Typesetting a context-free grammar
Which package can be used to write BNF grammars?
Can I get smaller tires without changing my rims? I currently ride a hybrid that came with 700x32c tires. Can I swap out a narrower tire without changing the rims, or do rims only fit a specific size of tire?   How small can I go before I have to buy new rims?  I hear having narrower tires makes you go quite a bit faster.  Does it really help that much? Command to run an application with Administrator privileges, but NOT AS administrator (prompt with UAC) SQL Join Differences What's difference between inner join and outer join (left join,right join), and which has the best performance of them?  Thanks! How can I attach a file to a question? The question is about the encryption equation in asymmetric encryption.  For $c\equiv m^e\bmod n$, prove that for every unique $m$ there is a unique $c$.  Here $n=pq$ where $p$ and $q$ are primes larger than $m$ and the exponent $e$, is $\geq3$ and coprime with $p-1$ and $q-1$.  Thanks! Did the verb “fire a weapon” exist before the actual introduction of firearms on battlefields?  More specifically, does it make sense for a creative work to have archers (or whatever ranged weaponry) be told to “fire!”, when the world they live in has not yet seen firearms?  It seems some kind of an anachronism to me, since before firearms, “fire” would never propel any projectile...  I've seen several movies do it; I can't remember them all, but for instance I verified it in The Lord of the Rings trilogy (left), Kingdom of Heaven (top right), and more recently Frozen (bottom right):    The first two do it several times (with Kingdom of Heaven also using it on ballistae, and The Return of the King even having Aragorn ask Legolas to ), whereas firearms never appear on the battlefields they feature (Kingdom of Heaven is set in the 12th century; as for The Lord of the Rings, apart from one occurrence of a witchcraft-ish bomb, it's archery and medieval siege engines all the way).  Frozen may be debatable, happening probably somewhere in the 18th or 19th century, but since we only see swords, spears and the like, it made me flinch to hear crossbowmen use “fire!”.  I've also noticed other movies avoiding this, using “loose!” instead, such as Troy (left) and Gladiator (right):    This makes me think it would be the right thing to say instead... although I'm still wondering why not use “shoot”, which seems simpler to me. Chrome browser will not connect to intranet web addresses How fast a (relatively) small black hole will consume the Earth? This question appeared quite a time ago and was inspired, of course, by all the fuss around "LHC will destroy the Earth".   Consider a small black hole, that is somehow got inside the Earth. Under "small" I mean small enough to not to destroy Earth instantaneously, but large enough to not to evaporate due to the . I need this because I want the black hole to "consume" the Earth. I think reasonable values for the mass would be $10^{15} - 10^{20}$ kilograms.   Also let us suppose that the black hole is at rest relative to the Earth.  The question is: How can one estimate the speed at which the matter would be consumed by the black hole in this circumstances? Deciding what answer to accept: the correct one-liner or the later elaborate one with code? Lately I've been getting some good answers to some specific questions. However, I've been hesitating on what answer I should accept.  Do I go with the short quick answer or do I accept a later answer where someone has elaborated on the previous answer with some code?  It's not really important but it kind of bugs me to not accept an answer when people have taken the trouble of replying.  Here's two examples: Where's the Rainbow Jetpack gone in Jetpack Joyride? Is Double-Annotating HREFLANG always alright? What if I remove one of them? How to make a redstone clock that I can turn on/off with a lever? I want to create a Redstone clock that can be enabled and disabled with a lever permanently placed down. How to use more than 10 tab stops in bmatrix or other amsmath matrix environments? I'm using bmatrix environment from the amsmath package, and I'm trying to do something like this:  \documentclass{article} \usepackage{amsmath}   \begin{document}      \begin{equation}          \begin{bmatrix}              1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 &amp; 7 &amp; 8 &amp; 9 &amp; 10 &amp; 11\\          \end{bmatrix}     \end{equation}  \end{document}   However, pdfLaTeX returns an error: "Extra alignment tab has been changed to \cr", and in the output the extra element is moved to the next row. (If I remove one element, then it works perfectly fine). I guess this means that I've hit some hard limit on the number of tab stops in a matrix. Is there a way to change this?  The elements in the matrix are short and could fit perfectly on a single row. Are keywords in URLs good SEO or needlessly redundant? A coworker and I are locked in a debate over the value of SEO keywords in the URL of a page. She wants to change all the filenames of the HTML pages of a fencing company so they look like residential-home-chicago.html, contact-chicago-contractor.html, and so on. She is convinced that because Google highlights keywords in URLs in search results, that means that putting keywords here is more valuable.  My position is that these do not improve SEO, since Google doesn't seem to give keywords in the URL any more weight than keywords in the body of the page, and . In the meantime, they make it harder for me to find the pages I want when its time to edit them, and the site as a whole looks cheap and spammy.   suggests to me that yes, keywords in URLs are useful, but not superior, and that they are more useful for human readability than search engine rankings.  I'm looking for authoritative sources that support either position, not blog articles from SEO optimization companies trying to promote themselves. I stupidly ran sudo rm /* followed by sudo rm -R /*  My sites still run, although the vast majority of commands like ls are now broken.  Is there a simple way to get data off the server or reverse some of damage? Is there a word for capitalizing in the middle of a word, as in "eBay"? Usually this sort of word is a trademarked name, such as eBay or BlackBerry.  Is there a word that describes capitalization mid-word? When I look in sys.sql_logins, I see a column called is_policy_checked. Can I trust that my password policy has been checked for all of the logins where this column value is 1? How do American English and British English use the definite article differently? I decided to make sure that I know this important difference between American and British English, so I wrote what I have found out so far and I would be grateful to anyone who reads this and tells me whether I am wrong, or not.  In British English when people say  to hospital or in hospital when  talking  about somebody being there as a patient they don't use the definite article : "I had to go to hospital", "She spent two weeks in hospital". And the meaning is that somebody was there as a patient.   If then for some other reasons British English speakers  will use the definite article which will change the meaning itself, I noticed that, in American English, native speakers often use the  the and if they need to show that somebody is in church to pray,  in school as a student, in hospital as a patient, in prison as a prisoner, they use 'in' and not 'at'. Do American English speakers use 'at' like British English speakers use 'the' to give the sentences a different meaning?    Are my sentences correct? Do they show American English usage?   He is in the school. (enrolled as a student) He is at the school. (for some different reasons) He is in the hospital. (as a patient) He is at the hospital. (visiting somebody) He is in the church. (to pray) He is at the church. (for some different reasons) He is in the university. (as a student) He is at the university. (not as a student) He is in the college. (as a student) He is at the college. (Not as a student) He is in the prison. (as a prisoner) He is at the prison. (not as a prisoner) Installing Ubuntu with Fake RAID 1 Suppose we're tying to differentiate the function $f(x)=x^x$. Now the textbook method would be to notice that $f(x)=e^{x \log{x}}$ and use the chain rule to find $$f'(x)=(1+\log{x})\ e^{x \log{x}}=(1+\log{x})\ x^x.$$  But suppose that I didn't make this observation and instead tried to apply the following differentiation rules:  $$\frac{d}{dx}x^c=cx^{c-1} \qquad (1)\\ \frac{d}{dx}c^x = \log{c}\ \cdot c^x \quad (2)$$  which are valid for any constant $c$. Obviously neither rules are applicable to the form $x^x$ because in this case neither the base nor the exponent are constant. But if I pretend that the exponent is constant and apply rule $(1)$, I would get $f'(x)\stackrel{?}{=}x\cdot x^{x-1}=x^x.$ Likewise, if I pretend that the base is constant and apply rule $(2)$ I obtain $f'(x)\stackrel{?}{=}\log{x}\cdot x^x$.  It isn't hard to see that neither of the derivatives are correct. But here's where the magic happens: if we sum the two “derivatives” we end up with $$x^x+ \log{x}\cdot x^x=(1+\log{x})\ x^x$$ which is the correct expression for $f'(x)$.  This same trick yields correct results in other contexts as well. In fact, in some cases it turns out to be a more efficient way of taking derivatives. For example, consider $$g(x)=x^2 = \color{blue} x\cdot \color{red} x.$$ If we pretend the blue $\color{blue} x$ is a constant we would get $g'(x)\stackrel{?}{=}\color{blue}x\cdot 1=x$. Now if we pretend the red $\color{red}x$ is constant we get $g'(x)\stackrel{?}{=}1\cdot \color{red} x=x$. Summing both expressions we end up with $2x$ which is of course a correct expression for the derivative.  These observations have led me to the following conjecture:     Let $f(x,y)$ be a differentiable function mapping $\mathbb{R}^2$ to $\mathbb{R}.$ Let $f'_1 (x,y)=\frac{\partial}{\partial x} f(x,y)$ and $f'_2 (x,y)=\frac{\partial}{\partial y} f(x,y)$. Then for any $t$ we have: $$\frac{d}{dt}f(t,t)=f'_1 (t,t) + f'_2 (t,t).$$   (I apologise for the somewhat awkward notation which I could not seem to get around without causing undue ambiguity.)  This formulation also seems to lend itself to following generalisation:     Let $f:\mathbb{R}^N \to \mathbb{R}$ be a function differentiable in each of its variables $x_1,x_2,\ldots,x_N$. For $n=1,2,\ldots,N$ define $f'_n(x_1,x_2,\ldots,x_N)=\frac{\partial}{\partial x_n}f(x_1,x_2,\ldots,x_N)$. Let $t$ be any real number and define the $N$-tuple $T=(t,t,\ldots,t)$. Then one has: $$\frac{d}{dt} f(T)=\sum_n f'_n(T).$$   Thus my question is:   Is this true? How can it be proven? (Specifically in the case $N=2$ but also in the general case.)