Quaternions: Why is the angle $\frac{\theta}{2}$?
half sine and half cosine quaternions
If the center of a group $G$ is of index $n$, prove that every conjugacy class has at most $n$ elements. (This question is from Dummit and Foote, page 130, 3rd edition.)  Here is my attempt: we have  $|G| =|C_G (g_i)| | G : C_G (g_i)| $  $|C_G (g_i)| =|Z(G)| \cdot  |C_G (g_i):Z(G)| $.  Then   $|G|= |Z(G)|\cdot |C_G (g_i):Z(G)|\cdot | G : C_G (g_i)| $.  Then  $n = |C_G (g_i):Z(G)|\cdot  | G : C_G (g_i)| $.  But $|C_G (g_i):Z(G)|$ is positive integar as  $ Z(G)$ is   subgroup of  $C_G (g_i)$.  If $| G : C_G (g_i)|$ bigger than $n$, then $|C_G (g_i):Z(G)|\cdot | G : C_G (g_i)|$ is bigger than $n $.  But   $|C_G (g_i):Z(G)|\cdot | G : C_G (g_i)|=n$, contradiction.  Then $| G : C_G (g_i)|\le n$ and this completes the proof.   Is this proof right or not ??!!! I have a very stupid question. I developed a website that has only one page at the public html directory(index.php).  all content are dynamically loaded as different pages.   How can i SEO my pages for google?  i mean like the what we type in this site are .............i believe stored in database and search engine find these pages like any static pages...  Just a little tip would be appreciable.......! Difference between logit and probit models Is there any way to prevent mc from taking 10-30 seconds to open? Before 1994 the last Triwizard Tournament had been held in 1792, where the three headmaster judges were injured by a rampaging cockatrice.  The tournament is held every 5 years, which would put the next one in , the year . Did they carry on with the tradition, or was it deemed too risky again? Maximum length of tags Possible Duplicate:        Is there a maximum limit on the length of tags? I noticed that the tag for SQL Server Management Studio is truncated to sql-server-management-stu.  It seems like this should be an easy thing to correct, if not to allow the creation of new longer tags, but to correct existing tags that have been truncated. Which is the correct use of these two words, and in which context should one be used rather than the other? Can I replace a 15 ampere circuit breaker with a 20 ampere breaker? When I plug-in a conventional heating pad or a hair dryer into any bedroom duplex, the lights dim; and/or trips the circuit breaker. The home is of new construction-- placed in service 31 Dec 2013. My question is: Can I safely replace the existing 15 AMP Breaker with that of a 20 AMP? . . . the service cable is of 14 Gauge Copper Conductor. Sphere inside a conical vessel touches the lid A conical vessel with semi-vertical angle 30 degrees and height 10.5 cm has a thin lid. A sphere kept in the vessel touches the lid. The radius of the sphere in cm is ... Question about Casella and Berger's proof of MLE invariance In Casella and Berger, p. 320, they have a proof of the invariance of the MLE. Let $g: \theta \mapsto \eta$ be a function. They define the induced likelihood as  $$ L^*(\eta \mid X) = \sup_{\{\theta: g(\theta) = \eta\}} L(\theta \mid X). $$  This ensures that $g$ is one-to-one with respect to the likelihood functions because if $g$ is not one-to-one, there may be multiple values of $\theta$ that map to a given $g(\theta) = \eta$.  Then they note  $$ \sup_{\eta} L^*(\eta \mid X) = \sup_{\eta} \sup_{\{\theta: g(\theta) = \eta\}} L(\theta \mid X) = \sup_{\theta} L(\theta \mid X). $$  The first equality just applies the definition of the MLE of $\eta$. However, the next inequality confuses me. They write,     The second inequality follows because the iterated maximization is equal to the unconditional maximization over $\theta$...   Can someone justify this claim or provide some intuition if the claim is supposed to obviously follow from definitions? Do "Best by: XX-XX-XX" foods have an expiration date? How long will foods that have "Best by: XX-XX-XX" on the product last past that date?  I have tried looking online, and all I can find is anecdotal evidence, which isn't very assuring for me since there seems to be no specific "x food will expire y weeks after its best by date".  Is there a rule of thumb or some guide out there that can tell me how long a "best by" food will last? Or is it safe to assume that they never expire, since it's not actually labelled on the product? How to relate combinatorial proof to bipartite graphs? The question is - Let $k \geq 1$ be an integer and consider a sequence $n_1,n_2,\cdots,n_k$ of positive integers. Use a combinatorial proof to show that  $\binom{n_1}{2} + \binom{n_2}{2} +\cdots+ \binom{n_k}{2} \leq \binom{n_1+n_2+\cdots+n_k}{2}$  How can i prove this without using induction?  Additionally, For each $i$ with $1 \leq i \leq k$,consider the complete graph on $n_i$ vertices. How many edges does this graph have?  I thought of making graphs and figuring out a relation but still wasnt able to. Suppose my speed is normally 30 feet. I get hit with a Slow spell, reducing my speed to 15 feet, and then I get knocked prone. How many feet of movement do I need to spend in order to stand back up from prone? How many feet of movement do I have left over once I've stood up?  What if I am also hit with a Ray of Frost spell while slowed, reducing my speed to 5 feet for 1 round? Am I able to stand up at all? I am running Xubuntu 11.10 and everything works nicely. I have selected a nice Xfce theme and everything looks beautiful, except Libreoffice.  While other programmes (Gimp, Inkscape, Thunar etc.) integrate with the selected Appearance &amp; Window Manager theme, Libreoffice looks like a Windows 98 programme.  Does anyone know how to fix this? I have already tried this solution, without success:   Thanks for your help! We're running a couple SQL servers in Azure that are set up with an Always On availability group and Windows Failover Clustering. The servers are Windows 2019 and we're running SQL Server 2019. When we set up the cluster, it was set up to use a Distributed Network Name instead of a static cluster IP address. Thanks to this we shouldn't need an internal load balancer according to these notes: .  I'm struggling to understand exactly how this works though. Based on what I read, it seems like our connection strings will point to the DNS name of the cluster (let's call it AgCluster). If I look in DNS, there is an A record for AgCluster pointing to sql1 and another pointing to sql2. When I use AgCluster in my connection string it seems to always connect me to the primary server, even if I have ApplicationIntent=ReadOnly set. When I query @@SERVERNAME I always get the same server.  So with the Distributed Network Name setup, what should I use in my connection strings to make sure read/write queries go to the primary and read only go to a secondary? Any guides on setting this up in general would be helpful. Thanks! Java GUI repaint() problem? I have a JFrame. This JFrame contains a JButton. I click the JButton and 10 JTextFields are created.  the problem: I cannot see them until "I force a repaint()" by resizing the window. Only then do I see the JTextFields created.  CODE:  JPanel points = new JPanel();  //Creating the JTextFields: for (int i=0; i&lt;10; i++) {     JTextField textField = new JTextField();     points.add(textField); }  repaint(); this.repaint(); super.repaint(); points.repaint();     THANK YOU - after the for loop, I just called points.validate() and it worked... How can I enable access to USB devices within VirtualBox guests? I cannot see any USB devices within my VirtualBox guest VMs from my host. How do I enable access for my guest VMs?  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; I would like to be able to copy and paste text in the command line in Bash using the same keyboard bindings that Emacs uses by default (i.e. using C-space for set-mark, M-w to copy text, C-y, M-y to paste it, etc.).  The  says that Bash comes with some of these key bindings set up by default.   For example, yanking (C-y) works by default on my terminal. However, I can't get the set-mark and copy commands to work, and they don't seem to be bound to any keys by default.  Usually, the way a user can define her own key bindings is to add them to .inputrc. So I looked and found the following bash functions in the documentation that I presume can help me define the Emacs-like behavior that I want (i.e. set-mark with C-space and copy with M-w).  copy-region-as-kill ()      Copy the text in the region to the   kill buffer, so it can be yanked right   away. By default, this command is   unbound.   and    set-mark (C-@)      Set the mark to the point. If a   numeric argument is supplied, the mark   is set to that position.   If I understand correctly, the above means that copy-region-as-kill is not bound to any keyboard sequence by default, while set-mark is bound to C-@ by default.   I tried C-@ on my terminal, but I don't think it runs set-mark because I don't see any text highlighted when I move my cursor. In any case, I tried adding  keyboard bindings (M-w and C-) to the functions  copy-region-as-kill and set-mark above in my .inputrc and then reloading it with C-x C-r, but this didn't work. I know that my other entries in .inputrc work because I have other user-defined keybindings defined in it.  Is there anything I am doing wrong? Am I missing anything? I completed the two contracts I was given (Scout and Heavy), and earned two of the 'Teufort' weapons for turning them in.  I expected to get some new contracts so I could continue to earn some more of the schmancy new weapons, but I can't seem to find out how to get more.  Do I need to buy the ticket again to unlock more contracts? If not, how are contracts assigned? How do I set up a Cron job? I want to schedule a task to run on a regular basis and have heard that Cron is the way to do this.  How do I add Cron jobs in Ubuntu?