Let each of n particles starting at the origin in Z perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set A(n) of n occupied sites is (with high probability) close to a disk Br of radius r = √ n/π. We show that the discrepancy between A(n) and the disk is at most logarithmic in the radius: i.e., there is an ...

The Riemann zeta function ζ(s) is defined by ζ(s) = ∑∞ n=1 1 ns for R(s) > 1 and can be extended to a regular function on the whole complex plane deleting its unique pole at s = 1. The Riemann hypothesis is a conjecture made by Riemann in 1859 asserting that all non-trivial zeros for ζ(s) lie on the line R(s) = 12 , which is equivalent to the prime number theorem in the form of π(x)−Li(x) = O(x...

In previous work of Gow, Ohmori, Lusztig and the author, the Schur indices of all unipotent characters of finite groups of Lie type have been explicitly determined except for six cases in groups of type F4, E7 and E8. In this paper, we show that the Schur indices of all cuspidal unipotent characters for type F4 and E8 are 1, assuming that the group is defined over a field of “good” characterist...

In this paper, we prove the Spanne-Guliyev type boundedness of the generalized fractional integral operator Iρ from the vanishing generalized local Morrey spaces V LM {x0} p,φ1 to V LM {x0} q,φ2 , 1 < p < q < ∞, and from the space V LM {x0} 1,φ1 to the weak space VWLM {x0} q,φ2 , 1 < q < ∞. We also prove the Adams-Guliyev type boundedness of the operator Iρ from the vanishing generalized Morrey...

For two fixed graphs T and H let ex(G(n, p), T,H) be the random variable counting the maximum number of copies of T in an H-free subgraph of the random graph G(n, p). We show that for the case T = Km and χ(H) > m the behavior of ex(G(n, p),Km, H) depends strongly on the relation between p and m2(H) = maxH′⊂H,|V (H′)|′≥3 { e(H′)−1 v(H′)−2 } . When m2(H) > m2(Km) we prove that with high probabili...