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]
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Sums are a beautiful notion which are sometimes taken for granted. 2 + 2 = 4, the sum from 1 to infinity of 1/2n 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]
Let k ≥ 0, and let n = k(2k + 1). Then we have: n2 + (n+1)2 + … + (n+k)2 = (n+k+1)2 + … + (n+2k)2. For example, |
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Let {qn} be an enumeration of the rationals, and for each n, let Un be an open interval of length 1/2n centered at qn. Denote the union of all Un by U. Then U is a dense open 