Prove that $GL_2(\mathbb{R})$ is open in$ M_2(\mathbb{R})$    My solution : Define the map $\phi:M_2(\mathbb{R})\rightarrow \mathbb{R}$ by sending $A$ to $\det(A)$ .  I guess $\phi$ is continuous . As $\mathbb{R}-\{0\}$ is open  $GL_2(\mathbb{R}) =\phi^{-1}(\mathbb{R}-\{0\})$ is open also open .   I would like to verify if my solution is correct .
let $n\gt1$  be a fixed natural number, $S:=${$A $: $M_n (\mathbb{R})$ be all real matrix,define this meter for all $A=[a_{ij}]$ $B=[b_{ij}]$   $d(A,B)$:=max{|$a_{ij}-b_{ij}$|:i,j=1,2,2...,n} and $GL(n,\mathbb R)$ is set of all $n\times n$ non singular matrix how to prove that $GL(n,\mathbb R)$ is  not connected and open subset of $S$ ?  thanks in advance
How is the optimal probability cut-off in a ROC defined by the R package Epi? The plot below was created by R package Epi::ROC for a binary classification problem. There is an optimal cut off or threshold for the classification probability at which sensitivity and 1-specifitiy are balanced in some optimal way (called ps3 in the plot). My question is how this point is defined mathematically in this case and if there are other ways to define it. Who was the colleague that Erik was going to contact for help? After S.H.I.E.L.D confiscates Jane's work in Thor, the researchers discuss what they'll do next.  Erik mentions two colleagues. The first is reasonably implied to be Bruce Banner, but who is the second colleague he mentions? Transcribed from the film:  Jane Who are these people? Erik I knew this scientist... a pioneer in gamma radiation. S.H.I.E.L.D showed up and... umm. He wasn't heard from again. Jane They're not going to do that to us. I'm going to get everything back. Erik mumbles Please. Let me contact one of my colleagues. He's had some dealings with these people before. I'll email him and maybe he can help. Darcy They took your laptop, too. Ugly 'post date' alignment on user profile when the date wraps into multiple lines I have a sentence that contains a number of long dataset names. I would like to allow LaTeX to break these names at certain positions that I would like to specify somehow just like I can use \- to allow a word break with hyphenation. But: I would like LaTeX not to insert a dash (which could be mistaken as belonging to the dataset name). How can I do this?   (Something similar is realized in the url package so I guess I could just read that code :) I would not like to use the url package here to be able to make a syntactical (and typographic) difference between urls and datset names.) What is the proper way to explain the twin paradox? How do I create a very long delay with Redstone? The only things I've found so far are a cart method (which ceased working in 1.6), a dispenser method (utilizing the decay on dropped items, but this requires occasional feeding), and a binary counter (which won't work for a single input but rather a fluctuating one).  I want to have a circuit that when I apply an input signal, the output signal doesn't change for 2 to 5 minutes.  When the input turns off, I again want a 2 to 5 minute delay.  I could accomplish this with (roughly) 240 to 600 redstone repeaters, but this is far in excess of being useful.  The only method I can think of, but would require a XOR gate (on the button-input and final output) to start and stop a clock attached to a binary counter.  While I can make room for a device of this size, I was hoping for something simpler. How to encrypt folders (not in Home)? Is there a way to encrypt a folder, which is not in Home (/home/user/) but even in a different partition, so only my user can access/read contained files?  Alternatively, I would like to understand if it is possible to turn a complete ext4 partition into an encrypted volume, which would be mounted at user login.  If possible, I would like to make the change without reinstalling Ubuntu.  My PC has (mount output):  /dev/sda1 on / type ext4 (rw,errors=remount-ro) /dev/sda3 on /home type ext4 (rw) /dev/sda4 on /home/igor/Personale type ext4 (rw)   sda4 is the partition containing folders I would like to protect. Loading OpenStreetMap data in QGIS? iOS 11-13 - unable to paste into comments I’m able to copy-paste from any application to any application. I’m able to copy from a comment or from a question or from an answer. I’m able to paste to a question or answer. However, if I attempt to paste into a comment, nothing happens. I am able to subsequently paste into another application, confirming that the text is actually on the clipboard.   However, copying SE links will paste them successfully, but all other text - including other links - still fails. This leads me to believe that it’s a SE issue and not an Apple one.   This has been an issue since 11.0 beta and persists through iOS 13 release. When do relativistic effects need to be explicitly included? Firefox browser opens but URL is not launched Product rule intuition Can anyone think of an intuitive explanation of the product rule? I'm not asking for a derivation. To me it seems like quite an un-untuitive result, as apposed to the chain-rule (which is ironically harder to derive). I think it would make sense to split up the flag "This is not an answer" into three flag choices:   This should be a comment This should be an edit This should be a new question   If we had that, it would make it easier (and therefore more efficient) for mods to figure out what to do with answers flagged thusly. Today I bought an ibook G4 from a garage sale. The people who sold it to me didn't know the password for the 2 accounts on it. Apparently it was the guys late father's laptop. Any suggestions on how to get into the laptop and set a new admin account and password? Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b&gt; 0$ and $c\geq 0$ ?   Is there a general proof for that relation ? Why can a function modify some arguments as perceived by the caller, but not others? I'm trying to understand Python's approach to variable scope. In this example, why is f() able to alter the value of x, as perceived within main(), but not the value of n?  def f(n, x):     n = 2     x.append(4)     print('In f():', n, x)  def main():     n = 1     x = [0,1,2,3]     print('Before:', n, x)     f(n, x)     print('After: ', n, x)  main()   Output:  Before: 1 [0, 1, 2, 3] In f(): 2 [0, 1, 2, 3, 4] After:  1 [0, 1, 2, 3, 4] Is there a tool in Ubuntu to automatically merge two images having a common strip? I need to scan a large image, which will not fit at once in the scanner. I was thinking about scanning it in two parts and hand-editing the result in Gimp to join them. But probably an automatic tool can do the same more quickly and better.  EDIT I think I may have not made my point clear. My difficulty is not aligning two images. My problem is that one has to compensate for the rotation or distortion introduced by the manual process of putting the sheet into the scanner, and an automatic tool should obtain more precise results at analyzing common patterns in the two images to rotate and align them seamlessly. Undergraduate research field different from PhD field every permutation is either even or odd,but not both How we can show every permutation is either even or odd,but not both......I can't arrive at a proof for this ..... Can anybody give me the proof...  Thanks in advance... Individual community preferences vs. SE network policy: who wins? In the recent question , the OP asks where the following question would be appropriate:     What would be the best way to approach someone that works for me about bad odor?   CHAOS team member Abby Miller  by suggesting the proposal  &#8212; which is in commitment as I'm writing this &#8212; because     One of the up voted example questions there is about dealing with an irritating coworker. This question will be perfectly on topic for that site.   In my opinion,  qualifies for the not constructive close reason*, which applies to all sites in the network, and therefore, the question shouldn't be allowed, no matter how much it has to do with workplaces.  I know Abby isn't the only one who disagrees with me about this. I once had a talk with a mod of some beta about the presence of poll/GTKY questions on his site. His position, in short, was that his community had different rules from SO and its members enjoyed such questions. I don't remember which site it was, it might even be graduated now, but I never did end up participating there.  Anecdotal examples are nice and all, but here's the real question: to what extent can individual communities/sites override network policies? To what extend should they be able to do so?  *: not constructive comes with the description "This question is not a good fit to our Q&amp;A format. We expect answers to generally involve facts, references, or specific expertise; this question will likely solicit opinion, debate, arguments, polling, or extended discussion."