Let G1 and G2 be two regions.  Suppose that G1 ∩ G2 ≠ 0.  Show that G1 U G2 is connected.  I know I have to first show that G1 U G2 is open and then show it is connected but I have no idea where to start or how to even prove this.
Let $\mathscr{F}$ be a collection of connected subsets of a metric space $M$ such that $\bigcap\mathscr{F}\ne\emptyset$. Prove that $\bigcup\mathscr{F}$ is connected.   If $\bigcup\mathscr{F}$ is not connected, then it can be partitioned into two disjoint, non-empty subsets $A,B$. Let $x$ be a point in $\bigcap\mathscr{F}$. Then either $x\in A$ or $x\in B$. I don't know where to go from here.
JavaScript get clipboard data on paste event (Cross browser) Is it possible for many domain names to share one IP address? Is it possible for many domain names to share one IP address? For example, www.xjtl.com, www.fmie.com, sdmfl.com, etc.  Hundreds of domain names share just one IP – when you do a DNS query for each of them, the same IP is returned.  Are there any laws or Internet rules that disallow this? Scale settings for exporting to STL (for 3d printing) Cross platform Hard Disk encryption? Do I have to enter the UK on my UK passport? I'm a British National (Overseas), which basically means that while I don't have the right of abode in the UK, I am permitted to visit the UK without a visa. However, my passport has expired, and while it's being renewed, the schedule is a bit tight.  At the same time, I'm an US Citizen, and since I'm only planning to visit the UK, I don't need a visa if I enter on my US passport.  I'm not familiar with UK immigration law, and given that this probably isn't too common a situation, the UK Border Agency website doesn't have an answer that I can can find, nor do they seem to have a contact form that fits my situation, so here's hoping someone knows—given that I'm a British National (Overseas), am I required to enter the UK on my UK passport, or can I just use my (valid) US passport? Electrostatics and tv screens Compute $\int_0^\infty\frac{\cos(xt)}{1+t^2}dt$ Let $x \in \mathbb R$  Find a closed form of $\int_0^\infty\dfrac{\cos(xt)}{1+t^2}dt$ .   Let me give some context: this is an exercise from an improper integrals course for undergraduates. My teacher was not able to give a valid solution without  resorting to Fourier series.   He actually wanted to prove first that the function is a solution of the ODE $$ y-y'' =0$$ without succeeding in proving it.   Anyway the result sould be $\dfrac{\pi}{2}e^{-x}$  Thanks for any help on this subject. I'm trying to factorise $$ x^3z - x^3y - y^3z + yz^3 + xy^3 - xz^3 $$ into four linear factors.  By plugging it into WolframAlpha I've learned that it's $$-(x-y)(x-z)(y-z)(x+y+z)$$  My question is: what are the steps involved in factorising the expression? Is there a method I don't know about that I'd have access to with my limited maths?  Really appreciate any help! I am running Ubuntu 14.04 from Windows, using virtualbox 5.0. I tried several options to make size of Ubuntu in virtual box larger, but none helped so far.   I am using latest virtual box with latest guest add on installed. sudo apt-get install virtualbox-guest-dkms virtualbox-guest-utils virtualbox-guest-x11 I went to Ubuntu 'software and Updates' and made sure that under 'Additional Drivers' tab, 'using x86 visualization solution...' is active. In virtualbox window, under view->virtual screen 1, all 'resize to' options are disabled.   None of these options helped. This is frustrating. Any comments or help? Can the problem be I am using 64 bit system? Does this problem occurs in earlier/later versions? If not, please suggest a 'good' version of Ubuntu for virtualbox. Is there a good way to determine the value of magic items in 5e? In 5e, magic items aren't as common or as varied as in older editions, but that also means it is hard to place prices on them.  As DM, NPC's can't really sell a magic item because we don't know the prices for these items. While it is stated that they are rare and usually unpurchasable and unsellable, but I still have to have some basis on what to price them.  So is there a way to find buying and selling prices for magic items in 5e? What is difference between a ring and a field? Reading the reviews of the  on Google Play,  said that since Chrome supports HTML5, he had managed to persuade the BBC iPlayer website to play him streams meant for iPad/iPhone devices by spoofing an IOS user agent string on his Nexus 7.  How might I go about getting the same thing to work myself?  I have looked at  and  but both seem to concentrate on the Dolphin browser. Also, please mention if a solution will require me to root my Nexus-7.  Incidentally, I also tried using  with Firefox, but although Firefox supports HTML5 and the iPlayer website shows pages without either install flash or use external app messages, it still won't play the streams (it just sits swirling its progress wheel), hence looking for a solution for Chrome. How do I calculate spoke length? I want to use input from a user as a regex pattern for a search over some text.  It works, but how I can handle cases where user puts characters that have meaning in regex? For example, the user wants to search for Word (s): regex engine will take the (s) as a group.  I want it to treat it like a string &quot;(s)&quot; .  I can run replace on user input and replace the ( with \( and the ) with \) but the problem is I will need to do replace for every possible regex symbol. Do you know some better way ? How can I draw outlines around 3D models? How can I draw outlines around 3D models? I'm referring to something like the effects in a recent Pokemon game, which appear to have a single-pixel outline around them: I installed the ubuntu 12.04 server edition for my server pc . i had installed lamp server. i need to change the var/www location to my secondary hard disk location. i was configured so many time to at gedit /etc/apache2/sites-available/default here is my code  &lt;VirtualHost *:80&gt;     ServerAdmin webmaster@localhost      #DocumentRoot /var/www     DocumentRoot /media/myserver/     &lt;Directory /&gt;         Options FollowSymLinks         AllowOverride None     &lt;/Directory&gt;     #&lt;Directory /var/www/&gt;         &lt;Directory /media/myserver/&gt;         Options Indexes FollowSymLinks MultiViews         AllowOverride None         Order allow,deny         allow from all     &lt;/Directory&gt;      ScriptAlias /cgi-bin/ /usr/lib/cgi-bin/     &lt;Directory "/usr/lib/cgi-bin"&gt;         AllowOverride None         Options +ExecCGI -MultiViews +SymLinksIfOwnerMatch         Order allow,deny         Allow from all     &lt;/Directory&gt;      ErrorLog ${APACHE_LOG_DIR}/error.log      # Possible values include: debug, info, notice, warn, error, crit,     # alert, emerg.     LogLevel warn      CustomLog ${APACHE_LOG_DIR}/access.log combined      Alias /doc/ "/usr/share/doc/"     &lt;Directory "/usr/share/doc/"&gt;         Options Indexes MultiViews FollowSymLinks         AllowOverride None         Order deny,allow         Deny from all         Allow from 127.0.0.0/255.0.0.0 ::1/128     &lt;/Directory&gt;  &lt;/VirtualHost&gt;   and also used  sudo chown -R var/www /media/myserver/   and  chmod -R 755 /media/myserver/   still i was not able to connect my /media/myserver and my browser show the following message  Forbidden  You don't have permission to access / on this server.   Please tell anyone how to mount myserver at my var/www, thanks advance I find it intuitive enough that the radical of $\mathfrak{gl}_n\mathbb F$ is the scalar matrices, but I have trouble finding an easy, but complete proof:  Proof. Let $\mathfrak s$ denote the scalar matrices. Clearly $\mathfrak s\subset\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$. Suppose that $\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$ is generated by more than one element, so that $X\in\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$, but $X\notin\mathfrak s$. We can change basis such that $\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$ is upper-triangular. (The change of basis leaves the scalar matrices invariant.) Then $X$ is upper-triangular.  I want to conclude that there exists a $Y\in\mathfrak{gl}_n\mathbb F$ s.t. $[X,Y]$ is not upper-triangular. How can I see that $Y$ always exists, except by waving hands? The only good way of creating circles in ArcMap is using true arc, (compound curves) but I need to convert those to vertex (densifying) because of a compatibility problem. I need info for geodatabase and SDO_geometry (arcsde)  and yes there is a way. I use:  UPDATE layer1 a SET arctype = 'compound' WHERE has_compound_curves(a.shape) &lt;&gt; 0;   and then using ET Geotools to densify  layer by layer.  But there should be easier way of doing this straightforward in Oracle or ArcMap, and I just want to densify the features that have true arc. I don't want any unnecessary vertices on straight lines and so on.  How can I identify true arcs in ArcMap, and how can I densify them as fast and easy as possible?  Same for Oracle SQL. How do I rename all folders and files to lowercase on Linux? I have to rename a complete folder tree recursively so that no uppercase letter appears anywhere (it's C++ source code, but that shouldn't matter).  Bonus points for ignoring CVS and Subversion version control files/folders. The preferred way would be a shell script, since a shell should be available on any Linux box.  There were some valid arguments about details of the file renaming.   I think files with the same lowercase names should be overwritten; it's the user's problem. When checked out on a case-ignoring file system, it would overwrite the first one with the latter, too. I would consider A-Z characters and transform them to a-z, everything else is just calling for problems (at least with source code). The script would be needed to run a build on a Linux system, so I think changes to CVS or Subversion version control files should be omitted. After all, it's just a scratch checkout. Maybe an "export" is more appropriate. Is there a way to have a copy of any message sent from a Gmail account sent to another address?  I've got any messages coming to my address copied to another (see  for one solution, setting up filters is another), but I haven't yet found a way to have outbound messages automatically copied.  Is there a way?