Let n 2 and let˛2 V n be an element in the Higman-Thompson group V n. We study the structure of the centralizer of˛2 V n through a careful analysis of the action of h˛i on the Cantor set C. We make use of revealing tree pairs as developed by Brin and Salazar from which we derive discrete train tracks and flow graphs to assist us in our analysis. A consequence of our structure theorem is that element centralizers are finitely generated. Along the way we give a short argument using revealing tree pairs which shows that cyclic groups are undistorted in V n .

Examples can play a critical role in the exploration of conjectures and in the subsequent development of proofs. Although proof has been an object of extensive study, there is more to learn about the precise ways in which mathematicians leverage examples as they formulate proofs. In this paper, we present results from surveys and interviews with mathematicians that targets the role of examples in mathematicians' proof-related activity. Their responses shed light on specific example-related activity (including strategic example selection and use) and on the overarching ways in which they engage in such activity (including a focus on generalization and metacognition). We share illustrative excerpts from the surveys and interviews and discuss educational implications of the results.

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Let K denote a field and let X denote a finite non-empty set. Let Mat X (K) denote the K-algebra consisting of the matrices with entries in K and rows and columns indexed by X.In this paper, we give a linear algebraic characterization of a Cauchy matrix. To do so, we introduce the notion of a Cauchy pair. A Cauchy pair is an ordered pair of diagonalizable linear transformations (X, X) on a finite-dimensional vector space V such that X − X has rank 1 and such that there does not exist a proper subspace W of V such that XW ⊆ W and XW ⊆ W . Let V denote a vector space over K with dimension |X|. We show that for every Cauchy pair (X, X) on V , there exists an X-eigenbasis {v i } i∈X for V and an X-eigenbasis {w i } i∈X for V such that the transition matrix from {v i } i∈X to {w i } i∈X is Cauchy. We show that every Cauchy matrix arises as a transition matrix for a Cauchy pair in this way. We give a bijection between the set of equivalence classes of Cauchy pairs on V and the set of permutation equivalence classes of Cauchy matrices in Mat X (K).

The equitable presentation of U q (sl 2 ) was introduced in 2006 by Ito, Terwilliger, and Weng. This presentation involves some generators x, y, y −1 , z. It is known that {x r y s z t : r, t ∈ N, s ∈ Z} is a basis for the K-vector space U q (sl 2 ). In 2013, Bockting-Conrad and Terwilliger introduced a subalgebra A of U q (sl 2 ) spanned by the elements {x r y s z t : r, s, t ∈ N, r + s + t even}. We give a presentation of A by generators and relations. We also classify up to isomorphism the finite-dimensional irreducible A-modules, under the assumption that q is not a root of unity.