How to show two elements in a group are (not) conjugate?
Why are two permutations conjugate iff they have the same cycle structure?
Show that both mixed partial derivatives exist at the origin but are not equal $$f(x,y) = \begin{cases} \displaystyle \frac{xy(x^2-y^2)}{x^2+y^2} &amp; \text{if } (x,y) \neq (0,0), \\ 0 &amp; \text{if } (x,y) = (0,0). \end{cases}$$ I tried finding both mixed partial derivatives but they ended up being the same for that function. I must be failing to take into account something dealing with the fact that it is piece-wise. I still need to show the mixed partial derivatives exist. How can I do all of this? Take the following script:  #!/bin/sh sed 's/(127\.0\.1\.1)\s/\1/' [some file]   If I try to run this in sh (dash here), it'll fail because of the parentheses, which need to be escaped. But I don't need to escape the backslashes themselves (between the octets, or in the \s or \1). What's the rule here? What about when I need to use {...} or [...]? Is there a list of what I do and don't need to escape? Under the  tag I have seen many questions on earth-616. Are there any other versions of earth as well? How do they differ from each other? How do I get a tourist visa (or a tourist card) for Cuba?  I live in the Netherlands, but I'm not a Dutch citizen, I have an Italian passport. fontawesome and pdflatex When using pdflatex and the fontawesome package the below error shows up.   !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! Fatal fontspec error: "cannot-use-pdftex" !  ! The fontspec package requires either XeTeX or LuaTeX to function. !  ! You must change your typesetting engine to, e.g., "xelatex" or "lualatex" ! instead of plain "latex" or "pdflatex". !  ! See the fontspec documentation for further information. !  ! For immediate help type H &lt;return&gt;. !...............................................     Wish to keep on using "pdflatex" instead of "xelatex" or "lualatex".   In fontawesome Version 4.4.0 manual it says "When using the (pdf)(LA)TEX engine, the fontawesome package doesn’t require any external package".  Tried solutions proposed for this question () without success.   MWE:  \documentclass{article} \usepackage{fontawesome} \begin{document} {\faAdjust} \end{document} Is there a word for happiness made possible by a tragic situation? I frequently find myself needing a word to express happiness that acknowledges a tragic or unfortunate circumstance underpinning that happiness. Preferably the word could have both adjective and verb forms. For example, "Gregory [verb-ed | felt adjective about] the balmy February day, suspecting it was yet another consequence of catastrophic and irreversible climate change."  "Bittersweet" is close, but not precisely right. It implies ambivalence rather than genuine gladness. "Schadenfreude" is also in the neighborhood, but I don't want to imply that the tragic part of the situation is what causes the pleasure. (Also, I recognize that the word may not quite exist in English, and I don't know if this is forbidden for this forum, but I'd welcome a loan word or phrase if there's a better one.) force long text inside \item[{\texttt{}] to wrap I'm a teacher and I teach at/in school OR at/in a school There are a lot of posts on this topic, but I can't find any answers to my question.  I came across the following two definitions of 'school':        [uncountable] (used without the or a) the process of learning in a   school; the time during your life when you go to a school (British   English)   [uncountable] (used without the or a) the time during the day when   children are working in a school      Source:   I know there are better ways to convey the message, but which sentence would a teacher utter if he was a school teacher teaching students, adults, etc.? What about a janitor, director, and secretary?     I teach at/in school OR I teach at/in a school   The above definitions concern students, and not working individuals like teachers or janitors.  I would appreciate if you also stated whether you're American, Canadian, British, etc. Taking advantage of one-time pad key reuse? How to generate a random alpha-numeric string If every prime ideal is maximal, what can we say about the ring? Suppose $R$ is a ring and every prime ideal of $R$ is also a maximal ideal of $R$. Then what can we say about the ring $R$? On June 13th 2017 all SharePoint calculated columns with HTML markup stopped working.   Basically the fields now render text (regardless of setting the data type to number). The gist of the notice from Microsoft () is:     In Microsoft SharePoint lists and libraries, calculated fields that   are running in the classic UI mode can be used to display results   based on Excel-like formulas. This is a long standing capability, and   is documented in Calculate data in lists or libraries.      Some users have added HTML markup or script elements to calculated   fields. This is an undocumented use of the feature, and we will begin   blocking execution of custom markup in calculated fields in SharePoint   Online from June 13, 2017 onwards. We are also providing this as a   configurable option for on-premises in SharePoint Server 2016 and   SharePoint Server 2013 via the June 2017 and subsequent Public   Updates.   As we don't have on premise SharePoint (online), we have to find an immediate work around, given that this impacts our processes and thus, business.   Is anyone aware of third party applications that will allow for HTML, script markup (that works with SP online)? Why does HTML require that multiple spaces show up as a single space in the browser? 'The average person' or 'an average person'? How does C compute sin() and other math functions? In addition to showing reputation scores, SO should rank users. In the users section, users are ordered by the reputations they have.  Each user should also have a unique rank, since many users can have the same reputation score.  Rank must follow the order laid out by reputation, but for those having similar reputation, an additional formula should be defined which can rank users based on tags, time since registration, frequency of activity and/or helpfulness of their answers.  What are your thoughts on this? Standard definition of graph says that it is an ordered pair G=(N,L) where N is set of nodes and L is set of lines which connect the nodes.   From what I've read, the set L can be empty, but can set N be empty too?  I'm mainly asking this because for my exam, I found numerous problems that go like this: Graph X is given. How many subgraphs with property Y exist? Often the number depends on the definition of graph. I'm trying to show that a finite sum of eigenspaces (with distinct eigenvalues) is a direct sum.   I have $ \alpha : V \to V $. The eigenspaces are $ V_{\lambda_i} = \ker(\alpha - \lambda_i id_V )$ for $ 1 \leq i \leq n $. My attempt at a proof:  $ A + B $ is a direct sum iff $ A \cap B = \{0\} $. If $ v \neq 0 \in V_{\lambda_i} \cap V_{\lambda_j} $ for some $i,j, i \neq j $, then $ \alpha(v) = \lambda_i v $ and $ \alpha(v) = \lambda_j v $. So $(\lambda_i - \lambda_j)v = 0 $, and so $ \lambda_i = \lambda_j $. This is a contradiction, so any pair of the eigenspaces have trivial intersection. Therefore $ \cap_{i=1}^n V_{\lambda_i} = \{0\} $, and so we have a direct sum.  Is this ok?  Thanks Could humans choose to establish fibre digesting colonies in our guts? Humans don't digest fibre, and nor do any animals, but some creatures have micro-organisms (bacteria, I'm guessing) in their gut which digest fibre for them.  My question is, could humans take some sort of pill and establish our own fibre-digesting colonies in our intestines?  Has this been tried?  Supposing it could be done, would you suppose it would have any negative effects on our health?  (i.e.. would those whose gut digested fibre for them be stuck on a low-fibre diet?).    Purpose of question: just wondering.  It would be cool to be able to eat bark. Can you add new statements (like print, raise, with) to Python's syntax?  Say, to allow..  mystatement "Something"   Or,  new_if True:     print "example"   Not so much if you should, but rather if it's possible (short of modifying the python interpreters code)