I just happened across , which has one close vote suggesting that it be closed as an exact duplicate of .  This suggestion is perfectly correct: these are exactly identical questions asked by the same user.  However, there's not much point in closing question A as a duplicate of B, since question B has already been closed as a duplicate of A.  Yet the close vote remains, doing nothing but wasting reviewers' time.  This is not the first time I've seen this happen, so I'd like to suggest that the Stack Exchange software should do something about it.  In particular, I'd like to suggest the following feature:     When a question Y is closed as an exact duplicate of another question X, any votes or flags to close X as a duplicate of Y should be automatically discarded.   Actually, I'm kind of surprised that the SE software doesn't do that already.  If anyone else thinks that this constitutes an actual bug, feel free to retag it.
It looks like this issue has , but it just happened again on Travel.SE. Perhaps this is another corner case? These two questions are closed as duplicates of each other.           I had the last vote on each, although in both cases, my vote was for "Unclear what you're asking", but the weight of votes in both cases were for the duplicate status. So if the existing checks are only in place at the time someone casts a close vote, that would explain how this one got in under the radar. Perhaps another check needs to be done when the question is actually closed, not just when close votes are cast?  And as @gnat points out, low-rep/logged-out users get a Redirect loop when trying to load either question.  NOTE: A moderator has already fixed this particular circular duplicate situation by closing one of the questions with a different close reason.
Using right-hand Riemann sum to evaluate the limit of $ \frac{n}{n^2+1}+ \cdots+\frac{n}{n^2+n^2}$ I'm asked to prove that $$\lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)=\frac{\pi}{4}$$ This looks like it can be solved with Riemann sums, so I proceed:  \begin{align*} \lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)&amp;=\lim_{n \to \infty} \sum_{k=1}^{n}\frac{n}{n^2+k^2}\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}(\frac{1}{n})(\frac{n^2}{n^2+k^2})\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}(\frac{1}{n})(\frac{1}{1+(k/n)^2})\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}f(\frac{k}{n})(\frac{k-(k-1)}{n})\\ &amp;=\int_{0}^{1}\frac{1}{1+x^2}dx=\frac{\pi}{4} \end{align*}  where $f(x)=\frac{1}{1+x^2}$. Is this correct, are there any steps where I am not clear? I have been looking at this for hours and it isn't making anymore sense than it did in the first hour.   If $a$ und ${x_{0}}$ are positive real numbers and ${x_{k}}$ defined as follows, prove that ${x_{k}}$ is monotone decreasing and bounded, then calculate the limit.  ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$  What I though I had to do was pick an ${x_{0}}$ and solve for ${x_{k}}$, so I picked ${x_{0}}$. Then I wanted to put the result back into the function to get ${x_{k+1}}$, which I still believe is what I'm supposed to be doing, but I don't understand what I am getting as a result.  I get that I should prove it is decreasing, then that it is bounded,  then address the limit, but the how is missing. How does light speed up after coming out of a glass slab? How to run Bash script on startup on Linux? I am running Linux 4.9.11-1.0.0 on my iMX7 dual development board. I have a test.sh file that I would like to run at startup. It currently only has a bunch of echo statements in it (just for testing purposes).   When I flick the switch on the board to turn it on, I see a bunch of initialization stuff come up. I would like the echo statements in my test.sh file to also be displayed in this initialization sequence on startup. Eventually once this is working, the plan is to do real stuff in place of the simple echo statements. I just want to make sure the script is running on startup at this point.  So far I have created the test.sh file and I am able to run it manually after the board has booted up. I have added the following to my /etc/rc.local file before the exit statement at the end:  sh '/home/root/test.sh'   I have verified that this rc.local file has #!/bin/sh -e at the top. I have also done:  $ chown root /etc/rc.local $ chmod 777 /etc/rc.local   Once this was all done, I verified that it should be working properly by typing:  $ /etc/init.d/rc.local start   Everything is working as expected up until this point and I am seeing the echo statements when I do that last step. However, when I go and try to reboot the board, I do not see the echo statements at all in the printed initialization stuff.  What am I missing? Maybe it is actually running on startup but I just don't see the output for some reason? If I have a PATH variable that (when exploded onto multiple lines) contains something like this (with a few notes added by me):  /Users/brandon/.rvm/gems/ruby-2.0.0-p353/bin /Users/brandon/.rvm/gems/ruby-2.0.0-p353@global/bin /Users/brandon/.rvm/rubies/ruby-2.0.0-p353/bin /opt/local/bin /opt/local/sbin /usr/local/sbin /usr/local/bin   # &lt;&lt;--- notice this path appears again later /usr/bin         # &lt;&lt;--- notice this comes AFTER /usr/local/bin /bin /usr/sbin /sbin /usr/local/bin   # &lt;&lt;--- here it is again! /usr/local/git/bin /Users/brandon/bin /Users/brandon/.homebrew/bin /usr/local/mysql/bin /usr/local/share/npm/bin /Applications/Postgres93.app/Contents/MacOS/bin /Users/brandon/.rvm/bin   is it possible for the presence of the second /usr/local/bin/ to cause the /usr/bin to take precedence over /usr/local/bin?  This seems crazy (to me: I've never heard of such a thing) but that's exactly the behavior I'm seeing.  I've just installed PHP with Homebrew, and yet the system PHP (from /usr/bin) is taking precedence:  $ which php /usr/bin/php   BUT, after manually altering the PATH (rather than digging through all the files sourced by my .zshrc to try to figure out why &amp; where the duplication is coming from) to remove the second instance of /usr/local/bin, I find that /usr/local/bin is taking precedence over /usr/bin as I would have expected it to in the first place:  $ which php /usr/local/bin/php   What would cause this?  Is this normal? Makeindex error when ! used in \index in align environment See MWE below. The symbol ! in the \index causes an error. This is only the case in align environments (not in equation). So it seems to have to do something with amsmath. If ! is put in an \mbox there is no error. But then of course the font size is not adjusted to subindices. Curiously enough, \index{$C_{K!!}$} (two exclamation marks) doesn't cause any error.    \documentclass{article} \usepackage{amsmath} \usepackage{imakeidx} \makeindex \begin{document} \begin{align} y\index{$C_{K!}$} \end{align} \printindex \end{document}     Any solutions suggested? Comparing and contrasting, p-values, significance levels and type I error I was wondering if anybody could give a concise rundown as to the definitions and uses of p-values, significance level and type I error.  I understand that p-values are defined as "the probability of obtaining a test statistic at least as extreme as the one we actually observed", while a significance level is just an arbitrary cutoff value to gauge if the p-value is significant or not. Type I error is the error of rejected a null hypothesis that was true. However, I am unsure regarding the difference between significance level and the type I error, are they not the same concept?  For example, assume a very simple experiment where I flip a coin 1000 times and count the number of times it lands on 'heads'. My null hypothesis, H0, is that heads = 500 (unbiased coin). I then set my significance level at alpha = 0.05.  I flip the coin 1000 times and then I calculate the p-value, if the p-value is > 0.05 then I fail to reject the null hypothesis and if the p-value is &lt; 0.05 then I reject the null hypothesis.  Now if I did this experiment repeatedly, each time calculating the p-value and either rejecting or failing to reject the null hypothesis and keeping a count of how many I rejected/failed to reject, then I would end up rejecting 5% of null hypotheses which were in actuality true, is that correct? This is the definition of type I error. Therefore, the significance level in Fisher significance testing is essentially the type I error from Neyman-Pearson hypothesis testing if you performed repeated experiments.  Now as for p-values, if I had gotten a p-value of 0.06 from my last experiment and I did multiple experiments and counted all the ones that I got a p-value from 0 to 0.06, then would I also not have a 6% chance of rejecting a true null hypothesis? When to pronounce ‹s› as /z/ in the middle of words? Is there a way to make a true toon shader material in Cycles?  Here's, in order of importance, what would be ideal to find in such a shader:   Must use Cycles Must be an actual shader Must not rely on lighting Must be independent of camera position Must react directly to lighting rather than manually having to change it to emulate it Should be easily made into a node-group Should have the following configurability:   Main color Darkness color Shadow depth (how far the shadow goes from the unlit side)    Below, I'll be listing some already-found methods and their flaws:    Cycles's Toon BDSF:  Source: Blender  Result:    Node setup:    Cons:   Not a clear-cut two-color shading Reacts to diffusion of surrounding objects      Mask &amp; Ramp:  Source:   Result:    Node setup (Compositor):    Cons:   Very grainy Does not support colour textures Not easy to change shadow depth in a nodegroup Not a material, requires compositor Requires use of object indexes     Shadeless Fresnel:  Source:   Result:    Node setup:    Cons:   Uses camera's direction Ignores lighting Not easy to change shadow depth in a nodegroup     Shadeless Normal:  Source:   Result:    Node setup:    Cons:   Ignores lighting Direction of shadow is manually changed Not easy to change shadow depth in a nodegroup   However, this one is easily the best one. It's pretty great, and would work still images very well.    That's all I've got. Does anyone else have any methods that fall under most of the criteria I listed above? Also note that OSL is totally fine, if it works. Title pretty much sums it up.  The external style sheet has the following code:  td.EvenRow a {   display: none !important; }   I have tried using:  element.style.display = "inline";   and  element.style.display = "inline !important";   but neither works. Is it possible to override an !important style using javascript.  This is for a greasemonkey extension, if that makes a difference. Now that tags have this erm...  which appears when you move your mouse over them (try it! ), one question remains: What in the billion blue blazes does the 'subscribe' button do?   It's not RSS, since a traditional RSS feed link appears beside it, and clicking on it seems to increment the number of subscriber by one, but other than that scouring over the Stack Overflow UI does not appear to reveal anything. Surely this feature would have more use than serving as a popularity contest for the tags? $\lim_{x\to 0} \frac{\sin x - x}{x^2}$ without L'Hospital or Taylor How can I compute the distance between two coordinates in TikZ? Given two points (defined, for instance using nodes), I want to compute the distance between them.   Is there some build in functionality in tikz to do this? If not, how can it be done using the mathematical engine?   The application I have in mind is to draw an circular arc centered at (a) and passing through some second point (b), where only (a) and (b) are known. I have some stuff in settings.py that I'd like to be able to access from a template, but I can't figure out how to do it. I already tried  {{CONSTANT_NAME}}   but that doesn't seem to work. Is this possible? I have a service (kado-demo.service) that a normal user (kado) should be able to restart without entering any password.   I created /etc/sudoers.d/user_restart and added  kado ALL=NOPASSWD: /bin/systemctl restart kado-demo.service   Permissions:  -r--r-----   1 root root   585 Mar  5 10:47 user_restart   As user k, I now should be able to execute the given command, but this happens:  /bin/systemctl restart kado-demo.service  ==== AUTHENTICATING FOR org.freedesktop.systemd1.manage-units ===  Authentication is required to restart 'kado-demo.service'.   Multiple identities can be used for authentication:   (...)   Which is the same result as before I added the file.   The /etc/sudoers contains  #includedir /etc/sudoers.d   And I even get the same result when I add content of the user_restart file directly to /etc/sudoers.  I'm running out of ideas... Proof of Equivalence Relation $R=\{(a,b)\in\Bbb N^2: 5\mid (2a+3b)\}$ Given a set of natural numbers ($\Bbb N$), and the relation $R=\{(a,b)\in\Bbb N^2: 5\mid (2a+3b)\}$. Is $R$ an equivalent relation?   My answer: Yes, because $2a + 3b$ is divisible by $5$, if and only if, $2a + 3b − 5b$ is divisible by $5$.  Is my reasoning right?   If it is not, then how do I prove that this is an equivalent relation? Can't unhide objects How do I get past the "do not enter" traffic sign that stops me from installing "Sierra" instead of "High Sierra"?  I booted into rescue mode and deleted the original "High Sierra" installation, but booting from the USB stick with the "Sierra" installer on it still gives me the "no". What does a two overlapping circles symbol mean in an electrical schematic/diagram? In the datasheet for a A3503 Hall Effect Sensor, I saw a circuit diagram labeled 'Functional Block Diagram', and it shows a symbol I have not seen before. It's shown here between 3 - Output and 2 - Ground.  What is this supposed to represent?    The full datasheet is available Is it possible to disable CPU throttling on a MacBook Pro with a completely dead (not recognized) battery? I have a MBP late 2011 running El Capitan, recently my battery died (x over the icon and "no battery available" in the dropdown.) I've noticed since then that CPU intensive tasks are very slow, and I read that without a battery macs will throttle the CPU to half capacity.  So now I'm looking for one of two solutions:  1) Either somehow disable the throttling, or  2) Trick the computer into thinking it has a battery.  In regard to the first one I've tried a few things such as described , removing a plist file that supposedly controlled it, but that hasn't helped. Any help would be greatly appreciated, TIA.  Edit: I've looked over the question about a 2006 MacBook, it's similar but I've tried everything outlined there and nothing has worked. I'm curious if there are any new ideas. I think the difference may be El Capitan.