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Does this seemingly elementary question require König's theorem? Let $f\colon X\to\overline{\mathbb{R}}_{\geq 0}$ be a measurable function. Show that     $$ \int f\, d\mu=\int\limits_0^{\infty}\mu(\left\{x\in X: f(x)&gt;t\right\})\, dt. $$     (The right integral is to be read as an Lebesgue-integral.)      Hello, my idea is to approximate $f$ by the functions $$ f_n:=2^{-n}\sum\limits_{k=1}^{\infty}\chi_{\left\{x\in X: f(x)&gt; k/2^n\right\}}. $$  With this approximation by simple functions I got $$ \int\limits_X f\, d\mu=\lim\limits_{n\to\infty}\int\limits_X f_n\, d\mu=\lim\limits_{n\to\infty}\sum\limits_{k=1}^{\infty}\frac{k}{2^n}\cdot\mu\left(\left\{x\in X: f_n(x)=k/2^n\right\}\right)\\=\lim\limits_{n\to\infty}\sum\limits_{k=1}^{\infty}\frac{k}{2^n}\cdot\mu\left(\left\{x\in X:f(x)&gt;k/2^n\right\}\right) $$  This is the point where I do not come along anymore... is that right to this point and if yes: How can I continue in order to get the desired right side?  With regards PHP generate file for download then redirect I have a bunch of raw data values that are dollar amounts and I want to find a confidence interval for a percentile of that data.  Is there a formula for such a confidence interval? FastBoot recovery failed: partition 'recovery' not support flash How to find the square root of a permutation Observe that  $$\begin{pmatrix} 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\  2 &amp; 4 &amp; 1 &amp; 5 &amp; 3 \end{pmatrix} \begin{pmatrix} 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\  2 &amp; 4 &amp; 1 &amp; 5 &amp; 3 \end{pmatrix} =  \begin{pmatrix} 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\  4 &amp; 5 &amp; 2 &amp; 3 &amp; 1 \end{pmatrix},$$  so $\begin{pmatrix} 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\  4 &amp; 5 &amp; 2 &amp; 3 &amp; 1 \end{pmatrix}$ is the square of a permutation. More generally, how do I check if a permutation is square and how can I find its root? problems with ellipse TikZ I'm working on a mac, I use the package TikZ and it works. But I have problems when I want use ellipse or diamond ex :  \documentclass{article} \usepackage{graphicx} \usepackage{tikz} \begin{document} \begin{tikzpicture} \node[ellipse] {Ellipse}; \end{tikzpicture} \end{document}   If somebody could help me thank you ! Statistical inference when the sample "is" the population How to create and design hoops net and attach ot to a hoop? I'tm trying to show in SQLcmd window the results of these queries:  SELECT SYSDATETIME() AS "date";  PRINT 'EXTRAÍDO POR:' SELECT SUSER_NAME() AS "Owner"  PRINT 'PASSWORD SETTINGS'; exec xp_loginconfig;  PRINT 'PASSWORD SETTINGS'; select * from sys.sql_logins;   And using this command in SQLcmd:  sqlcmd -S myServer -i inputpath -s, -o outputPath(like C:\log.xls) -e   So the sqlcmd output the file and does not show nothing in sqlcmd screen. How can i print these "Print" commands in the sqlcmd screen? (command promt).  When I run it on SSMS works well, like:  (1 row affected) OWNER:  (1 row affected) PASSWORD SETTINGS  (8 rows affected) PASSWORD SETTINGS  I would like to show in the prompt command this, like in the SSMS.  Thanks! Why are the longer edges being beveled more? Integral $\int\csc^3{x} \ dx$ I found these step which explain how to integrate $\csc^3{x} \ dx$. I understand everything, except the step I highlighted below.  How did we go from: $$\int\frac{\csc^2 x - \csc x \cot x}{\csc x - \cot x}\,dx%$$ to  $$\int \frac{d(-\cot x + \csc x)}{-\cot x + \csc x} \quad?$$  Thank you for your time! $$ \int \csc^3 x\,dx = \int\csc^2x \csc x\,dx$$ To integrate by parts, let $dv = \csc^2x$ and $u=\csc x$. Then $v=-\cot x$ and $du = -\cot x \csc x \,dx$. Integrating by parts, we have: $$\begin{align*} \int\csc^2 x \csc x \,dx &amp;= -\cot x \csc x - \int(-\cot x)(-\cot x\csc x\,dx)\\ &amp;= -\cot x \csc x - \int \cot^2 x \csc x\,dx\\ &amp;= -\cot x\csc x - \int(\csc^2x - 1)\csc x\,dx &amp;\text{(since }\cot^2 x = \csc^2-1\text{)}\\ &amp;= -\cot x \csc x - \int(\csc^3 x - \csc x)\,dx\\ &amp;= -\cot x\csc x - \int\csc^3 x\,dx + \int \csc x\,dx \end{align*}$$ From $$\int \csc^3 x\,dx = -\cot x\csc x - \int\csc^3 x\,dx + \int \csc x\,dx$$ we obtain $$\begin{align*} \int\csc^3x\,dx + \int\csc^3 x\,dx &amp;= -\cot x \csc x + \int\csc x\,dx\\ 2\int\csc^3 x\,dx &amp;= -\cot x\csc x + \int\csc x\,dx\\ \int\csc^3x\,dx &amp;= -\frac{1}{2}\cot x\csc x + \frac{1}{2}\int\csc x\,dx\\ &amp;=-\frac{1}{2}\cot x\csc x + \frac{1}{2}\int\frac{\csc x(\csc x - \cot x)}{\csc x - \cot x}\,dx\\ &amp;= -\frac{1}{2}\cot x \csc x + \frac{1}{2}\int\frac{\csc^2 x - \csc x\cot x}{\csc x - \cot x}\,dx\\ &amp;= -\frac{1}{2}\cot x \csc x + \frac{1}{2}\int\frac{d(-\cot x+\csc x)}{-\cot x +\csc x}\\ &amp;= -\frac{1}{2}\cot x\csc x + \frac{1}{2}\ln|\csc x - \cot x|+ C \end{align*}$$ Can I transfer saved data from my old PS3 hard drive to my new PS3? A while ago, my PS3 completely broke. Every time I turn it on, it just completely shuts down a few seconds later. I decided to buy a new ps3 to play my old games. The problem is that I don't have my saved data, but I still have the original hard drive. Is it possible to somehow transfer the saved data from the old hard drive to my new PS3? I was thinking maybe I could clone the hard drive, but I don't have a spare hard drive to do it on. I do however have a 2 TB external hard drive. Would it be possible to clone it on the external? Are there any alternative methods? How to solve configure: error: readline library not found? I have an Ubuntu 10.04 LTS server with a minimal install that I want to compile postgres on. With the minimal install the server did not have gcc so I had to do an apt-get install gcc but now I keep getting this error from the postgres ./configure.  configure: error: readline library not found   I can't seem to find the name of the package that contains the readline library. So I have two questions.   What is the name of the package that contains readline? Is there some way given the error message above for me to find the associated package from its standard name? Input 0  echo foo | xargs -L 1 -I '{}' echo '{}'   Output 0  foo   Input 1  echo foo | xargs -I '{}' -L 1 echo '{}'   Output 1  {} foo   Why changing the order of options of xargs changes the output?  Version: xargs (GNU findutils) 4.6.0 Definition of "pepper" in hash functions Usage of "as if" as interjection My question concerns the usage of "as if" as an idiomatic interjection. The references I have consulted (etymonline, online dictionaries, etc.) seem to agree that "as if" is used in this context to convey incredulity or doubt. This is how I have always thought of it.  The other day, I was with friends who used "as if" to express their surprise and joy about a wedding proposal. They weren't being sarcastic or incredulous; they were genuinely happy about the proposal. Here is a loosely paraphrased version of what they said:  "As if they're getting married!"  Is my friends' usage of "as if" standard? Is it a primarily regional distinction? (I am in Ontario, Canada right now.) The closest example we could come up with is someone says "I can't believe that you won first prize!" Of course, the person can and does believe it, so their words ought not to be taken literally. Is it the same idea with "as if"? Getting the class name of an instance? create multilingual node programmatically i'm trying to create a multilingual node using a php script.  I'm doing this:   $node = new stdClass();  $node-&gt;title = "Test ".mktime();  $node-&gt;type = "job";  $node-&gt;language = 'de'; // Or e.g. 'en' if locale is enabled  $node-&gt;uid = $user-&gt;uid;   $node-&gt;status = 1; //(1 or 0): published or not  $node-&gt;promote = 0; //(1 or 0): promoted to front page  $node-&gt;comment = 1; //2 = comments on, 1 = comments off  $node-&gt;path = array('alias' =&gt; 'das-ist-ein-test-'.mktime());  $node-&gt;body[$node-&gt;language][0]['value']   = "Body Body Body";  $node-&gt;body[$node-&gt;language] = text_summary("Body Body Body");  $node-&gt;body[$node-&gt;language]  = 'filtered_html';  node_save($node);   As long as i set $node->language to 'und' everything is working fine. But as soon as I set the language to 'de' or 'en' (both languages are activated in my Drupal installation) the body text doesn't get saved.  So my question is: can anybody provide me with a working example of how to store a multlingual node (de,en) into Drupal using program code.  Thank you Login screen loops unless you login as Guest I am running a VMWare Player with a Ubuntu 12.04 LTS Precise Pangolin as Guest on my Windows 7. Sometimes I get the shutdown blue screen error in Windows, this time it happened when I was running the Player. When I restarted everything Ubuntu gave me the &quot;Login Loop&quot; on adminstrator login. I log in and there's this black screen where I can only read:  piix4...smbus:0.0.0.07.3 Host Smbus controller not enabled .    When I go to the prompt in root mode it fails to update and only upgraded, specially some plugins ( I think graphic plugins) which also appear in one an error message after quitting the prompt, but they´are successfully installed. They are not the error message. After that I have been working with the Fail/safe Mode recovery panel. When I try to update via Root I get errors like this: W:failed to fetch http://extras.ubuntu.com/ubuntu/dists/precise/release.gpg could not        resolve 'extras/ubuntu.com  There are 8 more like this referring to areas like: -archive/canonical.com -ppa.Launchpad.net -security.Ubuntu.com -Us.archive.ubuntu.com - release.gpg                         precise-updates/release.gpg                         precise_backport/release.gpg   Final Message: some index files failed to download.....they have been ignored or old files are used. The black screens most of the time pass by too fast for me to pick up any information. But in general I think I have done everything I was able to in the recovery panel including updating network and graphic packages and recovering filesystem packages and the basic stuff ( I am a beginner regarding Linux ) in the root prompt. Now I am stuck in this screen with graphic options: - Run in low-graphics mode just for one session - Reconfigure Graphics - Troubleshoot the error - Exit to console login  I am trying to choose to reconfigure graphics but the mouse disappears in the virtual machine screen and sometimes when options change it´s only the first and last option. But this happens from the blue without messages. This particular option menu is in the regular GUI style against a black screen in Terminal style. Really strange.