How does greatest common divisor effected by field extension
GCD in polynomial rings with coefficients in a field extension
Prove the identity $\binom{2n+1}{0} + \binom{2n+1}{1} + \cdots + \binom{2n+1}{n} = 4^n$ I've worked out a proof, but I was wondering about alternate, possibly more elegant ways to prove the statement. This is my (hopefully correct) proof:  Starting from the identity $2^m = \sum_{k=0}^m \binom{m}{k}$ (easily derived from the binomial theorem), with $m = 2n$:  $2^{2n} = 4^n = \binom{2n}{0} + \binom{2n}{1} + \cdots + \binom{2n}{2n-1} + \binom{2n}{2n}$  Applying the property $\binom{m}{k} = \binom{m}{m-k}$ to the second half of the list of summands in RHS above:  $4^n = \binom{2n}{0} + \binom{2n}{1} + \cdots + \binom{2n}{n-1} + \binom{2n}{n} + \underbrace{\binom{2n}{n-1} +\cdots \binom{2n}{1} + \binom{2n}{0}}_{\binom{m}{k} = \binom{m}{m-k} \text{ has been applied}}$  Rearranging the above sum by alternately taking terms from the front and end of the summand list in RHS above (and introducing the term $\binom{2n}{-1} = 0$ at the beginning just to make explicit the pattern being developed):  $4^n = (\binom{2n}{-1} + \binom{2n}{0}) + (\binom{2n}{0} + \binom{2n}{1}) + \cdots + (\binom{2n}{n-1} + \binom{2n}{n})$  Finally, using the property $\binom{m}{k} + \binom{m}{k-1} = \binom{m+1}{k}$ on the paired summands, we get the desired result:  $4^n = \binom{2n+1}{0} + \binom{2n+1}{1} + \cdots + \binom{2n+1}{n}$ Is there a difference here:        It happened that he did not go to work today.   As it happened, he did not go to work today.      Do "it happened that" and "as it happened" have different meanings? Counterexamples to the avoidance lemma for arbitrary ideals Let $A$ be a commutative ring with $1$. Let $I$ and $J_k$, $k=1,\dots,n$ be ideals of $A$ with $I\subseteq \cup _{k=1}^n J_k$. Then I have obtained the following:  (1) If $J_k$, $k=1,\dots,n$, are prime ideals, then there exists some $j$ such that $I\subseteq J_j$.   (2) If $A$ is a principal ideal ring, then there exists some $j$ such that $I\subseteq J_j$.   I want some counterexamples that for all $k=1,\dots,n$, $I$ is not contained in $J_k$. How to get the counterexamples? So basically as it stands I have a laptop which has no cd/dvd drive, and I don't have a usb drive.  The laptop has windows 7 installed with ubuntu 11.04 installed through wubi.  What I want to do is remove windows completely, and make ubuntu the only OS installed on the system.  Is there a way to do this without re-installing ubuntu? (i.e. can I take my wubi install away from windows?)  or is there a way to from inside ubuntu have it run the ubuntu iso somehow so I can just wipe the system and install it fresh? (even if it means I need to have an e.g. 2gb partition just for the image to reside in). Using Custom Labels in a Class I have the below If statements using some hard coded strings and I want to make them use as custom labels. I have created custom labels with the same String names as given below. So how can I make these statements to use the Custom Labels I created ?  if(monitoringFrequency == 'daily'){             return dateValue.addDays(1);         }else if(monitoringFrequency == 'weekly'){             return dateValue.addDays(7);         }else if(monitoringFrequency == 'monthly'){             return dateValue.addMonths(1); How can Cooper communicate with his daughter before the time he went to space? Tikzcd conflict with babel and cd I have this problem: when I try to compile a diagram I get the following message:     I think the culprit is a tikzcd arrow in ell 1-1   But I have not problem when, in the tikzlibrary, I just use cd, matrix and backgrounds. The problem is when using babel and cd at the same time.  \documentclass{article} \usepackage{amsfonts,amssymb,amsmath,tikz} \usepackage[spanish]{babel} \usepackage[latin1]{inputenc} \usepackage[mathcal]{euscript}  \usetikzlibrary{babel,cd,matrix,backgrounds}  \begin{document} \begin{center}   \begin{tikzcd}     X\rar[mapsto]{T}&amp;Y   \end{tikzcd} \end{center}  \end{document} iOS app 1.4.3, iOS 9.2.1  While viewing a post, the app somehow ended up in a state where it refuses to do much of anything.  What does work:   Swipe-scrolling vertically Jumping to the previous / next post using the up / down buttons    Everything else, though, has no effect:   Swipe-scrolling horizontally will not return to the previous view Tapping on a code block will not zoom to code-block view Tapping Delete button to delete a post will show a confirmation dialog, but saying yes will not cause deletion The Add Comment button brings up the text field, but I cannot type anything in it; I can only dismiss it None of the other buttons (Share, More, etc.) do anything Cannot view user profiles or revision history Tapping on links in a post or comment will not show a web page Tapping on a tag shows some visual feedback with a temporary color change, but doesn't do anything else   I still receive push notifications, but tapping on them doesn't take me anywhere. Why do ssh operations always ask for passphrase? (with ed25519) I just updated my ssh key and added a passphrase, however it constantly nags me for it now, such as when I do a git (or mercurial hg) push.  ssh-agent is running:  xxx  1303  1118  0 09:16 ?  00:00:00 /usr/bin/ssh-agent    /usr/bin/dbus-launch --exit-with-session     /usr/bin/im-launch mate-session   I tried ssh-add:  &gt; ssh-add Identity added: /home/…/.ssh/id_rsa (/home/…/.ssh/id_rsa) Enter passphrase for /home/…/.ssh/id_ed25519:  Could not add identity "/home/…/.ssh/id_ed25519":      communication with agent failed   What is missing?  16.04 Ubuntu Mate, in case it matters. Already the early papers on supergravity were written using computer algebra software to do some calculations. What modern packages do people normally use for doing such calculations? Of course Mathematica and presumable a lot of other programs can be made to do such calculations, I am wondering if there are special packages or tutorials how to do such calculations in them. Present Perfect with the word "ago"? First, I check that my native language is not English, and the problem described below met on one of the exams.   With this sentence:      "The most important news is that my parents [...] a new restaurant a few   weeks ago."   Is the correct tense to use the Present Perfect (have opened) or Past Simple (opened)?   I know that the word "ago" but rather that time Past Simple, but is it also possible to use the present perfect in some cases?  I would be grateful for clarification. Force alt + tab to switch only on current workspace in GNOME shell Validation Error less than training error? I found two questions  and  about this issue but there is no obvious answer or explanation yet.I enforce the same problem where the validation error is less than training error in my Convolution Neural Network. What does that mean? Similar Matrices and Linear Transformations How can I add a new user as sudoer using the command line? After I add a user using adduser, I can't see it via System > Administration > Users and Groups unless I log out and then log in again. Is that normal?  Also, can I set a newly added user as a sudoer or do I have to change that only after adding it? How can I do that via the shell?  Finally, can I delete the original user that was created upon initial installation of Ubuntu, or is this user somehow 'special'? Is there any way to get real-time compilation for LaTeX? I would really love to have a software or tool which provided real time compilation of my LaTeX document (to be able to see the final pdf or dvi document as I type the latex code), especially for when I am drawing pictures using tikZ. So far, I know of two things, but none of them work for me. They are:   : This is only for tikZ, which would be perfect for me, but unfortunately as far as I know it only works with Ubuntu and Debian, and I use Mac OS X (so I have never actually tried it, though I did dig a lot to see whether I would be able to install on my mac and from what I see it is really what I want - I am even considering installing Ubuntu just to use it!). : This is a minor mode for Emacs. It was extremely hard to get it to work on the mac (see my quest ), and it is not all that fast. The speed is not all that big of a deal for me, but the fact that it does not work with tikZ pictures is (the nodes all collapse in one point so that the words are all one on the top of the other).   So my question is: is there a software ideally like KtikZ or another type of editor which provides real-time compilation and that I can use on my mac? I guess if you know of a Windows/Linux option I would also like to know, and the same for a way to set up a script to get the document compiled every 5 seconds or so (I am sure I have seen this written somewhere, but I don't know whether I can do it with a mac or whether it is editor/pdf viewer dependent), but what I really want is a software/tool which would work with a mac.  To be clear, I am not after something like LyX, that is, I am not after a WYSIWYG-type thing, but rather something where I can type real LaTeX and see my code, but have at the same time another window showing me the pdf (or dvi) file compiled. Can a goalkeeper chip the ball to the head of a teammate from a goal kick to circumvent the backpass restriction? I have this:  If String.IsNullOrEmpty(editTransactionRow.pay_id.ToString()) = False Then     stTransactionPaymentID = editTransactionRow.pay_id 'Check for null value End If   Now, when editTransactionRow.pay_id is Null Visual Basic throws an exception. Is there something wrong with this code? update-alternatives --install says it needs link name path priority? I typed:  sudo update-alternatives --install "/usr/bin/java" "java" "/usr/lib/jvm/ jdk1.8.0/bin/java" 1   It's says alternatives --install needs link name path priority Given the Python function:  def a_method(arg1, arg2):     pass   How can I extract the number and names of the arguments. I.e., given that I have a reference to func, I want the func.[something] to return ("arg1", "arg2").  The usage scenario for this is that I have a decorator, and I wish to use the method arguments in the same order that they appear for the actual function as a key. I.e., how would the decorator look that printed "a,b" when I call a_method("a", "b")?