Biholomorphic map between two annulus.
When can we find holomorphic bijections between annuli?
Quadratic Formula in Complex Variables How to create quizzes with macqtex? Case insensitive regular expression without re.compile? Monitor cell value and copy it to another cell Asynchronous shell exec in PHP It is well established that:   'In quantum mechanics, the behavior of an electron in an atom is described by an orbital, which is a probability distribution rather than an orbit.   There are also many graphs describing this fact: : (hydrogen atomic orbital - one electron) In the figure, the shading indicates the relative probability to "find" the electron, having the energy corresponding to the given quantum numbers, at that point.  My question is: How do these symmetries shown in the above article occur?                 What about the 'preferable' axis of symmetries? Why these? Can you upgrade the RAM of an Android device? I'm sure many people like me are aware and annoyed by the extremely limited amount of RAM in Nexus S (512MB but only 300+ is usable), I have searched all over google for a solution to increase the amount of RAM available and came up empty (I tried swap file applications but they just broke my system's SD card partitions causing a complete reset).  I wonder if there is any hardware solution to this problem. Is it possible to swap the chip out for a bigger capacity one? I realize that it is not as simple as swapping out PC RAM from a dimm but I haven't seen any schematics for the chip layout on the ARM boards in the smartphones so I really hope that it is somewhat possible to increase the amount of RAM on the board. How do I prove $F(a)=F(a^2)?$ Let $E$ be an extension field of $F$. If $a \in E$ has a minimal polynomial of odd degree over $F$, show that $F(a)=F(a^2)$.  let $n$ be the degree of the minimal polynomial $p(x)$ of $a$ over $F$ and $k$ be the degree of the minimal polynomial $q(x)$ of $a^2$ over $F$.   Since $a^2 \in F(a)$, We have $F(a^2) \subset F(a)$, then $k\le n$  In order to prove the converse:  $q(a^2)=b_0+b_1a^2+b_2(a^2)^2\ldots+b_k(a^2)^k=0$   implies   $q(a)=b_0+b_1a^2+b_2a^4\ldots+b_ka^{2k}=0$ Then   $p(x)|q(x)$, because $p(x)$ is the minimal polynomial of $a$ over $F$.  If I prove that $n|2k$ we done, since $k$ is odd, we have $n|k$ and $n\le k$ and finally $n=k$.  So I almost finished the question I only need to know how to prove that $n|2k$ It should be only a detail, but I can't see, someone can help me please?  Thanks How to change the voltage polarity in an inverting amplifier? I want to implement a simple circuit which can map -10V  +10V voltage range to 0V 5V voltage range. Thus I'm planning to use two op-Amps, one for the slope and the other for the offset. So basically I'm planning to use an inverting amplifier and a summing amplifier consecutively.  But Inverting amplifiers multiplies the input voltage with a minus sign. I need a positive slope. How can I overcome this?  Also Could you gibe me any ideas for a cheap op-Amp models for this purpose? how to *completely* turn off automatic updates in Windows 10 I have disabled the services &quot;Windows Update&quot; and &quot;Windows Update Medic Service&quot; in using services.mcs. Furthermore, I made changes to the registry, creating a key HKEY_LOCAL_MACHINE\SOFTWARE\Policies\Microsoft\Windows\WindowsUpdate\AU with a DWORD (32-bit) Value &quot;AUOptions&quot; with value set to 2. I still see frequent queries to ctldl.windowsupdate.com sent to my DNS server. Which services need to be disables to completely shut this down? Is there a limit to the number of magic rings a character can wear on one hand? Is there any example of a non-measurable set whose proof of existence doesn't appeal to the Axiom of choice? Is there any example of a non-measurable set whose proof of existence doesn't appeal to the Axiom of choice?  What would it imply if there was such an example?  EDIT: For instance, maybe this will help understand the kind of example I had in mind, it is known that an important feature to determine that the AC is needed in the Banach-Tarski case is the non-transitivity of rotations on Euclidean space, one might find an example of a transitive group for some given space keeping the rest equal and make the AC unnecessary?  I guess then it might be said that this transitivity will be in this particular case equivalent to the AC or some amount of it, but I guess that it would be important to show this if it hadn't been realized before. How accurate is the GPS on the iPhone's Find My iPhone app?  I used Find My iPhone app to track a friend. The first 3 stops were dead on but when it refreshed it showed them to be several streets over from actual location. What could cause that? I am using Sql Server 2008 on Small Business Server 2003; client is using WinXP.   I added a user to my Active Directory Security Group; Why can't this user immediately access the database? It seems there is a delay before the user is recognized in Sql Server.   I am using AD Security Groups for permissions expressly so that I don't need to add individual users in Sql Server. So effectively I don't need to do anything but add the user to the AD Security Group in order to grant access.  But for some reason Sql Server doesn't immediately recognize the addition. I've seen this a number of times. I add the user to the group but that user can't access data until the next day. It seems that it doesn't query the Active Directory in real time. Can you confirm that is so?  What do I need to do so that Sql Server "refreshes" the list of users from Active Directory? I am using Google Chrome 48.0 32-bit. Today when I launched Google Chrome, I got this message:    This computer will soon stop receiving Google Chrome updates because this Linux system will no longer be supported   I thought Ubuntu 14.04 is supported for five years, what's exactly happening here?  And on 64-bit Ubuntu, I get an error from Update Manager and apt-get update:  Failed to fetch http://dl.google.com/linux/chrome/deb/dists/stable/Release  Unable to find expected entry 'main/binary-i386/Packages' in Release file (Wrong sources.list entry or malformed file)   Some index files failed to download.   They have been ignored, or old ones used instead. Why is empty set an open set? I thought about it for a long time, but I can't come up some good ideas. I think that empty set has no elements,how to use the definition of an open set to prove the proposition. The definition of an open set: a set S in n-dimensional space is called open if all its points are interior points. When is unbalanced data really a problem in Machine Learning? Some improvements to blockquotes Update  Sorry for the delay! I’ve been thinking about this feedback in between my other projects, and I’ve landed where I’ve started 😕  I’ve made sure to continue highlighting spoilers in gray. The differentiation from normal blockquotes is super important since their behavior is so different. I explored keeping spoilers yellow, but that still has the same theming issues throughout the network. Red seemed too powerful, with its error connotations.  I understand completely that folks are using blockquotes to highlight images. However, I don’t think that’s a  we want to formally pave. I’m ok with being opinionated here—blockquotes should be used for quoting, not as a way to highlight images. I understand that this will be frustrating to Machavity who made excellent points about differentiation in his answer, but I don’t think image highlighting ought to dictate how blockquotes are styled.  I’m also cautious of adding borders to images as explored by user1306322 for the same reasons and for the additional fear of being too prescriptive in how we display images there.  It’s a tough one! I went back and forth with the design team and I still think this is the best path forward.  Original post  We’re making a few changes to blockquotes soon. Historically, we had a bit of feedback that the yellow backgrounds were overpowering, reducing the contrast of the quote itself. The yellow border also lacks contrast with the background. Others have indicated that there should be  and between .  These yellows also compete with theming in our communities, especially those with backgrounds that aren’t perfectly white like English Language &amp; Usage and Role-Playing Games.   We’re proposing we switch to a single gray bar in our blockquotes. This should fix those theming issues and make it easier to distinguish between quotes, spoilers and code.  This will also improve blockquote nesting, since our current nesting is a mess because the darker yellow bar is barely noticeable on the paler yellow background.  Spoilers should pretty much behave exactly the same way. More room for later improvements there.  Here’s how our current blockquotes look. You can click to expand:    Here’s what we propose:    There is a lot of prior art out there for this approach. GitHub has styled theirs this way for a long time. So has Slack.     Let me know what y’all think. I think these changes will be an improvement, and I hope you do too.  Huge thanks to Catija for gathering some meta posts on blockquotes and creating an awesome blockquote playground post for testing. Resolve build errors due to circular dependency amongst classes Why do Americans have to file taxes, even if their only source of income is from a regular job?