Confusion with how partial fractions work
Derivation of the general forms of partial fractions
Matching a space in regex I need to match a space character in a PHP regular expression. Anyone got any ideas?  I mean like "gavin schulz", the space in between the two words. I am using a regular expression to make sure that I only allow letters, number and a space. But I'm not sure how to find the space. This is what I have right now:  $newtag = preg_replace("/[^a-zA-Z0-9s|]/", "", $tag); Why is there something instead of nothing? A simple but fundamental question.    The "something" means the whole Universe (known and unknown), it could be represented as the reality version of the set of all sets, which is itself . It includes all the  and such.  A better version of the whole Universe could be described with a .  The "nothing" means an absence of the whole Universe, not a particular void or emptiness in the whole Universe but a nonexistent whole Universe. Prove the divisibility test by $7,11,13$ for numbers more than six digits An explanation of Hawking Radiation Could someone please provide an explanation for the origin of ? (Ideally someone who I have been speaking with on the h-bar)  Any advanced maths beyond basic calculus will most probably leave me at a loss, though I do not mind a challenge! Please assume little prior knowledge, as over the past few days I have discovered that much of my understanding surrounding the process as virtual particle pairs is completely wrong. How would one create an iterative function (or iterator object) in python? How can I save application settings in a Windows Forms application? What I want to achieve is very simple: I have a Windows Forms (.NET 3.5) application that uses a path for reading information. This path can be modified by the user, by using the options form I provide.  Now, I want to save the path value to a file for later use. This would be one of the many settings saved to this file. This file would sit directly in the application folder.  I understand three options are available:   ConfigurationSettings file (appname.exe.config) Registry Custom XML file   I read that the .NET configuration file is not foreseen for saving values back to it. As for the registry, I would like to get as far away from it as possible.  Does this mean that I should use a custom XML file to save configuration settings?  If so, I would like to see code example of that (C#).  I have seen other discussions on this subject, but it is still not clear to me. Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane that is perpendicular to $\alpha$ the elements of  $\Phi$ in $V$.   I want to show that:   The Weyl chambers are open, convex and connected.   My attempts:   A single hyperplane $H_\alpha$ separates the space $V$ into two half-spaces: positive half-space $H_\alpha^+=\{x \in V \mid (\alpha,x)&gt;0\}$ and negative half-space $H_\alpha^-=\{x \in V \mid (\alpha,x)&lt;0\}$. It is clear that $V \setminus H_\alpha = H_\alpha^+\cup H^-_\alpha.$ As the result, the Weyl chambers can be regarded as components of  $$V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}= \bigcap_{\alpha \in \Phi} \left( V \setminus H_{\alpha} \right)= \bigcap_{\alpha \in \Phi} \left( H_\alpha^+\cup H_\alpha^- \right).$$ By the discussion above, a Weyl chamber is simply a $\textbf{intersection}$ of finite half-spaces. It is the case since one may write $$\left( H_\alpha^+\cup H_\alpha^- \right) \cap \left( H_\beta^+\cup H_\beta^- \right) =\left( H_\alpha^+\cap H_\beta^+ \right)  \cup\left( H_\alpha^+\cap H_\beta^- \right)  \cup \left( H_\alpha^-\cap H_\beta^+ \right)  \cup\left( H_\alpha^-\cap H_\beta^- \right).$$ Therefore openness and convexity of the Weyl chambers are straightforward since half-spaces are open and convex, so are their intersections. Moreover, we would replace $V$ by $\Bbb R^{\dim V}$ since they are isomorphic. Now we can prove that the Weyl chamber is connected by showing that every convex set in $\Bbb R^{\dim V}$ is connected. And I'm sure that proof of this statement existed. So am I done? What do we mean when we say "for all n where n is an element of..." When we write "for all n where n is an element of A, P(n) is true." do we mean for all n, if n is in A then Pn or for all n, n is in A and Pn? How to prepare files for rsync on a case insensitive filesystem? I am transferring a large number of files on a HFS+ filesystem.  The files are currently on ext2 partitions.  I have conflicts due to case insensitivity of the target partition (HFS+).  I want to identify the files that have duplicates filenames once they are in lower case, and delete them if they are actually duplicates.  I also found that I will have duplicate folder names if I convert everyhing to lower case. Basically these hard drives contain years of unsorted data, and I happen to have this problem with folder names too.   Does this seem reasonable:  find . -type f | while read f; do echo $f:l; done | sort | uniq -d    $f:l is ZSH for convert to lower case.  Now I want to keep only one instance of each file that have duplicates. How to do this efficiently ?  I do not want to find files with duplicate content, unless they have the same lower case filename. I will deal with duplicates later. Imported CSV points are not visible I am trying to import  CSV file and get no reported errors but there are no points visible. If I click on "zoom to extent" nothing happens (even if I move away from the location of the points before doing this.  I have shut off all raster layers but they were not hiding the points.  I have tried to find an answer here but none seem to fit my problem. I have followed the steps in this tutorial:   Here is an image of the CSV file:    here is the import dialog box:   and here is the QGIS attribute table after import:   I am saving in WGS84.  I tried the local projection (NAD83 Zone 19N) but that made no difference.   I am using QGIS 2.12.1 on Windows 10. Any suggestions on what I am doing wrong? If $ar + bs =1$, then $\gcd(a,s) = \gcd(r,b) = \gcd(r,s) = 1$ I am trying to teleport a player to certain coordinates once they enter an area.  The idea is that they walk into a really small building, but once they turn a corner, they get teleported into a larger building.   Please refer to the adventure map named '5 Cubed', if you need help understanding what I am asking. How can I simplify the use of flags with Views Bulk Operations? I have an administrative display (made with views) that lists content that is pending approval.  In the case of my site, approval means flagging with the approval_for_pub flag (which triggers a custom rule).   allows nodes to be flagged/unflagged all at once by clicking a handy button:    However, upon clicking this button, by default it shows all possible flags and the flag/unflag option:    This is a problem on my site because some of the people responsible for approval get confused by the names of the flags and select the wrong option.  So, what I would like to do is create a VBO option that, when the button is pressed, automatically flags all applicable content with the approval_for_pub flag.  How can I approach this?  It seems like a small detail, but this step happens several times a day, so the chance of making a mistake is quite high. I would like to open all chapters of my PDF ebooks in the same window but neither evince nor okular seems to be capable of doing that (at least not out-of-the-box).  Is there a way to work this out? Add days to JavaScript Date How to add days to current Date using JavaScript. Does JavaScript have a built in function like .Net's AddDay? Abount linear functional: If $T(B)$ is bounded, is $T$ bounded? Let $H$ be a infinite dimensional Hilbert space and let $B$ be a basis of $H$ that $H=\overline{span(B)}$  moreover, let $T : H \rightarrow K$ be a linear functional   If $T(B)$ is bounded, is $T$ bounded? Which is correct, "neither is" or "neither are"? Using competitors domain with different ending to boost traffic alias or bash function does not work When I create  alias wd='ps -ef | grep java | awk {'print $2 " " $9'} | egrep "(A|B|C|D)"'   or  function wd () {     ps -ef | grep java | awk '{print $2}' ... }   in my .bashrc file, I get errors.  Interestingly, if I source my .bashrc file with the function, it 'compiles', but when executing, gives me:   context is      &gt;&gt;&gt; \ &lt;&lt;&lt; {\print     missing }   Can someone help me with this, and also answer when its better to put something in a function versus in an alias? How to negate a specific symbol? I want to negate the symbol whose code is \models (this would be specific to Formal Logic). How can I do that? I tried a number of combination, but didn't find how this could be done. I tried \neqmodels, \nemodels, \notmodels, \nmodels  but all of them are wrong.  Is there a general way to negate symbols?