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Why Is Pi Less Than Twice Phi?
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| Pengarang | : | Alejandro H. Morales |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 8) |
| Halaman | : | 715-723 |
| Abstrak | : | We give a proof of the inequality in the title in terms of Fibonacci numbers and Euler numbers via a combinatorial argument and asymptotics for these numbers. The result is motivated by Sidorenko’s theorem on the number of linear extensions of a partially ordered set and its complement. We conclude with some open problems. |
Viète’s Formula, Knar’s Formula, and the Geometry of the Gamma Function
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 8) |
| Halaman | : | 704-714 |
| Abstrak | : | The gamma function was originally discovered by Euler to solve an interpolation problem for the factorials. However, its ubiquity cannot be due to its algebraic nature alone. This article presents the close connection between the volumes of various spaces and the values of the gamma function to better understand its frequent appearance in many mathematical formulas. This connection produces a surprising similarity between Viète’s formula for π and Knar’s formula. |
Rational Right Triangles of a Given Area
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 8) |
| Halaman | : | 689-703 |
| Abstrak | : | Starting from any given rational-sided, right triangle, for example, the (3,4,5)-triangle with area 6, we use Euclidean geometry to show that there are infinitely many other rational-sided, right triangles of the same area. We show further that the set of all such triangles of a given area is finitely generated under our geometric construction. Such areas are known as “congruent numbers” and have a rich history in which all the results in this article have been proved and far more. Yet, as far as we can tell, this seems to be the first exploration using this kind of geometric technique. |
On the Surjectivity of Smooth Maps into Euclidean Spaces and the Fundamental Theorem of Algebra
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 10) |
| Halaman | : | 941-943 |
| Abstrak | : | In this note, we obtain the surjectivity of smooth maps into Euclidean spaces under mild conditions. As an application, we give a new proof of the fundamental theorem of algebra. We also observe that any C1-map from a compact manifold into Euclidean space with dimension ????≥2 has infinitely many critical points. |
The Lindemann Theorem for Matrices
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 10) |
| Halaman | : | 939-940 |
| Abstrak | : | In this note, we prove a Lindemann theorem about the transcendence of the exponential of square matrices. |
Roots of Polynomials and the Derangement Problem
| Pengarang | : | Lior Bary-Soroker |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 10) |
| Halaman | : | 934-938 |
| Abstrak | : | We present a new killing-a-fly-with-a-sledgehammer proof of one of the oldest results in probability which says that the probability that a random permutation on n elements has no fixed points tends to ????−1 as n tends to infinity. Our proof stems from the connection between permutations and polynomials over finite fields and is based on an independence argument, which is trivial in the polynomial world. |
A Partition Identity Related to Stanley’s Theorem
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 10) |
| Halaman | : | 929-933 |
| Abstrak | : | In this note, we use the Lambert series generating function for Euler’s totient function to introduce a new identity for the number of 1’s in the partitions of n. A new expansion for Euler’s partition function p(n) is derived in this context. These surprising new results connect the famous classical totient function from multiplicative number theory to the additive theory of partitions. |
Everywhere Differentiable, Nowhere Continuous Functions
| Pengarang | : | J. A. Hocutt |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 10) |
| Halaman | : | 923-928 |
| Abstrak | : | We introduce and analyze a notion of Cesàro differentiability that corresponds to the notion of Cesàro continuity studied by Halmos. |
Ducci Matrices
| Pengarang | : | Achim Clausing |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 10) |
| Halaman | : | 901-921 |
| Abstrak | : | For a real square matrix A, its Ducci map takes a vector x to |?????????|, where the absolute value is meant elementwise. We study the class of matrices with the property that for almost all x their orbits under the Ducci map suddenly “stops” at the zero vector, sometimes after a few iterations, sometimes after thousands of steps, in a seemingly unpredictable way. This generalizes the well known Four Number Game of E. Ducci. |
On the Lengths of Level Curves
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 10) |
| Halaman | : | 885-900 |
| Abstrak | : | This article establishes several remarkably simple identities concerning the lengths of level curves of a two-variable function, with the most notable one relating length to curvature. Some geometric and analytic applications of the results are considered. |