Dan Kalman

Six Ways to Sum a Series

The concept of an infinite sum is mysterious and intriguing. How can you add up an infinite number of terms? Yet, in some contexts, we are led to the contemplation of an infinite sum quite naturally. For example, consider the calculation of a decimal expansion for 1/3. The long division algorithm generates an endlessly repeating sequence of steps, each of which adds one more 3 to the decimal expansion. We imagine the answer therefore to be an endless string of 3’s, which we write .333· · ·. In essence we are defining the decimal expansion of 1/3 as an infinite sum

Polynomial Equations and Circulant Matrices

1. INTRODUCTION. There is something fascinating about procedures for solving low degree polynomial equations. On one hand, we all know that while general solutions (using radicals) are impossible beyond the fourth degree, they have been found for quadratics, cubics, and quartics. On the other hand, the standard solutions for the the cubic and quartic are complicated, and the methods seem ad hoc. How is a person supposed to remember them? It just seems that there ought to be a simple, memorable, unified method for all equations through degree four. Approaches to unification have been around almost as long as the solutions themselves. In 1545, Cardano published solutions to both the cubic and quartic, attributing the former to Tartaglia and the latter to Ferrari. After subsequent work failed to solve equations of higher degree, Lagrange undertook an analysis in 1770 to explain why the methods for cubics and quartics are successful. From that time right down to the present, efforts have persisted to illuminate the solutions of cubic and quartic equations ; see [21]. In this paper we present a unified approach based on circulant matrices. The idea is to construct a circulant matrix with a specified characteristic polynomial. The roots of the polynomial thus become eigenvalues, which are trivially found for circulant matrices. This circulant matrix approach provides a beautiful unity to the solutions of cubic and quartic equations, in a form that is easy to remember. It also reveals other interesting insights and connections between matrices and polynomials, as well as cameo roles for interpolation theory and the discrete Fourier transform. We begin with a brief review of circulants, and then show how circulants can be used to find the zeroes of low degree polynomials. Succeeding sections explore how the circulant method is related to other approaches, present additional applications of circulants in the study of polynomial roots, and discuss generalizations using other classes of matrices.

An Elementary Proof of Marden's Theorem

I call this Marden’s Theorem because I first read it in M. Marden’s wonderful book [6]. But this material appeared previously in Marden’s earlier paper [5]. In both sources Marden attributes the theorem to Siebeck, citing a paper from 1864 [8]. Indeed, Marden reports appearances of various versions of the theorem in nine papers spanning the period from 1864 to 1928. Of particular interest in what follows below is an 1892 paper by Maxime Bocher [1]. In his presentation Marden states the theorem in a more general form than given above, corresponding to the logarithmic derivative of a product (z − z1)1(z − z2)2(z − z3)3 where the only restriction on the exponents m j is that they be nonzero, and with a general conic section taking the place of the ellipse. For this discovery he credits Linfield [4], who obtained it as a corollary to an even more general result “established by the use of line coordinates and polar forms.” Marden asserts the desirability of a more elementary proof, and proceeds to give one based on the optical properties of conic sections. Interestingly, Marden’s proof, which appears in basically the same form in both his paper and his book, is incomplete for reasons that will be made clear below. A closely related argument in Bocher’s paper is also incomplete, although in a different way. By combining the two arguments, a complete proof of Marden’s Theorem is obtained. Moreover, the proof is completely elementary, requiring very little beyond standard topics from undergraduate mathematics. To be honest, there are rather a lot of these topics required, spanning analytic geometry, linear algebra, complex analysis, calculus, and properties of polynomials. To me, the way all of these topics weave together is part of the charm of the theorem, and presenting the proof is the primary motivation for this paper. In an on-line paper [3] a more completely self-contained exposition is provided, with hypertext links to discussions of many of the necessary background topics, as well as animated graphics dramatizing some of the geometric ideas of the proof. Before proceeding further, some additional observations may be illuminating. First, it is possible that the unique inscribed ellipse mentioned in the theorem is actually a circle. In this case the foci coincide, indicating that p′(z) has a double root. This case can only occur if the circumscribing triangle is equilateral. In fact, this special case is easy to verify by assuming that p′(z) has a double root, and deducing the form of p. The roots of p are then seen to be vertices of an equilateral triangle centered at the repeated root of p′.

A Generalized Logarithm for Exponential-Linear Equations

Dan Kalman (kalman@email.cas.american.edu) joined the mathematics faculty at American University in 1993, following an eight year stint in the aerospace industry and earlier teaching positions in Wisconsin and South Dakota. He has won three MAA writing awards, is an Associate Editor of Mathematics Magazine, and served a term as Associate Executive Director of the MAA. His interests include matrix algebra, curriculum development, and interactive computer environments for exploring mathematics, especially using Mathwright software.

The Most Marvelous Theorem in Mathematics

Picture the graph of a cubic, with three x intercepts (also known as the roots). We know from calculus that the derivative of the cubic will have two roots, one between each pair of consecutive roots. This situation extends in an amazing way to complex numbers, where the three roots of the cubic become vertices of a triangle in the complex plane. The roots of the derivative are within the triangle. But where? FEATURED SPEAKER Dr. Dan Kalman American University Dan Kalman Dan Kalman has been writing about and teaching mathematics for 30 years. A graduate of Harvey Mudd College (BS, 1974) and the University of Wisconsin (PhD, 1980) he is a Professor of Mathematics at American University, Washington, DC. He previously held faculty positions at the University of Wisconsin, Green Bay, and Augustana College, Sioux Falls, among other institutions, and worked for several years as an applied mathematician at the Aerospace Corporation. He also served for one year as an Associate Executive Director of the MAA. Kalman has been an invited speaker at numerous national and regional mathematics conferences, and has spoken to student clubs and PME chapters many times. His mathematical writing has been recognized with multiple MAA awards: a Ford Award in 2009, Allendoerfer Awards in 1998 and 2002, Polya Awards in 1994 and 2002, and an Evans Award in 1997. He is the author of two books published by the MAA. Kalman has served on the Editorial Boards for several MAA publications and is currently on the board for Math Horizons. The Most Marvelous Theorem in Mathematics The 13th Annual Michigan Undergraduate Mathematics Conference Saturday, October 9, 2010 DeVos Center, GVSU Grand Rapids Campus Abstract Submissions Deadline for Student Presentations: September 27, 2010 at 5:00 PM (ET) Registration Deadline: October 1, 2010 at 5:00 PM (ET) www.mumc2010.orgSubmissions Deadline for Student Presentations: September 27, 2010 at 5:00 PM (ET) Registration Deadline: October 1, 2010 at 5:00 PM (ET) www.mumc2010.org Sponsors: Major Funding for the 2010 MUMC is provided by Grand Valley State University and by NSF grant DMS–0846477 through the MAA Regional Undergraduate Mathematics Conference Program (see www.maa.org/rumc). Other support has been generously provided by:

Teaching Linear Algebra: Issues and Resources

Jane Day (day@sjsumcs.sjsu.edu) has been at San Jose State University since 1982. She participated in the Linear Algebra Curriculum Study Group in 1990 and has been particularly aware since then of how diverse the audience is for linear algebra. In 1994 she received the annual Award for Distinguished College or University Teaching from the MAA Northern California-Nevada Section. Her research interest is matrix theory, and she is presently on the Editorial Board of the American Mathematical Monthly.

Fractions with Cycling Digit Patterns

A branch of number theory that has long captured the curiosity of both profes? sional and amateur mathematicians centers on the representations of numbers. There are innumerable interesting connections between inherent properties of the integers (such as divisibility and prime factorization) and the patterns revealed by the base 10 representation of the numbers. As a very simple example, there are divisibility tests that depend on the digit sequence of the base 10 representation. A number is divisible by 3 if the sum of the digits is divisible by 3; a number is divisible by 11 if the alternating digit sum is divisible by 11. In this note, I present a curious result about the patterns of repeating decimals for fractions with certain denominators. In particular I will focus on denominators, like 7, for which any numerator produces essentially the same pattern of repeating decimal digits, differing only in the starting point for the pattern. For example, 1/7 = 142857 142857...,2/7 = .2857 142857 142857..., and the repeating deci? mals for the other fractions, 3/7, 4/7, 5/7, and 6/7, all behave similarly. This is the cycling digits property. Other denominators with the cycling digits property include 17 and 19. Throughout the paper, only proper fractions will be considered, so when a fraction m/n appears it is assumed that 1 < m < n. Before proceeding, I should renounce any claim to being the first to discover the results below. Very similar results are presented in [2, pp. 147-155] and [3]. These treatments make use of standard tools of number theory, notably modular arith? metic. My approach, in contrast, rests mainly on understanding the basic algo? rithms of arithmetic, especially long division. This somewhat naive view is widely accessible and leads to some interesting insights. A charming reference for the interested reader is [6]. By combining several theorems that are scattered through its pages, it is possible to obtain the results proven here (see the Midy property section), as well as much more. This subject strikes me as particularly suitable for investigation by students at the secondary and undergraduate level. With the current emphasis on applicability as the secret to making mathematics interesting, we should not lose sight of the whimsical side of the subject. Perhaps this topic will help some students to experience the fascination of order unlooked-for and the puzzle-like quality of mathematics motivated by pure intellectual curiosity.

Another Way to Sum a Series: Generating Functions, Euler, and the Dilog Function

Abstract It is tempting to try to reprove Eulers famous result that using power series methods of the sort taught in calculus 2. This leads to , the evaluation of which presents an obstacle. With two key identities the obstacle is overcome, proving the desired result. And who discovered the requisite identities? Euler! Whether he knew of this proof remains to be discovered.

An Underdetermined Linear System for GPS

Dan Kalman (kalman@american.edu) joined the mathematics faculty at American University in 1993, following an eight year stint in the aerospace industry and earlier teaching positions in Wisconsin and South Dakota. He has won three MAA writing awards and served a term as Associate Executive Director of the MAA. His interests include matrix algebra, curriculum development, and interactive computer environments for exploring mathematics, especially using Mathwright software.

A recursive approach to multivariate automatic differentiation

In one approach to automatic differentiation, the range of a function is generalized from a single real value to an aggregate representing the values of the function and one or more derivatives. The operations and functions of elementary analysis are extended to these aggregates so as to preserve the validity of the derivatives. In this paper we develop a recursive approach to defining the necessary operations in the context of functions of several variables. Formally, the definitions are essentially the same as those needed in the single variable case. The resulting system provides automatic propagation of values of all partial derivatives up to a prespecified order for a function of several variables