The action of a relativistic point particle is its negative rest energy along its worldline, the parameter being its own proper time. $$ S = - mc^2 \int d\tau $$    Is there a particular reason why the action is considered to be negative?
In the special relativistic action for a massive point particle, $$S=\int_{t_i}^{t_f}\mathcal {L}dt,$$ why is the Lagrangian $$\mathcal {L}=-E_o\gamma^{-1}$$ a negative number?
Renaming attributes/fields in shapefile attribute table using QGIS I would like to rename some fields in my attribute table:  Like rename &quot;XRF_N3&quot;in something else. It seems quite a easy issue but I don't find any information about how can I do that. Let $X$ be an infinite cofinite space and let $A \subset X$ be infinite. Then $\overline A = X$  I am having a tough time in trying to understand this. Why is every infinite set in an infinite cofinite space dense ? Permanently removing apache2 Can I use an inner tube that's too narrow for the tyre? I recently got a puncture and when I went to replace the inner tube I discovered that I'd bought the wrong size tube. My tyre is 700 x 40c, the tube is 700 x 28-35c.  What is likely to happen if I use this tube? Will the tube explode when I attempt to inflate it? Graphics horribly corrupted on Ubuntu 16.04 in VirtualBox, showing random strips of pixelated colors Inside a VirtualBox VM, I have installed myself an Ubuntu 16.04.1 LTS Server installation. However, when I installed the ubuntu-desktop package and rebooted, this happened:    Exact command I used: sudo apt-get update; sudo apt-get install ubuntu-desktop ..and then a reboot command.  This just seems like a corrupted top-left portion of the screen, as my cursor can appear in view (but corrupted):    This is a purely graphical issue because I can use the keyboard to successfully progress to a desktop:    Note: On startup it shows correct graphics for a split-second then reverts to glitch.    Any help on this? Integral of $\sqrt{1-x^2}$ using integration by parts How to automatically fit many bars in barchart using pgfplots? I am trying to use PGFPlots to display a large amount of bars, and I run into the problem that it cuts off the last of them.  MWE:  \documentclass[crop,tikz]{standalone}  \usepackage{pgfplots} \usepackage{pgfplotstable} \pgfplotsset{compat=1.13}  \begin{document}  \pgfplotstableread{ row label runtime 0 1 5264 1 2 2971 2 4 1866 3 8 1384 4 16 1205 5 32 1144 6 64 1109 7 128 1095 8 256 1108 9 512 1178 10 1024 1335 11 2048 1667 12 4096 1588 13 8192 2843 14 16384 5541 15 32768 10985 16 65536 21873 17 131072 43687 18 262144 87064 19 524288 174119 }\mytable   \begin{tikzpicture}   \begin{axis}[         ylabel={runtime (${\mu}s$)},         xticklabels from table={\mytable}{label},         xtick=data,         ymin=0,         ymax=10000,         ybar,         xticklabel style={           anchor=north west,           align=left,           rotate=-45,           font=\tiny,           inner sep=1pt,         }, ] \addplot+ [] table [y=runtime, x expr=\coordindex] {\mytable}; \end{axis} \end{tikzpicture}  \end{document}   Result:    By adding the following to the axis options, I can make all the bars fit. But I need to manually adjust bar width (and xmax)  xmin=-1, xmax=20, bar width=7pt,     Is there a way to handle many bars in a barplot automatically? Why won't OS X trust GitHub's SSL certificate? Is there a difference between "" and {} in BibTeX? I've been having this issue for a while where I come to open the Terminal and I'm met with this:  Login incorrect login: _   I've already deleted the com.apple.terminal.plist to no avail. The really interesting thing is that sometimes it works, maybe 1 in 50 times. I literally have no idea what might be causing it and after searching the web for the better part of half a day it seems everyone else's issues are fixed when they delete the plist file.  How do I sort this out? I've tried running different shells (bash and zsh), but I still get this log in screen.  Also when I use Activity Monitor, I can see there is no shell running just the login. How can I find the product key of Office 2011 after restoring from backup? I had to restore my iMac from a Time Machine backup.  Everything seemed fine until I tried to start Office 2011 and it is asking me to re-enter my product key. I am trying to locate my key (I believe I downloaded this version originally and it should be in my email but I can't find it)  I tried a program that is supposed to "find the keys for products installed on your computer" (Mac Product Key Finder) and it found my very old Office 2008 key, but not 2011.  I am surprised that if I had a totally working install before recovering from back-up that this wasn't restored as well - do I have any options to find it hiding somewhere on my drive?  Note: The recovery was after a hard drive was replaced by the Apple store - if that makes a difference, so this is a new internal drive. The internal storage on my phone is almost full. The storage screen tells me    2.77GB / 2.91 GB taken Apps 0.89GB Other 1.89 GB Available 138MB   What is this "other" and how can I clean it up? I already followed the steps of "make more space", it barely makes a difference.   Phone Storage then tells me 43.00MB / 1.09GB is taken.  And on the SD card I have 1.86GB / 14.72GB taken.  I have plenty of storage in various places yet when I try to install apps I'm unable to do so cause of "insufficient storage". How can I fix this? Value of $x=\sum\limits_{n=1}^\infty {\frac{1}{10^n-1}} $ and location of second digit $1$ of $x$ What is the value of the series   $\sum\limits_{n=1}^\infty {\frac{1}{10^n-1}}$   and can one find closed forms for the sums   $\sum\limits_{n=1}^N {\frac{1}{10^n-1}}$?   I found that $\sum_{n=1}^N {\frac{1}{10^n-1}} $  =  $\sum_{n=1}^N (\lim_{m\to\infty  }{\frac{10^{-n}-10^{-nm}}{1-10^{-n}}})$  =  $0.111\ldots+0.0101\ldots+\ldots$ but I can't go farther.     The original question is: "Where does the second '1' appear?" If every ideal is principal show the same is true for the image of a homomorphism. I have an input type="image". This acts like the cell notes in Microsoft Excel. If someone enters a number into the text box that this input-image is paired with, I setup an event handler for the input-image. Then when the user clicks the image, they get a little popup to add some notes to the data.  My problem is that when a user enters a zero into the text box, I need to disable the input-image's event handler. I have tried the following, but to no avail.  $('#myimage').click(function { return false; }); Touchpad stopped working 20.04 How can one cite theorems of references? Alternating binomial sum with intervals of two Fix integer $n\geq 1$. Consider the number $$1-\binom{n}{2}+\binom{n}{4}-\binom{n}{6}+\cdots$$where the sum continues as long as the lower number in the binomial is $\leq n$. Is there a way to simplify this sum?   The first few values are $1, 0, -2, -4, -4$. We have 10 PC with some version of Ubuntu and only remote access. While doing some upgrades to custom software I did not notice that the line endings in some scripts were Windows version (CR+LF) and not the Unix version (LF). So now when I want to launch the script it gives an error:   bash: /usr/local/bin/portsee: /usr/bin/python^M: bad interpreter: No such file or directory   Is there a way to change all line endings in a script from terminal. The thing is that I can not install any new software to this group of PC-s. What could cause this? f12:    Viewport:    (the f12 has fewer samples, but even when the viewport is at lower samples it looks different..)  I know about color management, but that is at the defaults..    EDIT: oops.. I know 'log' is not a default, but that is because I was fiddling with it. The image above is rendered with 'default'    The material is quite simple, only a translucent shader. (for testing) I even tried appending everything into a new file to reset all the settings, but no difference.  EDIT2: Uploaded blend