Prove or disprove that $f$ is an identity function if $f(f(f(x))) = x$ and $f$ is continuous
3rd iterate of a continuous function equals identity function
I am trying to define a proof for  $A \cup B = B$ if and only if $A \subset B$  I started with:  $$A \cup B$$ $$x \in A \lor x \in B$$   but I am unsure on how to continue. Custom macro for multicolumn with & without xparse The following example has two normal LaTeX \newcommand macros and two xparse \NewDocumentCommand for specifying multicolumn text. The only difference between each pair of commands is one has the alignment character as part of the macro and one does not.  However, the xparse version that requires the alignment character to be outside of the macro does NOT work and results in Misplaced \omit. error.  \documentclass{article} \usepackage{xparse}  \newcommand{\MulticolumnA}[3]{&amp;\multicolumn{#1}{#2}{#3}} \newcommand{\MulticolumnB}[3]{ \multicolumn{#1}{#2}{#3}}  \NewDocumentCommand{\MulticolumnX}{m m m}{&amp;\multicolumn{#1}{#2}{#3}} \NewDocumentCommand{\MulticolumnY}{m m m}{ \multicolumn{#1}{#2}{#3}}  \begin{document} \[ \begin{tabular}{lllllll}     {a} \MulticolumnA{4}{l}{A-abcdefghijkli}\\     {a}&amp;\MulticolumnB{4}{l}{B-abcdefghijkli}\\     {a} \MulticolumnX{4}{l}{X-abcdefghijkli}\\     %{a}&amp;\MulticolumnY{4}{l}{Y-abcdefghijkli}\\     {col1}&amp;col2&amp;{col3}&amp;{col4}&amp;{col5}\\ \end{tabular} \] \end{document}   Above compiles as is, but does not compile if the one line is uncommented. Should black holes be considered two-dimensional or three-dimensional? Google Chrome: Import/Export Passwords? I recently migrated from Windows7 to Kubuntu 10.0.4. In many ways, I'm loving the change. (I never knew it could be such a seemless process to write/test scripts!)  One of the few things that is causing me any hangup is that none of the passwords that were saved in my browser are available (obviously, since it's a completely separate installation). Is it possible to export my passwords from the Windows instance of Chrome and then import them into the Linux version? I cannot boot to the desktop because I filled the file system. How do I reclaim some space? My memory is completely full when I tried following the data recovery and copied the wrong file using command line tool Test Disk. As it was taking so long to copy, I turned it off and I realized I copied the wrong file. Now my laptop memory is now full and lead me to more problems.  I have had similar problems to :       Now my screen keeps blinking:   Can anybody help? Many thanks. Wish you all the best. My Visual Studio (2008) Editor has stopped to underline Errors (this nifty wavy red lines). I can't really tell when, but it can be related to the installation of .Net Framework 3.5 SP 1 or the MVC Beta (which I guess is unlikely). Furthermore have I installed and uninstalled both CodeRush and Resharper for evaluation purposes (decided not to keep either one of them).  Does anyone know the problem and how to restore this functionality again? Assume I have a dataset for a supervised statistical classification task, e.g., via a Bayes' classifier. This dataset consists of 20 features and I want to boil it down to 2 features via dimensionality reduction techniques such as Principal Component Analysis (PCA) and/or Linear Discriminant Analysis (LDA).  Both techniques are projecting the data onto a smaller feature subspace: with PCA, I would find the directions (components) that maximize the variance in the dataset (without considering the class labels), and with LDA I would have the components that maximize the between-class separation.  Now, I am wondering if, how, and why these techniques can be combined and if it makes sense.  For example:   transforming the dataset via PCA and projecting it onto a new 2D subspace transforming (the already PCA-transformed) dataset via LDA for max. in-class separation   or    skipping the PCA step and using the top 2 components from a LDA.   or any other combination that makes sense. I'm sure its a simple question and I suppose it was asked many times, but I just can't figure it out from other answers, sorry.  I use recent versions of PostgreSQL and MySQL. I have 2 tables:  CREATE TABLE authors (     id INT,     name VARCHAR )  CREATE TABLE posts (     id INT,     author_id INT,     text VARCHAR,     date DATE )   I need to select one most recent post for each author. Thanks!  UPDATE  Thanks, both links provide answers to my question with some exception. All the following queries give the same result (which one is the most efficient btw?) The issue is when there is more than one post from the same author with the same date. Then the returned result set contains all such posts. How should I modify these queries to return exactly one post per author?  SELECT p1.* FROM posts p1 LEFT JOIN posts p2 ON p1.author_id = p2.author_id AND p1.date &lt; p2.date WHERE p2.author_id IS NULL ORDER BY p1.author_id;  SELECT p1.*  FROM posts p1 INNER JOIN (   SELECT author_id, MAX(date) AS max_date   FROM posts   GROUP BY author_id) p2   ON p1.author_id = p2.author_id AND p1.date = p2.max_date ORDER BY p1.author_id;  SELECT * FROM posts p1 WHERE date = (SELECT MAX(p2.date)               FROM posts p2               WHERE p1.author_id = p2.author_id) ORDER BY author_id;  SELECT * FROM (     SELECT author_id, MAX(date) date     FROM posts GROUP BY author_id ) p1 INNER JOIN posts p2 USING (author_id, date) ORDER BY author_id; Prove $n-m \mid n^r - m^r\,$ [Factor Theorem, monomial case] In respect to a larger proof I need to prove that $(n-m) \mid (n^r - m^r) $ (where $\mid$ means divides, i.e., $a \mid b$ means that $b$ modulus $a$ = $0$). I have played around with this for a while now but to no avail and would be grateful if you could help me. Working on a future method to post values to external web service. But since there is a limit of 50 callouts per invocation, I am getting error "Too many future calls : 51". Cases are loaded into salesforce from external ETL in bulk. So is there a way to workaround this error.   Class  public with sharing class caseTrigger {      //method invoked by trigger after insert     public void CaseInsert(List&lt;Case&gt; caseIns){           list&lt;Id&gt; caseIds = new list&lt;id&gt;();             List&lt;Case&gt; caseList = [Select id from case WHERE Id IN :caseIds];                 for(Case c : caseList){                     if(c.Status != 'closed'){                         caseIds.add(c.id);                     }                          if(f.caseId_User.size()&gt;0){PostCases(caseIds);}      }     //postCases (future method)     @future(callout = true)     public static void PostCases(List&lt;Id&gt; Ids){         //prepare Json to post to webservice    } } What is bracketing? I have seen this term a lot and I was wondering what, in basic terms, what bracketing/autobracketing is.  Also, why would one want to 'bracket'?  P.S. - there are other questions asking what DoF and WB bracketing is but none asking quite the same as this. Now that experience does something besides render regions of a server uninhabitable due to  orb lag, what's the most efficient and safe way to get those excellent enchants?  A mob farmer may rank highly, but what about without tons of setup? Express $1 + \frac {1}{2} \binom{n}{1} + \frac {1}{3} \binom{n}{2} + \dotsb + \frac{1}{n + 1}\binom{n}{n}$ in a simplifed form I need to express $$1 + \frac {1}{2} \binom{n}{1} + \frac {1}{3} \binom{n}{2} + \dotsb  + \frac{1}{n + 1}\binom{n}{n}$$ in a simplified form.  So I used the identity $$(1+x)^n=1 + \binom{n}{1}x + \binom{n}{2}x^2 + \dotsb   + \binom{n}{n}x^n$$ Now on integrating both sides and putting $x=1$.  I am getting $$\frac{2^{n+1}}{n+1}$$ is equal to the given expression.But the answer in my book is  $$\frac{2^{n+1}-1}{n+1}.$$ Where does that -1 term in the numerator come from? Show that $\int_0^\pi f(\sin x)\,\mathrm{d}x = 2\int_0^{\pi/2}f(\sin x) \, \mathrm{d}x$ "Thus" vs. "so" in formal English What IDE to use for Python? Determining true mean from noisy observations How does the tmp folder work? It stores lots of junk files and takes space My tmp folder is set to /sites/default/files/tmp/ and I checked what's inside and for example there is folder inside called /update-cache-b2903cc4/ it keeps all modules in .zip that I installed via 'Install from a URL'. So why are those modules .zip files are kept there in tmp folder, isn't it a waste of storage? I don't really like it when the site is full of unnecessary files.  Am I supposed to delete these by myself all the time? If $f:\mathbb R \to \mathbb R$ satisfies $f'(x) <L$ for some $L<1$, then $f$ has a unique fixed point Can a 3rd party app *technically* take better pictures? I've seen in several forums that some phones have very good cameras and the photos  taken are compressed too much and the details are lost.  I was wondering whether a 3rd party application (forgetting the post processing done by the application) have access to the full resolution image without any compression.  So can another application access the raw data from the camera or all applications gets the same raw data when they access the camera?