From Finan:  Let $\sigma\in{S_n}$, define the $\mathbf{order}$ of $\sigma$ to be the smallest positive integer such that $\sigma^m=(1)$.  Prove that if $\sigma$ has order $m$ then $\tau\sigma\tau^{-1}$ has order $m$ for all $\tau\in{S_n}$.  So $\tau$ can have its own order, so let $order(\tau)=p$.  Thus $\tau^p=(1)$.  If $(m,p)=1$, then $order(\tau\sigma)=mp$.  But doesn't $order(\tau^{-1})=p$?  Thus $order(\tau\sigma\tau^{-1})=pmp$.  So them this implies $(\tau\sigma\tau^{-1})^{pmp}$...  Not sure where to go with this.
I dont understand the following very simple statement:  If $\tau \in S_n$ has order $m$, then $\sigma \tau\sigma^{-1}$ has also order $m$.  The proof is:  Suppose $\tau$ has order $m$.   $(\sigma \tau \sigma^{-1})^{m} = \sigma \tau^{m}\sigma^{-1}= (1)$  Then (using  exercise 17) they say that the order of $\sigma\tau\sigma^{-1}$ cannot be less than $n$ (remember we are in $S_n$), since $(\sigma\tau\sigma^{-1})^k=1 $ implies $\sigma\tau^k\sigma^{-1} =(1)$, and then $\tau^k=\sigma^{-1} \sigma = (1)$  I don't understand the last part.... Who can help me?
Rename all the files within a folder with prefix “Unix_” i.e. suppose a folder has two files a.txt and b.pdf than they both should be renamed from a single command to Unix_a.txt and Unix_b.pdf I know Newton's third law of motion might be the answer for this but still I am wondering how the rockets could thrust in the empty space and move in the opposite direction. I guess an astronaut wouldn't be able to push in the empty space with his hands or legs to move himself, but with a rocket engine it's possible. How? What might be the explanation for this in General Relativity? I thought of this a while ago, but can't make up a proof or a counterexample. Does anyone know more about this?  $$\sqrt{a+b}  \le \sqrt a + \sqrt b , \forall a,b \in \mathbb R_+$$  Moreover, what happens with more variables? Say:  $$\sqrt{x_1+x_2+...+x_n} \le \sqrt x_1 +\sqrt x_2 + ... + \sqrt x_n $$  with $ x_i \in\mathbb R_+ \forall i \in \{1,2,...,n\} $  Or when $i = \mathbb N$  PS: I tried looking on the internet for this but I don't know how I would call this. Using single quote by curly braces in String.format in Dynamic SOQL I'm struggling to use String.format to replace a token with a value when the token is within single quotes:  Example:  String query = String.format('SELECT Id FROM Availability WHERE Person__c = \'{0}\'', personId);   String query is equal to:  SELECT Id FROM Availability WHERE Person__c = {0}   Rather than:  SELECT Id FROM Availability WHERE Person__c = '234567890' Why is there conservation of kinetic energy in elastic collision and not in inelastic collision? It seems that KNN is a discriminative learning algorithm but I can't seem to find any online sources confirming this.  Is KNN a discriminative learning algorithm? I have this HW where I have to calculate the $74$th derivative of $f(x)=\ln(1+x)\arctan(x)$ at $x=0$. And it made me think, maybe I can say (about $\arctan(x)$ at $x=0$) that there is no limit for the second derivative, therefore, there are no derivatives of degree grater then $2$. Am I right? I'm an idiot, I was writing an image to a USB drive using dd when instead of typing /dev/sdb I typed /dev/sdd and wrote the 135mb image to my 750gb External Hardrive full of files. Is there ANY possibility of recovering the files or did dd completely zero out the drive? Please help, as this is one of the lowest points of my life EVER.  dd finished writing the 135mb image in 5 seconds, so I didn't have a chance to stop it. By the time I knew what happened to was too late. The command was as follows:   sudo dd if=generic.img of=/dev/sdd bs=1M   EDIT2: The External Harddrive's filesystem was FAT32(I've had the drive for awhile and FAT32 was the most Universal FS at the time). I also have a 2TB drive I just got which I was planning on moving most of the files to from the 750gb, If needed I can use it to transfer the Recovered files to. For some reason, when I try to run any programs (I am trying to run a rootkit hunter: rkhunter) as root on the command line with sudo like this:  sudo rkhunter   Then I get this message:  You must be the root user to run this program   and then I get these lines:  sudo: error in /etc/sudo.conf, line 0 while loading plugin 'sudoers_policy' sudo: /usr/lib/sudo/sudoers.so must only be writable by owner sudo: fatal error, unable to load plugins   I try typing in just sudo but that comes up with the same three errors as above. Also in /etc/ sudo.conf does not exist.  I cannot run any other programs as root through sudo. It was also strange because I could run as root a day ago. I would like it to work.  Just so you know, I am the only user of this computer, and the administrator, running Lubuntu 14.04.1. At the end of this nice video (), she says that  is a chain reaction of electric and magnetic fields creating each other so the chain of wave moves forward.  I wonder where the  is in this explanation. What is the relation between electromagnetic wave and photon? Is there a way to determine what version (distribution &amp; kernel version, I suppose) of Linux is running (from the command-line), that works on any Linux system? Is there any way to get real-time compilation for LaTeX? I would really love to have a software or tool which provided real time compilation of my LaTeX document (to be able to see the final pdf or dvi document as I type the latex code), especially for when I am drawing pictures using tikZ. So far, I know of two things, but none of them work for me. They are:   : This is only for tikZ, which would be perfect for me, but unfortunately as far as I know it only works with Ubuntu and Debian, and I use Mac OS X (so I have never actually tried it, though I did dig a lot to see whether I would be able to install on my mac and from what I see it is really what I want - I am even considering installing Ubuntu just to use it!). : This is a minor mode for Emacs. It was extremely hard to get it to work on the mac (see my quest ), and it is not all that fast. The speed is not all that big of a deal for me, but the fact that it does not work with tikZ pictures is (the nodes all collapse in one point so that the words are all one on the top of the other).   So my question is: is there a software ideally like KtikZ or another type of editor which provides real-time compilation and that I can use on my mac? I guess if you know of a Windows/Linux option I would also like to know, and the same for a way to set up a script to get the document compiled every 5 seconds or so (I am sure I have seen this written somewhere, but I don't know whether I can do it with a mac or whether it is editor/pdf viewer dependent), but what I really want is a software/tool which would work with a mac.  To be clear, I am not after something like LyX, that is, I am not after a WYSIWYG-type thing, but rather something where I can type real LaTeX and see my code, but have at the same time another window showing me the pdf (or dvi) file compiled. Why is apt-get always using proxy, although no proxy is configured? Probability of sampling with and without replacement In sampling without replacement the probability of any fixed element in the population to be included in a random sample of size $r$ is $\frac{r}{n}$. In sampling with replacement the corresponding probability is $\left[1- \left(\frac{1}{1-n}\right)^r\right]$.  Please help me show how this is proved. For instance if me and my friend had a Pidgeot and we powered them all the way up. Could mine end up with like 525 and his at like 500? Comparing two Lists and returning the distinct values and the differences More and more I see, especially in activist communities, Spanish-derived words ending in x or @ in order to neuter the gendering inherent in the original language. For example: latinx (or latin@) as opposed to latina and latino. How, though, are these endings pronounced? Please provide evidence of how this orthography is most commonly pronounced. Note 1: I recognize this is in an English Language site, rather than a Spanish language site, but this phenomenon is very much happening in English, albeit with Spanish loan words. Therefore the question is relevant to non-Spanish-speaking English speakers. Note 2: A question like this can lead to politically charged or, at least, sarcastic comments. Please refrain from bringing in your biases for or against. Note 3: This question is not seeking opinions as to how some people pronounce the suffixes. Answers should address evidence of a developing orthodoxy in pronunciation. Auto update specifick plugin I've created a plugin and I want to auto-update this plugin.  I've been looking in the WordPress Codex and found this following piece of code:  add_filter( 'auto_update_plugin', '__return_true' );   I added this code into a new Must-have Plugin but it doesn't work. It does not auto-update all my plug-ins..  Anyone that have experience with auto updating? I am reading through Folland's Real Analysis: Modern Techniques and Their Applications, and they have the following proposition and proof:     Proposition: If $\{f_{j}\}$ is a sequence of $\bar{\mathbb{R}}$- valued measurable functions on $(X, \mathcal{M})$, then the functions:      $$g_{1}(x)=\sup_{j} f_{j}(x)$$ $$g_{2}(x)=\inf_{j} f_{j}(x)$$ are   measurable.       Proof:      We have       $$g_{1}^{-1}((a,\infty])= \bigcup_{j=1}^{\infty}  f_{j}^{-1}((a,\infty])$$ $$g_{2}^{-1}([-\infty,a))=  \bigcup_{j=1}^{\infty} f_{j}^{-1}([-\infty,a))$$      Where the result follows from the fact that $g_{1}^{-1}((a,\infty])\in  \mathcal{M}$ for all $a \in \mathbb{R} \iff g$ is measurable, and   $g_{2}^{-1}([-\infty,a))\in \mathcal{M}$ for all $a \in \mathbb{R}  \iff g$ is measurable.   I understand the last part of the proof, but how do we know $g_{1}^{-1}((a,\infty])= \bigcup_{j=1}^{\infty}f_{j}^{-1}((a,\infty])$ and $g_{2}^{-1}([-\infty,a))=\bigcup_{j=1}^{\infty} f_{j}^{-1}([-\infty,a))$? I would appreciate any help with this. I added some extra repositories with the Software Sources program. But when I reload the package database, I get an error like the following:     W: GPG error:  trusty InRelease: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY 8BAF9A6F   I know I can fix it using apt-key in a terminal, according to the official Ubuntu documentation. But I would have liked to do it graphically. Is there a way to do this without using a terminal?