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College Mathematics Journal Contents—May 2017

Given a polygon with an even number of vertices, translate every other vertex by a fixed vector—why does the area stay the same? Read about the corset theorem in the May issue of The College Mathematics Journal to find out. Other article topics include a pivot point that arises from repeating a data point in linear regression and extensions for calculus classrooms, one involving prisms and one exploring examples similar to the standard Peano–Genocchi function which is not continuous at the origin. On the applied side, you can analyze a medieval assault weapon and develop a simple model for global mean sea level rise. The Classroom Capsules in this issue offer several new proofs of the infinitude of primes. The Media Highlights help keep you up to date with a wide range of recent publications, and Bonnie Gold reviews a book proposing a new way to understand mathematical knowledge. And, as always, there are five new problems and five solutions for your enjoyment. —Brian Hopkins

Vol. 48, No. 3, pp. 161-240


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Mathematical Models for Global Mean Sea Level Rise

p. 162.

Stephen Kaczkowski

Global mean sea level rise (GMSLR) is one consequence of global warming. We develop simple, deterministic, and concise models for GMSLR. We consider just two factors, ocean thermal expansion and glacier melt. The resulting calculus-based models give values within the margin of error of official estimates of these two major components of GMSLR.

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Water Mathematics

p. 170.

Donald Illich

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Finding Polygonal Areas with the Corset Theorem

p. 171.

Stuart M. Anderson and Owen D. Byer

We explore how one can interpret the base and altitude of a triangle relative to a coordinate system in order to generalize the standard formula for the area. This generalization leads to an elementary proof of a formula (similar to the shoelace formula) for the area of a general polygon in terms of the coordinates of its vertices. A surprising corollary is that if every other vertex of a polygon is translated by a fixed vector, then the area of the polygon is unchanged.

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A Lagrangian Simulation of the Floating-Arm Trebuchet

p. 179.

Eric Constans

The floating-arm trebuchet, a modern variation of a medieval weapon, is a popular project for physics and engineering instructors.We develop and analyze a model of this device as an introduction to constrained Lagrangian dynamics.

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Proof Without Words: Sum of a Row in Pascal’s Triangle

p. 188.

Ángel Plaza

Using Pascal’s identity, we visually demonstrate that the sum of entries in a row of Pascal’s triangle is a power of two.

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A Curious Feature of Regression

p. 189.

Carl V. Lutzer

In this article, we investigate how repeating a data point affects the least-squares regression. While it is well known that the regression line approaches the repeated point, the fact that it pivots surprises many people. The article uses linear algebra to explore this behavior, develops relevant formulas for the pivot point, and explains the essential connection between the pivot point to the set of data.

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Optimizing Prisms of All Shapes and Dimensions

p. 199.

Maria Nogin

We generalize the standard calculus problem of finding the optimal shape of a square prism to allow bases of any shape and then extend to arbitrary dimension. In all cases, the prism with the smallest possible surface area given the volume has height twice the apothem of the base (where the definition of apothem has been suitably generalized).

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Proof Without Words: Nested Square Roots

p. 204.

Roger B. Nelsen

We evaluate some nested square roots by computing the area of a square in two ways.

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On a Genocchi–Peano Example

p. 205.

Krzysztof Chris Ciesielski and David Miller

We characterize the simple rational functions of arbitrarily many real variables that are discontinuous but continuous when restricted to any hyperplane. The characterization is expressed by simple inequalities with respect to the exponents of each variable. Examples include two infinite families of such Genocchi–Peano examples.We also investigate the smallest degree of the denominators of such examples.

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Classroom Capsules

Two Short Proofs of the Infinitude of Primes

p. 214.

Sam Northshield

We present two new proofs of the infinitude of primes. The first proof uses the basic idea of Furstenberg’s celebrated topological proof but without using topology. The second proof uses probability theory.

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Partitioning the Natural Numbers to Prove the Infinitude of Primes

p. 217.

Arpan Sadhukhan

In this note we will introduce a partition of the natural numbers and use it to give two proofs of the infinitude of primes. The first proof is a slight variant of the various known combinatorial proofs. The second is similar to Euler’s proof but it makes no use of Euler’s product formula.

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Problems and Solutions

p. 219.

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Book Reviews

Mathematical Knowledge and the Interplay of Practices  By José Ferreirós

p. 226.

Reviewed by: Bonnie Gold

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Media Highlights

p. 233.

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