June 2011 Contents
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Concentrating on applications, John Adam investigates branching blood vessels; Olofsson and Daileda use a Fibonacci-like sequence to explain growth rates of yeast colonies; Ray Rosentrater's application is to computer design; and the application by Eisenberg and Sullivan is within mathematics. —Walter Stromquist, Editor
Budding Yeast, Branching Processes, and Generalized Fibonacci NumbersPeter Olofsson and Ryan C. Daileda
An application of branching processes to a problem in cell biology is described, in which the generalized Fibonacci numbers known as k-nacci numbers play a crucial role. The k-nacci sequence is used to obtain asymptotics, computational formulas, and to justify certain practical simplifications of the biological model. Along the way, an explicit formula for the sum of k-nacci numbers is established.
A Modification of Sylvester’s Four Point ProblemBennett Eisenberg and Rosemary Sullivan
In 1865 Sylvester posed his famous four point problem “What is the probability that a random quadrilateral is convex?” This somewhat ill-defined question led to the problem of finding the minimum and maximum of the expected area of a triangle whose vertices are chosen with a uniform distribution over a convex region of area one. We modify this problem to that of finding the normalized expected area of a triangle whose vertices are chosen at random with an arbitrary probability distribution in the plane. The normalizing constant is the expected squared length of a line segment between two random points with the given distribution. We solve this modified problem in many important cases and conjecture that the maximum value of the normalized expected area occurs when the probability distribution is the uniform distribution on the circumference of a circle.
Representational EfficiencyC. Ray Rosentrater
When data with a nonuniform distribution (such as the leading digits of real numbers) is represented using fixed length codes, the representation is inefficient. How is the distribution of leading digits related to the distribution of real number mantissas and what implications does this have for the efficiency of the standard computer representation of real numbers?
Blood Vessel Branching: Beyond the Standard Calculus ProblemJohn A. Adam
Calculating the optimal angle for blood vessel branching is a standard calculus problem. However, optimality in that setting is judged by a cost functional that turns out not to give realistic results. We study a sequence of improvements to the cost functional, finally arriving at one that passes an important modeling test: From this last functional, we derive three empirical laws of blood vessel branching, originally proposed by German zoologist Wilhelm Roux.
Stirred, Not Shaken, by Stirling’s FormulaPaul Levrie
In this note an elementary proof of Stirling’s asymptotic formula for nW is given. The proof uses the Wallis formula for π and the trapezoidal rule for the calculation of a definite integral, with error estimate.
A Note on Disjoint Covering Systems—Variations on a 2002 AIME ProblemJohn W. Hoffman, W. Ryan Livingston, and Jared Ruiz
A covering system is a system of k arithmetic progressions whose union includes all integers. It is a disjoint covering system (or exact covering system) if the progressions are also pairwise disjoint, so that each integer is covered exactly once. This paper presents upper bounds on the number of consecutive integers which need to be checked to determine whether a covering system is a disjoint covering system. The bounds depend only on the number of congruences in the system. The results provide an analog of a theorem by R. B. Crittenden and C. L. Vanden Eynden from 1969 and are presented as solutions to some variations of a 2002 AIME Problem about painting a picket fence.
Convexity and Center of MassZsolt Lengvarszky
We show that convexity is preserved by centers of mass by considering regions between two convex functions both in a discrete and continuous sense. The proofs are partly based on theorems in the classical literature, however, elementary arguments are given as well. An example is presented to demonstrate that while the results are also true for 0th and 1st derivatives (convexity being tied to second derivatives), a simple generalization to higher derivatives is not possible.
Eigenvalues in Filled Julia SetsJonathon E. Fassett
We show how Julia sets can be introduced very naturally in a junior-level linear algebra course, as a way of exposing students to the contemporary area of complex dynamics. The standard definition of the filled Julia set of a polynomial is generalized to the setting of polynomial iteration of matrices. We prove that the eigenvalues of any matrix bounded under iteration by a polynomial must lie in the corresponding filled Julia set. A partial converse is obtained if the matrix is assumed to be diagonalizable. Still another partial converse is proven by assuming the spectrum of the matrix is contained in the interior of corresponding filled Julia set.