Now showing items 8-20 of 20

    • 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 non-regular 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 root-finding 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 two-by-two 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.