Showing papers in "The Mathematical Gazette in 1996"
71 citations
TL;DR: Banach algebras and the general theory of algebraic structures have been studied extensively in the literature, see as discussed by the authors for an account of some of the most important results.
Abstract: banach algebras profhugodegaris, eudml banach algebra techniques in the theory of, banach algebras and the general theory of algebras, general theory of banach algebras book website of veuqayo, banach algebras and the general theory of algebras, banach algebras and the general theory of algebras, banach algebra encyclopedia of mathematics, newest banach algebras questions mathoverflow, banach algebra wikipedia, banach algebra structure and amenability of a class of, banach algebras and the general theory of algebras vol, theory of operator algebras i m takesaki springer, stable perturbation in banach algebras journal of the, banach algebras and the general theory of algebras, banach algebras and the general theory of algebras, banach algebras and the general theory of algebras by, annihilators in universal algebras a new approach, rickart general theory of banach algebras 1960 citeseerx, banach algebras and automatic continuity h garth dales, banach algebras generated by n idempotents and, banach algebras and the general theory of algebras, general theory of banach algebras charles e rickart van, banach spaces volume 1 1st edition, palmer theodore w banach algebras and the general theory, introduction to banach algebras and the gelfand naimark, banach algebra in nlab, some examples in c algebras and banach algebras, banach algebras and the general theory of algebras, yood review charles e rickart general theory of, banach algebras and the general theory of algebras, derivations from banach algebras sciencedirect, 0521366372 banach algebras and the general theory of, banach algebras and the general theory of algebras, banach algebras and compact operators volume 2 1st edition, a characterization of jordan homomorphism on banach algebras, fundamentals of the theory of operator algebras volume i, palmer theodore w banach algebras and the general theory, banach algebras and the general theory of algebras vol, k theory for frchet algebras international journal of, banach algebras and the general theory of algebras vol, banach algebra techniques in operator theory ronald g, banach algebras and c algebras springerlink, banach algebras and the general theory of algebras vol, math5615 banach algebras, de gruyter proceedings in mathematics ser banachbanach theory of linear operations unfree banach theory of the integral unfree cembranos mendoza banach spaces of vector valued functions unfree constantinescu c algebras vol 1 banach spaces unfree dales et al introduction to banach algebras operators and harmonic analysis unfree, classical banach spaces in the general theory commutative banach algebras and commutative topological algebras 46j99 none of the above but in this section multiplicative number theory 11n37 asymptotic results on arithmetic functions 11n56 rate of growth of arithmetic functions inversion theorems 40e10 growth estimates, this is the first volume of a two volume set that provides a modern account of basic banach algebra theory including all known results on general banach algebras this account emphasizes the role of algebraic structure and explores the algebraic results that underlie the theory of banach algebras and algebras, book information and reviews for isbn 9780521366373 banach algebras and the general theory of algebras volume 1 algebras and banach algebras encyclopedia of banach algebras and the general theory of algebras volume 2 rickart on amazon com e
33 citations
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TL;DR: The human body is a complicated machine whose movements involve many different joints, operated by a great many muscles as mentioned in this paper, and it is easy to get bogged down in detail when thinking about walking and running from a mathematical point of view.
Abstract: The human body is a complicated machine whose movements involve many different joints, operated by a great many muscles. For that reason it is easy to get bogged down in detail when thinking about walking and running from a mathematical point of view. Any position of the human body (or of any other jointed mechanism) can be described by giving the angles of joints. The number of angles needed for an unambiguous description is the number of degrees of freedom of the mechanism. For example, the position of a hinge joint is described by just one angle: a hinge allows only one degree of freedom. The human knee is a hinge. The ankle, however, allows rotation about two axes – you can tilt your foot toes up or toes down, and you can also rock it sideways so that the sole faces inwards towards the other foot – so it gives two degrees of freedom. The hip is a ball and socket joint allowing rotation about any axis through the centre of the ball, but any position can be described by just three angles (measured, for example, in three planes at right angles to each other), so it allows three degrees of freedom. In total, there are six degrees of freedom in each leg, making twelve in all, and suggesting that we need twelve equations of motion to describe walking. If we took account of the flexibility of the foot and the movements of the arms, we would need more.
21 citations
18 citations
TL;DR: The condition for the quadratic ax 2+bx+c to have two equal roots, i.e. to have a repeated root, is that its discriminant ∂2 = b2−4ac should be zero.
Abstract: EVERYONE knows that the condition for the quadratic ax 2+bx+c to have two equal roots, i.e. to have a repeated root, is that its discriminant ∂2 = b2−4ac should be zero. We should remark, at the outset, that we are concerned only with ordinary polynomials whose coefficients are complex numbers. Indeed, little is lost if a reader assumes that all our polynomials are real, i.e. have real numbers for all their coefficients, though their complex roots must be considered as well as their real ones. Though less at one’s finger-tips nowadays, it has been known since the sixteenth century that a cubic ax3 +3bx2 +3cx+d has a repeated root, i.e. two or three equal roots, if and only if ∂3 is zero, where ∂3 = G2 +4H3
17 citations
TL;DR: For example, the Machin's Formula as mentioned in this paper approximates the value of π using Gregory's series, which converges for values of tan θ between -1 and +1.1.
Abstract: 1. Background From the end of the 17th century until the 1980s, most approximations to the value of π made use of Gregory’s series: which converges for values of tan θ between -1 and +1. Since , the simplest, and most slowly convergent, series of this kind is Writing x = cot θ , the series becomes Expressions for π /4 have been devised which involve the inverse cotangents of larger numbers, causing the series to converge much faster; of such expressions (referred to herein as ‘identities’) probably the most celebrated is ‘Machin’s Formula’:
17 citations
15 citations
TL;DR: In this article, the Brocard points P and P' of the triangle ABC were established in terms of the lengths of the sides of the triangular triangle, and the solution given by various solvers was essentially the same as appeared in a paper by H. E. Piggott in the Gazette in 1924.
Abstract: The Brocard points P and P' of the triangle ABC , illustrated in Figure 1, are such that ∠ PAB = ∠ PBC = ∠ PCA = ω and ∠ P'BA = ∠P'CB = ∠P'AC = ω. Problem 78.A in the March 1994 edition of the Mathematical Gazette required a proof that the maximum length of PP' is R/2 , where R is the circumradius of triangle ABC . The solution given by various solvers in the November 1994 edition of the Gazette is essentially the same as appeared in a paper by H. E. Piggott in the Gazette in 1924. In the present paper various properties of the triangle and its Brocard points are established in terms of the lengths of the sides of the triangle. Some of the results may be new, but the methods used encompass ‘Mathematics Ancient and Modern’ in the guises of homogeneous areal coordinates and computer algebra!
12 citations
TL;DR: In the March 1996 issue of the Gazette, [p. 36] the following conjecture by Sir Bryan Thwaites appeared: ‘Take any set of n rational numbers. Iterate.
Abstract: In the March 1996 issue of the Gazette, [p. 36] the following conjecture by Sir Bryan Thwaites appeared: ‘Take any set of n rational numbers. Form another set by taking the positive differences of successive members of the first set, the last such difference being formed from the last and first members of the original set. Iterate. Then in due course the set so formed will consist entirely of zeroes if and only if n is a power of two.’
11 citations
9 citations
TL;DR: A reception hosted by the Cambridge University Press celebrated the fiftieth anniversary of the initiation of Godfrey and Siddons' first book: Elementary Geometry as discussed by the authors, which marked a turning point in the teaching of mathematics, and set a standard for mathematical education which remains relevant as we approach the end of the century.
Abstract: In 1952 I was invited, with teachers of mathematics from many other schools, to a reception hosted by the Cambridge University Press. The guest of honour was A. W. Siddons, and the purpose was to celebrate the fiftieth anniversary of the initiation of Godfrey and Siddons’ first book: Elementary Geometry. There can be few authors who survive to enjoy such an occasion; fewer still whose book remains in print and continues to sell well after fifty years. But this was a book which marked a turning-point in the teaching of mathematics, and set a standard for mathematical education which remains relevant as we approach the end of the century.
TL;DR: The Simson line property is associated with points on the circumcircle of a triangle as mentioned in this paper, and it is embodied by the following theorem: given any triangle ABC and a point P in the plane of the triangle, if perpendiculars from P on to the sides BC, CA, AB meet those sides at L, M, N respectively then L, M, N are collinear if and only if P lies on the circle of triangle ABC.
Abstract: The Simson line property is normally associated with points on the circumcircle of a triangle. It is embodied by the following theorem. Given any triangle ABC and a point P in the plane of the triangle, if perpendiculars from P on to the sides BC , CA , AB meet those sides at L , M , N respectively then L , M , N are collinear if and only if P lies on the circumcircle of triangle ABC . The line LMN is then known as the Simson line of P .
TL;DR: In this article, the authors use the term Ceva-type to describe any result of this general kind: one that specifies a configuration in affine space of n dimensions, defined only by incidences, about which one can make an assertion about a product of ratios of lengths, areas, etc.
Abstract: The classical theorems of Ceva and Menelaus make assertions about the value of certain products of ratios of lengths in configurations in the affine plane. We shall use the term Ceva-type to describe any result of this general kind: one that specifies a configuration in affine space of n dimensions, defined only by incidences, about which one can make an assertion about a product of ratios of lengths, areas, etc. Several results of this kind are known. Apart from the classical results there are, for example, Ceva's and Menelaus' Theorems for n -gons, Hoehn's Theorem for pentagrams [1], and the Selftransitivity Theorem of [2].
TL;DR: In the preface to the first edition (1897) of his book on the theory of finite groups, William Burnside wrote: ‘The subject is one which has hitherto attracted but little attention in this country; it will afford me much satisfaction if, by means of this book, I shall succeed in arousing interest among English mathematicians in a branch of pure mathematics which becomes the more fascinating the more it is studied as mentioned in this paper.
Abstract: In the preface to the first edition (1897) of his book [1] on the theory of finite groups, William Burnside wrote: ‘The subject is one which has hitherto attracted but little attention in this country; it will afford me much satisfaction if, by means of this book, I shall succeed in arousing interest among English mathematicians in a branch of pure mathematics which becomes the more fascinating the more it is studied.’ He returned to this point in his presidential address delivered to the London Mathematical Society on 12 November 1908.
TL;DR: In the Mathematical Gazette as discussed by the authors, an article with the above heading appeared shortly after the centenary of the birth of Charles Lutwidge Dodgson, better known and admired universally as the author of the Alice books, Lewis Carroll.
Abstract: Over 60 years ago an article with the above heading appeared in the Mathematical Gazette , shortly after the celebration in 1932 of the centenary of the birth of Charles Lutwidge Dodgson, better known and admired universally as the author of the Alice books, Lewis Carroll. At that time a literary critic expressed his opinion that ‘only a mathematician could have written the Alice books’, perhaps because they contain several references to arithmetic. In Wonderland, Alice met the irascible Ugly Duchess. ‘If everybody minded their own business’, the Duchess said in a hoarse growl, ‘the world would go round a great deal faster than it does’.
TL;DR: For example, in the Sunday supplement of the Spanish newspaper El Pais, the following quiz was posed: Check that it does not matter how you take any four numbers from the following array; provided that no two of them lie in the same row or column, their sum is always the same.
Abstract: Not long ago I came across the following quiz in the Sunday supplement of the Spanish newspaper El Pais: Check that it does not matter how you take any four numbers from the following array; provided that no two of them lie in the same row or column, their sum is always the same. This reminded me of magic squares: square matrices where the sum of the elements in each line (row or column) and diagonal is always the same. Magic squares are a topic in recreational mathematics. Usually one looks for magic squares with elements that are integer, positive and different from each other, e.g. 1,2,…, n 2 for a n × n as in According to a Chinese legend, matrix A appeared engraved on the carapace of a turtle about 2000 BC. On the other hand, matrix B appears in Albrecht Durer’s engraving Melencholia (1514) [1].
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TL;DR: In his 1982 Presidential Address to the Mathematical Association I tried to explain the general role of Geometry in Mathematics so it seems appropriate, on this centenary occasion, that I should move beyond the confines of Mathematics and discuss the interrelation of geometry and physics as mentioned in this paper.
Abstract: In my 1982 Presidential Address to the Mathematical Association I tried to explain the general role of Geometry in Mathematics so it seems appropriate, on this centenary occasion, that I should move beyond the confines of Mathematics and discuss the interrelation of Geometry and Physics. There are two very good reasons for doing this. One is historical and arises from the close ties between the two subjects in their early evolution. A second and more topical reason is that, over the past two decades, there has been a remarkable burst of interaction of a quite unexpected kind between Geometry and Physics.
TL;DR: Since there is far more mathematics even quite elementary mathematics than anyone can carry in their working memory, it must follow that results which are unused will be forgotten, and are then often rediscovered after many years as mentioned in this paper.
Abstract: Since there is far more mathematics even quite elementary mathematics than anyone can carry in their working memory, it must follow that results which are unused will be forgotten, and are then often rediscovered after many years. A tour of some old haunts which are now being opened up again would therefore seem not out of place in a centenary number of the Gazette. Tourists, however, will need some preliminary briefing if they are to enjoy the tour fully.
TL;DR: This article pointed out that the repeated putting into question of mathematicians' arguments at length led some people to think these arguments contained a hidden flaw which meant that a proof could never be found unassailable by any criticism.
Abstract: Mathematicians of every age have been seen to criticize the proofs of their predecessors or of their contemporaries as ‘not being rigorous’; and often those which they proposed as replacements for the defective proofs were in their turn considered inadequate by the succeeding generation. This apparent repeated putting into question of mathematicians' arguments at length led some people to think these arguments contained a hidden flaw which meant that a proof could never be found unassailable by any criticism.