How do I center my subscript?
How can I define a math operator behaving like \lim or \sum with limits?
I installed Ubuntu 14.04.1 on one of my laptops and now I want distribute that exact installation to a few of my friends. I tried making an iso with Gparted on a live session but it makes a 500 GB file (I've only used around 8 GB) .   Thanks in Advance ! Assume a p x n data matrix  (e.g. p independent samples of n variables). The n variables are correlated according to a correlation matrix $\rho$. Now if you consider the data matrix as some state t (t=time)  and based on this state I would like to calculate the next timestep (t+1). Therefore the correlation matrix would be allowed to have values &lt; 1 in the diagonal, which means that the diagonal elements would indicate the influence from each variable $x_{p,i}[t]$ on $x_{p,i}[t+1]$. So I would be interested to find a function f like this: $x_{p,i}[t+1] = f(x_{p,i}[t],\rho)$ so that the p x n data matrix  consisting of the $x_{p,i}[t+1]$ replicates the $\rho_{ij}$ (except the diagonal element). f should be something like this: $x_{i,j}[t+1] = \sum_{(k=1\ldots n)}\rho_{i,k} \cdot x_{kj}[t] + e_{i,j}.$ This means specifically, I not only want to calculate a data matrix, that replicates the correlation matrix as such, but also to let each observation be depending on its own state and linear dependent on the previous state of all the intercorrelated variables of its own state. For ease one can assume that the p variables are distributed normal and $e_{ij}$ are independent and distributed with mean$(e_{i,j}) = 0 $. Then for a numerical simulation it would be interesting  to find a formula to calculate the variance/standard deviation of each of the j variables (sd($e_{[.,j]} = ?).$ Thank you for any help! Definition of the Infinite Cartesian Product Where can I find a free, very quick, and reliable implementation of FFT in C#?  That can be used in a product? Or are there any restrictions? Suppose $f(x)$ is linear i.e. $f(x + y) = f(x) +f(y) $ and monotone on $[-\infty, +\infty]$, then $f(x) = ax$, a real.  We know $f(0)=0$, and if one assumes differentiability it is quite easy.      But how does monotone make up for differentiability? Was the significance of Arya's Braavosi coin ever explained? Jaqen H'ghar gives Arya the iron coin, she can apparently give it to any Braavosi and say Valar Morghulis and that gets her where she needs to go, but why?    Where do these coins come from and what do they indicate to a Braavosi? Rename a file to parent directory's name in terminal I have to deal with a large number of files nested inside of directories (one file per directory), resembling something like this:  fred/result.txt john/result.txt mary/result.txt ...   I am currently using the following command to process each of those files:  find . -maxdepth 1 -type d \( ! -name . \) -exec bash -c \"cd '{}' &amp;&amp; processResult result.txt\" \;   I am looking for something I can add into the end of this command that will change the filenames to fred.txt, etc. and then later move the file into the parent directory to eliminate the extra layer of directories.  What would be the best way to do this? When I create a UV sphere and add the subdivision surface modifier, the following wrinkle appears.    How can I avoid it?  I've read the article  but there seems to be no clear solution.  UPDATE 1  I've used the word 'apply' in the first sentence of the initial version of this question. It's misleading since I want to preserve the modifier without 'applying' it. Rather, what I meant is 'adding' the modifier.  UPDATE 2  I followed josh sanfelici's solution, but still there is an artifact at the pole.  Here is the screen: (A rendered view with a sun lamp)    Maybe it's the limitation of the general-purpose subdivision modifier.  BTW, when the cast modifier is added after the subdivision modifier, the sphere becomes perfect (even for my original wrinkled sphere).  But since I want to edit the resulting sphere (for example, scaling X, Y, etc) while modifier is added but not applied, it's not usable. How to cite lecture notes? (and should I even do this?) 2-10x Slower Deployments with Winter '15? Switching from MTB to road handlebars Difference between "@id/" and "@+id/" in Android Proving a sequence converges by defining the sequence recursively. Let $a &gt; 0$. Choose $x_1 &gt;$ $\sqrt a$. Define a sequence {$x_n$} recursively as $x_{n+1} = 1/2(x_n + a/x_n)$ for $n &gt; 1$.   Prove that lim$x_n = \sqrt a$.  I think I first want to prove that {$x_n$} is bounded below by $\sqrt a$. But, I am having trouble with this. Could someone help me out a little? One definition of an  is:     An abbreviation formed by (usually initial) letters taken from a word or series of words, that is itself pronounced as a word, such as RAM, radar, or scuba; sometimes contrasted with initialism.   An  is:     A term formed from the initial letter or letters of several words or parts of words, but which is itself pronounced letter by letter.   If I pronounce an initialism, as a word, and not just as letters, does that 'make' it an acronym? I'm trying to set up file transfer via FileZilla and it doesn't seem to matter what details I put in, I always get the following error:     Status: Connection attempt failed with "ECONNREFUSED - Connection   refused by server". Error: Could not connect to server   My Dad has tried on his computer (with FileZilla on Windows) and it connects and everything is fine.. When I try on mine with all the same details that he put in, it comes up with that error.  I read that a firewall might be blocking it so I double checked and my firewall through system preferences and it is off.  Anybody know why this might be???  Additional info:  Macbook Pro running OS X El Capitan version 10.11.6  FileZilla version 3.23.0.2 I am trying to prove that if I have $f:\mathbb{R} \to \mathbb{R}$ satisfying $\forall x,y\in\mathbb{R},f(x+y) = f(x) + f(y)$. Which is assumed continuous at $0$, that $f$ is continuous on $\mathbb{R}$  I am fairly sure it is, for that property seems to be a property of polynomials, and we know polynomials are continuous where defined(all reals)  Any ideas for rigor?  $|x-a| \lt \delta \implies |f(x+a)-f(x)-f(a)| \lt \epsilon$  No idea where to go from here, this is my first time doing $\epsilon-\delta$ stuff. What is the distinction between "among" and "amongst"? In Italian, when I write a sentence about an action (e.g. eating fish, playing), I would use the infinitive, such as  mangiare pesce fa bene alla salute (which literally is "to eat fish is healthy"), or giocare aiuta ad imparare (which literally is "to play helps to learn").   I would use the gerund in giocando si impara, but that would roughly mean "When playing, you learn." and in this case the infinitive would not be correct. (Giocare si impara is not what you would say in Italian.)  Should I use the infinitive, or the gerund, in sentences similar to the following ones?   To play with fire is dangerous. To win the lottery is just a matter of luck. To escape from Alcatraz is impossible. To practice makes you perfect.   Is the infinitive always wrong, when the verb is the subject of a sentence? Path connectedness of the set $\{(x,y):(x+1)^{2}+y^{2}\leq 1\}\cup\{(x,y):y=x\sin(\frac{1}{x}),x>0\}$ Let $A$ be the following subset of $\mathbb{R}^{2}$:$$A=\{(x,y):(x+1)^{2}+y^{2}\leq 1\}\cup\{(x,y):y=x\sin(\frac{1}{x}),x&gt;0\}$$ Then  $1.$ $A$ is connected.  $2.$ $A$ is compact.  $3.$ $A$ is path connected.  $4.$ $A$ is bounded.  It is clear that $A$ is unbounded as the graph of $x\sin(\frac{1}{x})$, so option $2$nd and $4$th are not correct. Again $A$ is connected by using the fact that closure of connected set is connected. But i am confused about path connectedness of $A$. Please help me about path connectedness of $A.$ Thanking you. How can I determine how much bleed to use?