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RSA encryption with private key and decryption with a public key When using the RSA cryptosystem, does it still work if you instead encrypt with the private key and decrypt with the public key? What about in the case of using RSA for sender authentication? Proving that an additive function $f$ is continuous if it is continuous at a single point Suppose that $f$ is continuous at $x_0$ and $f$ satisfies $f(x)+f(y)=f(x+y)$. Then how can we prove that $f$ is continuous at $x$ for all $x$? I seems to have problem doing anything with it. Thanks in advance. So i was solving exercises in propositional logic lately and stumbled upon a puzzle, that goes like this:     Each inhabitant of a remote village always tells the truth or always lies. A villager will only give a "Yes" or a "No" response to a question a tourist asks. Suppose you are a tourist visiting this area and come to a fork in the road.    One branch leads to the ruins you want to visit; the other branch leads deep into the jungle. A villager is standing at the fork in the road. What one question can you ask the villager to determine which branch to take?    I intuitively guessed the answer is "If I asked you this path leads to the ruins, would you say yes?". So my questions are:   What is the name and/or source of this logical riddle? How can i corroborate my answer with mathematical rigor? Big tilde in math mode Show images related to features in QGIS? Say for example, if there's a point data with old trees, then when you use the identify tool, you can see the photograph of the tree  Or even one level further, the labels displaying these images right away without any clicking needed  I believe the term is file streaming? How to scale a tikzpicture including texts? It seems that the scale option scales only the length of lines, but not the size of texts. For instance, 1 and true and not scaled in the following code.  \begin{tikzpicture}[thick, scale=0.6] \draw [dashed] (1,12) -- (11,12); \node[above] at (4,11) {1}; \node[above] at (8,11) {true}; \end{tikzpicture}   Could anyone tell me how to scale everything together within a tikzpicture? Thank you very much!  PS: It is a picture in a presentation with Beamer. Cardinality of a totally ordered union Let $(X_{\alpha})_{\alpha \in A}, (Y_{\alpha})_{\alpha \in A}$ be families of sets such that for $\{X_{\alpha} \mid \alpha \in A \}$ and $\{Y_{\alpha} \mid \alpha \in A \}$ are totally ordered by inclusion. We have $(X_{\alpha},Y_{\alpha}) \neq (X_{\beta},Y_{\beta})$ for $\alpha \neq \beta$. Suppose that $|X_{\alpha}| \leq |Y_{\alpha}|$ for any $\alpha \in A$. It is true that $|\bigcup_{\alpha} X_{\alpha}| \leq |\bigcup_{\alpha} Y_{\alpha}|$? Why does Math.Floor(Double) return a value of type Double? What is the formula for percent of frame filled at a specific distance and focal length? I am thinking about buying a new lens specifically to take photos of distant targets (birds, bears, etc). I currently own the Canon 70-300 IS USM, which is a nice lens, but I find that if I want to take a photo of anything further away than about 10 meters, the target does not fill a large amount of the frame (meaning I have to significantly crop in post production).  I am currently looking at the Canon prime 400mm, however I would like to determine at what distance an object will fill a reasonable proportion of the frame.  Is there a formula or rule of thumb I can apply that will help me in this situation? I realise the size of the target I am shooting will play a role here, so if we need to make an assumption about the size of the target please let me know. Is it possible to retrieve a previous un-evaluated, un-normalized input? What is the "exposure triangle"? Custom function won't refresh as inputs are changed A similar question was asked , but it was specific to .NET 3.5. Specifically, I'm looking for the following:   What is the correct way to determine which .NET Framework versions and service packs are installed? Is there a list of registry keys that can be used? Are there any dependencies between Framework versions? How do I parse command line arguments in Bash? Lebesgue density theorem in the line Here is my problematic code:  \begin{tikzpicture} \draw  (0,0)  to [bend left=20] (3,3) node[midway] {midway label}; \end{tikzpicture}   I wanted to put a label midway of the line, but instead the label is at (0,0).  If I remove [bend left=20], the line is not bent, and the label still misplaced. If I remove to and replace it by --, then it works fine, but my line is not bent any more.  \begin{tikzpicture} \draw (0,0) -- (3,3) node[midway] {midway label}; \end{tikzpicture}   Works fine.  The problem occurs with MacTeX 2011 and MacTeX 2012. Value of momentum of a particle in 1D box prepared in a particular state? What is the value of momentum of particle in 1D box in state $\sin(10\pi x/a)$? My understanding Standing waves representing particle in 1D box is not an momentum eigenstate so if we measure the momentum of particle in this state we may get any value of momentum which gives a distribution whose expectation value tends to zero. Answer key In the answer key it's given that the value of momentum is 5h/a and -5h/a.If we measure the momentum for this state we may get any of these two values.But is this a wrong reasoning because if these was the case,these should mean that 5h/a and -5h/a are the eigenvalue of the following wavefunction which is clearly not. What is the correct answer? Consider the sequence defined as  $x_1 = 1$  $x_{n+1} = \sin x_n$  I think I was able to show that the sequence $\sqrt{n} x_{n}$ converges to $\sqrt{3}$ by a tedious elementary method which I wasn't too happy about.  (I think I did this by showing that $\sqrt{\frac{3}{n+1}} &lt; x_{n} &lt; \sqrt{\frac{3}{n}}$, don't remember exactly)  This looks like it should be a standard problem.   Does anyone know a simple (and preferably elementary) proof for the fact that the sequence $\sqrt{n}x_{n}$ converges to $\sqrt{3}$? Capacitor with different charges on each plate I am confused as to whether/how capacitance changes when each plate has a different charge.  For example, consider a coaxial cable and put $20Q$ on the outer cable, and $-Q$ on the inner. Or how about concentric spheres, grounding either the inner or the outer? Seo: Change the title of a page Lets say my website looks like this:  News -Hello world -Foo -Bar  So the parent page will list the last 3 news messages. For each news message there is a sub-page. For the parent page I was thinking of setting the page title to: News - Hello world, Foo, Bar.  Now if somebody adds a new message the title tag will also change. I googled this and read a few blog posts where people said that changing the title actually hurts the ranking of that page.  So is this true? Should I just set a static title tag for all my pages?