Help remembering a Putnam Problem
Find all functions $f(x)$ such that $f\left(x^2+f(y)\right)$=$y+(f(x))^2$
When & how did Yoda come to know about the Rule of Two? At the very end of Star Wars Episode I: The Phantom Menace, Mace Windu and Yoda had this conversation:     Mace Windu: There is no doubt. The mysterious warrior was a Sith.   Yoda: Always two there are....no more...no less. A master and an   apprentice.   When did he come to know about the secret ? And, how? Any evidence from ? How to run scripts without typing the full path? I'd like to know if there is a way to run  program/shell script without typing full path:  /opt/idea/bin/idea.sh I want to see if my process makes a lot of context switches. I also want to see how manpulating task groups affects the number of context switches. Let $D \subset \mathbb{C}$ be the open unit disk, $f:D \to \mathbb{C}$ a holomorphic function and $d=\sup \{|f(z)-f(w)|:z,w\in D \}$. How can I show that $|f'(0)|\leq \frac{d}{2}$?  I tried using Schwarz's lemma on the function $g:D \to \mathbb{C}$ defined by $g(z)=\frac{f(z)-f(0)}{d}$, but I am only getting $|f'(0)|\leq d$. Student missing exam for “medical reasons” Is it possible to run Snow Leopard on a LATE 2011 MacBook Pro? I want to buy a new MacBook Pro, from Apple. However I want to run Snow Leopard instead of Lion.  I did some Google searches and found a few success stories of people who got SL to run on their early-2011 MBPs. I didn't find much of any experiences of people doing the same on a late-2011 MBP though, so I'm wondering (and really hoping!) if it's possible? How to replace one char with another in all filenames of the current directories? Apparently,     the Fisher analysis aims at simultaneously maximising the   between-class separation, while minimising the within-class   dispersion. A useful measure of the discrimination power of a variable   is hence given by the diagonal quantity: $B_{ii}/W_{ii}$.     I understand that the size (p x p) of the Between (B) and Within-Class (W) matrices are given by the number of input variables, p. Given this, how can $B_{ii}/W_{ii}$ be a "useful measure of the discrimination power" of a single variable? At least two variables are required to construct the matrices B and W, so the respective traces would represent more than one variable.  Update: Am I right in thinking that $B_{ii}/W_{ii}$ is not a trace over a trace, where the sum is implied, but the matrix element $B_{ii}$ divided by $W_{ii}$? Currently that is the only way I can reconcile the expression with the concept. Using Fermats prime numbers to prove that there is infinitely many prime numbers Local collection of Python packages: best way to import them? How to provide translations for a WordPress TinyMCE plugin? I know that I can get my translations into JavaScript doing:      $MyTranslations = array(         'translation1' =&gt; __("Some String 1", "MyTranslations"),         'translation2' =&gt; __("Some String 2", "MyTranslations")     );     wp_localize_script( 'jquery', 'my_translations', $MyTranslations );   But looking at some of WordPress TinyMCE plugins they use calls like this:  { title : ed.getLang('advlink.link_desc'), ... }   How is WordPress getting their translations into getLang()? Am I supposed to do it this way or do I just use my first bit of code and access the variables directly like:  { title : my_translations.translation1, ... } Prove that the functions $f(z)$ and $\overline{f(\overline{z})}$ are simultaneously holomorphic.   I take this to mean that $f(z)$ is holomorphic if and only if $\overline{f(\overline{z})}$ is holomorphic.  Let $g(z)=\overline{f(\overline{z})}$. Note that $\overline{g(\overline{z})}=f(z)$. So it suffices to prove that if $f(z)$ is holomorphic, then $g(z)$ is holomorphic.  Write $f(z)=u(z)+iv(z)$. Since $f(z)$ is holomorphic, the real and imaginary parts satisfy the Cauchy-Riemann equations:  $$\frac{\partial{u(z)}}{\partial{x}} = \frac{\partial{v(z)}}{\partial{y}}, \frac{\partial{u(z)}}{\partial{y}} = -\frac{\partial{v(z)}}{\partial{x}}.$$  We have $g(z) = u(\overline{z})+i(-v(\overline{z}))$. To prove that $g(z)$ is holomorphic, we must prove that its real and imaginary parts satisfy the Cauchy-Riemann equations:  $$\frac{\partial{u(\overline{z})}}{\partial{x}} = \frac{\partial{(-v(\overline{z}))}}{\partial{y}}, \frac{\partial{u(\overline{z})}}{\partial{y}} = -\frac{\partial{(-v(\overline{z}))}}{\partial{x}}.$$  How can we obtain this from the above relations? Getting new files to inherit group permissions on Linux I am having a problem with permissions on a Linux server. I am used to BSD. When a directory is owned by a group the user who owns it isn't in such as www-data, files created in it will be owned by that group. This is important because I want files to be readable by the webserver (which I will not run as root) but so a user can still put new files in the directory. I can't put the users in www-data because then they can read every other users websites.   I want the webserver to read all websites, I want users to be able to change their own.  The permissions are set like this on the folders at the moment....  drwxr-x--- 3 john www-data 4096 Feb 17 21:27 john   It is standard behavior on BSD for permissions to work this way. How do I get Linux to do this? At the moment I'm working with the . For this I read the following paper: .  But I have some mathematical problems with this formalism. I don't really see why this is a variation principle since it is opartor-valued.   The action principle is defined by: $\delta S  = G_1 -G_2$, where $S$ is the action operator $S = \int_{t_1}^{t_2} dt ~L$ and $G$ the generator operators of the transformations.  What does it mean that the action principle is operator-valued? For the classical action where $S$ is real we search for a stationary point but for operators? I also have a problem that $\delta S \neq 0$. What interpretation has this additional term $G_1 - G_2$? How do I remove top bar on secondary monitor?? Infectious chicken pox My 21 year old son has chicken pox and has been home for a few days now. The spots are just starting to crust over.  I work with elderly people but have had chicken pox as a child - should I carry on working while he is here (hes usually away at Uni) and infectious? Let $X$ ~ $Bin(n,p)$ and $Y$ ~ $Bin(m,p)$ be two independent random variables. Find the distribution of $Z=X-Y$.   see also   I figured this out:  $$ P(Z=z)=\cases{\sum_{i=o}^{min(m,n)} Bin(k+i,n,p)*Bin(i,m,p), &amp;if $z\ge0$;\cr \sum_{i=0}^{min(m,n)} Bin(i,n,p) * Bin(i-z, m, p),&amp;otherwise. \cr}$$  I also validated it by Monte Carlo simulation. For $n=30$, $m=20$ and $p=0.5$, I get the following distribution, where the circles are the analytical probabilities and the line connects the MC estimates.    Because that looked to me pretty much like a binomial distribution, I gave it a try and figured out that its actually a binomial, just shifted by m to the left. This can be simply written as $P(Y=y) = Bin(y+m, m+n, p)$. Hence, given equal success probabilities, the sum of two independent binomially distributed random variables is binomial, but also their difference, just shifted to the left.  This question here  is actually the same as mine, but received no answer and only the comment that there would be no simple formula. But the above formula looks pretty simple to me.   Is it correct that for the case of equal success probabilities, the above equations actually describes the distribution of $Z=X-Y$? I read in a book that $Z$ could not be binomial distributed because it had negative support. Is it right to call it a shifted binomial? How can you check for a #hash in a URL using JavaScript? Summing the power series $\sum\limits_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1}\prod\limits_{k=1}^n\frac{2k-1}{2k} $ I'd like to determine the function corresponding to the following power series: $$x + \sum_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n} \frac{x^{2n+1}}{2n+1}, $$ where $|x|&lt;1$. How strong is Superman on Krypton? When Superman goes back to visit his home planet (using his ability to travel through time, I guess), is he:    an exceptionally strong and athletic Kryptonian? just an average Kryptonian male? a weakling on account of spending a lifetime in the comparative microgravity of Earth?   A possibly related question: why is Superman so muscular? What kind of weights does he lift to stay in shape?  The answers to  are all about Superman's superpowers on Earth compared with other visitors from Krypton; I'm asking about Kal El without his superpowers.  Clarification: I'm asking about the "classic"  Superman, say late Golden to early Silver Age, roughly 1945-1960. (Yes, I know the concept evolved during those years.) I'm not sure what that would be in terms of "numbered Earths", those came along after my time. I'm pretty sure the Kryptonians of that era had no superpowers in their native milieu, nor would Kal El if he were deported in space-time to his native world.