I have a question:  Let $F$ and $K$ fields such that $F\subset K$. Can we conclude that $1_K=1_F$?  I found any counterexaples when $F$ and $k$ are rings, but not necessarily fields. However when these are fields I don't get conclude nothing...
If $R$ is an integral domain and a subring $S$ has identity $1_S$, how would you show that $1_S=1_R$ (here $1_R$ is the identity of the ring $R$)? I am unsure about what an integral domain really is and how the subring comes into play here.
Can you change your email for Pokemon Go? Where can the viewport clipping parameters be changed? Zooming in to inspect texture details in 3D view clips some closer faces like shown in the image. It's not the camera view so changing camera settings doesn't help.   Where can the distance to the clipping plane be adjusted? There has been some recent behavior change on SE (around July 2015).  I have two accounts I use, one Google openId based for one set of sites (call it Account-A) and another SE login based for a different set of sites (Account-B).  Used to be that I could be logged into one set of sites with the first user and a different set with the other users.    But now I am logged in with the Account-A on Site-1.  When I go to one of the sites I use the other account for and click "log in" under the SE menu, I go to a page where I only have an option to confirm creation of an account based on my Google login.  It does not present a way to log in with an SE account.    How can I log in with my other account here?  (and before you flag this as a duplicate, please ensure the other question is recent enough and that the answer is relevant.  This behavior has definitely morphed over the years... and I am having trouble finding something that is relevant.  For instance this question  simply points out how to merge the accounts which is NOT what I want  .  This question  talks about presenting different user info on each site but that's not what I want -- I want totally separate accounts). I have been developing in the windows space with Visual Studio for a while now with work, but I have also been using Ubuntu for a while and am keen to get into some software development for linux.  I should also note. I am not looking for .NET and I am aware of mono. I am also familiar with c++ development and some python, so the language isn't so much relevant as the "all in one" aspect.  I was interested to know if there is a useful all in one code/debug/design(gui) IDE similar to something like Visual Studio but for linux? How many numbers of 6 digits, that can be formed with digits 2,3,9. And also divided by 3? How many numbers of 6 digits, that can be formed with digits 2,3,9. And also divided by 3?  $$$$ I was trying to to add 2^6 (when there is no 2)+ C(6,2)2^2 (when 2 can be in two places)+C(6,3)2^3 (when 2 can be in three places)... Ubuntu 12.04 comes with Gimp 2.6, so I installed Gimp 2.8 using the instructions below:   sudo apt-get purge gimp*   sudo add-apt-repository ppa:otto-kesselgulasch/gimp   sudo apt-get update   sudo apt-get install gimp    When try to open help, I get the message below. How do I install and configure the help for 2.8 on Ubuntu? How to locate a font in Ubuntu? Ubuntu desktop with 12.04 is about to crash.  Installed 13.04 on a new machine, want to transfer Book Antiqua font from old machine to the new one.  How do I find it on the old machine?  My /usr directory is empty. My fonts.conf file has these lines: &lt;!-- Font directory list --&gt; &lt;dir&gt;/usr/share/fonts&lt;/dir&gt; &lt;dir&gt;/usr/x11R6/lib/X11/fonts&lt;/dir&gt;  &lt;dir&gt;/usr/local/share/fonts&lt;/dir&gt; &lt;dir&gt;~/.fonts&lt;/dir&gt;  Don't know what this means, but my /usr directory has nothing in it. optimality of the Carnot cycle Check if table exists in SQL Server I would like this to be the ultimate discussion on how to check if a table exists in SQL Server 2000/2005 using SQL Statements.  When you Google for the answer, you get so many different answers. Is there an official/backward and forward compatible way of doing it?  Here are two possible ways of doing it. Which one among the two is the standard/best way of doing it?  First way:  IF EXISTS (SELECT 1             FROM INFORMATION_SCHEMA.TABLES             WHERE TABLE_TYPE='BASE TABLE'             AND TABLE_NAME='mytablename')     SELECT 1 AS res ELSE SELECT 0 AS res;   Second way:  IF OBJECT_ID (N'mytablename', N'U') IS NOT NULL     SELECT 1 AS res ELSE SELECT 0 AS res;   MySQL provides the simple   SHOW TABLES LIKE '%tablename%';    statement. I am looking for something similar. This is relating to the movie. Frodo finally relinquishes his desire for the ring, and it melts in the fires of Mount Doom. The next scene shows the eruption of Mount Doom and the destruction of Barad Dur. Gandalf looks up with tears in his eyes.  I have read the books, and can't seem to remember this being explained or described. I'm just wondering why they depicted Gandalf as being so emotional about the event, when throughout the story it depicts Gandalf as having unparalleled wisdom, faith and knowledge. Out of all the Ainur he listened to Eru's song the longest and contemplated it's meaning.   It strikes me that Tolkien portrayed Gandalf as all knowing, and has ultimate faith that Frodo would carry out the task at hand. So why was he so emotionally surprised that it happened? Especially when he had seen far more surprising things happening throughout the War of the Ring and in the movies he was depicted as emotionally immovable, an almost cold hearted awareness of the evil that lurks within the world.   I understand this is probably going to be down-voted as it has alot of opinion based statements. The TL:DR is: Was Gandalf described in the book as being so emotional at Sauron's fall? Or was this simply for effect in the screenplay? Do there Exist Proper Classes that aren't "Too Big" Some proper classes are "too big" to be a set in the sense that they have a subclass that can be put in bijection with $\alpha$ for every cardinal $\alpha$. It is implied in  post that every proper class is "too big" to be a set in this sense, however I have been unable to prove it. It's true if every two proper classes are in bijection, but it's consistent with ZFC for there to be a pair of non-bijective classes.  So, is the following true in ZFC:     For all proper classes, $C$, and $\alpha\in\mathbf{Card}$, $\exists S\subset C$ such that $|S|=\alpha$?   If not, is there something reasonable similar that preserves the intuition about classes that are "too big to be sets"? Show that $11^{10} \equiv 1 (\mod 100)$ $11^{10} \equiv 1 \pmod{100}$  I tried to solve by using euler's theorem, But I got stuck.  $\gcd(11, 100) = 1$  $11^{φ(100)} \equiv 1 \pmod{100}$  $11^{40} \equiv 1 \pmod{100}$  I don't know how to go on as $11^{40}$ is bigger than $11^{10}$ GroupBy a list of associations with their keys Finding where source has branched from git Given that $W$ is a subspace of $V$, prove: $\dim W=\dim V \implies W=V$ Suppose $W$ is a subspace of the finite-dimensional vector space $V$ and $\dim W = \dim V=n$  Let $\{\overrightarrow w_1,\ldots,\overrightarrow w_n\}$ be a basis for $W$ and $\{\overrightarrow v_1,\ldots,\overrightarrow v_n\}$ be a basis for $V$.  The set $\{\overrightarrow w_1,\ldots,\overrightarrow w_n,\overrightarrow v_1,\ldots,\overrightarrow v_n\}$ still spans $W$. Since $W$ is a subspace of $V$, $W\subset V$ and all the $w_i$'s can be written in terms of $\{ \overrightarrow v_{1},\ldots,\overrightarrow v_n\}$.  $\therefore$ We can remove one by one all the $w_i$'s from the set $\{\overrightarrow w_{1},\ldots,\overrightarrow w_n,\overrightarrow v_1,\ldots,\overrightarrow v_n\}$ while still having it span $W$, which leaves us with the set $\{\overrightarrow v_1,\ldots,\overrightarrow v_n\}$. We know it to be linearly independent since it is a basis for $V$. Therefore the basis of $V$ is also a basis of $W$. From there on, it is trivial to prove double inclusion $W\subset V$ and $V\subset W$ which gives us $W=V$.  Is my approach correct? Travelling from India to canada via Dubai and London. Do I need a transit visa Video editing software options? What video editing software would you recommend on Ubuntu (or Linux in general) and why?  It can be for either beginners or experienced users.  Please only place one piece of software per response and include details to what makes it great! Launch shell scripts from anywhere Is it possible to run my scripts located in ~/scripts from anywhere?  Say i want to run ./scriptname regardless of my current working directory. How can I escape special HTML characters in JSP? An .mov video that is upside down in Windows appears right side up in OSX