MONTHLY, January 2005
Touring the Calculus Gallery
by William Dunham
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
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 C2 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
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
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
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.
Double Integrals for Euler’s Constant and ln 4/π and an Analog of Hadjicostas’s Formula
by Jonathan Sondow
Roots of the Derivative of a Polynomial
by Dan Romik
Quaternions and Rotations in R4
by Joel L. Weiner and George R. Wilkens
Another Proof of the Fundamental Theorem of Algebra
by José Carlos Santos
Riemann’s Geometric Ideas
by B. F. Kagan
Problems and Solutions
Turing: A Novel About Computation
by Christos H. Papadimitrious
Reviewed by Steve Kennedy
The Magic of Numbers
by Benedict Gross and Joe Harris
Reviewed by Marion D. Cohen