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An Impossibility Theorem for Gerrymandering
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 10) |
| Halaman | : | 878-884 |
| Abstrak | : | In 2018, the U.S. Supreme Court considered a proposed mathematical formula to help detect unconstitutional partisan gerrymandering. We show that in some cases, this formula only flags bizarrely-shaped districts as potentially constitutional. |
Geometry in the Courtroom
| Pengarang | : | Noah Giansiracusa |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 10) |
| Halaman | : | 867-877 |
| Abstrak | : | There has been a recent media blitz on a cohort of mathematicians valiantly working to fix the democratic system in the United States by combatting gerrymandering with geometry. While statistics commonly features in the courtroom (forensics, DNA analysis, etc.), the gerrymandering news raises a natural question: in what other ways has pure mathematics, specifically geometry and topology, been involved in court cases and legal scholarship? In this survey article, we collect a few examples with topics ranging from the Pythagorean formula to the ham sandwich theorem, and we discuss some jurists’ perspectives on geometric reasoning in the legal realm. One of our goals is to provide mathematics educators with engaging real-world instances of some abstract geometric concepts. |
Cubic Equations from an Analytic Point of View
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 9) |
| Halaman | : | 845-849 |
| Abstrak | : | Abstract–About five hundred years ago, the first algebraic solution of a class of cubic equations was discovered by Scipione del Ferro. In The Great Art (1545), Girolamo Cardano devoted 13 chapters to an attempt to solve all cubics, which was unsuccessful in precisely those cases in which the cubic possessed three (real) solutions. About 50 years later, François Viète discovered a trigonometric method for dealing with the cases that had eluded Cardano. It turns out that the method of Viète can be used to solve all cases and provide analytic insight into Cardano’s bewilderment. |
A Characterization of Convex Functions
| Pengarang | : | Paolo Leonetti |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 9) |
| Halaman | : | 842-844 |
| Abstrak | : | Let f be a real-valued radially lower semicontinuous function defined on a convex subset D of a real vector space. It is shown that f is convex if and only if, for all there exists such that . |
Another Proof of the Famous Formula for the Zeta Function at Positive Even Integers
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 9) |
| Halaman | : | 839-841 |
| Abstrak | : | In this note, we present an elementary proof of the classical formula for the zeta function at the positive even integers. This proof could also have been given by Euler in the 18th century since it uses many of his (sneaky) analytic techniques, namely the infinite product representation of the sine function. |
When the Cauchy Inequality Becomes a Formula
| Pengarang | : | Davit Harutyunyan |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 9) |
| Halaman | : | 835-838 |
| Abstrak | : | In this note, we revisit the classical geometric–arithmetic mean inequality and find a formula for the difference of the arithmetic and the geometric means of n given nonnegative numbers . The formula yields new stronger versions of the geometric-arithmetic mean inequality. We also find a second version of a strong geometric-arithmetic mean inequality and show that all inequalities are optimal in some sense. Another striking novelty is that the equality in all new inequalities holds not only in the case when all n numbers are equal, but also in other cases. |
Stokes’s Theorem, Data, and the Polar Ice Caps
| Pengarang | : | Yuliy Baryshnikov |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 9) |
| Halaman | : | 830-834 |
| Abstrak | : | Geographers and climate scientists alike sometimes need to estimate the area of a large region on the surface of the Earth, such as the polar ice caps. Doing so using only a sequence of latitude–longitude data points along a piecewise-linear approximation of the boundary of the region can be accomplished via a novel use of Stokes’s theorem that generalizes classical and contemporary applications of Green’s theorem to data. The polar ice caps are seen to hide a complication that makes their estimation from data singularly interesting. |
The Mahler Conjecture in Two Dimensions via the Probabilistic Method
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 9) |
| Halaman | : | 820-828 |
| Abstrak | : | The Mahler volume is, intuitively speaking, a measure of how “round” a centrally symmetric convex body is. In one direction, this intuition is given weight by a result of Santaló, who in the 1940s showed that the Mahler volume is maximized, in a given dimension, by the unit sphere and its linear images, and only these. A counterpart to this result in the opposite direction is proposed by a conjecture, formulated by Kurt Mahler in the 1930s and still open in dimensions 4 and greater, asserting that the Mahler volume should be minimized by a cuboid. In this article, we present a seemingly new proof of the two-dimensional case of this conjecture via the probabilistic method. The central idea is to show that either deleting a random pair of edges from a centrally symmetric convex polygon, or deleting a random pair of vertices, reduces the Mahler volume with positive probability. |
Quadratic Forms, Chebyshev Polynomials, and Geometric Inequalities
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 9) |
| Halaman | : | 811-819 |
| Abstrak | : | We discuss a geometric inequality for 2n-gons proved in [Mushkarov, O., Nikolov N. (2006). Semiregular polygons. Amer. Math. Monthly. 113(4): 339–344] and show how the algebraic inequality it is based on can be proved by using the standard theory of quadratic forms. In addition, we prove an “odd” version of the geometric inequality which leads to some interesting geometric problems for polygons with an odd number of sides. |
Everything You Wanted To Know About ax2+by2+cz2+dt2 But Were Afraid To Ask
| Pengarang | : | Kenneth S. Williams |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 9) |
| Halaman | : | 797-810 |
| Abstrak | : | Lagrange’s theorem tells us that the quadratic form represents all positive integers. What about the more general form , where a, b, c, and d are positive integers? We provide a gentle introduction to this question by posing and answering some natural questions about the possible integers that can represent. |