Data Mining the Monthly’s Greatest Hits

June 24, 2011

The American Mathematical Monthly has an extraordinary history of publishing papers by leading mathematicians and expositors. With such a rich pool to draw from, the question of which papers in the Monthly are the most read and the most cited is a natural one.

Following is a list of the most accessed articles in the period February 2008 through February 2011 on JSTOR together with the number of times each paper was cited on Google Scholar on March 20, 2011.

Most Accessed Articles on JSTOR (2008-2011)

  1. Accessed 2040 times; cited 1818 times: College Admissions and the Stability of Marriage, D. Gale and L. S. Shapley, 69 (1962) 9-15.
  2. Accessed 1897 times; cited 86 times: What Does the Spectral Theorem Say?, P. R. Halmos, 70 (1963) 241-247.
  3. Accessed 1531 times; cited 793 times: Can One Hear the Shape of a Drum?, Mark Kac, 73 (1966) 1-23.
  4. Accessed 1161 times; cited 1912 times: Period Three Implies Chaos, Tien-Yien Li and James A. Yorke, 82 (1975) 985-992.
  5. Accessed 1016 times; cited 2 times: Biography: René Descartes, B. F. Finkel, 5 (1898) 191-195.
  6. Accessed 1015 times; cited 38 times: A One-Sentence Proof That Every Prime p ≡ 1(mod 4) Is a Sum of Two Squares, D. Zagier, 97 (1990) 144.
  7. Accessed 958 times; cited 276 times: Random Walk and the Theory of Brownian Motion, Mark Kac, 54 (1947) 369-391.
  8. Accessed 912 times; cited 20 times: The Fundamental Theorem of Linear Algebra, Gilbert Strang, 100 (1993) 848-855.
  9. Accessed 902 times; cited 54 times: What Is the Laplace Transform?, D. V. Widder, 52 (1945) 419-425.
  10. Accessed 864 times; cited 57 times: Very Basic Lie Theory, Roger Howe, 90 (1983) 600-623.
  11. Accessed 787 times; cited 12 times: The History of Calculus, Arthur Rosenthal, 58 (1951) 75-86.
  12. Accessed 776 times; cited 477 times: What is Cantor’s Continuum Problem?, Kurt Gödel, 54 (1947) 515-525.
  13. Accessed 774 times; cited 4 times: Galois Theory for Beginners, John Stillwell, 101 (1994) 22-27.
  14. Accessed 722 times; cited 38 times: The Strong Law of Small Numbers, Richard K. Guy, 95 (1988) 697-712.
  15. Accessed 675 times; cited 16 times: Hypatia and Her Mathematics, Michael A. B. Deakin, 101 (1994) 234-243.
  16. Accessed 635 times; cited 77 times: Tricks or Treats with the Hilbert Matrix, Man-Duen Choi, 90 (1983) 301-312.
  17. Accessed 609 times; cited 21 times: Differentiation under the Integral Sign, Harley Flanders, 80 (1973) 615-627.
  18. Accessed 600 times; cited 10 times: Calculating Sums of Infinite Series, Bart Braden, 99 (1992) 649-655.
  19. Accessed 568 times; cited 0 times: A Remark on the Rank of a Matrix, Itiro Murase, 67 (1960) 176-177.
  20. Accessed 561 times; cited 131 times: The Plane Symmetry Groups: Their Recognition and Notation, Doris Schattschneider, 85 (1978) 439-450.

Some comments on the above list are in order. The only person with more than one paper on the list is Mark Kac, and both of his are highly cited. Zagier’s paper is only one page and Murase’s is two pages long. Three of the most accessed papers have been cited only four times or fewer. Remarkably, Murase’s paper was viewed 568 times in a recent three-year period but never cited in 50 years. Finkel’s biography of Descartes was accessed 1016 times in three years but cited only twice in 103 years. Finkel was the founder of the Monthly in 1894.

Only two papers on the list were written by joint authors, partly because jointly authored papers were far less common in the Monthly in the past than they are now. The term “chaos” in a mathematical sense was introduced in the article by Li and Yorke.

Many of the authors of these articles have received prestigious honors. Zagier received the AMS Frank Nelson Cole Prize in Number Theory. Gale and Shapley won the John von Neumann Theory Prize. Gale, Shapley, Strang, and Howe were elected to the National Academy of Sciences.

Kac’s name is associated with two famous formulas: Feynman-Kac and Erdős-Kac. Yorke won the Japan Prize for “original and outstanding achievements in science and technology.” Halmos received the AMS Leroy P. Steele Prize for Mathematical Exposition. Gödel was a permanent member of the Institute for Advanced Studies and received the National Medal of Science.

Knuth received the National Medal of Science, the Kyoto Prize, the A. M. Turing Award, and the IEEE John von Neumann Medal. Strang received the IEEE John von Neumann Medal and is a past president of the Society for Industrial and Applied Mathematics and a Rhodes Scholar.

The list above is only a snapshot of a continually evolving compendium. For example, comparing a list of the top 20 accessed articles from the Monthly via JSTOR published in February 2009 with the February 2011 list shows an overlap of 12 articles. Clicking on the link “Most Accessed” on any Monthly page in JSTOR provides up-to-date data for the past three years.

Although there is a positive correlation between the most accessed articles in the Monthly and the most cited ones, the following list shows many exceptions. Five of the articles are only three pages in length.

Most Cited Articles according to Google Scholar (March 20, 2011)

  1. Cited 1912 times: Period Three Implies Chaos, Tien-Yien Li and James A. Yorke, 82 (1975) 985-992.
  2. Cited 1818 times: College Admissions and the Stability of Marriage, David Gale and Lloyd Shapley, 69 (1962) 9-15.
  3. Cited 1161 times: The Mathematics of Sentence Structure, Joachim Lambek, 1 65 (1958) 154-170.
  4. Cited 793 times: Can One Hear the Shape of a Drum?, Mark Kac, 73 (1966) 1-23.
  5. Cited 769 times: Semi-Open Sets and Semi-Continuity in Topological Spaces, Norman Levine, 70 (1963) 36-41.
  6. Cited 625 times: A Fixed Point Theorem for Mappings which do not Increase Distances, W. A. Kirk, 72 (1965) 1004-1006.
  7. Cited 482 times: On Devaney’s Definition of Chaos, J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, 99 (1992) 332-334.
  8. Cited 475 times: The Problem of Simplifying Truth Functions, W. V. Quine, 59 (1952) 521-531.
  9. Cited 447 times: What is Cantor’s Continuum Problem?, Kurt Gödel, 54 (1947) 515-525.
  10. Cited 370 times: Some Results on Fixed Points–II, R. Kannan, 76 (1969) 405-408.
  11. Cited 365 times: A Way to Simplify Truth Functions, W. V. Quine, 1955 62 (1955) 627-631.
  12. Cited 319 times: The 3x + 1 Problem and Its Generalizations, Jeffrey C. Lagarias, 92 (1985) 3-23.
  13. Cited 310 times: Hilbert’s Tenth Problem is Unsolvable, Martin Davis, 80 (1973) 233-269.
  14. Cited 276 times: Random Walk and the Theory of Brownian Motion, Mark Kac, 54 (1947) 369-391.
  15. Cited 275 times: Pseudo-Inverses in Associative Rings and Semigroups, M. P. Drazin, 65 (1958) 506-514.
  16. Cited 260 times: Shuffling Cards and Stopping Times, David Aldous and Persi Diaconis, 93 (1986) 333-348.
  17. Cited 216 times: On Sets of Distances of n Points, Paul Erdős, 53 (1946) 248-250.
  18. Cited 210 times: Review of Hilbert’s Tenth Problem by Yuri V. Matiyasevich, Martin Davis, 102 (1995) 366-369.
  19. Cited 206 times: The Number of Partitions of a Set, Gian-Carlo Rota, 71 (1964) 498-504.
  20. Cited 204 times: On Complementary Graphs, E. A. Nordhaus and J. W. Gaddum, 63 (1956) 175-177.
  21. Cited 204 times: Commuting Mappings and Fixed Points, Gerald Jungck, 83 (1976) 261-263.
  22. Cited 200 times: New Foundations for Mathematical Logic, W. V. Quine, 44 (1937) 70-80.
  23. Cited 199 times: Conway’s Tiling Groups, William P. Thurston, 97 (1990) 757-773.
  24. Cited 192 times: A Remark on Stirling’s Formula, Herbert Robbins, 62 (1955) 26-29.
  25. Cited 190 times: A Decomposition of Continuity in Topological Spaces, Norman Levine, 68 (1961) 44-46.

It is noteworthy that three articles on the most cited list were written by Quine and two by Levine. Quine, who was in the philosophy department at Harvard, was voted by his peers as one of the five most important philosophers of the past two centuries. Levine was an expert in point set topology.

The two lists also offer clear evidence of the longevity of mathematics papers. Every decade from the 1940s to the 1990s is represented on both lists. The most accessed list includes an article from 1898, and the most cited list has one from 1937.

The winners of Lester R. Ford Awards, established in 1964 and made annually to authors of outstanding expository papers published in the Monthly, include many prominent mathematicians. Interestingly, the roster of winners has only three papers from the most accessed and most cited lists (Kac, Guy, M. Davis), though some of these authors did win the prize for other articles.

The Monthly has had its share of notable mathematicians among its authors. These luminaries include Peter D. Lax, winner of the 2005 Abel Prize, and three Fields medalists: William P. Thurston, Stephen Smale, and John W. Milnor (who also won the Abel Prize in 2011).— Joseph A. Gallian

Joseph A. Gallian, a mathematics professor at University of Minnesota Duluth, was MAA President (2007-2008)

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