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The Circular Roulette of the True Catenary
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| Pengarang | : | Vincent E. Coll Jr. & George V. McIlvaine |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 6) |
| Halaman | : | 511-531 |
| Abstrak | : | In mathematics and physics, a catenary is the curve that an idealized hanging chain assumes when hanging under its own weight in a uniform gravitational field when supported only at its ends. However, standard calculations assume a flat Earth. A “true” catenary (“catenaria”) recognizes the inverse-square attractive gravitational forces of a central body (e.g., Earth). While the catenary is unique up to scale, catenaria are not. Following from a characterization of curves that maintain a circular roulette, we show that all catenaria, and their associated arches, have a circular roulette. |
Creating Decidable Diophantine Equations
| Pengarang | : | Robert Dougherty-Bliss, Charles Kenney & Doron Zeilberger |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 10) |
| Halaman | : | 947-959 |
| Abstrak | : | Hilbert’s famous 10th problem asked whether an algorithm exists to determine if a given Diophantine equation has a solution—in other words whether Diophantine equations are decidable. Yuri Matiyasevich proved that the answer is no, Diophantine equations are not decidable. However, it turns out that Matiyasevich’s ideas can be turned around and used to construct families of decidable Diophantine equations. All you need (to get started) are the Tribonacci numbers and a bit of calculus. |
Problems and Solutions
| Pengarang | : | Daniel H. Ullman, Daniel J. Velleman, Stan Wagon & Douglas B. West |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 5) |
| Halaman | : | 492-502 |
| Abstrak | : | Abstrak tidak tersedia. |
The Subseries are Dense in the Basel Problem
| Pengarang | : | George Stoica |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 5) |
| Halaman | : | 491 |
| Abstrak | : | Abstrak tidak tersedia. |
Mathematics professors’ evaluation of students’ proofs: A complex teaching practice
| Pengarang | : | David Miller & Sarah Hanusch |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 5) |
| Halaman | : | 485-490 |
| Abstrak | : | It is a common practice in upper-level mathematics proof courses for the professor to evaluate, grade, and comment on the proofs that students hand in for homework, quizzes, and exams. Their assessment consists of writing feedback on the students’ proofs, assigning a numerical score for the proof, and emphasizing good proof writing. We may ask: What do professors really want to see in the written proofs that students produce in upper-level mathematics courses? Do they want to see proofs written in complete sentences, using proper punctuation, having fluency and clarity in their proof writing, and demonstrating a good understanding of the material? Furthermore, do professors agree on how to score student proofs, and if not, how much do their scores vary? Moreover, what are the implications for undergraduate students if there are differences in how professors grade their proofs? |
On 2025 and Other Torn Numbers
| Pengarang | : | Mohammad Javaheri |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 5) |
| Halaman | : | 478-484 |
| Abstrak | : | 2025 is an example of a torn number, a perfect square that is a fixed point of the following split-add-square procedure: the number is torn in half (as 20 and 25), then the two parts are added together (20+25=45), and finally the result is squared (452=2025). We show that there exist at least 2?????(????)−1 2k-digit torn numbers, ????≥1, where ?????(????) is the number of prime factors of 10????−1. Moreover, we show that the split-add-square procedure has infinitely many periodic points of any arbitrary period. |
Testing Whether a Table Defines a Group
| Pengarang | : | Robert Beals |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 5) |
| Halaman | : | 472-477 |
| Abstrak | : | We consider the problem of testing whether a given ????×???? table for a binary operation ∗ defines a group. The existence of an identity element 1 is readily checked. Likewise, we can quickly verify that elements possess inverses. The apparent bottleneck is to test associativity, which naively seems to require checking (????∗????)∗????=????∗(????∗????) for all ????3 triples (????,????,????). We show that ????2+?????(????? log 2?????) carefully selected triples suffice to either prove that the table defines a group or to find a triple violating associativity. The bookkeeping cost of selecting which triples to check is ?????(????2), linear in the input length. |
A Simple Closed-Form Approximation to the Prime-Counting Function
| Pengarang | : | Kimihiro Noguchi |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 5) |
| Halaman | : | 471 |
| Abstrak | : | Abstrak tidak tersedia. |
Prevalence Theory: A Measure-Theoretic Approach to Large Sets in Infinite-Dimensional Spaces
| Pengarang | : | Samuel Nicolay |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 5) |
| Halaman | : | 466-470 |
| Abstrak | : | The concept of prevalence provides a rigorous framework for identifying “large” sets in infinite-dimensional spaces, extending measure-theoretic ideas of negligibility beyond finite-dimensional contexts. This paper introduces the foundational definitions of prevalence, exploring its key properties and illustrating its relevance through examples in functional spaces. We also investigate the interplay between prevalent sets and Baire categories, emphasizing that while these notions share certain structural parallels, they are fundamentally distinct. |
The Quadrilateral Nagel Point, The Behavior of Splitters, and a Special Line
| Pengarang | : | https://maa.tandfonline.com/author/Cooper%2C+Thomas+E |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 5) |
| Halaman | : | 452-465 |
| Abstrak | : | In this article, we use the four semi-excircles of a general convex quadrilateral to define a Nagel point, and we examine the behavior of certain segments that divide the perimeter of a convex quadrilateral in half. Most notably, we classify all quadrilaterals that have concurrent splitters, and we find that a certain set of intersections involving the splitters are collinear with the Nagel point and the intersection of the diagonals in any convex quadrilateral. |