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Browse Classroom Capsules and Notes

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Displaying 41 - 50 of 1214

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The author provides a combinatorial explanation of a "striking" result from Herbert Wilf`s book “generatingfunctionology,” equating the likelihood of certain permutations...

Every integer can be expressed in base \(2\) using the set \(\{-1, 0, 1\}\) as coefficients. Does one need to use this set, or might another set of numbers do as well? The author investigates...

The authors study period-\(3\) orbits of the logistic function \( f_r(x)=rx(1-x)\), and provide another derivation of the fact that \(r_0=1+\sqrt {8}\), where \( r_0\) is the smallest positive...

Andrew Wiles proves that Fermat's Last Theorem is false for integer exponents larger than \(2 \).  Using the Gelfond-Schneider Theorem on transcendental numbers, the author generalizes...

A standard square in the xy-plane is a unit square whose corners have integer coordinates.  This note shows that when a polyomino consisting of an odd number of standard squares is cut into...

This note shows that graphs with \(n\) vertices containing no complete graph with \(r \) vertices, have no more than \( (r - 2)n^2/(2r - 2) \) edges , for \( r \geq 2\).

Let \(X\) be a nonempty metric space without isolated points.  If \(G \) is a countable intersection of open sets, the author shows that there is a function \(\phi (x) \) that is continuous...

Using a result on periodic continued fractions, the author presents a rational function method of approximating square roots that is faster than Newton's method.

Let \(G\) be a group and \(C(a)\) be the centralizer of \(a\in G\).  The author studies the properties of the skew centralizer \(B(a)=\{x\in G :  xa=a^{-1}x \}\) and the reversing...

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