Find all ordered pairs $(a, b)$ such that $a$ and $b$ satisfy $a^b$ = $b^a$ and $a$ and $b$ are integers.   The only way I can think of solving this question is by trial and error, but there must be a more efficient way.
We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?
I am having an access problem to some files and directories that I have tracked down to the group assigned to access them.    When I manually change the name of the group in the properties/permissions menu for a directory from the default setting (MyGroup) to www-data used by my Drupal Website, the directory error messages I get are gone.    There are a large number of files and directories that require this change.  If believe using chmod will change the owner which is set correctly and being new to Ubuntu, I am reluctant to experiment without a better understanding of how owner and groups work in the permissions settings.   What command changes the Group setting for a directory? Importing DWG into QGIS project? I have a lot of DWG files like basemap, and water and wastewater network.  How do I import these files into my QGIS project? Mask and convert color in photo/video I am very new to blender still.  I want to simply change the color of an object in the video.  I want to know if there is good way to do this.  I am including what I have to test. Lower Semicontinuity Concepts Escaping quotes in zsh alias Following on from  about stripping newlines out of text, I want to turn this into a zsh alias as follows:  alias striplines=' awk " /^$/ {print \"\n\"; } /./ {printf( \" %s \",$0);}"'   I've tried escaping the quotes inside the awk script, but I'm getting this error:  awk: (FILENAME=bspsrobustness FNR=1) fatal: division by zero attempted   (The file is called bspsrobustness)  Is there a way to do what I want? I suppose I could turn this into an awk script rather than a zsh alias, is that my best option? Bold name in bib when having a prefix in last name How do you detect where two line segments intersect? Determining if a game is CPU- or GPU-limited What is a reliable, but not too time-consuming way to determine if a certain game is limited by the graphics card or by the processor speed?  I have some preconceptions about whether I'm mostly GPU or CPU limited, but I'd like to verify if I'm right about that. I have some ideas how I would go around and test that but performing accurate benchmarks is notoriously tricky and I don't have much experience with it.   So I'm wondering what would be an easy way to determine the bottleneck, having only one computer with a certain configuration available? What tools would I use for that purpose?  It would also be interesting to determine if the amount of VRAM on my graphics card is a limiting factor. If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$? Let $f(x)$ be a continuous real-valued function on $[a,b]$ and $M=\max\{|f(x)| \; :\; x \in [a,b]\}$. Is it true that: $$ M= \lim_{n\to\infty}\left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n} ? $$  Thanks! I understand that faster-than-light communication is impossible when making single measurements, because the outcome of each measurement is random. However, shouldn't measurement on one side collapse the wave function on the other side, such that interference effects would disappear? Making measurements on "bunches" of entangled particles would thus allow FTL communication, by making observed interference effects appear or disappear. How does such an experiment not:  1) Clearly imply that faster-than-light communication is possible?  or  2) (if #1 is rejected) Imply that measurement of one half of an entangled pair does not cause the collapse of the other half's wave function.  Why doesn't this thought experiment clearly show that if we maintain that FTL communication is ruled out, we must also rule out "universal collapse" in the Copenhagen interpretation?  EDIT: Here is an example of an explicit experiment (though I think experts could come up with something better):  You can entangle a photon with an electron such that the angle of the photon is correlated with the electron's position at each slit of a double slit experiment. If the photon is detected (it's outgoing angle measured), then which-path information is known, and there is no interference. If the photon is not detected, the interference remains.   The experiment is designed such that the photon and electron go in roughly opposite directions, apart from the tiny deflection which gives which-path information. You set up a series of photon detectors 100 ly away on one side, and your double slit experiment 100 ly away in the opposite direction. Now you produce the entangled pairs in bunches, say of 100 entangled pairs, each coming every millisecond, with a muon coming between each bunch to serve as a separator.   Then the idea is that someone at the photon detector side can send information to someone watching the double-slit experiment, by selectively detecting all of the photons in some bunches, but not in others. If all of the photons are detected for one bunch, then the corresponding electron bunch 200 ly away should show no interference effects. If all of the photons are not detected for one bunch, then the corresponding electron bunch 200 ly away would show the usual double-slit interference effects (say on a phosphorus screen). (Note that this does not require combining information from the photon-detector-side with the electron-double-slit side in order to get the interference effects. The interference effects would visibly show up as the electron blips populate the phosphorus screen, as is usual in a double-slit experiment when which-path information is not measured.)  In such a way the person at the photon detectors can send '1's and '0's depending on whether they measure the photons in a given bunch. Suppose they send 'SOS' in Morse code. This requires 9 bunches, and so this will take 900 milliseconds, which is less than 200 years. The point is that such an experiment would only work if you assume that the measurement of the photon really does collapse the wave function nonlocally. Downloading US Census data? How can I write a spiral symbol? How can I get a spiral like    There seems to be a spiral-like unicode codepoint:   I tried  \documentclass[varwidth=true, border=2pt]{standalone} \usepackage[ngerman]{babel}  \usepackage[utf8]{inputenc} % Color \usepackage{color} \definecolor{lightgray}{gray}{0.95}  \DeclareUnicodeCharacter{43612}{\spiral}  %document \begin{document}{\textcolor{lightgray}{a}}\spiral{\textcolor{lightgray}{b}} \end{document}   but I got  ! Undefined control sequence. l.13 ...document}{\textcolor{lightgray}{a}}\spiral                                               {\textcolor{lightgray}{b}}   edit: I know . None of the answers gave me a spiral symbol. I am getting this message every time I do something like starting or stopping a service.  perl: warning: Setting locale failed.    perl: warning: Please check that your locale settings:            LANGUAGE = "en_US:en",            LC_ALL = (unset),            LC_MESSAGES = "en_US.UTF-8",            LANG = "en_US.UTF-8"        are supported and installed on your system.    perl: warning: Falling back to the standard locale ("C").    locale: Cannot set LC_CTYPE to default locale: No such file or directory    locale: Cannot set LC_MESSAGES to default locale: No such file or directory    locale: Cannot set LC_ALL to default locale: No such file or directory    (Reading database ... 21173 files and directories currently installed.)   Removing bind9 ...    * Stopping domain name service... bind9                                        [ OK ] Processing triggers for man-db ...    locale: Cannot set LC_CTYPE to default locale: No such file or directory    locale: Cannot set LC_MESSAGES to default locale: No such file or directory    locale: Cannot set LC_ALL to default locale: No such file or directory      How do I fix this error ? How do I expire a PHP session after 30 minutes? I need to keep a session alive for 30 minutes and then destroy it. What is the name of this game involving racing to the other side of the board and putting down barriers? Does anyone recognize this game, where players take turns in a square board, each turn a player can move his pawn or place a wall and the objective is to reach the other side of the board before the other player.  Here is a picture of a near end game. Here is the documents needed.     Here are the documents needed.       Which is grammatically correct and why? My guess would be the second one because of the plural form. How are Python's Built In Dictionaries Implemented? Does anyone know how the built in dictionary type for python is implemented?  My understanding is that it is some sort of hash table, but I haven't been able to find any sort of definitive answer. Easy way to change apache www location Can anyone hint me to prove:  $\forall n\in \mathbb{N}: \exists$ Fibonacci numbers $ F_{i_1},\ldots,F_{i_k}$ such that: $$\sum F_{i_k}=n$$    Note: Every Fibonacci number can appear only once. Thanks What is the name of the character $\wp$? What is the name of the character whose code is $\wp$ ?  Thanks