Surjection of ring homomorphism give surjection of group of units?
Does a surjective ring homomorphism have to be surjective on the unit groups?
How to find chromatic number of the $n$-dimensional hypercube $Q_n$? Allow people to answer protected questions if they have earned at least some amount of combined SE rep The purpose of protected questions is, according to the message presented on such a question, "to prevent 'thanks!', 'me too!', or spam answers by new users." The more verbose  agrees on the purpose. And the  says this is why protected questions exist:     Some questions are protected because they are expected to attract   either spam or users -- often new users -- who may mistake the site as   a traditional forum, posting "noisy" answers such as "Thank you" or   "This worked for me" or "I'm also having this problem".   Considering this, shouldn't a user be able to answer protected questions if they have gained at least 10 rep on any stack exchange site? If they've already demonstrated one site that they are familiar enough with the SE Q/A format to not be a drive-by useless answerer, shouldn't that confidence in them carry over to other SE sites?  Or maybe at some level of combined SE rep this restriction should be waived. I find it silly that at 15k+ combined network rep, I'm unable to answer a protected question on a SE site I've just started using. Problem  I have various footnotes, which are hyperlinked from the text to the bottom of their respective pages. When I click on the hyperlinks (in the main text), they always take me to the first page of my document. I also have footnotebackref which puts a hyperlink at the footnote to take the reader back to the main body of text and these links for fine.   Edited MWE  After a lot of deletion, I have managed to reduce the document to the following lines. One might say that I have found the problem, since the removal of \usepackage[hang, flushmargin]{footmisc} stops the problem, but I would still like to have non-indented footnotes.  \documentclass{article}  \usepackage[usenames,dvipsnames]{color} \usepackage[colorlinks=true]{hyperref}  \usepackage[hang, flushmargin]{footmisc}         %Problem line. \usepackage{footnotebackref}  \begin{document} \tableofcontents             \newpage                 \section{Section} This is some text\footnote{This is a footnote.}. \end{document} Another (perhaps humorous) term for "The Gold Standard" Gold Standard as in:     an example against which others of the same type are compared   -    Any other humorous ways to refer to this?  Something like "the holy mother"... Add lines to cron from script Is there a way to add lines to a user's cron via script?  I usually do it using crontab -e, but I would like to automate this task with a shell script. Who can use magic scrolls? I'm confused about who can use scrolls in 5e. In the DMG, p139 under Magic Item Categories: Scrolls, it states that:     Any creature that can understand a written language can read the   arcane script on a scroll and attempt to activate it.   This implies that anyone can try to cast a spell using a scroll. However, on p.200 in the Spell Scroll entry it states that:     If the spell is on your class's spell list, you can use an action to read the scroll and cast its spell without having to provide any of the spell's components. Otherwise, the scroll is unintelligible.   Then there is the bit about a DC 10 + scroll spell level ability check if the spell is higher than a spell slot that you can cast.     If the spell is on your class's spell list but of a higher level than you can normally cast, you must make an ability check using your spellcasting ability to determine whether you cast it successfully. The DC equals 10 + spell's level. On a failed check, the spell disappears from the scroll with no other effect.    So, which is it? Can anyone who can read a language use a scroll (like a fighter) if they make an arcana check, or just a PC with that specific spell in their spell list (only spell casters)? Can't find wifi drivers for 0bda:f179 Realtek Semiconductor Corp What is the difference between currying and partial application? I quite often see on the Internet various complaints that other peoples examples of currying are not currying, but are actually just partial application.  I've not found a decent explanation of what partial application is, or how it differs from currying. There seems to be a general confusion, with equivalent examples being described as currying in some places, and partial application in others.  Could someone provide me with a definition of both terms, and details of how they differ? What are the alternatives to OS X's Spotlight? I'm calculating the covariance of a distribution in parallel and I need to combine the distributed results into on singular Gaussian. How do I combine the two?  Linearly interpolating between the two almost works, if they are similarly distributed and sized.   provides a forumla at the bottom for combination but it doesn't seem right; two identically distributed distributions should have the same covariance, but the formula at the bottom of the page doubles the covariance.  Is there a way to combine two matrices? Why are Bitmap-Fonts used automatically? I have the following LaTeX-document:  \documentclass[a4paper,11pt]{article} \usepackage[utf8x]{inputenc} \usepackage[ngerman]{babel}  \usepackage[T1]{fontenc}  \title{Sample} \author{Litb}  \begin{document}  \maketitle  \begin{abstract}  This is a report about my praxis phase at the company foo bar which I enjoyed. \end{abstract}  \end{document}   Much to my frustration, compiling with pdflatex creates a document with Bitmap-fonts. The reader's font-list shows that it uses Type3 fonts.  If I remove the fontenc loading, then I get a PDF using vector-fonts, but I heard that I should keep it, for having native German Umlauts working.  Does anyone know how I can get vector fonts with T1, and why it keeps using bitmap fonts if I have that package-load line in it? Print programs with its proper syntax I have defined my own language whose syntax is very close to the one of Pascal. I would like to type the following program for instance. Many people use the packages algorithmicx or algorithm2e to print programs, but it seems that we have to use their specific keywords of its syntax like procedure, end while.  I just want to indent spaces when needed, and bold some words when I want (sorry that I could not bold letters in the following figure). Do I have to use  texttt for each line? Also, with verbatim, I could not bold words...  Also, as the program is not large, I would need to put it on the left side of the page, and leave space for some explanasion text on its right hand side. So I  guss I need to get them all together, and use multicols.  Could anyone help?  program test                                      some text here var   i : integer;                                    ...    b : boolean; begin    i := 1;                                        ...    b := true;    while i &lt; 15 do       i := i  + 1;       b := not b                                  some text here    od end   Edit1: Following @Werner's tabbing solution, I have made the code a minipage:  \begin{minipage}[t]{0.48\linewidth}% \begin{tabbing} Werner's solution \end{tabbing} \end{minipage}\hfill \begin{minipage}[t]{0.48\linewidth} a paragraph \end{minipage}   Actually as the code shows, I want the texts to be another separated minipage on the right hand side of the code. I have then 2 questions:   is it reasonable to make 0.48\linewidth instead of 0.5\linewidth? 0.48\linewidth is a conventional choice? is it possible to make a frame around the code, and use \caption and \label to make it a reference (I prefer figures)? Separate labels in cases As much as I have tried, it doesn't seem possible to use separate labels when using a casesconstruction (from the amsmath package).  Ideally, I would like to be able to write something like  \documentclass{article} \usepackage{amsmath} \begin{document} \begin{equation} f(x)=\begin{cases}   1&amp;x\geq0\label{positive}\\   0&amp;x&lt;0\label{negative} \end{cases} \end{equation} \end{document}   and later refer to both cases at different places. What would be the best way to achieve this result? How can I formalize that the function $$f:\mathcal{P}(A)\to(A\to \{0,1\})$$ is bijective? It seems clear to me but I don't know how to write a proof. Thanks in advance. At the end of the game, both remaining players have a Baron.   The sum of all cards discarded by both players is equal.  Who wins? Is it possible to change the linkage of Nether portals? Kernel of composition of linear transformations Let $f : U \to V$ and $g : V \to W$ be linear transformations on the vector spaces $U$, $V$, and $W$. Supposedly,  $$ \dim(\ker(g \circ f)) = \dim(\ker(f)) + \dim(\ker(g) \cap \operatorname{im}(f)). $$  How might I go about proving that?    (Attempt:) The $\dim(\ker(g)\cap\operatorname{im}(f))$ term suggests to me that I should define a vector space $V' = \ker(g) + \operatorname{im}(f)$ to invoke the theorem that  $$ \dim(V') = \dim(\ker(g)) + \dim(\operatorname{im}(f)) - \dim(\ker(g) \cap \operatorname{im}(f)), $$  but I don't really see where to go from there. How can you use optional parameters in C#? Note: This question was asked at a time when C# did not yet support optional parameters (i.e. before C# 4).  We're building a web API that's programmatically generated from a C# class. The class has method GetFooBar(int a, int b) and the API has a method GetFooBar taking query params like &amp;a=foo &amp;b=bar.   The classes needs to support optional parameters, which isn't supported in C# the language. What's the best approach? How do I restore iPhone 6s stuck on connect to iTunes screen/Apple logo I'm not really sure how, but after I woke up today, my phone was just displaying the loading spinner and doesn't start up anymore. It's weird because my phone turned off last night and suspecting that it was probably out of charge, I thought charging it overnight would fix the problem.   So, my iPhone is currently stuck on the iTunes/Apple Logo loop. When I put it into DFU/Recovery Mode and connect it to iTunes, it isn't able to restore it. It keeps throwing "Network Connection Problems" -- it downloads iOS 10 for a bit and suddenly stops downloading, throwing the error. I've also tried downloading a custom .ipsw from  but it seems that the file is corrupted whenever iTunes tries to unpack it.   Any thoughts on how I could go about fixing this?   Note - I don't have a jailbroken iPhone. What is the final price if the bid price exceeds the ask price in an order driven market?