Show that:  $$\left(\frac 3p\right)= \begin{cases} p \equiv 1, 11 \pmod {12} \\ p\equiv 5, 7 \pmod {12} \end{cases}$$  where $\left(\frac 3p\right)$ is Legendre symbol  I have done this for $\left(\frac {-3}p\right)$ and reach $\left(\frac 3p\right) = 1$ if $p=1$ and $\left(\frac 3p\right)= -1$ if $p=2$. in the case where $p=1$ the next step was to see that $p=3k+1=6l+1$ so then $p=12j+1$ and $p=12j+11$ give $\left(\frac {-3}p\right)=1$.   I'm very confused about all of this so if someone could help it'd be greatly appreciated. it would probably be better to start from the beginning with $\left(\frac 3p\right)$ or to  use the same logic used to find $\left(\frac 2p\right)$ and Gauss lemma
$(\frac 3p)$ is the Legendre symbol here, not a fraction, if that wasn't clear.  This is what I have so far:  We know by Gauss's Lemma that $(\frac qp) = (-1)^v$ where $$v = \#\left\{1 \le a \le \frac{p-1}{2} \mid aq \equiv k_a \pmod{12}),\space -p/2 &lt; k_a &lt; 0\right\}$$  And we also know that multiples of $(\dfrac{p-1}{2}) \times 3$ must be between $0$ and $p/2$, between $p/2$ and $p$, or between $p$ and $3p/2$.   Representing $p$ as $12k + r$ seems like the next step, but I'm not sure where to go from there.
How do I remove Windows 7 from my dual boot system with Ubuntu? There are 11 letters in the word.   M - 1 I - 4 S - 4 P - 2  so the number of different permutations is $\dfrac{11!}{1! 4 !4!2!}$  Is this correct solution? Since arrays and hashes can only contain scalars in Perl, why do you have to use the $ to tell the interpreter that the value is a scalar when accessing array or hash elements? In other words, assuming you have an array @myarray and a hash %myhash, why do you need to do:  $x = $myarray[1]; $y = $myhash{'foo'};   instead of just doing :  $x = myarray[1]; $y = myhash{'foo'};   Why are the above ambiguous?   Wouldn't it be illegal Perl code if it was anything but a $ in that place? For example, aren't all of the following illegal in Perl?  @var[0]; @var{'key'}; %var[0]; %var{'key'}; The new Top-Bar does not show reputation changes from Area 51. Hide iPhone HTML5 video play button How can I make a material only apply to a side of a plane? In my cover letter I want to write that I am a graduate student in statistics. But I am not sure about the "in". So what is correct?     I am a graduate student in statistics.      I am a graduate student of statistics. Change bounding box of math glyphs in LuaTeX I'm trying to use the Garamond-Math font in LuaTeX (despite the documentation saying I should use it with XeTeX only :-). It's working more or less OK with LuaTeX, except for some unfortunate glyph combinations, mostly at the end of inline math, such as “where $T$ is the period”.  I used \fbox to draw the bounding box of the glyph and noticed that the bounding boxes are different in LuaTeX:        and in XeTeX:        In XeTeX the glyph fits almost entirely in the bounding box, but in LuaTeX it overshoots far too much.  I'm not a font guy, so I have no idea why this happens. I thought the bounding box was a property of the font, so in my imagination the result should be the same. Thus the first question is: what actually happens that the bounding boxes are different?  The second question, in the title, is: can I change the bounding box of specific glyphs (using LuaTeX) so that the output looks more like XeTeX's?  I tried manually adding some spacing after the glyphs, but this looks hideous and will go down the drain if I happen to change the font or engine, so I'd really like not to do that. I also tried using the italic correction \/ but it seems to have no effect in math.  Here's the code to produce the pictures above:  \documentclass{standalone} \usepackage{unicode-math} \setmathfont{Garamond-Math.otf} \begin{document} \fboxrule=1sp \fboxsep=0pt In Lua\TeX{} this \fbox{$T$} looks horrible, but the \fbox{$X$} looks OK. \Huge\fbox{$T$}\quad. \end{document}   I'm using TL 2018:  This is LuaTeX, Version 1.07.0 (TeX Live 2018)  (format=lualatex 2019.3.29)  13 APR 2019 12:43  restricted system commands enabled. **test.tex (./test.tex LaTeX2e &lt;2018-12-01&gt; Lua module: luaotfload-main 2019-02-14 2.96 luaotfload entry point Can we opt for fast track service after the refusal? Last Saturday, I got a refusal from BHC due to lack of funds and my reasons to come back to India. I am planning to visit my family who are settled in the UK. I am married and planning to visit my family along with my husband. My father is sponsoring our whole trip. But we got a refusal as we didn't show much funds in our account. We thought that if my father is sponsoring, we should show his documents only. But now I have applied again with a fast track service. Will BHC will consider it under the fast track process? This time I have cleared all their doubts which they mentioned in the refusal letter. Analogy of Pearson correlation for 3 variables Shut down computer from keyboard Is there a keyboard shortcut available to shutdown the computer?  I know I can use the Power button, but I would like to be able to it from the keyboard.  If such a shortcut does not exist already, how do I set one? Unity Tweak tool fails to start with the error "com.canonical.unity.webapps schema is missing" com.canonical.unity.webapps   In order to work properly, Unity Tweak Tool recommends you install the necessary packages I think we can all agree, this sucks:  If you've been around a little while, you've probably encountered hundreds of answers like this in various forums, some of them even marked as &quot;The Answer&quot; by well-meaning1 forum admins looking to close a thread. We could try to enumerate the commonly-observed problems with it...  Link is probably broken If link isn't broken, it's spam If link isn't spam, it's to some broadly-related resource that the answerer found by pressing &quot;I'm feeling lucky&quot; when googling the question's title.  ...but the real issue here is that we don't know! Even if the link was originally pointing directly to the most awesomeest answer ever for exactly the question being asked, there's nothing in the answer itself to even hint at what direction we're being pointed in. Strip the markup, and this is what you're left with:  i think you should take The tutorial HERE! This will help you a lot  Is there even one tiny speck of information there? No. Even the title of the page being linked to would have provided something to go on, but that was entirely too much work. So let me be clear: this sort of response is not an answer. If you see this, flag it. Moderators, if you see it flagged, delete it. There's really only one valid exception to this rule, and that's when the question is:  Suggest me some tutorials where i can learn quick.  Don't get me wrong, it's still a bad answer - but when the question is kinda asking for bad answers this is to be expected. Close or flag the question and move on - this is much more efficient than hanging around to babysit it by deleting every crap answer that inevitably shows up. If you're still reading, you're probably getting really bored by now. &quot;Gee, Shog - answers that contain ZERO information are terrible, horrible, very-bad things? What shocking revelations will you share next?&quot; Yeah. It's really simple... Except, . See, the problem is that once you step away from black-and-white &quot;answers / doesn't answer&quot; decision-making, things get... Interesting. And all those other answers with links in them See, this is an answer:  And so is this:  Yes, they're both very short, and yes, they contain links. But strip the markup, and you still get at least a little bit of useful information. Does that mean these answers should forever hang around the site? No, not necessarily - if it turns out they're just not that useful, they should probably still be removed - or at very least, down-voted so that they rank below other answers. Just remember: if the text of the post contains an honest attempt at answering the question, then it is an answer - so don't flag it otherwise, and if you do, don't complain if your flag gets declined. Answers &amp; Apples If you're still confused, forget about answers. Think about apples instead:  Answers are just like that, but less tasty in pies.  1 By which I mean to say, &quot;criminally lazy&quot; In Microsoft Word, the following sentence is flagged.  It tells me to use "was" instead of "were"     There were half a dozen books strewn about the floor.   I would think that you would use "were" since it's a quantity more than one.  You wouldn't say, "There was twelve books strewn about the floor."  Does the use of the "half" modify it somehow? How to use inotifywait to watch a directory for creation of files of a specific extension I have seen .      You should consider using inotifywait, as an example:  inotifywait -m /path -e create -e moved_to |     while read path action file; do         echo "The file '$file' appeared in directory '$path' via '$action'"         # do something with the file     done    My question is that, the above script watches a directory for creation of files of any type, but how do I modify the inotifywait command to report only when a file of certain type/extension is created (or moved into the directory) - e.g. it should report when any .xml file is created.  WHAT I TRIED:  I have run the inotifywait --help command, and have read the command line options. It has --exclude &lt;pattern&gt; and --excludei &lt;pattern&gt; commands to EXCLUDE files of certain types (by using regEx), but I need a way to INCLUDE just the files of a certain type/extension. I just recently bought an arduino and requires gcc-avr/avrdude to compile the software. I installed avr for another microprocessor component a while ago but is obviously an outdated version (gcc version 3.3 20030512 (prerelease)) so I went ahead to update these but it didn't work.   (Please note that before hand I also broke my aptdaemon through an incomplete wine installation (couldn't get passed font installation agreement) but I fixed that via a re-installation then accepting the agreement.)   I am trying to update these by running  but the dependencies it requires cannot be installed due to avr... terminal reports   Reading package lists... Done Building dependency tree        Reading state information... Done build-essential is already the newest version. libncurses5-dev is already the newest version. libncurses5-dev set to manually installed. libusb-0.1-4 is already the newest version. libx11-dev is already the newest version. You might want to run 'apt-get -f install' to correct these: The following packages have unmet dependencies: arduino-core : Depends: gcc-avr but it is not going to be installed avr-libc : Depends: gcc-avr (&gt;= 1:4.3.4) but it is not going to be installed Depends: binutils-avr (&gt;= 2.20) but it is not going to be installed binutils-dev : Depends: binutils (= 2.21.0.20110327-2ubuntu3) but 2.21.0.20110327-2ubuntu2 is to be installed libcwidget-dev : Depends: libcwidget3 (= 0.5.16-3ubuntu2) but it is not going to be installed Depends: libsigc++-2.0-dev but it is not going to be installed Depends: libncursesw5-dev but it is not going to be installed libmpfr-dev : Depends: libgmp3-dev (&gt;= 4.2.dfsg-1) but it is not going to be installed python-dev : Depends: python (= 2.7.1-0ubuntu5.1) but 2.7.1-0ubuntu5 is to be installed Depends: python2.7-dev (&gt;= 2.7.1-1~) but it is not going to be installed tk8.4-dev : Depends: tk8.4 (= 8.4.19-4) but it is not going to be installed E: Unmet dependencies. Try 'apt-get -f install' with no packages (or specify a solution).   But running   apt-get -f install   results in   (Reading database ... 163021 files and directories currently installed.) Unpacking binutils-avr (from .../binutils-avr_2.20.1-1ubuntu2_i386.deb) ... dpkg: error processing /var/cache/apt/archives/binutils-avr_2.20.1-1ubuntu2_i386.deb (--unpack): trying to overwrite '/usr/bin/avr-size', which is also in package avr-binutils 2.13.90.030512-2 dpkg-deb: error: subprocess paste was killed by signal (Broken pipe) Unpacking gcc-avr (from .../gcc-avr_1%3a4.3.5-1_i386.deb) ... dpkg: error processing /var/cache/apt/archives/gcc-avr_1%3a4.3.5-1_i386.deb (--unpack): trying to overwrite '/usr/bin/avr-g++', which is also in package avr-gcc-c++ 3.2.90.20030512-2 dpkg-deb: error: subprocess paste was killed by signal (Broken pipe) Errors were encountered while processing: /var/cache/apt/archives/binutils-avr_2.20.1-1ubuntu2_i386.deb /var/cache/apt/archives/gcc-avr_1%3a4.3.5-1_i386.deb E: Sub-process /usr/bin/dpkg returned an error code (1)   I have tried running update manager and updating my system via it but all I get is an error message  The following packages have unmet dependencies: arduino-core: Depends: gcc-avr but it is not installed avr-libc: Depends: gcc-avr (&gt;= 1:4.3.4) but it is not installed Depends: binutils-avr (&gt;= 2.20) but it is not installed   which then tells me to try running apt-get -f install which just results the same as last time.  So how can I fix my system, I really need the new avr, please :) BTW, my system is Ubuntu 11.04 Finding the Transform matrix from 4 projected points (with Javascript) If $a$, $b$, and $c$ are sides of a triangle, then $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$. Let $a,b,c$ be the lengths of the sides of a triangle. Prove that $$\sum_{\text{cyc}}\frac{a}{b+c}=\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}&lt;2\,.$$  Attempt.  By clearing the denominators, the required inequality is equivalent to $$a^2(b+c)+b^2(c+a)+c^2(a+b)&gt;a^3+b^3+c^3\,.$$ Since $b+c&gt;a$, $c+a&gt;b$, and $a+b&gt;c$, the inequality above is true.  Is there a better, non-bruteforce way? How do I protect random people pointing their domain to my server? What does (obj.length === +obj.length) compare?