Browsing by Author "Northshield, Sam"
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Amenability and superharmonic functions
Northshield, Sam (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 ... 
Associativity of the Secant Method
Northshield, Sam (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 ... 
Cogrowth of Regular Graphs
Northshield, Sam (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 ... 
Complex Descartes Circle Theorem
Northshield, Sam (American Mathematical Monthly, 2014)We present a short proof of Descartes Circle Theorem on the curvaturecenters of four mutually tangent circles. Key to the proof is associating an octahedral configuration of spheres to four mutually tangent circles. We ... 
Geodesics and Bounded Harmonic Functions on Infinite Graphs
Northshield, Sam (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 ... 
Integrating across Pascal's triangle
Northshield, Sam (Mathematical Analysis and Applications, 2011)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 ... 
A Lyness equation for graphs
Northshield, Sam (Journal of Difference Equations and Applications, 2012)The Lyness equation, x(n+1)=(x(n)+k)/x(n1), can be though of as an equation defined on the 2regular 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 ... 
A New Parameterization of Ford Circles
Northshield, Sam; McGonagle, Annmarie (Pi Mu Epsilon Journal, 2014)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 ... 
Not mixing is just as cool
Northshield, Sam (Mathematics Magazine, 2007)Newton's law of cooling, a staple of the Calculus curriculum, is an empirical law not meant for mathematical proof. However, we show it is mathematically equivalent to the intuitively appealing principle that the average ... 
A note on the Zeta Function of a Graph
Northshield, Sam (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 integral Apollonian circle packings
Northshield, Sam (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 ... 
On Iterates of Moebius transformations on fields
Northshield, Sam (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 ... 
On Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,...
Northshield, Sam (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 ... 
On the Commute Time of Random Walks on Graphs
Northshield, Sam; Palacios, Jose Luis (Brazilian Journal of Probability and Statistics, 1995)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 ... 
On the spectrum and Martin boundary of homogeneous spaces
Northshield, Sam (Statistics and Probability Letters, 1995)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 ... 
On two types of exotic addition
Northshield, Sam (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 ... 
A rootfinding algorithm for cubics
Northshield, Sam (Proceedings of the American Mathematical Society, 2013)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 ... 
A short proof and generalization of Lagrange's theorem on continued fractions
Northshield, Sam (American Mathematical Monthly, 2011)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 ... 
Square Roots of 2x2 Matrices
Northshield, Sam (2010)This paper is designed to pique the interest of undergraduate students who are familiar with the concepts of linear algebra. We investigate five methods of computing square roots of twobytwo matrices. Each method gives ... 
Sums across Pascal's triangle modulo 2
Northshield, Sam (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.