How can I use this fact to show that f(y)=y for every real number y, given that f is continuous
Prove that any continuous map on $\mathbb{R}$ which is the identity on $\mathbb{Q}$ is the identity map, hence Aut$(\mathbb{R}/\mathbb{Q})= 1.$  proof: Suppose $\sigma$ is continuous on $\mathbb{R}$ which is an identity on $\mathbb{Q}$. let $a \in R $. Then since $\sigma$ is continuous, then by definition there is $\delta_1 &gt; 0$ such that $|a - b| &lt; \delta_1$.  And assume $|\sigma(a) -\sigma(b)| &lt; \epsilon/2$ .  Let $\delta $= min{$\epsilon/2, \delta_1$}.  And now let $b \in \mathbb{Q}$ such that $|a - b| &lt; \epsilon$. Then  $|\sigma(a) - a| = |\sigma(a)  - b + b - a| \leq |\sigma(a) - \sigma(b) | + |b - a| &lt; \epsilon/2 + |b-a|$.  Can someone please help me? I am not sure . Can I say $|b - a| &lt; \epsilon/2 $?
How do I disable the drum beat sound on the login screen? I would have expected System ▸ Administration ▸ Login Screen ▸ Play login sound to control this, but unchecking it does not prevent the drum beat sound from playing. I've been building my own emacs because I want to fix some issues in it and get the bug fixes upstream. However, Ubuntu keeps automatically removing my own built emacs and I have to keep rebuilding and reinstalling it.  Here's my output of apt-cache policy emacs when I went to bed last night:  ~/emacs$ sudo apt-cache policy emacs emacs:   Installed: 26.0.50.aaron-1   Candidate: 46.1   Version table:      46.1 500         500 http://us.archive.ubuntu.com/ubuntu xenial/main amd64 Packages         500 http://us.archive.ubuntu.com/ubuntu xenial/main i386 Packages  *** 26.0.50.aaron-1 100         100 /var/lib/dpkg/status   Here it is when I wake up:  ~/emacs$ sudo apt-cache policy emacs [sudo] password for excelsiora: emacs:   Installed: 46.1   Candidate: 46.1   Version table:  *** 46.1 500         500 http://us.archive.ubuntu.com/ubuntu xenial/main amd64 Packages         500 http://us.archive.ubuntu.com/ubuntu xenial/main i386 Packages         100 /var/lib/dpkg/status   Why is this package being versioned at 46 instead of 24 or 25? 26 is the currently the highest major emacs version you can get.   I can see that this is a meta-package:   Why is Ubuntu updating software without asking?  Main Question: What am I supposed to do - assign my package version a number higher than 46 just to make it stick?   That seems like a horrible hack, or maybe that's how it's supposed to work? Why don't the Ubuntu repositories have the latest versions of software? Why with two classes, LDA gives only one dimension? I am working with dimensionality reduction algorithms. Linear Discriminant Analysis (LDA) is a supervised algorithm that takes into account the class label (which is not the case of PCA for example). I am using Python to do a comparative study between some algorithms.  Why with two classes (k = 2), regardless of the data dimensionality, LDA gives one dimension, i.e. the new subspace is composed of only one dimension? For example if I try to project my dataset (of 100 attributes and 2 classes) to 10-dimensional space by using LDA(n_components=10) I only obtain 1 dimension?  For k&gt;2 I can obtain the number of discriminant vectors that I specified as n_components. How can I get n_components dimensions for k=2? I hope I'm right in thinking this is the site where you post general computer-related problems...  A problem I have when playing games (since you won't be pressing multiple keys simultaneously anytime else), is that when I press up/left and then another key, it fails to 'register' as if that last key is never pressed. This doesn't happen when I'm pressing any other two arrow keys, e.g. down/left or up/right. The weird thing is that when I'm playing with WASD instead of the arrow keys, i still have this problem! Meaning W+A+another button doesn't work.  If anyone knows what I should, I'd appreciate the help. (My reflexes are fast and I can always circumvent this problem, but it's quite annoying and shouldn't exist to begin with!) How do I get a selected vertex and it's corresponding uv vertex?  A vertex I can get with  bpy.context.active_object.data.vertices[0]   A uv vertex I can get with  bpy.context.active_object.data.uv_layers['UVMap'].data[0]   But the index is not the same for uv and mesh vertex. How do comments work? Across the Stack Exchange network you may leave comments on a question or answer.  How do comments work? What are comments for, and when shouldn't I comment? Who can post comments? Who can edit comments? How can I format and link in comments? My comment doesn't contain some of the text I typed in it; why? My comment appears to be long enough, but I can't post it because it's too short; why? Who can delete comments? When should comments be deleted? Why can't I comment on specific posts even if I have enough reputation to comment? When can comments be undeleted? What are automatic comments? How can I link to comments? Anything else I should know about comments?  Also see the FAQ articles on  and . The optical density of a medium is quantified by its refractive index $n$. Does it have any correlation with the mass/number density of the medium? In other words is an optically denser medium must have a greater mass density compared to an optically rarer medium? I want to type Arabic text in Photoshop, but the text appears disjointed and the letters do not connect at all. For example محمد appears as م ح م د.   I'm using Photoshop CS6 on OSX Mountain Lion. Make HttpURLConnection load web pages with images "Communicate over" vs. "communicate through" Also schemas can communicate over database links to schemas in other remote databases.    Could one replace the preposition over with through? Will it change the meaning, and if so, when should we use over instead of through? What are the best ways to improve a site's position in the Google search results? Can Ubuntu really run all Windows programs? The ideal $I= \langle x,y \rangle\subset k[x,y]$ is not a principal ideal.   I don't know how to consider it. Any suggestions? "For security reasons, framing is not allowed; click OK to remove the frames." on Imgur upload Ok first, there's that lugubrious use of the semicolon; not necessary.  Second, I didn't open a frame!  You did!  Alls I tried to do is upload a 5MB animated .gif file via Imgur.    Looking further into this, Imgur actually posts an error when you upload an animated .gif larger than 2MB.    Go figure! $T$ self-adjoint on Hilbert space $H$, $T^n=T^{n+1}$ will $T$ be an orthogonal projection operator? I have seen several answers and proofs here at MSE that a self-adjoint operator $T$ on a Hilbert space $H$ where $T^2=T$, implies $T$ being an orthogonal projection operator.   The thought struck me; what would happen if the condition $T^2=T$ is replaced by $T^3=T^2$ or instead by $T^4=T^3$ or why not more generally by $T^n=T^{n+1}$ for some $n\in\mathbb{N}$? Would this still imply $T$ to be an orthogonal projection operator?  I guess someone must have thought of this before me? In that case perhaps there is a reference in the literature for this or perhaps the proof is way simpler than I might expect? Induction and electromagnetic fields I've got a few questions on induction and electromagnetic fields. My current understanding of induction and electromagnetic fields is that, when electricity/current flows through a wire, it creates an electromagnetic field. If you twist that wire into a coil then you can concentrate that field. You can then induce current into a different coil by placing it inside of the electromagnetic field created by the first coil.  So here are my questions:  Q1. Does the electromagnetic field contain any energy?  EDITED QUESTION  Q2. Electromagnetic fields are the same as waves, so does that mean that when current is induced, they get the energy from electromagnetic waves being exchanged, or do they induce current directly from the electromagnetic field?   Q3. If electromagnetic fields and waves are the same, then doesn't that mean that you can induce current from waves? Therefore transferring electricity over long distances?  Q4. What spectrum of light are the electromagnetic waves that are in a field created by electricity?  Q5. Finally, why is it not possible to induce current from the Earth's magnetic field? I thought Maxwell said that electric fields and magnetic fields were the same thing.   Thank you to anyone who answers.  Edited Q2: So my original question (although asked poorly) still stands. Where does the current that gets induced in the secondary coil come from? How does the electricity transfer from the primary coil to the secondary? For example, is it taken from the stored energy in the e.m field? Does the field just act as a bridge between the 2 coils? Or do the 2 coils exchange e.m waves which carry the electricity that is being transferred from coil 1 to coil 2? Or is it none of these things and something completely different? I want to implement the EM algorithm manually and then compare it to the results of the normalmixEM of mixtools package. Of course, I would be happy if they both lead to the same results. The main reference is Geoffrey McLachlan (2000), .  I have a mixture density of two Gaussians, in general form, the log-likelihood is given by (McLachlan page 48):      $$ \log L_c(\Psi) = \sum_{i=1}^g \sum_{j=1}^n z_{ij}\{\log \pi_i + \log f_i(y_i;\theta_i)\}. $$ The $z_{ij}$ are $1$, if the observation was from the $i$th component density, otherwise $0$. The $f_i$ is the density of the normal distribution. The $\pi$ is the mixture proportion, so $\pi_1$ is the probability, that an observation is from the first Gaussian distribution and $\pi_2$ is the probability, that an observation is from the second Gaussian distribution.   The E step is now, calculation of the conditional expectation:      $$ Q(\Psi;\Psi^{(0)}) = E_{\Psi(0)}\{\log L_c(|\Psi)|y\}. $$ which leads, after a few derivations to the result (page 49):      \begin{align} \tau_i(y_j;\Psi^{(k)}) &amp;= \frac{\pi_i^{(k)}f_i(y_j;\theta_i^{(k)}}{f(y_j;\Psi^{(k)}}  \\[8pt]                        &amp;= \frac{\pi_i^{(k)}f_i(y_j;\theta_i^{(k)}}{\sum_{h=1}^g \pi_h^{(k)}f_h(y_j;\theta_h^{(k)})} \end{align} in the case of two Gaussians (page 82):      $$ \tau_i(y_j;\Psi) = \frac{\pi_i \phi(y_j;\mu_i,\Sigma_i)}{\sum_{h=1}^g \pi_h\phi(y_j; \mu_h,\Sigma_h)} $$ The M step is now the maximization of Q (page 49):      $$ Q(\Psi;\Psi^{(k)}) = \sum_{i=1}^g\sum_{j=1}^n\tau_i(y_j;\Psi^{(k)})\{\log \pi_i + \log f_i(y_j;\theta_i)\}. $$ This leads to (in the case of two Gaussians) (page 82):      \begin{align} \mu_i^{(k+1)} &amp;= \frac{\sum_{j=1}^n \tau_{ij}^{(k)}y_j}{\sum_{j=1}^n \tau_{ij}^{(k)}}  \\[8pt] \Sigma_i^{(k+1)} &amp;= \frac{\sum_{j=1}^n \tau_{ij}^{(k)}(y_j - \mu_i^{(k+1)})(y_j - \mu_i^{(k+1)})^T}{\sum_{j=1}^n \tau_{ij}^{(k)}} \end{align} and we know that (p. 50)      $$ \pi_i^{(k+1)} = \frac{\sum_{j=1}^n \tau_i(y_j;\Psi^{(k)})}{n}\qquad (i = 1, \ldots, g). $$ We repeat the E, M steps until $L(\Psi^{(k+1)})-L(\Psi^{(k)})$ is small.        I tried to write a R code (data can be found ).    # EM algorithm manually # dat is the data  # initial values pi1       &lt;-  0.5 pi2       &lt;-  0.5 mu1       &lt;- -0.01 mu2       &lt;-  0.01 sigma1    &lt;-  0.01 sigma2    &lt;-  0.02 loglik[1] &lt;-  0 loglik[2] &lt;- sum(pi1*(log(pi1) + log(dnorm(dat,mu1,sigma1)))) +               sum(pi2*(log(pi2) + log(dnorm(dat,mu2,sigma2))))  tau1 &lt;- 0 tau2 &lt;- 0 k    &lt;- 1  # loop while(abs(loglik[k+1]-loglik[k]) &gt;= 0.00001) {    # E step   tau1 &lt;- pi1*dnorm(dat,mean=mu1,sd=sigma1)/(pi1*dnorm(x,mean=mu1,sd=sigma1) +            pi2*dnorm(dat,mean=mu2,sd=sigma2))   tau2 &lt;- pi2*dnorm(dat,mean=mu2,sd=sigma2)/(pi1*dnorm(x,mean=mu1,sd=sigma1) +            pi2*dnorm(dat,mean=mu2,sd=sigma2))    # M step   pi1 &lt;- sum(tau1)/length(dat)   pi2 &lt;- sum(tau2)/length(dat)    mu1 &lt;- sum(tau1*x)/sum(tau1)   mu2 &lt;- sum(tau2*x)/sum(tau2)    sigma1 &lt;- sum(tau1*(x-mu1)^2)/sum(tau1)   sigma2 &lt;- sum(tau2*(x-mu2)^2)/sum(tau2)    loglik[k] &lt;- sum(tau1*(log(pi1) + log(dnorm(x,mu1,sigma1)))) +                 sum(tau2*(log(pi2) + log(dnorm(x,mu2,sigma2))))   k         &lt;- k+1 }   # compare library(mixtools) gm &lt;- normalmixEM(x, k=2, lambda=c(0.5,0.5), mu=c(-0.01,0.01), sigma=c(0.01,0.02)) gm$lambda gm$mu gm$sigma  gm$loglik   The algorithm is not working, since some observations have the likelihood of zero and the log of this is -Inf. Where is my mistake? If column matches another file, print every line with match (awk/grep) I am facing problem to enable wifi on my Lenovo G50-30 Below are the command out put please help me out   lspci | grep -i wireless:  02:00.0 Network controller: Realtek Semiconductor Co., Ltd. RTL8723BE PCIe Wireless Network Adapter     nm-tool:  NetworkManager Tool  State: disconnected  - Device: eth0    Type:              Wired   Driver:            r8169   State:             unavailable   Default:           no   HW Address:        28:D2:44:81:36:0B    Capabilities:     Carrier Detect:  yes    Wired Properties     Carrier:         off   - Device: wlan0   Type:              802.11 WiFi   Driver:            rtl8723be   State:             unavailable   Default:           no   HW Address:        14:2D:27:21:8D:47    Capabilities:    Wireless Properties     WEP Encryption:  yes     WPA Encryption:  yes     WPA2 Encryption: yes    Wireless Access Points      nmcli nm:  RUNNING         STATE           WIFI-HARDWARE   WIFI       WWAN-HARDWARE   WWAN       running         disconnected    disabled        disabled   enabled         disabled       sudo rfkill list:  0: ideapad_wlan: Wireless LAN     Soft blocked: no     Hard blocked: yes 1: ideapad_bluetooth: Bluetooth     Soft blocked: no     Hard blocked: yes 2: hci0: Bluetooth     Soft blocked: no     Hard blocked: no 3: phy0: Wireless LAN     Soft blocked: no     Hard blocked: no