Probability—James Bernoulli's Theorem

On the Rate of Convergence of Krasnosel’skǐi–mann Iterations and Their Connection with Sums of Bernoullis

In this paper we establish an estimate for the rate of convergence of the Krasnosel’skǐı–Mann iteration for computing fixed points of non-expansive maps. Our main result settles the Baillon–Bruck conjecture [3] on the asymptotic regularity of this iteration. The proof proceeds by establishing a connection between these iterates and a stochastic process involving sums of non-homogeneous Bernoull...

Digital Borsuk-Ulam theorem

The aim of this paper is to compute a simplicial cohomology group of some specific digital images. Then we define ringand algebra structures of a digital cohomology with the cup product. Finally, we prove a special case of the Borsuk-Ulam theorem fordigital images.

Some Results on Baer's Theorem

Baer has shown that, for a group G, finiteness of G=Zi(G) implies finiteness of ɣi+1(G). In this paper we will show that the converse is true provided that G=Zi(G) is finitely generated. In particular, when G is a finite nilpotent group we show that |G=Zi(G)| divides |ɣi+1(G)|d′ i(G), where d′i(G) =(d( G /Zi(G)))i.

Pick's Theorem via Minkowski's Theorem

ACKNOWLEDGMENTS This note was inspired by Brian Davies, who challenged Nick Trefethen at Oxford by pointing out that the evaluation of the integral (7) would cause serious difficulties both analytically and numerically. One of the authors (T.S.) had the pleasure of attending Trefethen’s course in Oxford, where problems like Davies’s had to be solved on a weekly basis. Discussing the integral wi...

Lucas’s Theorem: a Great Theorem