American Mathematical Monthly - March 2016

March comes in like a lion in the Monthly. Read about the 2016 Gung and Hu Award winner George Berzsenyi in our lead article by Jennifer Quinn. Karen Parshall offers us a glimpse of the life of an African-American mathematician in the mid-20th century in her article "Mathematics and the Politics of Race: The Case of William Claytor (Ph.D., University of Pennsylvania, 1933)."

In "Inserting Plus Signs and Adding," Steve Butler, Ron Graham, and Richard Stong show us a provocative way to reduce a natural number down to a single digit. Craig Jackson reviews "Mathematics and Climate," by Hans Kaper and Hans Engler. Need something to do over Spring Break?  Our Problem Section will keep you busy. Stay tuned for our April Edition when the Monthly celebrate "polynomial month."

  - Scott T. Chapman, Editor

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Table of Contents

Yueh-Gin Gung and Dr. Charles Y. Hu Award for 2016 to George Berzsenyi for Distinguished Service to Mathematics

Jennifer J. Quinn

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.3.211

Mathematics and the Politics of Race: The Case of William Claytor (Ph.D., University of Pennsylvania, 1933)

Karen Hunger Parshall

William Claytor (1908–1967) entered the graduate program in mathematics at the University of Pennsylvania in 1930 after spending one year in the newly inaugurated master′s program at Howard University. A student of J. R. Kline who was himself a student of R. L. Moore, Claytor embraced the point set topology that was then quickly becoming an American area of expertise. By 1933, Claytor had earned his Ph.D. for what Kline praised as “a very fine thesis … perhaps the best that I have ever had done under my supervision” and had begun the process of trying to turn that promising beginning into a productive career as a research mathematician. This paper traces the efforts of Claytor and his supporters to realize such a career as it explores the politics of race that “colored” American academia in the 1930s.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.3.214

Correcting Three Errors in Kantorovich & Krylov′s Approximate Methods of Higher Analysis

John P. Boyd

The Nobel laureate Kantorovich and his collaborator produced a book full of treasures, a chrestomathy of accurate, explicit solutions to linear and nonlinear differential equations, both ordinary and partial, and also integral equations. We show that computer algebra systems (CAS) such as Maple, Mathematica, and Reduce compress weeks of paper and pencil calculations (“chirugery”) into a few lines of code. (Four Maple codes are included as tables.) However, modern number-crunching (“arithmurgy”) has offered new insights. Here, we improve upon three problems in Kantorovich and Krylov. First, the error in a Legendre–Galerkin explicit approximation to the solution of a linear integral equation is reduced by a factor of 35 by building the boundary conditions into the approximation. Second, we apply a perturbation series in powers of an artificial parameter ϵ to conformally map a complicated domain to or from the unit disk, and, unlike the book, we analyze corner singularities. Lastly, we apply Euler acceleration to obtain a rapidly convergent series in ϵ where the unaccelerated series is divergent or converges so slowly as to be useless. An important theme is that what are usually thought of as numerical algorithms can generate explicit, symbolic approximations. The methods of Kantorovich and Krylov are where the often antagonistic worlds of number crunching and of analytical expansions and approximations are blended into tools of great power through computer algebra.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.3.241

Milnor′s Lemma, Newton′s Method, and Continued Fractions

Boris M. Bekker, Oleg A. Ivanov and Alexander S. Merkurjev

Let f be a quadratic map of the Riemann sphere S² into itself. Such a map has three fixed points, counted with multiplicity. Let μ₁, μ₂, and μ₃ be the multipliers of f at these points. Milnor proved (see [2, Lemma 3.1] ) that μ₁, μ₂, and μ₃ determine f up to holomorphic conjugacy and are subject only to the restriction μ₁μ₂μ₃ − (μ₁ + μ₂ + μ₃) + 2 = 0. In the present paper, we consider arbitrary regular maps from a projective line over an arbitrary field K into itself and give an elementary proof of the fact that two such maps coincide if the maps have the same collection of fixed points and equal multipliers at the corresponding points. We apply this result to demystify the long-known link between Newton′s approximations and continued fractions (see [5] and [1]).

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.3.258

Slow Beatty Sequences, Devious Convergence, and Partitional Divergence

Clark Kimberling and Kenneth B. Stolarsky

Sequences (⌊nr⌋) for 0 < r < 1 are introduced as slow Beatty sequences. They and ordinary Beatty sequences (for which r >1) provide examples of sequences that converge deviously (which at first might seem to diverge), as well as partitionally divergent sequences (which consist of convergent subsequences).

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.3.267

Notes

Inserting Plus Signs and Adding

Steve Butler, Ron Graham and Richard Stong

For a fixed base b, we consider the following operation called “insert and add” on a natural number: Write the number as a string of digits and between some of the digits insert a plus sign; finally, carry out the indicated sum. If at least one plus sign is inserted, this results in a smaller natural number and so repeated application can always reduce the number to a single digit. We show that for any base b, surprisingly few applications of this operation are needed to get down to a single digit.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.3.274

Electromagnetism in a Multiconnected Universe

Tom Mertens and Jeff Weeks

Our raw intuition tells us that the topology of the universe shouldn′t affect the behavior of a current-carrying wire, but when the wire wraps all the way around a multiconnected space, our intuition is wrong. While an infinitely long perfectly conducting wire in Euclidean space may carry a constant current, a simple application of the Ampère–Maxwell circuital law shows that the analogous wire in a 3-torus may not. Instead, we find that the current spontaneously oscillates, and the wire acts like an antenna, gradually radiating away its energy.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.3.280

The Laplacian and Mean and Extreme Values

Jeffrey S. Ovall

The Laplace operator is pervasive in many important mathematical models, and fundamental results such as the mean value theorem for harmonic functions, and the maximum principle for superharmonic functions are well known. Less well known is how the Laplacian and its powers appear naturally in a series expansion of the mean value of a function on a ball or sphere. This result is proven here using Taylor's theorem and explicit values for integrals of monomials on balls and spheres. This result allows for nonstandard proofs of the mean value theorem and the maximum principle. Connections are also made with the discrete Laplacian arising from finite difference discretization.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.3.287

On the Smooth Jordan Brouwer Separation Theorem

Peter McGrath

We give an elementary proof of the Jordan Brouwer separation theorem for smooth hypersurfaces using the divergence theorem and the inverse function theorem.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.3.292

Problems and Solutions

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.3.296

Book Review

Mathematics and Climate By Hans Kaper and Hans Engler

Reviewed by Craig H. Jackson

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.3.304

MathBits

An Identity of Carlitz and Its Generalization

José L. Ramírez

100 Years Ago This Month in the American Mathematical Monthly

Edited by Vadim Ponomarenko