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MONTHLY, January 2005

**Touring the Calculus Gallery**

by William Dunham

wdunham@muhlenberg.edu

Here we visit an imaginary gallery whose exhibits trace the evolution of calculus from its beginnings with Newton and Leibniz up to the innovative results of Baire and Lebesgue. There are rooms featuring specific theorems of Euler, Cauchy, and other contributors to the enterprise that John von Neumann hailed as "the first achievement of modern mathematics."

**The Groups of Order Sixteen Made Easy**

by Marcel Wild

mwild@sun.ac.za

There are two types of group theory books. The first kind does not deal with the classification of groups of order sixteen, the second kind obtains it as a special case of a sophisticated theory of *p*-groups, and many details are left to the reader to verify. Here we give a complete classification of the fourteen groups of order sixteen that is based on *elementary* facts. More specifically, we look at extensions of order eight groups *N* by *C*_{2} and deal a lot with automorphisms. But no Sylow-theory, no theory of *p*-groups, no relatively free groups, no group actions, not even the structure of finite Abelian groups is invoked.

**A Harnack Inequality for Ordinary Differential Equations **

by Shif Berhanu and Ahmed Mohammed

berhanu@math.temple.edu, amohammed@bsu.edu

Harnack's inequality is a very powerful tool in studying the regularity of solutions for many elliptic and parabolic partial differential equations. We present an elementary proof of Harnack's inequality for nonnegative solutions of second-order ordinary differential equations with continuous coefficients.

**Figures of Constant Width on a Chessboard**

by Janko Hernández and Leonel Robert

janko@math.utoronto.ca, lrobert@math.utoronto.ca

A figure of constant width *w* on an *n* x *n * chessboard is a subset of squares of the chessboard such that every row, column, and diagonal of slope 1 or -1 contains either zero or *w* squares of the figure. It is easy to find a figure of width two for *n* as small as four. Notice that the solutions of the *n*-queens problem are figures of constant width one. Is it possible to find nonempty figures of constant width three or higher? As we show here, the answer turns out to be yes for every width *w* if *n* is suitably large.

**A Simpler Approach to Integration and the Fubini Theorem**

by H. S. Bear and Dale Myers

bear@math.hawaii.edu,dale@math.hawaii.edu

The paper outlines a simpler approach to integration, based on the upper and lower Darboux sums as in the calculus texts. The partitions are now countable families of disjoint measurable sets instead of finite sets of intervals. It is the change from intervals to measurable sets that provides the versatility of the Lebesgue integral. In addition, we adopt a weakened form of the axiom of choice that allows the assumption (as an additional axiom for the reals) that all sets are measurable. All plane sets are then measurable, and a simple proof of the Fubini theorem is possible.

**Notes**

**Double Integrals for Euler’s Constant and ln 4/π and an Analog of Hadjicostas’s Formula**

by Jonathan Sondow

jsondow@alumni.princeton.edu

**Roots of the Derivative of a Polynomial**

by Dan Romik

romik@wisdom.weizmann.ac.il

**Quaternions and Rotations in R**^{4}

by Joel L. Weiner and George R. Wilkens

grw@math.hawaii.edu, joel@math.hawaii.edu

**Another Proof of the Fundamental Theorem of Algebra**

by José Carlos Santos

jcsantos@fc.up.pt

**Evolution ofÂ….**

Riemann’s Geometric Ideas

by B. F. Kagan

**Problems and Solutions**

**Reviews**

**Turing: A Novel About Computation**

by Christos H. Papadimitrious

Reviewed by Steve Kennedy

skennedy@mathcs.carleton.edu

**The Magic of Numbers**

by Benedict Gross and Joe Harris

Reviewed by Marion D. Cohen

mathwoman199436@aol.com