I read a general principal of graph theory which says "A graph that can be traversed without lifting the pencil from the paper , while tracing each edge exactly once, can have not more than 2 odd vertices." Can I have any proof of it? I am new to graph theory
Is the below brief explanation correct and complete (at least in context)? Can it be improved/simplified without getting technical? I am especially concerned about the last sentence: Is that reasoning absolutely clear to a layman?            The problem can be viewed as drawing the above graph without lifting your hand and without retracing a line. Observe that each dot is connected by three lines. Also observe that you have to draw a line to arrive at a dot, and you have to draw a line to leave that dot. Because each dot is connected by three lines, each must be visited twice. But it is impossible to visit each dot twice (even number of times) and end up with each dot connected with three lines (odd number of times).
Consider a hypothetical situation: In which a time traveler travels in a spaceplane at close to speed of light circling the earth and I get a live telecast of inside of that plane to my television. How time dilation will effect him and what will I observe sitting next to the tv seeing him live. Considering time runs slower at speed of light, will he be looking like a slow motion video? (Or let he also watching live streaming of us at earth, what it will look to him) P.S. I'm not sure if it even makes some sense but if it does kindly explain in logical details. Sorry for the the bad english, I'm not a native speaker. How do I bypass restrictions on the length of the PATH variable System Properties      This environment variable is too large. This dialog allows setting values up to 2047 characters long.   I use/have used a lot of software, and my PATH has grown as a result. When I try to add new paths, I get the above message. As a workaround, I'm editing my user path, but prefer not to. So basically I want to show:  $\aleph_{\alpha}^{\aleph_{1}} = \aleph_{\alpha}^{\aleph_{0}}\cdot2^{\aleph_1}$ , Where $\omega\leq\alpha&lt;\omega_1$  There is a proof that I found;    But I don't understand his proof for the limit case of $\alpha$.  To be more specific, I dont understand why $ \left( \sup_{\omega \le \beta &lt; \alpha} \aleph_{\beta}\right)^{\aleph_1} \le \left( \prod_{\omega \le \beta &lt; \alpha} \aleph_{\beta}\right)^{\aleph_1}$  Any insights or help is deeply appreciated.  Cheers In Mootools, I'd just run if ($('target')) { ... }.  Does if ($('#target')) { ... } in jQuery work the same way? Label equations arbitrarily When I have equations, as in  \begin{equation} ... \end{equation}   Can I arbitrarily assign the numbers to the equations? Instead of the usual order (1), (2), (3), ..., I'd like something like (1), (2.1), (2.2), (2.3), (3.1), (3.2), (4), ... While talking about ships and countries, is it a good practice to use the feminine form?  For example:  "Her economy" - while referring to a country's economy "Her flag (or deck etc)" - while referring to a ship  Is this practice common? Is it used today? How to fill a mesh with solid color How do you distinguish between magic effects that bypass immunity and those that don't? This question was triggered by .  Basically, it goes like this: If you have a weapon with a certain magical effect, let's say the minor property unbreakable (can not be broken, requires special means) or temperate (no harm in temperatures between -20 to 120 F), it can bypass the immunity of something like a Werewolf because it is magical instead of mundane. This goes for any minor property because those properties are under the Magic Items portion of the DMG on pg.143.  How does this differ from a pair of gauntlets with the exact same effect being used as an improvised weapon (emphasis added for clarity) from dealing damage in the same manner to the same Werewolf? Or something that's clearly not a weapon like a Shield of Missile Snaring being used to bash an opponent? error: no such partition. Grub rescue Disable the default app key (super+num) functionality on Ubuntu 17.10 and later I like to switch workspaces via Super+Number. But unfortunately the default behavior in Ubuntu for Super+Number is to select applications from the dock.   Is it possible to disable that functionality? I've tried overriding the keyboard shortcuts but it doesn't appear to be working. How to return to the GUI? I am very new to Linux and Ubuntu. I use Ubuntu 11.10.  I was trying to exit the X server (in order to install some Nvidia drivers) and I typed in the terminal:  sudo service lightdm stop   How can I return to the Ubuntu interface? Hi Blender coders and artists.  I am updating one of my add-ons for Blender 2.8 and have run into some  issues finding the appropriate way to hide a collection that is added to my scene.  I want to deselect the eyeball on the toolbar with a python command in a script but I don't want to disable the collection in the view-port entirely as it is used as a collection in a particle system.   I just want to deselect the eyeball circled in green below.  Anyone know the python API for this? Gravitational waves as dark energy? Where do I get the key for the first house? I've 'completed' the game, but didn't find the key for the first house. (I got the one for the arcade machine house). Where can I find it? Windows 7 Automatically Connecting To Unsecured Wireless Networks On Startup Most of the questions on this topic related to folks connecting to somebody else's wireless network when their own was available and could remedy the situation by going to their connections and unchecking the "connect automatically" box.  See this: "  " as an example.  In my situation, I've noticed that Win 7 will automatically connect to any unsecured wifi network - even if I have never connected to it in the past.  If I am traveling and boot Win 7, it will start and connect to what appears to be the best signaled unsecured network without prompting me for confirmation (note: in the above link, "Naveen" seems to have same problem).  Obviously, that is a security concern to me.  Further, when I open "Network and Sharing" and "Manage wireless networks"  the network is not displayed (probably because I labelled it a public network).  Again, these are new, never connected with before, wireless networks.  I always promptly disconnect from them but don't want to have to be on constant guard for an auto connection to a malicious network.  This began about a month ago, as I recall, Win 7 did not behave like this in the past, I didn't monkey with wifi settings, and don't use a 3rd party connection manager.  I did have to download some internet security certificates for army website access but I don't think that should mess with network settings.  Any ideas how I can tell Win7 cease automatically connecting to networks or, at least, to prompt me for a confirmation before connecting? Help center still has the guidance about old Favorite tags Securing Cable when installing old work box I am installing an electrical outlet above a gas fireplace for TV.  I understand that electrical code requires securing the NM cable within 12" of outlet box.  I am curious how one goes about achieving this in a finished wall being that there is really no room to get hand, hammer, drill or screwdriver inside the wall through that box opening.   I am sure this is straightforward for experienced electricians but I have no idea how to go about this. How to get client's IP address using JavaScript? change citation style ( [1-4] to [1]-[4], and [5, 8-9] to [5], [8]-[9] ) with natbib? I have spent a whole day trying to solve the following problem. Please help if you know how to do this.  I am using natbib (\usepackage[sort&amp;compress,square,comma]{natbib}), and my current citation style looks like [1-4], [5, 8-9].  What is the easiest way to change the citation style to [1]-[4] and [5],[8]-[9]? Concatenate table without header I would like to concatenate several tables and just keeping the header in the first line (and thus not having a new header each time that a new table has been concatenate). Does anyone know how to do that?  I am currently using:  find "/dir/folder" name "*.dat" -exec cat {} + &gt;&gt; "/dir/folder/table.txt"   but I can get rid of the header with that.