Prove spectral radius of a primitive matrix is 1
Proof that the largest eigenvalue of a stochastic matrix is $1$
How does the Hogwarts Headmaster know whom to invite to school? So all wizards get an invitation to Hogwarts at 11... How do they know whom to invite?  If a wizard is Muggle-born, then nobody would know about their existence.  Or do they register some weird magical stuff being done by them before 11? One of my laptop's keys has fallen off. Is there any way I can remap another key to serve as that key?  I'll 'sacrifice' the other key because I never use it. I want a green for "correct answer" and a red for "wrong answer". My current choice is rgb(144, 238, 144) for the green and rgb(250, 128, 114) for the red. (See wikilearner.net/wiki for the colours in action.)  I have been told that these colors are very close in contrast, making it difficult for a person with red-green color-blindness. What would be a better choice? After a lot of messing around today I curiously observed that $a^2+b^2$ is only divisible by 3 when both $a$ and $b$ contain factors of 3.  I am trying to prove it without using modular arithmetic (because that would be way too easy), but finding it very difficult to do so.  Is there an easy way to prove this without using modular arithmetic?  I am also interested in a more general statement.  Namely, I want to find the values of $Z$ for which $Z \mid \left( a^2+b^2 \right)$ necessarily implies that $Z \mid \gcd \left( a,b\right)$.  We know that $Z$ cannot be 5, because $3^2+4^2=5^2$.  More generally, if $Z$ is the largest element of a pythagorean triple then the above implication does not hold. How to input open intervals I want to write open and half-open intervals using the following notation:  ]a, b[ ]a, b] ]-∞, b]   When writing them just like that in the LaTeX source, the spacing doesn't come out right. For example, for this equation:   \(X = ]-\frac{π}{2}, +\frac{π}{2}[\)   the minus sign is typeset as a binary operator. I can solve this by surrounding the whole value in braces, but then the spacing around the equal sign is still not correct. For other combinations of intervals and operators, different spacing inconsistencies arise.  What is the correct and easy  way to input intervals in order to avoid having to take care of this at all? Expanding the equation out gives  $(3^{2n}\times3)+(2^n\times2^2) \equiv 0\pmod{7}$   Is this correct? I'm a little hazy on my index laws.  Not sure if this is what I need to do? Am I on the right track? Why do some websites show the company name next to the URL? Some websites show the company name next to the URL (in Chrome) and some don't.  How do I set this up for my website? Is there a reason why or why I wouldn't want to do this? To hyphenate or not? How can I disable landscape mode in Android? How can I disable landscape mode for some of the views in my Android app? 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 ? I tried installing Office 2007 in Ubuntu with Wine but it was throwing an error after installing and also when I tried to open a PowerPoint document and save it. Has anybody gotten Office 2007 installed without any issues? Please help me. An alternative software to Live Writer? Composition of 2 involutions How can we prove that any bijection on any set is  a composition of 2 involutions ?  Since involutions are bijections mapping elements of a set to elements of the same set, I find it weird that this applies to any bijection.  Thanks for your help ! No processing tools in QGIS 3.2 How to link a variable of a material to another material? Using `\input` to insert a section of a document into another What is the difference between metonymy and synecdoche? I've sometimes seen this construct, used to avoid repetitive handwriting, but I don't even know if it's proper English.  If it is, what is the name of this construct? I have no idea what to search for when looking it up...        Don't forget to buy eggs.   &emsp;"&emsp;&emsp;      "&emsp;&emsp;"&emsp;"&emsp;yogurt.   &emsp;"&emsp;&emsp;      "&emsp;&emsp;"&emsp;"&emsp;milk. Getting hybrid graphics to work nvidia-prime GT650M I recently made the switch to Ubuntu. Everything on my laptop works fine except my hybrid graphics card. It is Intel graphics combined with Nvidia GeForce GT650M. I tried Bumblebee, but primusrun doesn't work (segmentation fault, core dumped) and optirun works with glxspheres64, but steam games look worse with optirun than with Intel graphics (e.g. Surgeon Simulator 2013 runs fine with Intel and hangs a lot with optirun)   I tried Nvidia Prime, but it gives me a black screen on startup after install, so I can't do anything and have to power off my PC with the power button. I also tried nvidia-prime 0.5 on Ubuntu 14.04, and then I get 'low-graphics mode' and I can't get to Unity. How can a path animation be used as an action in the game engine? I animated a rollercoaster using a Bezier Curve and a Follow Path Constraint affecting the vehicle.  Then I tried to use the automatically created action in the game engine but the Action Actuator has no effect .    How can this action be used in the BGE? Can the Path Animation alternatively be converted into keyframes?