Introducing Dan Kalman
“Wow! That is so cool!” That’s what I have felt so many times after discovering an unexpected or surprising mathematical result. Generally, these results are not what accomplished research mathematicians would consider deep. Indeed, almost every paper I have written is based on undergraduate mathematics and involves methods that never stray far from the standard curriculum. But these topics are so rich and interconnected that opportunities for new discoveries abound. It has been my great good fortune to uncover a few of them. Having made a discovery, I invariably want to share it. And it is another aspect of my great good fortune that I enjoy writing about discoveries almost as much as I enjoy making them (and talking about them).
One is sometimes asked, where do you get your ideas? For me, it has almost always been pure serendipity. Over and over again, some intriguing mathematical puzzle has unexpectedly dropped into my lap. Inspiration has come from preparing for classes, ideas and work of my students, suggestions of colleagues, and applied projects from my aerospace industry days. Here are some examples: I am trying to understand why a co-worker (a software pro) had coded an application in a certain way. In a sudden flash of insight, I realize that his programming approach can be extended to an entirely different problem. (Doubly Recursive Multivariate Automatic Differentiation) After giving a lecture to students, I receive an email message from an attendee pointing out a wonderful extension, complete with a reference to one of Euler’s papers. (Another Way to Sum a Series: Generating Functions, Euler, and the Dilog Function) My father is telling me about all of the references to numerology in a book he is reading. I mention number theory as an area of math that is somewhat related, and as an example describe a long-standing open problem. To my surprise, he instantly offers me a solution. (A Perfectly Odd Encounter in a Reno Cafe) Although there is not room here to do them justice, I must at least mention two other aspects of my mathematical meanderings: teaching and collaboration.
It is a great pleasure to be able to share some of my favorite articles in this MAA Virtual Special Issue. In light of the comments above, no one should be surprised that it is quite an eclectic collection. I hope that readers will derive as much enjoyment from these papers as I obtained in their development and writing.
Dan Kalman’s Chosen Articles
| Award |
Journal/Book |
Article |
Info |
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| George Pólya Award ● 1994 |
The College Mathematics Journal |
Six Ways to Sum a Series |
This paper began as a handout for a talk I gave at a meeting of CMC3-South – an affiliate of AMATYC and the California Math Council. Through MAA activities I had met Ann and Bill Watkins, LA area math professors at California State University, Northridge and co-Editors of the College Math Journal (1989 – 1993). Ann, who later became MAA President (2001-2002), suggested that the handout might make a good paper for CMJ. So I polished it up a bit and sent it in. |
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| Trevor Evans Award ● 1997 |
Math Horizons |
A Perfectly Odd Encounter in a Reno Cafe |
Although my father had little mathematical training, there was nothing wrong with his logic or his insight. Witness his nearly instantaneous argument for the impossibility of odd perfect numbers. But was it a proof? |
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| Carl B. Allendoerfer Award ● 1998 |
Mathematics Magazine |
Variations on an Irrational Theme—Geometry, Dynamics, Algebra |
Shahriar Shahriari, a fellow student in grad school, came to DC to visit a mutual friend. I picked Shahriar up at the airport, and driving back, described a project I was working on with Robert Mena, a friend from my days in LA. Next thing you know, Shahriar has joined the project. This paper is the result of that three-way collaboration.
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| George Pólya Award ● 2003 |
The College Mathematics Journal |
An Underdetermined Linear System for GPS |
At Aerospace Corp. I worked in the same department as Karl Rudnick (two years ahead of me at Harvey Mudd College) and Paul Massatt. They did a lot of work on the early development of the GPS satellite system, the rudiments of which Karl once explained to me. Years later I used what he taught me to formulate a nice example for my linear algebra class: an underdetermined linear system that arises in the GPS space-time triangulation problem. |
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| Carl B. Allendoerfer Award ● 2003 |
Mathematics Magazine |
Doubly Recursive Multivariate Automatic Differentiation |
One of my few experiences with the kind of out-of-the-blue mathematical insight that Poincare so artfully described. I was wondering what had inspired Bob Lindell, a top notch software engineer with whom I was working, to code a program in a certain way. Like a bolt of lightning it just hit me – Bob’s coding tactic could provide a novel approach to automatic differentiation! |
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| Lester R. Ford Award ● 2009 |
The American Mathematical Monthly |
An Elementary Proof of Marden's Theorem |
The seed was planted in 1973 during a summer student-research project at Harvey Mudd College. It lay dormant for a couple of decades, but suddenly germinated thanks to the work of a brilliant mathematics educator and good friend, James White. (Additional background here). After this paper appeared I discovered that I had unintentionally produced another example of Stigler's law. |
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| Beckenbach Book Prize ● 2012 |
Dolciani Mathematical Expositions |
Uncommon Mathematical Excursions: Polynomia and Related Realms. |
Originally, the book was conceived as a professional development resource for secondary math teachers. The idea was to reveal fascinating offshoots of the secondary curriculum that every teacher should want to know about. The second chapter, included below, is just what the title says: a potpourri of delicious morsels, each presentable in a page or two. |
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| Trevor Evans Award ● 2012 |
Math Horizons |
Harvey Plotter and the Circle of Irrationality |
Nathan Carter and I were both present when Ivars Peterson, then MAA Director of Publications, told the Math Horizons editorial board to find articles that made connections between math and the non-math interests of college math students. In an invited address at the same meeting, Manjul Bhargava presented a perfect gem for a MH article. Struck by sudden inspiration, I wrote a brief draft of this paper and immediately enlisted Nathan as a coauthor. Good decision: his revision was a big improvement! |
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| Paul R. Halmos – Lester R. Ford Award ● 2013 |
The American Mathematical Monthly |
Another Way to Sum a Series: Generating Functions, Euler, and the Dilog Function |
With this article we come full circle, returning to the topic addressed in Six Ways to Sum a Series (SWTSAS). The inspiration for this later article came from Ari Mark Turner, who attended the Student Lecture I presented at Mathfest 1999. The lecture covered much of the same ground as SWTSAS, and also discussed my failed attempt to sum the series using Calc II methods. A few weeks later Ari sent me an email showing how to make the Calc II approach work, and supplying a reference to one of Euler’s papers! When that brought to light a difficult historical puzzle, I turned to one of my favorite Euler scholars, Mark McKinzie, for help.
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