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Associativity of the Secant Method
(American Mathematical Monthly, 2002)Iterating a function like 1+1/x gives a sequence which converges to the Golden Mean but does so at a much slower rate than those sequences derived from Newton's method or the secant method. There is, however, a surprising ... 
On Iterates of Moebius transformations on fields
(Mathematics of Computation, 1997)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 Moebius transformation with fixed point ... 
A note on the Zeta Function of a Graph
(Journal of Combinatorial Theory Series B, 1998)The number of splanning trees in a finite graph is first expressed as the derivative (at 1) of a determinant and then in terms of a zeta function. This generalizes a result of Hashimoto to nonregular graphs. 
On Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,...
(American Mathematical Monthly, 2010)We investigate several of the many interesting properties of the title sequence. In particular, we focus on the combinatorics of the sequence (e.g., what the numbers count), some parallels with the Fibonacci sequence, some ... 
Sums across Pascal's triangle modulo 2
(Congressus Numerantium, 2010)We consider sums of the binomial coefficients C(i + j, i) modulo 2 over lines ai + bj = n. Many interesting sequences (old and new) arise this way. 
On two types of exotic addition
(Aequationes Mathematicae, 2009)We classify, under reasonable assumptions, all differentiable functions f for which the 'secant method' [xf(y) yf(x)]/[f(y) f(x)] is continuous and associative. Further, we classify all differentiable functions for which ... 
On integral Apollonian circle packings
(Journal of Number Theory, 2006)The curvatures of four mutually tangent circles with disjoint interior form what is called a Descartes quadruple. The four smallest curvatures of circles in an Apollonian circle packing form what is called a root Descartes ... 
Amenability and superharmonic functions
(Proceedings of the American Mathematical Society, 1993)Let G be a countable group and u a symmetric and aperiodic probability measure on G . We show that G is amenable if and only if every positive superharmonic function is nearly constant on certain arbitrarily large subsets ... 
Geodesics and Bounded Harmonic Functions on Infinite Graphs
(1991)It is shown there that an infinite connected planar graph with a uniform upper bound on vertex degree and rapidly decreasing Green's function (relative to the simple random walk) has infinitely many pairwise finitelyintersecting ... 
Cogrowth of Regular Graphs
(Proceedings of the American Mathematical Society, 1992)Let G be a dregular graph and T the covering tree of G. We define a cogrowth constant of G in T and express it in terms of the first eigenvalue of the Laplacian on G. As a corollary, we show that the cogrowth constant is ...