January 2013 Contents
For subscribers, read recent issues online (Requires MAA Membership)
In the first issue of The College Mathematics Journal for 2013, Marcio Diniz explains how linear algebra throws light on asset pricing in financial markets, and Nat Kell and Matt Kretchmar explain why they suspect that to enhance suspense the producers manipulate the tabulation of the ballots that determine which contestant wins a million dollars in the season finales of the pioneering and popular reality television show Survivor. Also, in Classroom Capsules, Michal Misiurewicz explains how to avoid a terrible pitfall in teaching about Irrational Square Roots, Kurk Fink and Jawad Sadek show how to evaluate Other Indeterminate Forms, and Neal Brand and John A. Quintanilla show how calculus students can perform an experiment in Modeling Terminal Velocity (of a Whiffle ball).
Asset Pricing, Financial Markets, and Linear Algebra
Concepts from asset pricing and financial markets theory are used to illustrate concepts of linear algebra and linear programming.
Suspense at the Ballot Box
Nat Kell and Matt Kretchmar
In the popular television show Survivor, the winner of a million-dollar prize is determined in a final election, where the votes are read aloud as the winner is announced. We hypothesize that the show’s producers purposely alter the order of the ballots in order to build audience suspense. We test our hypothesis using the Poisson binomial distribution, then turn to entropy to confirm that the ballot order is likely altered.
Proof Without Words: An Algebraic Inequality
Madeubek Kungozhin and Sidney Kung
A visual proof of an algebraic identity.
Using Differentials to Differentiate Trigonometric and Exponential Functions
Starting from geometric definitions, we show how differentials can be used to differentiate trigonometric and exponential functions without limits, numerical estimates, solutions of differential equations, or integration.
When Can One Expect a Stronger Triangle Inequality?
Valerii Faiziev, Robert Powers, and Prasanna Sahoo
In 1997, Bailey and Bannister showed that a + b > c + h holds for all triangles with where a, b, and c are the sides of the triangle, h is the altitude to side c, and is the angle opposite c. In this paper, we show that a + b > c + h holds approximately 92% of the time for all triangles with .
The Combinatorial Trace Method in Action
Mike Krebs and Natalie C. Martinez
On any finite graph, the number of closed walks of length k is equal to the sum of the kth powers of the eigenvalues of any adjacency matrix. This simple observation is the basis for the combinatorial trace method, wherein we attempt to count (or bound) the number of closed walks of a given length so as to obtain information about the graph’s eigenvalues, and vice versa. We give a brief overview and present some simple but interesting examples. The method is also the source of interesting, accessible undergraduate projects.
Polynomial Graphs and Symmetry
Geoff Goehle and Mitsuo Kobayashi
Most quadratic functions are not even, but every parabola has symmetry with respect to some vertical line. Similarly, every cubic has rotational symmetry with respect to some point, though most cubics are not odd. We show that every polynomial has at most one point of symmetry and give conditions under which the polynomial has rotational or horizontal symmetry with respect to that point.
A Family of Identities via Arbitrary Polynomials
Dong Fengming, Ho Weng Kin, and Lee Tuo Yeong
In this short article, we prove an identity from which a theorem of Katsuura and two conjectures previously posed in this JOURNAL follow directly.
Proof Without Words: Tangent Double Angle Identity
A visual proof of the tangent double angle identity, .
Old Tails and New Trails in High Dimensions
We discuss the motivation for dimension reduction in the context of the modern data revolution and introduce a key result in this field, the Johnson-Lindenstrauss flattening lemma. Then we leap into high-dimensional space for a glimpse of the phenomenon called concentration of measure, and use it to sketch a proof of the lemma. We end by tying this classical pure result to a current, revolutionary application.
Irrational Square Roots
If students are presented the standard proof of irrationality of , can they generalize it to a proof of the irrationality of , p a prime if, instead of considering divisibility by p, they cling to the notions of even and odd used in the standard proof?
Other Indeterminate Forms
Kurt Fink and Jawad Sadek
Seven indeterminate forms are usually presented in connection with L’Hôpital’s rule. We introduce several others and show how they may be evaluated.
Modeling Terminal Velocity
Neal Brand and John A. Quintanilla
Using a simultaneously falling softball as a stopwatch, the terminal velocity of a whiffle ball can be obtained to surprisingly high accuracy with only common household equipment. This classroom activity engages students in an apparently daunting task that nevertheless is tractable, using a simple model and mathematical techniques at their disposal.
PROBLEMS AND SOLUTIONS