Let $f$ be a holomorhic function on the open unit disk $D$. Suppose that there is an $r\in[0,1)$ such that restriction of $f$ to the annulus $$U=\left\{z\in C:r&lt;\left|z\right|&lt;1\right\}$$ is one-to-one. Prove that $f$ is one-to-one on $D$.  Firstly, $f$ is one-to-one if and only if derivative of $f(x)$ is not zero. So my aim is to use the boundary property of this function to prove the result. I tried Cauchy integral theorem. But it doesn't seem to work very well. Neither do the maximum and minimum principle.
Let $f: B(0,1) \rightarrow \mathbb{C}$ be holomorphic and suppose $\exists\ r \in (0,1)$ such that $f$ is injective in $A = \{z \in \mathbb{C} : r &lt; |z| &lt; 1\}$. Prove that $f$ is injective.  I tried using Rouché theorem, or the identity theorem, but I don't know what to do. Any hints? :)
Select an area in rep graph not working on Chrome The "select an area by dragging across the lower chart" thing is not working on Chrome.  Tested with Chrome 13.0.782.220 m, works with Firefox 6.0.1 and IE 8  Quirk: For some reason, if I open an incognito window, it works. Tried clearing the cache and cookies, still not working. Is there any practical difference between saving at 300ppi or 72ppi? I often see statements in discussions about web displayed image quality that go something like this: "I only upload pictures at 72 dpi and 1200 pixels on the long edge, so if someone copies and prints the image it won't look as good as if it were 300 dpi."  What? Have I missed something here?  Does an image edited and saved/exported with 1200x800 pixels at 300 ppi look any different online than the same image saved/exported with 1200x800 pixels at 72 ppi? Other than the metadata regarding ppi, is there any difference in the two images at all? If so, what is the difference? If I tell my printer to print the images at 4R (4X6 inches) will there be any difference at all in how the printer creates a print from the 1200x800 @ 300 dpi image versus the 1200x800 @ 72 dpi image? Spoilers for The Last Jedi:      Just before Snoke is slain,   he narrates Kylo Ren's thoughts/feelings (saying things like "he turns the lightsaber towards his enemy" and so on) we see the Skywalker lightsaber being manipulated through the Force to point towards and then slay Snoke. My question is, do we know for sure who was using to Force to use the Skywalker lightsaber in this scene?   It could have been   A) Kylo Ren was turning igniting the Skywalker lightsaber with the Force. Snoke assumed the thoughts/feelings he was sensing were directed at Rey but they weren't.   Or... B) Snoke was so preoccupied with Kylo's thoughts that he wasn't paying any attention to Rey's thoughts. Rey takes advantage of this and uses the Force to turn and ignite the Skywalker lightsaber to slay Snoke. This option is reinforced by the fact that  1) the Skywalker lightsaber then zooms into Rey's hand, not Kylo's.  2) after the ensuing scuffle, Kylo Ren reveals that he hasn't switched sides; he is still a believer in the First Order.  3) Kylo Ren even tells Hux that Rey slew Snoke. Should I use UUID as well as ID Brokerage accounts are seen as relatively tax-inefficient, since you invest with post-tax dollars, and growth is taxed upon realizing gains. The traditional IRA allows you to deduct contributions and defer taxes until you take distributions in retirement ...  Unless you are covered by a retirement plan from work and make over   $71,000/year if single, or $118,000/year if married filing jointly.   The numbers above come from . Above those thresholds, you can still make contributions to a traditional IRA, but those contributions will be post-tax. Distributions are taxed as ordinary income, excluding post-tax principal.  Similarly, eligibility to make direct Roth IRA contributions phases out to zero according to a :   $131,000 if single, or $193,000 if married filing jointly.   Looks to me that someone who cannot make direct Roth IRA contributions also cannot deduct traditional IRA deposits from income.  Assume   investment period is 20 years, is some garden variety ETF (VTI/VXUS/etc), those investments return 5%, the current marginal tax rate is 28%, the expected marginal tax rate at retirement is 25%, the long term capital gains rate is 15%, the entity is not eligible to make Roth IRA contributions, and 401k contributions are already maxed out.   A $5000 of gross income contribution into a traditional IRA at retirement might grow to an after-tax sum of:  (5000) * (1 - 0.28) * (1 + (1.05 ** 20 - 1) * (1 - 0.25)) = $8064   Instead, since contributions were going to be taxed anyway, why not put it in a brokerage account? Then you could claim the favorable long-term capital gains rate.  (5000) * (1 - 0.28) * (1 + (1.05 ** 20 - 1) * (1 - 0.15)) = $8659   All I found was , and the accepted answer says to still make a traditional IRA contribution, but I still don't see why that would be a good idea. What am I missing here? Need a MP3 ID3 tagger, and cover fetcher I need to tag my MP3 library, and have tried kid3 (which was manual tagging), when I used Kubuntu 9.10 (I now use Ubuntu Meerkat) Here are the features I am hoping for:   A good and clean UI. Tagging should be automatic, like Winamp's autotag feature, which rocks, btw! It should also embed the cover-art in the mp3, not copy a jpeg file in the folder, because now-a-days all players support displaying cover art. But acceptable if not possible. Rename the files as per some regular expression like %TrackNo - %Artist - %Title.  Should be accurate, and more importantly smart. I want to start tagging at night, and hopefully my collection should be done by the morning, w/o it being stuck at a user prompt at 1%.   If one app cant do all, I am willing to use 3, wouldn't mind exposure to a few more apps ;)  I have used picard or someting, and I didnt like it quite a lot. But wouldn't mind using it, if there is no other alternative.   Thanks for your time! Can Rails Routing Helpers (i.e. mymodel_path(model)) be Used in Models? When to use 'which' or 'in which' or 'that' (as relative pronouns)? Preferred languages: C/C++, Java, and Ruby.  I am looking for some helpful books/tutorials on how to write your own compiler simply for educational purposes. I am most familiar with C/C++, Java, and Ruby, so I prefer resources that involve one of those three, but any good resource is acceptable. Why is $\frac{d^n}{dx^n}(y(x))$ the notation for the $n$th derivative of $y(x)$, instead of $\frac{d^n}{d^nx}(y(x))$? I've always wondered why the numerator is $d^n$ while the denominator is $dx^n$ instead of $d^nx$ like the numerator.  I must be missing something very obvious or fundamental. Is this notation derived from the chain rule in some way? Combining multirow and multicolumn When combining multirow and multicolumn in the same cell of a table I am getting extra vertical lines:  \documentclass[plain]{article} \usepackage{multirow} \pagestyle{empty}  \begin{document}  \begin{tabular}{|l|l|l|l|}\hline   \multirow{10}{*}{numeric literals} &amp; \multirow{5}{*}{integers} &amp; in decimal &amp; \verb|8743| \\ \cline{3-4}   &amp; &amp; \multirow{2}{*}{in octal} &amp; \verb|0o7464| \\ \cline{4-4}   &amp; &amp; &amp; \verb|0O103| \\ \cline{3-4}   &amp; &amp; \multirow{2}{*}{in hexadecimal} &amp; \verb|0x5A0FF| \\ \cline{4-4}   &amp; &amp; &amp; \verb|0xE0F2| \\ \cline{2-4}   &amp; \multirow{5}{*}{fractionals} &amp; \multirow{5}{*}{in decimal} &amp; \verb|140.58| \\ \cline{4-4}   &amp; &amp; &amp; \verb|8.04e7| \\ \cline{4-4}   &amp; &amp; &amp; \verb|0.347E+12| \\ \cline{4-4}   &amp; &amp; &amp; \verb|5.47E-12| \\ \cline{4-4}   &amp; &amp; &amp; \verb|47e22| \\ \cline{1-4}   \multicolumn{3}{|l|}{\multirow{3}{*}{char literals}} &amp; \verb|'H'| \\ \cline{4-4}   &amp; &amp; &amp; \verb|'\n'| \\ \cline{4-4}   &amp; &amp; &amp; \verb|'\x65'| \\ \cline{1-4}   \multicolumn{3}{|l|}{\multirow{2}{*}{string literals}} &amp; \verb|"bom dia"| \\ \cline{4-4}   &amp; &amp; &amp; \verb|"ouro preto\nmg"| \\ \cline{1-4} \end{tabular}  \end{document}     Notice the vertical lines in the char literals and string literals cells (which should span three columns). How can they be avoided? My competitor is using Alexa to see what I am doing. I need to get stop Alexa from including my sites in their rankings. How can I do so?   Is it possible to deny access to any visitor with the Alexa toolbar and deny access to Alexa bot requests? Err http://archive.canonical.com natty InRelease     Err http://security.ubuntu.com oneiric-security InRelease                Err http://extras.ubuntu.com natty InRelease                             Err http://security.ubuntu.com oneiric-security Release.gpg   Temporary failure resolving ‘security.ubuntu.com’ Err http://archive.canonical.com natty Release.gpg   Temporary failure resolving ‘archive.canonical.com’ Err http://extras.ubuntu.com natty Release.gpg   Temporary failure resolving ‘extras.ubuntu.com’ Err http://gb.archive.ubuntu.com oneiric InRelease Err http://gb.archive.ubuntu.com oneiric-updates InRelease Err http://gb.archive.ubuntu.com natty-backports InRelease Err http://gb.archive.ubuntu.com oneiric Release.gpg   Temporary failure resolving ‘gb.archive.ubuntu.com’ Err http://gb.archive.ubuntu.com oneiric-updates Release.gpg   Temporary failure resolving ‘gb.archive.ubuntu.com’ Err http://gb.archive.ubuntu.com natty-backports Release.gpg   Temporary failure resolving ‘gb.archive.ubuntu.com’  Reading package lists... Done W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/oneiric/InRelease       W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/oneiric-updates/InRelease   W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/natty-backports/InRelease   W: Failed to fetch http://security.ubuntu.com/ubuntu/dists/oneiric-security/InRelease   W: Failed to fetch http://archive.canonical.com/ubuntu/dists/natty/InRelease   W: Failed to fetch http://extras.ubuntu.com/ubuntu/dists/natty/InRelease   W: Failed to fetch http://archive.canonical.com/ubuntu/dists/natty/Release.gpg  Temporary failure resolving ‘archive.canonical.com’ W: Failed to fetch http://security.ubuntu.com/ubuntu/dists/oneiric-security/Release.gpg  Temporary failure resolving ‘security.ubuntu.com’ W: Failed to fetch http://extras.ubuntu.com/ubuntu/dists/natty/Release.gpg  Temporary failure resolving ‘extras.ubuntu.com’ W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/oneiric/Release.gpg  Temporary failure resolving ‘gb.archive.ubuntu.com’ W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/oneiric-updates/Release.gpg  Temporary failure resolving ‘gb.archive.ubuntu.com’ W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/natty-backports/Release.gpg  Temporary failure resolving ‘gb.archive.ubuntu.com’ W: Some index files failed to download. They have been ignored, or old ones used instead.   This is what I'm seeing when I try to run sudo apt-get update. I did an update on my instance yesterday and am now experiencing this. How to install Eclipse J2EE IDE from a tarball? I have downloaded Eclipse 3.6.1 as a tar.gz file from eclipse site. Then I extract using  cmd: tar -zxvf eclipse-jee-helios-SR1-linux-gtk.tar.gz  after execute this cmd I got eclipse folder in this there is eclipse file. When I double click on this eclipse file I'm getting following error:     Could not display "/media/D-DEVELOPME/eclipse/eclipse".   There is no application installed for executable files   is there any solution for it? What is the "Bootable flag" option when installing a distro? Is the "bootable flag" needed in today's distributions? If not, then why is it still in the installers? What is it exactly? Remove Personal Options section from Profile I want to hide/remove the Personal Options in the Your Profile (wp-admin/profile.php) admin page.  I am aware that solutions for this exist, but I they use jQuery to do hide this section. This works, but when a user has JavaScript disabled in their browser, it will show up again. Therefore it is not a proper way to remove Personal Options.  Is there a way to remove the Personal Options section from the HTML source of the page? This means no jQuery or CSS hacks, or core file modification. I can follow the proofs in which the asymptotic normal-distribution of a maximum likelihood estimator $\tilde{\theta}_n$ is derived.   however, does this already imply that the maximum likelihood estimates are asymptotically unbiased, i.e. do we have $$E(\tilde{\theta}_n) \to \theta \text{ as } n \to \infty?$$  Since I am aware that in general it is not true that convergence in distribution implies convergence in moments, an explanation would be nice.  The result is often stated e.g.  as "...it means that the bias of the maximum likelihood estimator is equal to zero up to the order $n^{-1/2}$")  1.) Are  some kind of regularity conditions from the mle theory used to establish this result?  2.) Or is a $\sqrt{n}$-convergence of an estimator (to a normal distribution) in general already enough to establish convergence of its moments?  Note: the Wikipedia article mentions Cox, David R.; Snell, E. Joyce (1968).  , as a source where the order of the bias is derived (formula (12) or (20)).   However in this paper I can't follow the arguments 100%, since their Taylor approximation of $L'(\widehat{\beta})$ is lacking the remainder term. What is the argument used here to discard it completely? Does anyone know the major differences between the Java and .Net garbage collectors? A web search has not revealed much, and it was a question that came up in a test. Physically Identify the failed hard drive How can I completely remove a objects (e.g. particle system or groups used in particle systems) in blender using python?  The problem is this: I have a script, with which I create a scene (e.g. adding some normal objects and some group of objects with a particle system). For a research study I would like to repeat that that several times, without restarting blender. So I can create a scene, delete everything, e.g. with a script like this:  import bpy scn = bpy.context.scene for obj in scn.objects:     obj.select = True bpy.ops.object.delete()   If I rerun the script to recreate the scene with objects using particle system, the latter is not shown in the scene anymore (only the placeholders of the particle system). Now I'm not sure where exactly the problem lies, but I guess not everything is immediately deleted, running the script above. Only the startup of blender helps (but that's not what I want) or pushing the purge button. The purge button seems to start the command:  bpy.ops.outliner.orphans_purge()   But I can't get this piece of code to run in the console. It says that "the context is incorrect". Is there a way to purge the unused data by python script? Thanks a lot for any help.