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Alice and Bob on X: Reversal, Coupling, Renewal
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| Pengarang | : | Geoffrey R. Grimmett |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 5) |
| Halaman | : | 439-451 |
| Abstrak | : | A neat question involving coin flips surfaced on ????, and generated an intensive “storm” of “social mathematics.” In a sequence of flips of a fair coin, Alice wins a point at each appearance of two consecutive heads, and Bob wins a point whenever a head is followed immediately by a tail. Who is more likely to win the game? The subsequent discussion illustrated conflicting intuitions, and concluded with the correct answer (it is a close thing). It is explained here why the context of the question is interesting and how it may be answered in a quantitative manner using the probabilistic techniques of reversal, coupling, and renewal. |
The Prisoners and the Swap: Less than Half is Enough
| Pengarang | : | Uri Mendlovic |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 5) |
| Halaman | : | 423-438 |
| Abstrak | : | We improve on the solution of the classical prisoners and drawers riddle, where all prisoners can find their number using the pointer-following strategy, provided that the prisoners can send a spy to inspect all drawers and swap one pair of numbers. In the traditional approach, each prisoner may need to open up to half of the drawers. We show that this strategy is sub-optimal. Remarkably, a single swap allows all n prisoners to find their number by asymptotically opening only ???? ln  ln ???? ln ?????(1+?????(1)) drawers in the worst case. We show that no strategy can do better than this bound by a factor larger than two. Efficiently constructing such a strategy is harder, but we provide an explicit efficient strategy that requires opening only ?????(???? log  log ???? log ????) drawers by each prisoner in the worst case. |
Equi-Triangulation of Polygons
| Pengarang | : | Christopher J. Bishop |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 5) |
| Halaman | : | 407-421 |
| Abstrak | : | We prove that any two polygons with the same area can be triangulated using the same set of triangles. This strengthens the 200-year-old Wallace–Bolyai–Gerwien theorem that equal area polygons have an equi-dissection. |
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. 3) |
| Halaman | : | 290-300 |
| Abstrak | : | with the collaboration of Bonnie Amende, Paul Bracken, Hongwei Chen, Raluca Dumitru, Zachary Franco, George Gilbert, Leonid V. Kovalev, László Lipták, Rick Luttmann, Frank B. Miles, Lenhard Ng, Rajesh Pereira, Kenneth Stolarsky, Richard Stong, Lawrence Washington, and Li Zhou. |
Advanced mathematical knowledge in teaching practice: Perceptions of secondary mathematics teachers
| Pengarang | : | Elizabeth Arnold, Yvonne Lai & Carl Olimb |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 3) |
| Halaman | : | 285-289 |
| Abstrak | : | Each year, mathematics departments across the US may interact with more than 60,000 teachers preparing to teach K-12 mathematics.Footnote1 One role for mathematics faculty is teaching and creating courses that focus on developing teachers’ “mathematical knowledge for teaching” (MKT). |
Does the Convex Order Between the Distributions of Linear Functionals Imply the Convex Order Between the Probability Distributions Over ?d
| Pengarang | : | Iosif Pinelis |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 3) |
| Halaman | : | 281-284 |
| Abstrak | : | It is shown that the convex order between the distributions of linear functionals does not imply the convex order between the probability distributions over ????? if ????≥2. This stands in contrast with the well-known fact that any probability distribution in ?????, for any ????≥1, is determined by the corresponding distributions of linear functionals. By duality, it follows that, for any ????≥2, not all convex functions from ????? to ? can be represented as the limits of sums ∑????????=1????????°???????? of convex functions ???????? of linear functionals ???????? on ?????. |
On Subprime Recurrences
| Pengarang | : | Vadim Ponomarenko, Sam Rhinehart, Lara Schwarz & Jessie Tong |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 3) |
| Halaman | : | 275-280 |
| Abstrak | : | Mathematical legend John Conway invented a sequence that combines elements of Fibonacci numbers and the Collatz problem: after adding the previous two elements, we divide that sum by its least prime factor (but only if that sum is composite). This leads to some very interesting structure, which has yet to receive the widespread attention it deserves. We generalize from Fibonacci numbers to general second order recurrences, and prove a variety of results for such sequences. |
Three Proofs of the Spectral Theorem That You Might Not Know About
| Pengarang | : | Shuyang Cheng |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 3) |
| Halaman | : | 268-274 |
| Abstrak | : | In this article we present three different proofs of the spectral theorem. Compared to the standard proofs using determinants or Lagrange multipliers, the three proofs presented have their own pedagogical advantages and deserve to reach a broader audience. |
Nonzero Constant Wronskians of Polynomials and Laurent Polynomials
| Pengarang | : | Juan G. Alcázar & Carlos Hermoso |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 3) |
| Halaman | : | 250-267 |
| Abstrak | : | We characterize the polynomials ????1?(????),…,?????????(????) whose Wronskian ?????(????1,…,????????) is a nonzero constant. Then, we generalize our results to characterize the Laurent polynomials with the same property. Finally, for rational functions, we prove an impossibility result for ????=2, and pose the case ????≥3 as an open question, although we suggest an impossibility conjecture. Some geometric consequences are derived, especially in the case of polynomials. Furthermore, we show a connection between the results for Laurent polynomials and homogeneous, linear Euler differential equations. |
Perfect Powers Predominantly Produce Polygons
| Pengarang | : | Hamish Arora & Vadim Ponomarenko |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 3) |
| Halaman | : | 249 |
| Abstrak | : | Abstrak tidak tersedia. |