In this exposition, we explore the behavior of the Hardy–Littlewood maximal function of measures that are singular with respect to Lebesgue measure. We are going to prove that for every positive Borel measure that is singular with respect to Lebesgue measure , for all , where is a torus and is the Hardy–Littilewood maximal function… Continue reading On Hardy–Littlewood maximal function of singular measure
Month: June 2013
Alternating Fourier Coefficients
Suppose is a periodic function from to with period . Let be its Fourier coefficients, namely for all . Prove for all it is almost surely that function is in where is an infinite sequence of independent and identical random variables indexed by with equals either or with probability . I heard this problem from… Continue reading Alternating Fourier Coefficients
Number of non-isomorphic graphs
This expository essay is to test my understanding of the techniques used in More Bricks – More Walls?, Thirty-three Miniatures by Jiří Matoušek’s. We shall prove the sequence is unimodal, i.e., it is first nondecreasing and then, from some point on, non-increasing, where is the number of non-isomorphic graphs with vertices and edges. In particular,… Continue reading Number of non-isomorphic graphs
Fun with Hex
According to the folklore, The Hex game was invented by the Danish mathematician Piet Hein, who introduced it in 1942 at the Niels Bohr Institute. It was independently re-invented in 1947 by the mathematician John Nash at Princeton University. The rules are really simple. Each player has an allocated color, Red and Blue being conventional.… Continue reading Fun with Hex