prove $EO=OF$ is there an euclidean solution?
How to prove the midpoint of a chord is also the midpoint of the line segment defined by the points of intersection of other two chords with it?
How to convert light activated switch Can anyone teach me how to convert a light activated switch into  dark activated switch vice versa. Does the diameter of the aperture increase with increases in the focal length with constant F-number(F-stops)? What I have learned from the definition of F-number is that;  The F-number (sometimes also called focal ratio, f-ratio, f-stop, or relative aperture) of a lens is the ratio of its focal length divided by the diameter of the aperture. And F-numbers are expressed as f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, f/32 etc. Hence F-number (F-stop) depends on two variables; that are diameter of aperture and focal length of the lens.  Aperture diameter/size is one of the component which governs the amount of light that enters to the camera. As far as I known, at different F-stops values (f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, f/32) the amount of light allowed to enter the camera is fixed and are with either 1/2 or 1/3 stop increment.  This concept is very easy to understand, in case of prime lenses (with a  fixed focal length and variable aperture), where F-stop is inversely proportional to the aperture diameter. For lower F-number (F-stop) the aperture would be bigger allowing more light and for higher Hence F-number (F-stop) the aperture would be smaller allowing less light in to the camera.  F-numbers (F-stops) have been evolved to allow a definite/fixed amount of light in to the camera (As far as I known). In case of zoom lens (with variable aperture and variable focal length), I am not able to understand the what would be the relationship between F-stop and the diameter/area of aperture at different focal length.  I have two related questions  1 . In case of zoom lens,  for example, Canon EF-S 18-135mm f/3.5-5.6  (with variable aperture and variable focal length). At any particular F-number(F-stop), for example at f/5.6 , does the diameter of the aperture increase with increases in the focal length? I mean,  Is the diameter of the aperture at  18mm and f/5.6 would be smaller than that of at 135 mm and f/5.6?   In the above case (i.e. Canon EF-S 18-135mm f/3.5-5.6 ), What is amount of light that would enter the camera at 18mm and f/5.6 compared to that of at 135 mm and f/5.6? Will the amount of lights be same or different? User is able to edit a file using vim I have a file named file.txt with some contents located in /home/myuser directory and I change the permissions of that file to root.root and 644.   -rw-r--r-- 1 root root 1.4K Jan 15 07:09 file.txt   Then I switched to myuser and opened it with vim(without sudo) and tried to edit and save it :wq! and I got output like:  "file.txt" E949: File changed while writing Press ENTER or type command to continue   Then I hit enter and then again :wq! and the output was :  WARNING: The file has been changed since reading it!!! Do you really want to write to it (y/n)?   I hit y, the file got saved and exit from vim.  The question is why a file with ROOT privileges and read-only access for a user changed by the way?  PS: 1. myuser is a sudo user.   The user permissions after saving the file changed to myuser.myuser as well.  -rw-r--r-- 1 myuser myuser 1.4K Jun 27 18:57 file.txt Is ∅ equivalent to {∅}? I think they are, but I am not sure? If anyone could clarify, that would be great. Thank you! set css border to end in a 90 instead of a 45 degree angle How to call C++ function from C? If $\Sigma \vdash \varphi$ then $\Sigma \vdash \forall x \varphi$. Why when $(\varphi \implies \forall x \varphi)$ is not true? In the book of first order logic I'm reading they say that if $(\varphi_1,...,\varphi_n)$ is a proof from a set of formulae $\Sigma$, then so is $(\varphi_1,...,\varphi_n,\forall x \varphi_j)$ for all $1\leq j \leq n$. They assume that as an axiom but it doesn't make sense to me. Why if $\varphi$ is true then so is $\forall x \varphi$? For example if $\varphi$ is $x=1$ that wouldn't mean that $\forall x : x=1$ , or would it?  Maybe what I don't understand is: why $(\Sigma \vdash \varphi \implies \Sigma \vdash \forall x \varphi)$ is true but $(\varphi \implies \forall x \varphi)$ is not true?  The book is this: Sum and Product of Infinite Radicals In  (ignoring Hawking radiation), why is a  black? Why nothing, not even light, can escape from inside a black hole? To make the question simpler, say, why is a  black hole black? Can two people with different one time pads securely exchange a message like this? Alice has a message, generates a one time pad, encrypts her message and sends it to Bob. Bob generates his own one time pad, encrypts the message again, and sends it back to Alice. Alice then decrypts this message using her one time pad, sends it back to Bob who then decrypts it the last time using his one time pad.  The only problems I see with this technique:   It is vulnerable to a man in the middle attack. It requires sending the message back and forth twice between sender and recipient. It doesn't scale well if you want to use it to communicate with multiple people. Requires a way of generating cryptographically secure random numbers.   Does this method of encryption have a name?  If we ignore the fact that it's vulnerable to a man in the middle attack, is this method of encryption information-theoretically secure?  Why isn't this method used widely, i.e. which of the above listed problems or other problems makes this method infeasible for production use? I was wondering on the following and I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function itself. I can guess that it's probably not, because otherwise $e$ wouldn't be that special, but is there any proof of it? Box Cox Transforms for regression I'm trying to fit a linear model on some data with just one predictor (say (x,y)). The data is such that for small values of x, the y values give a tight fit to a straight line, however as x values increase, the y values become more volatile. Here is an example of such data (R code)  y = c(3.2,3.4,3.5,3.8,4.2,5.5,4.5,6.8,7.4,5.9) x = seq(1,10,1)   I'm curious to know if there exists any power transform (Box cox perhaps?) that lets me get a better fit for the data than simply doing a linear fit as shown below.  fit = lm(y ~ x) A piecewise $C^1$ curve has Jordan measure zero. $\newcommand{\Reals}{\mathbb{R}}\gamma:[0,1]\to \Reals^2$ is an injective parametrization of a curve $\Gamma$, which is piecewise $C^1$ and the length of the curve is $L(\Gamma_k)&lt;\infty$.   1.1.: Show that for every $n\in N$ there is a decomposition $0=t_0&lt;t_1&lt; \dots &lt;t_n=1$ of $[0,1]$ so that $L(\Gamma_k) = \frac{L}{n}$ for $k=1,\dots ,n$ and $\Gamma_k=\operatorname{Im}(\gamma_k)$, $\gamma_k=\gamma|_{[t_{k-1},t_k]}$.   1.2.: Show that you can cover $\Gamma$ with a system of spheres of radius $\frac{L}{n}$.   1.3.: Use 1.2. and 1.3. to show that $\Gamma$ has Jordan measure zero.   I wasn't there when they discussed when a function has Jordan measure zero  and how those things correlate with this exercise so if someone could give me some hints that would be great. How to reinstall Windows 10 on new hard disk Consider the function : $$f: \mathbb{R}^2 \rightarrow \mathbb{R} , (x,y) \mapsto   \begin{cases}         0 &amp; \text{for } (x,y)=(0,0) \\    \frac{x^3}{x^2+y^2}       &amp; \text{for } (x,y) \neq (0,0)   \end{cases} $$  Show that $f$ not differentiable at $(0,0)$ but all directional derivatives exist.  I don't know how to tackle this problem. Can someone give some hints or solve the problem? Thanks :) How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$ Uniform convergence of $f_n(z)=nz^n$ in the set $|z|<\frac{1}{2}$ Why java classes do not inherit annotations from implemented interfaces? How to customize my titlepage? what is the [count=\i] in \foreach for? Can someone explain what does [count=\i] in a foreach mean?  Antal S-Z's answer to  use this. It is also mentioned in , but I am not able to locate documentation on this in the TikZ-PGF manual (V2.10)