In special relativity formulas for changing coordinates x,t into x',t' are based on the relative velocity of the 2 frames. What is the base for assuming that if v is the velocity of frame O' relative to frame O, the velocity of O measured by somebody on O' will be -v?
In books the equation for length contraction is derived by supposing that the velocity of the spacecraft is the same for both observers. So the question is that, is the velocity really the same for both observers? Whereas there must be some trouble in the velocities noted by observers. I mean: If the observer on the spacecraft measures the velocity of the spacecraft to be $v$ then is it possible for the other observer to measure the same velocity from the Earth?
How can I remove a trailing newline? $\infty = -1 $ paradox Google Chrome PPA upgrade invalid signature Commutative non Noetherian rings in which all maximal ideals are finitely generated In commutative rings we have the following  Theorem. $R$ is Noetherian if and only if each prime ideal of $R$ is finitely generated.  From this Theorem I am looking for commutative rings $R$ in which every maximal ideal is finitely generated but $R$ is non Noetherian.  Question: Is there a straightforward example of a commutative ring $R$ so that each maximal ideal is finitely generated, but $R$ is non Noetherian?  Thank You A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$.  Is there a characterization of the groups in which this property holds?  If this question is too broad, I might ask if such a characterization exists for $p$-groups.    History: I originally posed the opposite question, regarding groups for which $\exists N\unlhd G\,:\, \not\exists H \unlhd G\, \text{  s.t. } H \cong G/N$, and crossposted this to .  I received an answer there to the (now omitted) peripheral question about probability, which shows that most finite groups probably have this property.  After this, I changed the question to its current state, as this smaller collection of groups is more likely to be characterizable. Intellij shortcut for quick call hierarchy Is there a shortcut key to bring up the call hierarchy of a method inline with the code, in the quick menu format, rather than bringing up the call hierarchy panel? What apps to sync movie subtitles easily are available for Ubuntu? I have a subtitle file that is about 3 to 4 seconds behind. What tools can I use to automatically move all the times in the subtitle file so they are correct with the time of the movie? Something that I can for example just add the time I want to add and it will add this time to the synced times in the subtitle files without having to edit each line individually. Zlib-compatible compression streams? How to restore Windows 7 MBR without a CD Sometimes a suggested edit is  because of a concurrent direct edit that takes precedence.  Since this is no fault of the edit suggestor, such rejections should not count against the suggestor when counting .  Here are some examples where it happened: Is Design School a Requirement for Graphic Design? Is it required for people to go to design schools to become a Graphic Designer? I've heard many cases where they have started from scratch without previous education in design and are doing fine as freelancers/working in studios.  Would it be better to go to a design school to become a Graphic Designer? How much leverage would it give you if you went to one? This question has an inverse:     What would be the most efficient way to translate a combination of $k$-tuple into its positions within the $\left(\!\!\binom{n}{k}\!\!\right)$ combinations? I need this to be fast for combinations of $\left(\!\!\binom{70}{7}\!\!\right)$ order of magnitude - very large, but not exceeding 2 billion (fits into int32 maximum value).  Below is an example of $\left(\!\!\binom{6}{3}\!\!\right)$, where the aim is to quickly translate (a, d, f) tuple to value 9, while in the real problem $k$ ranges between 5 and 8.  $$\begin{array} {cccccc|c} a&amp;b&amp;c&amp;d&amp;e&amp;f&amp;^{combination}/_{sort\_order}&amp; \\\hline x&amp;x&amp;x&amp; &amp; &amp; &amp;1\\ x&amp;x&amp; &amp;x&amp; &amp; &amp;2\\ x&amp;x&amp; &amp; &amp;x&amp; &amp;3\\ x&amp;x&amp; &amp; &amp; &amp;x&amp;4\\ x&amp; &amp;x&amp;x&amp; &amp; &amp;5\\ x&amp; &amp;x&amp; &amp;x&amp; &amp;6\\ x&amp; &amp;x&amp; &amp; &amp;x&amp;7\\ x&amp; &amp; &amp;x&amp;x&amp; &amp;8\\ x&amp; &amp; &amp;x&amp; &amp;x&amp;9\\ x&amp; &amp; &amp; &amp;x&amp;x&amp;10\\ .&amp;.&amp;.&amp;.&amp;.&amp;.&amp;.\\  &amp; &amp; &amp;x&amp;x&amp;x&amp;20\\ \end{array}$$  I know that I could pre-calculate all the combinations and reverse the lookup dictionary. However, such dictionary would not be efficient in terms of memory usage. Therefore I am looking for either calculation-based approach, or a more efficient data structure to perform this mapping. How to use PSTricks in pdfLaTeX? I thought that PSTricks package was not possible to use in pdfLaTeX but .   How can I force pdfLaTeX to use PSTricks then? How would I find a minimum weight spanning tree for W? Is there a way to detect if a browser window is not currently active? I am reading the book Darth Plagueis, and when an assassination attempt is made on him by use of a nuclear bomb, the book states that nuclear weapons are scarcely available.   I searched briefly but found nothing on this and wondered if an explanation is available. Why would nuclear weapons be rare in a universe with such advanced technology across the board? $S_3$ permutes the elements of $\mathbb{R}^3$, show that $\sigma (x_1, x_2, x_3) = ( x_{\sigma^{-1}(1)}, x_{\sigma^{-1}(2)}, x_{\sigma^{-1}(3)})$. As the title of the Question states, I want to know what are the tools available for Ubuntu which would allow me to take screenshots of "Scrolling Windows" ie; tall webpages, tall nautilus window which contains lots of files and folders etc. as a single image file.  For the sake of clarity, I have already seen . However, I am specifically looking for tools to capture Scrolling Windows.  As an example, my personal favorite for Microsoft Windows is . It's light on resources and helps me take beautiful screenshots. Is there anything close to it available for Ubuntu? How many cpu sockets does Windows 10 support? I would like to build a new computer (it's been 7 years since the last one). Does Windows 10 support more than one ? Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing.  $$\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$$  Perhaps someone can help me out I don't understand how to move forward from here: $$\sum\limits_{k=1}^{n+1}{\frac{1}{\sqrt{k}}+\frac{1}{\sqrt{n+1}}\ge \sqrt{n+1}}$$ proof and explanation would greatly be appreciated :) Thanks :)  EDIT sorry meant GE not = fixed :)