How do I find and enumerate the number of perfect matchings in directed complete graphs?
How to get all Perfect or Near-Perfect Matchings of a graph
How do I insert a backslash inside a mathematical statement? I'm trying to write a set subtraction operation in an AMS formatted template.  In the end, what I want should look like A\B.  But I'm doing this in math-mode $A\B$ and this gives me errors.  I've also tried using the \diagdown symbol, but it doesn't work either.  Is there another way to do this? Show a connected metric space (X,d) is $\epsilon$-chainable for $\epsilon &gt;0$       $\epsilon$-chainable Definition      (X,d) is $\epsilon$-chainable if given any two points $a,b \in X$, $\exists$ a finite set $\{ x_0,x_1,...,x_n\}$ of points in X with $x_0 = a$, $x_n=b$ and $d(x_{n-1},x_n)&lt;\epsilon$, $\forall$ n      Clopen Definition      A set is clopen $iff$ its boundary is empty      Connected Metric Definition      Let $(X,d)$ be a metric space. It is connected $iff$ it has no proper subsets which are clopen      The following are equivalent:         $X$ is connected   The only subsets of $X$ that are clopen are $\emptyset$ and $X$   $\nexists$ non-empty, disjoint, open sets $A$ and $B$ such that $X=A\bigcup B$   $\nexists$ non-empty, disjoint, closed sets $A$ and $B$ such that $X=A\bigcup B$         Theorem      Let $(X,d)$ be a metric space, and let $S$ be any subset of $X$. Then $S$ is closed $iff$ $S^c$ is open.     So how would someone show this exactly?  Let the set $A$ be the set of all elements in $(X,d)$ that can be reached from $a\in X$ in a finite number of steps with length less than $\epsilon$.  Does this mean that there should be no isolated points? (an isolated point would be at distance infinity from any other point, or distance zero from itself, since $\epsilon &gt;0$ we don't want this?)  Knowing that $X$ is connected $\implies$the only subsets of X that are clopen are $\emptyset$ and $X$  Also, by construction of A;   A is the set of all $\epsilon$-chains between any element and $a$  And, so $A^c = \emptyset$ if $A$ was $\epsilon$-chainable (because A has to have a path from all elements to a, for this to be true, the complement wouldn't have anything in it)  Leaving only $A$ as the remaining clopen set  Does this make any sense? Or should an alternative be used, and how to formalize it.  Thanks. Find a file when you know its checksum? Copyright of a set of colors? I often see color combinations on webpages, images, posters, products etc. that harmonize really well. Am I allowed to use those colors in my own design?   This is solely about the colors. I will not use images, logos or shapes from that source, and my content is not related to the original color source. I am also far from any commercial aspect here. The Complete Rate-Limiting Guide I noticed that I can only perform certain actions such as commenting a finite number of times in a given period of time. Obviously, rate limiting is in place to prevent accidental misuse or intentional abuse of certain features.  Where else is rate limiting applied on Stack Exchange sites, and what are the limits? Number of equivalence relations on a set Combining multiple textures with a white background I'd like to sort of "overlay" these shaders, but the only options are mix and add. Allow me to explain.  I made two images, both white background, one with red scribbles and one with blue scribbles, both 1920x1080. Image textures plugged into diffuse was my node setup. I want both scribbles to be added to the plane while the background stays the same color.  If I mix them I get the following:  The red and blue scribbles are now 50% dimmer than they would be on their own, and the white color hasn't changed in intensity.  If I add them I get the following:  The red and blue scribbles are still 50% dimmer than they normally would be (since the white background is being added over each). But the white background is now 2x brighter than normal.  Can I get a result where the white background is normal brightness and both the blue and red scribbles are at normal brightness? I am trying to get an Epson WF-2760 scanner that is network-connected working. I'm using Ubuntu 16.04.2 64-bit and the printer works a treat. I have installed the Epson drivers from their site, including the plugins. I have edited /etc/sane.d/dll.conf to enable epson, epson2, and epsonds. I have edited each of the respective .conf files to include the network address of the device (IPV4: 192.168.128.11).   The scanner is not detected in any software that I have tried (including Image Scan and Simple Scan).  Any help is appreciated! How can I delete an account on a specific site? My account has 20 (!) sites associated to it.  I only use it for 3-5 sites.  To reduce clutter, I would like to delete my profiles that I do not use much.  I have went through , but when I enter the profile URL as:  http://russian.stackexchange.com/users/5772/alex   it gives the error message:     oops! this isn't a link to a valid profile   What procedure do I go through to delete my profile only on certain sites? I originally read this book in the mid- to late '70s.   The premise was a bureaucracy whose job it was to manipulate events in human history by making the smallest possible change that would effect the desired outcome. Not time cops in the sense of going after law breakers.  The intent was to maintain a policy of non-interference as much as possible but when interference was warranted then to make the least possible change and to do so invisibly.  For example, they might take someone's keys off the dresser and put them just out of sight underneath the dresser where they plausibly might have fallen.  The subject would never know of the manipulation but would then be delayed getting out of the house and the ripples grew from there to avert a war or otherwise affect events on a large scale.  After reading that, whenever I would lose something - especially in an improbable way such as dropping a screw at my feet and then finding it on the other side of the room half an hour later - I would take some consolation in the thought that I had just averted some global catastrophe.  I was pretty scatterbrained as a kid (we call that ADHD nowadays) so I experienced this a lot.  Given that I've spent so much time pondering the ideas from the book, I'd love to re-read it. How do you convert a numerical number to an Excel column name in C# without using automation getting the value directly from Excel.  Excel 2007 has a possible range of 1 to 16384, which is the number of columns that it supports. The resulting values should be in the form of excel column names, e.g. A, AA, AAA etc. I am interested in calculating area under the curve (AUC), or the c-statistic, by hand for a binary logistic regression model.  For example, in the validation dataset, I have the true value for the dependent variable, retention (1 = retained; 0 = not retained), as well as a predicted retention status for each observation generated by my regression analysis using a model that was built using the training set (this will range from 0 to 1).  My initial thoughts were to identify the "correct" number of model classifications and simply divide the number of "correct" observations by the number of total observations to calculate the c-statistic. By "correct", if the true retention status of an observation = 1 and the predicted retention status is > 0.5 then that is a "correct" classification. Additionally, if the true retention status of an observation = 0 and the predicted retention status is &lt; 0.5 then that is also a "correct" classification. I assume a "tie" would occur when the predicted value = 0.5, but that phenomenon does not occur in my validation dataset. On the other hand, "incorrect" classifications would be if the true retention status of an observation = 1 and the predicted retention status is &lt; 0.5 or if the true retention status for an outcome = 0 and the predicted retention status is > 0.5. I am aware of TP, FP, FN, TN, but not aware of how to calculate the c-statistic given this information. What oil is best for seasoning a cast-iron skillet How is well-ordering in real line possible?  I know that the axiom of choice provides possible well-ordering, but intuitively, this does not seem to make sense.  How can you compare 1.111111.... and 1.111111... when we can never know the full digits of each number? There is a very bad problem with the "Blank screen" function in Ubuntu 17.10. I refer to the setting accessed via Settings->Power->Blank Screen. This seems to actually perform a screen lock, since I need to enter my password after un-blanking the screen. (However, given the behavior described below it's not clear exactly what this function actually does - maybe it just logs you out? maybe it reboots the machine?)  The problem is that a simple screen lock, which one would expect to lock the screen without affecting the state of any applications, in fact closes all applications that are open when the screen lock activates. In other words, if you have the screen lock set to 5 minutes, and you walk away from your machine to get a sandwich or whatever, when you come back and unlock the machine all your apps have closed without saving their state. So far, the only work-around I have found is to entirely disable screen locking in Ubuntu 17.10. My question is, how can this be fixed so that screen locking works properly - that is: the screen locks on schedule after a period of inactivity, and when unlocked all applications that were running are still running?  I am asking a new question for the following reasons:  1) The existing answers to (and comments on)  do not address the issue, and in fact misrepresent the actual problem in ways that make me think the comment writers have not understood what is happening;  2) I cannot usefully address that question because I do not have sufficient reputation to either upvote the question or comment on it, and I cannot answer the question because I have no idea why this problem occurs;  3) I will probably never achieve a reputation of 15, much less 50, on StackExchange. Which is fine, that's not among my life goals.  4) But this question is important and deserves to be addressed. It is a drastically unacceptable bug. C# Mono - Low Level Keyboard Hook I was watching the  tutorial by Andrew Price and I was able to follow along in Cycles until I hit the part where he mixes the day and night materials together so that the lights show up on the "dark side" of the earth.  How can I recreate this effect in cycles? I thought I could use the ray information to tell me if there was incoming light but I've found no such node to give me this information. Which is the rarest badge awarded/the least popular one/the one hardest to get? Question on groups of order $pq$ What to do when the correct answer to a question changes over time? My question concerns . I asked the question, and at the time selected the correct answer. However, time passes and libraries get updated, and now a different answer is actually correct.   I should probably select this new answer as the correct answer, but I feel bad for the person that will lose 15 reputation points even though is answer was the a good answer at that given point in time.  Any suggestions?