# College Mathematics Journal Contents—November 2013

The November issue of The College Mathematics Journal is a special, theme issue supporting the Mathematics of Planet Earth initiative, MPE 2013. The articles in this extra large issue discuss a wide range of earth science and environmental questions.

Charles Hadlock applies undergraduate mathematics (linear equations, interpolation, geometry) to modeling the movement of water underground. Meredith Greer, Holly Ewing, Kathleen Weathers, and Kathryn Cottingham describe a mathematical/ecological collaborative study of a cyanoobacterium (attractively named Gloeo) in New England lakes. Osvaldo Marrero describes a statistical method for detecting seasonal variation in an epidemic. (Such variation may betray an environmental influence.) And Christiane Rousseau describes the discovery of the Earth’s inner core by Inge Lehmann.

Four articles/Classroom Capsules/Student Research Projects concern climate change. Two papers (by, respectively, James Walsh and Richard McGehee, and Emek Köse and Jennifer Kunze) apply dynamical systems and differential equations to modeling global temperature; a Classroom Capsule by John Zobitz discusses carbon absorption in forests; and a Student Research Project by Lily Khadjavi describes how to coax students to model the rate of climate change.

The issue begins with a guest editorial by Mary Lou Zeeman relating mathematics to sustainability and concludes with a review by Ben Fusaro of the recent book Mathematics for the Environment by Martin Walter. Michael Henle

Vol. 44, No. 5, pp.346-460.

The November CMJ is available FREE of charge via ingentaconnect. Individual copies of the print journal can be purchased in the MAA Store.

Mary Lou Zeeman

## ARTICLES

### Modeling Climate Dynamically

James Walsh and Richard McGehee

A dynamical systems approach to energy balance models of climate is presented, focusing on low order, or conceptual, models. Included are global average and latitude-dependent, surface temperature models. The development and analysis of the differential equations and corresponding bifurcation diagrams provides a host of appropriate material for undergraduates.

### Underground Mathematics

The movement of groundwater in underground aquifers is an ideal physical example of many important themes in mathematical modeling, ranging from general principles (like Occam’s Razor) to specific techniques (such as geometry, linear equations, and the calculus). This article gives a self-contained introduction to groundwater modeling with examples of these themes.

### Collaborative Understanding of Cyanobacterial Growth in Lake Ecosystems

Meredith Greer, Holly A. Ewing, Kathleen C. Weathers, and Kathryn L. Cottingham

We describe a collaboration between mathematicians and ecologists studying the cyanobacterium Gloeotrichia echinulata and its possible role in eutrophication of New England lakes. The mathematics includes compartmental modeling, differential equations, difference equations, and testing models against high-frequency data. The ecology includes observation, field sampling, and parameter estimation based on observed data and the related literature. Mathematically and ecologically, a collaboration like this, progresses in ways it would never have if either group worked alone.

### Seasonal Variation in Epidemiology

Osvaldo Marrero

Seasonality analyses are important in medical research. If the incidence of a disease shows a seasonal pattern, then an environmental factor must be considered in its etiology. We discuss a method for the simultaneous analysis of seasonal variation in multiple groups. The nuts and bolts are explained using simple trigonometry, an elementary physical model, and several graphics. Two examples show how the method is applied.

### How Inge Lehmann Discovered the Inner Core of the Earth

Christiane Rousseau

The mathematics behind Inge Lehmann’s discovery that the inner core of the Earth is solid is explained using data collected around the Earth on seismic waves and their travel time through the Earth.

### Proofs without Words: Monotonicity of $\sin(x)/x$ and $\tan(x)/x$

Xiaoxue Li

The Aristarchus’ inequality states that for $0<x<y<\pi/2$, $\sin(y)/\sin(x)<y/x<\tan(y)/\tan(x)$. It follows that $sin(x)/x$ decreases and $tan(x)/x$ increases on the interval $(0,\frac{\pi}{2})$. What causes the different monotone behavior of sine and tangent when both functions are increasing on this interval? Using their differing concavities, we give visual proofs of their monotonicity.

### Descartes' Calculus of Subnormals: What Might Have Been

Gregory Mark Boudreaux and Jess E. Walls

René Descartes’ method for finding tangents (equivalently, subnormals) depends on geometric and algebraic properties of a family of circles intersecting a given curve. It can be generalized to establish a calculus of subnormals alternative to the calculus of Newton and Leibniz. Here we prove subnormal counterparts of the well-known differentiation rules of The Calculus.

### Forest Carbon Uptake and the Fundamental Theorem of Calculus

John Zobitz

Using the fundamental theorem of calculus and numerical integration, we investigate carbon absorption of ecosystems with measurements from a global database. The results illustrate the dynamic nature of ecosystems and their ability to absorb atmospheric carbon.

### CLimate Modeling in the Calculus and Differential Equations Classroom

Emek Köse and Jennifer Kunze

Students in college-level mathematics classes can build the differential equations of an energy balance model of the Earth’s climate themselves from a basic understanding of the background science. Here we use variable albedo and qualitative analysis to find stable and unstable equilibria of such a model, providing a problem or perhaps a research project for students interested in environmental studies and mathematics.

## STUDENT RESEARCH PROJECTS

### About the Pace of Climate Change: Write a Report to the President

This project allows students to understand better the scope and pace of climate change by conducting their own analysis. Using data readily available from NASA and NOAA, students can apply their knowledge of regression models (or of the modeling of rates of change). The results lend themselves to a writing assignment in which students demonstrate their understanding by explaining their findings to a non-expert.

### A Typology of Finite Groups

Eric Tou

This project classifies groups of small order using a group's center as the key feature. Groups of a given order n are typed based on the order of each group's center. Students are led through a sequence of exercises that combine proof-writing, independent research, and an analysis of specific classes of finite groups (including the dihedral, symmetric, and alternating groups) to produce a list of groups of each possible type. The project can serve as a capstone experience that unifies group theory with expository mathematical writing.

## PROBLEMS AND SOLUTIONS

Problems 1002 (correction), 1011-1015
Solutions 986-990

## REVIEW AND COMMENTARY

Mathematics for the Environment, by Martin Walter

Reviewed by Ben Fusaro

Ben Fusaro