We investigate decompositions of a graph into a small number of low diameter subgraphs. Let P (n, 2, d) be the smallest k such that every graph G = (V, E) on n vertices has an edge partition E = E0 ∪ E1 ∪ . . . ∪ Ek such that |E0| ≤ 2n and for all 1 ≤ i ≤ k the diameter of the subgraph spanned by Ei is at most d. Using Szemerédi’s regularity lemma, Polcyn and Ruciński showed that P (n, 2, 4) is...

We prove a variant of the abstract probabilistic version of Szemerédi’s regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random variables in Lp for any p > 1. Our approach is based on martingale difference sequences.

We present a regularity lemma for Boolean functions f : {−1, 1}n → {−1, 1} based on noisy influence, a measure of how locally correlated f is with each input bit. We provide an application of the regularity lemma to weaken the conditions on the Majority is Stablest Theorem. We also prove a “homogenized” version stating that there is a set of input bits so that most restrictions of f on those bi...

A complex algebra A is called ideally factored if Ia = Ca is a left ideal of A for all a ∈ A. In this article, we investigate some interesting properties of ideally factored algebras and show that these algebras are always Arens regular but never amenable. In addition, we investigate ρhomomorphisms and (ρ, τ)-derivations on ideally factored algebra.

Let G be a locally compact non-compact group. We show that under a very mild assumption on the weight function w, the weighted group algebra L1(G, w) is strongly Arens irregular in the sense of [Dal–Lam–Lau 01]. To this end, we first derive a general factorization theorem for bounded families in the L∞ ( G, w ) ∗ -module L∞ ( G, w ) .

We revisit regularity of SLE trace, for all κ 6= 8, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia–Rodemich–Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index min(1 + κ/8, 2), improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Vik...