Thursday, December 27, 2012

The problem of characterizing derivatives

I noticed that someone had asked a question not too long ago on the MathOverFlow site about characterizing derivatives.  The answers were somewhat helpful but none of the usual experts seems to have dived into the discussion, so there was only limited information. 

I did add a comment, but maybe I should post some stuff here for those who happen to have some interest in the problem.

The problem naively is to find some necessary and sufficient condition in order that a function  f: [a,b] --> R  would be the derivative everywhere in [a,b] of some other function.   (For example continuity would be sufficient but certainly not necessary,  Baire class 1 would be necessary but not sufficient.)

To research the problem you should consult at least the following:

Andrew M. Bruckner, Differentiation of real functions. Lecture Notes in Math., vol. 659, Springer-Verlag, Berlin and New York, 1978, x + 246 pp.
or the more recent edition:
Differentiation of real functions.
Second edition. CRM Monograph Series 5.  American Mathematical Society, Providence, RI, 1994. xii+195 pp. ISBN: 0-8218-6990-6 

Chapter seven contains an entertaining and accessible account of the problem, including the original formulation of the problem by W. H. Young.

Andy has updated his discussion of this problem in a 1995 survey article written for the Real Analysis Exchange:   

Bruckner, Andrew M.  The problem of characterizing derivatives revisited.  Real Anal. Exchange  21  (1995/96),  no. 1, 112--133.
 Since then there have been a few varied expressions of conditions that characterize derivatives in perhaps unusual ways:
  1. On Characterizing Derivatives,  D. Preiss and M. Tartaglia Proceedings of the American Mathematical Society, Vol. 123, No. 8 (Aug., 1995), pp. 2417-2420
  2. Freiling, Chris,   On the problem of characterizing derivatives.
    Real Anal. Exchange 23 (1997/98), no. 2, 805–812.
  3. Brian S. Thomson, On Riemann Sums, Real Analysis Exchange Vol. 37(1), 2011/2012, pp. 1–22

Tuesday, December 18, 2012

Chinese translations of mathematics articles?

We are working on a Chinese translation of our textbook Elementary Real Analysis.   Chinese students can already download a free PDF in the English language version from our web site, but they will have to wait a bit for our translated version.

Oddly enough I just learned that the Chinese have begun a journal dedicated to translating articles from English to Chinese.  This is part of the letter where they inform me that they have translated a Monthly paper of mine.  (They received permission from the MAA -- I was not in the loop.)

"I am writing this letter on behalf of the editorial office of "Mathematical Advance in Translation", a translated journal which was sponsored by the Academy of Mathematics and Systems Science (Chinese Academy of Sciences). The journal is non-profit targeting at providing a general information about the world mathematics advances. Many of the appearing articles are translated papers that are well selected from "Notices of AMS" and "Bulletin of AMS", "The Amer. Math. Monthly" and "Mathematics Magazine" of the Mathematical Association of American(MAA), “SIAM News" and " SIAM REVIEW", “EMS Newsletter" and et al. From the very first issue, it always states very clear the originations, including author, title, journal, year, volume and pages, to ensure the reader easily referring to the original. The distribution is in a small scale, only around 600-800 subscriptions, all of them at a cost price, but the journal was very welcomed among the mathematicians, especially those in the rural areas and the young school teachers to them such kind of material are not easily accessible."

For the curious (and the bilingual) here is a link to my paper in English:

   Thomson, Brian S.
   Monotone convergence theorem for the Riemann integral. 
   Amer. Math. Monthly  117  (2010),  no. 6, 547--550.

and here is a link to a scan of the Chinese version that they published  (ISSN 1003-3092):

   [Chinese Language Version

I have no idea how accurate and literate the translation is.  Perhaps someone will comment?