If there is a disc rotating about its centre, let the surface be frictionless and if a coin is placed anywhere on the disc (not at the center) why doesn't it fly off even though there is a centrifugal force acting on it with respect to the disc's frame?
Imagine that a pebble is placed on a uniformly rotating, frictionless disk.  What will happen to this pebble? Will the disk slide under it and the pebble stay as is? Or will there be a centrifugal force on the pebble and it'll be thrown off the disk?
Vectorization operator For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$ Hereditary properties with nested frames (mdframed and tikz) What is the max variance distribution with fixed mean and support Is there a upper bound on the variance for a distribution that is confined say, in $[0,1]$, and has fixed mean $\mu \in [0, 1]$ Order Bars in ggplot2 bar graph How can I prove this inequality in a triangle: $\frac{1}{(b+c-a)} +\frac{1}{(c+a-b)} +\frac{1}{(a+b-c)} \gt \frac{1}{a} +\frac{1}{b} +\frac{1}{c}$? If $a,b,c$ are the sides of a triangle then show that- $\frac{1}{(b+c-a)} +\frac{1}{(c+a-b)} +\frac{1}{(a+b-c)} \gt \frac{1}{a} +\frac{1}{b} +\frac{1}{c}$  I got this problem from an old book on algebra. I've been trying for a long time but can't reach the answer.  My try on the question-  We know that in a triangle, the sum of two sides is always greater than the third side  $b+c\gt a$ $\implies$ $b+c-a \gt 0$  $c+a\gt b$ $\implies$ $c+a-b\gt 0$  $a+b \gt c$ $\implies$ $a+b-c\gt 0$  Also,  $\frac {1}{(b+c)} \lt \frac{1}{a}$  $\frac{1}{(c+a)} \lt \frac{1}{b}$  $\frac{1}{(a+b)} \lt \frac{1}{c}$  Adding,  $\frac{1}{(b+c)} + \frac{1}{(c+a)} +\frac{1}{(a+b)} \lt \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$  How do I proceed further? All genuine answers are welcome :) How do I log in if my OpenID is not available, or maybe even no longer exists? Can I restore my lost account, or do I need to start over?  See also: How do I bake multiple texture maps into 1 UV, In Blender Cycles? Using:  Blender V2.79 Cycles Render Engine  After creating a complex interior scene consisting of 3 different texture types, each having 3 associated maps (Diffuse, Gloss, Normal) I noticed the texture and baking options will not successfully combine to create a texture for the exported mesh once I bring it into Unity 3D. The mesh is a single object with the different textures assigned to specified faces.  I've played around with a default cube mesh for testing purposes, using the same 3 textures split among the 6 sides, but have only been able to successfully pull over the color info by baking a diffuse map to my UV and saving the image. When using the other baking options, Blender does not successfully create a map for either my Glossy textures, nor my normal map. Even using the "Combined" baking option, I am provided a flat dull diffuse image for my mesh, no gloss or normal info is saved.  My Node process for textures is displayed here  Here is the result of baking my textures as a Normal  The result of adding other textures made from Blender's baking, results in flat gloss at best.  What can I do differently in Blender, to have my assigned UV Diffuse, Normal, and Gloss maps all appear in a single object in Unity? I'm having some troubles understanding the solution (going from step $2$ to step $3$) of this linear congruence equation. Here's the problem with the solution:   Solve the congruence $2x + 11 \equiv 7 \pmod 3$.   First, we reduce all the coefficients $\pmod 3$:  $$2x + 2 \equiv 1 \pmod 3. \tag{1}$$  Next, we add $1$ to both sides, using the fact that $2 + 1 \equiv 0 \pmod 3$:  $$2x \equiv 2 \pmod 3. \tag{2}$$  Finally, we multiply both sides by $2$, using the fact that $2 · 2 = 4 \equiv 1 \pmod 3$:  $$x \equiv 1 \pmod 3. \tag{3}$$  That is, any number in the set $\{\ldots, −5, −2, 1, 4, \ldots\}$ will solve the original congruence.    Like I said my problem is the step between $(2)$ and $(3)$. If I multiply both sides of $(2)$ by $2$ I get:  $$4x \equiv 4 \pmod 3.$$  Now, I understand that $4$ is congruent to $1 \pmod 3$ so the RHS "simplifies" to $1 \pmod 3$ but I don't understand why the LHS "simplifies" to $x$. Can someone help me with this? pkgloader is not finding l3regex Can someone tell me if $\frac{3}{5}$, $2+3\sqrt{-5}$, $\frac{3+8\sqrt{-5}}{2}$, $\frac{3+8\sqrt{-5}}{5}$, $i\sqrt{-5}$ are all quadratic integers in $\mathbb Q[\sqrt{-5}]$. And if so why are they in $\mathbb Q[\sqrt{-5}]$. How can I stop mobs from despawning? I am using Spigot, some plugins, and I want to make like a zoo on a map, with mobs behind glass cubes. The problem is that when I load my map, every time some of the mobs just vanish, empty cages. I put eggs again, they spawn, I go to my other map in the multiverse (/mv tp my_other_map) and when I come back, empty cages again, always the same ones...   I've  read it's a question of distance of me (distance in cubes unit), if I'm too far away, the mobs will despawn... Is there a way for them to stay? Dragging files into blender I'm trying to make a longtable that is wider than the textwidth at a certain moment. I've tried different things always ending up with a different problem:  First possibility: If I use the adjustwidth environment the longtabu does not align correctly. The left margin stays fixed and the right margin gets out of the page. This in contrast, if I do the exact same thing to make a tabu (instead of longtabu), it works perfectly. I think using the adjustwidth is the best way to go, but I can't manage to get it working.  Second possibility: I've found a solution using the fullwidth package. That package does the trick on small longtables, but freezes when I'm trying to do it on my big longtable. The command \setlength\LTleft{-2cm} didn't work as well.  The final option I tried was to use geometry. But that changes the margin of the whole page thus making a page break which I'm not interested in. The code I used for this was  \newgeometry{left=1.2cm} \begin{longtable} long table stuff \end{longtable} \restoregeometry   Here's an example of what I want contra what I get  \documentclass[a4paper]{article} \usepackage{longtable} \usepackage{chngpage} \usepackage{lipsum} \usepackage{tabu}  \begin{document} \section{This is the original textwidth.} \lipsum[1]  \section{This is the width I need for my longtabu} \begin{adjustwidth}{-1in}{-1in} \begin{tabu} to \linewidth {*{13}{|X[l]} |} 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ \end{tabu} \end{adjustwidth}  \section{This is what happens} \begin{adjustwidth}{-1in}{-1in} \begin{longtabu} to \linewidth {*{13}{|X[l]} |} 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ \end{longtabu} \end{adjustwidth}  \end{document} What is the best way to convert a JSON code as this:  {      "data" :      {          "field1" : "value1",          "field2" : "value2"     } }   in a Java Map in which one the keys are (field1, field2) and the values for those fields are (value1, value2).  Any ideas? Should I use Json-lib for that? Or better if I write my own parser? how to make clean without any modification? Linear independence of $\sin(x)$ and $\cos(x)$ In the vector space of $f:\mathbb R \to \mathbb R$, how do I prove that functions $\sin(x)$ and $\cos(x)$ are linearly independent. By def., two elements of a vector space are linearly independent if $0 = a\cos(x) + b\sin(x)$ implies that $a=b=0$, but how can I formalize that? Giving $x$ different values? Thanks in advance. Exclude menu item from the collapse of bootstrap 3 navbar How to install (old) packages for Ubuntu 9.04? Based on some excellent feedback by Mark here (), today I was able to use the vmware converter to clone my production server for a staging server.  However the nic won't come up no matter what I do. I attempted to inistall vmware tools, as I suspect that the fact that it is not installed might prevent the nic from working. (I have the nic set as a vmxnet3 card in the vm settings). The install failed because there were several dependencies missing as well as the Linux headers. Given that Ubuntu 9.04 has been EOL'd, the packages I need to install to get the vmware tools to install are no longer available. I doubt the ubuntu 9.04 install CD has the packages on it.   What are my options? I'd rather not upgrade the version of Ubuntu yet, as the point of the vm right now is to maintain parity with the production server. Might I have better luck resetting the driver to use vmxnet2 instead of the vmxnet3?  Thanks in advance!  Chris I've tried everything I can think of. Resetting the router, changing the channel number and band.   The internet connection is fine, and then just hangs for a minute or 2, then returns to normal.   The wifi icon remains connected, and I have the latest 10.10.4 update installed. At first I thought it was a DNS bug with Yosemite, and have played with the settings such as changing DNS provider, changing the IPV6 to IPV4 only etc.  Driving me nuts. Has anyone had a similar problem? I've checked the surroundings with wifi scanners and have chosen the best channel available.     One thing I have noticed, it seems a lot more reliable on 5GHz as opposed to 2.4Ghz It seems to affect my Macbook Pro Retina 13in Early 2015 a lot more than the other devices on the network. I'm using a Virginmedia  1.