Calculate the limit of spherical cap in spherical coordinate
Calculate the area on a sphere of the intersection of two spherical caps
Make back/forward work on paginated lists We all use mathematical induction to prove results, but is there a proof of mathematical induction itself? I just realized something interesting. At schools and universities you get taught mathematical induction. Usually you jump right into using it to prove something like   $$1+2+3+\cdots+n = \frac{n(n+1)}{2}$$  However.  I just realized that at no point is mathematical induction proved itself? What's the mathematical induction's proof? Is mathematical induction proved using mathematical induction itself? (That would be mind blowing.) Terminal method of formatting storage drive I want to know how to format a storage drive from the terminal. Helpful things to provide in the answer would be often used options for commands and base knowledge that one can use to extrapolate future uses. Specifically I would like to know how to format in the different file systems such as NTFS, FAT32, EXT4, etc. Information on how to partition the drive via terminal is also wanted.  I am trying to format a high capacity external hard drive (EHDD) into NTFS - from the terminal.   I know I can use gparted for this as well as other GUI programs, but I still want to now how to do it from the terminal. I recently started an account on , and went to add an answer to a question which was protected. I have enough rep on other sites that I have the "Association Bonus", so I was confused as to why I wasn't being allowed to answer the question.     This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have earned at least 10 reputation on this site.   After re-reading the message and reviewing , I realized that the bonus didn't count as points earned toward that site.  This is a really minor issue, but I think it's worth fixing (not sure if bug or feature).  I believe that users who have enough rep to receive the association bonus should generally already know not to post "thanks!" and "me too!" style answers, so if the message is accurate, users who receive the association bonus should be able to answer protected questions.  If there's another reason to keep users from answering protected questions, such as  claim that:     For protected questions, you are expected to have experience on the target site itself. Experience at other sites is not enough.   then the message should be updated to reflect this.  Option 1:  make protected questions open to anyone who has >10 rep on that site and update the message to read:     This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have at least 10 reputation on this site.   Option 2:  leave protected questions the way they are, but update the message to read:     This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have earned at least 10 reputation on this site to show that you have experience on this topic; experience on other StackExchange sites is not enough.   Option 3:  Answer with your own suggestions/improvements to these ideas. Double calculation producing odd result I am looking for the definition of when I am allowed to do forward declaration of a class in another class's header file:  Am I allowed to do it for a base class, for a class held as a member, for a class passed to member function by reference, etc. ? Can the ship's captain really perform your wedding when in international waters? On , Mark Mayo posted a comment that any marriage has to be on dry land or close to it.  But I'm sure all of us have heard of ships' captains marrying couples at sea.  From the movie Casablanca:     Say, why don't we get married in Marseilles?   ...   Well, let's see. What about the engineer? Why can't he marry us on the train?   ...   Well, why not? The captain on a ship can. It doesn't seem fair that...    So is this fiction after all? If not how does it fit with Mark's comment? What special rules and such should we know?  If I'm travelling at sea with a significant other and wish to tie the knot, can we do so with any captain on any ship on any sea? Must it be international waters? My online friend is asking for money in order to visit my home country. Is this a legit request or a scam? Find the first occurrence/starting index of the sub-array in C# Differences between pure/mixed/entangled/separable/superposed states Beamer, Babel and Tikz conflict I am preparing lecture notes in Turkish, and I have a digram for mathematical modelling procedure. But I cannot produce a good output for the following code. When I remove the first three packages, I get my desired output. However, I need to have keep them in my document. Would you help me in this direction?  \documentclass[10pt,notheorems]{beamer}  \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[turkish]{babel}  \usepackage{tikz} \usetikzlibrary{positioning,shapes,arrows}  \hypersetup{bookmarksdepth=4,bookmarksnumbered=true,bookmarksopen=true}  \usetheme{Warsaw} \usecolortheme{seahorse}  \begin{document}  \begin{frame}[fragile]{}{}  \begin{figure}   \centering   \begin{tikzpicture}     \node (rwp) [draw,fill=blue!20,text centered,minimum height=2em,rounded corners] {Ger\c{c}ek ya\c{s}am problemi};     \node (mm) [right=2cm of rwp,draw,fill=purple!20,text centered,minimum height=2em,rounded corners] {Matematiksel model};     \node (mc) [below=2cm of mm,draw,fill=green!20,text centered,minimum height=2em,rounded corners] {Matematiksel \c{c}{\i}kar{\i}m};     \node (rwq) [below=2cm of rwp,draw,fill=red!20,text centered,minimum height=2em,rounded corners] {Ger\c{c}ek ya\c{s}am tahminleri};     \path [draw,-&gt;] (rwp.east) -- node [above] {Form\"{u}lasyon} (mm.west);     \path [draw,-&gt;] (mm.south) -- node [right] {\c{C}\"{o}z\"{u}m} (mc.north);     \path [draw,-&gt;] (mc.west) -- node [below] {Yorum} (rwq.east);     \path [draw,-&gt;] (rwq.north) -- node [left] {Kontrol} (rwp.south);   \end{tikzpicture}   \caption{Modelleme s\"{u}reci}\label{s2ss2fig1} \end{figure}  \end{frame}  \end{document} How can I change the registered name in Windows 7 to my own? how do I change the registered name in Windows 7 to my own name? Using Elseif Conditional Statement in QGIS Field Calculator? WoeUSB Error Code 256 with NTFS formatted USB Does seam itself affect baking? workspace switching via ctrl + alt + arrow not working after upgrade to 13.04, how to fix this? Suppose I have a line segment of length $L$.  I now select two points at random along the segment.  What is the expected value of the distance between the two points, and why? "404. That’s an error." when viewing from Google cache My site is getting 404. That’s an error. message when I click on cached from Google search results.  This is the error message from Google:  That’s an error.   The requested URL /search?q=cache:8kxLAgFrhZQJ:www.mysite.com.....&amp;cd=8&amp;hl=en&amp;ct=clnk&amp;gl=uk was not found on this server. That’s all we know.  What I did before I was getting this error is that about 10 days ago, I moved my hosting from Ixwebhosting to Gearhost (cloud based). I also added 301 redirect from non www to www.  I have no idea why I am getting this error. Please advise what might be the reason if there is a solution for it.  All of my pages are indexed by Google though. Countdowntimer in minutes and seconds Combinatorics identity question