User dexter - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T13:51:47Z http://mathoverflow.net/feeds/user/6258 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25610/making-n-set-of-all-positive-integers-a-group Making N (set of all positive integers) a group dexter 2010-05-23T01:07:43Z 2012-04-10T07:12:13Z <p>Can anybody please give me an example of a binary operation under which N forms a group? More generally, how to find some operations to make possibly any set a group?</p> http://mathoverflow.net/questions/71843/maximum-of-traba Maximum of Tr$(ABA')$ dexter 2011-08-01T23:54:45Z 2011-08-02T11:02:38Z <p>Let $B$ be a fixed symmetric $M\times M$ matrix over the reals.</p> <p>Let $A$ be an arbitrary $N\times M$ matrix over the reals.</p> <p>I want to consider the problem of finding the extremal value of $\operatorname{Tr}(ABA^T) = \operatorname{Tr}(A^TAB)$ under the constraint that $AA^T$ is a fixed $N\times N$ matrix. </p> <p>Can you give me a hint?</p> http://mathoverflow.net/questions/71843/maximum-of-traba Comment by dexter dexter 2011-08-03T08:19:29Z 2011-08-03T08:19:29Z @Gjergji, +1...I just realize there should (and indeed there was a hidden) constraint on $A$. I have done the case for maximum, and hope will be able to find the min as well. http://mathoverflow.net/questions/71843/maximum-of-traba Comment by dexter dexter 2011-08-02T10:48:53Z 2011-08-02T10:48:53Z @ Mikael de la Salle, My expected answer (intuition) is not exactly but something like yours (the problem arises in a physical problem I'm working);...but why? http://mathoverflow.net/questions/25610/making-n-set-of-all-positive-integers-a-group/25728#25728 Comment by dexter dexter 2010-05-25T18:48:00Z 2010-05-25T18:48:00Z @ Pietro Majer, as Scott Carnahan suggested, your example is not so simple, at least I am not getting how N forms a group under this operation. what is meant by 0 here?....But I take the general statement as a theorem (of course without verifying it:)). http://mathoverflow.net/questions/25610/making-n-set-of-all-positive-integers-a-group/25612#25612 Comment by dexter dexter 2010-05-23T02:37:40Z 2010-05-23T02:37:40Z @Arturo Magidin, Thanks for the general part. http://mathoverflow.net/questions/25610/making-n-set-of-all-positive-integers-a-group/25613#25613 Comment by dexter dexter 2010-05-23T02:21:07Z 2010-05-23T02:21:07Z @Fran&#231;ois G. Dorais, nice to learn it. Thanks.