Comments on: Nilpotent Infinitesimals I http://www.arsmathematica.net/archives/2008/05/19/nilpotent-infinitesimals-i/ Dedicated to the mathematical arts. Thu, 08 Jan 2009 13:19:55 +0000 http://wordpress.org/?v=2.5.1 By: Ars Mathematica » Blog Archive » Nilpotent Infinitesimals II http://www.arsmathematica.net/archives/2008/05/19/nilpotent-infinitesimals-i/#comment-60355 Ars Mathematica » Blog Archive » Nilpotent Infinitesimals II Wed, 25 Jun 2008 07:15:48 +0000 http://www.arsmathematica.net/?p=662#comment-60355 [...] This is a follow-up to this post. [...] [...] This is a follow-up to this post. [...]

]]> By: Jacob Freeze http://www.arsmathematica.net/archives/2008/05/19/nilpotent-infinitesimals-i/#comment-59893 Jacob Freeze Fri, 23 May 2008 09:08:57 +0000 http://www.arsmathematica.net/?p=662#comment-59893 There's a general algebraic development of biplex numbers at http://www.rowan.edu/open/depts/math/osler/Biplex_Nos_Final_Corrected_Version_Aug_3.pdf including nilpotent infinitesimals in the form of dual numbers, along with the "perplex numbers," which were apparently invented by a group of freshmen at St. Olaf College, with a norm ||h||=–1. These exotic beasties are useful for extending the usual formalism of special relativity to the case ||v||>c, as described in the American Journal of Physics -- May 1986 -- Volume 54, Issue 5, pp. 416-422. You can <em>almost</em> define a useful nilpotent algebra in quantum mechanics, with exchange of coordinates as multiplication, and the product of superimposed fermions would vanish... but <b>nature abhors a nilpotence,</b> and the fermions remain stubbornly apart. There’s a general algebraic development of biplex numbers at http://www.rowan.edu/open/depts/math/osler/Biplex_Nos_Final_Corrected_Version_Aug_3.pdf including nilpotent infinitesimals in the form of dual numbers, along with the “perplex numbers,” which were apparently invented by a group of freshmen at St. Olaf College, with a norm ||h||=–1. These exotic beasties are useful for extending the usual formalism of special relativity to the case ||v||>c, as described in the American Journal of Physics — May 1986 — Volume 54, Issue 5, pp. 416-422.

You can almost define a useful nilpotent algebra in quantum mechanics, with exchange of coordinates as multiplication, and the product of superimposed fermions would vanish… but nature abhors a nilpotence, and the fermions remain stubbornly apart.

]]> By: Michael O'Connor http://www.arsmathematica.net/archives/2008/05/19/nilpotent-infinitesimals-i/#comment-59877 Michael O'Connor Thu, 22 May 2008 03:21:59 +0000 http://www.arsmathematica.net/?p=662#comment-59877 I wrote up some notes on it (the same notes that I mentioned in the comments on "Selling Infinitesimals") and put them at: http://arxiv.org/abs/0805.3307 There's also John Bell's "An Invitation to Smooth Infinitesimal Analysis" available at: http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf In addition to smooth infinitesimal analysis (which is essentially doing calculus with nilpotent infinitesimals) there is synthetic differential geometry (which is doing differential geometry with nilpotent infinitesimals. There's an exposition available at: http://www.math.uchicago.edu/~shulman/exposition/sdg/pizza-seminar.pdf There's also a book on synthetic differential geometry at: http://home.imf.au.dk/kock/sdg99.pdf I wrote up some notes on it (the same notes that I mentioned in the comments on “Selling Infinitesimals”) and put them at:
http://arxiv.org/abs/0805.3307

There’s also John Bell’s “An Invitation to Smooth Infinitesimal Analysis” available at:
http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf

In addition to smooth infinitesimal analysis (which is essentially doing calculus with nilpotent infinitesimals) there is synthetic differential geometry (which is doing differential geometry with nilpotent infinitesimals. There’s an exposition available at:
http://www.math.uchicago.edu/~shulman/exposition/sdg/pizza-seminar.pdf

There’s also a book on synthetic differential geometry at:
http://home.imf.au.dk/kock/sdg99.pdf

]]> By: John Sidles http://www.arsmathematica.net/archives/2008/05/19/nilpotent-infinitesimals-i/#comment-59876 John Sidles Thu, 22 May 2008 03:21:16 +0000 http://www.arsmathematica.net/?p=662#comment-59876 So are nilpotent infinitesimals is some sense the <i>opposite</i> of <a href="http://en.wikipedia.org/wiki/It%C5%8D%27s_lemma" rel="nofollow">Ito increments</a>? In the sense that in the former case, squares of infinitesimals are smaller than we expect; in the latter case larger than we expect. So are nilpotent infinitesimals is some sense the opposite of Ito increments?

In the sense that in the former case, squares of infinitesimals are smaller than we expect; in the latter case larger than we expect.

]]> By: Kenny Easwaran http://www.arsmathematica.net/archives/2008/05/19/nilpotent-infinitesimals-i/#comment-59872 Kenny Easwaran Thu, 22 May 2008 00:03:24 +0000 http://www.arsmathematica.net/?p=662#comment-59872 Are there any easily available references for the formal theory of nilpotent infinitesimals? Are there any easily available references for the formal theory of nilpotent infinitesimals?

]]> By: Domenic Denicola http://www.arsmathematica.net/archives/2008/05/19/nilpotent-infinitesimals-i/#comment-59857 Domenic Denicola Tue, 20 May 2008 09:29:43 +0000 http://www.arsmathematica.net/?p=662#comment-59857 Thanks for posting this! I look forward to part 2 :D. Thanks for posting this! I look forward to part 2 :D.

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