I tried this example in $S_3$ and saw that this held but I couldn't figure out how to prove it for all $S_n$. Any help would be appreciated.
I have heard that two permutations are conjugate if they have the same cyclic structure. Is there an intuitive way to understand why this is?
Random point uniform on a sphere How do I set the proper exposure for nighttime moon photos? All my attempts to get a good shot of the full moon with my DSLR result in an overexposed circle on a black background.  I've tried using a tripod, remote shutter release, low ISO, and long exposure, but nothing has worked so far.  What combination of ISO and exposure time that will produce good results?   I particularly want to catch a full moon with the reddish effect when it is close to the horizon. How do I resolve "WILL_NOT_PERFORM" MS AD reply when trying to change password in scala w/ the unboundid LDAP SDK? Suppose $f(x)$ and $g(x)$ are continuous function from $[0,1]\rightarrow [0,1]$, and $f$ is monotone increasing, then how to prove the following inequality: $$\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$$ What does limit actually mean? I have been in a deep confusion for about a month over the topic of limits! According to our book, the limit at $a$ is the value being approached by a function $f(x)$ as $x$ approaches $a$.  I have a doubt that in real number line we can never ever reach the closest value to $a$ because always a more closer value will exist.  Now when talking about our methods for calculating limit, what method comes in our mind when we have to calculate where the value approaches? Let's say we have to find what value $f(x)$ approaches when $x$ moves from $[0,a)$.  So we calculate the value of $f(x)$ at $x=0$, say $4$, and then at $x\to a$.   But the problem is that we don't know the value $x\to a$, so we say the value is $a-h$ where $h\to 0$ and calculate the value of $f(a-h)$, say $5-h$.  Now here is where my doubt starts! In the final step we put the value of $h=0$ say in that it is an infinitesimal quantity.  My doubt is that $h$ was tending to $0$ means that it was never equal to zero  maybe it is infinitesimally small, not a stationary value, not an imaginable value, but we know for sure that it is not  equal to $0$. Maybe it is the point closest to zero, but it is not equal to zero and when we use the result $5-h=5$ we are actually making an error which is tending to zero. Maybe the error is very small, but still, there is some error in that we cannot calculate it but we can see that there is this infinitesimal error present.  That means we don't get the exact limiting value or last value of $f(x)$.  $x$ belongs to $[0,a)$ but a value approximate to infinitesimal? Isn't it right! We get an approximated value? In a 'bulleted' list should items that happen to be complete sentences end with a period? Smaug first appeared in TA 2770 as a mature Fire-drake. I'm not sure if there are canonical sources which say his age or maturity, but he must have been quite large and quite old at the time in order to destroy the city of Dale and capture Erebor.  Since it is safe to assume that Smaug would have been attacking towns and stealing all of the gold and jewels he could get his claws on for most of his life, then he must have amassed a huge fortune prior to taking residence at Erebor. How did he transport his existing horde to Erebor? It seems unlikely that he would have simply abandoned all of his treasure that he had previously accumulated.  EDIT: It has been established that Smaug was not a mature drake at the time of his appearance. However, he may have already had a small horde (compared to Erebor). Retail boxed Steam activated games: can they be resold? I recently purchased retail boxed versions of   Dead Island (which is awesome) Warhammer 40k: Space Marine (not so much)   I finished both games and I was thinking of reselling them or giving them to a friend to play. However, these are both Steam activated games that had a required Steam registration code I had to enter -- and validate on Steam -- as part of the installation.  If I give these games to a friend or resell them, will they even work? Is the registration code single use and tied to a single person / computer / Steam account? How to make a safe in vanilla Minecraft that locks and unlocks with a key? How can I make a safe/vault or something to lock my door that is very easy (as I am not that advanced with redstone) to keep people out of my house? Without mods? Setting proxy for apt from terminal Long ago someone demonstrated the following card trick with a standard 52-card deck:  (1) A volunteer selects 5 cards from a shuffled deck, which the performer does not see.  (2) The assistant puts exactly 4 of those in a pile and passes it to the performer.  (3) The performer can then identify the 5th card.  There are a few ways to execute this trick, one of which goes as follows.  (1) The assistant finds a suit common to 2 of the 5 cards, with values $x$ and $y$ modulo $13$.  (2) He then removes one of them, say $x$, such that $x-y \in [1..6]$ modulo $13$.  (3) He then puts the other one at the top of the pile.  (4) And he puts the 3 other cards after that in a permutation corresponding to $x-y$ modulo $13$.  Since (1) works by pigeonhole, (2) works because $\frac{13-1}{2} = 6$, and (4) works because there are 6 possible permutations of 3 cards, the trick succeeds. My question is whether the same trick can work with more than 52 cards. It is not obvious to me how to do so, nor compute the maximum number of (pairwise distinct) cards possible. Also, it seems to me that even if the trick can be made to work for $n$ cards, it may not work for $n-1$ cards! Any ideas? I found this , but I am not sure if this is the same issue.  On , I did some reformatting of the indentation in the code block to make it a bit more readable and it was  for peer review. A few minutes later, it appears the OP made an edit that added some additional code. It looks like my suggested edit was applied when the OP saved his/her edit, and it looks like he got the "credit" for the edit.  I'm not really concerned about the 2 rep points; I'm more curious why this happened and if this is a bug (or limitation) in the editing feature, or if this is known/intended behavior?  Note that this is not a suggested edit that's overridden by a concurrent edit: the recorded edit starts from the post-suggested edit state (and in fact can be identical to it, as in  ). let q be an odd prime and p a prime dividing $1+n+\dots+n^{q-1}$. Prove that $p = 1 \mod q$ or $p = q$.  I don't actually know how to start, i thought of using fermat's little theorem or an application of eulers formula, but it didn't really give me any insights.   Any hints? Genitive case or attributive noun Is there any rule for when it is better to use genitive case or noun as adjective? I'm not sure if there is any difference in meaning in this example:     The department of accounting       The accounting department   Can I also use both forms in situations like:     The restaurants of Paris / The Paris restaurants      The room window / The window of the room Between vs. Among - "Difference [between | among] one thing and other things." Consider the following two versions of a question:1   Version 1: What is the difference in meaning between "[to be] not invited" and similar negation forms?  Version 2: What is the difference in meaning among "[to be] not invited" and similar negated forms?    Personally, I think Version 1 might subtly suggest a Star Topology (see below) with the target sentence "[to be] not invited"at the center.  And Version 2 might subtly suggest a more free-form comparison like the Mesh or Fully Connected, with the target sentence simply "one among many":   Between approx equal "Star" and among approx equal "Fully Connected"?  Don't get me wrong: I do think this is quite subtle, not always applicable, and I don't think anyone would intuit such a "network topology" meaning without further explanation.   So the main question is, are both grammatically correct?  If yes, is there a semantic difference to choose one over another, perhaps as I suggested?  Or does the plurality aspect of between and among trump other semantic and stylistic issues?    1. In my question , another user provided a very helpful edit. However, the change of the word between to among prompted this question. In the inbox of the app, there is no way to differentiate between comments and answers as is in the global inbox of the website.  Could this be added? Induced metric on a null hypersurface Proving that for a smooth function if $f(\frac 1 k)=0 :\forall k\in \mathbb N$ then $f(x)=0 :\forall x\in[-1,1]$ Why do we say the following?        I'll have an hour's wait at the airport      Today's lesson      Yesterday's programme    Two weeks' worth of groceries.      They are not possessive, or are they? Why is the possessive apostrophe used with time expressions and adverbs? Problem printing LaTeX from lua in lualatex