Elbow method not giving a proper curve
K-means - comparing solutions with SSwithin elbow-method: minimum "too early", or non-monotonic curve
I know that there is a precise solution of the equations:  \begin{align}     x + a &amp; = 0 \\     x^2 + a\cdot x + b &amp; = 0 \\     x^3 + a\cdot x^2 + b\cdot x + c &amp; = 0 \\     x^4 + a\cdot x^3 + b\cdot x^2 + c\cdot x + d &amp; = 0 \end{align}  Why there is no solutions for equation of degree $\ge 5$? Is LAN with StarCraft 2 definitely not possible? $$\int x^3\sqrt{4-x^2}dx$$ I tried changing the expression like this: $$\int x^3\sqrt{2^2-x^2}dx=\int x^3\sqrt{(2-x)(2+x)}dx$$ And that's where I got stuck. I tried integration by parts but it doesn't simplify the expression. How to check the count of .txt files in a folder (dir command)? How can use dir command to check how many .txt files there are in a folder (for example,  C:\Temp\)? I have a string which is like this:  this is "a test"   I'm trying to write something in Python to split it up by space while ignoring spaces within quotes. The result I'm looking for is:  ['this','is','a test']   PS. I know you are going to ask "what happens if there are quotes within the quotes, well, in my application, that will never happen. fleqn document class option, long text lines, and hyperref package When the fleqn option is used, the vertical spacing before an equation placed right after a "long" text line (exactly \linewidth I would say) is improperly set up. See below:    From the mathmode user guide, this could probably be fixed through the \topsep length. Is it the right way to proceed? The problem is reproduced by the code below:  \documentclass[12pt,fleqn]{book} \usepackage{amsmath} \usepackage{hyperref} \begin{document} \chapter{plo} \section{plo} eoiu oieu oeiuoe ueoi ueoi ueoiu eoiu oeu oe iupaêio umùaeoiuaem oeiu eoiu eoiu oeee. eoiu oieu oeiuoe ueoi ueoi ueoiu eoiu oeu oe iupaêio umùaeoiuaemm \begin{equation} \cos\pi=-1 \end{equation} eoiu oieu oeiuoe ueoi ueoi ueoiu eoiu oeu oe iupaêio umùaeoiuaem oeiu eoiu eoiu oeee. eoiu oieu oeiuoe ueoi ueoi ueoiu eoiu oeu oe iupaêio umùae \begin{equation} \cos\pi=-1 \end{equation} \end{document}   There is something wrong involving hyperref, amsmath, and fleqn. Removing one of those removes the problem. Meta's tag pages inaccessible due to self-redirect loop Tag pages (route /questions/tagged/tagname) redirect to themselves indefinitely:  $ curl -D - http://meta.stackexchange.com/questions/tagged/featured HTTP/1.1 302 Found Cache-Control: private Content-Type: text/html; charset=utf-8 Location: /questions/tagged/featured X-Frame-Options: SAMEORIGIN Date: Fri, 05 Jul 2013 16:47:42 GMT Content-Length: 143  &lt;html&gt;&lt;head&gt;&lt;title&gt;Object moved&lt;/title&gt;&lt;/head&gt;&lt;body&gt; &lt;h2&gt;Object moved to &lt;a href="/questions/tagged/featured"&gt;here&lt;/a&gt;.&lt;/h2&gt; &lt;/body&gt;&lt;/html&gt;   This only happens on Meta, so far, but only applies to any meta-tag that doesn't have a dash in it; those with a dash seem to be exempt, as do invalid tags:    redirects.  does not.  does not. I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but can anyone produce an explicit formula for such a bijection? How to keep vehicles upgrades equipped in Jetpack Joyride? I have Jetpack Joyride for Android, but every time I start the game I have to equip the magnetic upgrades to all vehicles, except the teleporter, which is the only one already equipped.  I had once a game crash that undid my last 2 magnets purchases, so I ended up with only the teleporter and 20k less in cash. After that, I've bought every vehicle magnet, but still the only one that will remain equipped is the teleport.  Anyone already found some similar bug? How could I fix this, it's kinda annoying have to mannually enable 5 different upgrades every time I want to play. There's an election going on. What's happening and how does it work? I'm trying to do a number checker in imessage. We all know that there is no such api, so I'm trying to use the tools at hand. The main took with past issues on this site on this topic.  I made a loop that inserts a number into a string, and by the color of the number (blue - registered, red - missing), using the standard utility Digital Colors Meter determines the color of the number. Then, depending on the color, it either adds a variable with a number from the initial txt file to the new one, or starts the script from the beginning. At least it should be.  But something is going wrong in loop. It checks the color, inserts numbers, everything is fine, but apparently the wrong condition, apparently the wrong data types. It inserts different numbers into the scatter. Some will insert, some will not, regardless of the initial condition. What could be the reason? Where is the error?      tell application "Digital Color Meter" to activate     tell application "System Events"         tell process "Digital Color Meter"             click menu item 1 of menu 1 of menu bar item 4 of menu bar 1         end tell         do shell script "/usr/bin/python -c \"import objc;bndl = objc.loadBundle('CoreGraphics', globals(), '/System/Library/Frameworks/ApplicationServices.framework');objc.loadBundleFunctions(bndl, globals(), [('CGWarpMouseCursorPosition', 'v{CGPoint=dd}')]);CGWarpMouseCursorPosition((" &amp; 1002 &amp; "," &amp; 222 &amp; "));\""         set rgbColors to words of (the clipboard)     end tell     if rgbColors is equal to {"77", "39", "38"} then         exit repeat     else         do shell script "echo " &amp; quoted form of n &amp; " &gt;&gt; /Users/oneone/Desktop/base2.txt"     end if   Only the color check condition for Digital Color Meter is presented here.  Through the do shell, the color is read at the mouse position and the variable is added to the new file. 14.04 --> 16.04 failed; apt (>= 1.0.1ubuntu2.13)' is not installed (This question was not resolved. Due to time constraints Computer was restored to factory settings.)  I was attempting to upgrade from 14.04 to 16.04, but apt threw this error:  The required dependency 'apt (&gt;= 1.0.1ubuntu2.13)' is not installed.   What does this mean, and how can I fix it?  When I type in the command apt-cache policy apt  Shageenth@shageenth-Inspiron-3451:~$ apt-cache policy apt apt: Installed: 1.0.1ubuntu2.11 Candidate: 1.0.1ubuntu2.11 Version table: *** 1.0.1ubuntu2.11 0 100 /var/lib/dpkg/status 1.0.1ubuntu2 0 500 http://us.archive.ubuntu.com/ubuntu/ trusty/main amd64  Packages   The error happens when I run the command:   sudo do-release-upgrade -d   and it produces the following:  Checking for a new Ubuntu release Get:1 Upgrade tool signature [198 B] Get:2 Upgrade tool [1,265 kB] Fetched 1,265 kB in 0s (0 B/s) authenticate 'xenial.tar.gz' against 'xenial.tar.gz.gpg' extracting 'xenial.tar.gz'  Reading cache  Checking package manager Reading package lists... Done Building dependency tree Reading state information... Done Building data structures... Done  Required depends is not installed  The required dependency 'apt (&gt;= 1.0.1ubuntu2.13)' is not installed. Random images appearing in my home directory... Where are they coming from? Here's an odd problem I've had for a while. Every so often, I find that I have several jpeg images sitting in my home directory that I didn't put there. I'm fairly sure this isn't a security breach but rather, there's something writing these files in there that I've forgotten about.  Here's what I'm looking at today:  oli@bert:~$ ls -l *.jpg -rw------- 1 oli oli  8061 2010-11-19 18:30 An-alien-from-Mars-Attack-002.jpg -rw------- 1 oli oli 24362 2010-11-19 18:30 An-alien-from-Mars-Attack-006.jpg -rw------- 1 oli oli  8147 2010-11-05 18:08 Avatar-Film---2009-002.jpg -rw------- 1 oli oli 35602 2010-11-05 18:08 Avatar-Film---2009-006.jpg -rw------- 1 oli oli  6058 2010-12-03 18:42 Britney-Spears-victim-of--002.jpg -rw------- 1 oli oli 28330 2010-12-03 18:42 Britney-Spears-victim-of--006.jpg -rw------- 1 oli oli  8135 2010-11-12 19:19 charlie-brooker-twitter-p-002.jpg -rw------- 1 oli oli 22115 2010-11-12 19:19 charlie-brooker-twitter-p-006.jpg -rw------- 1 oli oli  8059 2010-10-29 18:47 Leonardo-DiCaprio-002.jpg -rw------- 1 oli oli 29988 2010-10-29 18:47 Leonardo-DiCaprio-006.jpg -rw------- 1 oli oli  8064 2010-10-22 16:41 Spot-the-difference-.-.-.-002.jpg -rw------- 1 oli oli 29188 2010-10-22 16:41 Spot-the-difference-.-.-.-006.jpg   They all seem like they're TV or film related... But I honestly don't have a clue what could be writing them. They don't bother me enough to go on a full-scale witch hunt but if anybody knows of something that writes images in the home directory like this, please let me know. How do "Dummy Variables" work? I do not understand how dummy variables work in math.   Suppose we have:  $$I_1 = \int_{0}^{\infty} e^{-x^2} dx$$  How is this equivalent to:  $$I_2 = \int_{0}^{\infty} e^{-y^2} dy$$  How does this dummy variable system work?   Since $y$ is the dependent variable for $I_1$ How can $y$ itself be and independent variable for $I_2$  ??  Thanks! One problem I have found when learning calculus is that there are many different ways to denote the derivative. If $y=f(x)=x^2$, then we could write \begin{align} f'(x)&amp;=2x \\ y'&amp;=2x \\ \frac{df}{dx}(x)&amp;=2x \\ \frac{df(x)}{dx}&amp;=2x \\ \frac{d}{dx}f(x)&amp;=2x \\ \frac{dy}{dx}&amp;=2x \end{align} And this is just Lagrange and Leibniz's notations alone. What I find troubling is that they all seem to be suggesting subtly different things about what the derivative actually is. Is it a function, a limit of a quotient, or both? In the interests of keeping my post brief, I'll focus my attention on $f'(x)=2x$ and $\frac{dy}{dx}=2x$, as these seem to be the most common notations.  $$ f'(x)=2x $$  It does make sense to think of the derivative as the gradient function: $$ f'\colon x\mapsto\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x} $$ In this case the limit expression is equal to $2x$, and so we can write $$ f' \colon x \mapsto 2x $$ However, this notation seems a little counter-intuitive when we consider what it means to differentiate a function with respect to a variable other than $x$. If I ask what is the derivative of $f(x)$ with respect to $\frac{x}{2}$, does this question make sense? Is it simply $f'(\frac{x}{2})$? Or do we have to express $x^2$ in terms of $\frac{x}{2}$? And how can we can express this derivative using Lagrange's notation?  $$ \frac{dy}{dx}=2x $$  There are many things which are nice about Leibniz's notation, including the fact that it is explicit which variable you are differentiating with respect to. However, in this case, it is unclear whether we are talking about a function, or something else entirely. There are other issues. Some people say they dislike the Leibniz formulation of the chain rule $$ \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx} $$ saying that they find it to be inaccurate. I don't really understand why this is the case. Could someone please elaborate? The polymorph spell states, in part:     The target’s game statistics, including mental ability scores, are   replaced by the statistics of the chosen beast. It retains its   alignment and personality.   Does retaining its personality suffice to let it remember who its allies are?  Even if transformed into something stupid like a T-rex?  What does it even mean for a beast with Int 2 to have someone's "personality"? What is the difference between <%# and <%= in ASP.NET MVC? Please help solve 'oracle-java7-installer' missing. Error below  dpkg: warning: files list file for package !['oracle-java7-installer' missing;][1]  assuming package has no files currently installed (Reading database ... 302933 files and directories currently installed.) Removing linux-image-extra-3.8.0-19-generic ... /var/lib/dpkg/info/linux-image-extra-3.8.0-19-generic.postrm: 6: /var/lib/dpkg/info/linux-image-extra-3.8.0-19-generic.postrm: Syntax error: ")"  unexpected dpkg: error processing linux-image-extra-3.8.0-19-generic (--remove):  subprocess installed post-removal script returned error exit status 2  Errors were encountered while processing:  linux-image-extra-3.8.0-19-generic  E: Sub-process /usr/bin/dpkg returned an error code (1) A package failed to install.  Trying to recover: Don't show pending suggested edits to me after I run out of votes How many RSA keys before a collision? I was wondering how many possible private/public keys exist? If a million people – for whatever reason – would try to generate 5 keys each in the same minute (on the same date and time) is there a high chance of collision? I believe GUID would suffer from that problem as many bits are reversed for date/time (and GUID version) and isn't meant to be used in that way.  Would RSA suffer from collisions if many keys were to be generated in the same moment? Is the amount of possible keys known? I know RSA is based on prime numbers and small numbers are to be rejected. I’m sure values above a certain amount of digits/bits are rejected because software may not be able to support those large values?  So: How many RSA keys before a collision? And if you would try to make many at the same time, would that give you a high chance of collision?