We prove a Glivenko-Cantelli type strong law of large numbers for the pair correlation of independent random variables. Except for a few powers of logarithms the results obtained are sharp. Similar estimates hold for the pair correlation of lacunary sequences {nkω} mod 1.

1 School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, P.O. Box 14155-6455, Tehran, Iran [email protected] 2 Department of Mathematics, Sharif University of Technology, P.O. Box 11365-9415, Tehran, Iran [email protected] 3 Department of Mathematics, Statistics and Computer Science, Marquette University, P.O. Box 1881, Milwaukee, WI 53201, ...

In this paper, we construct a Hamiltonian Floer theory based invariant called relative symplectic cohomology, which assigns module over the Novikov ring to compact subsets of closed manifolds. We show existence restriction maps, and prove some basic properties. Our main contribution is identify natural geometric situation in cohomology two satisfy Mayer-Vietoris property. This tailored work und...

Directoids as a generalization of semilattices were introduced by J. Ježek and R. Quackenbush in 1990. We modify the concept of a pseudocomplement for commutative directoids and study several basic properties: the Glivenko equivalence, the set of the so-called boolean elements and an axiomatization of these algebras.