User hebert - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T11:31:56Z http://mathoverflow.net/feeds/user/16716 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89963/exponent-of-metacyclic-groups Exponent of metacyclic groups Hebert 2012-03-01T13:57:52Z 2012-03-01T18:45:46Z <p>I am interested in the following related questions in metacyclic groups of the form $\mathbb{Z}_n \ltimes_r \mathbb{Z}_m$, where $r^n \equiv 1 \pmod{m}$: </p> <ol> <li><p>The order of an arbitrary element $g = (\alpha, 0)*(0, \beta)$ - or some upper bound on the order - where * is the group operation. </p></li> <li><p>The exponent of the group</p></li> </ol> <p>I know that the first question reduces to finding the smallest integer $k$ such that: </p> <p>$k \alpha \equiv 0\pmod{n}$, and </p> <p>$\beta \frac{r^{k \alpha} - 1}{r^\alpha - 1} \equiv 0 \pmod{m}$, </p> <p>but that's about it. Thank you very much in advance. </p> http://mathoverflow.net/questions/86793/find-most-densely-located-k-points-among-n-nk-points-in-two-dimension/86803#86803 Answer by Hebert for Find most densely located K points among N (N>K) points in two dimension Hebert 2012-01-27T08:24:53Z 2012-01-27T08:24:53Z <p>First of all, your problem is ill-posed, since you have to define the 'area occupied by the points' more precisely: Is it the area of their convex hull? Is it the area of the smallest enclosing circle? These are two possible definitions of area, but there may be more. </p> <p>On the other hand, the problem has all the flavour of an NP-hard problem, though I cannot confirm that yet. If it were so, then basically the only guaranteed algorithm would be exhaustive search in the space of all subsets of $K$ points. </p> http://mathoverflow.net/questions/86723/properties-of-rational-functions Properties of rational functions Hebert 2012-01-26T14:32:23Z 2012-01-26T23:17:03Z <p>Hi everyone. It is well known that a polynomial of degree $n$ is completely determined by $n+1$ points. Now, is there any similar result for rational functions? </p> http://mathoverflow.net/questions/89963/exponent-of-metacyclic-groups/89985#89985 Comment by Hebert Hebert 2012-03-04T14:52:39Z 2012-03-04T14:52:39Z @GH: In our case, $k=n$, $m=m$, $l=m$, and $n=r$ (the left-hand sides are Hempel's variables, and the right-hand sides are mine). So, the value of $\lcm(n,m)$ follows from Lemma 2.1 of Hempel's paper. I would still like to obtain that same value by lowering $k$ in your argument. http://mathoverflow.net/questions/90123/np-hardness-of-a-graph-partition-problem Comment by Hebert Hebert 2012-03-03T21:54:00Z 2012-03-03T21:54:00Z Well, I don't have a formal proof now, but I think that it should be NP-hard. Take a look at problem GT12 of Garey and Johnson, which is quite similar. In that problem, the instance is a graph $G$, and a smaller graph $H$, and the question is whether you can partition the vertices of $G$ into isomorphic copies of $H$. Intuitively, the fact that the vertex set of both copies are not necessarily disjoint should make the problem harder. http://mathoverflow.net/questions/89963/exponent-of-metacyclic-groups/89985#89985 Comment by Hebert Hebert 2012-03-02T21:56:50Z 2012-03-02T21:56:50Z Well, it turns out that the exponent of the group is exactly $\mbox{lcm}(n,m)$. http://mathoverflow.net/questions/89963/exponent-of-metacyclic-groups Comment by Hebert Hebert 2012-03-02T13:31:02Z 2012-03-02T13:31:02Z I got it. Thanks again. http://mathoverflow.net/questions/89963/exponent-of-metacyclic-groups Comment by Hebert Hebert 2012-03-01T21:23:20Z 2012-03-01T21:23:20Z I still haven't got access to the paper, but thank you very much anyway. I'm looking forward to see it. http://mathoverflow.net/questions/89963/exponent-of-metacyclic-groups/89985#89985 Comment by Hebert Hebert 2012-03-01T20:53:25Z 2012-03-01T20:53:25Z It is indeed helpful; thank you very much. http://mathoverflow.net/questions/89963/exponent-of-metacyclic-groups Comment by Hebert Hebert 2012-03-01T16:57:21Z 2012-03-01T16:57:21Z This question was posted two days ago at MathStackExchange, and it hasn't got any answer so far. That's why I have posted it here. It is a legitimate research question. http://mathoverflow.net/questions/89963/exponent-of-metacyclic-groups Comment by Hebert Hebert 2012-03-01T16:30:30Z 2012-03-01T16:30:30Z It is not homework. If it looks so easy to you, could you please give me some hint? http://mathoverflow.net/questions/86793/find-most-densely-located-k-points-among-n-nk-points-in-two-dimension/86803#86803 Comment by Hebert Hebert 2012-01-27T11:37:57Z 2012-01-27T11:37:57Z The definition you give now for the area is precisely the convex hull of the $K$ points. http://mathoverflow.net/questions/86723/properties-of-rational-functions/86766#86766 Comment by Hebert Hebert 2012-01-27T08:36:21Z 2012-01-27T08:36:21Z Interesting. Thank you David. http://mathoverflow.net/questions/86723/properties-of-rational-functions/86729#86729 Comment by Hebert Hebert 2012-01-26T15:46:25Z 2012-01-26T15:46:25Z Thanks. I see that it can be generalized to the case when the numerator and the denominator have different degrees, right? http://mathoverflow.net/questions/86723/properties-of-rational-functions Comment by Hebert Hebert 2012-01-26T15:44:16Z 2012-01-26T15:44:16Z Thanks Darij. If you find or remember anything else, please let me know.