Meaning of Lie Bracket term in Riemann Curvature Tensor Formula
About the term $-\nabla_{[u,v]}w$ in the definition of Riemann curvature tensor
Figure refs stop working after I added a new counter I was using the amspset document class for generating exam practice problems. I had wanted to add some referencing to the problems so I started a new counter like this (adapted from some Tex Exchange post)  \newcounter{P} \newcommand{\plb}[1]{\refstepcounter{P}\label{#1} \textbf{P-\ref{#1}.}}  \begin{problem} \plb{p1} A parallel-wire transmission line is constructed of \#6 AWG copper wire (diameter $d=0.162$ in., conductivity $\sigma_c=5.8\times 10^7$ S/m) with a 12-inch separation in air. Assuming no leakage between the two wires, find $R'$, $L'$, $G'$, and $C'$. Assume a working frequency of 1 MHz. \end{problem}  \begin{solution} Using Table.~\ref{tab:tlparam} and Fig.~\ref{fig:tlempl},  \[ Rs = \sqrt{\pi f \mu_c/\sigma_c} = 2.61\times10^{-4}~ \Omega\] \[ R' = \frac{2R_s}{\pi d} = 4.04\times10^{-2} ~\Omega\text{/m} \] \[ L' = \frac{\mu_0}{\pi} \ln{\frac{2D}{d}}=2.0~\mu \text{H/m} \] \[ C' = \frac{\pi \epsilon}{\displaystyle\ln{\frac{2D}{d}}}=5.56 ~\text{pF/m} \] $G'=0$ because the problem states that there is no leakage between the two wires. \end{solution}  \begin{figure}[ht] \begin{centering}     \includegraphics{fig24}     \caption{(Fig.2-4 from FAE) A few examples of transmission lines.} \end{centering} \label{fig:tlempl} \end{figure}  \begin{table}[ht] \caption{(Table 2-1 from FAE)Transmission-line parameters $R'$, $L'$, $G'$, and $C'$ for three types of lines.}     \begin{tabular}{c}         \hspace{3cm}\includegraphics{tb21}     \end{tabular} \label{tab:tlparam} \end{table} ... and more   Now all the figure references stop working, I get "Fig. " instead of "Fig. 1". The table references are alright though. Any idea what the problem could be? How to sort a Google Sheets Query by a custom ordering If $a$ divides $bc$ and $\gcd(a,b) = d$ then $\frac a d$ divides c I'm trying to prove that if $a$ divides $bc$ and $\gcd(a,b) = d$ then $\frac a d$ divides c. I tried using Bezout identity but couldn't get anywhere. Why does input capacity of an opamp increase noise gain? On the image below you can see an inverting amplifier with input capacity C1 and capacity C_F for general stability.     Now my book shows me following frequency plot:  A_n ... Noise-Gain  A_D ... open-loop gain  A_U ... V_out / V_in --> inverting amplifier gain     You can clearly see that C1 is increasing the noise gain at some point. And C_f is stabilizing A_n and A_U.   But why exactly does C1 increase the noise gain exactly? In my case I'm talking about Assassin's Stance, but I'd like to know the answer for stances in general.  I feel like I might be missing something incredibly obvious caused by missing a word or two (or a paragraph), but I can't find anything that explicitly states whether or not I can use the Tome of Battle stances with ranged weapons like the bow (or crossbow).  I realize that half the classes have "blade" in their names so using ranged weapons is not in the spirit of things, but I'm looking for an explicit answer (or as close to one as possible) from any WotC official resource. If this has already been answered I'm sorry, but I could not find the answer myself from searching. Duplicity Backup Browse I have made a duplicity backup with Ubuntu 14.04 in another laptop of my home directory.  I copied all duplicity back files to my current laptop to "/home/user/Ubuntu-backup"  How do i browse the content of that backup see if it is worth restoring? I want to restore how do i restore from those files. It is a well known fact that Alex is more soft-spoken than (she/her).    Why would "her" be wrong? Why must the sentence end with "she"? Automatically login and lock I want to configure Windows 7 to login automatically after the computer is switched on. That's not because I'm lazy and don't like typing passwords. That's because I want programs that are configured to run on startup run automatically.  But I don't want everyone to see my desktop, so I want my computer to remain locked. When I'm ready to work, I just type my password and don't have to wait until all startup programs run. Note: I cannot specificly tell which version of which of the two conflicting packages fixed the issue, but as of 2016 mathpec and polyglossia no longer show the issue.   Basically, I have encountered the same error as  and the workaround suggested by Andrey Vihrov reenables me to compile the document. The problem now is, that any shorthand defined by the ngerman-package (most notably the non-breaking hypen "~) has now been disabled, and I see combinations of american style closing quotes followed by a non-break space where a non-breaking hyphen should be.  Is there any way possible of keeping the shorthand definition, or redefining it to something else (something like \nbhy pops up to mind)?  I am using XeLaTeX as supplied by TeXLive. Can I travel to the UK unaccompanied on an accompanied child visa now that I am over 18? This is on Stack Overflow. On Meta I'm able to see what the tags are on the right side of the screen. In fact, I think yesterday or some point earlier today I saw tags on the right side of the screen (I can't screenshot or remember correctly).  It's right above the asked and active heading.  I know this is minor but what happened to it? It might be useful for browsing instead of having to look at the bottom of the question post.    Found an image on Google and it shows the tags on the right side of the screen: I would like to create a minecraft server for me and and my boyfriend to play on.  How do I set it up? Do I need any additional software? Do you have some links to walkthroughs or tutorials?  I have tried to set this up a few times but have failed every time. I haven't been able to do this exercise:     Let $f: A \rightarrow B$ be any function. $f^{-1}(X)$ is the inverse   image of $X$. Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$ where $X \subseteq B$.   Since $X \subseteq B$, all the elements in $X$ belong to the codomain of $f$.  Since $f$ is surjective, it means that all elements in the codomain $B$ have some preimage in $A$. Given that $X \subseteq B$, all elements in $X$ must also have a preimage in $A$.  Have $\triangle = f^{-1}(X)$, $\triangle$ is now a set containing the preimages of the elements in $X$. Because of this, $\triangle \subseteq A$.  If we evaluate $f(\triangle)$, we...... nope, I don't know what I'm doing now.    What do you think? How to prove that $\mathbb{Q}$ ( the rationals) is a countable set I want to prove that $\mathbb{Q}$ is countable. So basically, I could find a bijection from $\mathbb{Q}$ to $\mathbb{N}$. But I have also recently proved that $\mathbb{Z}$ is countable, so is it equivalent to find a bijection from $\mathbb{Q}$ to $\mathbb{Z}$? Why are only certain shutter speeds or apertures available? How to build a single virtual machine on top of multiple computers? Is there a way to run Linux operating system on a virtual machine which aggregates the resources of physical machines on the local network and acts as a single super-computer? If possible, how?  Thanks. Linear Matrix Inequality of a matrix $T$ with $\| T \|_2 < \rho$ Assume $T \in \mathbb{R}^{d\times d}$. Let $\rho(T)$ be the spectral radius of $T$ and $\rho \ge 0$. Prove that  $\rho(T) &lt; \rho$ if and only if there exists a $P \succ 0$ satisfying $T^TPT - \rho^2 P \prec 0$.  I can prove the Necessity: If $P = I$, for any $x \in \mathbb{R}^d$,   \begin{equation} x^T T^T P T x - \rho^2 x^T P x = \| Tx \|^2 - \rho^2 \| x\|^2 &lt; 0 \end{equation} so $T^T PT - \rho^2 P \prec 0$  I get stuck in proving the Sufficiency. Commandline shortcut for current directory similar to ~ for home directory? When i use the cp or move command in a terminal window, i'm currently with bash in a certain folder like this.  NES@server:~/Desktop/dir1$   And now i wanna copy a file from here ~/anotherdir/dir2 into the current chosen folder in bash (dir1) i would use the command  cp ~/anotherdir/dir2/file ~/Desktop/dir1   does a shortcut string exist to refer to the current chosen directory? So that in this example i don't have to provide the full path to the target dir, but the command knows it should use the current chosen directory in bash? i.e. as ~ stands for the home directory? The man page for GNU find states:    -exec command ;     [...] The  string `{}'  is  replaced  by the current      file name being processed everywhere it occurs in the      arguments to the command, not just in arguments where      it is alone, as in some  versions  of  find.     Both  of  these  constructions might need to be escaped      (with a `\') or quoted to protect them from expansion      by the shell.     That's from the man to find (GNU findutils) 4.4.2.   Now I tested this with bash and dash, and both don't need to have the {} being masked. Here is a simple test:  find /etc -name "hosts" -exec md5sum {} \;    Is there a shell, for which I really need to mask the braces? Note, that it doesn't depend upon whether the file found contains a blank (invoked from bash):   find ~ -maxdepth 1 -type d -name "U*" -exec ls -d {} \;  /home/stefan/Ubuntu One   This changes if the found file is passed to a subshell:  find ~ -maxdepth 3 -type d -name "U*" -exec bash -c 'ls -d {}' \;  ls: cannot access /home/stefan/Ubuntu: No such file or directory ls: cannot access One: No such file or directory   which can be solved by:  find ~ -maxdepth 3 -type d -name "U*" -exec bash -c 'ls -d "$0"' {} \;   in contrast to:   find ~ -maxdepth 3 -type d -name "U*" -exec bash -c 'ls -d "{}"' \;  /home/stefan/Ubuntu One   but that's not what the man page is talking about, is it? So which shell treats {} in a different way? Why is spawning threads in Java EE container discouraged? One of the first things I've learned about Java EE development is that I shouldn't spawn my own threads inside a Java EE container. But when I come to think about it, I don't know the reason.  Can you clearly explain why it is discouraged?   I am sure most enterprise applications need some kind of asynchronous jobs like mail daemons, idle sessions, cleanup jobs etc.  So, if indeed one shouldn't spawn threads, what is the correct way to do it when needed?