Bijections from $A$ to $B$, where $A$ is the set of subsets of $[n]$ that have even size and $B$ is the set of subsets of $[n]$ that have odd size
Exactly half of the elements of $\mathcal{P}(A)$ are odd-sized
What happens if you are refused entry between two countries? Last month, I was travelling on a backpacking trip across Russia and the Baltic states (Schengen zone). I'm an Indian by nationality. So I had to get the respective single entry visa for Russia and Schengen Europe.  At the border crossing into Estonia by road (at Ivangorod-Narva), I exited the Russian border with an exit stamp (so now I can't go back to Russia since I had a single entry visa). When I was at the immigration checkpoint of Estonia, I was heavily questioned on the intent of my travel to Europe while entering Estonia by road.   Though I finally got into Europe, It got me wondering - what happens if a foreign national is denied entry into a country not at the airport but by land between 2 countries he cannot go into ?  Generally, if you are denied entry at the airport the airline has the obligation to fly you back to your country. But what happens when you are stuck between two countries and you cant get into either ? What happens to you then ? You can use a standard dot notation or a method call in Objective-C to access a property of an object in Objective-C.  myObject.property = YES;   or  [myObject setProperty:YES];   Is there a difference in performance (in terms of accessing the property)? Is it just a matter of preference in terms of coding style? Schadenfreude is the joy or pleasure derived from the misfortunes of others. What is the word for joy or pleasure derived from the happiness of others? Saving Arduino sensor data to a text file How can I save data retrieved from a sensor to a text file on the computer? What's is the origin of Orbital Angular Momentum of electrons in atoms? Consider the Hydrogen 1s electron. We know that, in the quantum picture, the electron isn't orbiting or rotating at all, rather we simply state that the electron is spread over the entire space with the probability of finding it being maximum a radial distance $R$ (=Bohr radius) away from the nucleus. This helps explain why the electron does not radiate EM radiation while in the atom.  But, with this understanding, I do not understand the source of the orbital angular momentum. Is it intrinsic like the spin? I am trying to prove that if I have $f:\mathbb{R} \to \mathbb{R}$ satisfying $\forall x,y\in\mathbb{R},f(x+y) = f(x) + f(y)$. Which is assumed continuous at $0$, that $f$ is continuous on $\mathbb{R}$  I am fairly sure it is, for that property seems to be a property of polynomials, and we know polynomials are continuous where defined(all reals)  Any ideas for rigor?  $|x-a| \lt \delta \implies |f(x+a)-f(x)-f(a)| \lt \epsilon$  No idea where to go from here, this is my first time doing $\epsilon-\delta$ stuff. Find the integer solutions: $$a·b^5+3=x^3,a^5·b+3=y^3$$ This is the first problem of today's USAJMO (has finished),I only find a trival result that $x\equiv y \pmod6$ and $abxy≠0 \pmod 3$.  Thanks in advance! I am trying to figure out this:  c = 1 def f(n):     print c + n  def g(n):     c = c + n  f(1) # =&gt; 2 g(1) # =&gt; UnboundLocalError: local variable 'c' referenced before assignment   Thanks! Beamer: \nonumber equivalent for slides? Is there an equivalent command to \nonumber when one doesn't want to include certain slides in the numbering? I would like to leave outline and title slides out of my numbering, and instead just number slides with actual content. Other workarounds would be nice to know about as well.  Edit: Since a MWE example was requested, here is one. Suppose I have a title page and two other slides. The code below gives a slide counter that goes up to 3, and the title slide is slide 1. I would prefer to have the title slide not be numbered, the counter go up to 2, and the the numbering start at the slide after the title. Here is some code.  \documentclass{beamer}  \mode&lt;presentation&gt;  {    \usetheme{Dresden}  }   \setbeamertemplate  {footline}{\quad\hfill\insertframenumber/\inserttotalframenumber\strut\quad}    \begin{document}  \begin{frame}  \titlepage \end{frame}  \begin{frame} Some actual content here \end{frame}  \begin{frame} Some more actual content here \end{frame}  \end{document} How to install mods for the new Minecraft launcher I've recently changed to the new Minecraft launcher, and I have noticed a change in the .minecraft folder layout. I can no longer install mods just by copying it over into the versions/1.6.1 jar file, as this file refreshes every time it is launched.  For those who don't know what I am talking about, here is the new layout:    I am specifically trying to install Optifine, which does work with 1.6.1. Where do I drag in the mod files now? I can't seem to find a jar file that is suited to it, except in the version folder, but as I stated above, this refreshes every time.  After copying a recent version and renaming it in the version folder, I get this error in the development console:   Unexpected exception refreshing version list java.lang.IllegalArgumentException: Version '1.6.1' is already tracked     at net.minecraft.launcher.updater.VersionList.addVersion(VersionList.java:91)     at net.minecraft.launcher.updater.LocalVersionList.refreshVersions(LocalVersionList.java:44)     at net.minecraft.launcher.updater.VersionManager.refreshVersions(VersionManager.java:47)     at net.minecraft.launcher.Launcher$2.run(Launcher.java:164)     at java.util.concurrent.Executors$RunnableAdapter.call(Unknown Source)     at java.util.concurrent.FutureTask$Sync.innerRun(Unknown Source)     at java.util.concurrent.FutureTask.run(Unknown Source)     at java.util.concurrent.ThreadPoolExecutor.runWorker(Unknown Source)     at java.util.concurrent.ThreadPoolExecutor$Worker.run(Unknown Source)     at java.lang.Thread.run(Unknown Source) What is the difference between "Have you seen this?" and "Did you see this?" What's the difference between these two phrases? Nsolve gives solutions with a small complex part which can't be eliminated by chop command Why are new sites classified as "uncategorized"? How to stop iPhone users from sending iMessages and send SMS instead I used to have an iPhone, but I changed to an android (shock, horror!). Unfortunately Apple refuses to let me go and whenever friends message me the messages get sent as iMessages (unless they remember to specifically chnge the format). I never get these messages. They never get resent as SMSs automatically as is supposed to happen. How can I get my phone number deleted from the iMessage list. I've turned off iMessage on the old iPhone. Is there a video that explains how Stack Overflow works and can be used? Is there a video that explains how Stack Overflow works and can be used? I.e. a short tutorial to get new users started. If there isn't, then the people behind this site or others who know how it works should make one, because it's overwhelmingly different from other sites and thus difficult for beginners. How to draw Brownian motions in tikz/pgf Problem Currently, I am playing on a server where hoppers transport items at the rate of 64 items a second, and this has proved very difficult to build a sorter for. I'm not at all proficient in Redstone building but have tried building basic sorters and ones that I have found on YouTube and I haven't gotten anything to work. Solution Hopefully one of you is really good at Redstone and can help me design a sorter that can work under these conditions. (I need the sorter that can sort items while I AFK kill witches with a looting 47 sword P.S. most of the problem is with the sorter itself because no items will stay in the sorter hopper (bc the whole stack goes into the chest so the hopper doesn't have anything to go off of to sort) so any solution to that would be helpful as well. Installing grub2 in efi-mode in usb flash drive I've two different OS, which one is ubuntu. I'm using a PC, NOT a Mac. I use Chimera bootloader that don't find Ubuntu. So, i'd like to install grub 2 in EFI-mode on flash-drive USB. In this way, i can load Ubuntu only if the usb flash-drive is connected.  I followed this guide  replacing "/dev/sda" in  sudo efibootmgr --create --gpt --disk /dev/sda --part 1 --write-signature --label "GRUB2" --loader "\\EFI\\grub\\grub.efi"   with the usb-pen directory. Then, i've copied the grub.cfg file that exists in Ubuntu partition.Result? Grub2 loads, but when i select the menu entries, grub2 doesn't find any command, or disks. How can i fix this problem?  P.S. Here, there are the photo of the error(when i select ubuntu entry in grub2), the usb-drive's contents and the grub.cfg file Why is it called the Millennium Falcon? I'm particularly interested in how the "Millennium" part pertains to the ship itself. It wouldn't hurt to explain the "Falcon" part either though. Cannot edit posts on iOS app When I try to submit an edit in the iOS app it returns this message:        Text:     Account is not allowed to suggest edits   What happened and can it be fixed please?