Not Borel measurable set
Lebesgue measurable set that is not a Borel measurable set
Forward an invocation of a variadic function in C How can one securely generate an asymmetric key pair from a short passphrase? Background info: I am planning on making a filehost with which one can encrypt and upload files. To protect the data against any form of hacking, I'd like not to know the encryption key ($K$) used for a file, so the user will have to asymmetrically encrypt $K$ and send it as well. The reason for this double encryption is that you can still share (individual!) files with other accounts by encrypting $K$ with the public keys of those users, too.  My problem is: How can a user can get a public and private key without the server saving/knowing the private key and without the user having to remember a passphrase of 250 hex characters?  Question: Is there a way to derive a key pair from a password, like PBKDF-2 does for symmetric encryption? Suggestions on rearranging the cryptographic steps in an easier way are welcome, too.  Alternatively: It is also possible to encrypt the private key with a PBKDF-applied password and store it on the server, but I'd like to keep things 'simple'. ;)  Note: At this point I don't care if it requires RSA, ECC or an obscure cryptography scheme. Search engines and browser accept-language I'm building a web portal where language content will generally depend on the "accept-language" sent by the browser. The same content-URI will thus serve different content to different users depending on their browser setting.  I'm very curious to know how this will affect search indexing. Does Google index using all languages, and is it handled well? I'm using post thumbnails to link to a page.  Is it possible to add a class name to the post thumbnail image.  &lt;li&gt;&lt;a href="&lt;?php the_permalink(); ?&gt;" &gt;&lt;?php the_post_thumbnail(); ?&gt;&lt;/a&gt;&lt;/li&gt; Proving $\sum_{k=1}^n\frac1{\sqrt k}<2\sqrt n$ by induction I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing. $$\sum_{k=1}^n\frac1{\sqrt k}&lt;2\sqrt n$$ Please help! Bird flying in a cage Assume that you are holding a cage containing a bird.  Do you have to make less effort if the bird flies from its position in the cage and manages to stay in the middle without touching the walls of the cage? Does it make a difference whether the cage is completely closed or it has rods to let air pass? "Users" folder shared with everyone Today I found something strange on my newly installed Windows 7 OS - my "users" folder C:\Users was shared and "Everyone" had full access... I disabled that and now it seems fine, however I'm a bit scared actually, I'd like to know what caused that and if my system is compromised. Show for prime numbers of the form $p=4n+1$,  $x=(2n)!$ solves the congruence $x^2\equiv-1 \pmod p$. $p$ is therefore not a gaussian prime. I need to show that for prime numbers of the form $p=4n+1$, $x=(2n)!$ solves the congruence $x^2 \equiv-1\pmod p$. I then need to show this implies p isn't a gaussian prime.   I have started to solve this using Wilson's theorem that a number $z$ is prime iff $(z-1)!\equiv-1\pmod z$. Therefore the endpoint of my proof should be that $(p-1)!\equiv-1\pmod p$.  As $p$ is of the form $p=4n+1$, I only need to prove that $4n!$ is congruent to $-1$ modulo $p$.   Here is my working so far: Starting with the congruence $x^2 \equiv -1\pmod p$:    $$\eqalign{x^2 = -1\pmod p&amp;\implies   x^2 + 1 = kp \implies (x-i)(x+i) = kp\cr &amp;\implies ((2n)!-i)((2n!)+i))=kp \implies 4n!- 1 =kp\cr}$$   This is where I start to run out of any ideas that seem to get me anywhere. Any tips would be greatly appreciated! Why don't metals glow from red to yellow to green to blue etc.? Why only red, then yellow and then white? Shouldn't all wavelengths be emitted one by one as the temperature of the metal increases?   If some metals do glow at with different colours, could you give me examples of such metals and the reason why this happens in specific cases? Behavior of $\|f\|_p$ when $p$ is near the endpoint of $I$ Given measure space $(X, \mu)$ and measurable function $f$. If $p \in (0, +\infty]$, below are the facts I know about $\Vert f \Vert_p$:  (log convex in $\frac 1 p$) $\forall p_1, p_2\in (0, +\infty] \text{ such that } p_1 &lt; p_2. \forall t \in(0, 1), \text{ let } \frac 1 p = \frac t {p_1} + \frac {1-t} {p_2}$, then $\Vert f \Vert_p \le \Vert f \Vert_{p_1}^t\Vert f \Vert_{p_2}^{1-t}$ ($0\cdot(+\infty)$ is defined to be $0$). (lower semi-continuity) $\forall p_0 \in (0, +\infty], \liminf_{p\to p_0}\Vert f\Vert_p \ge \Vert f \Vert_{p_0}$. If $0&lt;\mu(X) &lt; +\infty$, then $ \Vert f \Vert_p/\mu(X)^{\frac 1 p}$ is monotone increasing in $p$, combine it with (2) we get $\lim_{p\to p_0-}\Vert f\Vert_p = \Vert f \Vert_{p_0}$.  From (1) we know the set of $p$ such that $\Vert f \Vert_p &lt; +\infty$ is an interval $I$ (possibly empty or a singleton).  If $I \not\in a = \inf I$ and $a \neq 0$, from (2) we know $\lim_{p\to a+} \Vert f \Vert_p = +\infty$. If $I \not\in b = \sup I$, then $\lim_{p\to b-} \Vert f \Vert_p = +\infty$.  From (1) and the dominated convergence theorem we know that if $I \ni p_0 \neq \inf I$ and $p_0 &lt; +\infty$, then $\lim_{p\to p_0+} \Vert f \Vert_p = \Vert f \Vert_{p_0}$. If $I \ni p_0 \neq \sup I$, then $\lim_{p\to p_0-} \Vert f \Vert_p = \Vert f \Vert_{p_0}$.  If $I$ is nondegenerate and $+\infty \in I$, then $\lim_{p\to +\infty}\|f\|_p = \|f\|_{\infty}$.  If $\inf I = 0$, from (1) we know $\lim_{p\to 0+}\|f\|_p$ exists.   Question: How $\Vert f \Vert_p$ behaves when $p$ is the endpoint of $I$? To be more precise, assume $I$ is nonempty. Let $p_0 = \sup I$ (only right endpoint is considered for simplicity):  $p_0\in I$ or $p_0 \not\in I$? Can $I$ be a singleton? What if $p_0 = +\infty$? What if $0&lt;\mu(X) &lt; +\infty$?  Please also notify me if there is any mistake above. Decoupling capacitors: what size and how many? Is the use of a hyphen between "non" and an adjective strictly necessary? Do I need to put a "-" between "non" and an adjective? As an example in physics we say "a non isolated photon", "non tight photon"... The context is very formal (paper publications and similar). Is there a general rule? Are there some differences between countries? How to get file creation & modification date/times in Python? I have a script that needs to do some stuff based on file creation &amp; modification dates but has to run on Linux &amp; Windows.  What's the best cross-platform way to get file creation &amp; modification date/times in Python? Manually calculated $R^2$ doesn't match up with randomForest() $R^2$ for testing new data I know this is a fairly specific R question, but I may be thinking about proportion variance explained, $R^2$, incorrectly. Here goes.  I'm trying to use the R package randomForest. I have some training data and testing data. When I fit a random forest model, the randomForest function allows you to input new testing data to test. It then tells you the percentage of variance explained in this new data. When I look at this, I get one number.  When I use the predict() function to predict the outcome value of the testing data based on the model fit from the training data, and I take the squared correlation coefficient between these values and the actual outcome values for the testing data, I get a different number. These values don't match up.   Here's some R code to demonstrate the problem.  # use the built in iris data data(iris)  #load the randomForest library library(randomForest)  # split the data into training and testing sets index &lt;- 1:nrow(iris) trainindex &lt;- sample(index, trunc(length(index)/2)) trainset &lt;- iris[trainindex, ] testset &lt;- iris[-trainindex, ]  # fit a model to the training set (column 1, Sepal.Length, will be the outcome) set.seed(42) model &lt;- randomForest(x=trainset[ ,-1],y=trainset[ ,1])  # predict values for the testing set (the first column is the outcome, leave it out) predicted &lt;- predict(model, testset[ ,-1])  # what's the squared correlation coefficient between predicted and actual values? cor(predicted, testset[, 1])^2  # now, refit the model using built-in x.test and y.test set.seed(42) randomForest(x=trainset[ ,-1], y=trainset[ ,1], xtest=testset[ ,-1], ytest=testset[ ,1]) This is, how I download various master branches from GitHub, and I aim to have a prettier script (and maybe more reliable?).  wget -P ~/ https://github.com/user/repository/archive/master.zip unzip ~/master.zip mv ~/*-master ~/dir-name   Can this be shorten to one line somehow, maybe with tar and pipe?  Please address issues of downloading directly to the home directory ~/ and having a certain name for the directory (mv really needed?). Conveying "under the radar" in a less colloquial way I am searching for a word to contrast with iconic  I want to characterize views of the city as both iconic and under the radar, maybe "non-iconic."  I was thinking about saying that a view of the city was undiscovered, but I think that maybe that overstates what I'm think-- I more mean under-appreciated. Are there any other words that come to mind? My singleton accessor method is usually some variant of:  static MyClass *gInstance = NULL;  + (MyClass *)instance {     @synchronized(self)     {         if (gInstance == NULL)             gInstance = [[self alloc] init];     }      return(gInstance); }   What could I be doing to improve this? Winforms Textbox - Using Ctrl-Backspace to Delete Whole Word To display a sensitivity analysis I want to produce a . In the plot a specific base case value is compared to the values produced after changing parameters in the model. I started from this . With this I was able to get this MWE:   \usepackage{pgf} \usepackage{tikz} \usepackage{pgfplots} \pgfplotsset{compat=newest} \usetikzlibrary{positioning} \usetikzlibrary{calc} \usetikzlibrary{intersections,backgrounds} \usetikzlibrary{decorations.pathmorphing}  \pgfplotsset{     TornadoPlot/.style={% Overall settings         width = 1\textwidth,     line width=1pt,     tick style={line width=0.8pt},     xmin = -50, xmax = 50,     xtick       = {-50, -40, -30, -20, -10, 0, 10, 20, 30, 40, 50},     xticklabels = {-50, -40, -30, -20, -10, 0, 10, 20, 30, 40, 50},     xmajorgrids = true,     minor x tick num = 1,     tick align = outside,     xtick pos = left, ytick pos = left,     xbar, bar shift=0pt,     height = 7cm,     enlarge y limits = 0.1, ytick = {0,1,2,3,4,5,6},     nodes near coords={         \pgfkeys{/pgf/fpu=true}%         \pgfmathparse{\pgfplotspointmeta}%         \pgfmathprintnumber{\pgfmathresult}     },     nodes near coords align = {horizontal},     before end axis/.code={         \draw ({rel axis cs:0,0}-|{axis cs:0,0}) -- ({rel axis cs:0,1}-|{axis    cs:0,0});     }              } }  \begin{document} \begin{tikzpicture}     \begin{axis}[TornadoPlot, legend style={at={(0.975,0.25)}}]             \addplot[fill=green!40] coordinates{(-3.5,0) (-16,1) (-17.5,2) (-19,3) (-34,4) (-27,5) (-22,6)};             \addplot[fill=red!40] coordinates{(4,0) (19,1) (12.5,2) (28.5,3) (34,4) (82,5) (38,6)};             \legend{upside, downside}      \end{axis} \end{tikzpicture} \end{document}   However, one of the bars (82, 5) has a positive x value that is outside the range of the domain of the plot. Can I restrict the length of that bar and force a break, while still displaying the original value of x? I have seen  for vertical bar-charts, but so far I have not been able to reproduce this for my tornado plot. Elizabeth's Brooch - follow-up question