Showing papers in "The Mathematical Intelligencer in 1998"
TL;DR: Arnabels invitation is inspired in part by Hilbert's list of 1900 (see e.g. [Browder, 1976]) and I have used that list to help design this essay.
Abstract: V. I. Arnold, on behalf of the International Mathematical Union has written to a number of mathematicians with a suggestion that they describe some great problems for the next century. This report is my response. Arnold's invitation is inspired in part by Hilbert's list of 1900 (see e.g. [Browder, 1976]) and I have used that list to help design this essay. I have listed 18 problems, chosen with these criteria:
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TL;DR: The survey art icle [Thil] contains fur ther detai ls on the work of the n ine teenth-century tabula tors, but, above all, the r eade r is encouraged to consul t the original sources, in par t icu la r the excel lent ser ies of pape r s by Tait [Tail.
Abstract: inc lude all pr ime knots wi th 16 or fewer crossings. This r epresen t s more than a 130-fold increase in the number of t abu la ted knots s ince the last burs t of tabula t ion tha t t ook p lace in the early 1980s. With more than 1.7 mil l ion knots now in the tables, we hope that the census will serve as a r ich source of examples and coun te rexamples and as a genera l test ing ground for our collective intuition. To this end, we have wri t ten a UNIX-based compute r p rog ram cal led KnotScape which a l lows easy access to the tables. The account of our me thodo logy is p re faced by a br ie f h is tory of knot tabulat ion, concentra t ing most ly on events taking p lace within the last 30 years. The survey art icle [Thil] contains fur ther detai ls on the work of the n ine teenth-century tabula tors , but, above all, the r eade r is encouraged to consul t the original sources, in par t icu la r the excel lent ser ies of pape r s by Tait [Tail. Ki rkman 's papers make fascinat ing reading, as they abound with original ideas and ornate l anguage -h i s definit ion of the te rm "knot' is a single sentence of 101 words. Conway's landmark p a p e r [Con] is also highly r ecommended . An impor tant feature of our pro jec t is that we have worked in two complete ly separa te teams, producing two tabula t ions which were kept secre t unti l af ter they were complete. 1 Al though it would be foolhardy to claim with absolute cer ta in ty that our tables a re correct , we must repor t the gratifying exper ience of fmding that our lists of 1,701,936 knots were in comple te agreement! Moreover, we did not use exac t ly the same methods , the pr imary difference being the use of hyperbol ic geomet ry by Hoste and Weeks and the comple te absence of hyperbol ic invariants in Thist lethwaite 's approach. Nevertheless, our overall programs are similar in spirit and differ little from the methods of most tabula tors who precede us. As par t of our tabulation, using Weeks 's program SnapPea we were able to compute the symmetry groups of the knots; we have included a short in t roduct ion to this beautiful and intriguing topic.
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TL;DR: In the 1970's, the scienti c community recognized another revolution, called the theory of chaos, which dealt a death blow to the Newtonian pic- ture as discussed by the authors.
Abstract: A mathematician discussing chaos is featured in the movie \"Jurassic Park\". James Gleich's book \"Chaos\" remains on the best seller list for many months. The characters of the celebrated Broadway play \"Arcadia\" of Tom Stoppard discourse on the meaning of chaos. What is chaos? Chaos is a new science which establishes the omnipresence of unpredictability as a fundamental feature of common experience. A belief in determinism, that the present state of the world determines the future precisely, dominated scienti c thinking for two centuries. This credo was based on certain laws of physics, Newton's equations of motion, which describe the trajectories in time of states of nature. These equations have the mathematical property that the initial condition determines the solution for all time. Thus lies the mathematical and physical foundation for deterministic philosophy. One manifestation of determinismwas the rejection of free will and hence even of human responsibility. At the beginning of this century, with the advent of quantum mechanics and the revelations of the German scientists, Heisenberg, Planck, and Schrodinger, the great delusion of determinism was exposed. At least on the level of electrons, protons, and atoms it was discovered that uncertainty prevailed. The equations of motion of quantum mechanics produce solutions which are probabilities evolving in time. In spite of quantum mechanics, Newton's equations govern the motion of a pendulum, the behaviour of the solar system, the evolution of the weather and many situations of everyday life. Therefore the quantum revolution left intact many deterministic dogmas. For example, well after the second world war, scientists held the belief that long range weather prediction would be successful when computer resources grew large enough. In the 1970's the scienti c community recognized another revolution, called the theory of chaos, which deals a death blow to the Newtonian pic-
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TL;DR: The first occurrence of any 10-digit sequence in pi could not be determined as mentioned in this paper, and even at the time that was written, it seemed well-nigh impossible that the first occurrence in any 10digit sequence could ever be determined.
Abstract: The occurrence of the sequence 0123456789 within pi. In his primer on Intuitionism, Heyting frequently relies on the occurrence or non-occurrence of the sequence 0123456789 in the decimal expansion of pi to highlight issues of classical versus intuitionistic (or constructivist) mathematics. At the time that Brouwer developed his theory. (1908) and even at the time that was written, it seemed well-nigh impossible that the first occurrence of any 10-digit sequence in pi could ever be determined.
TL;DR: In this paper, the authors compare the trianglesTi with the isosceles triangles which would be obtained by slicing the circle by the same procedure, and they show that the latter triangles cover all of R1 (perhaps even with overlap and/or extension beyond R1).
Abstract: Finally, we compare the trianglesTi with the isosceles triangles which would be obtained by slicing the circle by the same procedure. Since they all have the same short base length and the same angle opposite the base, the isosceles triangles composing the circle have more area than the trianglesTi. Because the latter triangles cover all ofR1 (perhaps even with overlap and/or extension beyondR1), we see that the area of the quarter-circle is greater than that ofR1, and thus the circle's entire area is greater than that of our original regionR.
TL;DR: In this article, it was shown that the global canonical height of an algebraic point on an elliptic curve arises as an entropy, and the Mahler measure should be lifted to the elliptic setting.
Abstract: Mahler’s measure is alive and well in several quite diverse contexts. The differing points of view seem to generate a healthy friction. If the general level of health is measured by the quantity and quality of unsolved problems, then it may help to list these.
1.
Lehmer’s Problem.
2.
The elliptic analogue of Lehmer, at least in tractable special cases.
3.
An explanation of Boyd’s remarkable formulae. It seems thatK-theory should provide the conceptual framework. More generally, perhaps values of the elliptic Mahler measure will arise as values of L-functions of higher-dimensional varieties.
4.
It looks almost certain that the elliptic Mahler measure should arise as an entropy. This would form a fascinating bridge between two large areas of interest. Ward and I have begun to write about this [10]. At the very least, this would show that the global canonical height of an algebraic point on an elliptic curve arises as an entropy. But of what, and what does this mean?
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There are many other pretty results about the classical Mahler measure which could be lifted to the elliptic setting.
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TL;DR: In this paper, a teacher of mathematics is asked to help her students to see mathematics from various vantage points, one of these should be from a point high enough to afford a full, sweeping view of the mathematical valley below, missing the details we strive to convey in class.
Abstract: I often ponder on my duties as a teacher of the subject I love. I feel I am responsible for more than simply transmitting knowledge. I wish I could help my students see mathematics from various vantage points. One of these should be from a point high enough to afford a full, sweeping view of the mathematical valley below—maybe missing the details we strive to convey in class-but seeing thelandscape of mathematics. Claude Bragdon said, “Mathematics is the handwriting on the human consciousness of the very Spirit of Life itself.” I want my students to consider that such a bold statement might actually be true.
TL;DR: The article with valuable information on the achievements of James Stirling during his time as mine manager in Scotland appeared in the Glasgow Herald of August 3, 1886 and was reprinted in Mitchell's The Old Glasgow Essays of 1905.
Abstract: directions so that others may follow in your tracks. A lthough James Stirling enjoyed a substantial reputation as a mathematician among his contemporaries in Britain and in some other European countries, he published remarkably little after his book Methodus Differentialis: sive Tractatus de Summatione et Interpolatione Serierum Infinitarum in 1730. An article with valuable information on the achievements of James Stirling during his time as mine manager in Scotland appeared in the Glasgow Herald of August 3, 1886, see [8]. The Herald article was reprinted in Mitchell’s The Old Glasgow Essays of 1905 [7]. Detailed accounts of Stirling’s mathematical achievements and correspondences with his contemporaries can be found in [12, 13], and [14].