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An Interesting Construction Problem
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| Pengarang | : | Russell A. Gordon |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 3) |
| Halaman | : | 207-221 |
| Abstrak | : | We present a collection of simple construction problems and determine conditions for which the construction can be carried out with compass and straightedge. Since the problems reduce to finding the roots of quartic polynomials, it is not sufficient to show that the polynomials are irreducible to conclude that the roots are not constructible. Some interesting results appear as we discover general conditions that determine when the solutions are or are not constructible. |
The Worpitzky Numbers Revisited
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 3) |
| Halaman | : | 198-206 |
| Abstrak | : | We relate how the Worpitzky number triangle emerges in the setting of a novel counting problem that involves placing up- and right-arrows in a rectangular grid in a polite manner. We next survey various properties, relying on a mixture of combinatorial argument and the functional equation for Worpitzky polynomials. We conclude with a short proof of the Von Staudt–Clausen theorem, a context in which Worpitzky numbers arose. |
On Certain Dense, Uncountable Subsets of the Real Line
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 4) |
| Halaman | : | 339-346 |
| Abstrak | : | It is a challenge to define an uncountable set of real numbers that is dense in the real line and whose complement is also uncountable and dense. In this article we specify, for any positive integer k, a partition of the real line into k + 1 sets that are uncountable and dense; moreover, every real number is a condensation point for each of these sets. We show that one of these sets has measure one when restricted to the unit interval, while each of the others has measure zero. We then define a function that is continuous on the set of measure 1 but discontinuous elsewhere. |
A Decoding Approach to Reed–Solomon Codes from Their Definition
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 4) |
| Halaman | : | 320-338 |
| Abstrak | : | Because of their importance in applications and their quite simple definition, Reed–Solomon codes can be explained in any introductory course on coding theory. However, decoding algorithms for Reed–Solomon codes are far from being simple and it is difficult to fit them in introductory courses for undergraduates. We introduce a new decoding approach, in a self-contained presentation, which we think may be appropriate for introducing error correction of Reed–Solomon codes to nonexperts. In particular, we interpret Reed–Solomon codes by means of the degree of the interpolation polynomial of the code words and from this derive a decoding algorithm. Compared to the classical algorithms, our algorithm appears to arise more naturally from definitions and to be easier to understand. It is related to the Peterson–Gorenstein–Zierler algorithm. |
The Mensa Correctional Institute
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 4) |
| Halaman | : | 306-319 |
| Abstrak | : | We investigate the following puzzle and several variations, some of which have quite surprising answers. Alice and Bob are in prison under the care of warden Charlie. Alice will be brought into Charlie’s office and shown 52 cards, face-up in a row in an arbitrary order. Alice can interchange two cards. Charlie then turns all cards face-down in their places and Alice leaves the room. Then Bob is brought in and Charlie calls out a random target card. Bob can turn over cards, one after another, at most 26 times as he searches for the target. Both prisoners are freed if Bob finds the target. Find a strategy that never fails. |
Seeing Curvature on Specular Surfaces
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 4) |
| Halaman | : | 291-305 |
| Abstrak | : | Reflections on a specular surface distort their surrounding environments in ways that give visual clues about the surface's principal and Gaussian curvatures. This article mathematically examines this phenomenon by introducing a calibrated environment and using standard photography to record the distortions of that environment on the specular surface. We provide a novel approach to this examination which ultimately succeeds in demonstrating the availability of this curvature information in the reflections. |
On One Type of Generalized Vandermonde Determinants
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 5) |
| Halaman | : | 433-442 |
| Abstrak | : | An expression including symmetric polynomials for the determinant of one type of generalized Vandermonde matrix is presented leading to a handy formula to compute determinants of general Toeplitz band matrices. |
A Visual Model of the Lorentz Triangular Moduli Space
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 5) |
| Halaman | : | 426-432 |
| Abstrak | : | We present a schematic construction of the triangularly-shaped space in the Lorentz plane, i.e., the moduli space of all Lorentzian triangles up to similarity. This space turns out to be homeomorphic to a band where the different types of triangles may be located according to the length and the causal character of their sides. In particular, several kinds of equilateral and isosceles triangles are identified. |
Gamma and Factorial in the Monthly
| Pengarang | : | Wirojoedo, Soebijanto |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 5) |
| Halaman | : | 400-424 |
| Abstrak | : | Since its inception in 1894, the Monthly has printed 50 articles on the Γ function or Stirling's asymptotic formula, including the magisterial 1959 paper by Phillip J. Davis, which won the 1963 Chauvenet prize, and the eye-opening 2000 paper by the Fields medalist Manjul Bhargava. In this article, we look back and comment on what has been said, and why, and try to guess what will be said about the Γ function in future Monthly issues. 1 It is a safe bet that there will be more proofs of Stirling's formula, for instance. We also identify some gaps, which surprised us: phase plots, Riemann surfaces, and the functional inverse of Γ make their first appearance in the Monthly here. We also give a new elementary treatment of the asymptotics of n! and the first few terms of a new asymptotic formula for invΓ. |
Maximally Nontransitive Dice
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 5) |
| Halaman | : | 387-399 |
| Abstrak | : | We construct arbitrarily large sets of dice with some remarkable nontransitivity properties. In a sense made precise later, each set exhibits all possible pairwise win/loss relationships by summing different numbers of rolls. The proof of this fact relies on an asymptotic formula for the difference between the median and mean of sums of multiple rolls of dice. This formula is a consequence of a suitable Edgeworth series (an asymptotic refinement of the central limit theorem), for which we give a detailed sketch of a proof in the final section. |