Let $(X, d)$ be a compact metric space, and $f: X \to X $ is a map such that $d (f(x), f(y)) &lt; d(x, y)$ where $x \neq y$. Show that there exists a point such that $f(x) = x$.  My idea is to make compact sets where any finite intersection is non-empty and then use the fact that an infinite intersection will also not be empty to show existence. I don't want to use continuity as that has not been covered in my notes yet.  Thus take $A_n = \{x \in X:d(f^{n-1}(x), f^n(x)) &lt; a \}$, where $f^n$ is $f$ iterated $n$ times. Since $A_n \subset X$, they are compact and any finite intersection is non-empty. Thus an infinite intersection is also non-empty.   Since $a$ was arbitrary and distances are always positive, I can create a series of non-empty $A_n$'s between $0$ and $a$ no matter how small $a$ is. Thus there exists a point where $f(x) = x$.  How do I know that you can always find points with distance less than any $a$? Because $X$ is compact, it is bounded and so there is a maximum distance between any 2 points. So start there, iterate $f$ once and now you have points with distance less than $a$. Repeat as needed.  I feel like it's missing some rigor though, or at least could use some clean-up or an extra piece that I've overlooked.
A contractive mapping on $M$ is a function $f$ from the metric space $(M,d)$ into itself satisfying $$d(f(x),f(y))&lt;d(x,y)$$ whenever $x,y\in M$ with $x\ne y$. Prove that if $f$ is a contractive mapping on a compact metric space $M$, there exists a unique point $x\in M$ with $f(x)=x$.   Consider the function $g(x)=d(f(x),x)$. Note that   \begin{align*}d(f(x),x)-d(f(y),y)&amp;\leq (d(x,y)+d(y,f(x))-(d(y,f(x)-d(f(x),f(y)) \\ &amp;=d(x,y)+d(f(x),f(y)) \\ &amp;&lt;2d(x,y)  \end{align*}  And similarly for $d(f(y),y)-d(f(x),x)$, so we have $|d(f(x),x)-d(f(y),y)|&lt;2d(x,y)$. This means $g(x)$ is continuous.  So $g(x)$ is a real-valued continuous function on a compact space, so must be bounded and has a minimum $c$, i.e. there exists $x_0$ such that $d(f(x_0),x_0)$ equals the minimum $c$. What then?
Upgrade to the latest PHP version in ubuntu 16.04 I have a major problem that I hope someone can help with. On my Google Maps feature under my Location History it shows a location that I was never at at any point, but when my wife happened to see that location she doesn't believe me that the GPS could be wrong and pinpoint that location if I was never there.  I was first wondering myself how could that could be possible if I was never there or my phone was never at that location that was 35 miles away from me and second is there any way I could prove the GPS being wrong? Can I contact Google Maps and can they give an official printout of that certain day and every location I was? I really need to prove my honesty to her and hope there is a way. How can I find the number of partitions of $n$ into exactly $k$ distinct parts, where each part is at most $M$?  The number of partitions $p_k(\leq M,n)$ of $n$ into at most $k$ parts, each of size at most $M$, is given by the generating function: $$  \binom{M+k}{k}_{x} = \prod_{j=1}^{k}\frac{1-x^{M+k-j+1}}{1-x^j}= \sum_{n=0}^{kM} p_{k}(\leq M,n) x^n $$  For the number of the partitions $p_k(\mathcal{D},n)$ of $n$ into at most $k$ parts there is the recurrence relationship: $$   p_{k}(\mathcal{D},n) = p_{k}(\mathcal{D},n-k) + p_{k-1}(\mathcal{D},n) $$  But what, if I want to count only the partitions with distinct parts and restricted number of parts and restricted part size?  Update: Now I know the generating function for the number of distinct restricted partitions $p_k(\leq M, \mathcal{D},n)$ of $n$ into exactly $k$ distinct parts, all at most $M$ is $$    \prod_{j=1}^{M} (1+xq^{j}) = \sum_{k,n=0}^{\infty}p_k(\leq M, \mathcal{D},n)x^{k}q^{n} $$ and there is also a recurrence relation $$    p_k(\leq M, \mathcal{D},n) =         p_{k-1}(\leq M-1, \mathcal{D},n-k) +         p_k(\leq M-1, \mathcal{D},n-k) $$ How can I prove this? Could you recommend a book, where I could read about this? Properly discontinuous action: equivalent definitions Let us define a properly discontinuous action of a group $G$ on a topological space $X$ as an action such that every $x \in X$ has a neighborhood $U$ such that $gU \cap U \neq \emptyset$ implies $g = e$. I would like to prove that this property is equivalent to, having given $G$ the discrete topology and in the $X$ locally compact Hausdorff case, the map $G \times X \rightarrow X \times X$ given by $(g, x) \mapsto (x, gx)$ being proper (i.e. closed and preimage of compact sets is compact) plus the action being free.  I have managed to prove one direction, that is, if the action is proper and free with $G$ having the discrete topology then it is properly discontinuous. I'm having trouble though with the other direction. Here is an attempt: let's denote by $\rho : G \times X \rightarrow X \times X$ the map $\rho(g, x) = (x, gx)$. Suppose $K \subset X \times X$ is compact. We wish to show $\rho^{-1}(K)$ is compact. Let $(g_i, x_i)$ be a net in $\rho^{-1}(K)$. Then $\rho(g_i, x_i) = (x_i, g_i x_i)$ admits a convergent subnet, so passing to it we may assume $x_i \rightarrow x$ and $g_i x_i \rightarrow y$. Essentially we must now find a way to prove $g_i$ converges, but I can't seem to do this. Any hints? Kworker, what is it and why is it hogging so much CPU? Two layers, same coordinate system, wont show up I have vector two sets of vector layers that won't show up together. One set shows up together and the other set together etc but they are of the same area.  The layers I was given did not have a defined spatial reference which I manually projected as the same as the data frame I specified so all layers now have same CRS...yet not showing up together. I have noticed they appear to have different extents... What should I do? Key Derivation Functions vs. Password Hashing Schemes Key derivation functions, such as HKDF (standardized in ), are meant to stretch some initial keying material having enough entropy, like a Diffie-Hellman shared value, into one or more strong cryptographic secret keys.   Password-hashing schemes, such as the  winner , are meant to hash usually low-entropy passwords with the goal of making the hash digest inversion as costly as possible for an adversary with respect to CPU and memory consumption, as well as parallelization.  Is it exact to consider that password-hashing schemes are actually key derivation functions specialized for low-entropy inputs? Or is there any other essential difference of theoretical nature between these two types of cryptographic schemes? Congruent Modulo $n$: definition In an Introduction to Abstract Algebra by Thomas Whitelaw, he gives examples of the congruence mod operation, such as $13 \equiv5 \pmod4$,  and $9 \equiv -1 \pmod 5$. But when I first learned about the modulo operation my junior year, I would have told you that $13 \equiv 1 \pmod 4$, and that $9 \equiv 4 \pmod 5$.     Is this just a difference in the definition of modulo? Or is this pretty typical (to not take it to the largest factor, or one over)? This is what happens:  sudo apt-get install vlc Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package vlc   Please help. I would also like to know when this error occurs, I've seen it before too. Tikzexternalize not compatible with Miktex 2.9 abntex2 package Reference template variable within Jinja expression Is there a word for saying "um" or "uh", etc, during speaking?  Or a word for "um" and "uh", etc? How to print the longest line in a file? I'm looking for the simplest method to print the longest line in a file. I did some googling and surprisingly couldn't seem to find an answer. I frequently print the length of the longest line in a file, but I don't know how to actually print the longest line. Can anyone provide a solution to print the longest line in a file? Thanks in advance. Jump/Scroll to selected object in the outliner? $f \in {\mathscr R[a,b]} \implies f $ has infinitely many points of continuity. Is it possible to view the latex code after Lua expansion is made? When using Lua code inside Latex file, where Lua code sends to Tex Latex code to insert into the document at the location the lua code is made, it will be useful to look at the content of Latex file (internal) after the expansion is made but right before the file is compiled to pdf. This can help in debugging and in many other cases.  Is there an option to do this?  A small example will help explain what I mean.   Given this latex file  \documentclass{article} \usepackage{luacode} \begin{document}  The following is my table  \begin{luacode} i=6 j=8 tex.print("\\begin{tabular}{|l|l|}\\hline") tex.print(i.."&amp;"..j) tex.print("\\\\ \\hline") tex.print("\\end{tabular}") \end{luacode}  \end{document}   Compiling this gives the pdf file:    What I'd like to now see is the Latex file that would resulted after the Lua code expansion, which should be something as follows  \documentclass{article} \usepackage{luacode} \begin{document}  The following is my table  \begin{tabular}{|l|l|}\hline 6 &amp; 8 \\\hline \end{tabular}  \end{document}   I could not find an option to do this so far.  TL 2015    Update  Another example to make clear what is being asked here. I'd like to see not only the latex code that was generated from the lua bits of code, but the full Latex document that would result after all the expansion of all lua code in the original Latex. But this has to be done without modifying the original Latex document itself.   Assuming there is this virtual pipeline:  Latex file -&gt; Lua code expand -&gt; Updated Latex (internal)-&gt; lualatex -&gt;pdf    I'd like to get a copy of the full latex code/file just after all the lua code expanded.  Here is another example. Given this Latex file input:  \documentclass{article} \usepackage{luacode} \begin{document} The following is my table \begin{luacode*} i=6 j=8 tex.print("\\begin{tabular}{|l|l|}\\hline") tex.print(i.."&amp;"..j) tex.print("\\\\ \\hline") tex.print("\\end{tabular}") \end{luacode*}  Our annual budget for the company is \luaexec{tex.print(200*15^3)} dollars as  shown in this list of items \begin{luacode*}  tex.print("\\begin{itemize}")  for i=1,5 do    tex.print("\\item "..i^2)  end  tex.print("\\end{itemize}") \end{luacode*} And so on. \end{document}   The PDF is    I'd like to get a copy of the following latex file, which has all the Lua code expanded and gone and only the Latex code is left in its place. This file can go to a temporary file, or anywhere, as long as it is the complete Latex file as shown and not bits and pieces. This would be very useful in debugging when there is lots of Lua code in different places in the original Latex file.  \documentclass{article} \usepackage{luacode} \begin{document}     The following is my table  \begin{tabular}{|l|l|}\hline 6 &amp; 8 \\\hline \end{tabular}  Our annual budget for the company is 675000 dollars as  shown in this list of items  \begin{itemize} \item 1 \item 4 \item 9 \item 16 \item 25 \end{itemize}  and so on     \end{document}   I hope now the question is even more clear :) iOS App stuck in 'Loading...' I tried downloading Tumblr yesterday but it never completed. At the moment I have an App icon stuck at 'Loading...'.  I have tried restarting the iPhone (holding down both home and side buttons at the same time) and I have also tried shutting down the iPhone (holding the side button and swiping).  I have also tried deleting it as usual (holding the icon down and choosing delete but it doesn't go away).  Any ideas on what I can do to get rid of this icon? I have two Ubuntu Software Centers, one can be opened, another one can't Integrating radial free fall in Newtonian gravity Can't format my usb drive. I have already tried with mkdosfs and gparted I have a MicroSD card in a USB adapter (which plugs into a USB port on my machine, and acts like a USB flash drive). I was using Unetbootin to make this a bootable USB flash drive with Kubuntu. But I needed to cancel while it was working. So I killed the Unetbootin process from the console.  Since then, I can't format the MicroSD or delete the folder that Kubuntu made. I have also tried mkdosfs, with no success. I can't mount the drive anymore either? What can I do to make this drive usable again?  Trying to create a new partition table in GParted, as described in Eliah Kagan's answer, does not work. It fails with the error message "imposible crear tabla de particiones" ("unable to create a partition table").