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Browse Classroom Capsules and Notes

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The author relates a personal story of a post-WWII assignment in the U.S. Navy.  His analysis of data from a test of a harbor mine-detecting algorithm leads to a novel construction of the...

When is the average of sums of powers of integers itself a sum of the first \(n\) integers raised to a power? We provide all solutions when averaging two sums, and provide some conditions...

The Cantor subset of the unit interval \([0,1)\) is large in cardinality and also large algebraically, that is, the smallest subgroup of \([0,1)\) generated by the Cantor set (using addition mod...

The author presents a variety of double and triple integrals designed to exercise a student's ability to visualize the interplay between an integrand and the region over which it is...

Using two well known criteria for the diagonalizability of a square matrix plus an extended form of Sylvester's Rank Inequality, the author presents a new condition for the diagonalization...

Here is a technique for proving the fundamental theorems of analysis that provides a unified way to pass from local properties to global properties on the real line, just as ordinary induction...

A typical first course on linear algebra is usually restricted to vector spaces over the real numbers and the usual positive-definite inner product.  Hence, the proof that \(\dim(\mathcal{S...

Hölder’s inequality is here applied to the Cobb-Douglas production function to provide simple estimates to total production.

Mathematical elegance is illustrated by rewriting the product and quotient rules of basic calculus in strikingly parallel forms. Applications are given for which these forms suggest meaning....

The author provides a somewhat recreational application of limit interchange and L'Hopital's rule. The goal is to hint at the importance of limit/sum interchange in Analysis without,...

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