For loop doesn't change the variable being operated on
Can't modify list elements in a loop
Proof that disjoint cycles commute I'm trying to prove that in a symmetric group two disjoint cycles commute. But I suspect that something is not right about my proof (a sense of vagueness). Some hints would be appreciated.  Here's my proof:  Let $\sigma=(s_1 s_2 ... s_n)$, $\tau=(t_1 t_2 ... t_m)$ for some integers $m, n$. That is, $s_i \neq t_j$ for any $i \in [1, n]$, $j \in [1, m]$. Now, $\sigma\tau=(s_1 ... s_n)(t_1 ... t_m)$ is a cycle of disjoint permutations, which cannot be represented by any other disjoint permutations. Thus $\sigma\tau=\tau\sigma$. What can I do when getting "We are no longer accepting questions/answers from this account"? How do you write this in LaTeX: Why is this quotient space not Hausdorff? Foobar2000-like music player? Is 'clamp' a formally recognized mathematical function? I was surprised to find the clamp function absent from  and . Is this mainly a concept particular to computer programming? Is it known by another name in mathematics? I'll be editing the Wikipedia shortly to include a reference to this function but I want to make sure I'm not omitting some synonymous terms I'm currently unaware of.  $$   \operatorname{clamp}(x) = \max( a , \min( x , b ) ) \in [a,b] $$ What are good learning resources for a LaTeX beginner? Which book (free or otherwise) was the most useful to you when you started learning LaTeX?  I am frequently asked this question by friends who want to learn LaTeX, and I recommend the book which got me started, , but I feel that there might be better options around.  Also see . "To like for somebody/something to be something" vs. "to like somebody/something to be something" Are these plural or singular? I was surprised when I heard zero is plural, and even 1.00001 is plural.  Then, what about following numbers ?  (1) 1.00  which means the value measured  between 0.995-1.004, having possibility that it was exactly 1.  (2) 0.99999.....  which is mathematically equal to 1. Also, are others that are mathemattically equal to 1 than this, treated completely same as 1?  (3) -1  Is this singular? Or negative numbers are generally plural?  (4) i = square root of -1  I know it's weird to count things with imaginary numbers, nonetheless, if you were to count, which form, plural or singular, would you be to use?  That's it. Including section titles in fancyhf headers What is the proof that given a set of $n$ elements there are $2^n$ possible subsets (including the empty-set and the original set). I've heard the number 7% thrown around a bunch of times, but I am struggling to find reputable data to back it up.   All I found was a quote from Warren Buffet that I couldn't find an original source for and data assembled by individuals on their personal websites. Is there something from an organization such as Market Watch making a statement about this?  I'm hoping to specifically find what the 10 or 20 year return is for investing in the S&amp;P 500 index fund or another diversified portfolio option. How To Set Default Wallpaper As Login Screen Wallpaper [Automatically]? Prove $2^n-1$ is divisible by $(2^m-1)^2$ iff... Prove that for any $m,n\in\mathbb{Z}^+$, the number $2^n-1$ is divisible by $(2^m-1)^2$ iff the number $n$ is divisible by $m(2^m-1)$.  My work so far:  It's an if and only if statement, so the common approach is to split the proof into two "if then" proofs ($A\rightarrow B \text{ and } B\rightarrow A$).  Starting with $A\rightarrow B$, for a direct proof, I would assume $A$ and prove $B$. I can use the definition of divisibility:  Assume $n=m(2^m-1)k$ for some integer $k$  Then I need to prove that $2^n-1=(2^m-1)^2h$ for some integer $h$  So I'm thinking I need to start with the expression $2^n-1$ and transform it to $(2^m-1)^2h$, or vice versa, and I need to use the assumption $n=m(2^m-1)k$ somehow.  I notice that the LS of the equality contains the variable $n$ while the RS contains $m$, and the assumption relates $n$ to $m$, so that could be a way forward. For example, simple substitution gives $2^{m(2^m-1)k}-1$, but I'm not sure what I could do from here. Perhaps there's a divisibility theorem that could help?  Other things I notice: $2^n-1$ and $(2^m-1)^2$ are always odd. Also, the divisors in both cases are multiples of $2^m-1$.  Any help/hints appreciated. Proof of the uncountability of reals using the diagonal argument—problem? Add Word Limit to Posts Hovering over featured moves the questions down a bit When I hover over featured tab, both on the front page and in questions, the questions move down by one pixel. Here's a small screen capture demonstrating the issue:        I see it in Chrome 34 and Opera 20, but I don't see it on IE11 or Firefox 28. In an alternating current, do electrons flow from the source to the device? If electrons in an alternating current periodically reverse their direction, do they really flow? Won't they always come back to the same position? Light "diode" and 2nd law of thermodynamics If I had a light "diode" - an object that only allowed light (at least for a range of frequencies) to travel through it in one direction, would this necessarily allow violations of the 2nd Law of Thermodynamics?  Or would it be possible to restrict other properties of the device such that it could be okay.  Why I think it should violate 2nd law:   I can place such devices at temperature T in a bath of thermal radiation at temperature T, such that it forms a box which only lets the light in or out.  This would appear to always allow one to create a thermal gradient.  Once a thermal gradient is formed, useful energy can be extracted.  This seems to allow useful energy to always be extracted from a thermal bath and thus give perpetual motion. Release one such device at temperature T in a bath of thermal radiation at temperature T.  Since the photons are absorbed or reflected in one direction, and undisturbed in the other, the device will start moving.  It can use a thermal bath to directly generate useful motion.   Why I think I'm missing something and such a device could potentially co-exist with the 2nd Law:   The above arguments seem valid if the diode was only 'partially' rectifying as well.  However there exist sunglasses which largely reflect on one side, but not from the other side. Researchers have succeeded in using meta-materials to create a "blackhole" for microwaves.   Can anyone help sort this out? How are ssl certificates verified? What is the series of steps needed to securely verify a ssl certificate?  My (very limited) understanding is that when you visit an https site, the server sends a certificate to the client (the browser) and the browser gets the certificate's issuer information from that certificate, then uses that to contact the issuerer, and somehow compares certificates for validity.     How exactly is this done?  What about the process makes it immune to man-in-the-middle attacks? What prevents some random person from setting up their own verification service to use in man-in-the-middle attacks, so everything "looks" secure?