Sam Northshield

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Stern’s diatomic sequence is a simply defined sequence with an amazing set of properties. Our goal is to present many of these properties—those that have most impressed the author. The diatomic sequence has a long history; perhaps first appearing in print in 1858 [28], it has been the subject of several papers [22, 23, 5, 9, 12, 13, 14]; see also [27,(More)
Let & be a ¿-regular graph and T the covering tree of S. We define a cogrowth constant of & in T and express it in terms of the first eigenvalue of the Laplacian on S . As a corollary, we show that the cogrowth constant is as large as possible if and only if the first eigenvalue of the Laplacian on 9 is zero. Grigorchuk's criterion for amenability of(More)
Let p be a quadratic polynomial over a splitting field K, and S be the set of zeros of p. We define an associative and commutative binary relation on G ≡ K∪{∞}−S so that every Möbius transformation with fixed point set S is of the form x “plus” c for some c. This permits an easy proof of Aitken acceleration as well as generalizations of known results(More)
We give two proofs that, for a nite regular graph, the reciprocal of Ihara's zeta function can be expressed as a simple polynomial times a determinant involving the adjacency matrix of the graph. The rst proof is based on representing radial symmetric eigenfunctions on regular trees in terms of certain polynomials. The second proof is a consequence of the(More)
ABSRACT. For a graph G, the " growth " of G is the constant c such that metric balls in the universal covering tree T grow like c n. The " cogrowth " of G is the constant c such that the intersection of the preimage in T of a vertex of G with metric balls grow with order c n. For example, the growth of a d-regular graph is d − 1 and the cogrowth of G is(More)
1. INTRODUCTION. This paper has its genesis in a problem the author first came upon while in college. Although the areas covered here are well travelled and nothing here is guaranteed original, it covers a pleasant nexus of many mathematical strands. Furthermore, we show the value of good notation and of reading an old master for solving a problem. Consider(More)
Imagine a sphere with its equator inscribed in an equilateral triangle. This Saturn-like figure will help us understand from where Cardano’s formula for finding the roots of a cubic polynomial p(z) comes. It will also help us find a new proof of Marden’s theorem, the surprising result that the roots of the derivative p(z) are the foci of the ellipse(More)
We give three proofs that the reciprocal of Ihara's zeta function can be expressed as a simple polynomial times a determinant involving the adjacency matrix of the graph. The rst proof, for regular graphs, is based on representing radial symmetric eigenfunctions on regular trees in terms of certain polynomials. The second proof, also for regular graphs, is(More)