# 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 strikingresult from Herbert Wilf`s book “generatingfunctionology” , equating the likelyhood of certain permutations with the...

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...