Combinatorial proof of $\sum_{i=0}^n {n \choose i}^2={2n \choose n}$ by using a binary string of length $2n$
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
Sometimes a question I ask will get answers that don't help me, or maybe it doesn't receive any answers at all.  How about making a notification when somebody links to my question?   It could be something like this:     New information regarding your question might be available at {link}   It might help me getting an answer from related questions that weren't there when I first asked the question. Find and remove large files that are open but have been deleted How does one find large files that have been deleted but are still open in an application? How can one remove such a file, even though a process has it open?  The situation is that we are running a process that is filling up a log file at a terrific rate. I know the reason, and I can fix it. Until then, I would like to rm or empty the log file without shutting down the process.   Simply doing rm output.log removes only references to the file, but it continues to occupy space on disk until the process is terminated. Worse: after rming I now have no way to find where the file is or how big it is! Is there any way to find the file, and possibly empty it, even though it is still open in another process?  I specifically refer to Linux-based operating systems such as Debian or RHEL. Android app intermittently shows blank body in questions and answers (multiple sites) Note: the bug has returned in version 1.0.89 in a somewhat different form. See:      This possible bug report is similar to the bug in  that was deemed not-reproducible.  I can reproduce something similar with more recent app &amp; Android versions.   App Version: 1.0.34 (original report; is still present in 1.0.38, 1.0.42, 1.0.58) Android Version: 4.4 (up to 4.4.4) Device: Motorola Droid Maxx   In the examples below, you can see some blank answer bodies.  These are on different sites, and represent answers with both edited and non-edited text. In the first example, I clicked on the top answer tab to get to the answers, and in the second example I clicked on the bottom "View n Answers" button. (Trying to be a good little tester!)  Because I've seen this blank-answer-body intermittently over the past couple weeks (before the May 7 update), my process to test was just to go to the first site I frequent and then systematically click through the question list until I found a question with answers in which at least one answer was blank.  For all three sites shown below, it was within the first 5 or 7 questions sorted by newest.  Please note that I have seen many questions with several answers, none of which were blank. It is truly intermittent.  Example blank answer body at Workplace.SE:    Example blank answer body at Programmers.SE:    Example blank answer body at UX.SE:    And now here's the kicker, which makes me think this is some sort of content-loading latency issue rather than a rendering issue:  I returned to the left-side menu, selected the User Experience site, picked the same question as above, and then clicked the top answer tab to view answers again, and lo and behold the text is there:    I then went back to the other questions with blank answers, shown above, and the answer bodies were no longer blank.  I am happy to go through more tests/paths and do screenshots if it is helpful.  The same bug also appears in question bodies. How do you use the CSS content property to add HTML entities?  Using something like this just prints &amp;nbsp; to the screen instead of the non-breaking space:  .breadcrumbs a:before {     content: '&amp;nbsp;'; } If I want to print multiple figures in figure, I'm using subfigure in the figure environment. This work quit well and is looking smart, if all figures have the similar height. Now there is this figure which is more than two times higher than the other ones. I would like to order the subfigures like its shown in the following picture:     But I don’t know how to get this into Latex.   My regular code for four subfigures:  \begin{figure} \centering \begin{subfigure}{0.32\textwidth}    \centering    \includegraphics[width=\textwidth]{highone}    \caption{}    \label{fig:a} \end{subfigure} \begin{subfigure}{0.32\textwidth}     \centering     \includegraphics[width=\textwidth]{b}     \caption{}     \label{fig:b} \end{subfigure} \begin{subfigure}{0.32\textwidth}     \centering     \includegraphics[width=\textwidth]{c}     \caption{}     \label{c} \end{subfigure} \begin{subfigure}{0.32\textwidth}     \centering     \includegraphics[width=\textwidth]{d}     \caption{}     \label{fig:d} \end{subfigure} \caption{Blabla.} \label{fig:abcd} \end{figure}   Thank you for your help! More Efficient Spatial join in Python without QGIS, ArcGIS, PostGIS, etc Can a Rogue (D&D 3.5) use sneak attack on both the first and second attack? How do I activate Dauntless Cathar's ability? has the following ability:     {1}{W}, Exile Dauntless Cathar from your graveyard: Put a 1/1 white Spirit creature token with flying onto the battlefield. Activate this ability only any time you could cast a sorcery.   Please clarify. If the card is IN my hand, how can I exile if FROM the graveyard if it is not there? Did the writers mean SEND it to graveyard, or actually exile it? Confused, the writing does not make sense. Expectation inequality for a set and its subset. Given that a function $g(x)$ is a monotone increasing function. Its domain is an interval [c,d]. There are two sets $a_j \in A, j= 1,...,J$ and $b_k \in B,k=1,...,K$  on this interval, which satisfy $c_z \in C = B$\A and $c_z \geq a_j$ for all $a_j \in A$. For example, the interval $[c,d]=[0,8]$,$A=\{0,1,2,3,4,5\}$ and $B=\{0,1,2,3,4,5,6,7\}$. In this case, does the following inequality hold? If yes, could you give me some hints to prove that?  $$\frac{1}{K} \sum_{k=1}^{K}g(b_k) \geq \frac{1}{J} \sum_{j=1}^{J}g(a_j)$$ Proving by induction that $ \sum_{k=0}^n{n \choose k} = 2^n$ Does a subsequent charm effect override a previous one? In a recent session, I was confused about how multiple charm effects would interact with each other:  A vampire used its 'Charm' ability to charm a PC. It reads (paraphrased):     The target must succeed on a saving throw or be charmed by the vampire. It takes the vampire's requests in the most favourable way it can. It can repeat the saving throw if the vampire or its companions harm it. Otherwise, the effect last 24 hours or until the vampire is destroyed, is on another plane of existence, or takes a bonus action to end the effect.   Another PC then decided to attempt to wrest control back from the vampire by casting crown of madness on their friend. it reads (paraphrased):     The creature must succeed on a saving throw or be charmed. The charmed target must use its action before moving on each of its turn to make a melee attack against a creature other than itself that you mentally choose. The target can act normally on its turn if you choose no creature, or none are within its reach.   At the time, I ruled that both effects would take place simultaneously. This meant that Crown of Madness took effect first, and the vampire's charm afterwards.  Was this the correct ruling? Can a creature be charmed twice? Or does one charm effect override the other? How is the equation of motion on an ellipse derived? I would like to show that a particle orbiting another will follow the trajectory   \begin{equation} r = \frac{a(1-e^2)}{1 + e \cos(\theta)}. \end{equation}  I would like to do this with minimal assumptions. Any pointers? I've read that this is nothing but the equation for an ellipse, as defined from the focus of the ellipse. But I'm not satisfied in just taking the result, I mean why should I assume that orbits are elliptical? I guess it should be derivable from energy and angular momentum considerations... Google Sheets QUERY won't display cell text if other cells have numbers I'm not sure what is happening in my Google Sheet query and could really use some help. I have  that demonstrates and explains the issue.   Basically, I'm using a query to pull data from a Google Spreadsheet that is populated by a Google Form. If the form submits only numbers (100), there is no problem. However, if someone submits a string ($100 or 100 dollars), the query will not display the string. It will only display form submissions that are 100% numbers.   Also, if the 1st submission is a string (not 100% numbers) the query will work.   Here is my query  =query(data!B2:CL,"select B, C, D, E, F, G, H, I, J, CI, BZ, CA, BA  where J = 'Industry Certification'",false)   Any ideas on why this is happening?  Also, I'm aware that I can limit the form to numbers only. I'm trying to avoid that and learn 'why' this is happening. How do I evolve Kadabra to its final form without trading it? What is the best way to generate a GUID/UUID from Apex code (such as a trigger)? Preferably in the following format:     nnnnnnnn-nnnn-nnnn-nnnnnnnnnnnnnnnnn   Example:     13219ec0-3a81-44c5-a300-de14b7d0235f Best methods to fix broken tracks I've made a couple of PCBs which need the odd few 10 mil tracks, but are mostly 20/30 mil. Unfortunately the etching process isn't fantastic and there are numerous breaks in the traces, most of them small.  I've managed to convince solder lumps to go over the top of the smaller gaps, and have used components legs to bridge some of the larger ones. The solder lumps don't feel particularly reliable though, and there are some gaps which are too small and awkward to get a piece of wire across but too big to be bridged with just the iron.  Are there any standard ways to reliably fix up broken traces?  Update: Thanks for the suggestions everyone!  I found and tried out a silver conductive pen. As a first time user I found it difficult to dispense, and obviously I didn't use enough of it or mix it properly because it didn't even end up being conductive.  It seems like the main problem was that I was trying to use pieces of wire which are too short. I found some small gauge tinned wire and used longer lengths, running it over significant lengths of good track on either side of the fault. That makes it much easier to position, and you can tack them down one end at a time. I'm trying to write a simple bash function that takes, as its arguments, a number of files and/or directories. It should:   Fully qualify the filenames. Sort them. Remove duplicates. Print all that actually exist. Return the number of non-existent files.   I have a script that almost does what I want, but falls down on the sorting. The return value of the script as it stands is correct, but the output is not (unsorted and duplicates). If I uncomment the | sort -u statement as indicated, the output is correct but the return value is always 0.  N.B. Simpler solutions to solve the problem are welcome but the question is really about why is this occurring in the code I have. That is, why does adding the pipe seemingly stop the script incrementing the variable r?  Here's the script:  function uniqfile {     local r=0       for arg in "$@"     do           readlink -e "$arg" || (( ++r ))      done #| sort -u    ## remove that comment      return $r } Opening Chromium on Ubuntu VPS Ubuntu noob here. I recently got a VPS running Ubuntu, and decided to install Chromium, my preferred browser, on it. After installation completed, I got a message stating that I cannot open Chromium with root privileges. How would I go about opening Chromium? Localisation is isomorphic to a quotient of polynomial ring Why can't we downvote our own posts? Once in a blue moon, I've wanted the ability to downvote my own answer to some question.  ()  Obviously upvoting one's own posts is out of the question.  But down is different.  Voting my own answer down can help me improve the site by pushing my answer lower in the ranking than another answer with a better or more situation-appropriate alternative.  It's not something I'd do often, but I would use this feature if it were available.  If implemented, I'd expect this feature to operate just like downvotes on other user's posts.  That is:   I get a few seconds to undo the downvote;  I get to remove the downvote after an edit to the post;  I lose a point for placing the downvote (CW excepted); I lose two points for receiving the downvote (CW excepted).   If you're noticing that I'd lose 3 points for downvoting my own non-CW post, congratulations! You can add!  I can think of a couple of potential downsides to this change.  The first one may be a dealbreaker; the second is probably not that big a deal.   Strategic downvoting of oneself: since you can undo the vote after an edit, and you can always edit your own posts, someone could downvote themselves soon after posting (especially on a FGITW answer) to garner extra sympathy upvotes. Badge collecting: the Peer Pressure badge just got easier, because if you have a -2 score on an answer, just downvote and delete for an instant badge.