By bgillespie, on August 12th, 2010
As a final project at the 2010 University of Georgia mathematics REU, I wrote a manuscript discussing the properties and meaning of a mathematical notion called Fourier Pseudorandomness, which provides a quantitative measure of the randomness (in some sense) of a finite set of integers. Some of the more involved proofs require a first course . . . [Read More]
By bgillespie, on January 6th, 2010
Sums are a beautiful notion which are sometimes taken for granted. 2 + 2 = 4, the sum from 1 to infinity of 1/2^{n} is equal to 1, and so forth. In the case of countably infinite sums, questions of convergence become important. But what happens when you have a sum over uncountably many summands? It . . . [Read More]
By bgillespie, on November 23rd, 2009
Let {q_{n}} be an enumeration of the rationals, and for each n, let U_{n} be an open interval of length 1/2^{n} centered at q_{n}. Denote the union of all U_{n} by U. Then U is a dense open . . . [Read More]
By bgillespie, on November 21st, 2009
Let k ≥ 0, and let n = k(2k + 1). Then we have:
n^{2} + (n+1)^{2} + … + (n+k)^{2} = (n+k+1)^{2} + … + (n+2k)^{2}.
For example, 0^{2} = 0, 3^{2} + 4^{2} = 5^{2}, 10^{2} + 11^{2} + 12^{2} = 13^{2} + 14^{2}, 21^{2} + 22^{2} + 23^{2} + 24^{2} = 25^{2} . . . [Read More]

