How to show there exist a unique $x_o$ in X such that f($x_o$)=$x_o$?
Prove the map has a fixed point
Trying to plot the sin(x) function, I get this result:    The MWE is here:  \documentclass{standalone} \usepackage{ pgfplots} \begin{document} \begin{tikzpicture} \begin{axis} \addplot[samples=500]{sin(x)}; \end{axis} \end{tikzpicture} \end{document}   cos(x) looks like an inverted parabola and tan looks just like sin.  I'm using pdflatex and MiKTeX. Any ideas? Can you help me with my software licensing issue? This is a  about Licensing.    Questions on licensing are off-topic on Server Fault.  If your question has been closed as a duplicate of this question, then this is because we want to help you understand why licence questions are off topic rather than just telling you "it just is". In all likelihood, this question will not address your question directly, it was not meant to.    I have a question regarding software licensing. Can the Server Fault community please help with the following:   How many licenses do I need? Is this licensing configuration valid? What CALs do I need to be properly licensed? Can I run this product in a virtual environment? Can I downgrade this product to an earlier version? Am I entitled to feature X with license Y? Evaluating $\int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x$ Today I was informed by VFS that my tourist visa for UAE got approved. I'm travelling to UAE on 16th December 2014 and returning on 7th Jan 2015. But when I checked the "valid until" date on the visa it states 30 December 2014.    Does this date mean I should return before December 30th? Should I get it corrected from VFS? Is there any easy LINQ expression to concatenate my entire List&lt;string&gt; collection items to a single string with a delimiter character?  What if the collection is of custom objects instead of string? Imagine I need to concatenate on object.Name. Nautilus doesn't launch on Ubuntu What's the best way to check if a String represents an integer in Java? I normally use the following idiom to check if a String can be converted to an integer.  public boolean isInteger( String input ) {     try {         Integer.parseInt( input );         return true;     }     catch( Exception e ) {         return false;     } }   Is it just me, or does this seem a bit hackish?  What's a better way?    See my answer (with benchmarks, based on the  by ) to see why I've reversed my position and accepted  to this problem.  I think this original code will be used by most people because it's quicker to implement, and more maintainable, but it's orders of magnitude slower when non-integer data is provided. @online bibliography biblatex why can't i select two objects in edit mode?     I am trying to make a face from two different meshes but can't perform the action How can I determine if someone's SSH key contains an empty passphrase? Some of my Linux &amp; FreeBSD systems have dozens of users. Staff will use these "ssh gateway" nodes to SSH into other internal servers.  We're concerned that some of these people use an unencrypted private SSH key (A key without a . , because if a cracker ever gained access to their account on this machine, they could steal the private key and now have access to any machine which uses this same key. For security reasons, we require all users to encrypt their private SSH keys with a passphrase.  How can I tell if a private key is not-encrypted (e.g. Does not contain a passphrase)? Is there a different method to do this on an ASCII-armored key vs. a non-ASCII-armored key?  Update:  To clarify, assume I have superuser access on the machine and I can read everybody's private keys. ReaderWriterLockSlim vs. Monitor I have an IDictionary&lt;TKey,TValue&gt; implementation that internally holds n other Dictionary&lt;TKey, TValue&gt; and distributes that insertions by the HashCode of the key to the invidual sub-dictionaries. With 16 sub-dictionaries, the number of collisions is pretty low on a 4-core machine.  For parallel insertions, i locked the Add-method with a ReaderWriterLockSlim, locking only the individual sub-dictionary:    public void Add(TKey key, TValue value)         {             int poolIndex = GetPoolIndex(key);             this.locks[poolIndex].EnterWriteLock();             try             {                 this.pools[poolIndex].Add(key, value);             }             finally             {                 this.locks[poolIndex].ExitWriteLock();             }         }   When inserting items with four threads, i only got about 32% cpu usage and bad performance. So i replaced the ReaderWriterLockSlim by a Monitor (i.e., the lock keyword). CPU usage was now at nearly 100% and the performance was more than doubled.   My question is: Why did the CPU usage increase? The number of collisions should not have changed. What makes ReaderWriterLock.EnterWriteLock wait so many times? C++ preprocessor __VA_ARGS__ number of arguments How to influence the position of float environments like figure and table in LaTeX? How to influence the position of float environments like figure and table in LaTeX?   This is a general question and should collect useful answers for all users. I hope we can use this as a reference What do all those cryptic number and letter codes in a lens name mean? Has the Doctor’s memory been reset? At the end of The Time of the Doctor, it is shown that Matt Smith has received an entire new regeneration cycle and using it he regenerates into Peter Capaldi. Capaldi's first question to Clara is about how to fly the TARDIS. Does this mean there has been a memory reset for the Doctor?  If so, has he forgotten his true name?  If he has forgotten his true name, then he can't bring back the Time Lords now can he? MySQL: How to retrieve a random row or multiple random rows? Why the number of indexed pages decreased for a "site:" search Recently I created a website with hundreds of thousands of unique pages. A week ago, Google had indexed about 250,000 pages. It has now decreased to 90k URLs (using a site:domain.com query).   What should I do to have Google index all the pages again? Why did all these pages get de-indexed from Google?. What is this colored fiber in my chicken? I smoked a whole chicken. When I cut into the breast I found this:    What is it and is my chicken ruined? How can you handle an IN sub-query with LINQ to SQL? Direction arrows on a circle I would like to plot circle (or another curve) with positive direction (and show it wit arrows).  This plots the circle  \documentclass{article} \usepackage{tikz} \RequirePackage{pgfplots} \usetikzlibrary{shadows} \usetikzlibrary{shapes} \usetikzlibrary{decorations} \usepackage{verbatim} \usepackage[active,tightpage]{preview} \PreviewEnvironment{tikzpicture} \setlength\PreviewBorder{5pt} \begin{document} \begin{tikzpicture}     \begin{axis}[             xmin=-1.5,xmax=2.5,ymin=-1.5,ymax=1.5,             axis lines=center,             ticks=none,             unit vector ratio*=1 1 1,            ]                   \addplot [very thick, smooth, domain=(0:2*pi)] ({1.+cos(deg(x))},{sin(deg(x))});        \addplot[only marks,mark=*] coordinates{(1,0)};         \end{axis} \end{tikzpicture} \end{document}   And I want to add arrows like that    Would you advise me how to add such arrows?