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Solutions of a Polynomial Equation Modulo a Prime Power
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| Pengarang | : | Arnaud Bodin & Christian Drouin |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 3) |
| Halaman | : | 231-248 |
| Abstrak | : | How do you find the integer solutions of a polynomial equation modulo some integer? |
Polynomial versus Exponential Growth
| Pengarang | : | David M. Bradley |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 3) |
| Halaman | : | 230 |
| Abstrak | : | Abstrak tidak tersedia. |
The Triviality of Commuting Polynomials in Many Variables
| Pengarang | : | Arseny Mingajev |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 3) |
| Halaman | : | 220-229 |
| Abstrak | : | We consider the equation ?????(?????(????1,…,????????))=?????(?????(????1),…,?????(????????)). Here ????∈??[????] is a polynomial in one variable over the field of complex numbers and ????∈??[????1,…,????????],????≥2, is a polynomial in two or more variables, but not in any one of variables alone, i.e., ????∉??[????????],????∈{1,…,????}. We prove that if deg?(????)>1, then all solutions to the functional equation above are characterized by monomials. More specifically, we prove that this equation is solvable if and only if there exists an affine bijection ???? on ?, such that ????°????°????−1=???????? and ????°????°????−1=?????????????11…????????????????, where ????????−1=1. |
Retracing the Turns of the Archimedean Spiral
| Pengarang | : | John B. Little |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 3) |
| Halaman | : | 203-219 |
| Abstrak | : | This article will consider some of the properties of Archimedean spirals and their history, by retracing appearances of the spirals in various eras of the development of mathematics in reverse chronological order. The starting point will be the observation that the spirals currently serve as a fertile source of interesting and accessible examples for secondary and college courses. Their history and their intriguing properties also give various understandings of the motivation and context for considering these curves. As a by-product, ideas for some historically-based projects drawing on original sources, and suitable for undergraduate courses, will be suggested. Finally, it will become clear that the spiral itself serves almost as a visual metaphor for this story. |
Surprises from mathematics education research: Student (mis)use of mathematical definitions
| Pengarang | : | Amir H. Asghari |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 7) |
| Halaman | : | 691-696 |
| Abstrak | : | In moving from the first question to the second, as readers of Monthly, you can quickly switch from the everyday definition of a circle, also known as its concept image—“the mental attributes associated with [it]” [Citation1, p.152]—in which there is no distinction between a circle and a disk, to the mathematical definition of a circle, also known as its concept definition, which calls only for the set of points equidistant from a center. Thus your answer to the first question is 14, without any need to explicitly evoke the definition of a circle, or even to scrutinize your concept image; and your answer to the second question is 2, obtained by swiftly shifting to a concept image that is now in harmony with the formal concept definition of a circle. |
Every Jordan Curve Contains All Vertices of Uncountably Many Rhombi—A Short Proof
| Pengarang | : | Stephen E. Wright |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 7) |
| Halaman | : | 686-690 |
| Abstrak | : | A long-standing conjecture of Otto Toeplitz asserts that every Jordan curve contains the vertices of some square. Proofs addressing special classes of curves sometimes begin by demonstrating the existence of a suitable family of rhombi with all vertices on the given curve or an approximating curve. We provide a very short proof that, given a line in the plane, a general Jordan curve contains the vertices of a rhombus with two sides parallel to that line. |
On Carlson and Stolarsky Inequalities
| Pengarang | : | Lazhar Bougoffa & Bikash Chakraborty |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 7) |
| Halaman | : | 681-685 |
| Abstrak | : | The purpose of this short note is to demonstrate that the Arithmetic-Logarithmic-Geometric (AM-LM-GM) inequality, Carlson’s inequality, and the generalized form of the logarithmic mean (Stolarsky’s mean) can be straightforwardly obtained from the Hölder’s inequality and its reverse. |
Intersections of Random Chords of a Circle
| Pengarang | : | Cynthia Bortolotto & Victor Souza |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 7) |
| Halaman | : | 661-680 |
| Abstrak | : | Where are the intersection points of diagonals of a regular n-gon located? What is the distribution of the intersection point of two random chords of a circle? We investigate these and related new questions in geometric probability, extend a largely forgotten result of Karamata, and elucidate its connection to the Bertrand paradox. |
Some New Insights from Highly Optimized Polyhedral Passages
| Pengarang | : | Raj Gosain & Benjamin Grimmer |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 7) |
| Halaman | : | 649-660 |
| Abstrak | : | A shape possesses Rupert’s property if a hole can be cut through it such that a second identical copy of the shape can cleanly pass straight through the interior of the first. Such a passage proving cubes are Rupert was first shown more than 300 years ago. It remains open whether every polyhedron in three dimensions is Rupert. We propose a customized subgradient method providing high-accuracy local numerical optimization of the quality of a passage for a given polyhedron. From extensive numerical searches, we improve these best-known passages for more than half of the Platonic, Archimedean, and Catalan solids and for numerous Johnson solids. Our high accuracy solves support a new conjecture of a simple form for the tetrahedron’s optimal passage. Despite our computational search, three Archimedean and two Catalan solids remain open, providing further negative evidence against the conjecture that all polyhedra are Rupert. |
On Unsolvable Equations of Prime Degree
| Pengarang | : | Juliusz Brzezi?ski & Jan Stevens |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 7) |
| Halaman | : | 634-648 |
| Abstrak | : | Leopold Kronecker observed that either all roots or only one root of a solvable irreducible equation of odd prime degree with integer coefficients are real. This gives a possibility to construct specific examples of equations not solvable by radicals. A relatively elementary proof without using the full power of Galois theory is due to Heinrich Weber. We give a rather short proof of Kronecker’s Theorem with an argument that is slightly different from Weber’s. Several modern presentations of Weber’s proof contain inaccuracies, which can be traced back to an error in the original proof. We discuss this error and how it can be corrected. |