How to solve the iterated function $f(f(x))=x^2+x$?
How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$?
It is very easy to put a tilde over the symbol in math mode. Simply type $\tilde{a}$. But how to put a tilde under the symbol in math mode? Most instructions for equipment and software make a habit of shortening sentences by dropping words of certain parts of speech - usually definite and indefinite articles.  For example:     PULL (THE) LEVER TO OPERATE      CARS MAY BE TOWED WHEN (THEY ARE) PARKED ILLEGALLY      PLEASE USE (THE) OTHER DOOR   Is there a name for this custom? And what is the best way to describe it? Wrong ordering in generated table in jpa What are the ways to solve cubic equations? Cubic equations of the form $ax^3+bx^2+cx+d$ can be solved in various ways. Some are easy to easy to factor in a pair, for some the roots can be found out by trial-and-error, some are one-of-a-kind, some can be reduced to a quadratic equation. A compilation of all possible ways to solve cubic equations would be very helpful for students and learners. How to count derangements? I was going through .Suppose that there are n persons who are numbered 1, 2, ..., n. Let there be n hats also numbered 1, 2, ..., n. We have to find the number of ways in which no one gets the hat having same number as his/her number. Let us assume that the first person takes hat i. There are n − 1 ways for the first person to make such a choice. There are now two possibilities, depending on whether or not person i takes hat 1 in return:  A) Person i does not take the hat 1. This case is equivalent to solving the problem with n − 1 persons and n − 1 hats: each of the remaining n − 1 people has precisely 1 forbidden choice from among the remaining n − 1 hats (i's forbidden choice is hat 1).  B) Person i takes the hat 1. Now the problem reduces to n − 2 persons and n − 2 hats.  From this, the following relation is derived: !n = (n - 1) (!(n-1) + !(n-2))  I have a hard time understanding point B and the equation derived. I was trying to understand with an example of 4 people named A, B, C and D and the corresponding hats A, B, C and D. Everyone of them are supposed to wear any hat except their own matching hat i.e. A can wear B, C or D.  If suppose A wears B hat so we will be left with only 3 people and 3 hats. So I can understand "(n-1)!(n-1)" part but what i didn't understand is part B(marked above). Can anyone help? Getting user provided data out of a database and into R Show that the quotient space is homeomorphic to the n-disc After the latest Ubuntu updates, I have not been able to open virtualbox, on starting it gives me an error saying the kernel driver is either not loaded or set up properly and that I should reinstall virtualbox-dkms package. After removing virtualbox, and trying to reinstall I get the error below: ERROR: Cannot create report: [Errno 17] File exists: '/var/crash/virtualbox-dkms.0.crash' Error! Bad return status for module build on kernel: 5.8.0-34-generic (x86_64) Consult /var/lib/dkms/virtualbox/6.1.10/build/make.log for more information. dpkg: error processing package virtualbox-dkms (--configure):  installed virtualbox-dkms package post-installation script subprocess returned error exit status 10 dpkg: dependency problems prevent configuration of virtualbox:  virtualbox depends on virtualbox-dkms (&gt;= 6.1.10-dfsg-1~ubuntu1.20.04.1) | virtualbox-source (&gt;= 6.1.10-dfsg-1~ubuntu1.20.04.1) | virtualbox-modules; however:   Package virtualbox-dkms is not configured yet.   Package virtualbox-source is not installed.   Package virtualbox-modules is not installed.   Package virtualbox-dkms which provides virtualbox-modules is not configured yet.  dpkg: error processing package virtualbox (--configure):  dependency problems - leaving unconfigured dpkg: dependency problems prevent configuration of virtualbox-qt:  virtualbox-qt depends on virtualbox (= 6.1.10-dfsg-1~ubuntu1.20.04.1); however:   Package virtualbox is not configured yet.  dpkg: error processing package virtualbox-qt (--configure):  dependency problems - leaving unconfigured No apport report written because the error message indicates it's a follow-up error from a previous failure.                             No apport report written because the error message indicates it's a follow-up error from a previous failure.                                                         Processing triggers for menu (2.1.47ubuntu4) ... Processing triggers for bamfdaemon (0.5.3+18.04.20180207.2-0ubuntu2) ... Rebuilding /usr/share/applications/bamf-2.index... Processing triggers for desktop-file-utils (0.24-1ubuntu3) ... Processing triggers for mime-support (3.64ubuntu1) ... Processing triggers for hicolor-icon-theme (0.17-2) ... Processing triggers for gnome-menus (3.36.0-1ubuntu1) ... Processing triggers for systemd (245.4-4ubuntu3.3) ... Processing triggers for man-db (2.9.1-1) ... Processing triggers for shared-mime-info (1.15-1) ... Errors were encountered while processing:  virtualbox-dkms  virtualbox  virtualbox-qt E: Sub-process /usr/bin/dpkg returned an error code (1)  This happens even when I boot from the previous kernel using the grub menu, any help would be very appreciated. I have also tried to use aptitude to remove any conflicting versions of virtualbox from other repositories but still the same issue. Travelling to Italy without my spouse who is a UK citizen I am India citizen married to a UK national who has been issued a Schengen Visa from the Italian consulate as a spouse of a UK national. My husband was supposed to travel with me but now can’t. We have decided that I should still pursue the trip without him as I will be joining our friends there. Please can someone let me know if I will still be allowed to travel to the Schengen region with a valid Schengen Visa issued as a spouse of a EEA national but without my husband joining me there in person ? MT7610U - Unable to install wifi driver Calculate the uncertainty in velocity of electron I got this question in one of my periodic test in coaching, I (seemingly) solved the question but later recognised what mess I did.  So, the question is :      Calculate the uncertainty in velocity of an electron. If the uncertainty in its position is ${10^ {-15}} \; \mathrm m$.   By simple calculations one can see that the answer would be ${5.79\times{10^{10}}} \; \mathrm{m/s}$. However, as one can see, the uncertainty in velocity is greater than the velocity of light then how is that possible or there is some other way to handle such problems.  P.S. I know the concept of dynamic mass but I am not very familiar with it. (I think that would work here, but don't know how?) Converting WKT to SHP? Would this look natural:     StackExchange is a helluva site! I found out about this site at Facebook.    Or should we use the preposition on:     StackExchange is a helluva site! I found out about this site on Facebook.    Or are both okay?   (Inspired by a ) How can I check if /bin/x86_64/bin/ls is a directory in a bash shell script  Here is what I tried:  #!/bin/bash  if [ $# -eq 2 ]; then     if [[ "$1" = /* ]]     then     cd ./bin/x86_64/bin/ls         if [ -d "$1" ]; then             echo "ok"             i="$1"             echo $i         else             echo "error2"             exit         fi     else         echo "error"         exit     fi fi So, for example, when I type man ls I see LS(1). But if I type man apachectl I see APACHECTL(8) and if I type man cd I end up with cd(n).  I'm wondering what the significance of the numbers in the parentheses are, if they have any. VAR/VECM/ARDL optimal lag selection Question 1: Is it necessary to consider AIC and the BIC criteria when selecting the lag for a VAR, VECM or ARDL model OR can I use something else?  Example: Can I pick 12 lags because the model simply produce great results when I do or do I have to use the AIC/SIC criteria?  Question 2: A simple google search tells me that its possible to select lag 4 when quarterly data, 12 when monthly etc. Is that a valid approach ? Allow users to leave an anonymous comment when voting I don't know if people don't explain why they up or down vote because they don't care or because they don't want the person to know they voted on an answer (or question)... But, it might be nice to be able to leave a comment anonymously as to why it was voted.  It could give us more insight as to people's thoughts on posts.    Really, I think this would come into play a lot more dealing with down votes more than up votes, but I could see it being beneficial in both situations.   See also I think I have found the final answer $\left(4 \left[\frac{2t \cosh(4t) - \sinh(4t)}{t^2}\right]\right)$ but need to verify it. I need to use an alias in the WHERE clause, but It keeps telling me that its an unknown column. Is there any way to get around this issue? I need to select records that have a rating higher than x. Rating is calculated as the following alias:  sum(reviews.rev_rating)/count(reviews.rev_id) as avg_rating Finding the maximum value of $\int_0^1 f^3(x)dx$ Find the maximum value of $\int_0^1 f^3(x)dx$ given that $-1 \le f(x) \le 1$ and $\int_0^1 f(x)dx = 0$   I could not find a way to solve this problem. I tried to use the cauchy-schwarz inequality but could not proceed further  $$\int_0^1 f(x) \cdot f^2(x) dx \le \sqrt{\left(\int_0^1f^4(x)dx\right) \left( \int_0^1 f^2(x) dx\right)}$$  Any hints/solutions are appreciated.