I need help installing Ubuntu on a Laptop
How do I install Ubuntu alongside a pre-installed Windows with UEFI?
Prove the map has a fixed point Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$ such that $\rho (Ax,Ay) &lt; \rho(x,y)$ for $x\neq y$. Prove A have a unique fixed point in $K$.  The uniqueness is easy. My problem is to show that there a exist fixed point. $K$ is compact, so every sequence has convergent subsequence. Construct a sequence ${x_n}$ by $x_{n+1}=Ax_{n}$,$\{x_n\}$ has a convergent subsequence $\{ x_{n_k}\}$, but how to show there is a fixed point using $\rho (Ax,Ay) &lt; \rho(x,y)$? New to both Ruby and Rails but I'm book educated by now (which apparently means nothing, haha).  I've got two models, Event and User joined through a table EventUser  class User &lt; ActiveRecord::Base   has_many :event_users   has_many :events, :through =&gt; :event_users end  class EventUser &lt; ActiveRecord::Base   belongs_to :event   belongs_to :user    #For clarity's sake, EventUser also has a boolean column "active", among others end  class Event &lt; ActiveRecord::Base   has_many :event_users   has_many :users, :through =&gt; :event_users end   This project is a calendar, in which I have to keep track of people signing up and scratching their name out for a given event. I figure the many to many is a good approach, but I can't do something like this:  u = User.find :first active_events = u.events.find_by_active(true)   Because events don't actually HAVE that extra data, the EventUser model does.  And while I could do:  u = User.find :first active_events = [] u.event_users.find_by_active(true).do |eu|   active_events &lt;&lt; eu.event end   This seems to be contrary to "the rails way".  Can anyone enlighten me, this has been bugging me for a long time tonight (this morning)? What do you call someone who acts and thinks it is helpful, even though they do not know the context of what is going on? What is a name for a person who assumes they are doing something helpful, even though they are ignorant to the situation? Someone who reacts without bothering to understand what’s going on, and feels as though they are doing good? Is there such a word?   For instance, someone who pulls another out of the pool, and pats them self on the back for saving them from drowning...without ever asking if the person was drowning or merely swimming. $\mathbb R = X^2$ as a Cartesian product I wonder if it is possible to consider $\mathbb R$ as a Cartesian product $X\times X$ for some set $X$. From the point of view of the dimensionality, there are spaces with a Hausdorff dimension $1/2$ (sort of Cantor sets), but I guess there are other problems in this construction  unrelated to the dimension.  Edited: replying on the comment by Asaf. I want that for all $r\in\mathbb R$ there exists a unique representation $r=\langle x,y\rangle$ where $x\in X$ and $y\in X$. Also, what if we want to have $X$ to be a topological space and $h:X\times X\to\mathbb R$ to be a homeomorphism? Divergence of $\frac{ \hat {\bf r}}{r^2} \equiv \frac{{\bf r}}{r^3}$, what is the 'paradox'? How to disable "time travel" entirely? Which resources are available to self-study GAP? Background: This year I'll do another Group Theory course ( Open University M336 ). In the past I have used Mathematica's AbstractAlgebra package but (although visually appealing ) this is no longer sufficient (i.e. listing subgroups of $S_4$ takes ages). So, I want to learn more about GAP. I worked through beginner tutorials that I found via the . Currently, I am not making much progress with GAP. The  does not help me much at this stage.  Question: Which resources are available to self-study GAP? How does one become proficient in GAP? What ( books, tutorials ) should you study? Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$ I have no idea where to even start. WolframAlpha cant compute it either.  $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$  I think it can be done with series, but I am not sure, can someone help a little so I can get a start??  Thanks! The unique style of the diagrams at  has an informative but nice hand-drawn touch. I guess they are actually drawn by hand but just recently on our partner site for Mathematica someone asked how to draw a similar diagram such as this one with Mathematica's plot functions ().    Now without Mathematica this should be possible as well but I have a hard time getting the details right. Can such a diagram be created in TeX with a similar design? My first attempt does not look particularly great but here it is (the font is from ):      \documentclass{standalone}     \usepackage{tikz,color}     \usepackage{pgfplots}     \usepackage{fontspec}     \setmainfont{Humor-Sans}      \begin{document}     \begin{tikzpicture}     \pgfplotsset{every axis/.append style={line width=1pt}}      \begin{axis}[%         axis x line=bottom,         axis y line=left,         xtick={1.2, 2.9, 4.5},         xticklabels={-3 days, -1 day, 5 minutes},         yticklabels={}     ]     % 1st plot     \addplot [cyan!80!white, samples=200, domain=0:6] {0.05*rnd+3+(sin(deg(x))^2)/sqrt(x)*exp(-(x-2))};     % white background for 2nd plot     \addplot [white, samples=200, domain=0:6,line width=4pt] {0.05*rnd+0.4*x+2+x^2*sin(deg(x))^2*exp(-x)};      % 2nd plot     \addplot [red!80!white, samples=200, domain=0:6] {0.05*rnd+0.4*x+2+x^2*sin(deg(x))^2*exp(-x)};      \end{axis}       \node at (2.7,4.6) {downhill};     \draw  plot[smooth, tension=.7,line width=2pt] coordinates {(1.2679,5.5206) (1.9454,5.493) (2.4017,4.9261)};      \end{tikzpicture}      \end{document} Reading and searching long man pages I finally got fed up when wanting to read about bash's read and it's -s option with man bash. I found the right spot eventually (around line 4500), but it was a frustrating as usual, since both /read and even /\s-s\s searches have way too many matches.  So, the question is: How can I read long man pages efficiently, or get same information in other ways, locally? As a specific example, how to reach the relevant documentation after seeing read -s pwd in a shell script? A good answer could be a shell script snippet, or hint about some tool and how it is used, or something else entirely, as long as it helps in finding the right spot to read.  Note: I'm not tagging with  because I want the question to be about man page reading in general, even though that quite possibly is the most commonly encountered humongous man page. /usr/bin/sudo must be owned by uid 0 and have the setuid bit set I have run following command accidentally   sudo chown [username] -hR /   Now sudo su getting error:  sudo: /usr/bin/sudo must be owned by uid 0 and have the setuid bit set   How to Solve This? Cardinality of Borel sigma algebra It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, and actually hard to find a proof of it. Can anyone help me out? What does "if not" mean in the given sentence Let there be given this sentence (which came from an ):     The contest has become personalised, if not bitter.   Then what does the phrase if not mean?   Seeking after is a general guide or rule of such usage. For example, when I type ping 8.8.8.8 &amp;, why can we see ping process working? And when I type find / -name '*test*' &amp; , it's also seen on monitor.  Why is that? Is it not really in background? Does there exist a bijective $f:\mathbb{N} \to \mathbb{N}$ such that $\sum f(n)/n^2$ converges? We know that $\displaystyle\zeta(2)=\sum\limits_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ and it converges.        Does there exists a bijective map $f:\mathbb{N} \to \mathbb{N}$ such that the sum $$\sum\limits_{n=1}^{\infty} \frac{f(n)}{n^2}$$ converges.      If our $s=2$ was not fixed, then can we have a function such that $\displaystyle \zeta(s)=\sum\limits_{n=1}^{\infty} \frac{f(n)}{n^s}$ converges How do I completely uninstall Google Chrome and Chromium? I'm having a problem with Chrome and one of its extensions, so I want to completely uninstall everything. I've uninstalled Chrome through the Software Center, but when I reinstall it, all of my extensions and preferences are still there.   If I want to start over from scratch, what do I need to do? Conflict between tkz-fct and thmtools? find value of the $$\lim_{x\to0}\frac{e-(1+x)^{\frac{1}{x}}}{x}$$I use hospital law and can't find answer Is it possible now to use microtype with opentype fonts? Taking back turns, and forgotten privileges A generic question across most board games: once a player has taken their turn (taken their finger off the piece they are playing), should they be allowed to change their mind?  Similarly, if a player forgets to take a privilege, such as an occupied building in Puerto Rico giving an extra worker, or plantation, if they remember a few moves later, should they be allowed to take the privilege?  Very few rules explicitly express how these events should be dealt with, so they are generally left up to the group. However, my group's rules will change depending on the personality of the players playing (not all members are available each time we play). I would like to make this consistent, and would appreciate feedback on how others deal with this scenario.