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An Elementary Proof for the Krull Dimension of a Polynomial Ring
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 7) |
| Halaman | : | 623-637 |
| Abstrak | : | It is proved that, for every infinite field F, the polynomial ring F[t1, …, tn] has Krull dimension n. The proof uses only “high school algebra” and the rudiments of undergraduate “abstract algebra.” |
Another n-Dimensional Generalization of Pompeiu’s Theorem
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 7) |
| Halaman | : | 612-622 |
| Abstrak | : | A theorem of D. Pompeiu states that the distances from the vertices of an equilateral triangle to any point in its plane can serve as the side lengths of a triangle. In this paper, it is proved that the distances from the vertices of a regular tetrahedron to any point in its affine hull can serve as the areas of the faces of a tetrahedron. The analogous statement for higher-dimensional regular simplices is also established, and several questions are posed for further investigation. |
The Wild Goose Chase Problem
| Pengarang | : | Andrew Gard |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 7) |
| Halaman | : | 602-611 |
| Abstrak | : | The classical pursuit problem considers the path traced by a point in space as it charges directly toward a moving target. But what if the target has more lofty goals than mere escape? To what degree can it control the path taken by its pursuer? We prove that the pursuer can be led to any point in without being allowed to close more than an arbitrarily small distance en route. |
Lattice Point Visibility on Generalized Lines of Sight
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 7) |
| Halaman | : | 593-601 |
| Abstrak | : | For a fixed we say that a point (r, s) in the integer lattice is b-visible from the origin if it lies on the graph of a power function f(x) = axb with and no other integer lattice point lies on this curve (i.e., line of sight) between (0, 0) and (r, s). We prove that the proportion of b-visible integer lattice points is given by 1/ζ(b + 1), where ζ(s) denotes the Riemann zeta function. We also show that even though the proportion of b-visible lattice points approaches 1 as b approaches infinity, there exist arbitrarily large rectangular arrays of b-invisible lattice points for any fixed b. This work specialized to b = 1 recovers original results from the classical lattice point visibility setting where the lines of sight are given by linear functions with rational slope through the origin. |
Sacks of Dice with Fair Totals
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 7) |
| Halaman | : | 579-592 |
| Abstrak | : | A fair sack is a finite set of independent dice, not required to be fair and allowed to have any number of sides, for which all totals are equally likely. These have been studied for over 60 years. Most results restrict the possible orders of dice in such a sack and almost no examples were known. Building on a rather different approach due to Gasarch and Kruskal, we give a canonical construction of all such sacks. |
Sines, Cosines, Square Roots, and Binary Bits
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 8) |
| Halaman | : | 750-754 |
| Abstrak | : | A procedure due to L. D. Servi for evaluating sines and cosines of certain angles in terms of nested square roots of 2 is discussed. It is shown how it can be applied using the binary digits of the angles and extended to find the inverse functions. An equivalent procedure is also discussed. |
Infinitude of Primes Using Formal Languages
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 8) |
| Halaman | : | 745-749 |
| Abstrak | : | Formal languages are sets of strings of symbols described by a set of rules specific to them. In this note, we discuss a certain class of formal languages, called regular languages, and put forward some elementary results. The properties of these languages are then employed to prove that there are infinitely many prime numbers. |
Monsters in Calculus
| Pengarang | : | Krzysztof Chris Ciesielski |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 8) |
| Halaman | : | 739-744 |
| Abstrak | : | One of the strangest, most mind-boggling examples in analysis is that of a function from to that is everywhere differentiable but monotone on no interval. The graph of such a “monstrous” function is simultaneously smooth and very rugged. Although such examples have been known for over 100 years, so far the existing constructions are quite involved. In this note, we provide a simple example of such a map. It is presented in a broader context of other paradoxical examples related to differentiability of continuous maps from to , including a differentiable function that maps a compact perfect subset of onto itself even though its derivative vanishes on . |
Simple Proofs of the Uniform Convexity of Lp and the Riesz Representation Theorem for Lp
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 8) |
| Halaman | : | 733-738 |
| Abstrak | : | We give elementary and self-contained proofs of the facts that the Lp space with is uniformly convex and its dual space is isometrically isomorphic to . |
Social Balance and the Bernoulli Equation
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 8) |
| Halaman | : | 724-732 |
| Abstrak | : | Since the 1940s there has been an interest in the question of why social networks often give rise to two antagonistic factions. Recently a dynamical model of how and why such a balance might occur was developed. This article provides an introduction to the notion of social balance and a new (and simplified) analysis of that model. This new analysis allows us to choose general initial conditions, as opposed to the symmetric ones previously considered. We show that for general initial conditions, four factions will evolve instead of two. We characterize the four factions, and give an idea of their relative sizes. |