# Browse Classroom Capsules and Notes

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Displaying 41 - 50 of 1212

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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...