How do I turn off the FN key?
How to change Function (Fn) Key behavior on wireless Logitech keyboard
Transferring through US airport. Do I need traveling visa? Solution af a system of 2 quadratic equations I have a system of two quadratic equations with unknowns $x$ and $y$:  $$a_{1 1} x y + a_{1 2} x^2 + a_{1 3} y^2 + a_{1 4} x + a_{1 5} y + a_{1 6} = 0,\\ a_{2 1} x y + a_{2 2} x^2 + a_{2 3} y^2 + a_{2 4} x + a_{2 5} y + a_{2 6} = 0,$$  where $a_{i j}$ are arbitrary scalars.  Is there an algebraic solution of the above system? Isomorphism between $\Bbb R$ and $\Bbb R(X)$? My questions are:     $1.$ Is there a field morphism $\Bbb R(X) \hookrightarrow \Bbb R$ ?      $2.$ If the answer to $1.$ is "yes", are $\Bbb R$ and $\Bbb R(X)$ isomorphic as fields?   $ $    $ $  For $1.$, here are some comments:   This is true for $\Bbb C$, see . The key point here is the fact that $\Bbb C$ is algebraically closed, of cardinality $2^{\aleph_0}$, and of characteristic $0$. The same argument doesn't work for $\Bbb R$. We can actually find a subfield $F$ of $\Bbb R$ such that $F \cong F(X) \cong F(X,Y) \cong \cdots$, see .  that $\Bbb Q(X)$ does not embed in $\Bbb Q$ (as a field).  This is equivalent to: Is there an injective ring morphism $\Bbb R[X] \hookrightarrow \Bbb R$ ? (By taking the fraction fields).   As for 2. :   They are isomorphic as abelian groups, and actually as $\Bbb Q$-vector spaces. They are not isomorphic as $\Bbb R$-algebras, because they haven't the same transcendence degree over $\Bbb R$. Let's try to see how to construct such a field isomorphism. We could try$^{[1]}$ to find a subset $B$ of $\Bbb R$ such that $\Bbb R = \Bbb Q(B)$ and $C \subsetneq B \implies \Bbb R \neq \Bbb Q(C)$.  The set $B$ must contain transcendental elements, pick such an $x_0 \in B$.  Then $\Bbb R = \Bbb Q(B) = \Bbb Q(B \setminus \{x_0\})(x_0)$ and we could try$^{[2]}$ to find a field isomorphism $\Bbb Q(B \setminus \{x_0\}) \cong \Bbb Q(B)$, so that $\Bbb Q(B \setminus \{x_0\})(x_0) \cong \Bbb Q(B)(x_0) \cong \Bbb R(X)$.  For ${[1]}$, I tried to use Zorn's lemma on $\mathscr B = \{B \subset \Bbb R \mid \Bbb R = \Bbb Q(B)\}$ and the partial order $B_1 ≤ B_2 \iff B_1 \supset B_2$. But this is not clear to me why this should be an inductive set.  For ${[2]}$, my idea was to define $f : \Bbb Q(B \setminus \{x_0\}) \cong \Bbb Q(B)$ by choosing a bijection $b : B \setminus \{x_0\} \to B$ (these sets should be uncountable), and then trying to extend $f\vert_{B \setminus \{x_0\}} = b$ to a field morphism.     Edit for $^{[1]}$: in  article (MINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS, by LORENZ HALBEISEN, MARTIN HAMILTON, AND PAVEL RUZICKA) the theorem 2.4. proves that $\Bbb R$ has no minimal generating set as field (or $\Bbb Q$-algebra, i.e. as ring). The argument uses Artin-Schreier theorem. Minimal generating sets of modules are   .  I think that $B$ is a minimal generating set of a subfield $K \subset \Bbb R$ over $\Bbb Q$ if and only if any element of $B$ cannot be written as a rational fraction of other elements of $B$. Under these conditions and using (transfinite) induction, we can probably prove (and this was my first idea) that any function $f: B \to \Bbb C$ such that $b$ and $f(b)$ are conjugate over $\Bbb Q$ (i.e. have the same minimal polynomial) extends to a (unique) field morphism $\sigma : K = \Bbb Q(B) \to \Bbb C$.  Actually, the collection $\mathscr B$ defined above doesn't have to be an inductive set, at least if we replace $\Bbb R$ by $K =\Bbb Q(\sqrt 2, \sqrt[4]{2},\sqrt[8]{2},\dots)$, because the descending chain $\left(E_n = \{\sqrt[2^m]{2} \mid m \geq n\}\right)_{n \geq 1}$ satisfy $K = \Bbb Q(E_n)$, but $K \neq \Bbb Q\left(\bigcap_{n≥1} E_n\right)=\Bbb Q$, so the chain has no upper bound w.r.t. $≤$ in $\mathscr B$. What are the responsibilities of each Pseudo-Terminal (PTY) component (software, master side, slave side)? I am trying to figure out how a tty works1 (the work-flow and responsibilities of each elements). I have read several interesting articles about it, but there are still some blurry areas.  This is what I understand so far:   The emulated terminal makes different system calls to /dev/ptmx, the master part of the pseudo-terminal. The master part of the pseudo terminal allocates a file in /dev/pts/[0-N], corresponding to the obsolete serial port, and "attaches" a slave pseudo terminal to it. The slave pseudo terminal keeps information such as session Id, foreground job, screen size.   Here are my questions:   Has ptmx any purpose besides allocating the slave part? Does it provide some kind of "intelligence", or the emulated terminal (xterm for instance) has all the intelligence of behaving like a terminal? Why does xterm has to interact with the master part, as it only forwards the stdout and stdin of the slave part? Why can't it directly write and read from the pts file? Is a session ID always attached to one pts file and vice versa? Could I type a ps command and found 2 sessionId for the same /dev/pts/X? What other information does the pts store? Does Xterm update all fields by himself, or does the ptm add some "intelligence" on it?     1. I base my understanding on the , and the  posts, as on several other I flew a long range flight carrying my laptop, a quite heavy one, inside a neoprene briefcase in the passenger cabin. The recommendation by the aircrew was to place it in the overhead compartment or under the seat in front of me. However, during take-off and landing, the vibrations on the plane are quite strong and, as a result, it received minor damage.   Placing them in the overhead compartment might be equivalent to placing it on the floor under the seat, with the added issue of additional luggage in the compartment shifting.  I considered the alternative of holding the laptop by the handle while sitting, so it won't be any contact with the aircraft's floor. This is impracticable during cruise, but I thought it could be better during the take-off and landing segments. Another option I considered is to wrap it with completely with bubble wrap before boarding the plane.   Which is the safest way to transport a laptop during flight? I have put his on wolfram and obtained answer as follows:  $\sum_{n=1}^{\infty}\frac{n}{2^n} = 2$  And the series is convergent too because $\lim_{n\to\infty} \frac {n}{2^n} = 0$  However I am wondering if there is a convenient way to solve this; I don't think you can represent it by a geometric progression either. So how do we have to do it on paper? Where does the curve $x^y = y^x$ intersect itself? This problem is quite easily solved by using logarithms and derivative and forming the function $f(x,y) = x^y - y^x$, however there are assertions that this problem can be solved without using either of the two. I can not see how one would proceed to solve this problem in the absence of logarithmic manipulation and differentiation. Can anyone shed some light on this? JavaScript + Unicode regexes I have some Java executable (jar) that is run my some shell script from a cron job once every night. That executable does not print log statements &quot;as usual&quot; just by printing them out in a sequential manner like line after line (print after print), but while it processes its data its printing a single line with status data and then &quot;overwrite&quot; or &quot;update&quot; just that single line over and over again, until its done with this part of processing. Examples: A common program would output something line this: Program XYZ started..&lt;cr+lf&gt; processing 1&lt;cr+lf&gt; processing 2&lt;cr+lf&gt; processing 3&lt;cr+lf&gt; im done&lt;cr+lf&gt;  Simple! I can easily redirect its output to a file and I'm done. But the program I will have to deal with does it more like this: Program XYZ started..&lt;cr+lf&gt; processing 1&lt;cr&gt; processing 1 - part a&lt;cr&gt; processing 1 - part b&lt;cr&gt; processing 1 - part c&lt;cr&gt; processing 1 - part d&lt;cr&gt; &lt;cr+lf&gt; processing 2&lt;cr&gt; processing 2 - part a&lt;cr&gt; processing 2 - part b&lt;cr&gt; processing 2 - part c&lt;cr&gt; processing 2 - part d&lt;cr&gt; &lt;cr+lf&gt; processing 3&lt;cr+lf&gt; im done&lt;cr+lf&gt;  but much more intensive than I showed here. So it just not always overwrite status lines, but does it a lot while processing. So just redirecting or appending its output to a log file does not seem to make a lot of sense here, since it will just clutter the log file with millions of screen-updates that can not be viewed and understood that easily by humans. So my question is: is there a way or tool for programs that may behave like that to &quot;log&quot; or maybe &quot;screenshot every 30 seconds&quot; their output to a log file? What are "Community Wiki" posts? Some questions and answers are marked Community Wiki and are owned by a community wiki user:  (In fact, this very question is one of them)  Why have Community Wiki posts? How do Community Wiki posts work? How does a post become a Community Wiki post? How can the Community Wiki status be removed from a post? How can I find Community Wiki posts? Math.cos() gives wrong result Replicate the Fourier transform time-frequency domains correspondence illustration using TikZ "NG NS" on ticket after airport security Filter the score of the questions on the homepage before loading them At the moment questions on the homepage are loaded by the top N most recent and then ones with a score of -4 or less are filtered out. This doesn't make much difference for a user who is logged in but consider anonymous users; their home page is only 15 questions in length. If someone was to edit several posts that bump to the top with a score of -4 or less then the home page looks tiny. At the moment on SFF, for example, the homepage for anonymous users only shows 9 questions:  At one point it only showed 3 questions (a few weeks back):  Can the filtering logic be changed to before the questions are loaded to appear on the homepage? Residence has changed - Must I re-apply for ESTA? I have an old, but still valid ESTA, with the country of residence showing my home country. However, I have recently moved for education in the UK. Do I have to apply for a new one?  Nothing else has changed - country of citizenship, passport, names etc How to prove  $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$ How to move a file? How do you extract IP addresses from files using a regex in a linux shell? View history of changes Possible Duplicate:          How do I view the changes made to a question, in the SO-style visual diff?  I'm sure I've seen this before, but the "history" button eludes me. Clicking "edit" gives me a drop down of the revisions, but not the actual diff.    Edit  This edit is a demonstration of the revision history functionality. How to measure time taken by a function to execute