Timeline for Zeros of linear partial fractions
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 8, 2011 at 0:50 | comment | added | Gerry Myerson | I imagine $\lambda_1=0$ can be accommodated by considering $(1/z)+(-2/(z+9))+(1/(z+19))$. I don't know about the constraints on the $\alpha_i$. Maybe if you post a new question with all the constraints spelled out (and with a reference to the current question) someone will have an idea. | |
| Mar 7, 2011 at 9:33 | vote | accept | dan | ||
| Mar 7, 2011 at 8:49 | comment | added | dan | Hi Gerry, Sorry for the delay. Yes, your solution does provide a counter example for the problem as I stated. I guess, however, that my problem has more structure. For example, the poles $\lambda_i$ are the eigenvalues of the graph Laplacian, and therefore $\lambda_1=0$. Secondly, the constants $\alpha_i$ come from the eigenvectors of the Laplacian. In particular, given 2 nodes $r$ and $q$ in the graph (which are fixed), and the eigenvectors of the the Laplacian $u_k$, then $\alhpa_i = u_{i,r}u_{i,q}$ (where $u_{i,r}$ corresponds to the r-th element of the eigenvector $u_i$). | |
| Mar 5, 2011 at 3:16 | comment | added | Gerry Myerson | Dan, are you still here? Does my answer of 27 February meet the specifications, or did I miss the point? | |
| Feb 27, 2011 at 4:28 | answer | added | Gerry Myerson | timeline score: 2 | |
| Feb 25, 2011 at 13:09 | comment | added | Gjergji Zaimi | Sorry that was my comment earlier, and I deleted it because as you say it doesn't answer the question. I somehow read the condition as $\alpha_i\geq 0$. I think that by approximating $\alpha_i$ by rationals and rescaling makes it safe to assume that they are integers, bu this doesn't help as the only case I know how to put any sort of bound on the roots of $P$ is when $\sum \alpha_i \neq 0$. | |
| Feb 25, 2011 at 12:34 | comment | added | dan | I received a couple comments (that appear deleted now) that suggest the Gauss-Lucas theorem might be applicable. I guess this is not the case, as the constants $\alpha_i$ are not integer (and may not even be rational). Are there any generalizations of the Gauss-Lucas theorem that can handle this case? | |
| Feb 25, 2011 at 11:58 | comment | added | dan | Hi Emil, Yes, you are correct. $z_i \in [0, \; M]$ and the poles of the system are therefore in the interval $[-M, \; 0]$. | |
| Feb 25, 2011 at 11:23 | comment | added | Emil Jeřábek |
The poles are $-z_i\in[0,M]$, not in $[-M,0]$. Or did you want to write $\sum_{i=1}^n\frac{\alpha_i}{z-z_i}$?
|
|
| Feb 25, 2011 at 8:56 | history | asked | dan | CC BY-SA 2.5 |