Sums across the rows of Pascal's triangle yield powers of 2 while certain diagonal sums yield the Fibonacci numbers which are asymptotic to powers of the golden ratio. Sums across other diagonals yield quantities asymptotic ...

The Lyness equation, x(n+1)=(x(n)+k)/x(n-1), can be though of as an equation defined on the 2-regular tree T2: we can think of every vertex of T2 as a variable so that if x and z are the vertices adjacent to y, then x,y,z ...

Lester Ford introduced Ford Circles in 1938 in order to geometrically understand the approximation of an irrational number by rational numbers. We shall construct Ford circles by a recursive geometric procedure and by a ...

Newton's method applied to a quadratic polynomial converges rapidly to a root for almost all starting points and almost all coefficients. This can be understood in terms of an associative binary operation arising from 2 x ...

We present a short proof of Descartes Circle Theorem on the curvature-centers of four mutually tangent circles. Key to the proof is associating an octahedral configuration of spheres to four mutually tangent circles. We ...

We present a short new proof that the continued fraction of a quadratic irrational eventually repeats. The proof easily generalizes; we construct a large class of functions which, when iterated, must eventually repeat when ...

Given a conservative, spatially homogeneous Markov process X on an homogeneous spaces χ, we show that if the bottom of the spectrum of the generator of X is zero then the Martin boundary of contains a unique point fixed ...

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 ...

Given a simple random walk on an undirected connected graph, the commute time of the vertices x and y is defined as C(x,y) = ExTy+EyTx. We give a new proof, based on the optional sampling theorem for martingales, of the ...

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 finitely-intersecting ...