Wikipedia:Reference desk/Archives/Mathematics/2011 November 11

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November 11

Tensor algebra

Hello. The tensor algebra T(V) is abstractly constructed as a direct sum of tensor powers of the vector space V, with an associated multiplication that makes T(V) into an algebra. However other texts sometimes teach about tensors, covariance and contravariance, and multiplying tensors of mixed contravariance and covariance first, and they are probably talking about objects of A=T(V)⊗T(V*). So far, I've only seen this A=T(V)⊗T(V*) in a book by Mac Lane and Birkhoff, and I haven't been able to find it anywhere else since.

1. Can anyone explain why it seems that the usual tensor algebra T(V) is done "without contravariance"?
2. Does anyone have any more references for T(V)⊗T(V*), and any likely names it has been given?

Thanks, Rschwieb (talk) 15:57, 11 November 2011 (UTC)

A direct summand of T(V)⊗T(V*) would be an algebra of the form, for example, V⊗V⊗V⊗V*⊗V*. The elements are " tensors of mixed contravariance and covariance", as you say. (Taking the dual of vector spaces is a contravariant functor.) Mct mht (talk) 08:34, 15 November 2011 (UTC)

I'm looking forward to seeing the object appearing in a reference (other than Mac Lane and Birkhoff), if anyone has any luck finding one. Rschwieb (talk) 16:13, 15 November 2011 (UTC)

Smallest uninteresting number

According to QI, British tv series, 12407 is the smallest number of no interest (other than that). They said there is an online site listing the properties of all integers. What is that site? Kittybrewster ☎ 23:16, 11 November 2011 (UTC)

You do see, don't you, that the notion of a smallest uninteresting number is a paradox? There can be no such number. Having said that, I can't help you find the website you seek—PaulTanenbaum (talk) 23:18, 11 November 2011 (UTC)

See Interesting number paradox. There is obviously no site with an entry for all integers. There are lots of sites with entries for many of them. See for example the links at http://math.crg4.com/uninteresting.html. Wikipedia also has articles for some of them in Category:Integers. PrimeHunter (talk) 23:34, 11 November 2011 (UTC)

http://www.nathanieljohnston.com/2009/06/11630-is-the-first-uninteresting-number/ says: "Update [November 12, 2009]: It looks like 11630 is now listed in the OEIS. Additionally, 12067 was recently added, meaning that 12407 is now the first uninteresting number."

OEIS is the On-Line Encyclopedia of Integer Sequences. As their name says, they are for sequences and not individual integers, but I guess your references was about them. http://oeis.org/search?q=12407 currently only shows a sequence containing -12407. Lots of their sequences contain 12407 when more terms of the sequence are shown than the usual around three lines. For example, 12407 = 19×653 is a semiprime so it's eventually in oeis:A001358. If you click "Table of n, a(n) for n = 1..10000" there then you get a page showing it as term 3235 which admittedly doesn't sound very interesting, but neither does a lot of whole sequences in OEIS. PrimeHunter (talk) 23:48, 11 November 2011 (UTC)

They even have a sequence which contains all the uninteresting numbers oeis:A001477, now that has got to be an interesting sequence ;-) Dmcq (talk) 23:56, 11 November 2011 (UTC)

Yes, OEIS is the site they named in the programme. 81.98.43.107 (talk) 02:05, 12 November 2011 (UTC)

Somebody created 12407. That sounds like a poor idea to me. The number already has a brief mention in Interesting number paradox which seems sufficient. PrimeHunter (talk) 19:06, 12 November 2011 (UTC)

The Penguin Dictionary of Curious and Interesting Numbers is along these lines. Hut 8.5 12:27, 13 November 2011 (UTC)

But should it be 11630, 12067 or 12407 all three of which feature in the paradox page. PH you could always turn 12407 into a redirect, which would be BOLD. Kittybrewster ☎ 12:34, 13 November 2011 (UTC)

This number is interesting, because it is uninteresting. Plasmic Physics (talk) 00:04, 17 November 2011 (UTC)