Is the annulus of centre $0$ and radii $1$ and $2$ biholomorphic to the punctured disc $$\{ z \in \mathbb{C} \ | \ 0&lt;|z| &lt;1\}$$ ? Why ?  I know the Riemann mapping theorem but here we have  non simply connected domains.
I want to show that the two annuli $$A=\{r&lt;|z-z_0|&lt;R\} $$ $$A'=\{r'&lt;|z-z_0'|&lt;R'\} $$ are conformally equivalent (i.e. there exists a biholomorphic map between the two) iff $$\frac{R}{r}=\frac{R'}{r'}. $$  Sufficiency:  Suppose the ratios of the radii are the same. A suitable scaling (by $\frac{r'}{r}$) will make $A$ congruent to $A'$, and a translation will map the scaled $A$ onto $A'$. This is clearly a conformal homeomorphism.  Necessity:  The annulus $A$ has the following set of periods $$\left\{ \pm \frac{2 \pi}{\log (R/r)} \right\} $$ (this step required some computations which I will omit)  Similarly $A'$ has   $$\left\{ \pm \frac{2 \pi}{ \log(R'/r')} \right\}$$ as its set of periods. Since the set of periods is a conformal invariant. The ratios of the radii must indeed by the same.  Is this proof correct?   Thanks!
How do I duplicate a whole line in Emacs? Is the matrix diagonalizable for all values of t? Is the topological product $\Pi \{G_\alpha: \alpha < \kappa\} $ a topological group? Spinor dotted and undotted indices Outlining in Blender? I am currently working on converting my logo vectored as a SVG to a 3D model using the 2.72 version. I ran into a problem with my logo because it contains a black stroke and an orange stroke (see both images below), but currently all I have is the interior color of each shape. I was wondering whether or not there was a simple way to add in these strokes, and how exactly to go about creating the strokes. Thank you in advance community! In case, if I get a multiple-entry Schengen visa in one of the embassies in Ukraine, Kyiv - should my first trip be to the country which issued the visa?  Any there any requirements of this sort? Does this condition vary depending on embassy which issued the visa?  What are the possible consequences on not visiting that country first? Does it somehow depend on country where the embassy is (i.e. non-Schengen and not EU members)? I keep reading in economics journals about a particular result used in random utility models. One version of the result is: if $\epsilon_i \sim_{iid}, $ Gumbel($\mu, 1), \forall i$, then:   $$E[\max_i(\delta_i + \epsilon_i)] = \mu + \gamma + \ln\left(\sum_i \exp\left\{\delta_i \right\} \right),  $$  where $\gamma \approx 0.52277$ is the Euler-Mascheroni constant.  I've checked that this makes sense using R, and it does.  The CDF for the Gumbel$(\mu, 1)$ distribution is:  $$G(\epsilon_i) = \exp(-\exp(-(\epsilon_i - \mu)))$$  I'm trying to find a proof of this and I've had no success.  I've tried to prove it myself but I can't get past a particular step.   Can anyone point me to a proof of this?  If not, maybe I can post my attempted proof up to where I get stuck. At what stage should acknowledgements be inserted into the paper if review is not double-blind? When the review process is double-blind, I guess (although I have not read this anywhere) that acknowledgements should not be written in the submitted paper, as this might break anonimity. They should only be inserted into the camera-ready version, after the paper is accepted.  What is the custom when the review process is not double-blind? Should I write the acknowledgements in the submitted version, or wait to the camera-ready version? php String Concatenation, Performance In languages like Java and C#, strings are immutable and it can be computationally expensive to build a string one character at a time.  In said languages, there are library classes to reduce this cost such as C# System.Text.StringBuilder and Java java.lang.StringBuilder.  Does php (4 or 5; I'm interested in both) share this limitation?  If so, are there similar solutions to the problem available? I was wondering if there was a method to alter object settings at a set point in an animation in blender.  For example, I want to make the color of an object blue for the first 20 frames, but after 20 frames I want its color to change to green. Is there any known method of doing this? I could not find anything when I searched. I would like to look at my referral traffic over time on a per domain-basis in Google Analytics. This is what I have done:   Set New visits as advanced segment  Navigated to traffic-sources -> Referrals Checked a coulple of domains Clicked plot rows   The individual domains now show up in the graph, but the problem is that "all traffic" is also visible. All traffic is much bigger than the graphs for individual domains. This makes it very hard to see any changes as they look all values are Zero.  Is there any way to hide "Visists" from the graph in order to zoom in on the vertical axis? Why won't this work?  I'm trying to make an instance of a class delete itself.  &gt;&gt;&gt; class A():     def kill(self):         del self   &gt;&gt;&gt; a = A() &gt;&gt;&gt; a.kill() &gt;&gt;&gt; a &lt;__main__.A instance at 0x01F23170&gt; What desktop environments can group window icons in the taskbar like Windows 7? I find it much faster to identify a window in the taskbar (window list) by its icon,    rather than by its text:    The problem with most graphical interfaces I've  is that they don't group windows, and if you have more terminals or file managers, they'll just clutter the taskbar and you won't know which is which. Windows 7 offers a relatively elegant solution to this by grouping similar window icons together.  Question: What Linux  have this functionality? If addons exist that provide it, that's alright.  For reference, a Cinnamon applet that sort of does this is . Google chrome is reading characters backwards Google Chrome is my favorite browser, and I can't imagine going on without it. Suddenly (without closing and re-opening) the browser; chrome has been reading characters upside down, very weird:    Chrome version:    I tried restarting my browser and my MAC (running OS X Yosemite 10.10), but the error keeps on coming. $d = \gcd(c,m),\ ac≡bc\pmod{\! m}\Rightarrow a≡b\pmod {\!m/d}\ $ [Congruence Cancellation & Division Rule] How would you show that if $ac≡bc$ $\mod m$ and $\gcd(c,m)=d$, then  $a≡b$ $\mod \frac{m}{d}$?  Any help would be much appreciated! What Happens to electrons after Alpha Decay and Nuclear Fission? Where do the electrons go? In alpha decay do 2 electrons follow the alpha particle and make stable Helium or does the larger daughter nucleus become an anion?  Also what do the electrons do in the mixture of fission and alpha decay? With Beryllium-8 it decays into two alpha particles and 4 lonely electrons or do you get two Helium atoms?If someone could add an entry to Holocron it would be greatly appreciated, thanks. How to use aes-encryption in PostgreSQL? I'm automating a web application (the Mantis bug tracker) and I'm getting an interesting response header from it, called Refresh:  HTTP/1.x 200 OK ... Refresh: 0;url=my_view_page.php   It seems to be acting the same way that  does, and the meta refresh technique implies that it is an equivalent of a header in HTTP.  Problem is, I can't find any mention of the Refresh header in the  or any other definitive documentation on how it should be parsed and what the browser should do when it encounters it.  What's going on here? Once a world enters hardmode, corruption and hallows being spreading at an alarming rate, including the underground. Grass, which grows on dirt, succumbs easily to both. Jungles and glowing mushrooms grow on mud and provide unique and valuable items. Will corruption/hallows destroy these if left unchecked? If they do destroy mud-based life, how can I preserve these fragile ecosystems? At this time, Wubi does not work with Windows 8 default boot-loader. Thus at this point Wubi would not work on a new Windows 8 machine.      Citation fom    Question  Recently I've learned of , a way to install Ubuntu right from Windows.  Besides installing from Windows, I would like to know key differences between a regular "alongside Windows" or Dual-boot Ubuntu installation and one done with Wubi. Are there any disadvantages (for example a performance penalty) or incompatibilities I should look out for when using Wubi?