distance between the first and second particle(s)? I mean the distance between the first particle and the second particle. Should I say:  distance between the first and second particle  or  distance between the first and second particles  ?
The first and (the) second chapter(s) are difficult - which is correct? I'm a little confused about the following sentences. Could you please tell me which one(s) is/are correct?        The first and the second chapter are difficult.   The first and the second chapters are difficult.   The first and second chapters are difficult.   The first and second chapter are difficult.      I believe #1 &amp; #3 are correct, while #2 &amp; #4 are incorrect.  But when seeing the answers to the Quora question , I wonder why they say #2 is correct.
I want to find the $nth$ derivatives of the function $e^{f(x)}$ with respect to $x$, the first derivative is $$e^{f(x)}f^{\prime}(x).$$  The second derivative is $$\left( {\frac {d^{2}}{d{x}^{2}}}f \left( x \right)  \right) {{\rm e}^ {f \left( x \right) }}+ \left( {\frac {d}{dx}}f \left( x \right)   \right) ^{2}{{\rm e}^{f \left( x \right) }} .$$ The third derivative is $$\left( {\frac {d^{3}}{d{x}^{3}}}f \left( x \right)  \right) {{\rm e}^ {f \left( x \right) }}+3\, \left( {\frac {d^{2}}{d{x}^{2}}}f \left( x  \right)  \right)  \left( {\frac {d}{dx}}f \left( x \right)  \right) { {\rm e}^{f \left( x \right) }}+ \left( {\frac {d}{dx}}f \left( x  \right)  \right) ^{3}{{\rm e}^{f \left( x \right) }} $$  The fourth derivative is $$ \left( {\frac {d^{4}}{d{x}^{4}}}f \left( x \right)  \right) {{\rm e}^ {f \left( x \right) }}+4\, \left( {\frac {d^{3}}{d{x}^{3}}}f \left( x  \right)  \right)  \left( {\frac {d}{dx}}f \left( x \right)  \right) { {\rm e}^{f \left( x \right) }}+3\, \left( {\frac {d^{2}}{d{x}^{2}}}f  \left( x \right)  \right) ^{2}{{\rm e}^{f \left( x \right) }}+6\,  \left( {\frac {d^{2}}{d{x}^{2}}}f \left( x \right)  \right)  \left( { \frac {d}{dx}}f \left( x \right)  \right) ^{2}{{\rm e}^{f \left( x  \right) }}+ \left( {\frac {d}{dx}}f \left( x \right)  \right) ^{4}{ {\rm e}^{f \left( x \right) }} $$  My question is: Is there a general formula or a pattern for the nth derivative of $e^{f(x)}$. You may use the maple command diff(exp(f(x)), x$5) to do some experiments. Thanks a lot ^^ Is there a universal way to turn Windows 7 Aero desktop, all windows, etc into grayscale? I had an option to do this in Intel video drivers on my old laptop, now I have NVIDIA and don't know how to replicate this effect.   There might be a shader or filter that would cause Aero to render stuff into grayscale, but I was unable to find anything. Or must it be explicitly supported by video driver as a bonus? Remap Keys in Linux I'd like to switch my Ctrl and Alt keys under Linux (to make it more Mac-like).  How do I do this?  If it helps, I'm running Ubuntu 9.04. How to handle relative urls correctly with a nginx reverse proxy The difference between a poor or meh question and a stellar question can often simply be someone understanding it and providing it a great answer. I can't begin to count the number of times I've justified re-opening a question as a moderator by saying:     Look at the answer it got, though. This isn't something we want deleted, this is something we want fixed, because it's obviously valuable. Someone with domain knowledge can easily edit that question based on the answer it received.   This doesn't only apply to borderline-poor or poor questions – sometimes a good edit means the difference between 100 and 10,000 people finding something through typing things into Google. Good titles are hard.   We think we've come up with a way to provide incentive for good, opportunistic edits to happen more often, ideally without questions needing to go through closure or collecting a centimeter of dust before they do. What if we simply rewarded folks that answered and edited questions with a special, relatively difficult to earn badge?  Let's say that you answer something, and:   You edit the question 12 hours before or after answering it. This allows you to edit now, answer later &mdash; or answer now and edit later, when you have the time. Your edit isn't rolled back, or outright rejected if it was a suggested edit The question is not closed for any reason, even simply being a duplicate Your answer has a score of 1 or higher   &hellip; then you've done something that we probably want to recognize. You understood something, you provided the knowledge that you have, and then you provided an edit to make sure that more people in need of this knowledge can find it, while raising the overall quality of the site.  Here's the proposed tiers for the badge (within the above stated criteria):   Explainer (Bronze): Answered &amp; edited 10  questions Refiner (Silver): Answered &amp; edited 50 questions Illuminator (Gold): Answered &amp; edited 500 questions   We think this is going to be pretty hard to game. You can't go leaving streaks of terse snark all over the place on questions you know are going to be closed anyway, and junk edits aren't going to fly.  We're talking about rewarding folks that take a tiny bit more time out of their day just to write a more descriptive title so that folks can find the awesome answer that they wrote. Just doing that alone can make a big difference. Sure, it's a tad hard to get at the gold level, but it's supposed to be. If you routinely edit questions that you answer and have done so for some time, you'd probably get it the day we roll it out.  While this is technically something that we came up with on the quality initiative ( | ) &mdash; this badge could be earned on all of our sites.  What do you think? What did we miss? What could be better about the idea? Superman has been exploited by Poison Ivy in Hush, and he's definitely mad in Injustice. What about Batman? Has Batman ever had to be brought to justice/stopped/killed by his fellow superheroes? (That one time when Superman dreams of Batman snapping Joker's neck doesn't count)  Edit: The Dark Knight Returns doesn't count: Batman wasn't really mad, and he wasn't wrong either. What is this “(arg: 1)” appearing in terminal/tty after I pressed a combination of keys? In a virtual terminal I (accidentally) pressed Win+1 and got a new line consisting of  (arg: 1)   As long as I keep Win pressed, all numbers I press get appended to the number in the parentheses, e.g. Win and 1+4 yields (arg:14).  When I (after releasing Win) press any key, that character is printed as many times as the number reads and the (arg: ...) expression is replaced by the standard prompt.  Does anybody know what happens here? How to carry audio over SSH? Is there a way to carry audio over ssh?   Say after connecting to remote machine through ssh -X I opened a browser and want to watch a youtube video. Currently I could only able to watch the video without any audio. But I want to hear the audio too. How to do that? Was Voldemort allowed to visit Hogsmeade while in school? We know that in order for third-years and up to be allowed to visit the village of Hogsmeade, they must present a form duly signed by their parent or guardian. When Harry’s guardians, the Dursleys, miss their opportunity (!) to sign his, he is not allowed to go to Hogsmeade. Both Cornelius Fudge (as Minister for Magic) and Professor McGonagall (as head of Gryffindor House) refuse to sign the form on his guardians’ behalf—probably at least in part because they think there’s a mad killer on the loose looking for Harry, but also, as McGonagall specifically states, simply because:  “You heard what I said. No form, no visiting the village. That’s the rule. […] The form clearly states that the parent or guardian must give permission.” Prisoner of Azkaban, Bloomsbury/UK paperback, p. 113  Obviously, Tom Riddle did not have any parents or guardians ready to sign his form for him: his mother was dead, his father likely did not even wish to know of his existence, and if he did have a legal guardian at all (I don’t know how orphanages worked in Britain in the 1930s), it was likely to be  (warning: Wikia link) as the matron of Wool’s Orphanage where Riddle grew up. As Mrs. Cole was perfectly Muggle in every way and was never told by Dumbledore—nor, we must assume, by the boy himself—that Riddle was a wizard attending a wizardry school, it seems unlikely in the extreme that she ever signed Riddle’s form to visit Hogsmeade.1 I don’t recall any incidents in the books where Voldemort speaks specifically of him visiting Hogsmeade as a school boy (then again, why would he?); but being content to remain at the castle while his classmates have fun in the village seems quite out of character for him, too. Are there any (canonical) indications whether special provisions were made for orphans with no legal guardians to sign, so that they were allowed to visit Hogsmeade after all? Or, oppositely, whether such cases were played strictly by the book and unguardianed orphans, and thereby Riddle, were not allowed to visit Hogsmeade?    1 One might of course easily envision that Riddle simply Confounded Mrs. Cole to sign the form; but that would lead to detection of underage magic (assuming this was even a thing in 1940). Riddle may of course have forged Mrs. Cole’s signature, but that would be rather likely to be discovered as a forgery, since at least Dumbledore was aware that Mrs. Cole did not know of Riddle’s magical abilities. Discord suddenly very slow Teleporting a player from the overworld to the end Is there any way to teleport a player from the overworld to the end?   Or a way of teleporting a player to a random enderman in the end? I have a set of polygons representing administrative districts.  The districts were digitized by hand, and there are small spaces (slivers) between polygons where the polygons should be touching. How can I have the polygons snap together, removing the spaces? How can I disable Ctrl+Q for all applications (system wide)? A lot of applications like Firefox, Chrome etc, get closed when I accidentally press Ctrl+Q instead of Ctrl+W because of the proximity of the Q and W keys on the keyboard. Is there a way this shortcut can be removed or reset on a system wide basis? Manage SSH Users with MySQL backend Ubuntu not detecting second monitor I'm running Ubuntu 12.04 on a Lenovo x61s Thinkpad. As the screen's rather small and I want to do some video editing, I thought I'd plug in a monitor and use that. The monitor is Relisys JM777 (quite old).  When I plug it into my other computer, which is running Windows 7, it immediately mirrors the display; but when plugged into the Lenovo the monitor screen remains blank. The graphics card on the Lenovo is a "VGA compatible controller" according to SysInfo.   Anybody got any suggestions for getting this monitor to work? I'm quite new to Linux. How to obtain a confidence interval for a percentile? I have a bunch of raw data values that are dollar amounts and I want to find a confidence interval for a percentile of that data.  Is there a formula for such a confidence interval? How do I create silhouettes in a studio environment? The trick to creating silhouettes in the field, as far as I know, is to use spot-metering and to expose for the background. However, this is a bit more difficult in a studio environment. What is the key to creating "artificial" silhouettes?  One solution that I'm considering is placing a bright flash directly behind the subject - would that work? If $a, b$ are positive real numbers and $a+b = 1$, prove that :  $$\left(a+\frac{1}{a}\right)^2 + \left(b+\frac{1}{b}\right)^2 \geq \frac{25}{2}$$  I can see that the value $\frac{25}2$ is attained for $a=b=\frac12$. But I do not know how to show that this is the minimal possible value.  Thank you. I am stuck with the identity  $$ F_{2n} = \sum_{k=1}^n \binom{n}{k} F_k, $$  which happens to be formula . I am using induction, but so far without too much result.  $$ \sum_{k=1}^{n+1} \binom{n+1}{k} F_k = \\ F_{n+1} + \sum_{k=1}^n \binom{n+1}{k} F_k = \\ F_{n+1} + \sum_{k=1}^n \binom{n}{k} F_k + \sum_{k=1}^n \binom{n}{k-1}F_k =\\ F_{n+1} + F_{2n} + \sum_{k=1}^n \binom{n}{k-1}F_k, $$   I am stuck with the sum $ \sum_{k=1}^n \binom{n}{k-1}F_k$, because, on the other hand   $$ F_{n+1}+ \sum_{k=1}^n \binom{n}{k-1}F_k = F_{2n+1} $$  Which would conclude the proof. What am I missing? Say, I have foo-1.2.3.deb which depends on perl and python, however, running command:  dpkg -i ./foo-1.2.3.deb   won't install these dependencies. So I must apt-get install perl python by hand.  How to make dpkg -i install these dependencies for me automatically?