Timeline for A Fractional Linear Transformation Class Property
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 21, 2012 at 17:26 | vote | accept | Euplio M. | ||
| Apr 21, 2012 at 5:19 | answer | added | Vaughn Climenhaga | timeline score: 1 | |
| Apr 19, 2012 at 17:10 | answer | added | Robert Israel | timeline score: 1 | |
| Apr 19, 2012 at 16:01 | history | edited | Euplio M. | CC BY-SA 3.0 |
added 28 characters in body; deleted 6 characters in body
|
| Apr 19, 2012 at 15:48 | history | edited | Euplio M. | CC BY-SA 3.0 |
added 112 characters in body
|
| Apr 19, 2012 at 15:39 | comment | added | Euplio M. | @Vaughn: Thank you for the suggestion. I will edit the question to make it more clear. | |
| Apr 19, 2012 at 3:17 | comment | added | Vaughn Climenhaga | @Euplio: Do I understand correctly that the class $\mathcal{S}$ you consider consists of fractional linear transformations $f\colon x\mapsto \frac{ax+b}{cx+d}$, where $a,b,c,d\in\mathbb{R}$, such that $f([-1,0])=[-1,0]$ and $f'(x)>0$ on that interval? If so, you should edit the question to make that clear. | |
| Apr 18, 2012 at 22:37 | history | edited | Euplio M. | CC BY-SA 3.0 |
added 36 characters in body
|
| Apr 18, 2012 at 22:29 | comment | added | Euplio M. | Thank you to all those who responded so far. I apologize for not specifying that the maps in question should have strictly positive derivative $f'(x)>0$ on the real interval $[-1,0]$. Also, it is only necessary that the maps to be closed under composition, but additionally the class should also be closed under any affine change of variables (that is under composition of a linear map and its inverse) (Ideally I would like it to contain the fractional linear maps, but this isn't strictly necessary). I hope this clarifies matters. Let me know if you have any other questions. Thanks again. | |
| Apr 18, 2012 at 12:46 | answer | added | Lee Mosher | timeline score: 1 | |
| Apr 18, 2012 at 12:26 | comment | added | Lee Mosher | We also need a few other points to be resolved. What kinds of maps do you want, e.g. in the case of $\hat C$ do you want holomorphic bijections, or just holomorphic self-maps? You have restricted your class of maps $\mathcal{C}$ to be closed under composition; do you want them to be invertible and closed under inverse as well? The answers will have very different characters depending on how these points are resolved. | |
| Apr 18, 2012 at 3:06 | comment | added | Vaughn Climenhaga | I suppose we need the OP to clarify whether $\mathcal{S}$ refers to transformations on $\mathbb{R}\cup\{\infty\}$ or on $\hat{\mathbb{C}}$. That is, is this a question in real analysis or complex analysis? | |
| Apr 18, 2012 at 3:03 | comment | added | Vaughn Climenhaga | @Andreas: I see I've fallen into my usual trap of only thinking in terms of real numbers and neglecting the complex world. I interpreted the question to be dealing with fractional linear transformations $x\mapsto \frac{ax+b}{cx+d}$ on $\mathbb{R}\cup\{\infty\}$, where $a,b,c,d$ are all real, and so one can distinguish between FLTs with $f'>0$ (which I called orientation-preserving) and those with $f'<0$. Of course since rotation by $\pi$ in $\mathbb{C}$ is orientation-preserving the distinction vanishes if we think in $\mathbb{C}$, exactly as you say. | |
| Apr 18, 2012 at 2:04 | comment | added | Andreas Blass | @Vaughn: All fractional linear transformation (and all meromorphic functions) preserve orientation. | |
| Apr 18, 2012 at 1:41 | comment | added | Vaughn Climenhaga | In light of Robert Israel's answer, did you mean to restrict $\mathcal{S}$ to include only orientation-preserving FLTs? | |
| Apr 18, 2012 at 1:35 | answer | added | Robert Israel | timeline score: 1 | |
| Apr 18, 2012 at 1:03 | history | asked | Euplio M. | CC BY-SA 3.0 |