Here by $b(n,F)$ we mean the Lie algebra of all $n \times n$ upper triangular matrices. So, the first step of the question asks to prove that $L^m$ has a basis consisting of all matrix units $e_{ij}$ with $j - i &gt; m$.  So I thought I would try to compute $L^m$ for different $m$. Now, $e_{ij}$ where $i &lt; j$ form the basis for this Lie algebra. $[e_{ij}, e_{kl}]$ $=$ $\delta_{jk}e_{il}$ - $\delta_{il}e_{kj}$. So we have two cases, $j = k$ or $k = l$, but not both. These two cases lead to $L^1$ being the span of $e_{il}$ and $e_{kj}$. Now I don't know how to check if the claim in the question is true for $L^1$. Can anyone help me out?
I have been attempting this problem for a while, it is an assignment problem so I don't want somebody to just post the answer, I'm just looking for hints.  Let $\mathfrak{u}(n,\mathbb{C})$ be the Lie algebra of strictly upper triangular $n\times n$ matrices over $\mathbb{C}$.  I am trying to show that $\mathfrak{u}(n,\mathbb{C})$ is a nilpotent Lie algebra for $n\geq 2$.  This is what I have looked at so far.  I have done some explicit calculations (I shall omit the bulk of them and just summarise below) for $n=2,3,4$ and have come up with some ideas.  Let $L=\mathfrak{u}(n, \mathbb{C})$.  If $n=2$,  $L^1 = [L, L] = &lt;0&gt;_{\mathbb{C}}$  So $L$ is nilpotent.  If $n=3$,  $L^1 = [L, L] = &lt;e_{13}&gt;_{\mathbb{C}}$  $L^2 = [L, L^1] = &lt;0&gt;_{\mathbb{C}} = \left\{ 0 \right\}$  So, $L$ is nilpotent.  If $n=3$  $L^1 = [L,L] = &lt;e_{13},e_{14},e_{24}&gt;_\mathbb{C}$  $L^2 = [L,L^1] = &lt;e_{14}&gt;_{\mathbb{C}}$  $L^3 = [L, L^2] = &lt;0&gt;_{\mathbb{C}} = \left\{ 0 \right\}$  So, $L$ is nilpotent.  So my hypothesis so far is that for an arbitrary $m\in\mathbb{N}$, $L^{m-1}=\{0\}$  My instinct is to do this by induction. My base case, $n=2$ has already been shown. Now I assume that $\mathfrak{u}(n,\mathbb{C})$ is nilpotent for $n=k$ ($k\in\mathbb{N}$) and consider the case for n=k+1.  However, I have no idea how to link these two cases together. I have read in Introduction to Lie Algebras and Representation Theory - J.E. Humphreys that $L^k$ should be spanned by basis vectors $e_{ij}$ where $j-i=k+1$ - however in my calculations for $L^1$, I obtain an $e_{14}$ which does not satisfy 4-1=2.  Am I missing something completely obvious? Or have I made a mistake somewhere? (I have checked my calculations numerous times, but cannot find any errors!)  Thanks,  Andy.
Does transparency work in eevee I have been trying to create a transparent lens using the eevee rendering engine in 2.8 but the transparency slider has no effect. Has this feature not been implemented yet? I've looked all over the Web for an answer with no luck. I'm a noobee so am probably doing something wrong. Hide c# windows application source code I wrote a windows application using C# .Net 2.0 and i want to do something which hide the source code, so when any one use refactor tool can't see the source code. I used dotfuscator but it just changed the function names but not all the source code.  UPDATE: I want to hide the source code, not because of hiding the key, but to hide how the code is working.  Thanks, I have something like that  how do I convert this to a YouTube playlist? java.rmi.NoSuchObjectException: no such object in table I am writing a very simple RMI server, and I am seeing intermittent java.rmi.NoSuchObjectExceptions in the unit tests.   I have a string of remote method calls on the same object, and while the first few go through, the later ones will sometimes fail. I am not doing anything to unregister the server object in between.  These error do not appear always, and if I put in breakpoints they tend to not appear. Are those Heisenbugs, whose race conditions dissolve when looking at them through the slowed down execution of the debugger? There is no multi-threading going on in my test or server code (though maybe inside of the RMI stack?).  I am running this on Mac OS X 10.5 (Java 1.5) through Eclipse's JUnit plugin, and the RMI server and client are both in the same JVM.  What can cause these exceptions? Difference between $\Delta$, $d$ and $\delta$ Say I have a code to draw a flow chart, I want now a arrow from node 3 to node 1, how can I do that?  1&lt;--| |   | 2   | |   | 3----   MWE  \documentclass[class=article,border=0pt]{standalone}  \usepackage{tikz} \usetikzlibrary{shapes,arrows}  % Define block styles \tikzset{     block/.style={rectangle, draw, line width=0.5mm, black, text width=5em, text centered,                   rounded corners, minimum height=2em},     line/.style={draw, -latex} }% &lt;- if you insist in using this in the document add this % here. \begin{document} \begin{tikzpicture}[node distance = 1cm, auto]     % Place nodes     \node [block] (BLOCK1) {a};     \node [block, below of=BLOCK1] (BLOCK2) {b};     \node [block, below of=BLOCK2, node distance=1cm] (BLOCK3) {c};     % Draw edges     \path [line] (BLOCK1) -- (BLOCK2);     \path [line] (BLOCK2) -- (BLOCK3); \end{tikzpicture} \end{document} How to uninstall broken programs in windows 7 Could anyone tell me how to uninstall programs that their installation had a problem and therefore their uninstall doesn't work?  I removed all their references in the registry and their folder on the hdd but still don't know how to get rid of them completely from Add/Remove Programs list.  Thanks Where did object dimensions go in 2.8? Whilst I understand how the RSA algorithm works, I don't understand how the CPU operates when it needs to use the mod function with a huge number $n$, for example.  $n = 8192$ bits; $c = m^e \mod n$;  In essence my question is how does the CPU deal with this huge modulus? (Revised) Prove that $\phi$ is a group homomorphism and find the kernel. Quoting " Let $\phi : \Bbb Z \rightarrow \Bbb Z$ be given by $\phi(n) = 7n$. Prove that $\phi$ is a group homomorphism. Find the kernel, and the image of $\phi$."   My understanding:  Part 1:   Given $n,m \in \Bbb Z$, let's check that $\phi(n+m) =\phi(n)+\phi(m)$  $$\phi(n+m) = 7(n+m) = 7n + 7m = \phi(n)+\phi(m)$$  Therefore as the group operation in $\phi$ is preserved, $\phi$ is a group homomorphism.  Part 2 (revised):   $\phi$ is one-to-one as:  $\space \space \phi (n) = \phi(m) \Rightarrow 7n =7m \Rightarrow n=m  $. We also know that $\phi(e)=e$ by the property of homomorphism. Therefore, no other object than $0$ in the domain can map to $0$ in the codomain. It follows that: $$\ker(\phi)=\{ x \in \Bbb Z : \phi(x)=e\}=\{e\}$$   Part 3 (revised):  I claim that the only possible image-elements of $\phi$ are the multiples of $7$ in $\mathbb Z$ denoted $7\mathbb Z$.   To prove this I have to show that Im$(\phi) \subset 7 \Bbb Z$ and that $ 7 \Bbb Z \subset $ Im$(\phi)$.  Proving Im$(\phi) \subset 7 \Bbb Z$ :      $\forall y \in $ Im$(\phi)$ such that $  y =\phi(x)$ where $x \in 7 \Bbb Z$. It follows that $y \in 7 \Bbb Z$, therefore Im$(\phi) \subset 7 \Bbb Z$   Proving $ 7 \Bbb Z \subset $ Im$(\phi)$:   $\forall y \in  7\Bbb Z$ such that $  y =7x$ where $x \in \phi(x)$. It follows that $y \in \phi(x)$, therefore $ 7 \Bbb Z \subset $ Im$(\phi)$.   Any input to my understanding is much appreciated. How to understand Cauchy's proof of AM-GM inequality(the last step) The AM-GM inequality: $$a_1a_2\cdots a_n\leq\left(\frac{a_1+\cdots + a_n}{n}\right)^n$$  the trivial case: $a_1a_2 \leq \left(\frac{a_1+a_2}{2}\right)^2 $ is self-evident.  then cauchy use this fact repeatedly. He get: $$a_1a_2\cdots a_{2^m} \leq \left(\frac{a_1+a_2+\cdots +a_{2^m}}{2^m}\right)^{2^m}.\tag{$\ast$}$$  This is easy to understand, and natural, but next step:  he let  $$b_1 = a_1,\ b_2=a_2,\ \ldots,\  b_n=a_n,\  b_{n+1}=\cdots=b_{2^m}=\frac{a_1+a_2+\cdots a_n}{n}$$  replace $a_1, a_2,\ldots ,a_{2^m}$ with $b_1, b_2, \ldots b_{2^m}$ in $(\ast)$ and simplified both left and right. He get the answer. I can repeat his proof, but I don't know why Cauchy can got this idea. How does he got it? Give a sequence $(f_n)_{n\in \mathbb{N}}$ of differentiable functions which uniformly converge to $0$, but for which the seqeunce $(f_n')_{n\in \mathbb{N}}$ of the derivatives isn't even pointwise convergent.    I found this one in my textbook marked as "Fun things to solve" and although it may be fun, it's kind of hard to do.   To be honest I already failed at the first hurdle. I couldn't even find a sequence of functions which uniformly converges to $0$.   Is there a certain way of dealing with this kind of problem? Because it seems kind of hard for me to come up with sequences without a mathematical of way of doing so (or maybe there is a mathematical way I just don't know yet). How do you handle "that that"? The double "that" problem Stone-Weierstrass theorem with $p(1/x)$ Index of the center of a group in the group is not a prime number The question is to prove index of $Z(G)$ in $G$ is not a prime number.    We know that $|G:Z(G)|=|G:C_G (x)||C_G (x):Z(G)|$ where $C_G (x)$ means centralizer of $x \in G$  I want to mention that we do not want to use $G⁄Z(G)$ theorem.   We may assume that $|G:Z(G)|=p$ where $p$ is a prime. Connecting Asus USB-N13 Wireless Adapter I'm trying to connect my desktop (running 64-bit Ubuntu 12.04) to an Airport Extreme Base Station by using an ASUS USB-N13 wireless adapter.  I plugged it in, turned on the desktop, and the adapter was immediately recognized by the system. I found my home network listed in the available networks, typed my password (with the correct encryption) and hit connect. After that, the wireless icon just flashes (propagates? emits?) indefinitely, and every couple of minutes the "Wireless Authentication Required" box pops back up.   I don't know what to do. I've seen some solutions talking about an "rt2xxx" or "rt3xxx" driver or chipset. I don't know exactly what it is, but when I typed lsmod | grep rt I got:  parport_pc rtl8192cu rtl8192c_common rtlwifi mac80211 cfg80211 parport   I'm pretty much lost. Please AskUbuntu, what do? I make homemade cayenne pepper sauce every year. I do add some liquid pectin to slightly thicken it so it will stick to food rather than roll off. After a month or so on the shelf it seems to separate a bit. What can I add to this so that it won't or is there a method I can do to stop this. We are slowly working our way into marketing this product down the line and this won't work for that reason. We are using a water bath canning method, vinegar based, no sugar added. An init-script does not get called on shutdown I have set up an init-script to control the state of a VirtualBox VM:  #!/bin/sh #chkconfig: 35 99 5 #description: vTiger virtual machine  ### BEGIN INIT INFO # Provides: vtigervm # Required-Start: $local_fs # Requider-Stop: $stop_fs # Default-Start: 3 5 # Default-Stop: 0 1 2 6 # Short-Description: Manage vTiger virtual machine # Description: Utility to start and stop vTiger virtual machine on VirtualBox ### END INIT INFO  start() {     echo -n "Starting vTiger"     echo     su myuser -c '/usr/bin/VBoxManage startvm "vTiger" --type headless'     echo "Started virtual machine" &gt;&gt; /var/log/messages }  stop() {     echo -n "Shutting vTiger down..."     echo     su myuser-c '/usr/bin/VBoxManage controlvm "vTiger" acpipowerbutton'     while [ ! -z "`su - juhani -c '/usr/bin/VBoxManage list runningvms | grep vTiger'`" ]; do         echo -n "."         sleep 1     done     echo "Done."     echo "Stopped virtual machine" &gt;&gt; /var/log/messages }  status() {     echo -n "Running VMs: "     su myuser -c '/usr/bin/VBoxManage list runningvms'     echo     if [ -z "`su - juhani -c '/usr/bin/VBoxManage list runningvms | grep vTiger'`" ]; then         RETVAL=3         else         RETVAL=0     fi     echo "Queried virtual machine status" &gt;&gt; /var/log/messages }  echo "Called virtual machine management script with: $1" &gt;&gt; /var/log/messages case "$1" in   start)         start    ;;    stop)         stop    ;;    restart|try-restart|condrestart|reload)         stop         start    ;;    status)         status    ;;    *)         echo $"Usage: $0 {start|stop|restart|status}"         exit 1    ;; esac  exit $RETVAL   The system is CentOS 6.5. If I manually run service [start|stop|status] vtigervm, it works as I expect it to. If the VM is running, $? after the service vtigervm status returns 0, and 3 if it is stopped. I installed it with chkconfig and it created among a few others rc5.d/S99vtigervm and rc0.d/K05vtigervm.  The problem  When I start the system it starts the "service", but on shutdown it does not even run the scripts.  grep "virtual machine" /var/log/messages shows:  *[machine starting]* Called virtual machine management script with: start Started virtual machine *[shutdown -h now]* *[machine stopped]*   What I expect:  *[machine starting]* Called virtual machine management script with: start Started virtual machine  *[shutdown -h now]* Called virtual machine management script with: status Queried virtual machine status Called virtual machine management script with: stop Stopped virtual machine *[machine stopped]*   File permissions &amp; etc:  # ls -lah /etc/rc0.d/ lrwxrwxrwx. 1 root root 13 7.2. 17:49 /etc/rc0.d/K05atd -&gt; ../initd.d/atd lrwxrwxrwx. 1 root root 18 9.2. 00:06 /etc/rc0.d/K05vtigervm -&gt; ../initd.d/vtigervm Is there a way to reorganize the fields/attributes in a data extension?  I've added some extra fields that relate to other fields in that extension.   I haven't found a way to regroup those fields. Recently I have started playing with jQuery, and have been following a couple of tutorials. Now I feel slightly competent with using it (it's pretty easy), and I thought it would be cool if I were able to make a 'console' on my webpage (as in, you press the ` key like you do in  games, etc.), and then have it Ajax itself back to the server in-order to do stuff.  I originally thought the best way would be to just get the text inside the textarea, and then split it, or should I use the keyup event, convert the keycode returned to an ASCII character, append the character to a string and send the string to the server (then empty the string).  I couldn't find any information on getting text from a textarea, all I got was keyup information. Also, how can I convert the keycode returned to an ASCII character?