I'm writing a book, and I'm trying to describe that the girl is making faces as she's trying to remember something. I'm not quite sure how to describe that. (Seen from the point of view of another character in the book.) So far, I've got this, but I think my sentence doesn't make sense.  She was trying to remember where she’d seen my aunt. She was making faces while thinking, which made her look so cute.  I thought about &quot;grimaces&quot; but that's not it, since grimace is usually in disgust or disapproval.
If you look at this picture, how would you describe the mood and the eyes of the character?  I would not say she's angry. I would say she is concentrating to cut the object. However there are many kinds of concentration, and thus the eye expression can differ. This one gives me the feeling that she doesn't accept the existence of the object, and thus must kill it.
Let A and B be sets. Show that A is a subset of B if and only if for any set C, one has A union C is a subset of B union C. Can you verify my proof if it is right?  Let $A$ and $B$ be sets.  (a) Show that $A$ is a subset of $B$ if and only if for any set $C$, one has $A$ union $C$ is a subset of $B$ union $C$. (b) Show that $A$ is a subset of $B$ if and only if for any set $C$, one has $A$ intersect $C$ is a subset of $B$ intersect $C$.  Edited Solution :  (a)  (=>) Given A is a subset of B, this implies A U B = B. Given a set C, A U B U C = B U C. Since A is a subset of B, this implies A U C is a subset of B U C.      Solution 1 :(&lt;=) Given A U C is a subset of B U C , For A U C , x is a member of A U C . For B U C, x is a member of B U C. Suppose x is a member of A but not a member of B, meaning A is not a subset of B. Therefore , A U C is not a subset of B U C. Hence, A must be a subset of B.      Solution 2 :(&lt;=) Given A U C is a subset of B U C, For A U C, x is a member of A U C and for B U C, x is a member of B U C. Suppose C is a disjoint set. Since A U C is a subset of B U C, it follows that A is a subset of B.  (b) (=>) Since A is a subset of B , it follows that A U B = B. [A intersect C] is a subset of A and  [B intersect C ] is a subset of B by definition of intersection. Let x be member of A and member of C. Since [ A intersect C ] is a subset of A, x must be in A. Let x be a member of B and member of C. Since [ B intersect C ] is a subset of B, x must be in B. Therefore, A intersect C is a subset of B intersect C.  (&lt;=) Since A intersect C is a subset of B intersect C, suppose we let Set C be a subset of B and a subset of A as well. C is a subset of A implies A intersect C is equals to C. C is a subset of B implies B intersect C is equals to C as well. Since A intersect C is equals to B intersect C, A is a subset of B. Why is df missing 500MB of available space? I've just had a message today from Ubuntu 11.04 that I have only 100 MB left, so I cleaned up some files, and I got 200 MB. Then, after a couple of hours, suddenly I have only 26 MB?! I tried df, du via mount --bind, /forcefsck with reboot - nothing could should what the culprit is - finally I searched big files, realized /var/log/syslog is 100MB+ and /var/log/kern.log is 200MB+ - blanked them with sudo bash -c 'echo &gt; ...' and rebooted, and now I have some spare MB.  But now, I realize I have another problem with df:  $ df -h Filesystem            Size  Used Avail Use% Mounted on /dev/sda5             9,7G  8,8G  385M  96% / none                  963M  696K  962M   1% /dev none                  969M   12K  969M   1% /dev/shm none                  969M  252K  969M   1% /var/run none                  969M     0  969M   0% /var/lock /dev/sda6             9,7G  8,1G  1,1G  89% /media/disk  $ df -h --block-size M Filesystem           1M-blocks      Used Available Use% Mounted on /dev/sda5                9845M     8960M      385M  96% / none                      963M        1M      962M   1% /dev none                      969M        1M      969M   1% /dev/shm none                      969M        1M      969M   1% /var/run none                      969M        0M      969M   0% /var/lock /dev/sda6                9845M     8235M     1110M  89% /media/disk   Note so for /, it says there are 9845M total; and Used 8960M - then remaining would be 9845-8960 = 885 M, however, here I have only 385M Available.  Also, for /media/disk, it says total 9845M, Used 8235M - then remaining would be 9845-8235 = 1610, however, here I have only 1110M Available.  In both cases, there is exactly 500 MB difference.  Where did this difference come from - and can I reclaim it?  Here is also lsof | grep 'deleted' - can't see nothing suspicious here:  $ lsof | grep 'deleted' nautilus  1911 user   21u      REG        8,5   142760     260562 /home/user/.local/share/gvfs-metadata/home (deleted) nautilus  1911 user   22w      REG        8,5    32768     268807 /home/user/.local/share/gvfs-metadata/home-fe882154.log (deleted) python    1919 user    8u      REG        8,5     4096     392258 /tmp/ffiqRK968 (deleted) python    2165 user    5w      REG        8,5        0     132261 /home/user/.[SNIP].lock (deleted) python    2166 user    5w      REG        8,5        0     132261 /home/user/.[SNIP].lock (deleted) python    2185 user   21r      REG        8,5   142760     260562 /home/user/.local/share/gvfs-metadata/home (deleted) python    2185 user   22r      REG        8,5    32768     268807 /home/user/.local/share/gvfs-metadata/home-fe882154.log (deleted) gnome-ter 2279 user   27u      REG        8,5      640     392575 /tmp/vte5KDX2X (deleted) gnome-ter 2279 user   28u      REG        8,5     4936     392605 /tmp/vteKRDX2X (deleted) gnome-ter 2279 user   29u      REG        8,5      648     392947 /tmp/vteMZDX2X (deleted) ubuntuone 2544 user   17u      REG        8,5     4096     392335 /tmp/ffiMErq0V (deleted) bamfdaemo 3235 user   12r      REG        8,5   143868     269077 /home/user/.local/share/gvfs-metadata/root (deleted) bamfdaemo 3235 user   13r      REG        8,5    32768     272310 /home/user/.local/share/gvfs-metadata/root-18092a02.log (deleted) firefox   5291 user   59u      REG        8,5    33288     132262 /var/tmp/etilqs_YdeZiWSd5iQwJ4U (deleted) firefox   5291 user   60w      REG        8,5    32768     132271 /var/tmp/etilqs_MNLXhEaEqoXMm9b (deleted) firefox   5291 user   70u      REG        8,5    34952     132297 /var/tmp/etilqs_yXDdwVeMxmmdpNz (deleted) Why do Star Trek vessels always approach each other on parallel planes? Space is just open 3D space, right? In Star Trek, ships are always coming from somewhere else: different solar systems, galaxies, etc. However, when they meet, they always seem to be on parallel planes (facing each other).  In some episodes I recall seeing ships that were adrift (and they appeared facing a different trajectory). How can there possibly be some standardized reference plane in space? Do ships just choose to align with each other to be polite? Or perhaps when entering a solar system, the pilot, or navigation system aligns orthogonally with the local orbital axis? Creating a textarea with auto-resize There was , which I've tried. But there is one problem: the textarea doesn't shrink if you delete the content. I can't find any way to shrink it to the correct size - the clientHeight value comes back as the full size of the textarea, not its contents.  The code from that page is below:  function FitToContent(id, maxHeight) {    var text = id &amp;&amp; id.style ? id : document.getElementById(id);    if ( !text )       return;     var adjustedHeight = text.clientHeight;    if ( !maxHeight || maxHeight &gt; adjustedHeight )    {       adjustedHeight = Math.max(text.scrollHeight, adjustedHeight);       if ( maxHeight )          adjustedHeight = Math.min(maxHeight, adjustedHeight);       if ( adjustedHeight &gt; text.clientHeight )          text.style.height = adjustedHeight + "px";    } }  window.onload = function() {     document.getElementById("ta").onkeyup = function() {       FitToContent( this, 500 )     }; } Drawing a triangle from medians What fallacy dismisses a conclusion because supporters give invalid arguments for it? How can I keep Ubuntu updated automatically? I want help to  make Ubuntu 'update &amp;&amp; upgrade' itself automatically, select the settings for this automatic system.  How can I do it? Please advice about  where to find the menu (GUI) or file (command line) to do it, what settings that can be recommended (if different from the default).  As a bonus, please describe how different types of program packages are treated,  program packages from Ubuntu repositories and PPAs snap packages flatpak packages other kinds of programs/packages. Recently I noticed that many people post .htaccess files here with:  &lt;IfModule mod_rewrite.c&gt;   Sometimes this even appears several times in the file! It obviously checks to see if mod_write is in fact enabled but the statements it protects are necesary to those sites anyway. So if the IfModule fails, the website is broken anyway.  Is there an advantage in having the &lt;IfModule mod_rewrite.c&gt; fail rather than the RewriteRule statements below it?  mod_write now is so essential and ubiquitous that I do not check for it anymore. Might this cause a vulnerability of some kind? I tried to write , but I couldn't find any simple explanation page to link to. I ended up garbling a home-spun description of my own with a not-terribly-helpful image from 2012.  This  puts it better than I can:     As a new user in 2015 can there please be more information about what is WinterBash. The link took me to a bunch of stick figures running around making (I think) numbers (I think its a countdown clock). It doesn't explain the event or even what a hat is. – Ian Miller Dec 4 at 2:58   All I found on Meta was dead links. For example,  was downvoted to -2 and answered with a link to an FAQ page that no longer exists. Accidentally turned on talkback I accidentally turned on talkback in the Accessibility menu. When I turned it on I got some kind of walk through on how to use it. I couldn't cancel it. So I turned off my phone.   When I turned it back on again it took me 15 minutes to unlock my phone. Apparently I had to touch a button to give it focus and then double tap it to click it.   Now my question is: how do I turn this off? I can't reach the settings because I have don't know how to access a drop down and I can't scroll through all my apps. I have downloaded an ISO file recently.  How do I burn it to a CD or DVD or mount it? about how to open a URL in your default browser using the Terminal. Basically, you use the command   xdg-open http://google.com   My default browser is Firefox. But lets say that I wanted to open a specific website in Chrome, instead of Firefox. Is there a command to open it in Chrome instead of Firefox? When using sudo to allow edits to files, I regularly get 'permission denied'.  For example, my mouse is jittery and sluggish, so I want to disable polling:  sudo echo "options drm_kms_helper poll=N"&gt;/etc/modprobe.d/local.conf   I'm prompted for a password, and then get:  bash: /etc/modprobe.d/local.conf: Permission denied   So I tried to do a temporary change to disable polling by using:  sudo echo N&gt; /sys/module/drm_kms_helper/parameters/poll   Yet again the system responded with:  bash: /sys/module/drm_kms_helper/parameters/poll: Permission denied   Any ideas? In Ubuntu, I want to scroll up while a command is running in my terminal window and generates new output (the command that is running is rspec). I can scroll up, but each time rspec generates a new line of console output, it throws me back down to the bottom. Is there a way to scroll all the way up? I'm using terminator on Linux Mint/Ubuntu 12.04 LTS. I found a possible workaround.  says that with stty ixon  you can turn on &quot;XON/XOFF flow control for terminal output. Being able to pause output by typing ^S and resume it by typing ^Q [...]&quot; This it not totally what I wanted though, without having to pause the output (and maybe wondering when forgetting to turn it on again), I thought there might be a simple modifier key like shft-alt with mouse wheel scroll-up. Are there better solutions? As you can see from the image below, I have an issue with the text alignment of extra long chapter names. How can I fix it so that the text will not go further left than the start of the chapter name? That is, I don't want any text to be underneath the "Appendix A" part.  I added that red line just to show how I'd like it aligned, if possible.    Here's my MWE:  \documentclass{book} \let\cleardoublepage\clearpage  \RequirePackage{titletoc} \RequirePackage{tikz} \RequirePackage[explicit]{titlesec} \RequirePackage{DejaVuSansCondensed} \RequirePackage[normalem]{ulem} \RequirePackage[   left=2.25in,   right=0.75in,   top=1.25in,   bottom=1.25in,   marginparwidth=1.75in,   marginparsep=.25in,   asymmetric]{geometry}  \renewcommand*\familydefault{\sfdefault} % Set default font  %%% Table of Contents ---------------------------------------------------------- \setcounter{tocdepth}{1} % Show only Chapters and Sections  %%% Change font/color/layout of TOC \makeatletter \titlecontents{chapter}[-2em] {\color{cyan}\bfseries\LARGE\addvspace{3mm}} {\makebox[4.4em][r]{\@chapapp}\contentslabel{-0.25em}\hspace{1.9em}} {} {\hspace{3em}\contentspage} \g@addto@macro{\appendix}{%   \addtocontents{toc}{\protect\renewcommand*{\protect\@chapapp}{\protect\appendixname}}% } \makeatother  \titlecontents{section}[5.3em] {\color{cyan}\large\itshape\addvspace{3mm}} {\contentslabel{0em}\hspace{3.2em}} {} {\hspace{3em}\contentspage} [\addvspace{0mm}]  %%% Chapter Header ---------------------------------------------------------- \makeatletter \titleformat{\chapter}   {\normalfont\sffamily\Huge\scshape}   {}{0pt}   {\thispagestyle{empty}    % Remove page number on new chapters   %\setcounter{definition}{0}   \begin{tikzpicture}[remember picture,overlay]     \node[yshift=-3cm] at (current page.north west)       {\begin{tikzpicture}[remember picture, overlay]         \draw[fill=cyan] (0,-1) rectangle           (25cm,3cm);         \draw[fill=cyan] (0,-24) rectangle           (25cm,-25cm);         \ifttl@label% &lt;---------------------- Added condition on \ifttl@label           \node[anchor=west,xshift=.21\paperwidth,yshift=-.01\paperheight,rectangle]               {\color{white}\LARGE \MakeUppercase{\@chapapp} \Huge\thechapter};         \fi% &lt;------------------------------- end condition on \ifttl@label        \end{tikzpicture}       };    \end{tikzpicture}\endgraf   \vskip-.7cm   \color{cyan}\Huge\raggedright\leftskip-1cm    \noindent\MakeUppercase{#1}\endgraf   } \makeatother  \titlespacing*{\chapter}{0pt}{10pt}{0pt}{}  \titleformat{\section}{\color{cyan}\itshape\LARGE}{\llap{\thesection} #1}{1em}{}[\titleline{\color{cyan}\titlerule[1pt]}] \titleformat{\subsection}{\color{cyan}\normalfont\large}{\; #1}{1em}{}  \begin{document}   \tableofcontents   \chapter{Turing Machines}     \section{The Turing Machine}     \section{Variations of the Turing Machine}       %\subsection{Non-deterministic Turing Machines}     \section{The Church-Turing Thesis}   \chapter{Decidability}     \section{The Universal Turing Machine}    \appendix   \chapter{ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM} \end{document} I have to determine how many six-integer subsets of $\{1, 2, 3, \ldots, 20\}$ are possible if no two consecutive integers are in a set. Routing in home LAN I'm trying to understand some basic concepts of networking. I have following home LAN:     Is it right (like above) to have 2 NIC's each one having 2 different gateways on one station? There PC2 have 2 NICs - the first's one gateway is 192.168.0.1 the second's 192.168.1.1? Initially (just after system startup) the ip table is like that:  192.168.0.0    255.255.255.0         On-link      192.168.0.11    276 192.168.0.11   255.255.255.255       On-link      192.168.0.11    276 192.168.0.255  255.255.255.255       On-link      192.168.0.11    276    and when i run netcat:  nc -zv -s 192.168.0.11 192.168.0.11 80    to test if port 80 is open it works - the port 80 is indeed open. But when i remove the 192.168.0.11 route i got "TIMEOUT" on nc test. Shouldn't it pick 192.168.0.0 route and work? Moreover when i add the route manually:   route add 192.168.0.11 mask 255.255.255.255 0.0.0.0   it still doesn't work! I can't change the metric to desired value. When i run for eg. route change 192.168.0.11 mask 255.255.255.255 192.168.0.11 metric 500 the result metric is't 500 but 520! (no matter if auto-metric is enabled or not)? Is there any connection between -s flag (local source address) in netcat and routing table's Interface column? Consider $X = (a+b)^n-a^n-b^n$  where  $a$ and $b$ and $a+b$ are not divisible by $n$, $n&gt;=3$ is positive and prime, $a \ne b$ and both are greater than 1.  By the binomial expansion we know $X$ is divisible by $n$, but how can we study whether $X$ is divisible by $n^2$ or other powers of $n$? Specifically I am interested to prove $X$ is NOT divisible by $n^2$, but I am not sure how to prove or disprove it. Why does PHP consider 0 to be equal to a string? How is Anti Aliasing Implemented in Ray Tracing? After reading a few articles online I can confidently say I am clueless on how Anti-Aliasing works when using Ray Tracing.  All I understand is that A Single Pixel/Ray is split into 4 sub-pixels and 4 rays rather than 1.  Could somebody please explain how this is done (preferably with code)?