Show that $K_n$ is not the union of two planar graphs
How to prove that a simple graph having 11 or more vertices or its complement is not planar?
I mean, why is $F=ma$? Why not $m^{0.123}$, $a^{1.43}$ or some random non-integers or irrational?  I hope you understand that my question isn't limited just to force, energy, velocity, etc.; it also extends to the area of a square, circle, etc. and all other formulas.  I think the whole thing starts with direct proportionality.  Most of them tell about the area of a circle, $A = πr^2$, where π is 3.14159..... an irrational number! It's not about the constant. I am talking about the power of a physical quantity.  I know why it has pi. It's because we chose the constant for area square to be one. If we chose it to be 1 for the circle then the area of a square will have a constant, 1/pi.   I've edited the question to 'rational exponents' since all are giving me examples of decimal non-integers. Can I use variables inside {} expansion without `eval`? In terminal, how can I write 'sagi' instead of 'sudo apt-get install' launch an exe/process with stdin stdout and stderr? Should the first word after a colon be capitalized? Is the product of symmetric positive semidefinite matrices positive definite? I see on Wikipedia that the product of two commuting symmetric positive definite matrices is also positive definite. Does the same result hold for the product of two positive semidefinite matrices?  My proof of the positive definite case falls apart for the semidefinite case because of the possibility of division by zero... Programmatically download (seamless) NED tiles? I used to download 1/3 arcsecond elevation geotiffs from extract.cr.usgs.gov using web services.  But a few days ago, that stopped working, probably because USGS reorganized their site.  I tried emailing USGS and haven't heard back yet.    I could, of course, manually download states or degree tiles using the seamless server, but I want to do it programmatically so that users of my trip planner can get elevation data for their region without human intervention.  And SRTM isn't high enough resolution for bicycle/wheelchair trip planning.  Any ideas? Can I fly back to Italy if my Italian residency card has expired and is being renewed? Help needed with definite integral for $\lim_{n\rightarrow \infty}\sum_{k=1}^{n} \frac{1}{n+k}$ Here's my preamble:  \documentclass[11pt]{amsart} \usepackage{geometry}                % See geometry.pdf to learn the layout options. There are lots. \geometry{letterpaper}                   % ... or a4paper or a5paper or ...  %\geometry{landscape}                % Activate for for rotated page geometry %\usepackage[parfill]{parskip}    % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx} \usepackage{amssymb} \usepackage{epstopdf} \usepackage{float} \usepackage[normalem]{ulem}  \DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}  \title{\emph{Research Proposal:}\\Perceptual Effects of Cross-Contextualized Soundscapes within the Built Environment}   Here's what happens:    Any ideas? Blacklight CFL Bulb Effect? I have been searching everywhere, and I can't find an answer so I'm coming here!  I'm trying to create a CFL Blacklight bulb. I've attached a couple images below of what I've tried to make. I'd love to have an effect like this:    Does anyone know how to recreate this? It seems like the color is affected by time as the bulb heats up, eventually being all white-ish. That would be fine, too, but I can't even seem to get the lighting effect of the pink glow around it and whatnot. I can post pictures of my node setup if you need it, though I think I made it necessarily complicated.  Thanks in advance!  UPDATE: This is what I have now:   I want to adjust the colors, but that isn't the issue now... The way I made this is a glass spiral that acts as the bulb. Inside of that bulb (that has a glass-like shader) is a purple material that I made that mimics the phosphor coating blacklight light bulbs have to filter light. Then, a separate mesh spiral is inside of that, and it has the emission stuff on it.   As Duarte suggested, I used the node setup he had for my emission, but used a transparent node instead of glossy. Why isn't the emission light from inside my glass causing the glass to be glossy? If I put a point light in the scene, it looks glossy.   Thanks in advance! How can window snap be enabled in Openbox? I would like to have window snap behaviour in Openbox. For me, this would mean that dragging a window to the edge of the display using the mouse would change the size of the window such that it is docked in half or quarter of the display, as can be done in Unity7. How can this be done in Openbox? My gaming computer is currently downloading Napoleon: Total War and Supreme Commander 2 which I purchased on Steam, and which will take some time to complete. While this download is going on, I would like to play some of my other multiplayer games on my laptop, but I would rather not interrupt the download to do so.   Is it possible to have both computers logged in at once? I know if I was playing a single player game, I could set Steam into offline mode, then resume the download on the other computer, but this won't work for multi player games. does anyone know the following result? If it holds in this form and any source which presents it? Thanks a lot.  Consider metric space $(X,d_{X})$.  Let $f:A \subset (X,d_{X}) \rightarrow \mathbb{R}$ be a Lipschitz continuous function i.e. $\exists K &gt; 0$ such that  \begin{align*} |f(x)-f(y)| \leq Kd_{X}(x,y) ~~~~\forall x,y \in A \end{align*}  then define functions  $$\bar{f_{1}}(x) := \sup\lbrace f(y) - Kd_{X}(x,y): y\in A \rbrace ~~ \forall x  \in X$$ and $$\bar{f_{2}}(x) := \inf\lbrace f(y) + Kd_{X}(x,y): y\in A \rbrace ~~ \forall x  \in X$$ it follows that  \begin{align*} \bar{f_{1}}, \bar{f_{2}}: X \rightarrow \mathbb{R}  \end{align*} are Lipschitz continuous with  \begin{align*} |\bar{f_{i}}(x)-\bar{f_{i}}(z)| \leq Kd_{X}(x,z) ~~~\forall x,z \in X \text{ and } \forall  i = 1,2 \end{align*} and  \begin{align*} \bar{f_{i}}(x) = f(x) ~~~ \forall x \in A \text{ and } \forall i =1,2 \end{align*} Did this character even qualify for being Azor Ahai or The One Who Was Promised? In the books and to some degree in the show as well there is a prophesy about the legendary figure Azor Ahai. But in the latest Game of Thrones episode (S08E03) we see Arya      killing the Night King.   However in a  we discussed​ all possible candidates but nobody thought about Arya. Does she even fulfill the prophecy? If she does, how? Passive POE: Need 5V @ 2A at the destination I am aware that, to use a Spell Scroll, you must have the related spell on (at least one of) your spell list(s). Then, if you have the spell slot to cast it, you succeed automatically; else, you must make an ability check in order to cast it (failure resulting in the scroll being wasted).  Consider this example : A Cleric 5 / Wizard 4 finds a Raise Dead Spell Scroll. The spell is on the Cleric List, but is a fifth level spell. Considering that the cleric-wizard has fifth level slots, but can only pick up to third level cleric spells, would he have to make the ability check to successfully cast the Raise Dead in the Spell Scroll, or would he succeed automatically ? Should I use the datetime or timestamp data type in MySQL? Would you recommend using a  or a  field, and why (using MySQL)?   I'm working with PHP on the server side. What is the difference between a framework and a library?   I always thought of a library as a set of objects and functions that focuses on solving a particular problem or a specific area of application development (i.e. database access); and a framework on the other hand as a collection of libraries centered on a particular methodology (i.e. MVC) and which covers all areas of application development. How do I insert the total continuation count in the allowframbreaks frame title? I would like to change the numbering of allowframbreaks frame, for instance, from I II III to (1/3) (2/3) (3/3)， but fail to find out a command to get the total continuation count. Here is the MWE.  \documentclass[compress]{beamer} \usetheme{Warsaw}   %allowframebreaks numbering in the title \setbeamertemplate{frametitle continuation}{(\insertcontinuationcount/)} %\setbeamertemplate{frametitle continuation}{(\insertcontinuationcount/\inserttotalcontinuationcount)}  \title{title} \author{author}  %%%%% \begin{document}  \begin{frame}[allowframebreaks]{allowframebreaks title} frame 1 \framebreak  frame 2 \framebreak  frame 3 \framebreak \end{frame}  \end{document}