most recent 30 from http://mathoverflow.net 2013-05-24T18:16:34Z http://mathoverflow.net/feeds/user/2955 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10700/self-similar-matrices Igor Korepanov 2010-01-04T12:12:19Z 2010-01-05T22:28:53Z

<p>Does anyone know anything about self-similar (infinite) matrices, with more or less fractal(-like) structure and admitting meaningful matrix-algebra operations?</p>

http://mathoverflow.net/questions/359/a-reading-list-for-topological-quantum-field-theory/10575#10575 Igor Korepanov 2010-01-03T10:34:22Z 2010-01-03T17:18:24Z

<p>You might also be interested in new "fermionic" theories, with new and almost totally unexplored features. If yes, see e.g. arXiv:0907.3787 and arXiv:0911.1395. To understand these properly, you should read, however, at least something in the beginning of Turaev's book on torsions (but <em>not</em> his papers on quantum invariants) and, of course, some book on Grassmann-Berezin calculus of anticommuting variables (a few relevant pages in Berezin's book on Second Quantization will work).</p>

http://mathoverflow.net/questions/386/do-all-3d-tqfts-come-from-reshetikhin-turaev/10570#10570 Igor Korepanov 2010-01-03T09:10:29Z 2010-01-03T09:10:29Z

<p>There must be lots of <em>fermionic</em> TQFTs behaving in a very different way compared to RT. One feature of these fermionic (based on <em>Berezin integral</em>) theories is that they are quite straightforwardly generalized to any dimensions, not just 3 - which, by the way, provides you with a powerful and intriguing means of studying 3-manifolds using 4-manifolds, etc. These new theories are related to Lie groups (without any obvious restrictions as for semisimplicity etc.) and their homogeneous spaces, see e.g. <a href="http://arxiv.org/abs/0907.3787" rel="nofollow">http://arxiv.org/abs/0907.3787</a> .</p>

http://mathoverflow.net/questions/10700/self-similar-matrices/10854#10854 Igor Korepanov 2010-01-05T23:30:36Z 2010-01-05T23:30:36Z

Henry, yes, here I agree. Indeed, it was my oversight, I should have remarked about tensor products. Otherwise, I cannot see at this moment any more remarks overseen by me. But maybe this discussion with me was a bit entertaining for Scott? :)

http://mathoverflow.net/questions/10700/self-similar-matrices/10854#10854 Igor Korepanov 2010-01-05T23:14:10Z 2010-01-05T23:14:10Z

Henry, thanks for your comment. It you re-read my question, you will see that, essentially, I have already got the very clear answer, in the negative (nobody knows). My question was exactly what I wanted to ask: <i>anything vaguely resembling</i> self-similarity would work. So my question was very precise, including the fact that the definition of the terms was not fixed.

http://mathoverflow.net/questions/10700/self-similar-matrices/10854#10854 Igor Korepanov 2010-01-05T22:37:00Z 2010-01-05T22:37:00Z

Scott, thanks for you answer. I am just studying what this mathoverflow is, and your answer is helpful for this. Perhaps I will explain in the next version of my question that just tensor products are not &quot;fractal enough&quot; - or, otherwise, I may decide that this was a wrong site to ask serious questions. But thank you anyhow!

http://mathoverflow.net/questions/10700/self-similar-matrices Igor Korepanov 2010-01-05T07:33:18Z 2010-01-05T07:33:18Z

OK, Dmitri, I will do as you say: staying here, I must respect the rules of this forum. I was, however, a bit embarrassed having read, at the page &quot;How to ask a good MO question&quot;, about vague and specific questions. It is definitely very sad that &quot;vague&quot; questions are forbidden, because, first, creative questions are always vague and, second, they reflect the way of thinking of some minority of scientists (I think, not only professors but PhD students as well). Anyhow, as I say, I will prepare soon a more detailed question.

http://mathoverflow.net/questions/10700/self-similar-matrices Igor Korepanov 2010-01-04T20:19:08Z 2010-01-04T20:19:08Z

One more remark for Mariano and all the readers, after a little musing: posing a question in such a broad way also reflects my personal way of thinking. And, after having used this forum for just one day, I see that I like it so much, and don't want to quit... Maybe there is some way to settle this?

http://mathoverflow.net/questions/10700/self-similar-matrices Igor Korepanov 2010-01-04T19:55:10Z 2010-01-04T19:55:10Z

This depends on the aims of this site: if broad/general questions don't work here, then ok. Note however that this question has been for already almost 2 years on another math forum, and <i>nobody</i> could say <i>anything</i>. Note also that it is exactly such questions that stimulate research activity in people. Cheers

http://mathoverflow.net/questions/10700/self-similar-matrices Igor Korepanov 2010-01-04T16:15:47Z 2010-01-04T16:15:47Z

Indeed this paper consists in a collection of facts and looks to me as sort of an unfinished scientific work, this is why I am searching for new ideas. And that sequence of matrices is constructed in a self-similar way: the elements of a finite matrix are replaced with some blocks of the like elements, then once more and so on. Here is where the self-similarity appears, although (alas) the fractal dimension of the set of nonzero elements in the resulting infinite matrix is just one. Nevertheless, it does look like a fractal, at least a bit...

http://mathoverflow.net/questions/10700/self-similar-matrices Igor Korepanov 2010-01-04T13:29:26Z 2010-01-04T13:29:26Z

Well, John, give me some time to think how to explain this... Or, in the case if you can read some Russian, here are two short texts with examples of such matrices: <a href="http://csc.ac.ru/ej/file/4381" rel="nofollow">csc.ac.ru/ej/file/4381</a> and <a href="http://csc.ac.ru/ej/file/4641" rel="nofollow">csc.ac.ru/ej/file/4641</a> . And thanks to Dmitri for creating the fractals tag!

http://mathoverflow.net/questions/10700/self-similar-matrices Igor Korepanov 2010-01-04T12:45:57Z 2010-01-04T12:45:57Z

Also, could the moderators or/and high-reputation users kindly add here a tag &quot;fractals&quot; or something like that?

http://mathoverflow.net/questions/386/do-all-3d-tqfts-come-from-reshetikhin-turaev/10570#10570 Igor Korepanov 2010-01-03T09:34:05Z 2010-01-03T09:34:05Z

I would like to add also that, on a big scale, such great achievements of 20th century as Jones polynomial and related stuff, including RT invariants, fall within the framework of Yang-Baxter equation which, in its turn, goes back to Onsager's solution of 1940's of Ising model in math physics. Now fermionic formulas are my attempt to go outside that framework (think it looks natural in 21st century); see also a 4d paper <a href="http://arxiv.org/abs/0911.1395" rel="nofollow">arxiv.org/abs/0911.1395</a> (and I am of course its author, perhaps I will once make myself a better sort of login here so as this could be seen automatically). Cheers, I.K.