Variants of Karamata’s iteration theorem

Variants of Karamata’s Iteration Theorem

Karamata’s Iteration Theorem is used to refine the asymptotic behavior of iterates of a function, under a more restrictive assumption than Karamata’s, but still involving regular variation. A second result gives a necessary and sufficient integral condition for convergence of a series of iterates. Historical background to the idea of regularly varying sequence precedes a short concluding sectio...

On Satisfying the LL-Iteration Theorem

It is shown that the conditions in the iteration theorem t’or Ll.-languages arc not sufhcient to characterize these languages. Therefore the question raised by Reatty [I] can be answered negatively. 1. On satisfying the LL-iteration theorem In Beatty [l] the question is raised whether satisfying the conditions of the iteration theorem for LL(k) languages is also sufficient to ensure that a lang...

On variants of a theorem of Schur

In a paper [1] published in 2001, we modified a famous theorem of Issai Schur, which asserts that if G is a group with center Z, such that G/Z is finite, then the commutator subgroup G′ = [G,G] is also finite. Our modification was twofold; in the first place, we confined ourselves to nilpotent groups G, so that we could use effective localization methods at an arbitrary family P of primes, and,...

Variants of the Hajnal-Szemerédi Theorem

The Hajnal-Szemer edi Theorem 6] states that a graph with hk vertices and minimum degree at least (h?1)k contains k vertex disjoint copies of K h. Its proof is not algorithmic. Here, we present an algorithm which, for a xed h, nds in such a graph k?C(h) vertex disjoint copies of K h in polynomial time in k, C(h) being a constant depending on h only. The proof suggests a variant of the theorem f...

Van der Waerden’s Theorem: Variants and “Applications”