User alexander farrugia - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T09:44:45Z http://mathoverflow.net/feeds/user/24715 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111787/is-there-an-analytical-method-of-solving-general-square-root-equations Is there an analytical method of solving general square root equations? Alexander Farrugia 2012-11-08T07:17:43Z 2012-11-08T07:35:53Z <p>Equations such as $\sqrt{x+1}+\sqrt{x+2}=x+3$ are easily solvable by squaring both sides. But if we increase an extra square root, like if trying to solve $\sqrt{x+1}+\sqrt{x+2}+\sqrt{x+3}=x+4$ we encounter a problem. Squaring both sides doesn't work now, since the L.H.S. would still end up with three terms involving square roots, rather than only one as for the previous equation.</p> <p>That is, unless I'm missing something. Is there a way of solving equations such as the following analytically?</p> <p>$\sqrt{P_1(x)}+\sqrt{P_2(x)}+\cdots+\sqrt{P_n(x)}=Q(x)$ where $n>2$ and $Q(x), P_1(x), \ldots, P_n(x)$ are all polynomials.</p> http://mathoverflow.net/questions/104297/eigenvectors-of-asymmetric-graphs Eigenvectors of asymmetric graphs Alexander Farrugia 2012-08-08T18:36:01Z 2012-08-09T06:22:39Z <p>Let $G$ be an asymmetric connected graph. Then is it always the case that at least one of the eigenvectors of its adjacency matrix $A$ consists entirely of distinct entries?</p> <p>Thanks!</p> http://mathoverflow.net/questions/100749/is-there-any-relationship-between-a-treegraph-theory-and-semi-metric/100757#100757 Answer by Alexander Farrugia for Is there any relationship between a tree(graph theory) and semi-metric? Alexander Farrugia 2012-06-27T08:35:39Z 2012-06-27T08:35:39Z <p>Yes, it is possible. If we define the distance between two vertices in a graph as being the smallest sum of the weights of the edges connecting both, that would form a metric (if the graph is connected).</p> http://mathoverflow.net/questions/100674/simple-laplacian-versus-simple-adjacency-matrix-eigenvalues Simple Laplacian versus simple adjacency matrix eigenvalues Alexander Farrugia 2012-06-26T11:15:09Z 2012-06-26T13:14:17Z <p>If the eigenvalues of the Laplacian matrix of a graph G are all simple, is it always the case that the eigenvalues of the adjacency matrix of G are all simple as well?</p> <p>Thanks in advance!</p> http://mathoverflow.net/questions/111787/is-there-an-analytical-method-of-solving-general-square-root-equations/111788#111788 Comment by Alexander Farrugia Alexander Farrugia 2012-11-08T07:53:55Z 2012-11-08T07:53:55Z Thank you very much for this answer. http://mathoverflow.net/questions/111787/is-there-an-analytical-method-of-solving-general-square-root-equations Comment by Alexander Farrugia Alexander Farrugia 2012-11-08T07:33:04Z 2012-11-08T07:33:04Z You're right! Good insight. http://mathoverflow.net/questions/104297/eigenvectors-of-asymmetric-graphs/104307#104307 Comment by Alexander Farrugia Alexander Farrugia 2012-08-09T06:29:00Z 2012-08-09T06:29:00Z Funny that I had checked the Frucht graph before asking this question and hadn't noticed that all eigenvectors' entries were not distinct. Well spotted. http://mathoverflow.net/questions/104297/eigenvectors-of-asymmetric-graphs Comment by Alexander Farrugia Alexander Farrugia 2012-08-09T06:23:27Z 2012-08-09T06:23:27Z Yes, the graph has a trivial automorphism group. And yes I'm assuming a connected graph. I edited the question to reflect this. http://mathoverflow.net/questions/100674/simple-laplacian-versus-simple-adjacency-matrix-eigenvalues/100683#100683 Comment by Alexander Farrugia Alexander Farrugia 2012-06-26T13:54:42Z 2012-06-26T13:54:42Z Thank you. I had already found a graph with the spectrum of its adjacency matrix consisting entirely of simple eigenvalues, but whose Laplacian spectrum not entirely consisting of simple eigenvalues, which is why I did not ask the reverse question as well. The example I found happens to have 6 vertices as well.