I thought about the parameterizing the 4D sphere using $$x_1= r\cos(\theta_1), x_2= r\sin(\theta_1)\cos(\theta_2), x_3= r\sin(\theta_1)\sin(\theta_2)\cos(\theta_3),\\ x_4= r\sin(\theta_1)\sin(\theta_2)\sin(\theta_3),$$ where $x_1,x_2,x_3,x_4$ are my 4D space coordinates. But what will be the range of values of $\theta_1, \theta_2, \theta_3$ to make the mapping from $(r,\theta_1, \theta_2, \theta_3)$ to $(x_1,x_2,x_3,x_4)$ bijective ?
What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by  $$x_1=r \cos  \theta$$ $$x_2=r \sin  \theta$$  For $n=3$, the analogue would be the ordinary spherical coordinates $r,\theta ,\varphi$, related to the Cartesian coordinates $x_1,x_2,x_3$ by  $$x_1=r \sin  \theta  \cos  \varphi$$ $$x_2=r \sin  \theta  \sin  \varphi$$ $$x_3=r \cos  \theta$$  So these are my questions: Is there an analogue, or several, to spherical coordinates in $n$-dimensions for $n&gt;3$? If there are such analogues, what are they and how are they related to the Cartesian coordinates? Thanks.
Smaller parskip than half for KOMA-Script? I am using the documentclass scrbook. To mark paragraphs I need both vertical spacing and indentation.  I know that KOMA-Script supports the option parskip or halfparskip, however, this disables the indentation. In order to get it back I declared \setlength{\parindent}{1.5em} in the preamble.  Since I am using both vertical spacing and indentation, I would like to have even smaller spacing than halfparskip. Is it possible to get an even smaller parskip than half in KOMA-Script? I have decided to switch from Powerpoint to Beamer, and for my first presentation I'd like to use the , and the same colors as in this goggle slide template:  .  I tried to do some research about it, the closest answers I found are the question  and the . I tried this code:  \documentclass{beamer} \usepackage[frenchb]{babel} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc}  \usetheme{metropolis}  \definecolor{bluegreen}{rgb}{3, 166, 155} \definecolor{pitchblack}{rgb}{0, 0, 0} \definecolor{lightbeige}{rgb}{255, 251, 241} \definecolor{mediumgray}{rgb}{183, 183, 183}  \setbeamercolor{background canvas}{bg=pitchblack} \setbeamercolor{normal text}{fg=lightbeige} \setbeamercolor{headline}{bg=bluegreen, fg=mediumgray}  \begin{document} \begin{frame}     \frametitle{The title}     Lorem ipsum... \end{frame} \end{document}   I does everything I want, except that the title bar's background color is lightbeige:    What did I do wrong? Or what argument should I pass to \setbeamercolor to get it right? Thanks in advance :-). Single word which means "keep moving forward" I need a single word which means "keep moving forward". What are my options?  I need word as   persevere Be persistent, refuse to stop  cognizant (sometimes followed by 'of') having or showing knowledge or understanding or realization or perception  [] keep moving forward  I want to make meaningful name for firm  please help me How do I get the local machine name? When induction proofs need a choice function Reading the comment to  the user makes the argument that one can pick a dense subset of a separable vector spaces without relying the axiom of countable choices. Since the comment got a lot of up votes, I guess it's correct, but to me this seems very strange. My rule of thumb was/is the following: We can get by without a choice function if   There is a canonical choice at each iteration. The proof doesn't actually require the iteration process to terminate.   None of these seem to be satisfied in the present case, unless there is a canonical choice for the dense subset, or the set is dense at some finite iteration, so my question is:   When can I tell if an induction arguments relies on the existence of a countable choice function? jQuery document.createElement equivalent? Way to list "Contact Us" email address on web site, yet reduce likelihood of spam? What's a good way to list a "Contact Us" email address on a web site, while reducing the likelihood it will get spammed?  Is putting the email address in an image the best technique, or are there others? What do I do? I looked at a similar question to mine and there was an answer saying to make sure Java is allowed past the firewall. So I checked and it already is. I've tried uninstalling and installing both Minecraft and Java but nothing has changed.  says that all services are fine, so what's going on? I bought a 2nd account to use as an alternative in-game account and now I can't log in.  (I cleared the email and password boxes for obvious reasons in this screenshot: security purposes) My /boot partition is nearly full and I get a warning every time I reboot my system. I already deleted old kernel packages (linux-headers...), actually I did that to install a newer kernel version that came with the automatic updates.  After installing that new version, the partition is nearly full again. So what else can I delete? Are there some other files associated to the old kernel images?  Here is a list of files that are on my /boot partition:  :~$ ls /boot/ abi-2.6.31-21-generic         lost+found abi-2.6.32-25-generic         memtest86+.bin abi-2.6.38-10-generic         memtest86+_multiboot.bin abi-2.6.38-11-generic         System.map-2.6.31-21-generic abi-2.6.38-12-generic         System.map-2.6.32-25-generic abi-2.6.38-8-generic          System.map-2.6.38-10-generic abi-3.0.0-12-generic          System.map-2.6.38-11-generic abi-3.0.0-13-generic          System.map-2.6.38-12-generic abi-3.0.0-14-generic          System.map-2.6.38-8-generic boot                          System.map-3.0.0-12-generic config-2.6.31-21-generic      System.map-3.0.0-13-generic config-2.6.32-25-generic      System.map-3.0.0-14-generic config-2.6.38-10-generic      vmcoreinfo-2.6.31-21-generic config-2.6.38-11-generic      vmcoreinfo-2.6.32-25-generic config-2.6.38-12-generic      vmcoreinfo-2.6.38-10-generic config-2.6.38-8-generic       vmcoreinfo-2.6.38-11-generic config-3.0.0-12-generic       vmcoreinfo-2.6.38-12-generic config-3.0.0-13-generic       vmcoreinfo-2.6.38-8-generic config-3.0.0-14-generic       vmcoreinfo-3.0.0-12-generic extlinux                      vmcoreinfo-3.0.0-13-generic grub                          vmcoreinfo-3.0.0-14-generic initrd.img-2.6.31-21-generic  vmlinuz-2.6.31-21-generic initrd.img-2.6.32-25-generic  vmlinuz-2.6.32-25-generic initrd.img-2.6.38-10-generic  vmlinuz-2.6.38-10-generic initrd.img-2.6.38-11-generic  vmlinuz-2.6.38-11-generic initrd.img-2.6.38-12-generic  vmlinuz-2.6.38-12-generic initrd.img-2.6.38-8-generic   vmlinuz-2.6.38-8-generic initrd.img-3.0.0-12-generic   vmlinuz-3.0.0-12-generic initrd.img-3.0.0-13-generic   vmlinuz-3.0.0-13-generic initrd.img-3.0.0-14-generic   vmlinuz-3.0.0-14-generic   Currently, I'm using the 3.0.0-14-generic kernel. continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$ Let $g$ be a function on $\mathbb R$ to $\mathbb R$ which is not identically zero and which satisfies the equation $g(x+y)=g(x)g(y)$ for $x$,$y$ in $\mathbb R$.   $g(0)=1$. If $a=g(1)$,then $a&gt;0$ and $g(r)&gt;a^r$ for all $r$ in $\mathbb Q$.   Show that the function is strictly increasing if $g(1)$ is greater than $1$, constant if $g(1)$ is equal to $1$ or strictly decreasing if $g(1)$ is between zero and one, when $g$ is continuous. The ring $\Bbb Z\left [\frac{-1+\sqrt{-19}}{2}\right ]$ is not a Euclidean domain Let $\alpha = \frac{1+\sqrt{-19}}{2}$. Let $A = \mathbb Z[\alpha]$. Let's assume that we know that its invertibles are $\{1,-1\}$. During an exercise we proved that:     Lemma: If $(D,g)$ is a Euclidean domain such that its invertibles are $\{1,-1\}$, and $x$ is an element of minimal degree among the elements that are not invertible, then $D/(x)$ is isomorphic to $\mathbb Z/2\mathbb Z$ or $\mathbb Z/3\mathbb Z$.   Now the exercise asks:     Prove that $A$ is not a Euclidean Domain.   Everything hints to an argument by contradiction: let $(A, d)$ be a ED and $x$ an element of minimal degree among the non invertibles we'd like to show that $A/(x)$ is not isomorphic to $\mathbb Z/2\mathbb Z$ or $\mathbb Z/3\mathbb Z$.  How do we do that? My problem is that, since I don't know what this degree function looks like, I don't know how to choose this $x$!  I know that the elements of $A/(x)$ are of the form $a+(x)$, with $a$ of degree less than $x$ or zero. By minimality of $x$ this means that $a\in \{0, 1, -1\}$. Now I'm lost: how do we derive a contradiction from this? It is stated that if $\psi(x_0)=\psi'(x_0)=0$ at some $x_0$, then $\psi(x)=0,\forall x$.  Hence, we cannot have the wavefunction and its derivative vanishing at the same point.  My question is, how to prove the above statement? What is the first movie to show this hard work message at the very end? What is the first movie to show this hard work message at the very end? I know I can set the render border with Ctrl+B and dragging a box.  But is there  a way I can exactly specify the border dimensions? Let $V$ be a finite dimensional vectorspace over a field $\mathbb{ F}$. It's easy to show that if $U$ and $V$ are subspaces of $V$ then $U \cap V$ is a subspace. But what if there are an infinite number of them? I have seen sometimes in pure maths that you take an arbitrary finite collection of sets and show the statement you are trying to prove is true for them to deduce it is true for the whole infinite collection. Why is this technique allowed? Can I do it for this case? Why? How to avoid pressing Enter with getchar() for reading a single character only? A phonon is a quantized unit of sound; they are encountered when quantizing lattice vibrations in solids. Now, even an ideal gas supports sound waves, but in this case, interactions between atoms are weak. That makes it hard to imagine what a quantized vibration would look like, since at small scales, the particles are free!   Is there a phonon picture for sound in an ideal gas? Is it ever useful? I read this sentence today:  If I were five years old again, I would sleep in every Saturday while I still had the chance!  I am confused why is had used in the last part. When the first part is referring to a hypothetical present/future then how can the last part be in past tense. The answer I received from a native speaker is that :  in type 2 and type 3 conditional sentences, a subordinate clause within the protasis or apodosis goes in the past (sometimes past perfect) tense.  Is this true? If yes, where can I find this information ? These are very advanced grammar concepts for me and any resource suggestion can be very helpful for me. What is the problem with nested environments using \BODY (environ package)? This question led to a new package:      I want to define some environments, which place their content into multiple different other environments. To access the contents I used the environ package, which introduces the \BODY command. However at some point pdflatex seemed to be stuck in an infinite loop.  I reduced it to a minimal example:  \documentclass{article} \usepackage{environ}  \NewEnviron{assertion}{Assertion: \BODY} \NewEnviron{outerassertion}{\begin{assertion}\BODY\end{assertion}}  \begin{document}  \begin{assertion} test \end{assertion}  \begin{outerassertion} test2 \end{outerassertion} \end{document}   Any ideas how to work around this problem? I guess, the problem is with the \BODY command since everything works fine as long as I use it in only one of the new environments. What is the "canary thread" and why is it starving?