I'm reading a paper by V.S. Vladimirov and I.V. Volovich and they make a particular claim which is supposed to be discussed in:

N.M. Krylov, Dokl. Akad. Nauk SSSR, 60, 687 (1947).

I have not been able to reproduce the claimed result, so, it would be nice to see what precisely is done in the reference above.

The claim is that hyperderivatives of functions on $\mathcal{A} = \mathbb{R} \oplus j\mathbb{R} \oplus j^2\mathbb{R}$ with $j^3=1$ give three dimensional wave equations. V.S. Vladimirov and I.V. Volovich do show other examples that I've been able to reproduce, but the $j^3=1$ example is elusive.

Thanks in advance for any insights.

Update:(9/13/2012) I have tried my interlibrary loan service and contacting the author, but neither has been thus far successful. I need to find the title of the article and/or the page numbers for the interlibrary loan service to try to find it. The style of reference I list above is sadly insufficient for them to locate the article.

  • $\begingroup$ You're not looking for a reference, you're looking for a precise paper you don't have access to, right? $\endgroup$ – Fernando Muro Sep 6 '12 at 5:41
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    $\begingroup$ Did you try interlibrary loan? $\endgroup$ – Igor Belegradek Sep 6 '12 at 10:35
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    $\begingroup$ Most of Russian mathematical joournals are available in electronic form here: mathnet.ru/#, but this is not the one. You can try to contact I.V.Volovich directly, his address is here: mathnet.ru/php/person.phtml?option_lang=rus&personid=8846 I am sure he will help. $\endgroup$ – Sergei Akbarov Sep 6 '12 at 11:26
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    $\begingroup$ @James S. Cook: mathscinet (or google search) says that "Dokl. Akad. Nauk SSSR" is the abbreviation of "Doklady Akademii Nauk SSSR, Continued as Dokl. Akad. Nauk, Mathematics section translated in Soviet Math. Dokl., Soviet Phys. Dokl. and Soviet Physics Dokl." $\endgroup$ – Igor Belegradek Sep 6 '12 at 21:59
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    $\begingroup$ It appears that my library has this journal, in offsite storage. I've put in a request and should have the bound volumes tomorrow or early next week. Our catalog lists volume 60 with year 1948, so I also requested volumes 55-59 just in case (though from the catalog it appears 56-58 may be missing). I don't read Russian but hopefully I can transliterate well enough to tell if I'm looking at your article, and if so I'll send it to you. Is the yahoo.com address on your web page a good one to use? $\endgroup$ – Nate Eldredge Sep 14 '12 at 4:04

Does this sound like it?

Крылов Н.М. О кватернионах Роана Гамильтона и понятии моногенности// ДАН СССР.- 1947.- т.55, № 9. с.799-800.

In English: N.M.Krylov, On Rowan Hamilton quaternions and the notion of monogeny, Doklady Akademii Nauk SSSR (Proceedings of the Academy of Sciences of the USSR), 1947, vol. 55, no 9, pages 799-800

It disagrees with the data you have in the volume and the issue numbers but it is the closest one I could find as far as the other parameters are concerned. I do not have an access to the article either (I doubt any kind soul was patient enough to scan all old volumes of DAN) but if you confirm that the title sounds reasonable, I can try to get a scan of the article (alas, in Russian). Also, I doubt it'll have a full proof of anything there: most likely it is a short announcement (a la Comptes Rendus). However, it still may give some clues.

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    $\begingroup$ @James S. Cook: our library has this issue of DAN; I'll scan the article and email it to you this Monday (I looked before in volume 60, without finding it, so evidently I was looking in the wrong volume). $\endgroup$ – Carlo Beenakker Sep 14 '12 at 8:21
  • $\begingroup$ @fedja This does look like the right article. In Russian will do nicely. Since it sounds like Carlo has easy access, and in the interest of not duplicating the effort, If Carlo Beenakker could scan it and email it to classicalsuperfield@gmail.com then I would be grateful. Of course, I am thankful for the offers from Nate Eldrege and fedja. $\endgroup$ – James S. Cook Sep 14 '12 at 15:08
  • $\begingroup$ I'm sorry, this volume is missing from Leiden University library; what we have of Doklady Akademii Nauk was published through the French Academy of Sciences, and apparently the war years interrupted that. $\endgroup$ – Carlo Beenakker Sep 17 '12 at 9:28
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    $\begingroup$ I've just sent you an e-mail with the scan. $\endgroup$ – fedja Sep 24 '12 at 11:41
  • $\begingroup$ @fedja Success! However, that paper is apparently not about the three-dimensional real associative, unital algebra with $j^3=1$. Which is not surprising given the title you posted in this answer to start... so, I guess the right paper is somewhere... but I don't know it's title or year of publication. Fedja has kindly provided a list of all Krylov's publications so I will search that and see if I can make a less ambiguous request in the future. $\endgroup$ – James S. Cook Sep 25 '12 at 3:15

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