When are quotients of the Cantor set Hausdorff?
$X/{\sim}$ is Hausdorff if and only if $\sim$ is closed in $X \times X$
Copyright of a set of colors? I often see color combinations on webpages, images, posters, products etc. that harmonize really well. Am I allowed to use those colors in my own design?   This is solely about the colors. I will not use images, logos or shapes from that source, and my content is not related to the original color source. I am also far from any commercial aspect here. Prove upper bound for recurrence I am working on  problem 3 from MIT's Fall 2010 OCW class 6.042J. This is covered in  which is about recurrences.  Here is the problem:  $$A_0 = 2$$ $$A_{n+1} = A_n/2 + 1/A_n, \forall n \ge 1$$  Prove   $$A_n \le \sqrt2 + 1/2^n,  \forall n \ge 0$$  I have graphed the recurrence and the upper bound and they seem to both converge on $\sqrt2$.    Also, if you ignore the boundary condition $A_0 = 2$ then you find that $\sqrt2$ is a solution to the main part of the recurrence. i.e. $\sqrt2 = \sqrt2/2 + 1/\sqrt2$.  The chapter and videos on recurrences have a lot to say about a kind of cookbook solution to divide and conquer recurrences which they call the Akra-Bazzi Theorem. But this recurrence does not seem to be in the right form for that theorem. If it were in the form $A_{n+1} = A_n/2 + g(n)$ then the theorem would give you an asymptotic bound. But $1/A_n$ is not a simple function of $n$ like a polynomial. Instead it is part of the recurrence.  Also, the chapter has a variety of things to say about how to guess the right solution and plug it into an inductive proof, but I haven't had much success. I have tried possible solutions of various forms like $a_n = \sqrt2+a/b^n$ and tried solving for the constants $a$ and $b$ but to no avail.  So, if someone can point me in the right direction that would be great. I always assume that the problem sets are based on something taught in the videos and in the text of the book but I am having trouble tracking this one down.  Bobby Birthday Problem applied to collisions I'm trying to extend the birthday problem to detect collision probability in a hashing scheme. Here is my problem. I use the letters and numbers [A-Z][a-z][0-9] to make a set of keys by randomly choosing from that set 10 times and concatenating them all together. So A24cv30kf2 could be one key for example. This implies that the base set has $62^{10}$ possible keys. Next, I want to compute the probability that I would see a collision if I picked a key from that set and issued it to $n$ users. If $M$ represents the size of the set (= $62^{10}$) and $P(M,n)$ the probability then  $$P(M,n) = \frac{M!}{M^{n}(M-n)!}$$  This follows by replacing 365 in the well known birthday problem shown   What I want to understand is the following: I want to check and see if my choice of concatenating 10 is right, when I know that my $n$ is of the order of a few millions and my $1 - P(M,n)$ needs to be less than 0.0001 or some small number. The large values of M &amp; n make the above intractable to compute. Is there an easier way to compute this? A reflection question about a complex function Why do I get (author?) when I use \citet with natbib? I would like to show the author names using \citet but I get just the reference number. Here is a minimal code:  \documentclass{report} \usepackage[utf8]{inputenc} \usepackage[english]{babel} \usepackage{natbib}  \begin{document}   The author \citet{toschi09} has shown that ..  \bibliographystyle{plain} \bibliography{foo}  \end{document}   And here is foo.bib  @article{toschi09, author = {Toschi, F. and Bodenschatz, E.}, title = {Lagrangian Properties of Particles in Turbulence}, journal = {Annual Review of Fluid Mechanics}, volume = {41}, number = {1}, pages = {375-404}, year = {2009},  }   What I get is:     The author (author?) [1] has shown that ..   Any help will be more than welcome, thank you! Surjectivity implies injectivity and conversely Prove the statement 'World is not flat' In my graph theory book exercise, I found a problem that:     Prove that the World is not flat using Mathematics     This picture is used in the exercise, but no idea of applying it.  I would have used the angles don't add to $180^\circ$, but the author is using a fourth city. Prove that $f(x)$ is irreducible iff its reciprocal polynomial $f^*(x)$ is irreducible. This is what I'm trying to prove:     Let $f(x)\in\mathbb{Q}[x]$ and $\deg(f(x))&gt;1$. Prove that $f(x)$ is irreducible in $\mathbb{Q}[x]$ iff its reciprocal polynomial $f^*(x)$ is irreducible in $\mathbb{Q}[x]$.    Note: The reciprocal polynomial $f^*(x)=x^nf(1/x) \in \mathbb{Q}[x]$, where $n=\deg f$.   So my thoughts are to prove the contrapositive, i.e. $f^*(x)$ is reducible iff $f(x)$ is reducible.   So to prove the ($\Rightarrow$) direction I assume that $f^*(x)=g(x)h(x)$, so I get that $g(x)h(x)=x^nf(1/x)$ which implies $f(1/x)=(1/x^n)g(x)h(x)$. I want to say this somehow makes $f(x)$ reducible but I am unable to proceed as $f(1/x)\notin \mathbb{Q}[x] $. I thought about substituting $y=1/x$, which gives $f(y)=y^ng(1/y)h(1/y)$, but I am unable to show that this is irreducible in $\mathbb{Q}[y]$. If I could, then it would be irreducible in $\mathbb{Q}[x]$ since $\mathbb{Q}[y]\cong \mathbb{Q}[x]$   Any suggestions or hints will be appreciated. playing a sound/notification in finch over ssh on my local machine So I have finch, a command line chat client (think pidgin) running on a remote machine that I ssh into. finch has notification sounds and can run a command when certain events occur, letting you pass the sound file in as %s. e.g. mplayer %s  from the remote machine, when the notification occurs, what kind of command could I run to play whatever %s is on my local machine? Applying Dupli-Objects Is it possible to bake dupliverts into one mesh for the BGE? (I have a large ground landscape that I use as the parent to duplicate trees, and the BGE doesn't support most modifiers.) Possible Beamer bug: Frame number has different color than date in infolines footer I use TeX Live on Mac and I get my weekly updates through the TeX Live Utility. I am working on some slides for an upcoming talk. I am using the same Beamer templates and settings that I always use, so I was surprised to see that the page numbers that I always include at the bottom of my slides suddenly appeared in a different color. Here's an MWE:  \documentclass{beamer} \useoutertheme{infolines} \usecolortheme{beaver} \title{Title} \date{\today} %\setbeamercolor{page number in head/foot}{fg=darkred!60!black} \begin{document} \begin{frame} \end{frame} \end{document}     As you can see in the bottom right corner of the slide the page number appears but its color looks off. Until recently it would look exactly the same color as the date appearing next to it.  By Googling around a bit I managed to figure out how to get this to look normal again using the line commented out in the code above:  \setbeamercolor{page number in head/foot}{fg=darkred!60!black}   By uncommenting this line I got back the desired behavior:    So my question is not so much about how to fix this problem, but about why this is happening in the first place. Is this a bug that was introduced in a recent update, and if so will it be fixed? Is this a deliberate choice on the part of the developers (if so I can't imagine why)? Or am I doing something wrong or is something broken in my TeX setup?  I'd appreciate any insight on this. Thanks in advance! Problem with PDO MySQL For the last couple of days I've been getting the following error:     Fatal error: Undefined class constant 'MYSQL_ATTR_USE_BUFFERED_QUERY' in /home/mydomain/public_html/drupal/includes/database/mysql/database.inc on line 46   This comes after my Hosting did an upgrade and PHP is now version 5.2.17.  From what they have told me, I think I have to enable a PDO MySQL extension in php.ini.  The problem is I'm on a shared hosting and I don't have access to the file; also, my Hosting said they can't change it either.  My info.php shows:     PDO      PDO support enabled      PDO drivers sqlite, sqlite2        pdo_sqlite      PDO Driver for SQLite 3.x enabled      PECL Module version (bundled) 1.0.1 $Id: pdo_sqlite.c 293036 2010-01-03 09:23:27Z   sebastian $      SQLite Library 3.3.7     I tried modifying the .htaccess file and/or adding my own php.ini file to the root directory per some suggestions I found, but nothing I've tried has worked yet.  Can anyone please suggest what else I could do?  EDIT:  Once again I asked the Hosting to enable PDO_MySQL extension and their answer was that they could not, that cPanel (to which they migrated from Plesk, causing this whole problem) had this extension disabled by default and changing it would affect their servers.  Their solution?  That I would change my programming to adjust to these new features, which I wouldn't even know how to go about.  Are they right requesting that I do this? Is there any other way? How to show that the commutator subgroup is a normal subgroup It is suggested as an exercise in Serge Lang's book "Algebra" to show that the commutator subgroup $G^c$ of a group $G$ is a normal subgroup.   I'd like to do that but I am afraid I need help,   I think the first thing I need to figure out is how a general element in the commutator subgroup looks like, so that I can check that the defining condition for normality is satisfied.  That is, supposing for a moment that a general element in $G^c$ is denoted by $g$, I need to show that $aga^{-1} \in G^c,$ for all $a \in G$.  But here I get stuck, first because I am unsure how to write a general element in $G^c$ - a simple product in $G^c$ is of the form $xyx^{-1}y^{-1}aba^{-1}b^{-1}$ where $a,b \in G$. I cannot see a way to simplify this - I am sure there is one, but somehow I am blind today.  The second thing then is, even if one tries out the conjugation of a simple element like $xyx^{-1}y^{-1}$ in $G^c$, again not simplification offers itself easily I think .. what am I missing ?  An alternative would be to find a homomorphism of $G$ whose kernel is precisely $G^c$ - here I tried to think of this as a map $G \times G \to G$ but whatever I cook up is not a homomorphism.  Thanks for your hints !! How do I convert seconds to hours, minutes and seconds? I would like to know how many switch can be stacked with fiber channel to get a best transfer rate between computers.  I have 200 computers connected to 12 3Com switches. All switches have two fiber ports.  Is the best way to have fiber backbone switch and connect fiber channel from every switch to the backbone? Or connect switch 1 to switch 2 to switch 3 to ... switch 12 to switch 1 again?  Thanks! counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$ I was hoping someone could help me come up with a combinatorial proof, one with an easy to understand 'story' for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$. For instance I understand that $2^n$ is the number of possibilities when creating subsets of size $n$. For example, if you're given a choice of any topping on a pizza, and you can only have said topping once (decide to include or exclude) then you have $2^n$ choices. But I'm not sure how the $-1 $ would be incorporated, or why the LHS would be $1+2+4+8+...+2^{n-1}$ I did not know that you could find an answer for that. However, I can only use Excel so far to do it.  How to calculate 3.6! by hand? A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int\limits_0^\infty \frac{\sin x} x \,\mathrm  dx = \frac \pi 2$$ Well, can anyone prove this without using Residue theory?  I actually thought of using the series representation of $\sin x$: $$\int\limits_0^\infty \frac{\sin x} x \, dx = \lim\limits_{n \to \infty} \int\limits_0^n \frac{1}{t} \left( t - \frac{t^3}{3!} + \frac{t^5}{5!} + \cdots \right) \,\mathrm  dt$$ but I don't see how $\pi$ comes here, since we need the answer to be equal to $\dfrac{\pi}{2}$. I am attempting to search jobs by tag on StackOverflow Careers. On SO, this is done using square brackets, but that is simply doing a keyword search, and I still get results that don't have the tag I am looking for.  For example, when I search for the keyword "android", I get this:        which is the same as when I search for the keyword "[android]":        Is there a format I can use to search jobs by tag? If not, I believe this would be a nice feature (seems intuitive for the Stack Exchange invite-only job site). Let $S = \bigoplus_{n\ge 0} S_n$ be a graded commutative ring. Let $f$ be a homogeneous element of $S$ of degree $&gt; 0$. Let $S_{(f)}$ be the degree $0$ part of the graded ring $S_f$, where $S_f$ is the localization with respect to the multiplicative set $\{1, f, f^2,\dots\}$. Suppose $S$ is finitely generated algebra over $S_0$. Then $S_{(f)}$ is a finitely generated algebra over $S_0$.