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Compactness of $A:=${$f \in C[0,1], |f|_\infty \le K, |f'|_\infty \le M$} Here we use infinity norm as metrics for $C[0,1]$. The professor claims that this set is compact. I can show this set is relative compact by Arzela Ascoli, i.e., for each subsequence there is a further sub-subsequence that uniformly converges to a continuous function on $[0,1]$. But it is not intuitive to me why this set is compact, i.e. the limiting function also satisfies $|f'|_\infty \le M$. Is it true that $A$ is close and therefore compact? "Much more easy" versus "much easier" What is time dilation really? How to get address of cell which calls a VBA Functions in a Excel sheet I've been testing PCA via SVD to decompose a simple time series data matrix, $X$.  I have two signals $x_1(t)$ and $x_2(t)$ in a data matrix where $M$ rows represents each timepoint sample and each column represents $x_1$ and $x_2$.  The mean signal, $\hat{x}$, is defined as the mean along the row axis (average of $x_1$ and $x_2$ along each timepoint).  I normalize each column of $X$ by subtracting its mean and dividing by the standard deviation.  When I use [U S V] = svd(Xz) in matlab, regardless of how the variables are distributed (whether they are correlated or uncorrelated), one of the columns of the right singular matrix, V, always points in the same direction (to a multiplicative constant) as the mean vector $(1/2, 1/2)$.  But when add an additional third time series, this is never the case (where the mean vector is $(1/3, 1/3, 1/3)$.  Because I normalize the standard deviation for each vector, it does make sense that the direction of most variance given by PCA would be the diagonal 45 degree line.  But if both variables $x_1$ and $x_2$ are independent gaussians, couldn't the PCA direction be any direction since the distribution is radially symmetric?  MATLAB Code:  s = RandStream('mcg16807', 'Seed', 0); RandStream.setDefaultStream(s);  G = zeros(1000,2); G(:,1) = 40*randn(1000,1)-100; G(:,2) = tan(G(:,1)) + randn(1000,1); X = G - repmat(mean(G),[size(G,1), 1]); Xz = X./repmat(std(G),[size(G,1), 1]); [U S V] = svd(Xz); G_mean = mean(Xz,2); corrcoef(G_mean, U(:,1)) corrcoef(G_mean, U(:,2)) Find all number $n\in \mathbb N$ such that $\varphi (n)=14$ A classic exponential inequality: $x^y+y^x>1$ This is a classic problem, but I haven't seen it on the site before.     Suppose $x$ and $y$ are real numbers that both lie in the interval   $(0,1)$. Prove  $$x^y+y^x&gt;1.$$   The following spoiler box contains a hint that indicates the solution I know. Does anyone know a more natural solution?       Use Bernoulli's inequality to show $$x^y\ge \frac{x}{x+y}.$$   Edit: I recently learned the source is the 1996 French math olympiad. Inequality between $\ell^p$-norms windows bat file error I currently have Windows 7 on my computer and I want to install Ubuntu alongside Windows safely.  Could someone please provide clear instructions? How to create a rounded cone Are Stormtroopers paid or rewarded in any way for their service? Expanding GCDs Reasoning: $\gcd(a,c) \cdot \gcd(b,c) = \gcd(ab,bc,ac,cc)$ I'm just wondering the reason behind being able to do this:  $$\gcd(a,c) \cdot \gcd(b,c) = \gcd(ab,bc,ac,cc)$$  Is there an elementary proof to the reasoning behind this? Why won't Blender run on Windows XP? I downloaded Blender 64 bit and the OS indicated that the program was not compatible with my computer chip. I then downloaded the 32 bit version which seemed to allow me to install. However the blender Icon in windows when pressed informs me that the program is not a window application. I have an Intel(R) Core (TM)2 Duo CPU E7300 @2.66Ghz if that is important and I am running XP. I am fairly sure I am doing something wrong here- can anyone help please? Find all real numbers $x$ for which $$10^x+11^x+12^x = 13^x+14^x.$$   I thought about taking the logarithm of both sides, but didn't see how that would help. How can we make the equation simpler? Search results on https://stackexchange.com are broken because of "insecure content" The search results on  are blocked because of:     [blocked] The page at  ran   insecure content from .   I am using Chrome. chemformula: No auto-completion in TeXstudio? I have trouble getting auto-completion to work with chemformula in TeXstudio. I have other packages working just fine.   The .cwl-file is showing up in "Options > Configure TeXstudio > Completion", and I have it ticked. However, it doesn't work when typing.  I have contacted Clemens Niederberger (who maintains the package), however he recommended I ask here instead.  It is mainly the \ch command I would like to have auto-complete, but all commands would be ideal.  N.B. I'm running TeXStudio 2.9.4 TeXLive 2015 (both newest, at the time of writing) Where does Google Chrome automatically save pages? Get attention to a new answer to an old question It happened to me that I found a solution to an older question. This question was commented by many people with comments like "very interesting, important, ..." such that I thought that my answer has also some value. My answer was the only one. It was upvoted and accepted by the asker. However, I assume, that very few of the "commentators" saw my answer, because the system does not inform automatically about new answers.   I think it would have been helpful for me and the commentators when they would get informed. I could achieve this by adding my own comment to their comments, because this triggers a system message. My question is: would this action be considered admissible or not? Are there some other possibilities? I can't get grub menu to show up during boot