If $x$ and $y$ are irrational numbers then $x$ to the power of $y$ is irrational I am asked to prove or disprove this statement. To do so I got an idea to use the contra-positive, for that I need to prove if $x$ to the power of $y$ is rational then $x$ and $y$ are rational. I took $\ln$ for both sides, because keeping some element in power would make the sum more difficult, now I have $y\ln(x) = \ln(p/q)$. How could I show that $x$ and $y$ are rational?
Can an irrational number raised to an irrational power be rational?  If it can be rational, how can one prove it?
The number of solutions to an $n^{th}$ order differential equation. For an $n$th order differential equation, why are there always $n$ solutions? Why exactly $n$, not $n - 1, n+1$ or infinite many?    Addendum by LePressentiment :   This is motivated by P176 on Strang's Intro to Lin Alg, 4th Ed. An $n$th order differential equation, how many basis functions does it have? I'd guess $n$, because the diffl eqn have $n$ linearly independent solutions (predicated on Julián Aguirre's answer), all of which look to span the nullspace/solution space of the homogeneous ODE. Let n be an even integer. How many different ways can I choose subsets of the set {1,2...,n} Let n be an even integer. How many different ways can I choose subsets of the set {1,2...,n} if they all have to have odd size, but the intersection of any two of them has to have even size?  Please help me answer this question or lead me in the right direction. Thank you :) I see that Stack&nbsp;Overflow is very Linux specific. Doesn't Unix &amp; Linux and Ask&nbsp;Ubuntu cover much of the same ground? I see that Unix &amp; Linux and Server&nbsp;Fault also cater to many Linux server questions. Similarly, Ask&nbsp;Ubuntu and Super&nbsp;User seems to contain most like questions. I want to say "someone (or something) intends to help, but instead it makes things worse". Is there any succinct expression or phrase for this?  Thanks. I try to launch Firefox over SSH, using  ssh -X user@hostname   and then  firefox -no-remote   but it's very very slow.  How can I fix this? Is it a connection problem? Why split the  tag when writing it with document.write()? On our minecraft server, I've built a Nether portal in my basement. Which connected two-way (tested) to a location in the Nether.    This nether portal took me to an area in the nether which was a small cave. I dug upwards and surfaced close to a Nether Fortress. After clearing the fortress of Blaze spawners, I decided to 'move in'. So I went back to the portal, dismantled it, and reactivated in in a location in the Fortress.    The fortress would be within the same 128m radius I would assume as it's really close to the original place. Literally some steps to the surface, then a ladder up the nearest Nether Fortress bridge support.  Now, taking this portal back to my original doesn't work, I spawn deep underground in an Abandoned Mine Shaft.  I read somewhere that my Nether portal has found a 'better match' and this could be remedied by disabling the 'wrong' portal in the mine shaft.  However, upon multiple attempts, a new nether portal is dynamically created in the same area. See screenie for example. You'll see two 'stacked' portals on the left (from my different attempts at disabling) and then the current active one on the right.    My question is, how can I make sure these portals go to the correct place. Is it a matter of returning the portal to it's original location or am I stuck for returning to the basement? I can't solve this problem:  Suppose $f$ is polynomial function of even degree $n$ with always $f\geq0$.  Prove that $f+f'+f''+\cdots+f^{(n)}\geq 0$. What is a NumberFormatException and how can I fix it? Why does $1+2+3+\cdots = -\frac{1}{12}$? $\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) &gt; 1$.  Why should analytically continuing to $\zeta(-1)$ give the right answer? "Closed" form for $\sum \frac{1}{n^n}$ Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating series etc.  One series that popped into our discussion was $\sum\limits_{n=1}^{\infty} \frac{1}{n^n}$.  Proving the convergence of this series is trivial but finding the value to which converges has defied me so far. Mathematica says this series converges to $\approx 1.29129$.  I tried Googling about this series and found very little information about this series (which is actually surprising since the series looks cool enough to arise in some context).  We were joking that it should have something to do with $\pi,e,\phi,\gamma$ or at the least it must be a transcendental number :-).  My questions are:   What does this series converge to? Does this series arise in any context and are there interesting trivia to be known about this series?   I am actually slightly puzzled that I have not been able to find much about this series on the Internet. (At least my Google search did not yield any interesting results). How do I add a repository from behind a proxy? I've a problem at the office. We're behind a proxy (which is set and applied at ubuntu proxy settings) and when I try to add a repository from terminal, I get:      Error reading --some url here--: urlopen error [Errno 113] No route to host   I've tried with   Launchpad-getkeys script. I get this (image)      I've tried with another "hack", without luck   Press Alt-F2 and type gksu gedit /usr/lib/python2.6/dist-packages/softwareproperties/ppa.py Find line 88, change keyserver.ubuntu.com to hkp://keyserver.ubuntu.com:80 Save, close and reboot.    Does anyone know if I could solve this problem in any way? Thanks How can I access cloth properties via python? I want to manipulate options like Disk Cache with python:    Any Ideas? msbuild, defining Conditional Compilation Symbols I'm possibly just blind, but is there a command line to specify conditional compilation symbols in MSBUILD?  I currently have this Line in my buildscript:  SET MSBUILD=C:\WINDOWS\Microsoft.NET\Framework\v2.0.50727\MSBuild.exe SET CONFIG=Debug %MSBUILD% /p:Configuration=%CONFIG% /p:OutputPath=..\..\output source\MyProject\MyProject.csproj   And I'd like to add a condition. In Visual Studio, i can just go into Project Properties => Build => Conditional compilation symbols, but I have not seen that option for msbuild?  Bonus Karma if you know if I can completely override all symbols already specified in the .csproj files to make sure that only the conditionals from my Buildscript go in. Ignoring accented letters in string comparison I need to compare 2 strings in C# and treat accented letters the same as non-accented letters. For example:  string s1 = "hello"; string s2 = "héllo";  s1.Equals(s2, StringComparison.InvariantCultureIgnoreCase); s1.Equals(s2, StringComparison.OrdinalIgnoreCase);   These 2 strings need to be the same (as far as my application is concerned), but both of these statements evaluate to false. Is there a way in C# to do this? TeXnicCenter & Adobe Acrobat 10 Chain skip happening seemingly at random with a new chain What is the earliest work considered to be Science Fiction? How to graphically represent a language I read a YA novel in the early 2000s that I think takes place on Earth some time in the future. The human race has been enslaved by some other race, possibly aliens. If I remember correctly, humans were kept as slaves for some purpose and then exterminated at age 16, and I think some girls were kept to breed.  I remember scenes of a band of teenagers walking around sewers fighting these aliens who bled a blue "ichor." The only thing I distinctly remember about the book was that it was the first place I saw the word "ichor."  This is a pretty sorry description; but does anyone have a clue what this book was?