A Leonard pair is an ordered of diagonalizable linear maps on a finite-dimensional vector space, that each act eigenbasis for the other one in irreducible tridiagonal fashion. In present paper we consider type pair, said to have spin. The notion spin model was introduced by V.F.R. Jones construct link invariants. symmetric matrix over C satisfies two conditions, called II and III conditions. It...

The positive part Uq+ of Uq(slˆ2) has a presentation with two generators W0, W1 and relations called the q-Serre relations. algebra contains some elements, said to be alternating. There are four kinds alternating denoted {W−k}k∈N, {Wk+1}k∈N, {Gk+1}k∈N, {G˜k+1}k∈N. elements each kind mutually commute. A tridiagonal pair is an ordered diagonalizable linear maps A,A⁎ on nonzero, finite-dimensional...

We consider bipartite subgraphs of sparse random graphs that are regular in the sense of Szemerédi and, among other things, show that they must satisfy a certain local pseudorandom property. This property and its consequences turn out to be useful when considering embedding problems in subgraphs of sparse random graphs.

We consider the distribution of cycle counts in a random regular graph, which is closely linked to the graph’s spectral properties. We broaden the asymptotic regime in which the cycle counts are known to be approximately Poisson, and we give an explicit bound in total variation distance for the approximation. Using this result, we calculate limiting distributions of linear eigenvalue statistics...

We are interested in (ε)-regular bipartite graphs which are the central objects in the regularity lemma of Szemerédi for sparse graphs. A bipartite graph G = (A]B,E) with density p = |E|/(|A||B|) is (ε)-regular if for all sets A′ ⊆ A and B′ ⊆ B of size |A′| ≥ ε|A| and |B ′| ≥ ε|B|, it holds that |eG(A, B′)/(|A′||B′|) − p| ≤ εp. In this paper we prove a characterization for (ε)-regularity. That ...