Suppose a complete metric space $M$ and $f:M\rightarrow M$ such that $f^p = f\circ f\circ\ldots\circ f$ ($p$ factors) is a contraction, for some $p\in\mathbb{N}^*$. Show that $\exists!a\in M$ such that $f(a) = a$ (a fixed point) and $$\lim_{n\to \infty} f^n(x)= a,$$  $\forall x\in M$.   Does anyone know how to solve this exercise?
Consider a complete metric space $(X,d)$ and $T\colon X\to X$. Suppose there exists $n\in\mathbb{N}$ such that the n-th power of $T$ is $q$-contractive. Show that then $T$ has exactly one fixed point $\overline{x}\in X$.      The n-th power is defined inductively: $$ T^{n+1}(x):=T(T^{n}(x)), n\in\mathbb{N}. $$ And for a function $T\colon X\to X$, $q$-contractive means  $$ \exists 0\leq q&lt;1~\forall~x,y\in X: d(T(x),T(y))\leq q d(x,y). $$    Now to the proof.  I think I have to show, that $T$ is q-contractive, because then it follows with Banach, that $T$ does have exactly one fixed point. So I have to show, that there exists a $0\leq q&lt;1$ so that for all $x,y\in X$ it is $$ d(T(x),T(y))\leq q\cdot d(x,y). $$ And most likely I have to use the q-contractivity of $T^n$, i.e. that there exists a $0\leq q &lt;1$ so that for all $x,y\in X$ it is $$ d(T^n(x),T^n(y))\leq q\cdot d(x,y). $$  Can you help me?
When trying to install CrossOver deb I get "Error: Dependency is not satisfiable libpng12-0" Resize multiple layers around their respective anchor points I have 50 or so layers (imported images as smart objects) that I would like to scale down, while they stay in place. Is there a way of doing this? I know After Effects has, is there a similar Scale property in photoshop? With biblatex, the author can be replaced by a dash (or “idem”) in the bibliography. Is it possible to have this replacement also if there are multiple authors, and the first of these multiple authors is the same as the author of the book listed before? (It is somewhat difficult to describe, see the attached example instead, where “Buchautor, Hans-Wilhelm” should be replaced in book3 as well.)   I assume that it should be possible by customising \DeclareNameFormat, but I don’t know how.  Minimal example:  \documentclass{scrartcl} \usepackage[T1]{fontenc} \usepackage[latin1]{inputenc} \usepackage[english]{babel} \usepackage{csquotes} \usepackage{filecontents} \begin{filecontents}{\jobname.bib} @BOOK{book1,   author = {Buchautor, Hans-Wilhelm},   title = {Irgendein Buch},   location = {Buch am Wald},   date = {2000} } @BOOK{book2,   author = {Buchautor, Hans-Wilhelm},   title = {Ein anderes Buch},   location = {Berlin},   date = {2001} } @BOOK{book3,   author = {Buchautor, Hans-Wilhelm and Zweitautor, Zoran},   title = {Noch ein Buch},   location = {London and New York},   date = {2010} } \end{filecontents}  \usepackage[ style=authortitle, backend=bibtex8 ]{biblatex} \bibliography{\jobname}  \renewcommand*{\bibnamedash}{%   \bibstring[\mkbibbold]{idem\thefield{gender}}}  \begin{document} \nocite{*}  \printbibliography \end{document}   Edit 15/02/2011 17:14   I should have added that there was a feature request for biber which pointed back to biblatex. See .  Edit 21/06/2011 9:26  Is this somehow possible now with biblatex 1.4/1.5? I played a bit with uniquename/uniquelist, but I could not get it to work. In ArcGIS there is a Erase Point tool how can this be done in R Would Gollum qualify, as a Ringbearer, to go to the Undying Lands? Alternative to LibreOffice Base for Ubuntu What's interesting in latus rectum? I'm a maths teacher in Italian secondary school and I've been spending some time trying to construct "meaningful" problems about conic sections. I particularly like problems which focus on practical applications of the curves (in engineering, physics, finance, etc.), on their geometric properties, or both.  I know there's this really easy formula to find the latus rectum of a conic from its key parameters. It's quite nice and straightforward to derive it, but...  ... What's its use? Why should anyone be interested in knowing the latus rectum of a curve? Is there any physical phenomenon, any architectural trick, or anything else where its position and/or length plays a key role?   I know just the relation between angular momentum, standard gravitational parameter and latus rectum in an astronomical orbit ($l=\frac{h^2}{GM}$, where $l$ is half the latus rectum, and $h$ is the ratio between the angular momentum associated to the orbiting body and its mass). It's cool, but a bit tricky, and probably too far-fetched for secondary school students... Is there anything simpler? Or just anything else, at any difficulty level?  Many thanks! Invalid token 'while' in class, struct, or interface member declaration in very simple code I am not sure what the problem is but I keep receiving this error when I try to use a while statement in my code.     Invalid token 'while' in class,   struct, or interface member   declaration   I want to use a while loop to have something continuously update while a statement is true.  The rest of my code is rather long but whenever I type in the syntax:  while(a&lt;b) { //do whatever i want it to do here }   It gives me that compiler error right off the bat. Not quite sure what the problem is. I am doing this in a C# windows application under the Form1.cs file with all the other event handlers (for buttons and such).  Thanks!    I was unaware that loops had to be placed within a method (fairly new to c#), but I tried it and no errors were returned. Thanks for your help everybody!  Previously, I just had the loop within the main class of the program. In contemporary American English usage, I come across sentences like:   I ain’t got no money.  Ain’t no man like him.    Saying ain’t no sounds incorrect to me because it is a double negative.  What is the origin of this particular construction? I wasn’t able to find anything online related to the origins of this incorrect usage. Are partially charged wands generally purchasable? Ubuntu n00b here.  I have created several disasters of the first order already.  I am getting ready to try to install opera - it is a very long and arduous process.  I am afraid - no - I am absolutely certain that I will damage something important.  How do I create a system restore point?  Does such a thing exist? How to force Windows to rename a file with a special character? I have a song that Windows can't play because there is a question mark in the name of the file.  "Where Have All the Cowboys Gone?.ogg" // as an example   So I try to rename it and Windows complains whether I try it in Explorer or from command prompt.  Error I get when trying to copy, rename, or move is:     The Filename, directory name, or volume label syntax is incorrect   Is there a Windows way to force a rename in this case?   Update  I'll keep an eye on this question, but after 13 answers and many attempts (aside form 3rd party solutions) it seems that Windows can't do this (or at least my windows can't, no short names). So I'm accepting the answer which was my original solution anyway of using Linux. It would be nice to see Windows handle this somehow, so don't stop just because I've accepted this answer, the question still stands! Lemma vs. Theorem I've been using Spivak's book for a while now and I'd like to know what is the formal difference between a Theorem and a Lemma in mathematics, since he uses the names in his book. I'd like to know a little about the ethymology but mainly about why we choose Lemma for some findings, and Theorem for others (not personally, but mathematically, i.e. why should one classify a finding as lemma and not as theorem). It seems that Lemmas are rather minor findings that serve as a keystone to proving a Theorem, by that is as far as I can go.  NOTE: This  doesn't address my concern, so please avoid citing it as a duplicate. How could towns restrain a magic user? The modern concept of crime and punishment is very reliant on the ability to disarm and restrain and control an individual (think handcuffs, the dividers in police cars, and jails).  How can that be replicated in a world where even a wizard's apprentice has a much power as an armed man with a handgun?  I've seen stories where magical restraints are present, but you can't write that into every thorp and hamlet, even in Eberron.    How can you restrain a magic user in a fantasy setting?   Either answers that conform to modern concepts of crime and punishment or answers about restraining a magic-user in a psuedo-medieval setting are acceptable. I recently found a formula to help find a sequence given a generating function:  $(\frac{1}{1-x})^k=\sum_{n=0}^{\infty}\binom{n+k-1}{n}x^n, k \in \mathbb{N}$  I just wondered if there was a known proof for this, or if anyone knows of a way to prove it. How can I check the SMART status of a SSD or HDD on current versions of Ubuntu 14.04 through 20.04? How can I check the SMART status of a drive under 14.04 and beyond? I've seen  but it doesn't seem to apply under 14.04 and later. I need to label the four corners of a dataframe as shown in the image. Can this be done with dynamic text scripts? Removing las2dem streaks from raster DEM? I've created a series of DEM files using the  tool from LAStools; however since I don't have a licensed version they have black streaks running across them. Does anyone know a good method for removing these streaks? Solving functional differential equation $f'(x)=2f(2x)-f(x)$ mariadb Unknown system variable 'server_uuid' Im trying to setup replication between mariadb(10.1.31-MariaDB - MASTER) and mysql(5.7.12-log - SLAVE). After executed the start slave, I got the below error.     Fatal error: The slave I/O thread stops because a fatal error is   encountered when it tries to get the value of SERVER_UUID variable   from master.   Then I tried to get the value of master_uuid.  MariaDB [(none)]&gt; show variables like '%server_uuid%'; Empty set (0.00 sec)   Then tried to setup this in mysql prompt.  MariaDB [(none)]&gt; set global  server_uuid = 11123c25-cac0-3523-8089-4c54d9b46f33; ERROR 1193 (HY000): Unknown system variable 'server_uuid'   And also I tried to setup auto.cnf in data directory.  [auto] server_uuid = 11123c25-cac0-3523-8089-4c54d9b46f33  chmod 644 auto.cnf   But still im getting the fatal error is encountered when it tries to get the value of SERVER_UUID variable from master.