What does matrix transposition represent?
What's the meaning of the transpose?
Underfull \hbox in bibliography I am getting lots of "Underfull \hbox" warnings that point to the *.bbl file. It looks like these messages appear only when URLs are very long, like in the below example:  @techreport{nistguidesec,   author    = {Wayne Jansen and Timothy Grance},   title     = {Guidelines on Security and Privacy in Public Cloud Computing},   month    = {January},   note     = {Draft Special Publication 800-144. Available at \url{http://csrc.nist.gov/publications/drafts/800-144/Draft-SP-800-144_cloud-computing.pdf}}, }   In the produced PDF text is correctly hyphened and links are correctly split over several lines (I am using hyperref package)  How can I avoid annoying warning messages about "underfull \hbox" pointing to bibliography? Choice of $q$ in Baby Rudin's Example 1.1 First, my apologies if this has already been asked/answered.  I wasn't able to find this question via search.  My question comes from Rudin's "Principles of Mathematical Analysis," or "Baby Rudin," Ch 1, Example 1.1 on p. 2.  In the second version of the proof, showing that sets A and B do not have greatest or lowest elements respectively, he presents a seemingly arbitrary assignment of a number $q$ that satisfies equations (3) and (4), plus other conditions needed to show that $q$ is the right number for the proof.  As an exercise, I tried to derive his choice of $q$ so that I may learn more about the problem.  If we write equations (3) as $q = p - (p^2 - 2)x$, we can write (4) as   $$ q^2 - 2 = (p^2 - 2)[1 - 2px + (p^2 - 2)x^2]. $$  Here, we need a rational $x &gt; 0$, chosen such that the expression in $[...]$ is positive.  Using the quadratic formula and the sign of $(p^2 - 2)$, it can be shown that we need   $$ x \in \left(0, \frac{1}{p + \sqrt{2}}\right) \mbox{ for } p \in A, $$  or, for $p \in B$, $x &lt; 1/\left(p + \sqrt{2}\right)$ or $x &gt; 1/\left(p - \sqrt{2}\right)$.  Notice that there are MANY solutions to these equations! The easiest to see, perhaps, is letting $x = 1/(p + n)$ for $n \geq 2$.  Notice that Rudin chooses $n = 2$ for his answer, but it checks out easily for other $n$.  The Question:  Why does Rudin choose $x = 1/(p + 2)$ specifically?  Is it just to make the expressions work out clearly algebraically?  Why doesn't he comment on his particular choice or the nature of the set of solutions that will work for the proof?  Is there a simpler derivation for the number $q$ that I am missing? Why there is much interest in the study of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$? What is a good strategy to deal with lots of engineers turtling on the other team? Now that the Engineer update has come, there will be lots of Engineers building up everywhere.  How should this best be handled? I have been using LaTeX for a long time now, and I find myself often going to this website to solve my problems. This often includes copy pasting codes which include \newcommand, \newenvironment and \def. I think it is time I start to learn how to use them so I can solve my problems myself! It is always better if you actually know what is going on inside your code.  I am wondering if there is any turtorial or book out there that handles how to use those commands. Or is the best way to learn to just mess with a random file?   Thanks in advance!  (Also this is my first time I post here after using this website for so long, I really appreciate everyone who posts and tries to help other people here!) Error Code: 1215. Cannot add foreign key constraint (foreign keys) What is the logic behind the timer of the Sojourner Badge? How to convert an "object" into a function in JavaScript? JavaScript allows functions to be treated as objects--if you first define a variable as a function, you can subsequently add properties to that function.  How do you do the reverse, and add a function to an "object"?  This works:  var foo = function() { return 1; }; foo.baz = "qqqq";   At this point, foo() calls the function, and foo.baz has the value "qqqq".  However, if you do the property assignment part first, how do you subsequently assign a function to the variable?  var bar = { baz: "qqqq" };   What can I do now to arrange for bar.baz to have the value "qqqq" and bar() to call the function? Is it possible to show our custom message in QGIS Status bar using python? Can matter be described as the result of the curvature of space, instead of vice versa? Can matter be described as the result of the curvature of space, rather than the curvature of space being the result of matter, and energy being the cause of the curvature of space? Why are potential differences equal across two capacitors in parallel, but charge on each capacitor is not? Two or more capacitors are said to be connected in parallel if each one of them is connected across the same two points. In a parallel combination of capacitors potential difference across each capacitor is same but each capacitor will store different charge. Why is this true? Why is the potential difference across each capacitor in parallel the same and why will each capacitor store different charge?  Also, in series, why are Potential difference across each capacitor different, while charge is the same? If the universe is expanding, what is it expanding into? If the universe is expanding, what is it expanding into? When the big bang happened where did it occur?  When the big bang happened how did it occur?  Where did the energy come from? Energy can not be created or destroyed does that mean, energy has existed before the universe was here? Give an example of a sequence of continuous functions $f_n:\mathbb{R} \to \mathbb{R}$ such that the pointwise limit $f:\mathbb{R} \to \mathbb{R}$ exists but is discontinuous. My idea is $f_n(x)=\displaystyle\frac{x}{n}.$ Then the pointwise limit is $0$ but $f$ is discontinuous on $\mathbb{R}.$ Is that correct? I'm having a hard time understanding the definition of pointwise convergence. I have this definition: $\displaystyle\lim_{n\to \infty} f_n=f$ pointwise iff $\displaystyle\lim_{n\to \infty} f_n(x)=f(x)$ But then what is $f(x)$? Where does the $n$ go? In the example I gave, for instance, $\displaystyle\lim_{n\to \infty} f_n(x)=\displaystyle\lim_{n\to \infty} \frac{x}{n}=0.$ But does this equal $f(x)$? Is $f(x)$=0 a discontinuous limit? How to reduce the number of hyphenation? Sometimes, assume our document contain many long words, TeX may generate some hyphenation at the end of line. But it really hinder our smooth reading. Although I know we can add the hyphenation rule at the preamble. It can't reduce hyphenation number but effect the hyphenation position. It is said that microtype package can give us more compact output. But it require expert level to tuning the parameter. I'm currently working my way through Harvard's online abstract algebra lectures (if you're interested, you can find them ).  The lectures come complete with notes and homework problems.  Of course, since I don't actually go to Harvard, I can't hand in the homework assignments to find out if I'm doing them correctly.  So I've decided to try to crowd source the grading of my solutions.  I'll post individual questions as I finish them and wait for comments.  I would like to get critiques of not just my reasoning but the style of my write ups as well.  Also, any alternative approaches to the problem would be welcome.  I've looked for forums dedicated to these kinds of OCW courses but have not been able to find any.  This surprises me.  It seems like, with the advent of these free online educational resources, an online meeting place for those who take advantage of them would be a natural offshoot.  So this might be a bit of an experiment.  Unless I'm the 873rd person to post something like this here.  If that's the case, sorry for being so long winded.  Anyway, on with the question.  This one is assigned in the 4th lecture.     Let $V$ denote the Klein 4-group.  Show that $\operatorname{Aut}(V)$ is isomorphic to $S_3$.   (I've moved my solution below to keep this question from showing up in the unanswered list.)  Thanks... Effects of continuing after a total party wipe? I've had a couple of encounters so far where I stumbled into enemies that were higher level than expected and gotten wiped out. At the game over screen, I chose the option "Continue". After doing so, I didn't notice any negative consequences. I was plopped back just a few metres from the area I died, and was free to continue on as if nothing had happened. Am I missing something? Is there some negative effect from a party wipe? Bevel is not symmetrical Change folder permissions and ownership I would like the user to have full rights on this folder (as well as all sub-directories and files in it):  ~/.blabla   currently owned by root.  I have found numerous posts (in this forum and elsewhere) on how to do this for files but I can't find a way to do it for whole folders. Switching between windows with scroll wheel on Ubuntu Dock Sorry for the vague title--I don't want the title to be a spoiler.  I've seen the movie three times, and the last two times I looked specifically for clues to this question, but found none.  The final scene in  shows       Cobb's top spinning, as he goes off to be reunited with his children.  The top continues to spin, then the movie ends, without telling us if the top topples--and therefore revealing whether he's in the "real world" or still in a dream. Do we know if he was in a dream at the end of the movie?   I'm assuming the answer is that we simply cannot know.  But did I miss anything?  Were there clues that I missed?