What happens if the earth stops rotating?
If the earth would stop spinning, what would happen?
Negative Mass and gravitation The diophantine equation $x^2+y^2=3z^2$ I tried to solve this question but without success:     Find all the integer solutions of the equation: $x^2+y^2=3z^2$   I know that if the sum of two squares is divided by $3$ then the two numbers are divided by $3$, hence if $(x,y,z)$ is a solution then $x=3a,y=3b$. I have $3a^2+3b^2=z^2$ and that implies $$ \left(\frac{a}{z}\right)^2+\left(\frac{b}{z}\right)^2=\frac{1}{3} $$ so I need to find the rational solutions of the equation $u^2+v^2=\frac{1}{3}$ and I think that there are no solutions for that because $\frac{1}{3}$ doesn't have a rational root, but I dont know how to explain it.  Thanks \mathcal(O) and font size I use the following mystic TeX so as to display the Landau's notations for big and small O.  There is one partial solution to improve the scriptscriptstyle so as to not use a small o : see the secund answer below. The problemen with \smallOBis is the wrong spacing, contrary to \smallO. How can I do this ?  % Sources : %     1) http://forum.mathematex.net/latex-f6/bonnes-commandes-de-base-t12278.html %     2) http://tex.stackexchange.com/questions/30944/mathcalo-and-font-size  \documentclass{article}     \usepackage{amsmath}     \usepackage{amsfonts}     \usepackage{amssymb}      \usepackage{graphicx}      \newcommand{\bigO}[1]{\ensuremath{\mathop{}\mathopen{}\mathcal{O}\mathopen{}\left(#1\right)}}      \newcommand\smallO[1]{         \mathchoice             {% \displaystyle                 \ensuremath{\mathop{}\mathopen{}{\scriptstyle\mathcal{O}}\mathopen{}\left(#1\right)}             }             {% \textstyle                 \ensuremath{\mathop{}\mathopen{}{\scriptstyle\mathcal{O}}\mathopen{}\left(#1\right)}             }             {% \scriptstyle                 \ensuremath{\mathop{}\mathopen{}{\scriptscriptstyle\mathcal{O}}\mathopen{}\left(#1\right)}             }             {% \scriptscriptstyle                   \ensuremath{\mathop{}\mathopen{}{o}\mathopen{}\left(#1\right)}             }     }      \newcommand\smallOBis[1]{         \mathchoice         {% \displaystyle             \scriptstyle\mathcal{O}\left(#1\right)         }         {% \textstyle         \scriptstyle\mathcal{O}\left(#1\right)         }         {% \scriptstyle         \scriptscriptstyle\mathcal{O}\left(#1\right)         }         {% \scriptscriptstyle         \scalebox{0.8}{$\scriptscriptstyle\mathcal{O}$}\left(#1\right)         }     }   \begin{document}  \[     \renewcommand{\arraystretch}{1.2}%     \begin{array}{llll}          \textbf{Style}            &amp; \verb!\bigO!         &amp; \verb!\smallO!                          &amp; \verb!\smallOBis!         \\ \hline         \verb!\displaystyle!      &amp; \displaystyle      \bigO{\frac{1}{n}}         &amp; \displaystyle      1 + \smallO{\frac{1}{n}} - 2 &amp; \displaystyle      1 + \smallOBis{\frac{1}{n}} - 2         \\         \verb!\textstyle!         &amp; \textstyle         \bigO{\frac{1}{n}}         &amp; \textstyle         1 + \smallO{\frac{1}{n}} - 2 &amp; \textstyle         1 + \smallOBis{\frac{1}{n}} - 2         \\         \verb!\scriptstyle!       &amp; \scriptstyle       \bigO{\frac{1}{n}}         &amp; \scriptstyle       1 + \smallO{\frac{1}{n}} - 2 &amp; \scriptstyle       1 + \smallOBis{\frac{1}{n}} - 2         \\         \verb!\scriptscriptstyle! &amp; \scriptscriptstyle \bigO{\frac{1}{n}}         &amp; \scriptscriptstyle 1 + \smallO{\frac{1}{n}} - 2 &amp; \scriptscriptstyle 1 + \smallOBis{\frac{1}{n}} - 2     \end{array} \]  \end{document} Often when I mistype a command such as ls (e.g. I hit ENTER before I type 's') there is a long (~2s) delay after the terminal displays:  bash: l: command not found...   I can understand the reasons for a similar delay after an incorrect password is entered, per . But why delay after an unrecognized command? Does FAIL_DELAY in /etc/login.defs affect this also? Kinematic Viscosity Atiyah-Macdonald, Exercise 8.3: Artinian iff finite k-algebra. Atiyah Macdonald, Exercise 8.3. Let $k$ be a field and $A$ a finitely generated $k$-algebra. Prove that the following are equivalent:   (1) $A$ is Artinian.   (2) $A$ is a finite $k$-algebra.   I have a question in the proof of (1$\Rightarrow$2): By using the structure theorem, we may assume that $(A,m)$ is an Artin local ring. Then $A/m$ is a finite algebraic extension of $k$ by . Since $A$ is Artinian, $m$ is the nilradical of $A$ and thus $m^n=0$ for some $n$. Thus we have a chain $A \supseteq m \supseteq m^2 \supseteq \cdots \supseteq m^n=0$. Since $A$ is Noetherian, $m$ is finitely generated and hence each $m^i/m^{i+1}$ is a finite dimensional $A/m$-vector space, hence a finite dimensional $k$-vector space.  But now how can I deduce that $A$ is a finite dimensional $k$-vector space? How to run a specific program as root without a password prompt? Acquiring drive names (as opposed to drive letters) in Java On my Windows machine, my main hard drive has the letter C: and the name "Local disk".   To list the drive letters in Java on Windows, the File object has the static listRoots() method. But I can't find a way to acquire the drive names (as opposed to the drive letters) on Windows.  Has anyone tried this before? Showing $\gcd(2^m-1,2^n+1)=1$ A student of mine has been self-studying some elementary number theory. She came by my office today and asked if I had any hints on how to prove the statement      If $m$ is odd then $\gcd(2^m-1,2^n+1)=1$.    It's been a while since I took number theory and I'm not sure what to do. She said she is learning about congruences, primitive roots, and power residues.  She has not taken any group theory. How to add numbering to reference list but still use authoryear in natbib Is there an easy way to add numbers to the reference list generated by \usepackage[authoryear]{natbib} with whatever style file (using plainnat for ease)? Specifically, I use \cite and \citenum from natbib in order to have some references show up in the text as author-year and others just appear as numbers. Switching authoryear out for numbers in the natbib options generates a reference list with numbered references, but also causes the references in the text to all appear as numbers, which I don't want.  \documentclass[12pt]{article} % use larger type; default would be 10pt \usepackage[authoryear]{natbib} \begin{document}  \cite{A} said this, but other research \citenum{B,C} says otherwise.  \bibliographystyle{plainnat} \bibliography{MalariaBib,MST} \end{document} 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. Agent Smith, in the first Matrix movie (The Matrix) talks about how the machines originally created a paradise setting and humans didn't accept it, so it went back and created a more realistic one that humans would accept.  The human bodies, as we've seen, are kept in tanks and the body is kept asleep.  But why did the Matrix need to provide any kind of simulation for the brain to experience?  Why couldn't it have just kept all humans in a medically induced coma?  Or performed something equivalent to a lobotomy on every human?  Why did the Matrix need to spend the resources to keep the humans in good mental shape? Many other solutions of the Cauchy's Functional Equation By reading the  on the Wiki,  it is said that      On the other hand, if no further conditions are imposed on f, then (assuming the axiom of choice) there are infinitely many other functions that satisfy the equation. This was proved in 1905 by Georg Hamel using Hamel bases. Such functions are sometimes called Hamel functions.   Could anyone give a more explicit explanation of these many other solutions?   Besides the trivial solution of the form $f(x)=C x$, where $C$ is a constant, and the solution above constructed by the Hamel Basis, are there any more solutions existing? I have an old PowerMac G4 that I installed Lubuntu 16.04 on a while ago. I've been getting notifications in the terminal that 18.04 is available to upgrade, but the upgrader fails to run, saying that the package ubuntu-minimal is not available. I haven't seen any ISOs for PowerPC available, and ports.ubuntu.com doesn't have any bionic distributions available for architecture  powerpc. But while searching I found  that suggests that there is/was a version of Ubuntu 18.04 for PowerPC. Is this just a pre-generated page, or is there somewhere I can get Ubuntu Bionic for PPC? I created a interior scene and  i don't know how to bake a texture using cycles bake. How do I calculate areas of an area shapefile in square meters (m²) or in acres (ha)? I didn't find that functionality in the Vector Tools. How can one express frequentist p-value in terms of Bayesian concepts? Ubuntu 18.04 opening system settings logs me out How large is it possible for a PC to get?  For clarity, here are my parameters:   This is about size categories, not 'feet' or 'inches' or 'meters' or any other precise measurement. This is about Medium to Large to Huge to Gargantuan categories.  This is for an NPC charlatan using this to trick people. (It's an NPC but will have a full PC sheet, so the distinction between PC and NPC doesn't matter). Any means of size category enlargement works for this purpose, including magic spells, magic items, etc. Illusions do not count because it's about actually becoming larger, not just casting an illusion of a gargantuan giant version of yourself.  The character must remain the same physical structure. Polymorphing into a Gargantuan Dragon would not satisfy the need here. Consumable items are fine. This does not need to be permanent. How can't install a 338KB app when I have 474MB free? I have a Motorola phone, with 8GB of internal memory.  To my surprise, in a few days, I had 474MB of free space, but couldn't install a 338KB app!! So, clearly the limit of ~25MB referred to in some comments ( and ) are not always the case. A user just told me the following:     Actually it's "25MB or 10% of the storage", whichever is hit first. So   with e.g. 10 GB internal storage, it would hit you as soon as you drop   below 1 GB free. Crazy, but that's what it is.   Actually, that's not correct, because I didn't have to free up 1GB of internal memory. But anyway, half a gigabyte is enough to run a full operational system. I've seen similar complaints of hundreds of MB not enough to install apps ~ 1MB or similar small operations ( and ).  What's wrong here?