What does observation mean in two-slit electron diffraction experiment?
What constitutes an observation/measurement in QM?
Number of ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$. Question:      Find the number of non trivial ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$. ($f$ is not necessarily unitary, i.e., $f(1)$ need not be $1$.)    Suppose $f$ is a ring homomorphism from $\mathbb Z_{12}$ to $\mathbb Z_{28}$.  Consider $f$ as a additive group homomorphism.  Let $k= |\ker f|$ and  $ t = |\operatorname{im}(f)|$. Then $k\mid 12$ and $t\mid 24$ and $kt=12$, by first isomorphism  theorem of groups.  There are two possibilities $k=3$, $t=4$   and $k=6$, $t=2$.  For the first case $f$ should map $1$ to an element of the subgroup generated by $7$ as there is a unique subgroup of $\mathbb Z_{28}$ of order $4$ generated by $7$. For the second case $1$ has to map to $14$, for the same reasoning.  So there are at most two ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$. Question is how to check the possible maps which are ring homomorphisms.  Thanks. Getting inverse of a transposition key This question is inspired by my own answer to a  which I tried to answer and got stuck at one point.     The question was:     HI DARLING.      USE MY ATM CARD, TAKE ANY AMOUNT OUT, GO SHOPPING AND TAKE YOUR FRIENDS FOR LUNCH.      PIN CODE: $\displaystyle \int_{0}^{1} \frac{3x^3 - x^2 + 2x - 4}{\sqrt{x^2 - 3x + 2}} \, dx $      I LOVE YOU HONEY.      Anyone knows? Are we gonna get an integer number?     My attempt:  Does this help?     $$\frac{3x^3-x^2+2x-4}{x-1}=3x^2+2x+4$$   (long division)   \begin{align*} I&amp;=\int\frac{3x^3-x^2+2x-4}{[(x-1)(x-2)]^{1/2}} dx = \\ &amp;=\int\frac{(3x^2+2x+4)(x-1)^{1/2}}{(x-2)^{1/2}} dx = \\ &amp;=\int 3(u^4-4u^2-4)(u^2+1)^{1/2}du \times 2 \end{align*}   after the substitution   \begin{gather*} (x-2)^{1/2}=u\\ du=\frac1{2(x-2)^{1/2}}dx\\ u^2=x-2\\ (x-1)^{1/2}=(u^2+1)^{1/2} \end{gather*}   Update:  may help us proceed.    I tried to proceed:     $$6\int (u^4-4u^2-4)(u^2+1)^{1/2} du = 6\int ((t-3)^2-8)t \frac{dt}{2u}$$   after $u^2+1=t$ and $dt=2udu$   \begin{align*} u^4-4u^2-4 &amp;= (u^2+1)^2-(6u^2+5) \\ &amp;= (u^2+1)^2-6(u^2+1)+1 \\ &amp;= ((u^2+1)-3)^2-8 \end{align*}   I wonder whether this question can be solved from here?    Update:   This has been getting a lot of views, and I think most people came for the sort of problem mentioned in the title (where I got stuck) rather than the original problem itself.  Keepin this in mind, I'm reopening the question and here's the kind of answers I expect — Solutions to the original problem are good, but I'd prefer solutions that continue from the part where I got stuck — the polynomial in $u$ — that's the sort of problem mentioned in the title. Proof of a formula involving Euler's totient function: $\varphi (mn) = \varphi (m) \varphi (n) \cdot \frac{d}{\varphi (d)}$ Prevent mail being marked as spam This is a  about avoiding outgoing mail being classified as spam.   Also related:                    I'm wondering how to prevent my emails from my site being marked as spam? I'm using sendmail.  I'm trying to send emails through my ruby-on-rails application. The mails are all written in swedish (if that does make a difference?). I don't know why they keep getting marked as spam.  Are there any things that I can do to minimize the risk? Can you partition an infinite set, into an infinite number of infinite sets? How to show textures in the 3D View editor? Time ordering operator if commutator is $c$-number function Opening a context menu (with the Menu key) When does decision tree perform better than the neural network? Resuming and Finishing the flow Created a flow in the sandbox to capture a survey.  I have enabled the ability to Pause the flow and added the component to Home page.   All the screens for the flow is complete and when I test run below 2 issues are found.  1. In the last screen I do not find the Finish Button. There is a record update after last screen. When I add a screen after record update It shows the finish screen But if click the previous button present there it moves to the first and create a new record. Is there any possibility to finish without going on the previous screens   When I resume a Paused flow, after clicking on the last screen, instead of redirected to the Account page, the flow is restarting. Appreciate your help.   On pause going to first screen is normal functionality , Is there any modification that can be done. Elegant way of obtaining the envelope of oscillating function Quake-style terminal with splitting in Gnome other than Yakuake The most common Quake-style terminal in Gnome seem to be Guake and Tilda, but they can't split to show several shells in single tab like Yakuake can. Is there any alternative which can do so? If the distribution of test statistic is bimodal, does p-value mean anything? How to change boot priority? I have a laptop with Ubuntu installed as the only OS. I also have a USB stick with a Windows 7 installation on it. I want to install Windows from the USB stick on my laptop but when I try to restart the laptop to get access to BIOS, I can't seem to find the command for it, its like it skips everything and just says Ubuntu and then takes me to the login screen.  How do I change the boot priority to USB in Ubuntu? Can the system leave a deletion reason when it deletes a question? I want to clear a element from a vector using the erase method. But the problem here is that the element is not guaranteed to occur only once in the vector. It may be present multiple times and I need to clear all of them. My code is something like this:  void erase(std::vector&lt;int&gt;&amp; myNumbers_in, int number_in) {     std::vector&lt;int&gt;::iterator iter = myNumbers_in.begin();     std::vector&lt;int&gt;::iterator endIter = myNumbers_in.end();     for(; iter != endIter; ++iter)     {         if(*iter == number_in)         {             myNumbers_in.erase(iter);         }     } }  int main(int argc, char* argv[]) {     std::vector&lt;int&gt; myNmbers;     for(int i = 0; i &lt; 2; ++i)     {         myNmbers.push_back(i);         myNmbers.push_back(i);     }      erase(myNmbers, 1);      return 0; }   This code obviously crashes because I am changing the end of the vector while iterating through it. What is the best way to achieve this? I.e. is there any way to do this without iterating through the vector multiple times or creating one more copy of the vector? SQL Server - how to determine if indexes aren't being used? I have just booted my clean Ubuntu 18.04 LTS VM and tried to install some application with sudo apt install.  But got error message:     E: Could not get lock /var/lib/dpkg/lock-frontend - open (11: Resource temporarily unavailable)       E: Unable to acquire the dpkg frontend lock (/var/lib/dpkg/lock-frontend), is another process using it?   What process created the /var/lib/dpkg/lock-frontend file in Ubuntu? How to gently stop it and take manual control on APT? What is a referee in the manuscript submission process? Please submit, with the manuscript, the names, addresses (including   countries), and e-mail addresses of five potential referees who are   outside the authors' institutions and must not have conflict of   interest with the authors.   Do you think I should name 5 researchers from a similar area that are not affiliated with my university and co-authors? Is this what referees are?