Is `$(< file)` completely bash, or does `$(< file)` really depend on `cat`? From Bash Manual:     The command substitution   $(cat file) can be replaced by the equivalent but faster $(&lt; file).   cat is from coreutils, instead of a bash builtin command. So I am surprised that the bash manual would mention something outside bash, such as cat.  Is $(&lt; file) really a shorthand of $(cat file)?  Is $(&lt; file) completely bash, or does $(&lt; file) really depend on cat?   If $(&lt; file) is completely bash, and doesn't depend on cat,    is &lt; file a redirection or a command or both (a command with a redirection and an empty command name)? how can &lt; file (a command with a redirection and an empty command name) output to stdout?   Thanks.
Understanding Bash's Read-a-File Command Substitution I am trying to understand how exactly Bash treats the following line:  $(&lt; "$FILE")   According to the Bash man page, this is equivalent to:  $(cat "$FILE")   and I can follow the line of reasoning for this second line. Bash performs variable expansion on $FILE, enters command substitution, passes the value of $FILE to cat, cat outputs the contents of $FILE to standard output, command substitution finishes by replacing the entire line with the standard output resulting from the command inside, and Bash attempts to execute it like a simple command.  However, for the first line I mentioned above, I understand it as: Bash performs variable substitution on $FILE, Bash opens $FILE for reading on standard input, somehow standard input is copied to standard output, command substitution finishes, and Bash attempts to execute the resulting standard output.  Can someone please explain to me how the contents of $FILE goes from stdin to stdout?
move file by list in file (with leading whitespace) When does Hamiltonian equals to energy of the system? Suppose $G=\{a_1,...,a_n\}$ is a finite abelian group, and let $x=a_1a_2\dotsm a_n$.   Prove that if there is more than one element of order $2$ then $x=e$.   What I've done so far: (#1 is just for illustration, There should be at least $3$ such elements)   $G$ has an even number of elements with order $&gt;2$ (They are paired so: $(a,a^{-1})$ ). And then there are $e$ and $\{b|o(b)=2\}$. Also, if $G$ has at least one element of order 2 then $|G|$ is even (Lagrange). Thus $|\{b|o(b)=2\}|$ must be odd. If there is exactly one element $b$ of order $2$ then $x=b$. That's because $G$ is abelian, and we can write: $x=e\cdot b\cdot(a_1 a_{1}^{-1}\dotsm a_n a_{n}^{-1})=b\cdot(e\dotsm e)=b$. If there are exactly $3$ such elements $a,b,c\in G$, then, as shown above, $x=abc$ and thus $x=e$ (That's because $(abc)^2=a^2b^2c^2=e$ which means $abc\in \{a,b,c,e\}$. If $abc=a$ then $b=c^{-1}=c$ which is false, same goes for $abc=b,abc=c$.)   I haven't managed to find a reason why the claim must hold for $5, 7, ...$ (i.e. All other odd integers). Does anyone have an idea?  Thanks in advance. Please excuse my English.  edit: Not a canonical answer but a simple one (if such even exists). How do I set the default program? "Where" exactly are complex numbers used "in the real world"? I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be used/applied and hence am curious.  So what practical applications of complex numbers exist and what are the ways in which complex  transformation helps address the problem that wasn't immediately addressable?  Way back in undergrad when I asked my professor this he mentioned that "the folks in mechanical and aerospace engineering use it a lot" but for what? (Don't other domains use it too?). I'm well aware of its use in Fourier analysis but that's the farthest I got to a 'real world application'. I'm sure that's not it.  PS: I'm not looking for the ability to make one problem easier to solve, but a bigger picture where the result of the complex analysis is used for something meaningful in the real world. A naive analogy is deciding the height of tower based on trigonometry. That's going from paper to the real world. Similarly, what is it that is analyzed in the complex world and the result is used in the real world without imaginaries clouding the problem?  The question:  is nice but covers a more mathematical perspective on interim results that make solving a problem easier. It covers different ground IMHO. iOS: Compare two dates I am using extensively trigonometric functions when an angle is given in degrees. Some of these functions like sine or cosine have rational values, for example, the well   known example is that $\cos(\theta) =0.6 $ and $\sin(\theta) =0.8 $.       However besides the case of multiplicity of $90^\circ$ it seems there are no rational numbers $\theta$  with simultaneously rational  values of  sine and cosine.      Is it possible somehow to prove that for rational values of an angle given in degrees there are  no values simultaneously rational of sine and cosine functions, beside obvious case of multiplicities of $90^\circ$? A logic puzzle involving a balance. Possible Duplicate:          You have 12 balls and you know that they all weigh the same except for 1 which is heavier or lighter than all the others (you don't know which though). How can you make sure you know which ball is the heaviest/lightest in only 3 weighings?  The way I approached it was to split up the 12 balls into three sets of 4 and weigh two of the sets. If the sets balanced the scale, then I know the ball I am looking for must be in the set of 4 balls not weighed, else, I disregard said set and arbitrarily choose the heaviest set of 4 (as opposed to choosing the lightest set). I split the heaviest set of 4 balls into 2 and weigh that... etc.  Repeating this process until all 3 tries have been "used up", even if everything just so happened to be in your favor (the arbitrary choice you have in choosing the heaviest or lightest set happens to be the correct choice) in the end you still end up having to choose between 2 balls. A 50% chance is good, but I am wondering, is there a way to make sure 100%? Calculating peaks in DEM using GRASS Are commands complete sentences? seems to indirectly ask the question, but the upvoted answer says &quot;the actual sentence is...&quot;. So my question is this: Are commands considered to be grammatically sound, complete sentences in the English language? Some examples:  Go. Add a note. Do work. Assume: $p$ is a prime that satisfies $p \equiv 3 \pmod 4$  Show: $x^{2} \equiv -1 \pmod p$ has no solutions $\forall x \in \mathbb{Z}$.  I know this problem has something to do with Fermat's Little Theorem, that $a^{p-1} \equiv 1\pmod p$. I tried to do a proof by contradiction, assuming the conclusion and showing some contradiction but just ran into a wall. Any help would be greatly appreciated. Why are second order linear PDEs classified as either elliptic, hyperbolic or parabolic? Is there a geometric interpretation of second order linear partial differential equations which explains why they are classified as either elliptic, hyperbolic or parabolic, or is this just a naming convention? That is, do they have any relation with actual ellipses, hyperbolas and parabolas? Why are the lower 3 bits of curve25519/ed25519 secret keys cleared during creation? I am currently experimenting with ed25519 and I noticed that on secret key creation, bit 254 is always set and the lower 3 bits are always cleared. I found that bit 254 is always set to protect against timing attacks in this question:   But why are the lower 3 bits cleared. Obviously it has to do with the formula in the curve25519 paper:  The set of secret keys is defined to be $\{\underline{n} : n \in 2^{254} + 8\{0, 1, 2, 3,\ldots, 2^{251}-1\}\}$  It's because of the 8 in there, but why is that 8 there? I suspect it has something to do with theorem 2.1 in the curve25519 paper, but I am not sure, because I do not understand fully what is being proven there.   I am experimenting with ed25519 primitives in some cryptographic routines which need me to add scalars to the secret key (add secret keys). Even if I add two well formed ed25519 secret keys, the result will not always have bit 254 set and the lower 3 bits cleared. Is this a security problem? I understand the risk for bit 254 but not the lower 3. how to set camera to the same view as I turn my cursor? I want to turn the camera to get the view I want, but everytime I turn the camera, I need to go to render window to check if it's the view I want. I turn the view with my mouse cursor, but that does not affect the actual camera view in rendering, can I set the camera view the same as I turn my cursor? Computing the integral of $\log(\sin x)$ How to compute the following integral?  $$\int\log(\sin x)\,dx$$    Motivation: Since $\log(\sin x)'=\cot x$, the antiderivative $\int\log(\sin x)\,dx$ has the nice property $F''(x)=\cot x$. Can we find $F$ explicitly? Failing that, can we find the definite integral over one of intervals where $\log (\sin x)$ is defined? I tried to find a topic here on this (possible problem) but I didn't.   I reported it at  but Tim Hoffmann pointed me that it is not a problem of editor. After tested with nano and pdflatex I'm posting here.  The minimal code produces errors. OK, we can solve it using \\{}.   But is this a problem with TeXlive 2014 or there are some good reasons to have that behavior.  \documentclass{report} \begin{document} test\\ [foo]  test\linebreak [foo]  test\newline [foo] \end{document}      ! Missing number, treated as zero.                        f l.3 test\ [foo] how to show image in table cell through json parsing The Borel-Cantelli Lemma in  seems to be a sort of "corollary" of the .  It says:  "Let $(E_k)_{k=1}^{\infty}$ be a countable collection of measurable sets for which $\sum_{k=1}^{\infty} m(E_k) &lt; \infty$. Then almost all x $\epsilon \ \mathbb{R}$ belong to at most finitely many of the $E_k$'s"  I don't get three things:   "Then almost all x $\in \ \mathbb{R}$ belong to at most finitely many of the $E_k$'s"   What does that mean? Since the Borel-Cantelli Lemma comes after the "almost everywhere" definition, I guess: $\exists \ Z \ \subset \mathbb{R}$ s.t. $m(Z) = 0$ and $\forall x \ \in \ \mathbb{R} \setminus Z$, x belongs to $(E_{j}, E_{j+1}, E_{l}, ..., E_{m})$.   In the proof, one can see the $m(\limsup \ E_k) = 0$ part, stated as $m(\cap_{n=1}^{\infty} [\cup_{k=n}^{\infty} E_k]) = 0$. For some reason, $m(\cap_{n=1}^{\infty} [\cup_{k=n}^{\infty} E_k]) = 0$ implies that "Therefore almost all x $\in \ \mathbb{R}$ fail to belong to $\cap_{n=1}^{\infty} [\cup_{k=n}^{\infty} E_k]$ and therefore belong to at most finitely many $E_k$'s"   Okay, why? What is the contradiction if almost all $x \epsilon \ \mathbb{R}$ belong to all the $E_k$'s? Or a countably infinite subcollection?   What is the significance of this? Why not just state the Borel-Cantelli Lemma as "Let $(E_k)_{k=1}^{\infty}$ be a countable collection of measurable sets for which $\sum_{k=1}^{\infty} m(E_k) &lt; \infty$. $m(\cap_{n=1}^{\infty} [\cup_{k=n}^{\infty} E_k]) = 0$"? The residue at $\infty$ How many injective functions $f:[1,...,m]\to{[1,...,n]}$ has no fixed point? $(m\le n)$