Escape velocity of Kerr black Hole
Escape velocity from a rotating black hole
I have an itemized list:  \begin{itemize}     \item Lorem     \item Ipsum     \item Sit     \item Dolor     \item Amet \end{itemize}   I want to create an accolade on the right side of 'Lorem' and 'Ipsum', and add a bit of text at the point of the accolade. The level of this text should be between the first and the second line. How can I get this done? Omitting the hypotheses of finiteness of the measure in Egorov theorem I want to prove that if I omit the fact that $\mu (X) &lt; \infty$ in Egorov theorem and place instead that our functions $|f_n| &lt;g$ and $g$ is integrable, we still get the result of Egorov's theorem.  Fix $m$ a natural number.  I took $ w_{n} = |f_n-f|$ and thus by DCT $\int |f_{n} - f|$ goes to zero.  Then I took $\bigcup_n {( w_{n} \geq 1/m)}$. I need its measure to be finite. Its measure is less than the sum of the measures of each $ w_n\geq 1/m)$ varying $n$, and by Tchebychev, this is less than $m\int|f_{n} - f|$.  But I got stuck here.  Any help is appreciated. Thanks! WinForms DataGridView font size How do I change font size on the DataGridView? I want to change the default placeholder text in my Custom Post Type title field. This isn't possible while registering a new post type.  The following screenshot describes it. Input file1 is:  dog 123 4335 cat 13123 23424  deer 2131 213132 bear 2313 21313   I give the match the  pattern from in other file ( like dog 123 4335 from   file2).  I match the pattern of the line is dog 123 4335 and after printing all lines without match line my output is:   cat 13123 23424 deer 2131 213132 bear 2313 21313   If only use without address of line only use the pattern, for example 1s how to  match and  print the lines? Order of the Rubik's cube group Why does $x$ divided by zero not equal $x$? Why does $x$ divided by zero not equal $x$? After all, $x$ is not being divided by anything. Call C++ library in C# I have a lot of libraries written in C++. I want to call these libraries from C#, however, I have met many problems. I want to know if there is a book or guideline to tell me how to do that. Showing $d(x,y) = \frac{|x-y|}{1+|x-y|}$ is a distance. Show that $(\mathbb{N}, d)$ is a metric space   with $$d(x,y) = \frac{|x-y|}{1+|x-y|}$$   My attempt:   let $x,y \in \mathbb{N}$,  1) $d(x,y) = 0 \implies |x-y| = 0 \iff x = y$  2) $d(x,y) = d(y,x)$  3) show for $z \in \mathbb{N}$, $d(x,z) \leq d(x,y) + d(y,z)$  Using the triangle ineuqality I managed to get:  $$d(x,z) \leq \frac{|x-y|}{1+|x-y| - |y-z|} + \frac{|y - z|}{1 + |x-y| - |y-z|}$$  but i assume here I must use the fact that $x,y,z \in \mathbb{N}$ but I'm not sure how.  Any hints please:  edit: improvements on solution - Using fact that $|x-y|$ is a metric we have that $$d(x,y) \leq \frac{|x-y|}{1+|x-y| + |y-z|} + \frac{|y - z|}{1 + |x-y| + |y-z|}$$ (Since $|x-z| \leq |x-y| + |y-z|$) Is this correct? From there I can use the fact that $$\frac{1}{1+|x-y|+|y-z|} \leq \frac{1}{1+|x-y|} $$ and $$\frac{1}{1+|x-y|+|y-z|} \leq \frac{1}{1+|y-z|}$$ to conclude? Bug? LaTeX misparses nested optional arguments LaTeX seems to be having problems handling nested optional arguments. I've included a minimal code example that generates the error:  \documentclass[12pt,oneside]{book} \begin{document}  \newcommand{\termbad}[2][]{#2#1} \newcommand{\termgood}[2][]{#2#1}  \newcommand{\up}[1]{#1} \newcommand{\dd}[1][]{#1}  \termgood{\up{Tart}}{Pop} \termbad[\up{Tart}]{Pop}      \termgood{\dd[Tart]}{Pop} \termbad[{\dd[Tart]}]{Pop} %\termbad[\dd[Tart]]{Pop}   \end{document}   If you run this code, it works. If you remove the % in the second to last line, it fails with:  Runaway argument?  Tart{Pop} \par \par \end {document}  !File ended while scanning use of \dd.   After further tinkering, I have discovered the problem: LaTeX is reading the ] for the inner optional argument as the end of the outer optional argument and then passing a malformed argument in. So it is passing in \dd[Tart as the optional argument to \termbad instead of \dd[Tart].  This still feels like a bug to me. Nesting braces in arguments works, brackets should as well. The following are workarounds:   Enclose the inner optional argument in braces (as in the line before %). Redefine your inner function to require its argument instead of using optional arguments. Redefine your outer function to require its argument instead of using optional arguments.   Any one of the three works around the issue.  (Reposted here; question was originally posted to stackexchange and then closed instead of being migrated.) How to calculate square root or cube root? I was reading Richard Feynman biography when I read that one time he was able to calculate the  cube root of large number in his brain by just using simple facts of everyday life. So my question is there a way to calculate square root or cube root by using just a pen a paper, I mean a systematic way not by trial and error. I have studied some of Einstein's Theory of General Relativity, and I understand that it states that gravity isn't a force but rather the effects of objects curving space-time. If this is true, then why are we instructed in middle school that it is a force? What are Click packages? I noticed that Ubuntu SDK recently had some updates and it installed a program named "Click". I would like to know what they are and how to use them.  Also, would they make Debian packaging easier? The difference between sys.stdout.write and print? What is the benefit of Krivine's notation? I saw some people uses Krivine's notation for function application when presenting the syntax for the $\lambda$-calculus.  For example, the $\lambda$-term $\lambda f . \lambda x . \lambda y . f\ x\ y$ (with the normal convention that function application associates to the left, so it actually means $\lambda f . \lambda x . \lambda y . ((f\ x)\ y)$) is written $\lambda f . \lambda x . \lambda y . (f)\ x\ y$ (with a similar convention that it actually means $\lambda f . \lambda x . \lambda y . ((f)\ x)\ y$).  I do not see the point of having another pair of parentheses around the innermost $f$.  Why do people use Krivine's notation instead of the usual one? tcpdump : lost packets VSE isn't multi-threading Whenever I render a video using the blender video editor only one core is ever used at any time. I have an 8 core CPU(FX-9590) yet regardless of whether blender is set to auto detect or a fixed 8 threads only one core ever sits at 100% at any given time. The rest of the cores sit and really low usage as if blender is completely ignoring them. I'm running blender 2.77a on Linux. I'm using Theora as the codec with an ogg container and Vorbis audio. How do I get blender to use all my cores? How to install wps office via terminal? It's pretty complicated and sometimes the system freezes when I try to download and install wps office. I think it is much more convenient to use the terminal. But I never used installed WPS Office from the command line. That would be great if I could update it using software manager or terminal. Third party deb files are from separate locations. I specifically want to know about WPS Let $f:\mathbb{R}\to\mathbb{R}$ continuous such that $\lim_{x\to\infty}f(x+1)-f(x)=l$.  How can I prove that $\dfrac{f(x)}x$ converges for $x\to+\infty$ ?  I am just trying to prove the convergence, not that it converges to $l$ (that part be deduced immediately by Stolz-Cesaro and the sequential characterization of limits) What is the Hausdorff dimension of the set of irrationals on $\mathbb R^1$ I am sorry if this is a dump question but I can't find any reference for this. What is the Hausdorff dimension of the set of irrationals on $\mathbb R^1$? I speculate this is $1$ but don't know how to prove it rigorously.