Szemerédi’s regularity lemma proved to be a powerful tool in extremal graph theory. Many of its applications are based on the so-called counting lemma: if G is a kpartite graph with k-partition V1∪ · · ·∪Vk, |V1| = · · · = |Vk| = n, where all induced bipartite graphs G[Vi, Vj ] are (d, ε)-regular, then the number of k-cliques Kk in G is d( k 2)nk(1± o(1)). Frankl and Rödl extended Szemerédi’s r...

We develop diagram techniques for proving confluence in abstract reductions systems. The underlying theory gives a systematic and uniform framework in which a number of known results, widely scattered throughout the literature, can be understood. These results include Newman's lemma, Lemma 3.1 of Winkler and Buchberger, the Hindley Rosen lemma, the Request lemmas of Staples, the Strong Confluen...

We give a randomized polynomial-time identity test for noncommutative circuits of polynomial degree based on the isolation lemma. Using this result, we show that derandomizing the isolation lemma implies noncommutative circuit size lower bounds. More precisely, we consider two restricted versions of the isolation lemma and show that derandomizing each of them implies nontrivial circuit size low...

We have seen the Lovász Local Lemma and its stronger variant, Shearer’s Lemma, which is unfortunately quite unwieldy in applications. Quite recently, researchers in mathematical physics discovered an intermediate form of the lemma, which seems to give results close to Shearer’s Lemma but it is much more easily applicable. First let us review a connection between Shearer’s Lemma and statistical ...

The so-called S-lemma has played an important role in optimization, both in theory and in applications. The significance of S-lemma is especially pronounced in control theory, robust optimization, and non-convex quadratic optimization. Hitherto, S-lemma is however established only in the domain of quadratic functions. In this paper we shall extend the notion of S-lemma to the class of univariat...

We develop diagram techniques for proving con uence in abstract reduc tions systems The underlying theory gives a systematic and uniform framework in which a number of known results widely scattered through out the literature can be understood These results include Newman s Lemma Lemma of Winkler and Buchberger the Hindley Rosen Lemma the Request Lemmas of Staples the Strong Con uence Lemma of ...