If $n$ is an odd integer, then $4\mid n^2 − 1$.
Proof problem: Show that $n^2-1$ is divisible by $8$, if $n$ is an odd positive integer.
Book with a girl whose grandma is a phoenix, cover depicts the emerald/green-eyed blonde girl I read this book in 2016, but I remember the cover being torn up, so it couldn't have been a new book. There was a girl's face on the cover. She had lightly colored skin, and beautiful emerald green eyes. Her hair was a dirty blond. This book was first in a trilogy and the cover was illustrated.  The girl's grandmother in the story was actually a phoenix (I think this was revealed in a different book in the trilogy though). The book started out in a small village, and I can't remember exactly where in the village chapters (the chapters that took place in the village) but she sneaked her deceased mother's green dress out of a trunk she wasn't supposed to go in, and I think she wore it to some event. Search in favorited questions? How to uncover a table column wise in Latex beamer I have a table with 7 columns and want to show the first 5 at once, and then  the 6th and 7th on consecutive slides.  This is the code I have for now:  \begin{frame}{Наслов}     \begin{itemize}         \item Ставка.     \end{itemize}         \begin{tabular}{ccccc&lt;{\onslide&lt;2-&gt;}c&lt;{\onslide}c}             $p_1$ &amp; $p_2$ &amp; $\ldots$ &amp; $p_{9}$ &amp; $p_{10}$ &amp; $W$  &amp; $D$\\             $1$   &amp; $1$   &amp; $\ldots$ &amp; $1$      &amp; $1$     &amp; $w_1$&amp; $d_1$\\             $1$   &amp; $1$   &amp; $\ldots$ &amp; $1$      &amp; $0$     &amp; $w_2$&amp; $d_2$\\             $1$   &amp; $1$   &amp; $\ldots$ &amp; $0$      &amp; $1$     &amp; $w_3$&amp; $d_3$\\             $1$   &amp; $1$   &amp; $\ldots$ &amp; $0$      &amp; $0$     &amp; $w_4$&amp; $d_4$\\             $\vdots$ &amp; $\vdots$ &amp; $\vdots$ &amp; $\vdots$ &amp; $\vdots$ &amp; $\vdots$ &amp; $\vdots$\\             $0$   &amp; $0$   &amp; $\ldots$ &amp; $0$      &amp; $1$     &amp; $w_{1023}$&amp; $d_{1023}$\\             $1$   &amp; $1$   &amp; $\ldots$ &amp; $0$      &amp; $0$     &amp; $w_{1024}$&amp; $d_{1024}$         \end{tabular} \end{frame}   This produces 2 slides: the first one with p1, p2, ..., p9, p10 and D columns, and the second with W added, while I want them to appear in the following order:   1st slide: p1, p2, ..., p9, p10  2nd slide: p1, p2, ..., p9, p10, W 3rd slide: p1, p2, ..., p9, p10, W, D   I looked in the manual and found this example:  \begin{frame}     \begin{tabular}{l!{\vrule}c&lt;{\onslide&lt;2-&gt;}c&lt;{\onslide&lt;3-&gt;}c&lt;{\onslide&lt;4-&gt;}c&lt;{\onslide}c}     Class &amp; A &amp; B &amp; C &amp; D \\     217     X &amp; 1 &amp; 2 &amp; 3 &amp; 4 \\     Y &amp; 3 &amp; 4 &amp; 5 &amp; 6 \\     Z &amp; 5 &amp; 6 &amp; 7 &amp; 8     \end{tabular} \end{frame}   which does what is supposed to. However, I don't seem to be able to apply its logic to my problem. :( Is anyone willing to help?   P.S. What are the meanings of &lt; and ! in columns specification? I run a job every minute to reindex my site's content.   Today, the search engine died, and when I logged in there were hundreds of orphan processes that had been started by cron.  Is there another way using some kind of existing software that will let me execute a job every minute, but that won't launch another instance if that job doesn't return (i.e. because the search engine process has failed)? "Have been having classes" or "have had classes for three months"? Which sentence is correct:  "I have been having classes for three months."  or  "I have had classes for three months." An init-script does not get called on shutdown I have set up an init-script to control the state of a VirtualBox VM:  #!/bin/sh #chkconfig: 35 99 5 #description: vTiger virtual machine  ### BEGIN INIT INFO # Provides: vtigervm # Required-Start: $local_fs # Requider-Stop: $stop_fs # Default-Start: 3 5 # Default-Stop: 0 1 2 6 # Short-Description: Manage vTiger virtual machine # Description: Utility to start and stop vTiger virtual machine on VirtualBox ### END INIT INFO  start() {     echo -n "Starting vTiger"     echo     su myuser -c '/usr/bin/VBoxManage startvm "vTiger" --type headless'     echo "Started virtual machine" &gt;&gt; /var/log/messages }  stop() {     echo -n "Shutting vTiger down..."     echo     su myuser-c '/usr/bin/VBoxManage controlvm "vTiger" acpipowerbutton'     while [ ! -z "`su - juhani -c '/usr/bin/VBoxManage list runningvms | grep vTiger'`" ]; do         echo -n "."         sleep 1     done     echo "Done."     echo "Stopped virtual machine" &gt;&gt; /var/log/messages }  status() {     echo -n "Running VMs: "     su myuser -c '/usr/bin/VBoxManage list runningvms'     echo     if [ -z "`su - juhani -c '/usr/bin/VBoxManage list runningvms | grep vTiger'`" ]; then         RETVAL=3         else         RETVAL=0     fi     echo "Queried virtual machine status" &gt;&gt; /var/log/messages }  echo "Called virtual machine management script with: $1" &gt;&gt; /var/log/messages case "$1" in   start)         start    ;;    stop)         stop    ;;    restart|try-restart|condrestart|reload)         stop         start    ;;    status)         status    ;;    *)         echo $"Usage: $0 {start|stop|restart|status}"         exit 1    ;; esac  exit $RETVAL   The system is CentOS 6.5. If I manually run service [start|stop|status] vtigervm, it works as I expect it to. If the VM is running, $? after the service vtigervm status returns 0, and 3 if it is stopped. I installed it with chkconfig and it created among a few others rc5.d/S99vtigervm and rc0.d/K05vtigervm.  The problem  When I start the system it starts the "service", but on shutdown it does not even run the scripts.  grep "virtual machine" /var/log/messages shows:  *[machine starting]* Called virtual machine management script with: start Started virtual machine *[shutdown -h now]* *[machine stopped]*   What I expect:  *[machine starting]* Called virtual machine management script with: start Started virtual machine  *[shutdown -h now]* Called virtual machine management script with: status Queried virtual machine status Called virtual machine management script with: stop Stopped virtual machine *[machine stopped]*   File permissions &amp; etc:  # ls -lah /etc/rc0.d/ lrwxrwxrwx. 1 root root 13 7.2. 17:49 /etc/rc0.d/K05atd -&gt; ../initd.d/atd lrwxrwxrwx. 1 root root 18 9.2. 00:06 /etc/rc0.d/K05vtigervm -&gt; ../initd.d/vtigervm In the Landau theory of phase transitions, is the order parameter a thermodynamic variable of state? In the Landau theory of phase transitions, one typically considers a "free energy" $F$ as a function of the temperature $T$ and the "order parameter" $\psi$: $F(T, \psi)$.  For the sake of clarity, let's consider the liquid-gas transition, in which the order parameter is usually taken to be the difference of densities between the ordered phase and the disordered phase.  In this case, it looks that the order parameter $\psi$ is closely related to the the volume $V$, a global variable of state.  What is the correct interpretation of $\psi$ as a thermodynamic variable?   I consider several logical possibilities, but all interpretations look troublesome to me:    $\psi$ is a variable of state replacing the volume. In this case I have a few related concerns:   Is the "free energy" $F$ the Helmholtz or the Gibbs free energy?    If $F$ is the Gibbs free energy, $F$ should depend on the temperature $T$ and the pressure $p$ (the conjugate variable of the volume), so it shouldn't depend on $\psi$.  If it is the Helmholtz free energy, it's ok that it depends on $\psi$, but the experiments are usually carried out at constant pressure, not constant volume.  Usually, state variables are something one can select at will [for instance, think of the phase diagram $(p, T)$]. However, the order parameter $\psi$ is not selected at will, but it is selected by the system as a result of the minimization process according to the Landau theory.  $\psi$ is yet another variable of state in addition to volume and temperature.    Why the pressure is not usually taken into account as a variable of state, and only the temperature is considered? What is the conjugate variable to $\psi$? Can really $\psi$ and $V$ be taken as independent?  $\psi$ is not a variable of state, is something else.   What else can $\psi$ be from the thermodynamic point of view? Same question as above, why is pressure not usually taken into account in Landau theory? Clerical mistake in UK visa application: should I make a pen-and-ink correction or electronically edit the PDF? I've gone through all similar questions but I believe my specific question doesn't have an answer yet.  I've applied for a Standard Visitor visa. After I completed the application, made an appointment and paid the fee, I noticed that the way I spelled my name on the application differs from how it is spelled in the passport by 1 letter (this is partially because of myself being an idiot, but mainly because my new passport has an incorrect transliteration of my name, whereas my old passport and any other document in my life has the correct one).  So here are the options I see:   Print the application and make a pen-and-ink correction with a black ballpoint pen by crossing the printed name/letter and writing the correct one - as answers to other similar questions suggests. Edit the PDF electronically and print it so that there are no visible corrections. However, as the form is saved at their end, there will be a discrepance between the printed form and saved form with no visible signs of correction.   Which way has better chance for me to have my application accepted?  Also (probably doesn't matter much but still): if it's the pen-and-ink correction, is it better to cross-out the whole name or only the wrong letter? How to write a good PhD literature review? What is the justification for locking votes? Why am I not allowed to change my vote after some period of time? There are times when I go back to things I voted on, and decide they didn't deserve that specific vote. (I get wiser as I grow older) Would you please explain the difference between these two sentences?     Spain ceded (or gave) Puerto Rico to the United States. Since then, Puerto Rico has been a U.S. territory.      Spain ceded(or gave) Puerto Rico to the United States. Since then, Puerto Rico was a U.S. territory.   I am stuck between 'has been' and 'was' in the above sentences. On a crop sensor camera, e.g. Canon EOS 7D, what would be the best kit to travel with around Europe for a few weeks? My shooting style will focus on landscape, architecture, and general travel shots (of note would be my lack of interest for portraits).     Limit to around 1-3 lenses, and I prefer light ones.   Crop sensor camera chosen to save on weight over a full frame option. Question is open to all brands for general guidance, but I will be using a Canon body.   Cost is not a factor, I will rent the lenses I do not already own.   Quality is of paramount importance. I will not have a tripod.   I understand some may consider this subjective, but I do believe that for a crop sensor body, with weight as a consideration, and a limit to the amount of lenses, a true best kit can be determined. Gel circular polarising filter I am contemplating buying a lens that only takes rear gel filters. Is it possible to buy a circular polarising filter in gel format? If so, who can supply them (preferably UK based)? Where to put my code: plugin or functions.php? Compact way to assign values by slicing list in Python Why do both cron and systemd execute automatic upgrades? Nginx Subdirectory returning 404 I currently have a bunch of working static files at the domain name khairulslt.me (from NameCheap). Recently, I've tried setting up a subdirectory (khairulslt.me/RGBGame) as seen in the code below; However, I keep getting 404 errors. What am i missing out?  server {    server_name khairulslt.me www.khairulslt.me;    autoindex off;   location / {     root /var/www/khairulslt.me;     index circles.html;   }   listen 443 ssl; # managed by Certbot ssl_certificate /etc/letsencrypt/live/khairulslt.me/fullchain.pem; # managed by Certbot ssl_certificate_key /etc/letsencrypt/live/khairulslt.me/privkey.pem; # managed by Certbot include /etc/letsencrypt/options-ssl-nginx.conf; # managed by Certbot ssl_dhparam /etc/letsencrypt/ssl-dhparams.pem; # managed by Certbot  location /RGBGame {     root /var/www/khairulslt.me/RGBGame;     index colorGame.html;     try_files $uri $uri/ /var/www/RGBGame/colorGame.html?q=$uri&amp;$args;     autoindex off;   }  location /robots.txt { return 200 "User-agent: *\nDisallow: /\n";    }  }   server { if ($host = www.khairulslt.me) {     return 301 https://$host$request_uri;   } # managed by Certbot   if ($host = khairulslt.me) {     return 301 https://$host$request_uri;   } # managed by Certbot   listen 80; server_name khairulslt.me www.khairulslt.me; return 404; # managed by Certbot }   I've tried all the different settings like (alias vs root), try_files, autoindex(on, off). My main website www.khairulslt.me works fine and is serving the html/css/js files in /var/www/khairulslt.me perfectly fine. I'm just not sure why I can't replicate the settings for my other app (html/css/js files in folder /RGBGame). Could it be that I've missed out something that isn't related to the nginx config? How to get the CPU cycle count in x86_64 from C++? How to increase body height? I am an adult, and I want to be taller. I know about many medicines which claim to treat stunted height.  Is there any truth behind these? Prove that the additive groups of $\Bbb Z$ and $\Bbb Q$ are not isomorphic.    It is hard to create a map that will show homomorphism.