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The Horizontal Chord Set
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 131 (No. 6) |
| Halaman | : | 519-525 |
| Abstrak | : | We study the set of lengths of the horizontal chords of a continuous function. We give a new proof of Hopf’s characterization of this set, and show that it implies that no matter which function we choose, at least half of the possible lengths occur. We prove several results about functions for which all the possible lengths occur. |
On the Tower Factorization of Integers
| Pengarang | : | Jean-Marie De Koninck |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 131 (No. 6) |
| Halaman | : | 511-518 |
| Abstrak | : | Under the fundamental theorem of arithmetic, any integer n > 1 can be uniquely written as a product of prime powers pa; factoring each exponent a as a product of prime powers qb, and so on, one will obtain what is called the tower factorization of n. Here, given an integer n > 1, we study its height h(n), that is, the number of “floors” in its tower factorization. In particular, given a fixed integer ????≥1, we provide a formula for the density of the set of integers n with h(n) = k. This allows us to estimate the number of floors that a positive integer will have on average. We also show that there exist arbitrarily long sequences of consecutive integers with arbitrarily large heights. |
A Worked out Galois Group for the Classroom
| Pengarang | : | Lorenz Halbeisen |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 131 (No. 6) |
| Halaman | : | 501-510 |
| Abstrak | : | Let ????=????6−3?????2−1∈?????[????] and let ???????? be the splitting field of f over ????. We show by hand that the Galois group Gal?(????????/????) of the Galois extension ????????/???? is isomorphic to the alternating group A4. Moreover, we show that the six roots of f correspond to the six edges of a tetrahedron and that the four roots of the polynomial ????4+18?????2−72?????+81 correspond to the four faces of a tetrahedron, which allows us to determine all eight proper intermediate fields of the extension ????????/????. |
Enumeration of Payphone Permutations
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 131 (No. 6) |
| Halaman | : | 491-500 |
| Abstrak | : | The desire for privacy significantly impacts various aspects of social behavior as illustrated by people’s tendency to seek out the most secluded spot when multiple options are available. In particular, this can be seen at rows of payphones, where people tend to occupy an available payphone that is most distant from those already occupied. Assuming that there are n payphones in a row and that n people occupy payphones one after another as privately as possible, the resulting assignment of people to payphones defines a permutation, which we will refer to as a payphone permutation. In the present study, we consider different variations of payphone permutations and enumerate them. |
Double Balanced Quadrilaterals
| Pengarang | : | Allan Berele |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 131 (No. 6) |
| Halaman | : | 479-490 |
| Abstrak | : | The k-centroids Gk of a polygon, for k = 0, 1, 2, are the centroids of the polygon when the mass is equally distributed respectively between the vertices, along the perimeter, or across the area. A fundamental theorem by Al-Sharif, Hajja, and Krasopoulos in [Citation1] asserts that the quadrilaterals with either G0 = G1 or G0 = G2 are precisely all parallelograms. Our main result describes the non-parallelograms with G1 = G2 by providing formulas for their diagonals in terms of the sides, as well as formulas for the ratios determined on the diagonals by their intersection point. In this way, we complete a fifteen-year-old problem by these three authors on characterizing all double balanced quadrilaterals. As an application, we show how our main theorem can be used to deduce their characterizations of double balanced circumscribed and cyclic quadrilaterals. |
"The Riddle of the Ages”: James Joseph Sylvester and the Transcendence of ?
| Pengarang | : | Adrian Rice |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 131 (No. 6) |
| Halaman | : | 463-478 |
| Abstrak | : | The first proof of the transcendence of π by Ferdinand Lindemann in 1882 prompted the publication of a wave of further proofs, all attempting to elucidate, simplify, or generalize this epoch-making result. Modifications by the likes of Weierstrass, Gordan, and Hilbert all entered the literature. But one such proof sank without a trace and was almost completely ignored. This is surprising as, not only did it appear in a major journal with a large international readership, but it was also written by one of the most famous mathematicians of the time—James Joseph Sylvester. This paper tells its story. |
On Nonnegativity of the Generalized Discriminants of Quadratics
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 131 (No. 5) |
| Halaman | : | 425-431 |
| Abstrak | : | Discriminants of quadratics have been recently generalized as Δ????=?????(????−????)?????2????−????−(????+1)?(????−????+1)?????????−????−1?????????−????+1 for a polynomial ?????(????)=?????????????????+?+????1?????+????0 of degree ????≥2 for 1≤????≤????−1 and it has been shown that Δ1≥0 if f has real roots only, [Citation1]. In this article we extend this result to Δ????. Namely, we show that Δ????≥0 if f has real roots only for 1<????≤????−1. As an application, we also demonstrate how to graph a plane quartic without using any calculus tools other than continuity. |
Lucky Cars and the Quicksort Algorithm
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 131 (No. 5) |
| Halaman | : | 417-423 |
| Abstrak | : | Quicksort is a classical divide-and-conquer sorting algorithm. It is a comparison sort that makes an average of 2?(????+1)?????????−4????? comparisons on an array of size n ordered uniformly at random, where ????????:=∑????????=11???? is the nth harmonic number. Therefore it makes ????![2?(????+1)?????????−4?????] comparisons to sort all possible orderings of the array. In this article, we prove that this count also enumerates the parking preference lists of n cars parking on a one-way street with n parking spots resulting in exactly ????−1 lucky cars (i.e., cars that park in their preferred spot). For ????≥2, both counts satisfy the second order recurrence relation ????????=2??????????????−1−?????(????−1)?????????−2+2?(????−1)! with ????0=????1=0. |
Dubious Identities: A Visit to the Borwein Zoo
| Pengarang | : | Jesús A. De Loera |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 131 (No. 5) |
| Halaman | : | 400-416 |
| Abstrak | : | We contribute to the zoo of dubious identities established by J.M. and P.B. Borwein in their 1992 paper, “Strange Series and High Precision Fraud” with five new entries, each of a different variety than the last. Some of these identities are again a high precision fraud and picking out the true from the bogus can be a challenging task with many unexpected twists along the way. This work is dedicated to the Borwein family, mathematicians extraordinaire with a propensity for the implausible. |
Playing Mastermind with Wordle-like Feedback
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 131 (No. 5) |
| Halaman | : | 390-399 |
| Abstrak | : | We introduce an extension of Mastermind called Clear Mastermind with enhanced feedback inspired by that from Wordle. The only difference between Clear Mastermind and Mastermind is a rule that provides more precise feedback, as found in Wordle. In Clear Mastermind, the feedback contains the positions of the colors the codebreaker guessed correctly and the positions of colors that appear in the answer but in different positions. We explore the fewest number of guesses that a codebreaker requires to find the answer in Clear Mastermind according to its two parameters: the number of colors and the length of the answer. |