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Vol. 83, No. 4, pp. 242-328

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ARTICLES

**Structured Population Dynamics: An Introduction to Integral Modeling **

Joseph Briggs, Kathryn Dabbs, Michael Holm, Joan Lubben, Richard Rebarber, Brigitte ten Humberg, and Daniel Riser-Espinoza

pp. 243–257

A single species is often modeled as a structured population. In a matrix projection model, individuals in the population are partitioned into a finite number of stage classes. For example, an insect population can be partitioned into egg, larva, pupa and adult stages. For some populations the stages are better described by a continuous variable, such as the stem diameter of a plant. For such populations an integral projection model can be used to describe the population dynamics, and might be easier to use or more accurate than a matrix model. In this article we discuss the similarities and differences between matrix projection models and integral projection models. We illustrate integral projection modeling by a Platte thistle population, showing how the model is determined by basic life history functions.

**What Do We Know at 7 pm on Election Night?**

Andrew Gelman and Nate Silver

pp. 258–266

We use a probability forecasting model to estimate the chance of different branches on the tree of state-by-state outcomes on election night. Forecasting models can use data from pre-election surveys as well as extrapolation based on previous election results. We implement conditional probability calculations numerically using a matrix representing 10,000 simulations of the outcomes in the 50 states.

**Putzer’s Algorithm for ***eAt* via the Laplace Transform

William A. Adkins and Mark G. Davidson

pp. 267–275

A method due to E. J. Putzer computes the matrix exponential *eAt* for an *n*x *n* matrix *A* without transforming *A* to Jordan canonical form. A variation of Putzer’s algorithm is presented. This approach is based on an algorithmically produced formula for the resolvent matrix that is combined with simple Laplace transform formulas to give a formula, similar to Putzer's, for *eAt*.

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NOTES

**The Geometry of the Snail Ball**

Stan Wagon

pp. 276–279

The snail ball is a device that rolls down an inclined plane, but very slowly, repeatedly coming to a stop and staying motionless for several seconds. The interior of the ball is hollow, with a smaller solid ball inside it, surrounded by a very viscous fluid. We show how to model the stop-and-start motion by analyzing the cycloidal curve that would correspond to the motion of the center of gravity as the ball rolls down an inclined plane.

**Another Morsel of Honsberger**

Mowaffaq Hajja

pp. 279–283

In two of his books, Ross Honsberger presented several proofs of the fact that the point *A* on the circular arc for which is maximum is the midpoint of the arc. In this note, we give three more proofs and examine how these proofs and those of Honsberger are related to propositions in Euclid’s *Elements* and, less strongly, to other problems in geometry such as the broken chord theorem, Breusch’s lemma, Urquhart’s theorem, and the Steiner-Lehmus theorem.

**Cantor's Other Proofs that Is Uncountable**

John Franks

pp. 283–289

This expository note describes some of the history behind Georg Cantor’s proof that the real numbers are uncountable. In fact, Cantor gave three different proofs of this important but initially controversial result. The first was published in 1874 and the famous diagonalization argument was not published until nearly two decades later. We explore the different ideas used in each of his three proofs.

**Nothing Lucky about 13 **

B. Sury

pp. 289–293

Gauss sums were introduced by Gauss in 1801, when he stated some of their properties and used them to prove the quadratic reciprocity law in different ways. The determination of the sign of the Gauss sum was a notoriously difficult question; Gauss recorded the correct assertion in his mathematical diary in May 1801, but could find a proof only in 1805. This note uses the Gauss sums to evaluate certain sums of trigonometric functions.

**Proof Without Words: The Alternating Harmonic Series Sums to ln 2**

Matt Hudelson

p. 294

**Period Three Begins**

Cheng Zhang

pp. 295–297

By exploiting the geometry of the cobweb plot, we provide a simple and elementary derivation of the parameter for the period-three cycle of the logistic map.

**Stacking Blocks and Counting Permutations**

Lara K. Pudwell

pp. 297–302

This paper explores a surprising connection between a geometry problem and a result in enumerative combinatorics. First, we find the surface areas of certain solids formed from unit cubes. Next, we enumerate multiset permutations which avoid the patterns {132, 231, 2134}. Finally, we give a bijection between the faces of the solids and the set of permutations.

**Counting Ordered Pairs**

David M. Bradley

p. 302