Let $S^n\subset \mathbb{R}^{n+1}$ be the unit sphere centered at $0$. Prove that the isometris of $S^n$ is $O(n+1)$, the orthogonal maps of $\mathbb{R}^{n+1}$. Certainly all elements of $O(n+1)$ are isometries, but how to prove they are all the isometries? I want a rigorous proof.
Got this as homework and I don't know how to tackle this. Help please!  Prove that the isometries of $\mathbb{S}^{n} \subset \mathbb{R}^{n+1}$, with the induced metric, are restrictions to $\mathbb{S}^{n}$ of the linear orthogonal transformations.
Who are all of the companions and how do I get them? Debugging a custom-bib bst file that crashes on book items I created my own bibliography file with custom-bib (makebst), but when I run it with a reference to a book I get:  $ bibtex PhD [...normal output] Database file #1: PhD.bib You can't pop an empty literal stack for entry oecd2001brc while executing---line 1504 of file wdsphd.bst   The only thing on that line is a high-level ITERATE call.   The style was created with these options:  %% merlin.mbs  (with options: `head,exlang,annote,seq-no,nm-rvvc,dt-beg,yr-par,xmth,tit-qq,qx,bt-qq,vol-bf,volp-sp,num-xser,pub-date,isbn,issn,doi,english,pp,ord,url,url-nl,nfss,,{}') %% english.mbs  (with options: `exlang,annote,seq-no,nm-rvvc,dt-beg,yr-par,xmth,tit-qq,qx,bt-qq,vol-bf,volp-sp,num-xser,pub-date,isbn,issn,doi,english,pp,ord,url,url-nl,nfss,,{}') %% merlin.mbs  (with options: `tail,exlang,annote,seq-no,nm-rvvc,dt-beg,yr-par,xmth,tit-qq,qx,bt-qq,vol-bf,volp-sp,num-xser,pub-date,isbn,issn,doi,english,pp,ord,url,url-nl,nfss,,{}')   The code for book items looks like:  FUNCTION {book} { output.bibitem   author empty$     { format.editors "author and editor" output.check     }     { format.authors output.nonnull       crossref missing$         { "author and editor" editor either.or.check }         'skip$       if$     }   if$   format.btitle "title" output.check   crossref missing$     { format.bvolume output       new.block       format.number.series output       new.sentence       format.publisher.address output     }     {       new.block       format.book.crossref output.nonnull     }   if$   format.edition output   format.isbn output   format.doi output   new.block   format.note output   fin.entry   write.url }   And this is the entry:  @book{oecd2001brc,  author = {OECD},  doi = {10.1787/9789264193550-en},  isbn = {9789264186903},  publisher = {OECD Publishing},  title = {{Biological Resource Centres: Underpinning the future of life sciences and biotechnology}},  url = {http://www.oecd-ilibrary.org/science-and-technology/biological-resource-centres\_9789264193550-en},  year = {2001} }   Is there a way to get a better stacktrace from bibtex? What I have now doesn 't really tell me what is going on. If it can be fixed simply, help would also be appreciated. I need a quick algorithm to select 5 random elements from a generic list. For example, I'd like to get 5 random elements from a List&lt;string&gt;. How can I find such a $z$? (Injectivity of $T(z) = \lambda z + \mu\bar{z}$) We consider $T:\mathbb{C}\to\mathbb{C}$ defined by $T(z)=\lambda z+\mu\overline{z}$, where $\lambda ,\mu\in\mathbb{C}$. I want to prove that if $T$ is injective, then $\lambda\cdot\overline{\lambda}\neq\mu\cdot\overline{\mu}$.  Clearly, if $\mu =0$, then $\lambda\neq 0$ because $T$ is injective.  Now, if $\lambda\cdot\overline{\lambda}=\mu\cdot\overline{\mu}$ and $\mu\neq 0$, then I wanted to find a $z\in\mathbb{C}\setminus\{0\}$ such that $z=-\dfrac{\overline{\lambda}}{\overline{\mu}}\overline{z}$, because we would have that:  $T(z)=-\lambda\dfrac{\overline{\lambda}}{\overline{\mu}}\overline{z}+\mu\overline{z}=\overline{z}(-\lambda\dfrac{\overline{\lambda}}{\overline{\mu}}+\mu)=0$  ('Cause this would prove that $\lambda\cdot\overline{\lambda}=\mu\cdot\overline{\mu}$ implies $T$ is not injective.)  So, the question is, can we always find such a $z\in\mathbb{C}\setminus\{0\}$? Finding local maximum and minimum Coming from SketchUp one of the tools I miss most is the circle draw tool which lets you draw a circle on any face and then you could extrude it in/out. The only tool that I know of that comes close to this is the knife tool but this would be very impractical to cut out with the knife  Example:    Is there any tool in blender that can match this function? also how could you do this with squares? Someone has asked for answer to these topics. However, I still want someone to provide me with simple and universal answers. I recently read a sentence from the English-speaking person. It is in an article. It is, ' The available options for mitigating pad cratering are not universally effective (see link below), but they may be helpful on a case by case basis until more crack resistant laminates are developed and become available.  '  If I am to write this, I will use 'a case-by-case basis until more crack-resistant laminates'   Can anyone tell me when to use a hyphen to coin a new word? Why would the command "ls *" generate an error about invalid options? There is a thread that talks about ls "*" not showing any files, but I actually wonder why the simple  ls *   command doesn't output anything in my terminal other than  ls: invalid option -- '|' Try 'ls --help' for more information.   while   ls   will list all the files in the current directories as    1                  ferol                      readme.txt  2                  fichier                    sarku  2018               GameShell                  Templates  22223333          '-|h4k3r|-'                 test  3                  hs_err_pid2301.log         test2  CA.txt             important.top.secret.txt   toto.text  CA.zip             JavaBlueJProject           tp1_inf1070  countryInfo.txt    liendur                   'tp1_inf1070_A19(2) (1)'  currency           liensymbolique             tp1_inf1070_A19.tar  curreny            LOL                        Videos  Desktop            Longueuil                 'VirtualBox VMs'  Documents          Music                      words  douffos            numbers                    Zip.zip  Downloads          Pictures  examples.desktop   Public   Any ideas as to why the globbing doesn't take effect here? I'm on Ubuntu, working in the terminal, I don't know if it makes a difference.  Thanks. Chat arrangement question: Left or right? I'm building a small chat. Now I see but in other chats (whatsapp, facebook mobile chat, sms ...) that the arrangement is always the same:  Left: the chat partner. right: myself      Why is that? If a motherboard doesn't explicitly support a RAM frequency, is still compatible? For example, my  says this:   Supports DDR3 2133(OC)/1866(OC)/1600/1333/1066 non-ECC, un-buffered memory   Is RAM clocked at 2400 MHz compatible? Will the performance be downgraded or it just won't work?  I found other questions but they were related to the CPU cap at 1600 MHz. This cap is related to the motherboard, specs says 2133(OC) max. What is the difference between char * const and const char *? Simulate bad cell phone reception in the lab HTML element does not fade in using jquery If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$. If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$.   I want to use induction to prove this:  It is trivial when $G=p(n=1)$, assume $G=p^n$, $N$ is a normal subgroup in $G$ with order $p$. Since $G$ is p-subgroup, $G$ has a normal subgroup $H$ with order $p^{n-1}$,according to assumption, $N \subset C(H)$. But I don't know how to go on, is $C(H)=C(G)$?Can induction work in this case? What is the process for a moderator to request reinstatement or appeal a removal? I've tried latexdef which does not handle --- as the operand. And \texemdash is defined as a character. So that leaves me once again leaning on the collective wisdom of the group :) Proof of Equivalence Relation $R=\{(a,b)\in\Bbb N^2: 5\mid (2a+3b)\}$ Given a set of natural numbers ($\Bbb N$), and the relation $R=\{(a,b)\in\Bbb N^2: 5\mid (2a+3b)\}$. Is $R$ an equivalent relation?   My answer: Yes, because $2a + 3b$ is divisible by $5$, if and only if, $2a + 3b − 5b$ is divisible by $5$.  Is my reasoning right?   If it is not, then how do I prove that this is an equivalent relation? An integer $a \pmod m$ has inverse if and only if $\gcd(a,m)=1$? An integer $a \pmod m$ has inverse if and only if $\gcd(a,m)=1$?  Why is this? I tried understanding it from my notes but I don't get it.  A thorough explanation would be greatly welcome.  Thanks for all the answers so far. I think my biggest problem is understanding why $\gcd(a,m)\neq1 \Longrightarrow \not\exists a^{-1}$. How to format date and time in Android? How to format correctly according to the device configuration date and time when having a year, month, day, hour and minute? Insert only half a pdf page I'd like to include papers of mine which can easily be done with the pdfpages packages:  \usepackage{pdfpages} \includepdf[pages=-]{mypaper.pdf}   However, my thesis has an A5 format. Which means that each page is "half" an A4 page. Basically, I'd like two include each PDF page on two pages, like illustrated below:    While I found how to rotate the PDF using the angle parameter, I did not find how to split it...