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Composite Numbers in an Arithmetic Progression
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| Pengarang | : | Hùng Vi?t Chu, Steven J. Miller & Joshua M. Siktar |
| Nama Majalah/Jurnal | : | Mathematics Magazine |
| Volume / Edisi | : | 99 (No. 3) |
| Halaman | : | 224-232 |
| Abstrak | : | One challenge (or opportunity!) that many instructors face is how varied the backgrounds, abilities, and interests of students are. In order to simultaneously instill confidence in those with weaker preparations and still challenge those able to go faster, an instructor must be prepared to give problems of different difficulty levels. Using Dirichlet’s theorem as a case study, we create and discuss a family of problems in number theory that highlight the relative strengths and weaknesses of different ways to approach a question and show how to invite students to extend the problems and explore research-level mathematics. |
A Synthetic Geometry Proof of Heron’s Formula
| Pengarang | : | Colin Beveridge |
| Nama Majalah/Jurnal | : | Mathematics Magazine |
| Volume / Edisi | : | 99 (No. 3) |
| Halaman | : | 220-223 |
| Abstrak | : | Heron's formula, which dates back to at least the first century CE, links the area of a triangle to the lengths of its sides. This paper offers a proof of the formula using the triangle's incircle and the properties of similar triangles. |
Optimization and Learning via Stochastic Gradient Search
| Pengarang | : | Marissa Masden |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 6) |
| Halaman | : | 604-608 |
| Abstrak | : | What is stochastic gradient search, and who is interested in it? Optimization and Learning via Stochastic Gradient Search by Felisa Vásquez-Abad and Bernd Heidergott is contemporary and timely in answering these questions. This book (henceforth referred to as OLSGS) details some of the mathematics that underpins modern algorithms in fields such as computer science, economics, and control. It is pitched both as a textbook for graduate learners and as a reference for practitioners, aiming for an intuitive, algorithm-first pedagogy that is then supported by theory. The book advertises itself as one which sits at the boundary between theory and application. This balance is extremely difficult to strike well; my interest in finding books that succeed at such a task is the primary lens through that I am writing this review. |
An Intriguing Limit
| Pengarang | : | Lorenzo Fornari, Enrico Laeng & Vittorino Pata |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 6) |
| Halaman | : | 589-592 |
| Abstrak | : | The authors thank the anonymous referees for their valuable comments. |
Classification of d for Which the Period Length of the Continued Fraction Expansion of ?d is Odd
| Pengarang | : | Jun Ho Lee |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 6) |
| Halaman | : | 584-588 |
| Abstrak | : | Let d be a non-square positive integer. In this paper, we show that if the period length of the continued fraction expansion of √???? is odd, then ????≡1 or 2 (mod?4). This provides a simple classification criterion for d based solely on congruence conditions. Our approach relies only on basic properties of continued fraction expansions, offering an elementary perspective on a classical problem. |
Minkowskis Theorem on Archimedean Lattices
| Pengarang | : | Dana Paquin & Julian Schennach |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 6) |
| Halaman | : | 578-583 |
| Abstrak | : | Minkowski’s celebrated theorem, bounding the area of centrosymmetric convex sets intersecting only one point of a lattice, has broad implications from number theory to geometry. But what about discrete sets generated by symmetries other than translations? Here we explore extensions for all lattices constructed from Archimedean tilings of the Euclidean plane and observe how symmetry constraints and optimality interact in intricate ways. |
A Short Path to Konigs, Berges, and Halls Theorems via Edge Stability
| Pengarang | : | Daniel A. Jaume & Kevin Pereyra |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 6) |
| Halaman | : | 571-576 |
| Abstrak | : | We introduce a simple structural property of bipartite graphs—edge-stability—which states that removing an edge whose endpoints lie in a minimum vertex cover does not change the vertex cover number. This property admits an elementary proof and serves as a bridge between several classical theorems in matching theory. We show that edge-stability follows directly from K?nig’s Theorem and, conversely, can be used to rederive it. Moreover, we employ edge-stability to obtain short proofs of Berge’s characterization of maximum independent sets and Hall’s Marriage Theorem. Thus within the framework of Reichmeider’s equivalence of Mengerian theorems, edge-stability, K?nig’s, Berge’s, and Hall’s theorems are all equivalent. The approach highlights new conceptual and pedagogical connections between these cornerstone results in matching theory. |
Transcending the Alternating Harmonic Series
| Pengarang | : | Hannah Dempsey, Dominic Klyve, Sooie-Hoe Loke & Vincent Nguyen |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 6) |
| Halaman | : | 560-570 |
| Abstrak | : | In this work, we examine rearrangements of the alternating harmonic series, in which terms are grouped into blocks sharing a sign. The work extends and generalizes earlier results on similar rearrangements, including some published in the Monthly. We conclude by demonstrating (conditionally on Schanuel’s conjecture) that the sum of any such rearrangement is a transcendental number. |
Block Stacking, Airplane Refueling, and Robust Appointment Scheduling
| Pengarang | : | Simon Gmeiner & Andreas S. Schulz |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 6) |
| Halaman | : | 540-559 |
| Abstrak | : | How can a stack of identical blocks be arranged to extend beyond the edge of a table as far as possible? We consider a generalization of this classic puzzle to blocks that differ in width and mass. Despite the seemingly simple premise, we demonstrate that it is unlikely that one can efficiently determine a stack configuration of maximum overhang. Formally, we prove that the Block-Stacking Problem is NP-hard, partially answering an open question from the literature. Furthermore, we demonstrate that the restriction to stacks without counterweights has a surprising connection to the Airplane Refueling Problem, another famous puzzle, and to Robust Appointment Scheduling, a problem of practical relevance. In addition to revealing a remarkable relation to the real-world challenge of devising schedules under uncertainty, their equivalence unveils a polynomial-time approximation scheme, that is, a (1+????)-approximation algorithm, for Block Stacking without counterbalancing and a (2+????)-approximation algorithm for the general case. |
Stretching Newton Polygons Using Pure Polynomials
| Pengarang | : | Rylan Gajek-Leonard & Uri Tomer |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 133 (No. 6) |
| Halaman | : | 532-539 |
| Abstrak | : | The p-adic Newton polygon is a visual tool that encodes information about the roots and factorization of a polynomial relative to a prime p. In this article, we investigate how the Newton polygon changes under polynomial composition. If f and g are polynomials with rational (or p-adic) coefficients and the Newton polygon of g is pure (has only one segment), we show under some mild conditions that the Newton polygon of ????°???? is the same as that of f, but stretched horizontally by deg?????. When ????=????, this implies that all iterates of certain pure polynomials are irreducible, recovering a classical result of Robert Odoni on the irreducibility of iterated Eisenstein polynomials. |