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Getting arguments underneath \min in an align environment Prove that for for all $p,q\in \mathbb{Z}$, $q&gt;0$ we have: $$ \left| \sqrt{2}-\frac{p}{q} \right| &gt; \frac{1}{3q^2}. $$  To be honest, I do not know where to start - any help would be appreciated. What is the main difference between an inner class and a static nested class in Java? Does design / implementation play a role in choosing one of these? Was Thanos really committed or just a hypocrite? The title is extremely vague, because this question contains HEAVY spoilers for Avengers: Infinity War, specifically the very last scenes of the movie. I can't reasonably put the entire question in spoiler tags, so consider this your warning.  At the end of the movie, Thanos pretty much wins. He gets all the Infinity Stones, Gamora and Vision have been brutally murdered, and the rest of the cast completely beaten. Nobody could stop him. So he executes his plan.  As far as I know (unless I missed something), his plan is to eradicate exactly half of all mortals in the universe, and from his own words, in a fair lottery (random chance), without taking into account a person's wealth, social status, or anything else. Pure chance decides whether you are sacrified for the greater good or live through this greater good (according to him).  During the movie, Thanos and one of his underling (Ebony Maw) say themselves that being sacrified is an honor and should be appreciated. As you would expect, most people do not appreciate being sacrified.  At the end of the movie, Thanos uses all 6 stones to do a finger snap, deleting half the universe.  The question is: Did he exclude himself from the targets? Was there a 50% chance that Thanos himself would disappear right there?  As far as I know, this is not explicitly answered in the movie itself. I am looking for similar situations in the comics (or in the movie if I happened to miss something) where you could know, or have an educated guess, as to Thanos being the kind of person who follows through to the end, even if it means dying, or the kind of person who gets scared of death when it comes down to it, and comes off as an hypocrite.  Some people also mentioned his "deal" with Strange would make him spare Tony from the snap too. I'm also wondering about that, if there is a link between the two, but it's not the focus of the question.  The reason I'm wondering is that Thanos seems to me like a very extreme utilitarist. Even though he might be wrong about the value of his plan (Do the remaining half really become happy?), in-universe it seemed successful at least once (Gamora's home planet), so his reasoning is not bad, even if the premise might be horribly flawed. Excluding himself from the snap would make it into a straight villain, while considering himself equal to the rest would make him into a wannabe hero with very very evil methods. This makes a huge difference in the interpretation of the movie and his character. The qtree package seems to facilitate the drawing of simple trees only. Is there a way to incorporate:   Multiple roots &amp; Nodes with multiple parents   in a tree in LaTeX? How can edge rings be selected in a predictable way? When trying to select multiple edge-rings Shift-Alt-RMB sometimes loops get selected in the one (green) way or the other (red) way.  I couldn't derive a general rule.   How can two or more face-rings following one direction be selected ?  (Easy to reproduce with a default torus mesh.) Under Windows, most applications and application data are stored in a special directory known as C:\Program Files (and occasionally C:\Program Files (x86)). What is the Ubuntu/Linux equivalent to this path? Is there even one? when I look at my Nexus 5 with android 5.1. battery screen, I can see something like this:    Screenshots (click images for larger variants)  Sum of all per-app percentages in list is 10. However as you can see, there is 60% battery consumed (40% remaining). So, what is the deal here? Who consumed 50% battery? Or what do the percentages next to apps mean?  As you can see, screen was on just for few short times, the same for awake, and yet battery life about 18 hours? I dont get it.  Also, I found this issue reported on AOSP, but noone from google seems to care: How to include SVG diagrams in LaTeX? I'm making all my diagrams in SVG format using Inkscape and then I export them to some other format (e.g. PDF, EPS or PNG). I wrote a short script that does this for me automatically, but I'd like to avoid this step, if possible --- sometimes the conversion from SVG converts the text labels into vectors, and that's annoying. Hence the question:  Is there a way to include SVG diagrams directly without conversion to an intermediate format?  Package suggestions, or instructions for specific LaTeX distribution are welcome. Tips and personal experience on which tools you have found to be working reliably would be also appreciated. How to format a USB flash drive? In Windows there is a straightforward "Format" option in a flash drive's right-click pop-up menu. Where's that in Ubuntu? Checking to see if a DateTime variable has had a value assigned With the announcement of the detection of gravitational waves, questions about the implications proliferate.  Some relate to the possible existence of gravitons.  The analogous relationship between gravitons/gravitational waves and photons/electromagnetic waves is frequently mentioned.  The detection of individual photons required experiments of very low intensity light, yet their existence was inferred (prior to their actual detection) by Planck and Einstein (among others) using the properties of experimental black body radiation and the photo-electric effect.  If the prospects for detection of gravitons requires similar study of very low intensity gravitational waves, then those prospects are very dim indeed. My question: are there similar indirect "experimental" methods for inferring the existence of gravitons? Recently I've noticed more and more sign-in / sign-up workflows defaulting to the sign-up option first and providing a minimal "Sign-In / Login" action somewhere else on the page. What's the reason behind this practice?  Examples where I've seen this recently: I remember watching this one show when I was a kid in the 80's/early 90's in Canada. It was a live-action show about these two kids and had some puppet/muppet-like characters, but it kind of had a dark/mysterious element to it. I can't remember too much about it except portions of some episodes, but the main characters were kids, and they had some mysterious artifact (kind of like a compass?) that took them to this other world where there were giants and other mysterious beings that we rarely saw. The two kids weren't the only one's going to this place either, they find out kids from all around the world that also had this same artifact (and they all fit together like a round puzzle?). I seem to recall that the giants in this other world were the ones who made the artifacts, and were dying/or in desperate need of the help of the kids.  A couple episodes I remember:   There was one where the leaves were falling, and the world was doomed to end as soon as the last leaf touched the ground, so they're trying to figure out how to prevent all the leaves from falling, but after a long and suspenseful episode (for a kid) the last leaf dramatically falls, but out of no where the main girl catches the leaf and saves the world (it was very exciting for my young self, and I remember being very impressed with the pretty girl for thinking to catch the leaf).     There was another episode where the kids meet these two old guys with long white hair (muppets) sullenly working in a mine/tunnel, they're in tattered old clothes working away with their pick axes or hammers and they are connected to each other at the beard. This long old white beard grows out of one of their faces right into the other's. The kids save the day by cutting the old guy's beard in the middle, freeing them from each other. They celebrate being free from their bonds and excitedly thank the kids, but after the kids leave they continue to work away, this time in much brighter spirits. Seemingly unbeknownst to them though, as they work, their beards slowly and subtly grow back together.   Anyone else remember this show? vertical align middle in Who created the veil in the death chamber, and why? Was the veil created by someone or was it always there in HP? What is its purpose? If one can't communicate with the dead through the veil, then what/how do unspeakables learn from the the veil? (except investigating how does it flutter,I suppose,,,,) Did all Headmasters get a portrait in Hogwarts? Throughout the Harry Potter series, we see that deceased Headmasters of the school get a semi-sentient portrait of themselves in the Headmaster's Office, at the service of the current Headmaster. Do all Headmasters get a portrait? What are the conditions of getting one? For instance, there are a couple of times when Dumbledore is temporarily deposed as Headmaster: Dolores Umbridge takes over the job in Order of the Phoenix, and Minerva McGonagall takes the job in Chamber of Secrets and Half Blood Prince. Will Umbridge and McGonagall have portraits in the Headmaster's Office when they expire? Fred and George stumbled upon the Marauder's map in their first year. They passed it on to Harry while in their fifth year. Also note that, while talking to Harry on the train, Ron described Scabbers as being "Percy's old rat".   Hence, we can safely assume that the twins had the Marauder's Map for about five years. How could they not see Peter Pettigrew on the map, with Percy for first two years and with Ron for the next three years?  How come Harry didn't see Peter? How was Lupin the only one able to see Peter on the map? python3 is not supporting gtk module Shortcut for toggling between different windows of same app?