Let $(X,S,\mu)$ is a measure space and $\mu(X)<\infty$. Define $d(f,g)=\int\frac{|f-g|}{1+|f-g|}d\mu$ is a metric on the space of measurable functions
A pseudometric on the space of the measurable functions is complete
I have seen many articles that talk about a star's "habitable zone" as the ring around it which is just the right temperature for planets to have liquid water, which is necessary for life. My question is, why does a planet need to have liquid water in order to sustain life? Who says that extraterrestrial life can't live off of some other kind of sustenance? Only Fixed and Random Options are Working in Illustrator CC 2015 Callipraphy Brushes App updated but still the same. Only fixed and random options are working in Calligraphy Brushes. I have searched but can't find a solution. Packaging a bike for flying with a bike bag I am doing a cycle tour of Japan starting from Fukuoka and ending in Tokyo. I will be flying with the bike and as my start and end points differ I will not have the opportunity to use a box (unless anyone has any ideas?). I will most likely use one of  bags and I have been told by the airline that nothing is necessary by way of packing the bike but I would naturally like to see my bike in one piece at the other end. The bike is a Specialized Tricross with disc brakes. Does anyone have any pointers as to the safest way to package something like this. I think I will definitely take the rear dérailleur off and wrap bubble wrap along the length of the bike to protect the wheels, pedals etc but other than that I'm not sure what lengths to go to. Does anybody have any experience or ideas? Expanding code in etoolbox's \ifblank I am trying to use the etoolbox \ifblank (and its cousin \notblank), but I want to expand any macros in its test statement.  For example:  \documentclass{article} \usepackage{etoolbox}  \begin{document}  \newtoggle{longformt} \newcommand{\longform}{\toggletrue{longformt}} \newcommand{\iflong}[2]{\iftoggle{longformt}{#1}{#2}}  \newcommand{\quoteifnotblank}[1]{%     \notblank{#1}{``#1''}{}% }  % Try commenting this line. \longform  This should be completely blank if the form is short: \quoteifnotblank{\iflong{Some long quoted text}{}}  \end{document}   The problem here is that \notblank sees its argument of \iflong{Some long quoted text}{} as a string, not as code to be expanded, so the quotes always appear.  How do I expand arguments before they are passed to \notblank? Show that the free group on $n$ generators is a finite index subgroup of $F_2$ Using covering spaces, prove that for each integer $n \geq 2$, $F_n$ is a finite index subgroup of $F_2$, where $F_n$ is the free group on $n$ generators.  I get how the cayley graph of $F_n$ would be a wedge of $k$ circles and the cayley graph of $F_2$ would be a graph of a vertex and two edges, but I don't understand how we can create an injective map from $F_n$ to $F_2$ and show its a subgroup. I know how to compute modular multiplicative inverses for co-prime variables $a$ and $b$, but is there an efficient method for computing variable $x$ where $x &lt; b$ and $a$ and $b$ are not co-prime, given variables $a$, $b$ and $c$, as described by the equation below?   $ a x \equiv c \mod b $   For example, given  $ 154x \equiv 14 \mod 182 $, is there an efficient method for computing all the possibilities of $x$, without pure bruteforce?  Please note that I'm not necessarily asking for a direct solution, just a more optimized one.    I do not believe that the Extended Euclidean Algorithm will work here, because $a$ and $b$ are not co-prime.   Edit:  Follow up question, since the first one had a shortcut:  Could the be computed efficiently as well?  $12260x \equiv 24560 \mod 24755$.  $107$ needs to be one of the computed answers. Is there any built-in software or terminal method allowing me to view the hardware profiles on my system? Windows equivalent of such a feature would be Device Manager. If $\int_{0}^{x}f(t)dt\rightarrow \infty$ as $|x|\rightarrow \infty\;,$ Then every line $y=mx$ Intersect $ y^2+\int_{0}^{x}f(t)dt=2$ If $f$ is a continuous function with $\displaystyle \int_{0}^{x}f(t)dt\rightarrow \infty$ as $|x|\rightarrow \infty\;,$ Then show that every line $y=mx$ Intersect the curve $\displaystyle y^2+\int_{0}^{x}f(t)dt=2$  $\bf{My\; Try::}$ Put $y=mx$ in $$\displaystyle y^2+\int_{0}^{x}f(t)dt=2\;,$$ We get $$\displaystyle m^2x^2+y^2+\int_{0}^{x}f(t)dt=2.$$ We can write it as $$\displaystyle 2\int_{0}^{x}2m^2tdt+\int_{0}^{x}f(t)dt=2$$ So $$\displaystyle \int_{0}^{x}\left[2m^2t+f(t)\right]dt-2 =0$$ Now Let $$g(x) = \displaystyle \int_{0}^{x}\left[2m^2t+f(t)\right]dt-2.\;,$$ When $x\rightarrow \infty\;,$ Then $g(x)\rightarrow \infty.$ (Where $g(x)$ is an Continuous function.) Now I did not Understand How can i solve that, Help me Thanks What is the purpose of delayed suspend (Ctrl-Y) in Bash? I am looking for a transparent screensaver. Or a way for there to be no screen saver, but instead of locking, it disables the mouse and most of the keyboard, until I hit an unlock key. Which one is correct?     There is an apple and an orange..   or     There are an apple and an orange?" Is there a sandbox to post test questions? I have noticed a behaviour that may or may not be a  bug. It is a discrepancy between the preview and the rendered result, which is why I would need to post a question to check it out.  Is there an official place to do that? Something I did in the first Animal Crossing was make a house on an alternate character solely to showcase bugs/fish I felt were too cool to only leave in the museum. However bugs can't be mailed or dropped, so I'm at a loss as to how to do this.  Is there a way I can give bugs/fish to a second character or will that character have to find them themselves? Unique constraint that allows empty values in MySQL I am having trouble getting onto a server in Minecraft PE on iOS. When I tap on the person's server it just says invalid name:    I have tried turning off and restarting the iPad, going to home screen and logging back into the server but I have had no luck and no change. Is there a reason why this is so? Zero module of differentials implies finite extension? I am a Bangladeshi citizen. Back in 2013, I went to India with a tourist visa and overstayed for four days. It was due to a gout attack in my feet but I don't have my medical papers to prove that.  I applied again in 2015 for a visa, at Dhaka, and they refused my visa and blacklisted my passport, writing some numbers and date on it. Now I am not able to get an Indian visa.  How I can remove my passport from the blacklist and get a new Indian visa? Do I need a Indian immigration lawyer to do this? I'm trying to find a good way to print leading 0, such as 01001 for a . While the number would be stored as 1001, what is a good way to do it? I thought of using either case statements or if to figure out how many digits the number is and then convert it to an char array with extra 0's for printing, but I can't help but think there may be a way to do this with the printf format syntax that is eluding me. Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$ Why is feed-forward mechanism used in hash functions? The compression function of SHA-1 when used in Davies-Meyer mode adds its input to the chaining values at the final step. For the first message block, the IV is used as the input and in the next step, the previous block's output (as specified by Merkle-Damgård) is used. These values are fed to the last round of the compression function.  Why would removing the feed-forward part (i.e.,the final step where the input value is added to the result to get the final output) make the hash function insecure? Apart from the fact that the compression function would not satisfy desired randomness properties (being a permutation for each fixed message).