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 ...

(c0 + c1ζ + · · ·+ cp−2ζ) ≡ c0 + c1 + · · ·+ cp−2 mod p. The number p is not prime in Z[ζ], as (p) = (1 − ζ)p−1, so congruence mod p is much stronger than congruence mod 1− ζ, where all classes have integer representatives. Of course not every element of Z[ζ] that is congruent to a rational integer mod p is a pth power, but Kummer discovered a case when this converse statement is true, for cert...

This lemma was then used the following year as a crucial step in the proof of the well-known Bishop-Phelps theorem [1] that every Banach space is subreflexive; in other words, every functional on a Banach space E can be approximated by a norm-attaining functional on the same space. The original proof of this lemma uses the Hahn-Banach theorem and is therefore fairly abstract. In this note, we p...

The isolation lemma of Mulmuley et al [MVV87] is an important tool in the design of randomized algorithms and has played an important role in several nontrivial complexity upper bounds. On the other hand, polynomial identity testing is a well-studied algorithmic problem with efficient randomized algorithms and the problem of obtaining efficient deterministic identity tests has received a lot of...

We introduce a new variant of Szemerédi’s regularity lemma which we call the sparse regular approximation lemma (SRAL). The input to this lemma is a graph G of edge density p and parameters , δ, where we think of δ as a constant. The goal is to construct an -regular partition of G while having the freedom to add/remove up to δ|E(G)| edges. As we show here, this weaker variant of the regularity ...

Green [Geometric and Functional Analysis 15 (2005), 340–376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.

Three examples are provided which demonstrate that “convergence in probability” versions of the Toeplitz lemma, the Cesàro mean convergence theorem, and the Kronecker lemma can fail. “Mean convergence” versions of the Toeplitz lemma, Cesàro mean convergence theorem, and the Kronecker lemma are presented and a general mean convergence theorem for a normed sum of independent random variables is e...

In this paper, we state the positive real lemma and the strictly positive real lemma (KYP lemma) for non-minimal realization systems. First we show the positive real lemma for stabilizable and observable systems under only the constraint on the regularity of the systems, by using the generalized algebraic Riccati equation. Moreover we show that the solution of the Lyapunov equation in the posit...

Szemerédi’s Regularity Lemma is an important tool for analyzing the structure of dense graphs. There are versions of the Regularity Lemma for sparse graphs, but these only apply when the graph satisfies some local density condition. In this paper, we prove a sparse Regularity Lemma that holds for all graphs. More generally, we give a Regularity Lemma that holds for arbitrary real matrices.