User elena - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T15:05:07Z http://mathoverflow.net/feeds/user/6194 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86662/characteristic-polynomial-of-a-symmetric-integer-matrix Characteristic polynomial of a symmetric integer matrix Elena 2012-01-25T21:47:20Z 2012-01-25T21:47:20Z <p>I am wondering what, if anything, is known about the characteristic polynomials of integer symmetric matrices. I believe I read somewhere that not every polynomial with integer coefficients can be a characteristic polynomial for an integer matrix (correct me if I am wrong). What I would like to be true is that basically there aren't really any constraints in some sense.</p> http://mathoverflow.net/questions/63311/computing-index-of-a-subgroup-of-sl-2-in-sage Computing index of a subgroup of SL_2 in sage Elena 2011-04-28T16:51:16Z 2011-04-29T04:37:58Z <p>Suppose I have a subgroup of $\textrm{SL}_2(\mathbb Z)$ given by 3 generators, and it happens to be of finite index in $\textrm{SL}_2$. Is there a way (on Sage, since that is what I have access to) to figure out its index? I tried defining a matrix group via the generators and using index(), but this gave me an error. Perhaps if I define the group via permutations it would work? I am new to Sage, so any help would be appreciated.</p> http://mathoverflow.net/questions/59996/sum-of-log-p-p-for-p-equivalent-to-l-mod-d Sum of log p/p for p equivalent to l mod D Elena 2011-03-29T17:35:34Z 2011-03-29T18:42:05Z <p>It's fairly classical that for $D>1$ and $(D,l)=1$ one has $$\sum_{\stackrel{p\leq x}{p\equiv l\; (mod \; D)}}\frac{\log p}{p} = \frac{\log x}{\phi(D)} + \textrm{O}(1)$$ where if I understand correctly the dependence on $D$ in the $\textrm{O(1)}$ is captured by something like $$\frac{1}{\phi(D)}\sum_{\textrm{non-principal} \; \chi}\frac{-L'(1,\chi)}{L(1,\chi)}$$ which is $\ll_{\epsilon}D^{\epsilon}$ for any $\epsilon>0$. If this is correct, suppose I am interested in the sum $$\sum_{\stackrel{p\leq x}{\left(\frac{-D}{p}\right)=1}}\frac{\log p}{p}$$ where $\left(\frac{-D}{p}\right)$ is the Legendre symbol. There are $\phi(D)/2$ residue classes mod $D$ that $p$ can lie in, and so then this is just my previous sum $\phi(D)/2$ times, and I get that it's $1/2 \log x +\textrm{O}(1)$, where the dependence on $D$ in $\textrm{O}(1)$ is now something like $\textrm{O}(D^{1+\epsilon})$. Is this the best one can do? I was hoping I could get it to be $\textrm{O}(D^{\epsilon})$ but if that's not even correct perhaps I should stop trying.</p> http://mathoverflow.net/questions/25269/random-walks-and-lyapunov-exponents Random Walks and Lyapunov exponents Elena 2010-05-19T19:28:02Z 2010-07-04T00:27:26Z <p>Given a sequence $Y_1, Y_2, \dots$ of i.i.d. matrices in $GL_n(\mathbb R)$, there is a theorem of Furstenberg and Kesten which says that if $\mathbb E(\log||Y_1||)$ is finite, there exists a constant $\gamma$ (the Lyapunov exponent) such that $$\lim_{n\rightarrow\infty}\frac{1}{n}\log||Y_n\dots Y_1|| = \gamma$$ There are also versions of central limit theorems for this scenario. I'm pretty sure this is also known in a more general case (e.g. suppose we have a sequence of matrices $Y_i$ of order 2, and I don't want to consider sequences of length $n$ in which $\dots Y_i Y_i\dots$ appears). I am wondering if anyone knows a good reference for theorems regarding Lyapunov exponents and central theorems in this case.</p> http://mathoverflow.net/questions/86662/characteristic-polynomial-of-a-symmetric-integer-matrix Comment by Elena Elena 2012-01-26T04:43:51Z 2012-01-26T04:43:51Z Richard, thank you for the reference: somehow I couldn't find a relevant one through a usual search. Tom -- I can see that the 2 by 2 case is restrictive (though actually for my purposes the constraint isn't so strong, but that's not really relevant). My intuition though is that the constraints should be looser the higher the degree of the polynomial, but perhaps this is off. http://mathoverflow.net/questions/86662/characteristic-polynomial-of-a-symmetric-integer-matrix Comment by Elena Elena 2012-01-25T23:21:35Z 2012-01-25T23:21:35Z So if I give you a polynomial with integer coefficients and real roots, how do I know if it's a characteristic polynomial for an integer symmetric matrix or not? http://mathoverflow.net/questions/86662/characteristic-polynomial-of-a-symmetric-integer-matrix Comment by Elena Elena 2012-01-25T23:20:57Z 2012-01-25T23:20:57Z It's true, characteristic polynomials of $2\times 2$ symmetric matrices of trace 0 have constraints. For one, the coefficient in front of x then automaticlaly becomes 0 (or whatever else you restrict the trace to be). So clearly a constraint on the trace puts a constraint on the polynomial. But I'm not asking for trace 0. If you take ANY intger 2 by 2 symmetric matrix, then there are a lot fewer constraints. I am wondering what results are known in general, for characteristic polynomials of integer symmetric matrices. Of course it must have real roots. http://mathoverflow.net/questions/63311/computing-index-of-a-subgroup-of-sl-2-in-sage/63378#63378 Comment by Elena Elena 2011-04-29T16:14:44Z 2011-04-29T16:14:44Z Exactly! Which is why I made the comment that I thought to do what you said requires you to know that your group is finite index :) But the gap thing is something I should try. http://mathoverflow.net/questions/63311/computing-index-of-a-subgroup-of-sl-2-in-sage/63378#63378 Comment by Elena Elena 2011-04-29T15:43:56Z 2011-04-29T15:43:56Z I thought you meant that one could use the Schreier index formula for subgroups of free groups. I could post the generators but I think it will be useful for me to work this out for myself! http://mathoverflow.net/questions/63311/computing-index-of-a-subgroup-of-sl-2-in-sage/63313#63313 Comment by Elena Elena 2011-04-28T18:14:25Z 2011-04-28T18:14:25Z Igor -- but then I guess I would really have to be sure the group is finite index (in this case I just think it should be but haven't really confirmed it). That's why I was hoping to use some computer program to confirm it first. By the way, do you have any experience with the Todd-Coxeter algorithm wrt its slowness? Emmanuel -- I am aware of the Magma algorithm, but I don't have access to Magma at the moment. Thanks for the suggestion, though. http://mathoverflow.net/questions/63311/computing-index-of-a-subgroup-of-sl-2-in-sage Comment by Elena Elena 2011-04-28T17:18:55Z 2011-04-28T17:18:55Z No, I wasn't aware of it -- I'll try it there. http://mathoverflow.net/questions/59996/sum-of-log-p-p-for-p-equivalent-to-l-mod-d/60006#60006 Comment by Elena Elena 2011-03-29T18:51:17Z 2011-03-29T18:51:17Z Thanks, now that I see that it seems obvious. I had tried to work it out that way (since it did seem unnecessary to go through all of the characters to get information basically just about one) but for some reason I was failing. http://mathoverflow.net/questions/25269/random-walks-and-lyapunov-exponents Comment by Elena Elena 2010-05-19T20:34:43Z 2010-05-19T20:34:43Z Andrey, I would like to know what are the necessary conditions on a sequence of matrices to have such theorems. More immediately though, I was looking for a reference regarding, say, Markovian sequences. I will look at the book you suggested. rpotrie, I do know that book. It discusses the iid case very clearly, but I was having trouble finding references to more general cases there.