Can such a function be continuous? Let $f$ be a function from $\Bbb R$ to $\Bbb R$ such that $f(x)$ is rational when $x$ is irrational, and $f(x)$ is irrational when $x$ is rational. Can $f$ be continuous?   Thanks for your help.
No continuous function switches $\mathbb{Q}$ and the irrationals Is there a way to prove the following result using connectedness?  Result:  Let $J=\mathbb{R} \setminus \mathbb{Q}$ denote the set of irrational numbers. There is no continuous map $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(\mathbb{Q}) \subseteq J$ and $f(J) \subseteq \mathbb{Q}$.
Antonym of "phobia" If a phobia is to have an irrational fear of something, what is the word for having an irrational affinity for something?  For example a numerologist may fear the number 13, but be attracted to (or even have a love for) the number 8. (Some would say this is irrational.) Scalar Field Theories The Lagrangian density for a single real scalar field theory is \begin{equation}\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)\end{equation} I have often seen this written \begin{equation}\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\end{equation} which appears to differ in the kinetic term.  Specifically \begin{equation}(\partial_{\mu}\phi)^{2}=(\partial_{\mu}\phi)(\partial^{\mu}\phi)\neq\partial_{\mu}\phi\partial^{\mu}\phi=(\partial_{\mu}\phi)(\partial^{\mu}\phi)+\phi\partial_{\mu}\partial^{\mu}\phi\end{equation} where the product rule has been used in the last equality. Is this just sloppy notation (which is acceptable because only first derivatives in the field are considered anyway), or is my maths very bad? Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable over $\Bbb Z$.  My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this?  $(0)$.  $(f(X))$, where $f(X)$ is an irreducible polynomial.  $(p)$, where $p$ is a prime number.  $(p, f(X))$, where $p$ is a prime number and $f(X)$ is an irreducible polynomial modulo $p$. Pages documents created on Yosemite cannot be opened by Pages on Yosemite after latest update (5.5.2) Seemingly every single document that I created in the previous version of Pages (I update regularly whenever I see that a new update is available) on this same MacBook Pro with Yosemite will now not open with Pages after the latest update (5.5.2). I am still on Yosemite and nothing has changed aside from updating Pages, Numbers &amp; Keynote.  All of my documents are stored on my internal hard drive. When I attempt to open them I get a message stating the following:     This document can't be opened because it's too old. To open it, save it with Pages '09 first.   As I mentioned above, however, these documents were created in whichever version was right before the current version on the same computer with the same operating system (Yosemite).  Can anybody help? These are time sensitive documents. For now, I may try opening them with my iPad, but wow, if I knew this was going to happen I certainly would not have updated. Can Recovery Mode install OS X onto a blank SSD? Why does $a^n - b^n$ never divide $a^n + b^n$? UTF-8 all the way through Boolean and Texts I have used Hexagon for years but find that it sometimes crashes when trying to do Boolean operations with texts.  I have been trying to make the two tables of the Ten Commandments and have been trying to use Blender.  I did it about two years ago in Blender and lost everything on my computer so I am trying again with the latest version of Blender and am having difficulty.  I start with a cube and then get it close to the dimensions I would think the Ten Commandments might look like. I then bevel the top using the bevel modifier. But in doing it this way I don't have any cross sub divisions on the face of the object. I have tried sub dividing first but then I find that I can't bevel the top arches as far as I would like. I still don't know how to add divisions after I bevel the top. Yes I am still a novice at this program.  In any case I did figure out how to make text and turn them into an object. When I make the texts I do all four of the commandments for one side and then the other six for the other side. So this makes allot of texts to cut out.  When doing the Boolean operation in Difference sometimes nothing happens. I am thinking I need to have some sub - divisions on the face of the sized cube after it is beveled but am not really sure if this is the problem.  This whole thing about normal's is still Greek to me. I have watched some videos on how to cut letters into a cube but for some reason it does not work this easy after I bevel the top of the cube. Or it could be that Blender does not like a bunch of texts all at once.  Could someone show me a video or give me some simple steps to make my project work? Prove $\frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$. So I have to prove $$ \frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}.$$  I rearranged it   $$ a^2bc + ab^2c + abc^2 \leq b^2c^2 + a^2c^2 + a^2b^2 .$$  My idea from there is somehow using the AM-GM inequality. Not sure how though. Any ideas?  Thanks Template specialization based on inherit class How to make Unicode charset in cmd.exe by default? Workflow for developing add-on/script I'm trying to figure out a good workflow for developing scripts and add-ons in Blender. Here's what I'm currently doing as I try my luck with a PyNode add-on:   Open blender Create a text editor panel Open my script (external text file) Edit my script in a nice external editor Reload the script Save the .blend file Run the script   Now that I can see what my code does, this Blender instance has been modified, so I can't (make changes to my script and) run my script again. So I do this:   Close blender Open blender with the .blend from step 6 and repeat steps 4-7, over and over and over..   Is there a better workflow for this sort of thing? Hi is it possible to exclude traffic from a certain country in google analytics?  If yes, how do I do it? How to create & execute a script file Is there a way to create and execute a script file in Ubuntu 14.04?  Perhaps something like Windows batch files.  Specifically, I mean the default scripting language of the Ubuntu terminal (Bash).  How do you make a text file containing bash commands which will be run by the Ubuntu terminal from the top of the file to the bottom. Why are two universal objects isomorphic? Browsing some , it is claimed that two universal objects are isomorphic because they admit the identity morphism on themselves. But from two identity morphisms, how would you get an isomorphism between the two? Variational Derivation of Schrodinger Equation I'm trying to do something similar to this epsilon-delta graph I've found.  Specifically, I'd like to copy the placements and arrows, like the L+e and L-e coming out of L. Same thing with the x0 + d and x0-d coming out of the x0. Can someone help me with this? This is what I have so far.    \begin{tikzpicture} \begin{axis}[ ,         legend pos=north east,         axis x line=middle,         axis y line=middle,         x tick label style={/pgf/number format/fixed,                             /pgf/number format/fixed zerofill,                             /pgf/number format/precision=1},         y tick label style={/pgf/number format/fixed,                             /pgf/number format/fixed zerofill,                             /pgf/number format/precision=1},         grid = major,         width=8cm,           height=6cm,         grid style={dashed, gray!30},         xmin=0   ,     % start the diagram at this x-coordinate         xmax= 4,    % end   the diagram at this x-coordinate         ymin= 0,     % start the diagram at this y-coordinate         ymax= 11,   % end   the diagram at this y-coordinate %axis background/.style={fill=white},         xlabel=$x$,         ylabel=$f(x)$,         tick align=outside,         enlargelimits=false] % plot the stirling-formulae %\addplot[domain=0:200, red, ultra thick,samples=100] {K/(1 +    ((K-X0)/X0) * exp(r*x))    }; \addplot[domain=0:200, red, ultra thick,samples=100] {3*x-1 };  \addplot[line width=1pt, dashed, mark=none, black] coordinates {(0,8) (3,8)}; \addplot[line width=1pt, dashed, mark=none, black] coordinates {(3,0) (3,8)}; \addplot[line width=1pt, dashed, mark=none, blue] coordinates {(3.33,0) (3.333,9)}; \addplot[line width=1pt, dashed, mark=none, blue] coordinates {(2.666,7) (2.666,0)}; \addplot[line width=1pt, dashed, mark=none, blue] coordinates {(2.666,7) (0,7)}; \addplot[line width=1pt, solid, mark=none, blue] coordinates {(3.333,9) (0,9)}; \addplot[holdot] coordinates{(60,8.66)};  % 1\addlegendentry{$f(x) = 10 \sin (x)$}  \end{axis}   \end{tikzpicture}   which gives Point Release upgrade from 14.04.1 to 14.04.2? I wanted to upgrade to 14.04.2 point release from 14.04.  What I did was:  sudo apt-get update &amp;&amp; sudo apt-get upgrade &amp;&amp;  sudo apt-get dist-upgrade -y   Then I wanted to upgrade to 3.16 kernel, since it wasn't done with the the above command, so what I did was:  sudo apt-get install linux-generic-lts-utopic xserver-xorg-lts-utopic libegl1-mesa-drivers-lts-utopic xserver-xorg-video-all-lts-utopic xserver-xorg-input-all-lts-utopic sudo apt-get install linux-signed-generic-lts-utopic   Now when I do:   uname -a I get   Linux Y500 3.16.0-33-generic #44~14.04.1-Ubuntu SMP Fri Mar 13 10:33:29 UTC 2015 x86_64 x86_64 x86_64 GNU/Linux  cat /etc/os-release | grep VERSION= I get  VERSION="14.04.2 LTS, Trusty Tahr"   And when I do lsb_release -a I get  Distributor ID: Ubuntu Description:    Ubuntu 14.04.2 LTS Release:    14.04 Codename:   trusty   I know that uname -a and lsb_release -a both refer to different info, but why does  cat /etc/os-release | grep VERSION= show different version? Since a few months, on one of the desktop Ubuntu 14.04 systems I use, CPU has regularly been going up to 80-100% for no apparent reason. htop would usually attribute responsability to file system processes, such as nemo or ntfs. To get CPU back down I would kill these process, which in some cases would lead to loss of access to the file system and a consequent restart.  Days ago I noticed that whenever this situation happened another process was always present also using high CPU: mediascanner-service-2.0. Some research lead me to  that simply advised its removal. So I did.  The end result has been unexpectedly positive: not only are the high CPU load episodes gone, the system is altogether much faster. The speed up of the Lens is dramatic: finding an application like Calc now takes 2 seconds at most when previously it took some 20 seconds. Finding a file takes now some 5 seconds, whereas before it took in the order of 30 seconds. Heavy programmes like Eclipse are starting up faster and workspaces are more fluid. There seems to be also an overall reduction in the frequency of disk access.  What is exactly this mediascanner2.0 package? Is it part of Unity 7 or a dependency that gets installed a posteriori? Is it possible to prevent its installation?  Update: following  it is possible to track this package down to indicator-session:  $ sudo aptitude why mediascanner2.0 i   indicator-session         Recommends indicator-applet (&gt;= 0.2) | indicator-renderer i A unity8                    Provides   indicator-renderer                             i A unity8                    Recommends unity-scope-mediascanner2                      pi  unity-scope-mediascanner2 Depends    mediascanner2.0                                  Apparently, mediascanner2.0 (and the Unity 8 shell) are part of the base 14.04 system. Left to answer is the exact function of this rogue package. I'm wondering a point about N'Zoth. He will summon my dead deathrattle minions, but will he do it in a specific order (death order, mana cost order, ...) ? Or randomly ?  If there is already a few minions on my side of the board, I'd like to predict which minions will be summoned and which ones will not (because board is full). If it's by death order, this generally means lower cost minions (like fire bat) will be summoned before the interesting big drops (sylvanas, highmane or so).