作者：Logunov, Alexander
摘要：Let u be a harmonic function in the unit ball B(0, 1) subset of R-n, n >= 3, such that u(0) = 0. Nadirashvili conjectured that there exists a positive constant c, depending on the dimension n only, such that Hn-1 ({u = 0} boolean AND B ) >= c. We prove Nadirashvili's conjecture as well as its counterpart on C-infinity-smooth Riemannian manifolds. The latter yields the lower bound in Yau's conjecture. Namely, we show that for any compact C-infinity-smooth Riemannian manifold M (without boundary...
作者：Bamler, Richard H.
摘要：In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of codimension >= 4. We also establish a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. These results are based on a compactness theorem for Ricci flows with bounded scalar curvature, which states that any sequence of suc...
作者：Balister, Paul; Bollobas, Bela; Morris, Robert
摘要：Consider a random sequence of N integers, each chosen uniformly and independently from the set {1,..., x}. Motivated by applications to factorization algorithms such as Dixon's algorithm, the quadratic sieve, and the number field sieve, Pomerance in 1994 posed the following problem: how large should N be so that, with high probability, this sequence contains a subsequence, the product of whose elements is a perfect square? Pomerance determined asymptotically the logarithm of the threshold for ...
