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The Carl B. Allendoerfer Awards, established in 1976, are made to authors of expository articles published in *Mathematics Magazine*. The Awards are named for Carl B. Allendoerfer, a distinguished mathematician at the University of Washington and President of the Mathematical Association of America, 1959-60.

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The article by Ezra Brown is a story about a single object that is ?all at once a difference set, a block design, a Steiner triple system, a finite projective plane, a complete set of orthogonal Latin squares, a double regular round-robin tournament, a skew-Hadamard matrix, and a graph consisting of seven mutually adjacent hexagons drawn on the torus.? The intellectual picnic that Brown creates is written in a clear and engaging style and traces the connections prompted by (7, 3, 1).

From the first incidence of (7, 3, 1) in the set Q7={1, 2, 4} (the difference set S of 3 nonzero integers mod 7 such that every nonzero integer *n* mod 7 can be represented as a difference of elements of S in exactly 1 way) through connections in Galois Theory, Brown masterfully follows the combinatorial threads of (7, 3, 1). He leads the reader in a very natural way from one topic to the next and gives the needed definitions and hints for exploring further connections with references from Euler to R.A. Fisher.

Like many good stories, Brown ends his article with an additional question: ?Does (7, 3, 1) have any other names?? ?Where can I find out more about difference sets?? and clues about how to find the answers. Indeed, there are also suggestions for a sequel or two. The question ?Did the Reverend Thomas Kirkman ever get credit for anything?? contains a description of what is known as Kirkman?s Schoolgirls Problem. ?The solution to this problem is a particularly interesting Steiner triple system on 15 varieties, with parameters (35, 15, 7, 3, 1) - but that?s another story.? With the gifted telling of the story of (7, 3, 1) Brown makes us want another story.

**Biographical Note**

Ezra (Bud) Brown has degrees from Rice and Louisiana State, and has been at Virginia Tech since the first Nixon Administration, with time out for sabbatical visits to Washington, DC (where he has spent his summers since 1993) and Munich. His research interests include number theory, graph theory and combinatorics, and he particularly enjoys discovering connections between apparently unrelated areas of mathematics. He received the MAA MD-DC-VA Section Award for Outstanding Teaching in 1999 and MAA PÃ³lya Awards in 2000 and 2001.

He enjoys working with students who are engaged in research. Over the years, he has put together seventeen nomination dossiers which led to college and university recognition of his colleagues' superior teaching, advising and outreach. He enjoys singing (everything from grand opera to rock'n'roll), playing jazz piano, and talking about his granddaughter Phoebe Rose. With his wife Jo's help, he has become a fairly tolerable gardener. He occasionally bakes biscuits for his students.

**Response from Ezra Brown**

In my graph theory course in graduate school, Brooks Reid first introduced me to (7,3,1) as a difference set. ?And by the way,? he said, ?it is also a finite projective plane, a doubly regular tournament, and a symmetric balanced incomplete block design.? I marveled at the notion that the same mathematical object could have so many different names---or perhaps the rhythm of Brooks Reid's words got to me. At any rate, as time passed, my interest in (7,3,1) deepened with the discovery of each new name. The article was written with the hopes that its readers might share my fascination with (7,3,1) and combinatorial designs in general, and it is a privilege and a pleasure to be honored with the Allendoerfer Prize.

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?Automatic differentiation is a way to find the derivative of an expression without finding an expression for the derivative.? This opening sentence and the rest of Kalman?s model initial paragraph in which he explains what automatic differentiation is and is not, set the stage for the remainder of the article in which a rather intricate situation is masterfully explained. The automatic differentiation ?computation is equivalent to evaluating a symbolic expression for *f*?(X), but no one has to find that expression - not even the computer system.?

The author begins with a brief review of the one-variable, one-derivative case presented by L.B. Rall in *Mathematics Magazine *in 1986. He then discusses a beautiful and mathematically intriguing extension that can handle any number of variables and derivatives. By means of derivative structures and substructures (vectors for the one variable case, triangles for two variables, pyramids for three variables, and extending to derivative structures in general) and operations on them, the author shows how we can obtain an object containing the function value and the values of all partial derivatives through a particular order. The examples are well-chosen and the author?s style is friendly and clear throughout.

The paper ends with a discussion on implementation and efficiency in which Kalman points out some of the inefficiencies of the recursive approach and suggestions about how one might improve upon these inefficiencies. The final paragraph claims: ?More generally, as computational speed continues to increase, the importance of execution efficiency will continue to decline, particularly for problems with small numbers of variables. In these cases, the directness and simplicity of the current development offers an attractive paradigm for implementing an automatic differentiation system.? Indeed, Kalman?s presentation of double recursive automatic differentiation offers us an outstanding paradigm for masterful expository writing.

**Biographical Note**

**Dan Kalman** has been a member of the mathematics faculty at American University, Washington, DC, since 1993. Prior to that he worked for 8 years in the aerospace industry and taught at the University of Wisconsin, Green Bay. During the 1996-1997 academic year he served as an Associate Executive Director of the MAA. Kalman has a B.S. from Harvey Mudd College, and a Ph.D. from University of Wisconsin, Madison.

Kalman has been a frequent contributor to all of the MAA journals, and served as an Associate Editor for *Mathematics Magazine*. He has written one book, *Elementary Mathematical Models*, published in the MAA's Classroom Resources series. His current interests include creating interactive computer activities for mathematics instruction, using Mathwright software.

**Response from Dan Kalman**

I am very grateful to receive this Carl B. Allendoerfer Award. It is particularly gratifying because this paper reports my best mathematical work. The original inspiration occurred in one of those rare flashes of understanding which suddenly illuminates an entire development. At the moment of discovery I was actually thinking about something quite different. All at once I was struck by a new insight, in a completely unexpected direction. It was totally compelling: I knew with utter certainty that this idea would prove correct, before examining a single example, though it concerned a question I had never before considered. It is hard to convey the astonishment and euphoria of that experience.

My thanks go to Robert Lindell, a colleague at the Aerospace Corporation, whose collaboration paved the way for my discovery. Working with him was a privilege. I would also like to thank Frank Farris, editor of* Mathematics Magazine*, whose guidance greatly improved the structure of the paper. Finally, thanks to the MAA for publications and programs which so profoundly enhance our profession.

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If you have recently moved into new office space, with look-alike offices and office stations, you will appreciate Taalman and Hunsicker?s delightful article that relates the mathematics of tessellating the plane and space-filling solids to modular architecture. From Buckminster Fuller?s geodesic domes to modern space-efficient buildings, the authors show the simple mathematics behind modular architecture and speculate about how such architecture may help house the world!

**Biographical Note**

**Laura Taalman and Eugenie Hunsicker** first met at the University of Chicago when Laura was a senior undergraduate and Eugenie was a first year grad student. Several grueling years later, after Laura finished her PhD in algebraic geometry at Duke and had taken a job at James Madison University, and Eugenie finished her PhD in geometry at the University of Chicago and taken a job at Lawrence University, they were excited to find they were both participating in Project NExT (the red dot year). Unable to absorb all of the useful information being offered to them at MathFest in Los Angeles in the summer of 2001, they are ashamed to admit they played hooky one afternoon. Or, they would be ashamed, except for the stroke of luck that took them past Gregg Fleishman's studio, where they were intrigued by the geometric constructions in his window, and very warmly received.

Since writing this article, Laura has finished a calculus textbook and started a second one, Eugenie has begun working on a book about the mathematics of relativity and invariance, and Gregg has created sets of cool math toys based on his architectural designs, which are available through his website, http://www.greggfleishman.com.

**Response from Laura Taalman and Eugenie Hunsicker **

We would both like to thank Gregg immensely for teaching us so much. We could not have even begun to write this article without his help and inspiration. We would also like to thank Deanna Haunsperger and the Math Horizons folks, who encouraged us to write the article and helped with the editing. Thanks very much to the MAA for this honor. We regret that our busy book-writing schedules prevent our attending the ceremony.

**?The Instability of Democratic Decisions,? Math Horizons, April, 2002.**

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Philip Straffin?s compelling and sobering article concisely describes how unstable democratic decision-making can be. With clear mathematical prose and illuminating figures the reader gains deep insight into how voting outcomes can be cleverly and drastically manipulated. The results are both startling and beautiful. Furthermore, they are keenly relevant to all of us involved in our local and national democratic processes.

**Biographical Notes **

**Philip Straffin** is Professor of Mathematics at Beloit College. He holds B.A.?s from Harvard University and Cambridge University and a Ph.D. from the University of California at Berkeley, where his thesis was in algebraic topology. His books include *Topics in the Theory of Voting, Game Theory and Strategy*, and the edited collection* Applications of Calculus*. He has received the Allendoerfer Award for mathematical exposition and the Haimo Award for Distinguished Teaching from the MAA. He is the current editor of the MAA?s Anneli Lax New Mathematical Library.

**Response from Philip D. Straffin**

The geometric theory of voting uses simple but elegant geometry to gain insight into political questions that are central to our democratic society. It was my pleasure to present some of these fascinating and important ideas to students, first in Pi Mu Epsilon?s J. Sutherland Frame lecture, and then in *Math Horizons*.

**?Two Classical Surprises Concerning the Axiom of Choice and the Continuum Hypothesis,? The American Mathematical Monthly, June-July 2002, pp. 544-553.**

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The author examines two known, but still somewhat surprising, results in set theory: the trichotomy of infinite cardinals implies the axiom of choice (in fact is equivalent to it) (Friedrich Hartogs in 1915), and the General Continuum Hypothesis implies the Axiom of Choice (Waclaw SierpÃnski in 1947).

The article is written in a magisterial style showing firm command of all the necessary basic facts, which are assembled in clear and convincing order for the reader. The author operates in classical ?naive? set theory, and nothing relevant is omitted.

The proofs of the two results are linked via SierpÃnski?s version in *Cardinal and Ordinal Numbers*; in particular Harthog?s result that to every infinite cardinal is associated an aleph satisfying certain relations is crucial to both proofs.

**Biographical Note**

**Leonard Gillman** held a piano fellowship for five years at the Juilliard Graduate School, worked for nine years in naval operations research, and completed a Ph.D. at Columbia University in transfinite numbers directed by E. R. Lorch and unofficially by Alfred Tarski (UC Berkeley). He then taught for 35 years at Purdue, Rochester, and Texas, including two years on leave at the Institute for Advanced Study, the first year (silver anniversary of Juilliard) as a Guggenheim fellow. He is coauthor with R. H. McDowell of a calculus text and with Meyer Jerison of *Rings of Continuous Functions*.

Gillman was MAA Treasurer for thirteen years (1973-86), and President for the standard two (1987-89). He is the author of *Writing Mathematics Well*, an MAA booklet. He won a Ford Award in 1994 and the Gung-Hu Award for Distinguished Service to Mathematics in 1999.

Gillman has performed at the piano at five national meetings, three with Louis Rowen (cello) (San Antonio ? 1976, 1980; San Diego ? 1997) and two with William Browder (flute), the first of two being heralded as the *Presidents? Concert* (Gillman, MAA; Browder, AMS) (Phoenix ? 1989), and the second as the* Past-Presidents? Concert* (Baltimore ? 1992).

Gillman has also performed at the piano at a half-dozen MAA Section meetings (with local instrumentalists).

**Response from Leonard Gillman**

I thank the Committee on Ford Awards for selecting me for this honor. I have always applauded the Mathematical Association of America for its devotion to expository writing, and to be thus recognized for my own efforts is a source of immense pleasure and satisfaction.

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The opening sentence in this well-written article, which attracts the reader immediately, is: ?What is the *m*th derivative of a composite function?? One would almost expect a calculus student to be able to respond to this in a routine manner. Although the answer has been known since 1855 (special cases date even earlier), the topic is strangely absent from modern calculus texts and courses. FaÃ di Bruno?s elegant formula seems to have fallen into the niche familiar mainly to those who study combinatorics.

The author has done a great job of tracing the roots of this formula as it appeared in various forms over the years, and presenting them in a way that fully engages the reader regardless of their background. One of the nice features of the article is a thorough bibliography, with 69 references spanning a century and a half. This is a useful service for the casual reader, who would otherwise probably find it rather difficult to track down the older references. Another is the way in which the author weaves the various combinatorial threads together, illustrating in just a few pages the connections with Bell polynomials, Stirling numbers and set partitions, among other things.

The reader will probably come away from this article convinced that FaÃ di Bruno?s formula is a beautiful bit of mathematics that deserves to be more widely known.

**Biographical Note**

**Warren P. Johnson** grew up in Cinnaminson, New Jersey, and was an undergraduate at the University of Minnesota. He received his Ph.D. from the University of Wisconsin, under the direction of Richard Askey, in 1993. Since then he has gone from Penn State University to Beloit College to the University of Wisconsin, back to Beloit College and back again to the University of Wisconsin, and then to Bates College. He is interested in combinatorics, q-series, determinants and the history of 18th and 19th century analysis.

**Response from Warren P. Johnson**

I am deeply honored to receive the Lester R. Ford Award. To someone who enjoys reading and writing expository mathematics as I do, this means a great deal. I am very grateful to Roger Horn for going ahead with the paper when the referees were less than enthusiastic, and to Bruce Palka for fixing some little things at the last minute. I?d also like to thank all the people who wrote to me after the paper was published, most particularly Christopher Hammond, Steven Krantz, Hubert Kalf and Rich Neidinger.

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The article is described in line 3 as ?a pleasant nexus of many mathematical strands.? Using *x _{n}*for the

**Biographical Note**

**Sam Northshield** received his Ph.D. from the University of Rochester in 1989 and, apart from several years in Minnesota, has been at Plattsburgh State University since then.

**Response from Sam Northshield**

This paper was fun to research and to write. I am aware of the high quality of *Monthly *articles and was, in fact, surprised to have my paper accepted at all. Considering all the great *Monthly* articles that appeared last year, I was completely surprised (and pleased) to win an award.

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Plimpton 322 is one of the most famous of ancient mathematical artifacts. It contains (in base 60) fifteen Pythagorean triples; nothing like it has ever been found in other Mesopotamian mathematics. In an engaging and wittily written article the author examines the three main theories on the tablet?s function. Otto Neugebauer (apparently the first to discover its ?Pythagorean content?) developed a ?generating functions? theory of how the tablet came to be written (but left open the question of how the values in it were chosen). Other theories are that it is a kind of trigonometric table, and that it derives from considerations of reciprocal pairs. Robson shows that the trigonometric theory is anachronistic, and that the generating functions theory is organizationally implausible (when compared with other tables from Larsa where Plimpton 322 was found). The notion of reciprocal pairs seems historically and linguistically convincing.

Robson also demonstrates that it is unlikely that the author of Plimpton 322 was either a professional or amateur mathematician. More likely he seems to have been a teacher and Plimpton 322 a set of exercises.

Throughout the paper Robson makes useful and well-expressed observations about historical method, especially when treating periods that are generally unfamiliar.

**Biographical Notes**

Eleanor Robson is a historian of ancient Iraq whose research interests focus on mathematics, numeracy and literacy. *Mesopotamian Mathematics, 2100-1600 BC *was published by Oxford University Press in 1999; *Mathematics in Ancient Iraq: a Social History* is due out from Princeton soon. She teaches the archaeology, history, languages and literature of ancient Iraq (Mesopotamia) at the University of Oxford and is a Fellow of All Souls College, Oxford. In January 2004 she will take up a new university lectureship in the history of science at the University of Cambridge.

**Response from Eleanor Robson**

I am delighted and overwhelmed to be honored for my work on the mathematical cuneiform tablet Plimpton 322. Curiously enough, that tablet was my first introduction to Mesopotamian (ancient Iraqi) mathematics when, as an undergraduate math student, I gave a 10-minute presentation on it in my history of mathematics class. The article is addressed to my 19-year-old self and all the misconceptions and assumptions I then held about interpreting the mathematics of other cultures. I will need to write another upbraiding message to self in the years to come! Meanwhile I warmly thank Victor Katz, Bruce Palka, Karen Parshall, Fred Rickey, and Jim Tattersall for their practical and moral support on the long road from idea to workshop, lecture, and article.

This year has been truly terrible for Iraq and its history. My prize money will go to the British School of Archaeology in Iraq?s cultural heritage reconstruction fund.

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This is a well-written and detailed article, with 49 references, about the slippery concept of randomness and the pitfalls inherent in some intuitive approaches to it.

The author shows that defining random sequences as those that are not computable (in the sense of a Turing machine), would result in too large a set of random sequences. He also shows the inadequacy of a viewpoint introduced by von Mises and modified by Wald and by Church based on the notion of collectives or on frequency stability of digits. He then turns to the idea of ?incompressibility? due, independently to Solononoff, Kolmogorov and Chairin, the idea that a string is ?patternless? if it cannot be described more efficiently than by giving the whole string itself. Martin-LÃ¶f showed that there is a problem with this approach to randomness and introduced the notion of ?typicality? (in a measure theoretic sense) calling a sequence random if it belongs to all sets of ?effective measure one,? for some computable measure, the latter concept using the notion of a Turing machine. The author believes that there is a strong case for considering the Martin-LÃ¶f approach to be the best candidate for a mathematical definition of randomness.

The article can be read with enjoyment and profit by readers with different levels of knowledge.

**Biographical Note**

SÃ©rgio B. Volchan earned a M.S. in theoretical physics from the Pontificia Universidade Catolica do Rio de Janeiro (PUC-Rio), Brazil. He then turned to mathematics, receiving his Ph.D. from the Courant Institute of Mathematical Sciences in 1995, under the supervision of Professor Charles M. Newman. After spending two years as a researcher at Instituto de Mathematica Pura e Aplicada (IMPA), he moved to the mathematics department of PUC-Rio, where he is assistant professor. His interests are in probability theory and its applications in mathematical physics, and in the history and foundations of mathematics.

**Response from SÃ©rgio B. Volchan**

I am delighted and deeply honored to be one of the recipients of the prestigious Lester R. Ford Award of the year 2003. I would like to thank the MAA for this recognition but also for its long-standing generosity and open-mindedness in promoting the advancement and enjoyment of mathematical culture worldwide.

*Check back for this article.*

Binomial coefficients are integers. The number-theoretic implications of this seemingly innocent statement are the basis for this lovely article. The article begins with some surprising appearances of the factorial function in several now classical number-theoretic results. After pointing out the role played by the integers in the statement of each result, the author asks whether these results would remain valid if references to the integers were replaced by the author?s ?factorial function of S?. This intriguing generalization of the factorial function is motivated by the number-theoretic properties of *n*!, instead of by its well-known combinatorial interpretation. After discussing a number of interesting examples of generalized factorials, the author further extends his definition by permitting S to be an arbitrary subset of a Dedekind ring. With this extension, many classical generalized factorials may then be obtained as special cases. In addition, using his generalized factorials the author is able to definitely resolve an open question about integer-valued polynomials first posed by George PÃ³lya in 1919!

Following some further extensions of the author?s results, the article concludes with a number of interesting open questions. All readers, from those interested in research in this area, to those who simply want to learn about some fascinating extensions of the familiar factorial function, will find much to enjoy in this engaging, well-written article.

**Biographical Note**

Manjul Bhargava was born in Hamilton, Ontario, Canada but spent most of his early years in Long Island, New York. He received his A.B. summa cum laude in Mathematics from Harvard University in 1996, and his Ph.D. from Princeton University in 2001. His research interests are primarily in number theory, representation theory, and combinatorics, although he also enjoys algebraic geometry, linguistics, and Indian classical music.

In 1997, Bhargava received the AMS-MAA-SIAM Morgan Prize, and in 2000 was awarded the Clay Mathematics Institute's first five-year Long-Term Prize Fellowship. For the first three years of his fellowship he traveled extensively -- holding visiting positions at Harvard, Princeton, MSRI, and the Institute for Advanced Study -- before recently joining the permanent faculty at Princeton University as Professor of Mathematics.

**Response from Manjul Bhargava **

I am extremely grateful and honored to be the recipient of this year's MAA Hasse Prize. My *Monthly *article on generalizing the factorial function was such a joy to write, and there could be no greater reward for the hours of labor put into the article than to know that others are finding joy in reading it!

I would like to thank the MAA for establishing this award, and for publishing wonderful journals like the *Monthly*, where such a large and diverse audience of mathematicians can together learn and participate in exciting advancements in mathematics.

I must confess that since my article "The factorial function and generalizations" was primarily a research article, I was initially somewhat hesitant to submit it to the *Monthly*. But I am glad that I did -- the response has been wonderful, and receiving comments from people in all fields (not just number theorists, but topologists, analysts, representation theorists, people in industry...) was a most rewarding experience that I will never forget. I hope that my positive experience may encourage others to realize the *Monthly* as not just the best venue for publishing high quality expository articles, but also for publishing new research papers that are of broad interest and accessible to a wide range of mathematicians.

I wish to thank all those who encouraged me to write this article for the *Monthly*, and whose comments greatly improved the exposition of this paper. In particular, I would like to thank Professors Persi Diaconis, Joseph Gallian, Barry Mazur, Roger Horn (the then editor-in-chief of the *Monthly*), Jeffrey Lagarias, Gian-Carlo Rota, and Richard Stanley for all their inspiration and guidance.

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Even Sherlock Holmes, well known for his investigations into bicycle tracks, might not have suspected that it is possible to have a nonlinear track that could be made by either a bicycle or a unicycle. This paper shows how to construct such a track, and the details are remarkably simple.

Finn?s key observation is that a nonanalytic function can be used to start the track, which can then be continued to create a strange-looking curve that does allow the passage of a bicycle in such a way that the front and rear wheels both stay on the track. The example is one that intrigues calculus students as well as professional geometers. The construction fails when analytic functions are used, and so is a true application of a class of functions that are generally thought of as being pathological. They show their power here, since Finn required a function that was infinitely flat at two points. It is not often that an idea arises that is so novel, but yet is based on very elementary mathematics. The fact that it involves an object as familiar as a bicycle makes it even more attractive, and we are grateful to David Finn for exercising his imagination on this problem.

**Biographical Notes**

David L. Finn is currently an assistant professor of mathematics at Rose-Hulman Institute of Technology. He holds the M.S. and Ph.D. in mathematics from Northeastern University. His current mathematical interests lie in the interplay of analysis, geometry, and physics. His non-mathematical interests are centered around his family, specifically his wife Suzanne and son Avery.

**Response from David L. Finn**

I am deeply honored that the article was chosen for the George PÃ³lya award. This was not a pure individual effort. The referees deserve credit for vastly improving the exposition. The editor (Underwood Dudley) deserves credit for his suggestions to this article and for shortening the article. But, most of all James Tanton deserves credit for asking me the question, ?Can a bicycle create a unicycle track?? Without them, the article would not exist.

The construction of a unicycle track by bicycle in the article is to me an illustration of the Holmesian mantra that once you have eliminated the impossible whatever remains however improbably must be the truth. It is this, the seemingly impossible, that both intrigues and confounds students. I am honored to receive this award for an attempt to visualize and convey this aspect of mathematics. For I can truly say that the result surprised me as much as I hope it surprises one when reading the article, since the construction only came once I gave up trying to show that it was impossible to create a unicycle track with a bicycle.

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This paper is a very nice example of how to introduce a sophisticated, real world problem into an undergraduate course. Dan Kalman provides an example which not only shows very nicely how the problem of finding the solution to an underdetermined linear system can arise in practice, in this case in the context of Global Positioning Systems, but does so in such a way that his paper could be easily integrated into the early stages of a sophomore or junior course in linear algebra. Kalman also gently points out that in reality the problems are more difficult, showing the reader some of the subtleties that arise in passing from a theorem or toy application to real industrial mathematics.

**Response from Dan Kalman**

It is an honor and a great privilege to receive this PÃ³lya Award. I am grateful to the MAA for all its contributions to our profession, for the opportunities it has given me to participate in its programs, and most of all for the many friendships I have made through that participation. What could be more gratifying than receiving an award for something that I love to do and from the people I most respect and admire?

I am also grateful for the experience of working in industry at the Aerospace corporation. This paper owes much to that experience. I particularly want to mention Aerospace colleagues Karl Rudnick and Paul Massatt whose work on the GPS system is cited in the paper. They taught me most of what I know abut GPS.

The GPS system is an amazing technological accomplishment. The more you learn about how it works, the more impressed you will be. I highly recommend the references by Strang cited in the paper.

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