Showing papers in "The Mathematical Gazette in 2006"
TL;DR: One of the fundamental theorem of mathematics is the Descartes rule of signs as discussed by the authors, which states that a real polynomial of degree n has at most n real zeros.
Abstract: One of the bedrock theorems of mathematics is the statement that a real polynomial of degree n has at most n real zeros. Probably the best-known proof is the algebraic one, by factorisation. But there is also a pleasant analytic proof, by deduction from Rolle’s theorem. A slightly different question is how many positive zeros a polynomial has. Here the basic result is known as ‘Descartes′ rule of signs’. It says that the number of positive zeros is no more than the number of sign changes in the sequence of coefficients. Descartes included it in his treatise La Geometrie which appeared in 1637. It can be proved by a method based on factorisation, but, again, just as easily by deduction from Rolle’s theorem.
69 citations
TL;DR: In this article, the authors consider the cubic equation and describe how the French mathematicians Francois Viete (1540-1603) and Rene Descartes (1596-1650) related the three-real-roots case (casus irreducibilis) to circle geometry.
Abstract: An appreciation of the geometry underlying algebraic techniques invariably enhances understanding, and this is particularly true with regard to polynomials. With visualisation as our theme, this article considers the cubic equation and describes how the French mathematicians Francois Viete (1540–1603) and Rene Descartes (1596–1650) related the ‘three-real-roots’ case (casus irreducibilis) to circle geometry. In particular, attention is focused on a previously undescribed aspect, namely, how the lengths of the chords constructed by Viete and Descartes in this setting relate geometrically to the curve of the cubic itself.
55 citations
TL;DR: In this article, the authors introduced the generating function g (t, z) for polynomials in the form of the following formal series (for the theory of generating functions, see for example [2]).
Abstract: Pn (t) (of degree ri) has exactly n simple zeroes which are all real and lie in the interval [0,1]. It is seen that, excepting 0 and 1, all other zeroes of P„ (t) (n > 3) interlace corresponding zeroes of the polynomial Pn _ i (f). We introduce the generating function g (t, z) for our polynomials in the form of the following formal series (for the theory of generating functions, see for example [2])
15 citations
TL;DR: In this article, the concept of symmetry with a linoprint of Blake's tyger was introduced and a sample page from Longman's Modern Mathematical Series, which was not a commercial success but was seminal importance in developing the school mathematics of the last century.
Abstract: Oliver Byrne in 1847, showing the Pythagorean dissection illustrated in red, yellow, blue and black. The blind Lucasian Professor, Nicholas Saunderson, is shown holding an spherical frame in the frontispiece of his treatise on algebra. On the booklet's cover there is a net of a cuboctahedron which was produced in fold-out form in 1765. From 1913, we see both the full list and a sample page from Longman's Modern Mathematical Series, which was not a commercial success but was of seminal importance in developing the school mathematics of the last century. A pioneering SMP textbook of 1964 illustrates the concept of symmetry with a linoprint of Blake's tyger. And, to bring us not quite up to date, there is Neill and Quadling's text for the OCR modular AS level.
8 citations
TL;DR: In this article, the Hero's formula was used to obtain the area of the triangles XCY, YAZ, ZBX and XYZ, and then the areas of the triangle XYZ were added to the triangle YCY to obtain a hexagon.
Abstract: Proof. Let X, Y, Z be the reflections of P in BC, CA, AB respectively. Then, the triangles XCY, YAZ and ZBX are each isosceles with an angle of 120°. Also, the hexagon AZBXCY has area double that of ABC (because LAZB = AAPB, etc.), which is xV3/4. The triangle XYZ has sides a\\/3, b\\Jl>, c\\fi and therefore using Hero's formula, has area 3\\Js (s a)(s b)(s c). Finally, by adding the areas of XCY, YAZ, ZBX and XYZ to obtain the hexagon, we get
7 citations
TL;DR: In this paper, the Van Aubel theorem was extended to the case of four similar rhombi and four similar rectangles on the sides of a quadrilateral, and it was shown that these two results are in fact both special cases of a further extension of the original theorem.
Abstract: A theorem of Van Aubel, which first appeared in concerns the figure obtained by erecting squares on the sides of a quadrilateral, and considering various line segments such as the joins of the centres of opposite squares. In parts of the theorem are extended to the case of (i) four similar rhombi, and (ii) four similar rectangles erected on the sides of a quadrilateral. Further, these two results are described as in some sense ‘dual’ to each other. Here we show that they are in fact both special cases of a further extension of the original theorem (see Proposition 6, and the remarks that follow it), and we also find several extra squares in the original Van Aubel figure (Proposition 11).
7 citations
TL;DR: Tem Temperley as discussed by the authors derived recurrence relations for the odd terms, u n = F 2 n + 1, n ≥ 0, in the sequence of Fibonacci numbers, defined by
Abstract: Dedicated to H. N. V. Temperley on the occasion of his ninetieth birthday 4 March 2005 The authors of the recent Note [1] exhibit an odd preference. They derive recurrence relations for the odd terms, u n = F 2 n +1, n ≥ 0, in the sequence of Fibonacci numbers, F n , defined by
7 citations
TL;DR: Stopes-Roe as mentioned in this paper describes the early twenties of the last century, when there were seven postal deliveries each day and post offices were thick on the ground, and it is skilfully and unobtrusively narrated by the author.
Abstract: (p. 69), Barnes actually divides u + du by v + <5v directly. This must have taxed v Molly's algebra. There is an error of sign in the remainder which doesn't disturb the argument. Elsewhere there are a couple of typos; it seems churlish even to mention them. This is a lovely, charming book. It is skilfully and unobtrusively narrated by Mary Stopes-Roe. It affords a vivid snapshot of the early twenties of the last century. This was a time when there were seven postal deliveries each day and post offices were thick on the ground. As for nuances of language, states of affairs are 'awfully' this or that and 'ripping' means roughly what has become the ubiquitous 'absolutely fantastic'. With the exception of Bloxham, everyone was delighted with the engagement of the couple on Molly's 20th birthday. In a leather bound notebook, Barnes headed a list of significant dates in their relationship with the motto 'IT'S DOGGED AS DOES IT'. It must be very rare that mathematics has featured significantly in a courtship but, for his efforts here alone, Barnes deserved to win his heart's desire.
6 citations
TL;DR: This book is meant to be an object-lesson in the difficulty of talking to a computer, but it ends up being a bit tedious and better covered by Maple's own documentation.
Abstract: syntactical points. This is meant to be an object-lesson in the difficulty of talking to a computer, but it ends up being a bit tedious and better covered by Maple's own documentation. Likewise the Java section is part physics text, part introduction to programming, part introduction to object-oriented programming and part introduction to numerical methods. It's all too much for one book and this chimera ends up being clumsy and inadequate for all of its roles. The very last part of the book is a 28-page introduction to L A T E X . I can only ask why? Anyone seriously intending to use the product will buy a proper book on it or get online help. L A T E X cannot be done justice in 28 pages so I see no reason to put it there at all.
5 citations
TL;DR: In this paper, Santos et al. discuss the trisection problem in the context of basic algebra, and propose a solution to the problem of irreducibility theorem of A. O'Olyzko.
Abstract: References 1. N. Jacobson, Basic algebra I (2nd edn), W. H. Freeman (1985). 2. I. Stewart, Galois theory, Chapman and Hall (1973). 3. R. C. Yates, The trisection problem, The National Council of Teachers of Mathematics (1971). 4. J. Brillhart, M. Filaseta, and A. Odlyzko, On an irreducibility theorem of A. Cohn, Can. J. Math. 33 (1981) pp. 1055-1059. JOSE CARLOS de SOUSA OLIVEIRA SANTOS Departamento de Matemdtica Pura, Faculdade de Ciencias, Rua do Campo Alegre 687, Portugal
TL;DR: In this article, the moments of the parent distribution and the geometric mean of the sample geometric mean are compared to obtain a general description of how the distribution from which a sample is drawn will affect the distribution of its geometric mean and how this will vary with sample size.
Abstract: Although a number of earlier researchers had used the geometric mean as a convenient statistic to summarise observational data, Gallon is usually credited with being the first to consider its sampling distribution. At Gallon’s request, in 1879 McAlister undertook a pioneering mathematical study, which eventually led to the modern large-sample theory. However, some sixty years elapsed before much attention was paid to small samples from particular parent distributions. Since about 1960, new techniques have made it possible to derive exact sampling distributions for a much wider class of parent distributions. Some work has been done on producing approximate general relationships between the moments of the parent distribution and those of the sample geometric mean but they are of very limited value for small samples and even now it is difficult to find any general description of how the distribution from which a sample is drawn will affect the distribution of its geometric mean and how this will vary with sample size.
TL;DR: Koshy et al. as mentioned in this paper studied the family of subsets of a finite set and showed that the family can be expressed as a set of classes of finite sets, where the courses of mathematical analysis can be divided into four classes.
Abstract: References 1. T. Koshy, On the family of subsets of a finite set, Math. Gaz. 88 (March 2004) pp. 118-119. 2. D. E. Knuth, Fundamental algorithms (2nd edn.), Addison Wesley (1973) p. 70. 3. A. G. Howson, A history of mathematics education in England, Cambridge University Press (1982) pp. 265-266. 4. E. T. Whittaker & G. N. Watson, A course of mathematical analysis (4th edn.), Cambridge University Press (1927). NICK LORD Tonbridge School, Kent TN9 UP
TL;DR: In this article, it was shown that the behaviors of odd and even numbers m were quite different and that the odd number s is Lucasian while the even number m is not Lucasian.
Abstract: In [1, Chapter 3, Section 2], we collected together results we had previously obtained relating to the question of which positive integers m were Lucasian, that is, factors of some Lucas number L n. We pointed out that the behaviors of odd and even numbers m were quite different. Thus, for example, 2 and 4 are both Lucasian but 8 is not; for the sequence of residue classes mod 8 of the Lucas numbers, n ⩾ 0, reads and thus does not contain the residue class 0*. On the other hand, it is a striking fact that, if the odd number s is Lucasian, then so are all of its positive powers.
TL;DR: In this article, it was shown that the Hooke's limit can be broken by using a string with longer natural length and a correspondingly smaller value of m in order to preserve the specified value for /.
Abstract: This is the standard equation for three-dimensional simple harmonic motion with O as its attractive centre. It is known that the corresponding general form of the orbit of m will be an ellipse with O as its centre and with detailed parameters determined by the initial position and velocity of m (see, for example, [1]). Further, it follows from (4), that whatever the values of these parameters, the corresponding period for traversing the orbit will be given by equation (1), where now I = OQ. It follows therefore from the above analysis that the system we have described can be used in a similar way to the usual pendulum as a device for measuring time, but with the advantage that the initial displacement and velocity given to m can be 'large', as long as the limit for the application of Hooke's law is not exceeded. In practice it may well be that the simplest form of motion for m results from initially displacing it horizontally from O and then releasing it from rest. This will result in straight-line horizontal S.H.M., rather similar to the motion in the case of a simple pendulum but now, of course, without any corresponding constraint on the amplitude. Finally we make the point that, if, for specified values of / and amplitude, the 'Hooke's limit' is exceeded, the situation can be remedied by using a string with longer natural length and a correspondingly smaller value of m in order to preserve the specified value for /.
TL;DR: In this paper, it was shown that the coefficients of Jacobsthal and Fibonacci polynomials are the same as those of Fn (x), but in the reverse order.
Abstract: Both Jacobsthal and Fibonacci polynomials are generalisations of Fibonacci numbers, and they satisfy a number of elegant properties [1]. Interestingly, there is a close relationship between Jacobsthal and Fibonacci polynomials, which allows us to derive additional properties of the former from the latter. Notice that the coefficients of Jn (x) are the same as those of Fn (x), but in the reverse order. This suggests a relationship between the two polynomials, given explicitly in Theorem 1. We shall establish it by induction.
TL;DR: In this paper, Bender and Orszag present a mathematical method for scientists and engineers using Fourier analysis and generalised functions, which is based on the work of Lighthill et al.
Abstract: References 1. P. Dennery and A. Krzywicki, Mathematics for physicists, Harper (1969). 2. M. J. Lighthill, Introduction to Fourier analysis and generalised functions, Cambridge University Press (1962). 3. C. M. Bender and S. A. Orszag, Advanced mathematical methods for scientists and engineers, McGraw-Hill (1978). STUART SIMONS School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London El 4NS
TL;DR: A number of theorems about hexagons whose three pairs of opposite sides are parallel are known in the late nineteenth century or early twentieth century, as well as Tucker circles.
Abstract: This paper presents a number of theorems about hexagons whose three pairs of opposite sides are parallel. The first of these is a well-known result that the vertices of such a hexagon lie on a conic. Theorems 3 and 4 show how such conies are related to the Cevians of a triangle, and which Cevians lead to such conies being circles. When they are circles they are called Tucker circles. None of the results is at all obvious, yet it seems that some of the results presented here were known in the late nineteenth century or the early twentieth century. They seem to be of more than passing interest which should not get lost by the passage of time.
TL;DR: A handbook of integer sequences, available on the internet or CD ROM or book as discussed by the authors, can be used as a good starting point for a discussion of the secret Santa problem.
Abstract: References 1. A. C. Robin, The wife-swapping distribution, Math. Gaz. 69 (October 1985) pp. 175-178. K. M. McQuire, G. Mackiw and C. H. Morrell, The Secret Santa problem, Math. Gaz. 83 (November 1999) pp. 467-472. R. Pinkham, The Secret Santa problem revisited, Math. Gaz. 85 (March 2001) pp. 96-97. M. Griffiths, The 'Self Santa' problem, Math. Gaz. 86 (November 2002) pp. 487-489. N. Sloane, A handbook of integer sequences, available on the internet or CD ROM or book. 6. E. Lucas, Theorie des Nombres, Gaulthier-Villars, Paris (1891). 7. Takacs, The probleme des menages, Discrete Mathematics 36 (1981) pp. 289-297. A. Barbour & S. Janson, Poisson approximation, Oxford University Press (1992). ANTHONY C. ROBIN 29 Spring Lane, Eight Ash Green, Colchester C06 3QF
TL;DR: The author does not have access to the NAG library and so, when he was asked recently to calculate the value of the integral correct to 10 decimal places his first reaction was to try several different calculators as well as several mathematical software packages.
Abstract: In common with, I suspect, many people the author does not have access to the NAG library and so, when I was asked recently to calculate the value of the integral correct to 10 decimal places my first reaction was to try several different calculators as well as several mathematical software packages. On doing so it was disappointing to find they either gave widely differing values such as 7.9065200767, 4.1317217452 or 0.9174196842 or an error message indicating that the method had not converged.
TL;DR: The fourth chapter is unsuitable for a general readership as mentioned in this paper and is built around the Taussky Todd Celebration in 1995, a conference commemorating her life which reproduced papers on her own work, and also papers outlining the research of her successors.
Abstract: The fourth chapter is unsuitable for a general readership. It is built round the Taussky Todd Celebration in 1995—a conference commemorating her life which reproduced papers on her own work, and also papers outlining the research of her successors. Since these women were working at the cutting edge in their field, it is to be expected that the mathematics will only be appreciated by people with some level of specialist knowledge.
TL;DR: This chapter works simple examples of the travelling salesman and knapsack problems by branch and bound, but again there is no hint of the fact that these are really challenging problems, both mathematically and computationally.
Abstract: or of the computational difficulty of integer programming, important though these are in practice. Students could well get the impression that integer programming is no harder than linear programming: you just use branch and bound instead of simplex, and out comes the answer. The same chapter works simple examples of the travelling salesman and knapsack problems by branch and bound, but again there is no hint of the fact that these are really challenging problems, both mathematically and computationally.
TL;DR: In this paper, the authors used modulo checks for small primes, utilizing Fermat's little theorem, cf = a mod/?, to show that there are no more powerful numbers other than those listed earlier.
Abstract: then L < N < U and so in each case (a, = 0 to 9 for i = 2 to 10) the possibilities for a,, i = 11 to 19, can be found and they will be few in number. Then N can be tested to see if N = f (TV). This approach makes a computer search for each n < 22 feasible. Within the computer searches, since the size of the numbers being dealt with was large, modulo checks for small primes were used, utilizing Fermat's little theorem, cf = a mod/?. These computer searches showed that there were no more powerful numbers other than those listed earlier. So we have proved:
TL;DR: The authors argue that science is at least as much about explanation as it is about prediction, and that theory can make fundamental contributions to biology just as it has to physics, but they do not discuss the role of theory in biology.
Abstract: Those of us who, like Murray, have been in the field long enough to remember some of what used to pass for mathematical biology, will understand why he feels it necessary to insist that we tackle real problems and look for results that actually say something about the real world and can be tested. All the same, my own view is that science is at least as much about explanation as it is about prediction and that theory can make fundamental contributions to biology just as it has to physics.
TL;DR: In this article, the authors considered the infinite series formed by taking the sum of the reciprocals of the pyramidal numbers with alternating signs, i.e. 1, 5, 14, 30, 55, 91, 140, 204, 285.
Abstract: 90.67 A series for the 'bit' This note deals with yet another series for jr. It involves the reciprocals of square pyramidal numbers and Gregory's series for ill A. A square pyramidal number is a number that corresponds to a configuration of points which form a pyramid with a square base. The nth pyramidal number is the sum of the squares of the first n positive integers. So the formula for the nth pyramidal number is n(n + l)(2n + 1) 6 ' The first few pyramidal numbers are 1, 5, 14, 30, 55, 91, 140, 204, 285. Let us now consider the infinite series formed by taking the sum of the reciprocals of the pyramidal numbers with alternating signs, i.e. 1_ J 1_ J 1_ J 1_ 1 5 + 14 ~ 30 + 55 91 + 140 ~ 204 + 285 \" ' \" '
TL;DR: In this article, the problem of finding defining sets (preferably smallest defining sets) for various combinatorial structures such as latin squares and block designs was investigated, and it seemed interesting to try to solve the same problem for magic squares of small size.
Abstract: A topic of current interest is that of finding defining sets (preferably smallest defining sets) for various combinatorial structures such as latin squares and block designs, so it seemed interesting (though of no practical use) to try to solve the same problem for magic squares of small size. (The germ of this idea was suggested in the Gazette Note [1] of 1988 under the sub-heading ‘Squares from clues’.)