Every square of an infinite chessboard contains a positive number. Given that the number in each square is equal to the arithmetic mean of the numbers in the four neighboring squares, is it true that all numbers are equal to each other?  I would expect that this is a well-known problem, but I was unable to find it on the Web.  Notice that the question has an immediate solution if we drop the positivity assumption, and also if we insist on all numbers to be integers.  The problem can also be stated in terms of the adjacency operator of the infinite grid graph: we want to show that it does not have eigenvectors with all components positive.  (Please, feel free to re-tag this question.)
If a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}^{+} $ satisfies the following condition  $$\forall x, y \in \mathbb{Z}, f(x,y) = \dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, y - 1)}{4}$$  then is $f$ constant function?
I was doing graph theory homework. I was counting the spaning trees of graph G. Such that G is two complete graphs on n vertices connected by two edges. I got to this expression: $$2n^{2\left(n-2\right)}+2\left(n^{\left(n-2\right)}\sum_{i=0}^{n-2}\binom{n-2}{i}\left(1+i\right)^{\left(i-1\right)}\left(n-i-1\right)^{\left(n-i-3\right)}\right)$$ I know that this expression gives the correct result, however, I was told by my teacher that it is possible to express that number in closed form e.g. without the summation. However, I have yet to find a solution. Sleeping in a batch file When writing a batch file to automate something on a Windows box, I've needed to pause its execution for several seconds (usually in a test/wait loop, waiting for a process to start). At the time, the best solution I could find uses ping (I kid you not) to achieve the desired effect. I've found a better write-up of it , which describes a callable "wait.bat", implemented as follows:  @ping 127.0.0.1 -n 2 -w 1000 &gt; nul @ping 127.0.0.1 -n %1% -w 1000&gt; nul   You can then include calls to wait.bat in your own batch file, passing in the number of seconds to sleep.   (at last!). In the meantime, for those of us still using Windows&nbsp;XP, Windows 2000 or (sadly) , is there a better way?  I modified the sleep.py script in the , so that it defaults to one second if no arguments are passed on the command line:  import time, sys  time.sleep(float(sys.argv[1]) if len(sys.argv) &gt; 1 else 1) Alignment issue in Area 51 main site - site switcher, inbox and achievements What "exactly" happens when someone turns to the Dark Side? When I saw the original Star Wars saga as a kid, the concept of the Dark Side imprinted in me was quite simple. You did a bad thing, so you're a bad person so you go to the Dark Side and keep doing bad things.  Now, obviously this doesn't hold true in reality. In my mind it is perfectly reasonable for Luke to kill the Emperor and Vader and then just flee the Death Star and rejoin the Alliance. After all they are at war, those are enemy officers that have captured him and are taunting him by threatening with death of all loved ones and the whole operation was intended to kill the Emperor anyway.   The same would hold true for Anakin. He could have killed Mace Windu to save the Chancellor and Padme but that doesn't mean he's going to then go mad and kill all Jedis including children. It doesn't add up from a merely psychological point of view. He had a set of objectives and that made him want to keep the Chancellor alive (quite valid by the way, Mace Windu is quite clumsy on that) but just after killing him he goes into "Yes master I'll do whatever you say" which from a normal point of view will NEVER ever make sense.  In both cases it seems like the assumption is once you use the Dark Side you completely turn to the Dark Side and become a different person that you were before. That would be coherent with Anakin "dying" and Vader being born.  Is that the way the Dark Side works? Is there any canon information about what happens when someone turns to the Dark Side? Specifically to whether they become a different person or how he or she actually changes?  Note: Vader is angry after he finds out Padme is dead, even though he's already turned.  Example of what I mean. In Buffy they explain how people that turn into Vampires lose their soul and it's taken over by a demon, and thus they are, in all effects, a different person with their memories. What does it mean to say "a divides b" I am not a number theorist and I am learning about relations.  I encountered a relation that says      $a \leq b$ if $a$ divides $b$   Can someone clarify what it means to a number to divide another number?  Does it mean what I think? $a$ divides $b$ if $a | b \in \mathbb{Z}$?     So given a set $S = \{a, a^2, a^3, \ldots\}$, with relation $a | b \leftrightarrow a \leq b$, does the relation hold   going from left to right or right to left? i.e. $a|a^2, a^2|a^3, \ldots$ The cardinality of the set of all finite subsets of an infinite set Let $X$ be an infinite set of cardinality $|X|$, and let $S$ be the set of all finite subests of $X$. How can we show that Card($S$)$=|X|$? Can anyone help, please? What is the optimum button size of touch screen applications? There was a not so recent blog post about the ideal button size of touch screen control sizes (sorry there is no link, the website is now a spam trap) that was based on the  form the MIT touch lab that gives some measures of the size and sensitivity of the finger. If found that averages for finger pads are between 10–14mm and fingertips are 8–10mm. This would make 10mm x 10mm a good minimum touch target size.  As the author of the blog pointed out, this ideal contact size is much larger than the suggested size for Windows and Apple devices that is in their design guides. I am wondering why there is such a difference, and whether it makes more sense to increase the size of controls on touch screen applications, or if this is not really a usability issue at all? Also, it seems that many mobile websites and apps seem to completely ignore these principles. I am sure the screen size poses a limitation to how big the interface controls can be, so is this simply a trade-off between being able to fit in as much information versus how easy it is to interact with the control? It feels as if the stylus is not something that is in much use. A similar question has been raised on the minimum/smallest size that you can get away with, but I would consider this to be a different question to what the optimum size should be, because the smallest size accommodates the minimum usability requirement, whereas the optimum size caters for the best usability requirement. Archived link of the blog (original domain shows spam)  And here is another reference it to (just to show that it is not the problem with the link but the website () UPDATE #1: another new reference that has a very precise figure of a minimum of 44 x 44 points, not sure how it is derived exactly but interesting (). Read Section 2. Provide Large Clickable Elementss. UPDATE #2: another reference, this time from  citing touch target size to 9 mm square or greater (48x48 pixels on a 135 PPI display at a 1.0x scaling plateau). They also suggest avoid using touch targets that are less than 7 mm square. UPDATE #3: something from the  that I referenced recently citing 44 x 44 points. UPDATE #4: NN/g has published an article on  and the information/conclusion remains unchanged. UPDATE #5 A really nice write-up on the need to enhance the  of common UI elements, citing the  that the &quot;minimum target size for touch or mouse should be 44×44 pixel&quot; (2.5.5 - Target Size - Level AAA). Unbiased estimator of a uniform distribution Does changing your Compute Device, change all computes of Blender? So, I have recently moved from using Blender, on my family's Nettop PC, to my personal, gaming PC. And with this move, came to option to use CUDA instead of CPU.  Which leaves the question, if I move to CUDA, does that change all computing, for like physics baking, or only for rendering?  Just a nooby question that I wanted to know, and yes, I know you can specify GPU rendering separately. Deriving the conditional distributions of a multivariate normal distribution Non-uniform distribution of p-values when simulating binomial tests under the null hypothesis I heard that under the null hypothesis the p-value distribution should be uniform. However, simulations of binomial test in MATLAB return very different-from-uniform distributions with mean larger than 0.5 (0.518 in this case):   coin = [0 1]; success_vec = nan(20000,1);  for i = 1:20000     success = 0;     for j = 1:200         success = success + coin(randperm(2,1));     end     success_vec(i) = success; end      p_vec = binocdf(success_vec,200,0.5);     hist(p_vec);   Trying to change the way in which I generate random numbers didn't help.  I would really appreciate any explanation here. How do I change my username? "ImportError: No module named _struct" configuring python-minimal When I run sudo apt-get upgrade, the dpkg error appears and interupts the process.  Error details:  Setting up python-minimal (2.7.3-0ubuntu7.1) ... Traceback (most recent call last):   File "/usr/local/lib/python2.7/runpy.py", line 162, in _run_module_as_main     "__main__", fname, loader, pkg_name)   File "/usr/local/lib/python2.7/runpy.py", line 72, in _run_code     exec code in run_globals   File "/usr/local/lib/python2.7/compileall.py", line 16, in &lt;module&gt;     import struct   File "/usr/local/lib/python2.7/struct.py", line 1, in &lt;module&gt;     from _struct import * ImportError: No module named _struct dpkg: error processing python-minimal (--configure):  subprocess installed post-installation script returned error exit status 255 dpkg: dependency problems prevent configuration of python:  python depends on python-minimal (= 2.7.3-0ubuntu7.1); however:   Package python-minimal is not configured yet.   And this problem leads to a series of dependency problems when processing the following packages:    python  gwibber-service  libgwibber3  libgwibber-gtk3  gwibber  gwibber-service-facebook  gwibber-service-identica  gwibber-service-twitter  python-all  python-dev  python-all-dev  python-apt  python-problem-report  python-apport  python-libxml2  unity-lens-gwibber  unity-scope-video-remote   Finally, nothing I can do for this problem. And several days later, the root directory is full and I even could not uninstall any software because of this problem. Is there a simple way to convert a string to Title Case? E.g. john smith becomes John Smith. I'm not looking for something complicated like , just (hopefully) some kind of one- or two-liner. Intersection of the $p$-sylow and $q$-sylow subgroups of group $G$ Why does QgsSpatialIndex.nearestNeighbor(point, 1) sometimes return 2 lines? Dual citizenship passports and visas This is about entering the USA. I have   Hungarian citizenship with a 10 year B1/B2 visa in my passport I got before Hungary became part of the VWP. Real soon now, I will become a Canadian citizen (my oath is on April 5th) and will get a passport.   Since I, as a person, have a visa, am I obliged to enter into the USA with that? Or can I just forget it and use my Canadian passport? I know I could do whatever I want if I didn't have a visa but with that in the picture I am unsure.  describes a number of situations but not mine. Is there a way to buy SO reputation? Block computer from joining a domain if it has the same name as another computer object in ADUC What's the added value of SD line over regression line when examining association between 2 variables? I'm trying to incorporate different practices to use when exploring a new data set. Especially, how to examine the association between two variables.  Steps for example (not necessarily by order):   plot a y-by-x scatter plot of the raw data to see the relationship visually. compute summary statistics for each variable (mean and sd) compute correlation coefficient r draw the OLS regression line, compute its slope and intercept etc....   I've come across the "SD line" in Freedman's Statistics book, which is defined as:     "the line that goes through the point of averages and climbs at the rate of one vertical SD for each horizontal SD" Freedman, D., Pisani, R., &amp; Purves, R. (2007). Statistics (4th edn).   Since this book ("Statistics") is a canonical textbook, I consider its choice to discuss the SD line as an indication for the line's importance. However, a simple google search for the term "SD line" doesn't yield as many independent results. Most of them come directly from Freedman's book. This tells me it's not a central concept in bivariate analyses in general.  When comparing the SD with the OLS regression line, it seems like the regression line is more informative (than the SD line) for predicting y from x. Therefore, I'm wondering if bothering to plot the SD line has any benefit or added value that I would not already have when plotting the regression line.  Example using mtcars dataset, focusing on association between weight and mpg  data(mtcars)  ## calculate means mean_wt &lt;- mean(mtcars$wt) mean_mpg &lt;- mean(mtcars$mpg)  ## calculate standard deviations sd_wt &lt;- sd(mtcars$wt) sd_mpg &lt;- sd(mtcars$mpg)  ## scatter plot plot(x = mtcars$wt, y = mtcars$mpg)  ## add the "point of averages" points(mean_wt, mean_mpg, col = "red", cex = 1.5, pch = 16)  ## calculate the slope of the sd line slope &lt;- -1*sd_mpg/sd_wt  ## plot the sd line curve(expr = x*slope + (mean_mpg - slope*mean_wt), add = TRUE, col = 'blue', lwd = 2, type = "l", lty = 2)  ## plot the regression line model &lt;- lm(mpg ~ wt, data = mtcars) abline(model, col = "orange", lwd = 2)  ## legend legend("topright",        legend = c("Regression line", "SD line"),        col = c("orange", "blue"),        lty = c(1, 2),        lwd = c(2, 2))     Thus, my question: how can the SD line increase one's understanding about the relationship between two variables, in a way that's either adding or complementing on what the regression line already tells?