Aditive functor characterization. Aditive iff right/left exact.
Half exact functor which is neither right exact nor left exact
$3\times3$ linear system organization Numbering rows within groups in a data frame Rewrite code to make it faster I'm rotating an object on two axes, so why does it keep twisting around the third axis? I see questions come up quite often that have this underlying issue, but they're all caught up in the particulars of a given feature or tool. Here's an attempt to create a canonical answer we can refer users to when this comes up - with lots of animated examples! :)    Let's say we're making a first-person camera. The basic idea is it should yaw to look left &amp; right, and pitch to look up &amp; down. So we write a bit of code like this (using Unity as an example):  void Update() {     float speed = lookSpeed * Time.deltaTime;      // Yaw around the y axis using the player's horizontal input.             transform.Rotate(0f, Input.GetAxis("Horizontal") * speed, 0f);      // Pitch around the x axis using the player's vertical input.     transform.Rotate(-Input.GetAxis("Vertical") * speed,  0f, 0f); }   or maybe  // Construct a quaternion or a matrix representing incremental camera rotation. Quaternion rotation = Quaternion.Euler(                         -Input.GetAxis("Vertical") * speed,                          Input.GetAxis("Horizontal") * speed,                          0);  // Fold this change into the camera's current rotation. transform.rotation *= rotation;   And it mostly works, but over time the view starts to get crooked. The camera seems to be turning on its roll axis (z) even though we only told it to rotate on the x and y!    This can also happen if we're trying to manipulate an object in front of the camera - say it's a globe we want to turn to look around:    The same problem - after a while the North pole starts to wander away to the left or right. We're giving input on two axes but we're getting this confusing rotation on a third. And it happens whether we apply all our rotations around the object's local axes or the world's global axes.  In many engines you'll also see this in the inspector - rotate the object in the world, and suddenly numbers change on an axis we didn't even touch!    So, is this an engine bug? How do we tell the program we don't want it adding extra rotation?  Does it have something to do with Euler angles? Should I use Quaternions or Rotation Matrices or Basis Vectors instead? I need rules to drop some malicious Apache connections I drop all traffic on ports except on 80 for my web server.  I have some rules like this on iptables:  iptables -A INPUT -p tcp -m tcp --dport 80 -m string --string "cgi" --algo bm --to 1000 -j DROP   Someone who have more can share? I know always the bad hackers still updating, but some of them always begin with the same code. I need to Drop the connection based on some criteria. Here are some Apache log (I remove ips but each atack come from the same):  Attack 1: This I don't know what are trying to do, but make it 50 times from the same ip  GET / HTTP/1.1  301 224 -   Mozilla/5.0 (X11; Linux i686) AppleWebKit/537.22 (KHTML, like Gecko) Chrome/25.0.1364.152 Safari/537.22 GET / HTTP/1.1  302 3387    -   Mozilla/5.0 (X11; Linux i686) AppleWebKit/537.22 (KHTML, like Gecko) Chrome/25.0.1364.152 Safari/537.22   Attack 2: this try to get information about server only.  GET / HTTP/1.1  301 224 http://myip:80/ Go-http-client/1.1 GET / HTTP/1.1  302 3228    http mywebsite  Go-http-client/1.1 GET /es/ HTTP/1.1   200 40947   https mywebsite Go-http-client/1.1   Attack 3: they try get access to a login page vulnerability  GET /userlogin/login.aspx HTTP/1.1  302 186 -   -   Attack 4: this try to access a cgi at first request, (see my first iptables rule to drop this)  GET /hndUnblock.cgi HTTP/1.0    302 186 -   Wget(linux) GET /tmUnblock.cgi HTTP/1.0 302 186 -   Wget(linux)   I'm very new with server this 4 attacks are from only last 12 hours... Have thousands per week. File names for JSP include directive to avoid compilation of them How to find the roots of $x^4 +1$ I'm trying to find the roots of $x^4+1$. I've already found in this site solutions for polynomials like this $x^n+a$, where $a$ is a negative term. I don't remember how to solve an equation when $a$ is a positive term as the equation above.  Thanks list all suggested packages for currently installed packages How do I copyright my website? My website has been in development for a long time and cost a lot of money.  How do I go about securing an enforceable copyright on my website design and what is the proper way to post a copyright notice on the site so others know the work is copyrighted? A friend asked me last week how to enumerate or list all variables within a program/function/etc. for the purposes of debugging (essentially getting a snapshot of everything so you can see what variables are set to, or if they are set at all). I looked around a bit and found a relatively good way for Python:   #!/usr/bin/python                                                                                                                                                                                                                            foo1 = "Hello world" foo2 = "bar" foo3 = {"1":"a",         "2":"b"} foo4 = "1+1"  for name in dir():     myvalue = eval(name)     print name, "is", type(name), "and is equal to ", myvalue   which will output something like:   __builtins__ is &lt;type 'str'&gt; and is equal to  &lt;module '__builtin__' (built-in)&gt; __doc__ is &lt;type 'str'&gt; and is equal to  None __file__ is &lt;type 'str'&gt; and is equal to  ./foo.py __name__ is &lt;type 'str'&gt; and is equal to  __main__ foo1 is &lt;type 'str'&gt; and is equal to  Hello world foo2 is &lt;type 'str'&gt; and is equal to  bar foo3 is &lt;type 'str'&gt; and is equal to  {'1': 'a', '2': 'b'} foo4 is &lt;type 'str'&gt; and is equal to  1+1   I have so far found a partial way in PHP (courtesy of ) but it only lists all variables and their types, not the contents:   &lt;?php // create a few variables $bar = 'foo'; $foo ='bar'; // create a new array object $arrayObj = new ArrayObject(get_defined_vars()); // loop over the array object and echo variables and values for($iterator = $arrayObj-&gt;getIterator(); $iterator-&gt;valid(); $iterator-&gt;next())         {         echo $iterator-&gt;key() . ' =&gt; ' . $iterator-&gt;current() . '&lt;br /&gt;';         } ?&gt;   So I put it to you: how do you list all variables and their contents in your favorite language?    Edit by : I propose this question follows the spirit of a little "". If you do not agree, just edit and remove the tag and the link. Boundary of Ball of radius R has zero measure "Always on top" option missing for Firefox/Chrome in 18.04 It seems that the question has eventually become a series of questions....    An example goes as:     The 1st and (the) 2nd paragraphs of the article are extremely long.   Another example:     What are the situation, (the) task and (the) result of your story?   Is it necessary to use the in ()? I wonder a generalization of Jensen's inequality: let $\mathbf{X,Y}$ be two positive definite matrices, can we obtain the following Jensen like inequality $$(1-\lambda)\mathbf{X}^{-1}+\lambda\mathbf{Y}^{-1} \succeq((1-\lambda)\mathbf{X}+\lambda\mathbf{Y})^{-1}$$ for any $\lambda \in (0,1)$, where the notation $\mathbf{A}\succeq \mathbf{B}$ represents matrix $(\mathbf{A-B})$ positive semi-definite. Have red shifted photons lost energy and where did it go? prevent "action" if top of page I have a manual list of stuff that are sometimes separated by an hrule(or my own custom ruler) for visual purposes.  Sometimes the hrule ends at the top or bottom of a page which makes it useless.  Is there a way to prevent it from being added if it is at the top or bottom of a page?    MWE of problem:  \documentclass{book} \usepackage{tikz} \begin{document} \newcommand\rulesep{%   \par % make sure we end a paragraph   \vskip5pt % space above the rule   \leaders\vrule width \textwidth\vskip0.4pt % rule thickness is 0.4pt   \nointerlineskip % disable interline glue here   \vskip5pt % space below the rule } \def\dotikzcircle#1{\begin{tikzpicture}\draw(0,0)circle(#1);\end{tikzpicture}} \rulesep \noindent This is a test\noindent \rulesep \noindent End of test \rulesep \dotikzcircle3 \rulesep \begin{center}\dotikzcircle3\end{center} \rulesep \begin{center}\dotikzcircle3\end{center} \rulesep \dotikzcircle1 \rulesep \dotikzcircle2 \rulesep \dotikzcircle3 \rulesep \dotikzcircle4 \rulesep \dotikzcircle5 \rulesep  \end{document} How to read "service --status-all" results I need to identify the currently running services but I can't figure out the results of running service --status-all, I mean what does ?, - and + mean ?  $ service --status-all  [ + ]  acpid  [ + ]  anacron  [ + ]  apparmor  [ ? ]  apport  [ ? ]  atieventsd  [ + ]  avahi-daemon  [ ? ]  binfmt-support  [ + ]  bluetooth  [ - ]  brltty  [ + ]  console-font  [ + ]  console-setup  [ + ]  cron  [ + ]  cups  [ + ]  cups-browsed  [ - ]  dbus  [ ? ]  dns-clean  [ + ]  friendly-recovery  [ - ]  grub-common  [ ? ]  irqbalance  [ - ]  kerneloops  [ ? ]  killprocs  [ + ]  kmod  [ ? ]  lightdm  [ - ]  lm-sensors  [ ? ]  mysql  [ ? ]  networking  [ ? ]  ondemand  [ ? ]  pppd-dns  [ - ]  procps  [ - ]  pulseaudio  [ ? ]  rc.local  [ + ]  resolvconf  [ + ]  rfkill-restore  [ + ]  rfkill-store  [ - ]  rsync  [ + ]  rsyslog  [ + ]  saned  [ ? ]  sendsigs  [ + ]  setvtrgb  [ ? ]  speech-dispatcher  [ - ]  sudo  [ + ]  timidity  [ + ]  udev  [ ? ]  umountfs  [ ? ]  umountnfs.sh  [ ? ]  umountroot  [ - ]  unattended-upgrades  [ - ]  urandom  [ + ]  virtualbox  [ - ]  x11-common   And would running this command with sudo make any difference ? I tried it and in my case it didn't make any difference, but may it differ in other setups ? In general, in most places, it is said that using TeX defaults is the best. However, TeX does provide things like \, and \quad etc. When one should use them? Both in general and in math mode in particular. Speed of light in a gravitational field? Definite Integral $\int_{-\infty}^{\infty}\frac{x\sin x}{(x^2+a^2)(x^2+b^2)}\,\mathrm{d}x$ $$\int_{-\infty}^{\infty}\frac{x\sin x}{(x^2+a^2)(x^2+b^2)}\,\mathrm{d}x$$  This is easy to evaluate with complex analysis but is there an elementary way (substitution, partial fractions, integration by parts)?