In 1991, Persi Diaconis and Daniel Stroock obtained two canonical path bounds on the second largest eigenvalue for simple random walk on a connected graph, the Poincaré and Cheeger bounds, and they raised the question as to whether the Poincaré bound is always superior. In this paper, we present some background on these issues, provide an example where Cheeger beats Poincaré, establish some suf...

ni2+ supported on hydroxyapatite-core@shell γ-fe2o3 nanoparticles (γ-fe2o3@hap-ni2+) was found to be a useful catalyst for the synthesis of benzimidazole derivatives from o-phenylenediamine and aldehydes under solvent and solvent-free conditions at 80 °c. this reaction affords the corresponding benzimidazole derivatives compared with the classical reactions this method consistently gives a high...

A γ-quasi-clique in a simple undirected graph is a set of vertices which induces a subgraph with the edge density of at least γ for 0 < γ < 1. A cover of a graph by γ-quasi-cliques is a set of γ-quasi-cliques where each edge of the graph is contained in at least one quasi-clique. The minimum cover by γ-quasi-cliques problem asks for a γ-quasi-clique cover with the minimum number of quasi-clique...

These are notes from lectures given in the Queen Mary Combinatorics Study 1 Introductory Definitions In these notes, Γ = (V Γ, EΓ) denotes a graph, as we will use G to denote a group. All graphs considered will be simple, finite, connected and undirected. Definition An automorphism of Γ is a bijective function g : V Γ → V Γ such that v ∼ w if and only if g(v) ∼ g(w). The set of all automorphism...

Let Γ be a finitely generated group having the property that any action of any finite-index subgroup of Γ by homeomorphisms of the circle must have a finite orbit. (By a theorem of É. Ghys, lattices in simple Lie groups of real rank at least 2 have this property.) Suppose that such a Γ acts on a compact manifold M by automorphisms of a codimension-one C foliation, F . We show that if F has a co...

The perturbation Γ may be either random or deterministic. We will focus on two complementary situations. The first regime is when L is very mixing but non-dissipative, while Γ is dissipative. Their interaction leads to a dramatically increased dissipation, that can not be achieved by either of them alone. This phenomenon is known as relaxation enhancement. The second set of problems involves ra...