Given that $a > 0$, $b > 0$ and $b > a$, prove the solution of $a^b = b^a$ I have this question in my mathematics textbook:  If $a$ and $b$ are positive integers such that $b &gt; a$, when will $a^b = b^a$?  I had no clue where to start. So I looked at the answer:  If $a$ and $b$ are positive integers such that $b &gt; a$, $a^b = b^a$ when $a = 2$ and $b = 4$. It is not easy to prove that this is the only solution; students are only expected to use guess and check to find a solution.  The answer provides no proof for the solution, or insight into the method for finding the solution, apart from a vague mention of &quot;guess and check&quot;. I wish to find the (algebraic, if possible) proof for the question. So to sum up:  Given that $a &gt; 0$, $b &gt; 0$ and $b &gt; a$, prove that the only solution of $a^b = b^a$ is that $a = 2$ and $b = 4$.
Given that $x^y=y^x$, what could $x$ and $y$ be? It's not too difficult to figure out that $x$ and $y$ can both be 1, and also $x$ can be 2 and $y$ can be 4 (and vice versa). But I can't rule out if there are other solutions. Does it have anything to do with inverse functions? Is there a way to see the solutions by graphing, or algebraically?
Can you choose which version of a game to play on Steam? Where are the Unity desktop wallpapers located? given a polynomial $p(x,y)$ from $\mathbb Z[x,y]$. I want to substitute $x=1$ leaving $y$ as it is. The command Value(p,[1,y]) does not work. Can you give me a hint?  After that it gets a little more complicated. Let  $$p(x,y)=x^6+y^6$$ After substituting $x=1$ we have $p(1,y)=1+y^6$. Given a natural number $j$. I want to find all expressions $y^j$ in $p$ and substitute these $y^j$ by $1$ and the others by $0$. In this case (and for $j=6$) we would get $1+1=2$. If $$p(x,y)=x^4y^2+x^2y^6$$ we should get $0+1=1$. Is there any way to do this?  Thanks for reading! edit: Well, it turns out I used the function Value wrong. The right code for my problem would be Value(p,[x,y],[1,y]). So my first question is solved. Algebraic proof that if $a>0$ then $1+a^9 \leq \frac{1}{a}+a^{10} $ Prove that if $a&gt;0$ then $1+a^9 \leq \frac{1}{a}+a^{10} $.   Using the fact that $a \gt 0$, multiply by $a$ on both sides and get everything to one side we have; $a^{11}-a^{10}-a+1 \geq 0$. By factoring $(a^{10}-1)(a-1) \geq 0 $.  I am not sure how to proceed any further. Is there any canon evidence that FN-2187 actually worked in sanitation? Finn tells Han that he worked in "Sanitation" on Starkiller.  FINN The flooding tunnels are over that ridge. We'll get in that way. HAN What was your job when you were based here? FINN Sanitation. HAN Sanitation? Then how do you know how to disable the shields? FINN I don't. I'm just here to get Rey   It just occurs to me - the "I was in sanitation" line given to Han seems to be the only actual canon evidence of "Sanitation" job.   But was it the truth"?!?!  Finn is always trying to create a positive impression of himself with the Resistance, from the get-go:   He tells Poe that he's rescuing him because that's the "Right thing to do" (with Poe quickly seeing through the fib: "You need a pilot!") He tells Rey that he's Resistance (she buys that because she's a romantic 19 year old still playing with X-Wing pilot helmet) He tells Han that he's Resistance (that one lasts about a minute - Han was not a romantic 19 year old even at 19 :)   In reality, we know that he was 1% of his class, model stormtrooper cadet, personally groomed for promotion by Captain Phasma (; ).  However, I am pretty sure telling Han "Hey, I was 1% of my class, model stormtrooper cadet, groomed for promotion by Captain Phasma"  isn't high on his list of brand management plans.  Far better to pretend to be a lowly "sanitation" worker than top-flight cadet that he really was.  So, my question is - do we have any direct canon evidence proving that the "sanitation" job was real and not a fib to make himself look better to Han/Resistance?  Knowing where trash compactor was isn't direct evidence, though it is circumstantial one. Gave up waiting for root device on Ubuntu What is the difference between "lay" and "lie"? Make the network profile editable on the main Stack Exchange page Currently, the Stack Exchange Network Profile links to all the network accounts and acts as a centralized profile page. Seeing that the ability to edit the profile is available from the profile pages of all *.SE sites, I think the ability to edit profiles should be available on that Network Profile page as well.  It will be consistent with the nature of the centralized profile page and will better the user experience.   EDIT: As the current Network profile, is actually the oldest profile on the Network, providing an edit option on the Network profile which links to the oldest profile edit page, would be one way to implement this. Movie Identification: People record their experiences on reel-to-reel I know that this sounds like Strange Days, which I haven't seen, but from the Wikipedia description I don't think that's it.  I saw this back in the early-to-mid-90's on TV, pretty sure it was a premium channel. It follows a research group who develops a way to record experiences. Not thoughts, but at least the sights, sounds, smells, touch, etc. They record it onto this reel-to-reel tape, it looked striated and golden in color. The protagonist is the head of the research team.  I don't remember a great deal of the plot, but there were conspiracies surrounding the invention. The lab clown decides to give it a whirl by putting the bike-helmet-sized apparatus on his head and going out to have a little fun, in the classic wine, women, and song fashion. Once he returns, an older member of the team finds the tape and finds a section where the clown is having sex. Later on, the protagonist, who is the lead researcher, finds the older member shut up in his house seemingly for days and nearly comatose. The older researcher had the playback device over his head and is non-responsive. Worried, the protagonist removes the helmet and puts it on himself, only to find that the older researcher had snipped a few seconds of tape and had them running on a non-stop loop; the moment of orgasm for the clownish researcher. The older researcher eventually comes to and awkwardly apologizes.  I think there were also experiments in recording the sensorum of a lab monkey, and that played into the plot. I remember the plot seemed fairly thin and dull for what seemed to be a pretty neat idea. I don't desperately feel the need to watch the movie, but I'm mildly curious if my impressions would differ 20 years later now. Modelling with more variables than data points I'm fairly new to Machine Learning/Modelling and I'd like some background to this problem. I have a dataset where the number of observations is $n&lt;200$ however the number of variables is $p\sim 8000$. Firstly does it even make sense to consider building a model on a dataset like this or should one consider a variable selection technique to start with such as ridge regression or Lasso? I've read that this situation can lead to over-fitting. Is that the case for all ML techniques or do some techniques handle this better than others? Without too much maths a simple explanation on why the maths start to breakdown for $p&gt;n$ would be appreciated. It's taken me almost 10 years of Linux usage to ask this question. It was all trial and error and random late-night internet surfing.  But people shouldn't need 10 years for this. If I were just starting out with Linux, I'd want to know: When to alias, when to script, and when to write a function?  Where aliases are concerned, I use aliases for very simple operations that don't take arguments.   alias houston='cd /home/username/.scripts/'   That seems obvious. But some people do this:  alias command="bash bashscriptname"   (and add it to the .bashrc file)   Is there a good reason to do that? I'm trying really hard, but I genuinely can't think of any circumstances in which I'd want to do that. So, if there is an edge case where that would make a difference, please answer below.  Because that's where I would just put something in my PATH and chmod +x it, which is another thing that came after years of Linux trial-and-error.  Which brings me to the next topic. For instance, I added a hidden folder (.scripts/) in the home directory to my PATH by just adding a line to my .bashrc (PATH=$PATH:/home/username/.scripts/), so anything executable in there automagically autocompletes.  If I needed to.  I don't really need that, though, do I? I would only use that for languages that are not the shell, like Python.  If it's the shell, I can just write a function inside the very same .bashrc:  funcname () {   somecommand -someARGS "$@" }   As I stated, I found a lot of this out through trial and error. And I only truly saw the beauty of functions when my computer died and I was forced to use the computers of the people around me when they weren't using them.   Instead of moving a whole directory of scripts from computer to computer, I ended up just replacing everyone else's .bashrc with my own, since they had never even made a single modification.  But did I miss anything?   So, what would you tell a beginning Linux user about when to alias, when to script, and when to write a function?   If it's not obvious, I'm assuming the people who answer this will make use of all three options. If you only use aliases, or only use scripts, or only use functions—or if you only use aliases and scripts or aliases and functions or scripts and functions—this question isn't really aimed at you. An ideal whose radical is maximal is primary I've got to prove that an ideal $Q$ whose radical is a maximal ideal is a primary ideal. That is, I want to prove that if $xy\in Q$, then $x\in Q$ or $y^n\in Q$ for some $n&gt;0$.    I've been trying for a while and I'm not sure where to begin. All I've got is that if $\text{Rad}(Q)$ is maximal and by definition it's the intersection of all prime ideals $P_i$ containing $Q$, then it must be equal to each of these $P_i$. So there is only 1 prime ideal containing $Q$, namely $\text{Rad}(Q)$.  Could anyone point me in the right direction? Thanks for any replies. I'd never heard of anycast until a few seconds ago when I read "".  The wikipedia "" article on it is quite formal and doesn't really evoke a mental picture of how it would be used.  Can someone explain in a few informal sentences what "anycast" is, how you configure it (just in a general sense), and what its benefits are (what does it make easier)? Adding x and y values while creating a label QGIS I'm trying to create a label on a map in QGIS.  I can open the setting where there are an X and Y button and a few other properties, like opacity and distance.  I'm pretty sure the problem is that I need to enter the proper X and Y coordinate here, but I don't know how to get to where it lets you enter the coordinates. For which real values of t does the following system of linear equations:  $$ \left\{  \begin{array}{c} tx_1 + x_2 + x_3 = 1 \\  x_1 + tx_2 + x_3 = 1 \\  x_1 + x_2 + tx_3 = 1 \end{array} \right.  $$  Have:  a) a unique solution?  b) infinitely many solutions?  c) no solutions?  I haven't done linear algebra in almost a year, so I'm really rusty and could use some pushes in the right direction. I am currently facing a "performance problem" while using grep. I am trying to locate the occurrences of many (10,000+) keywords in many (think Linux kernel repository size) files. The objective is to generate a kind of index for each keyword:   keyword: my_keyword filepath 1 filepath 2 ------ keyword: my_keyword2 ...   Being fairly new to shell scripting, I have tried a few naive approaches:  while read LINE; do     echo "keyword: "$LINE &gt;&gt; $OUTPUT_FILE     grep -E -r --include='Makefile*' "($LINE)" . &gt;&gt; "$OUTPUT_FILE"     echo "-----" done &lt; $KEYWORD_FILE   This takes about 45 minutes to complete, with the expected results.  Note: all they keywords I search for are located in a file ("KEYWORD_FILE", so fixed before anything starts), and the files in which to search can be determined before the search starts. I tried to stored the list of files to search in before the search like that:  file_set=( $(find . -name "Kbuild*" -o -name "Makefile*") )   and then replace the grep call by  echo {$file_set[*]} | parallel -k -j+0 -n 1000 -m grep -H -l -w "\($LINE\)" &gt;&gt; "$OUTPUT_FILE"   It takes about an hour to complete&hellip;  Question: considering I can use whatever technique I want, provided it's sh compliant, how can this be done "faster" and/or more efficiently ?  Perhaps grep is not the tool to use, and my use of parallel is wrong&hellip; any ideas are welcome! Checking to see if a user has a licence for a managed package So I'm aware that there is a method on UserInfo called isCurrentUserLicensed which lets you check if the user running the code has a licence for a namespace that you specify, i.e. for a particular managed package.  What I want to know is whether there is a way I can test to see if some other user (assume I have their ID) has a licence for a particular package. Has anybody found a way to do this? What is the 64-bit Firefox Beta PPA? First time using bevel tool, doesn't bevel evenly. What am i doing wrong? First off I'll let you know I'm a total noob at blender I've used 3ds Max up until this point, but the company I'm working for at the moment needs me to use blender so I'm trying to make the transition over.  Literally all I'm making at the moment is a simple book and cannot understand how the bevel tool isn't working similar to how the chamfer tool would in 3ds Max.  I want a nice curved edge for the spine of the book however when I press control + B and try to bevel it bevels the edge unevenly with the new bevel pushing forward.    Is there a way I can get it to bevel evenly like it does in Max? How do I reinstall Unity? I tested out Unity 5, and of course, it broke things, so I tried downgrading by    sudo apt-get install ppa-purge sudo ppa-purge ppa:unity-team/staging   It removed Unity completely.    apt-get install unity   seemed to work, but the login screen now offers only Gnome.  How do I get back to Unity 4?