A. H. Lightstone

Albert Harold Lightstone (1926–1976)[1] was a Canadian mathematician. He was one of the pioneers of non-standard analysis, a doctoral student of Abraham Robinson, and later a co-author with Robinson of the book Nonarchimedean Fields and Asymptotic Expansions.[2]

Biography

Lightstone earned his PhD from the University of Toronto in 1955, under the supervision of Abraham Robinson; his thesis was entitled Contributions To The Theory Of Quantification.[3] He was a professor of mathematics at Carleton University[4] and Queen's University.[5]

Research

Decimal hyperreals

In his article "Infinitesimals" in the American Mathematical Monthly in 1972,[6] Lightstone described an extended decimal notation for the hyperreals. Here there is a digit at every hypernatural rank rather than merely a digit for every rank given by a natural number. Such a hyperreal decimal is written as

a.a_1 a_2 \ldots ; \ldots a_{H-1} a_H a_{H+1} \ldots\,.

Here the digit a_H appears at rank H, which is a typical infinite hypernatural. The semicolon separates the digits at finite ranks from the digits at infinite ranks. Thus, the number 0.000...;...01, with digit "1" at infinite rank H, corresponds to the infinitesimal 10^{-H}.

The difference 1 - 0.000...;...01 is 0.999...;...9, with an infinite hypernatural's worth of digits 9. An alternative notation for the latter is

0.\underbrace{999\ldots9 }_H \,

where H is an infinite hypernatural. The extended decimal notation provides a rigorous mathematical implementation of student intuitions of an infinitesimal of the form 0.000...01. Such student intuitions and their usefulness in the learning of infinitesimal calculus were analyzed in a 2010 study by Robert Ely in the Journal for Research in Mathematics Education.[7]

Other research

Lightstone's main research contributions were in non-standard analysis. He also wrote papers on angle trisection,[4] matrix inversion,[8] and applications of group theory to formal logic.[9]

Books

Lightstone was the author or co-author of several books on mathematics:

Awards and honours

Queen's University annually awards the Albert Harold Lightstone Scholarship, named for Lightstone, to a fourth year honors undergraduate student majoring in mathematics or statistics.[20][21] The scholarship was established by Lightstone's wife after his death.[22]

References

  1. "Mathematical Concepts and Methods in Science and Engineering". www.faqs.org: Plenum. Retrieved March 31, 2011.
  2. Nonarchimedean fields and asymptotic expansions. Lightstone, A. H. and Robinson, Abraham. North-Holland Pub. Co. (Amsterdam and New York), 1975.
  3. Albert Harold Lightstone at the Mathematics Genealogy Project
  4. 1 2 Lightstone, A. H. (1962), "A Construction for Trisecting the Angle", Mathematics Magazine 35 (2): 99–102, JSTOR 2688331, MR 1571175
  5. Queen's University Academic Calendar, Mathematics and Statistics, retrieved 2011-03-31.
  6. Lightstone, A. H. (March 1972), "Infinitesimals", American Mathematical Monthly 79 (3): 242–251, doi:10.2307/2316619, JSTOR 2316619, MR 0300889
  7. Ely, Robert (2010), "Nonstandard student conceptions about infinitesimals" (PDF), Journal for Research in Mathematics Education 41 (2): 117–146. This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for "0.999..." falling short of 1 by an infinitesimal 0.000...1.
  8. Lightstone, A. H. (1968), "Two methods of inverting matrices", Delta (University of Wisconsin) 41: 1–7, doi:10.2307/2687951, MR 0231832
  9. Lightstone, A. H. (1968), "Group theory and the principle of duality", Canadian Mathematical Bulletin 11: 43–50, doi:10.4153/cmb-1968-006-9, MR 0229507
  10. Review of The Axiomatic Method by R. L. Goodstein, Mathematical Reviews, MR 0163834.
  11. Review of The Axiomatic Method by Peter Andrews (1966), Journal of Symbolic Logic 31 (1): 106–108, JSTOR 2270630.
  12. Review of Concepts of Calculus by D. R. Dickinson (1966), Mathematical Gazette 50 (373): 329–330, JSTOR 3614713.
  13. 1 2 Review of Symbolic Logic by Burrowes Hunt (1969), American Mathematical Monthly 76 (6): 716–717, doi:10.2307/2316722.
  14. 1 2 Review of Symbolic Logic by G. Cuthbert Webber (1966), Science (New Ser.) 153 (3735): 519, doi:10.1126/science.153.3735.519, JSTOR 1719891, Bibcode: 1966Sci...153..519L.
  15. 1 2 Review of Symbolic Logic by R. L. Goodstein (1967), Mathematical Gazette 51 (375): 78, JSTOR 3613660.
  16. 1 2 3 Review of Nonarchimedean Fields by I. Fenyo, Mathematical Reviews, MR 0414354.
  17. 1 2 Review of Nonarchimedean Fields by Peter A. Loeb (1977), Bulletin of the American Mathematical Society 83 (2): 231–235, doi:10.1090/S0002-9904-1977-14277-8.
  18. 1 2 Review of Mathematical Logic by J. M. Plotkin (1980), Mathematical Reviews, MR 0497355)
  19. Review of Mathematical Logic by J. N. Crossley (1979), Bulletin of the American Mathematical Society 1 (6): 1003–1005, doi:10.1090/S0273-0979-1979-14718-9.
  20. "The Albert Harold Lightstone Scholarship". www.canadian-universities.net. 2010. Retrieved March 31, 2011.
  21. "Mathematics & Statistics Specific Awards". www.queensu.ca: Queen's University. Retrieved March 31, 2011.
  22. "The Albert Harold Lightstone Scholarship". www.queensu.ca: Queen's University. Retrieved March 31, 2011.
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