Let $f$ be a $C^{\infty}$-function from an open subset $U$ of $\mathbb{R}^n$ to $\mathbb{R}$. If $f(\mathbb{Q}^n\cap U) \subseteq \mathbb{Q}$ can we conclude that $f$ is a rational function?
What is known about $\mathcal C^\infty$ functions $\mathbb R\to\mathbb R$ that always take rationals to rationals? Are they all quotients of polynomials? If not, are there any that are bounded yet don't tend to a limit for $x\to +\infty$? If there are, then can we also require them to be analytic?  (This is basically just a random musing after I found myself using trigonometric functions twice in unrelated throwaway counterexamples. It struck me that it was kind of conceptual overkill to whip out a transcendental function for the purpose I needed: basically that it had to keep wiggling forever. I couldn't think of a nice non-trancendental function to do this job, however, and now wonder whether that is because they can't exist).
Are functions of independent variables also independent? CPT Template Not Showing - Getting 404 What is the fastest way to create a 90° joint? I often want to create joints like this:    Since I end up making these more than a little, I'd like a fast way to do it (i.e. on the order of &lt; 10 seconds)  I've found a couple tools which come close to being the ideal solution, but I've had some problems with them as listed below:   When using the spin tool I usually end up having to fiddle with the location for a while in order to to get a natural curve:    Are there any techniques to expedite this process of fiddling? The bridge tool is nice because I can precisely specify where both ends of the joint will be, however I often end up with subtly uneven joints, probably due to my initial mesh being not quite square.    Is there a fast way to correctly line up the initial edge loops in a way that will result in an even joint? How to remove a repository? Read Value from URL in PHP Reading Excel files from C# How to sort a list of objects based on an attribute of the objects? How to do a Point to Polygon analysis with multiple variables in QGIS? I have a database of almost 10k crimes, which are divided in to 10 categories and plotted on a map  I also have a database of 12k addresses, which are also plotted on the map.   I aim to perform a multiple regression to determine whether there are correlations between types of crime and house price.  I have created a buffer of 250 meters around each of the addresses, and would like to list a count of each of the crimes which are located within this buffered zone for each of the addresses, however, I can only get QGIS to give me a count of all types of crime, rather than the specifics.   Does anyone know how to do a point to polygon analysis of this type? I am in the middle of breaking the crime databases to analyse each one separately, which is a bit difficult! Careers Stack Overflow - Tag "ruby-on-rails" returns jobs that don't contain the tag Efficiently replace all accented characters in a string? Mathematica9: NDSolve slows down after repeated calls Understanding the Basic Context of Trusted Hosts I got the trusted host settings error.      The trusted_host_patterns setting is not configured in settings.php. This can lead to security vulnerabilities. It is highly recommended that you configure this. See Protecting against HTTP HOST Header attacks for more information.    I read the  but I'm not understanding some of the basic context.   Are the "trusted hosts" me and other administrators who want to be able to administer the site remotely?  If so, would I add my PC's domain to the trusted host list? Who decides which half of the universe should be destroyed? Although my question is based on the comic mini-series Infinity Gauntlet issues #1 through 6, I'm open to any and all answers (i.e Movies, other comic books etc. etc.) across the Marvel multi-verse.   It was elegantly presented in the Infinity Gauntlet series how Thanos, after collecting all the soul gems snaps his fingers and destroys half the universe.  Additionally, in the recent movie Avengers: Infinity War,       Thanos does it again! He snaps his fingers and destroys half the universe.   However, one thing is unclear to me. In both cases,      Half the universe is destroyed (or obliterated, or erased from existence).   But who decides which half of the universe should not exist anymore? Does the wielder of the Gauntlet choose, or is the Gauntlet (along with the stones) sentient enough to make the decision? If $\frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1$, then what's the value of $4\left(\frac{\cos^3y}{\cos x}+\frac{\sin^3y}{\sin x}\right)$? Normal + Connected -> Uncountable Here's a solution (to the exercise "prove that if $X$ is a space with more than one element, and is normal and connected then $X$ is uncountable"):   by Urysohn's lemma, given $A$ and $B$, closed and disjoint in $X$, there exists a continuous function from $X$ into $[0,1]$ such that $f(A)={0}, f(B)={1}$.   If $X$ were countable, so it would be $f(X)\subset [0,1]$; choose $r\in (0,1)\setminus f(X)$, then $X=f^{-1}([0,r))\cup f^{-1}((r,1])$, so $X$ is disconnected.  But the full power of the lemma wasn't used, all that was used is that there exists a continuous function $X\longrightarrow [0,1]$, so the condition of being normal seems too much.  So my question is, what are the weakest conditions on $X$ for the existence of such continuous functions into $[0,1]$ (or $\mathbb{R}$, for that matter)? "Congratulation" vs. "congratulations" Does an classical charge that is stationary in a gravitational field radiate? The well known Larmor formula tells us that an accelerated classical charge radiates a power proportional to the square of the acceleration. It has been  that a charge in free fall in a gravitational field does not radiate. Since gravity is locally equivalent to acceleration, one may wonder whether a charge at rest in a gravitational field radiates. The  is that it does. However this radiation is not  to an observer in the same reference frame! Apparently, the radiation does not carry energy or momentum, or it would be observable, in this reference frame and this would mean that charged matter behaves differently from neutral matter in gravity.    How does unobservable EM radiation without energy and momentum make sense (Q1)?  Does the physical origin of charge acceleration determine whether radiation is observable or not (Q2)?  Is the answer to the question in my title "It depends?"?   Note added:  I did not find the discussion at  and  sufficiently enlightening. I found this pretty good small research, but I am still concerned about "regular usage" in life, say, when I am talking in the office, especially because of this comment:     To me: Stone is something you can pick up with one hand (for   throwing). A rock takes two hands. How do I run a program in the background of a shell, with the ability to close the shell while leaving the program running? Lets say my UI is having problems or for some reason, I need to boot up a program from the terminal window, say, nm-applet:  nm-applet   When it's started, it occupies the foreground of the terminal window.  Is there any simple way to run the program in the background without needing to leave the terminal open or have it occupy the whole terminal?  On that note, I did find a way to run programs from the terminal and have it allow for other inputs, by appending an ampersand (&amp;) to the command as such:  nm-applet &amp;   But this isn't much use as any processes started in the terminal are killed once the terminal is closed. What's the difference between "these" and "those"? First of all, I'm not a native English speaker, but in school I learned that these is used if referring to something near, and those is used when referring to something far away (temporally or locally). But now I'm sometimes watching English movies and notice sentences like "Have you seen those boots?" while the boots were only centimeters away. Is this just not correct, or have I learned the wrong rules?  Another question is, what if I'm referring to something abstract such as ideas or thoughts so that I can't say if they are far away or not. Should I use those or these?  Edit  So it can't be really wrong in any case if I use these respectively those?