I am trying to find a &quot;fair&quot; matchup system for a competition where $16$ players participate in a 4- player game. I would like to emulate the advantages of a round-robin system For example if my game were a 2-player one, I could easily make a $15$-round tournament, with each player facing a different opponent on each round. At the end of the tournament, each player would have faced all opponents exactly once, so none of the players faced a stronger opposition than any other player. A total of $120$ games would have been played in total. This simply doesn't work in a game with more players. If I wanted to go through every possible matchup in a 3-player game, I'd need $560$ games (more than $100$ rounds), and with four players, I'd need $1820$ games (in other words, each player would participate in $455$ games). I've considered switching to something based on the &quot;Swiss&quot; format used in chess, but that also has its downsides. My question is, what would be the shortest pairing method that would make each player on the competition face every opponent the same number of times? Does it apply to a different number of competitors? And no a game of $n$ players? It still wouldn't be totally &quot;fair&quot;, as a cetain combination of opponents could be tougher than another, but it'd definitely be an improvement. PS: As three opponents are played at each round and there are $15$ opponents for each player. The number of rounds played should  therefore be a multiple of $5$, but I don't know if it would be possible to do it with exactly $5$ rounds (after all, if there were $20$ rather than $16$, you'd defintely need at least $19$ rounds
I'm looking to schedule 16 players to play a round robin tournament with each other such that there are 4 players at each table.  I'd like for each player to play with each other player exactly once over the course of 5 rounds.  I just took a look at this post:  , and I was wondering if anyone knows if the case n=m=4 has been solved, and if so what that solution is.
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. Is it feasible to convert math in a picture, say written by hand, into LaTeX?   For example, I'd deem it faster to write down (de novo)  a 10 x 10 matrix with long, complex entries, rather than typing it in LaTeX or MathType. If I then take a picture or scan my writing, how can it be rendered or transformed into LaTeX? I've heard that Lubuntu consumes less resources than Ubuntu. Is it because of the fact that Lubuntu uses different desktop environment and window manager?  If so, would it be equivalent to install lubuntu-desktop on top of Ubuntu rather than doing a fresh install? What are the functions that are their own inverse?  (thus functions where $ f(f(x)) = x $ for a large domain)  I always thought there were only 4:  $f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ f(x) = \frac {-1}{x} $   Later I heard about a fifth one     $$f(x) = \ln\left(\frac {e^x+1}{e^x-1}\right) $$ (for $x &gt; 0$)  This made me wonder are there more? What are conditions that apply to all these functions to get more, etc. Can an irrational number raised to an irrational power be rational?  If it can be rational, how can one prove it? Does a finite $[E:K]_s$ imply that $[E_s:K]$ is finite? Let $E/K$ be an algebraic extension of fields and denote by $E_s$ the separable closure of $K$ in $E$, i.e. the set of all elements of $E$ that are separable over $K$ (which can be shown is a (separable) subfield of $E$).  I would like to show that the implication     $\left[E:K\right]_s &lt; \infty \implies \left[E_s:K\right] &lt; \infty$   holds, where $\left[E:K\right]_s$ denotes the separable degree of $E/K$ and $\left[E_s:K\right]$ the 'ordinary' degree of $E_s/K$.  In the textbook I am learning with, it is said that one could use the Primitive Element Theorem to do this, but I would also be fine with any other approach, as long as it's not getting too fancy and staying in the realm of 'common' textbook theorems known when dealing with separable extensions (and I didn't see how the Primitive Element Theorem would help).  This is not my focus (I want to write a Bachelor's thesis about polynomial factorizing elements of $\mathbb{Z}[X]$), but I'm in the process of reading into commutative algebra after about 1.5 years break from it.  My idea was something like assuming $\left[E_s:K\right] &lt; \infty$ and $\left[E_s:K\right] = \infty$ to construct infinitely many $K$-homomorphisms $E_s/K \longrightarrow C/K$, where $C$ is some algebraic closure of $E_s$, which would then contradict $\left[E_s:K\right]_s = \left[E_s:K\right] &lt; \infty$. Of course, an infinite basis would yield infinitely many $K$-vector space homomorphisms, but getting (infinitely many) ring morphisms seems to by an entirely different ting (unsurprisingly so...). Thought of using something like     Given two roots of the minimal polynomial of one of the infinitely many elements of a $K$-basis of $E_s/K$, there is a $K$-morphism $E_s/K \longrightarrow C/K$ that sends one to the other.   but this didn't seem very fruitful. I admit I haven't spent hours on this, I just didn't get very far and should focus on moving on, while still getting an answer on the 'howto' along the road would be really nice. Keeping the time stamp when syncing with Ubuntu One? Is there a way to change the process group of a running process? Is there a way to change PID, PPID, SID of a running process? It would make sense for the answer to be no, but I'd like to make sure. Which form should be used for at­tribu­tive nouns like “stu­dent union”: sin­gu­lar or plu­ral, or pos­ses­sive sin­gu­lar or pos­ses­sive plu­ral? When should a noun that’s used at­tribu­tively to de­scribe an­other  noun be plu­ral, and when should it be sin­gu­lar?  And when should it be pos­ses­sive, like bak­er’s dozen and when should it be plu­ral pos­ses­sive, like farm­ers’ mar­ket?  In other words, why do we say teach­ers union rather than teacher  union?  And why do we say wed­ding plan­ner rather than wed­dings plan­ner?  Which of these vari­ants is or are cor­rect?        stu­dent union   stu­dents union   stu­den­t’s union   stu­dents’ union      What about com­mu­nity val­ues ver­sus com­mu­ni­ty’s val­ues?  Please note I am look­ing for a gen­eral rule or at least some tips.  These are only ex­am­ples.  Up­date: It seems even na­tive speak­ers fol­low their per­sonal  style to write such com­pound words. I won­der why in the IELTS lis­ten­ing sec­tion, the lan­guage learner has to write a spe­ci­fic form and there is no rule for this. Matlab - Function taking no arguments within a class In newer Google Maps I can make area available offline.  Do other applications that rely on Google Maps (utilize Google Maps Android API) use offline maps? Why does Linux require that a user be root/using sudo/specifically authorized per mount in order to mount something? It seems like the decision as to whether to allow a user to mount something should be based on their access rights to the source volume/network share and to the mount point. A couple of uses for non-root mounting are mounting file-system images to a user owned direction and mounting a network share to a user owned directory. It seems like if the user has control over both sides of the mount equation everything should be cool.  Clarification of access restriction:  I feel I should be able to mount anything that the user otherwise would have access to to a mount-point that the user is the owner of.  For instance, on my computer /dev/sda1 is owned by user root and group disk with permissions brw-rw----. Therefore non-root users can't mess with /dev/sda1 and clearly mount shouldn't allow them to mount it. However if the user owns /home/my_user/my_imagefile.img and mount point /home/my_user/my_image/ why shouldn't they be able to mount that image file on that mount point with:  mount /home/my_user/my_imagefile.img /home/my_user/my_image/ -o loop   As kormac pointed out there is a suid problem. So some restrictions would have to be added to prevent suid from being a problem as well as potentially some other issues. Perhaps one way to do this would be to make the OS treat all files as belonging to the user that did the mounting. However for simple read/write/execute, I don't see why this would be a problem.  Use case:  I have an account in a lab where my home space is restricted to 8GB. This is tiny and very very annoying. I would like to mount an nfs volume from my personal server to essentially increase the amount of room that I have. However, because Linux does not allow such things I'm stuck with scp'ing files back and forth to stay under the 8GB limit. #!/bin/bash echo "$(curl -s -I https://google.com|grep Server)" abc   returns:  abcer: gws   and bash -x shows:    ++ grep Server ++ curl -s -I https://google.com ' abco 'Server: gws  abcer: gws   but    echo abc "$(curl -s -I https://google.com|grep Server)"`    returns  abc Server: gws   In other words, if I add text AFTER the curl command, it gets mangled, but it's ok if I add it before curl. What is happening? I cannot figure it out. Large text files load not working on android Quitting a job in the US on a work visa, going to Canada and re-entering on a tourist visa (ESTA waiver) I have been working in the USA on an E-3 visa and intend to return home to Australia, but I want to travel around the US before I do so. I am under the impression that after leaving my current company, I will have 10 days to leave the country.   Will I be able to go into Mexico or Canada and re-enter with an ESTA waiver as a tourist? Let $X_1, \cdots, X_n$ be a random sample from the PDF: $f(x;\theta) = (\theta + 1) x^{\theta}$ with $0&lt;x&lt;1$ and $\theta &gt; -1$.  The likelihood function is: \begin{align} L(\theta) &amp;= f(x_1, \cdots, x_n; \theta) \mathbb{1}\{0&lt;x&lt;1\} \mathbb{1}\{\theta &gt;-1\}\\ &amp;= \prod_{i=1}^{n}{f(x_i ; \theta)}\mathbb{1}\{0&lt;x&lt;1\} \mathbb{1}\{\theta &gt;-1\}\\ &amp;= \prod_{i=1}^{n}{(\theta+1)x_i^{\theta}}\mathbb{1}\{0&lt;x&lt;1\} \mathbb{1}\{\theta &gt;-1\} \\ &amp;=(\theta+1)^n \left( \prod_{i=1}^n{x_i}\right)^\theta \end{align}  We look at the log likelihood $l(\theta)$ and take the derivative of this since it is easier to deal with and it is allowed because the log is monotonic:  \begin{align} l(\theta) = n\log(\theta+1) + \theta \left( \sum_{i=1}^n{\log(x_i)} \right) \end{align}  $\implies \frac{d}{d\theta} = \frac{n}{\theta+1} + \sum_{i=1}^n{\log(x_i)} = 0$  $\implies \hat{\theta}_{MLE} = -\frac{n}{\sum_{i=1}^n{\log(x_i)}} - 1$  However, this doesn't look right. I'm running an Ubuntu 14.04 and want to install the Eclipse IDE for both Java and C/C++ programming. I installed the separate installation files from their official website. I followed this thread . (The top answer)   The steps helped me install the Java IDE but I don't know what changes I should make to the procedure to install the C/C++ IDE.   Another thread , recommended to install eclipse-cdt but the Eclipse version installed is outdated. (Doesn't allow me to install Pydev)   I have TWO questions:   Can the C/C++ libraries be downloaded from within eclipse, like Pydev? If not, how do I install both IDEs without interfering with each other?   P.S. I commented this query in the first thread but got no reply, that is why I'm asking a fresh question! Let $R \simeq S$ be isomorphic commutative rings with unity. Prove the following:  a). If $R$ is an integral domain then $S$ is an integral domain.  For this, I said that If I let $f: R \to S$ be an isomorphism and $(R, + , \circ), (S, \oplus, \ast)$, then let $x, y\in R$,  such that $x \circ y = 0_R$, because $f$ is an isomorphism, $f(x \circ y) = f(x) \ast f(y) = f(0_R)$. If I let $R$ be an integral domain then, as per the definition $R$ is commutative, has unity, and no zero divisors.   So $f(x \circ y) = f(x) \ast f(y) = f(0_R)$ where we have either $f(x \circ y) = f(0_R) \ast f(y) = f(0_R)$ or $f(x \circ y) = f(x) \ast f(0_R) = f(0_R)$.   Since $f$ is an isomomorphism, let $s = f(x), t = f(y)| s,t \in S$. Then $s \ast t = f(0_R)$. By definition, $S$ also has unity so $0_S$ exists. If we map $0_R \to 0_S$. Then either $s = 0_S$ or $t = 0_S$.   So then, since $S$ has unity, is commutative, and has a no zero divisors as we showed above, it is also an Integral domain.  I'm not sure I have this correctly.  b). If $R$ is a field then $S$ is a field. I don't really understand how to do this one. Since $S$ is an integral domain, it has a $0_S$. So I need to show that $S$ distributes $\ast$ over $\oplus$.  Any help with these is appreciated I want to list a company name in the author field of a bibliographic entry:  @book{CUDA,   author    = {NVIDIA Corporation},   title     = {NVIDIA CUDA Compute Unified Device Architecture Programming Guide},   publisher = {NVIDIA Corporation},   year      = {2007} }   But the entry keeps showing up "N. Corporation".  Is there a way to get the full author name to show up?  I'm currently using the siam BibTEX style. The number of (non-equal) forests on the vertex set V = {1, 2, ...,n} that contains exactly 2 connected components is given by The number of (non-equal) forests on the vertex set V = {1, 2, ...,n} that contains exactly 2 connected components is given by $\sum_{k=1}^{n-1} {n-1 \choose k-1} k^{k-2} (n-k)^{n-k-2}$.  I am unsure how to approach this question. Could someone guide me through the proof? If G is a spanning tree and it is disconnected then I know it is a spanning forest. Also, I know by Cayley's thm that there are $n^{n-2}$ nonequal tree on a vertex set V = {1, 2, ...,n} which is where i'm assuming the $k^{k-2}$ comes from but I do not understand where the rest of it is coming from.