Can anyone give me a hint in this problem? Please.  Let $X$ be a topological space. If $f,g$ are continuous function of $f,g \colon X\to S^n$  which $f(x)\neq -g(x)$ for all  $x \in X  $, show $f$ and $g$ are homotopics.
I'm new here. Can anyone give me a hint in this problem? Please.  Let $X$ be a topological space. If $f,g$ are continuous function of $f,g\colon X\to S^n$  which $f(x)\neq -g(x)$ for all $x$ in $X$, show $f$ and $g$ are homotopics. I try to show a homotopy using $$q(x) = \frac{f(x)+g(x)}{\|f(x)+g(x)\|}$$ it's well defined, but i'm stuck.
Modelling a spiral around a sphere I'm trying to figure out the best way to model the sphere-spiral here:    Is it possible to make this shape directly? Or do you have to make it as a cylinder and wrap it to a sphere? Should I use single or double colon notation for pseudo-elements? Trying to install Atom. Unable to start Clangd language server I'm following the , but on launching the upgrade tool I get this response:  Checking for a new ubuntu release No new release found   Am I doing something wrong? Is there a workaround? Are there attacks that break collision resistance but not preimage resistance? Are there any examples of attacks on hash functions which:   break collision resistance and second preimage resistant, or break collision resistance and preimage resistant?   I have looked at Rogaway's paper1, but it seems complicated, hence was wondering if a succinct example exists.  1)  Phillip Rogaway and Tom Shrimpton. Fast Software Encryption (FSE) 2004, LNCS vol. 3017, pp. 371&ndash;388, Springer, 2004. It turns out that removing a virtual package with apt-get only remove the package itself,  leaving behind the real packages that were installed by it  So how do you remove a virtual package, along with all the real packages that it installs? My wife is an Indian national with an Indian passport. She holds an indefinite leave to remain in the UK (permanent residence card). Does she need to apply for a Schengen visa before travelling to France or any other Schengen countries? I'm a bit confused about what Gstreamer does.   Does it come installed by default with Ubuntu ?  If so which version  is the default for 14.04?  Does it provide hardware accelerated video playback (decoding) ?   If so does it work with VA-API and VDPAU or is it an alternative to those?    Thanks for your help Git is automatically merging changes from a different branch to master In the small amount of physics that I have learned thus far, there seems to be a (possibly superficial pattern) that I have been wondering about.  The formula for the kinetic energy of a moving particle is $\frac{1}{2}mv^2$.  The formula for kinetic rotational energy is $\frac{1}{2}I\omega^2$.  The formula for energy stored in a capacitor is $\frac{1}{2}C \Delta V^2$.  The formula for energy delivered to an inductor is $\frac{1}{2}LI^2$.  Finally, everyone is aware of Einstein's famous formula $e=mc^2$.  I realize there are other energy formulas (gravitational potential energy, for example) that do not take this form, but is there some underlying reason why the formulas above take a similar form? Is it a coincidence? Or is there a motivation for physicists and textbook authors to present these formulas the way they do? 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When you view a Stack Exchange site without being logged in, you see a friendly banner, prompting you to take a tour and understand the features of the site:    This links off to a high quality resource that can help new users understand how the site works. However, the moment a user signs up to the site, this banner disappears from immediate view and hides itself as a link under the help menu.  My proposal is to continue showing this banner until one of the following conditions are met;   The user gets the "Informed" badge The user reaches a set amount of reputation (250 or 500 should be sufficient) The user chooses to hide the banner   This should increase the exposure of the tour to new users, who are the people who need the tour, rather than hiding it the moment a user signs up. X,Y are independent exponentially distributed then what is the distribution of X/(X+Y) Been crushing my head with this exercise. I know how to get the distribution of a ratio of exponential variables and of the sum of them, but i can't piece everything together.  The exercise goes as this:  If X,Y are independent exponentially distributed with beta = 1 (parameter of the exponential distribution = 1) then what is the distribution of X/(X+Y)  Any ideas?  Thanks a lot. Possible Duplicate:          This is something I haven't quite figured out yet, but I have been using function(){}() just because my VIM syntax highlight screws up if I add the parenthesis, although I've seen (function(){})() around many times, maybe its an IE thing?  edit:  var singleton = function() {     // code }();  var singleton = (function() {     // code })(); What is Deviance? (specifically in CART/rpart) What is "Deviance," how is it calculated, and what are its uses in different fields in statistics?  In particular, I'm personally interested in its uses in CART (and its implementation in rpart in R).   I'm asking this since the  seems somewhat lacking and your insights will be most welcomed. Prove $\lim\limits_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=0$ Can't login with Stack Exchange OpenID when HTTPS-Everywhere plugin is active - apparent protocol mismatch