-294. 6. Kenneth H. Rosen, Elementary number theory, 2nd edn., Addison Wesley (1988). PETER A. BRAZA JINGCHENG TONG Department of Mathematics and Statistics, University of North Florida, 4567 St John's Bluff Road, Jacksonville, Florida, 32224-2645, USA e-mail: Pbraza@unf.edu e-mail: jtong @unf edu

Journal ArticleDOI
TL;DR: In this article, it was shown that in any scalene triangle the two Fermat points, the nine-point center and the circumcentre of the triangle are concyclic.
Abstract: The object of this note is to draw readers’ attention to a very new theorem in the Euclidean geometry of the triangle and to provide a straightforward Cartesian proof. This remarkable theorem, due to Lester, asserts that in any scalene triangle the two Fermat points, the nine-point centre and the circumcentre are concyclic. Lester’s original computer-assisted discovery and proof make use of her theory of ‘complex triangle coordinates’ and ‘complex triangle functions’ as expounded in, and . A proof has also been given by Trott using the advanced concept of Grobner bases in the reduction of systems of polynomial equations to ‘diagonal’ form. Trott’s work uses the computer algebra system Mathematica as an essential tool and he also provides an animation of the Lester circle as one vertex of the triangle is varied. Further information on the configuration is given in.

Journal ArticleDOI
TL;DR: In this paper, the authors show that if we select any specifiable pair of generations and find that the position of the dead heats of one generation mirrors that of the other, then somewhere, as Na -> °° this will cease to be so.
Abstract: handwritten asides, underlining in different colours etc. Paul wrote: 'I'll give you here the tiny \"extra\". It came to me in a flash (this is the truth) when looking at a full rainbow & being a very superstitious person, I know it is \"true\" (it also aesthetically is very pleasing). Of course, on the face of it, it IS wildly extravagant. Let us make it so that all the subsets of every generation always have the same number of (ordered) members—call this Na— then the difference between the sum of occurrences where the count of primes in each subset of a particular generation is the same for any other subset of that generation—and—the sum of occurrences of the same phenomenon in any other generation will approach zero as N -» «>.' Paul uses the term subsets of a generation to refer to the collection of residue classes modulo 2\" for a given value of n. I therefore interpret his tiny extra as follows: let Ma = Na x 2\"~' be a multiple of a power of n and, for m = 1, 2, ... , n, let Em(Ma) be the number of occasions with M < Ma for which nm <*>. A more realistic 'wild extravagance' would probably have S,(Ma) Ej(Ma) as Ma —» °°, but either version is probably some years (or centuries) from being established! I sent Paul the typeset version of this note and, true to form, he wrote back, 'This also I believe to be true (and structurally not insignificant):taking some given Na, if we select any specifiable pair of generations and find that the position of the \"dead heats\" of one generation mirrors that of the other, then somewhere, as Na -> °° this will cease to be so.'

Journal ArticleDOI
TL;DR: In this paper, the authors present an introduction to the mathematical theory of the design of articulated mechanical systems, or linkages as the title of the book, and the area, label them.
Abstract: This book presents an introduction to the mathematical theory of the design of articulated mechanical systems, or linkages as the title of the book, and the area, label them. The author seeks to combine two approaches: a consideration of the geometric configurations of points and lines generated as a moving body is displaced through a finite set of positions and the solution of non-linear constraint equations that characterise a mechanical connection. Using the basic language of vectors and matrices, the author guides the reader through from simple to more complex linkages. Additional mathematics that the author judges might be unfamiliar to the reader is provided within appendices and each chapter concludes with a summary of the material covered in the chapter, further references directly related to the chapter and exercises. The text is supported by very clear and explicit diagrams the greatly aid the reader in coming to terms with the material. As a novice in the area of linkages but with the requisite mathematical knowledge, I found the author had provided an excellent text that enabled me to come to terms with the subject. Readers with an interest in the area will find the volume rewarding. TOM ROPER School of Education, University of Leeds LS2 9JT

Journal ArticleDOI
Nick Lord1
TL;DR: In this paper, the authors define the discrete and the fast Fourier transforms, the latter being described as the most valuable numerical algorithm of our lifetime, and introduce wavelet analysis, which employs orthonormal bases other than sines and cosines in order to approximate a function.
Abstract: essential precursor to wavelets but has an elegance which clearly delights the authors. The next, short chapter defines the discrete and the fast Fourier transforms, the latter being described as the most valuable numerical algorithm of our lifetime. The final chapter introduces the growing subject of wavelet analysis, which employs orthonormal bases other than sines and cosines in order to approximate a function. The book is quite formidable in that it does not shirk any of the analytic rigour needed for a firm foundation. The authors are, however, aware of the need to sustain motivation and there are plenty of worked examples and exercises, with hints and solutions. There is also an extensive list of references for readers who wish to investigate further. This is a serious and scholarly work which should be in the library of every mathematics department.

Journal ArticleDOI
TL;DR: In this paper, it was shown that/(2) = 0,/(4) = 1 and/(6) = 2, and that/(8) = 5) is the number of different octagon loops for arbitrary even n.
Abstract: is easy to see that/(2) = 0,/(4) = 1 and/(6) = 1. A little bit of playing around should convince you that/(8) = 5. Is it possible to find a formula or generate an algorithm that would allow you to calculate the number of different octagon loops for arbitrary even n ? It may also be worth looking at the area that each loop encloses, especially when a particular number of octagons (eight, for example) gives rise to several different loop shapes. Readers interested in other polygon loops might like to consult [1].

Journal ArticleDOI
TL;DR: The book is nicely produced, well-illustrated (in monochrome) and effectively translated (by Katrin Gygax), but it does not have even a limited version of Mathematica, so if you do not have access to it on your computer, you cannot do the examples or follow the hyperlinks.
Abstract: The book is nicely produced, well-illustrated (in monochrome) and effectively translated (by Katrin Gygax). The CD-ROM that comes with the book can be used with MacOS, Windows 95/98/NT or UNIX. I tried it with Windows 95, found it was not self-loading, and that the instructions for setting it up are rather vague. The disk contains all the text, examples and exercises in the form of Mathematica notebooks, and there are hyperlinks to the on-line documentation of Mathematica. However, it does not have even a limited version of Mathematica, so if you do not have access to it on your computer, you cannot do the examples or follow the hyperlinks, merely read the text and with UNIX it seems that you cannot even do that. I suppose I have put the cart before the horse, since the book is really a printout of the contents of the disk. But as the author says, 'Books are still the most ergonomic medium for the sequential display of texts'.


Journal ArticleDOI
TL;DR: In this paper, the authors try to please three different kinds of readers at the same time, but, on the whole, this book does succeed, at the expense of some repetition and chapters which are written in quite different styles.
Abstract: beginners. Many n websites are listed on the CD-ROM. One site which lists current records is http://www.lacim.uqam.ca/pi/records.html (unfortunately misspelt in the reference given on page 12 of the book) and another general website is http:// www.cecm.sfu.ca/pi/pi.html. It is not an easy matter to please three different kinds of reader at the same time, but, on the whole, this book does succeed, at the expense of some repetition and chapters which are written in quite different styles. More casual readers may also be put off by the plethora of formulae whereas in fact there is much interesting material for them too. Of course one should ask why such a book is interesting at all—what is it about n which merits such a large amount of time and effort by computers and people? It is common in mathematics for a challenging problem to focus the attention of professionals on developing methods which are then applied in a wide variety of other situations. The techniques used to analyse n are very useful indeed, and calculation of n is merely one of their more glamorous applications. It is to be hoped that this is the general perception of the tax-paying public, who might otherwise look askance at huge efforts to calculate billions of decimal places of n\ We should all take care, as this book does, to keep the record straight.

Journal ArticleDOI
TL;DR: BASIC ARITHMETRIC and ALGEBRA as discussed by the authors introduces arithmetic and algebra, numbers, units and physical quantities, and Graphs and Graph Sketching with Scalars and Vectors.
Abstract: BASIC ARITHMETIC AND ALGEBRA. Introducing Arithmetic and Algebra. Numbers, Units and Physical Quantities. Functions and Graphs. Solving Equations. Trigonometric Functions. Exponential and Logarithmic Functions. Hyperbolic Functions. BASIC GEOMETRY. Introducing Geometry. Coordinate Geometry. Conic Sections. BASIC VECTOR ALGEBRA. Introducing Scalars and Vectors. Working with Vectors. BASIC DIFFERENTIATION. Introducing Differentiation. Differentiating Simple Functions. Differentiating Composite Functions. Stationary Points and Graph Sketching. BASIC INTEGRATION. Introducing Integration. Integrating Simple Functions. Integrating by Parts and by Substitution. Appendix. Answers and Comments. Index.

Journal ArticleDOI
TL;DR: Sato et al. as mentioned in this paper introduced the concept of Levy processes and infinitely divisible distributions (IVDPs) to mathematical probability theory and showed that they are one of the most important classes of stochastic processes both for theoretical and practical reasons.
Abstract: Levy processes and infinitely divisible distributions, by Ken-iti Sato. Pp. 486. £50.1999. ISBN 0 521 55302 4 (Cambridge University Press). Levy processes are one of the most important classes of stochastic processes, both for theoretical and practical reasons. Named after the great French probabilist, Paul Levy, who initiated their study in the 1930s, they were quickly seen to lie at die heart of the newly emerging discipline of mathematical probability theory. After two or three decades of relative neglect, the 1990s saw a renaissance in their systematic investigation, with new exciting theoretical developments and novel applications within such diverse areas as mathematical finance and quantum field theory. Indeed this is the third major book in the subject to be published during the last ten years.

Journal ArticleDOI
TL;DR: A latin square of order n as discussed by the authors is an n × n array (or matrix) containing n distinct symbols (which are often taken to be the symbols 0, 1, …, n - 1, though any n symbols will do) arranged in such a way that all n of these symbols occur in the n cells of each row and also in each column.
Abstract: A latin square of order n is an n × n array (or matrix) containing n distinct symbols (which are often taken to be the symbols 0, 1, …, n - 1, though any n symbols will do) arranged in such a way that all n of these symbols occur in the n cells of each row and also in the n cells of each column. Such squares occur in many guises: for example, as appropriate field layouts in the statistical design of experiments, as the addition or multiplication tables of groups, as a coding theory device and as a means of representing a finite geometry algebraically.

Journal ArticleDOI
TL;DR: In this paper, the authors introduce complex algebra and complex numbers, and introduce the concept of vector products of vectors, which can be used to represent complex numbers and define differential equations.
Abstract: COMPLEX NUMBERS. Introducing Complex Numbers. Polar Representation of Complex Numbers. Demoivrea s Theorem and Complex Algebra. VECTOR ALGEBRA. Scalar Products of Vectors. Vector Products of Vectors. DETERMINANTS AND MATRICES. Determinants. Matrices. DIFFERENTIATION, EXPANSION AND APPROXIMATION. Expansions and Approximations. Taylor Expansions and Polynomial Approximations. Hyperbolic Functions and Differentiation. INTEGRATION, SUMMATION AND AVERAGING. Areas, Volumes and Averages. Special Integration Techniques. DIFFERENTIAL EQUATIONS. Formulating and Classifying Differential Equations. Solving First--Order Differential Equations. Solving Second--Order Differential Equations. Waves and Partial Differential Equations. VECTOR CALCULUS. Differentiating Vectors. Integrating Vectors. Appendix. Answers and Comments. Index.

Journal ArticleDOI
TL;DR: The President of the new millennium shares a vision for the Association that has been developed by Council with the support of active members and relies heavily on the ability to attract more 'rank and file' members and to persuade more of them to become active members.
Abstract: The President has only two official duties: to preside over the AGM and to give a Presidential Address. Furthermore, since both of these duties come at the end of the year of office, I have had two years to think of a clever title for this address. It was sneakily chosen to allow me to give two talks in one, but I want to apologise in advance and unreservedly to any members who are offended by my use of the term 'rank and file' to refer to them. I will explain . . . . As the first President of the new millennium, I feel obliged to look to the future of The Mathematical Association. I want to spend a little time sharing a vision for the Association that has been developed by Council with the support of active members. This vision relies heavily on our ability to attract more 'rank and file' members and to persuade more of them to become active members. However, I also want to talk about mathematics itself, guided by the conference theme of visualisation.

Journal ArticleDOI
John Mason1
TL;DR: This article showed that the reason why Liouville's result works is because the numbers generated are element-by-element products of sequences of the form {1, 2,..., t }, which are well known to have the sum of their cubes equal to the square of their sum.
Abstract: David Pagni drew attention to a result which is ascribed by Dickson [2, p. 286] to Liouville (1857), that the sum of the cubes of the number of divisors of each of the divisors of an integer, is equal to the square of their sum. For example, the divisors of 6 are 1, 2, 3, and 6, which have 1, 2, 2, and 4 divisors respectively, and Pagni observed, as have others, including Mason et al . [4, p. 179], that the reason Liouville’s result works is because the numbers generated are element-by-element products of sequences of the form {1, 2, ..., t }, which are well known to have the sum of their cubes equal to the square of their sum.