Proof of AM-GM Inequality (setting $a_n$ in the last step)
How to understand Cauchy's proof of AM-GM inequality(the last step)
Why does du and quota results not match? dropping ticklabel on pgfplot axis How to get straight quotation marks? How do I get straight quotation marks, i.e., " and ' in TeX?  (No, \verb changes the font and this is unacceptable.)  [Edit]  I forgot to mention that I’m using XeLaTeX, and fontspec is a necessity.  The first two proposed answers did work, without fontspec. I've got the new Canon Pixma MG 3150 and so far, I can't seem to find a driver anywhere. I do have the CD for both Windows and Mac, but this doesn't help me, as I have Ubuntu 11.10. I'd prefer to have it running on the wifi.  I'm really new to this, so I need for you to explain this to me as basically as possible. Closure of a certain subset in a compact topological group How to make a grid-to-sphere animation like this? Concatenating numbers and adding in leading zeros? I have three text fields populated as follows: 1, 9, 1 that I need to concatenate into a field that looks like 01.09.01.  Getting to 1.9.1 is easy but can anyone help out with how to formulate it so that each value has two places (i.e. add in those zeroes).  I used to do this in Excel but can't figure out how to work in ArcMap 10. Open CSV File with Different Delimiter This is a follow-up to an earlier question regarding .  Having achieved the left-alignment goal, I would now like the caption of the subfigure to be bottom-aligned, while having the subfigure itself be top-aligned. As an example, in the figure below, subfigures (a) and (b) would share an uppermost edge, and their captions would share a lowermost edge, and similarly for subfigures (c) and (d). This is most appropriate for the overall layout style of my document and also the content of the subfigures themselves.    \documentclass{article} \usepackage[font=footnotesize]{subcaption} \usepackage[draft]{graphicx}  \begin{document} \begin{figure}[!ht] \captionsetup[subfigure]{justification=justified,singlelinecheck=false} \centering \begin{subfigure}[b]{0.3\textwidth} \includegraphics[scale=0.8]{fig/workflow-S} \caption{workflow $S$} \end{subfigure} \begin{subfigure}[b]{0.5\textwidth} \includegraphics[scale=0.8]{fig/workflow-S-run} \caption{run of $S$} \end{subfigure} \vskip 8pt \begin{subfigure}[b]{0.3\textwidth} \includegraphics[scale=0.8]{fig/workflow-S'} \caption{workflow $S'$} \end{subfigure} \begin{subfigure}[b]{0.5\textwidth} \includegraphics[scale=0.8]{fig/workflow-S'-run} \caption{run of $S'$} \end{subfigure} \caption{Here's a figure} \end{figure} \end{document}   I've experimented with the minipage settings for subfigure, specifically using [t] instead of [b], but it seems to have no effect. I'm unsure as to whether a position of [t] would produce the right effect anyway, or whether it would shift the captions up as well.  Why is [t] not working here, and what's the right way to produce the layout I need?   The solution should generalise to an arbitrary number of subfigures per row, subfigures which are text or tables rather than images, and should not require any hard-coded sizes. What is the biggest display of power through the Force in Star Wars? What are some alternative launchers that can be used to launch programs in Ubuntu's default Unity desktop? Better yet, are there any Spacewalk-like software packages that work for both RedHat-based and Debian-based systems? Is the $ L^{p}$$[0,1]$ norm continuous in p? I ran into the following problem when I was doing my homework, and I have no thoughts on where I should start with:  (1) If $f\in L^{2}$, show that $\displaystyle \lim_{p \rightarrow 1^{+}}\int_{[0,1]}|f|^{p}=\int_{[0,1]}|f|$  (2) If $0&lt;p$, show that $\displaystyle \lim_{q\rightarrow p^{-}}||f||_{q}=||f||_{p}$  My first thought was Generalized LDCT, but it didn't seem to work. I also made some other attempts but none of them were successful... Can anybody give me some hints on how I should look at this question?  Also, I know if $p\rightarrow\infty$ then $||f||_{p}\rightarrow||f||_{\infty}$ on $[0,1]$, but does similar continuity in p holds for other $L^{p}[0,1]$ norms in general?  Thank you!  Edit:   Sorry if I did not make it clear enough in the question. All $L^{p}$ refers to $L^p[0,1]$.  The first question is found  (thanks to t.b.), but the second question remains, mainly because $f$ is not guaranteed to be in any $L^{p}$. Which OpenType Math fonts are available? What are the integer solutions to $a^{b^2} = b^a$ with $a, b \ge 2$ If I take the $k \in \mathbb{N}$ power of 10 and mod it by a large prime, I notice that the remainders repeat at some point.  For instance $10^k mod~7$ seems to repeat every $8$th value of $k$.  Given $k$ (the power of ten), and a large prime number $p$, how can I compute the size of the cycle?  I tried writing a python script that computes the first instance of a repeat. However, I don't know if this works since a single instance of a repeat may not necessarily mean a full cycle. Let $f(x)$ and $g(x)$ be two continuous functions on $[0,1]$ and  $$\int_{0}^{1}f(x) dx= \int_{0}^{1}g(x)dx = 1$$   Show that there exist  $[a,b]\subset [0,1]$, such that  $$\int_{a}^{b}f(x) dx= \int_{a}^{b}g(x)dx = \frac{1}{2} $$  The question can be solved by considering the fundamental group of $S^1$, now I am wondering if we can solve it by real-analysis.  Here is the topological solution:  Assume that for any $[a,b]\subset[0,1]$, we always have  $$(\int_{a}^{b}f(x) dx\neq \frac{1}{2}) \quad \vee \quad(\int_{a}^{b}g(x)dx \neq \frac{1}{2}) $$ Consider mapping $$\phi:D\rightarrow\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\},\,\,(x,y)\mapsto(\int_{y}^{x}f(t) dt,\int_{y}^{x}g(t) dt)$$ where $D=\{(x,y)|0\leq x\leq y\leq 1\}$.  Let $a$ be path from $(0,0)$ to $(0,1) $in $D$ and $b$ be path from $(0,1)$ to $(1,1) $in $D$, then $ab$ is a path from $(0,0)$ to $(1,1) $in $D$ and $$\phi\circ (ab):[0,1]\rightarrow\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\} $$ Notice that $$(\phi\circ (ab))(0)=(\phi\circ (ab))(1)=(0,0)$$  so $\phi\circ (ab)$ is a loop based on $(0,0)$ in $\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\}$.  For any $t\in [0,\frac{1}{2}]$, it's not hard to get that $$(\phi\circ (ab))(t+\frac{1}{2})=(1,1)-(\phi\circ (ab))(t)$$ is equivalent to $$(\phi\circ (ab))(t+\frac{1}{2})-(\frac{1}{2},\frac{1}{2})=-((\phi\circ (ab))(t)-(\frac{1}{2},\frac{1}{2}))$$  Define retraction  $$r:\mathbb{E}^2\backslash\{(\frac{1}{2},\frac{1}{2})\}\rightarrow S^1,\quad (x,y)\rightarrow\ \frac{(x,y)-(\frac{1}{2},\frac{1}{2})}{||(x,y)-(\frac{1}{2},\frac{1}{2})||}$$  Then $r\circ \phi\circ (ab)$ is a loop on $S^1$, such that $$(r\circ \phi\circ (ab))(t+\frac{1}{2})=-(r\circ \phi\circ (ab))(t),\quad\forall t\in [0,\frac{1}{2}]$$  So $&lt;r\circ \phi\circ (ab)&gt;$ is not trivial in $\pi_1(S^1)$.   However, $ab$ is path homotopic to $c$ in $D$, where $c$ is the path between $(0,0)$ and $(1,1)$ in $D$. In this case, $r\circ \phi\circ (ab)$ is a point-path in $S^1$ by $$(\phi\circ c)(t)=\phi(t,t)\equiv (0,0)$$  which leads to contradiction. Why is "high" pronounced "hiy" but is not spelled that way? I have two partitions, one of them has a file system and the other is unallocated.  How to merge two partitions? What is an uninterruptible process? Sometimes whenever I write a program in Linux and it crashes due to a bug of some sort, it will become an uninterruptible process and continue running forever until I restart my computer (even if I log out).  My questions are:   What causes a process to become uninterruptible? How do I stop that from happening? This is probably a dumb question, but is there any way to interrupt it without restarting my computer?