Could we travel faster than the speed of light in a medium, but slower than the speed of light in vacuum?
What happens if light/particles exceeded the speed of light for a particular medium?
List all files in all subfolders Deriving Sum of a Geometric Progression Why does it take so long for a computer monitor to change resolution? Why does it take 2-3 seconds rather than, say, 50 ms?  E.g. my current Windows 7 with strong GPU and 24 inch monitor takes 2-3 seconds. Why so long? I know some might say "it's JUST 2 seconds", but come on... Games produce hundreds of FPS, and monitors work at 60 Hz, so what is it that takes so much time?  I remember this was the same in just about any machine I ever worked with. Could someone explain the muon $g-2$ experiment problem? Show that there's no continuous function that takes each of its values $f(x)$ exactly twice. I need to prove the following:  There's no continuous function $f:[a,b]\to \mathbb{R}$ that takes each of its values $f(x)$, $x\in [a,b]$  exactly twice.  First of all, I didn't understand the question. For example $x^2$ takes $1$ twice, in the interval $[-1,1]$. Is it saying that it does not occur for all $x$ in the interval? But what about $f(x) = c$? Is it saying that it does not occur only exactly $2$ times, then? I have no idea about how to prove it. I know that for $f(x)$ such that $f(a)&lt;f(x)&lt;f(b)$, if $f$ is continuous then there is a $c\in [a,b]$ such that $f(c) = f(x)$.  Now, there's the following proof in my book and I really wanted to understand it, instead of just getting a new proof  Since the interval $[a,b]$ has only $2$ extreme points, then the maximum or minimum of $f$ must be in a point $c\in int([a,b])$ and and in another point $d\in [a,b]$. Then, there exists $\delta&gt;0$ such that in the intervals $[c-\delta, c), (c,c+\delta)$ (and if $d$ is not extreme of $[a,b]$, $[d-\delta, d]$) the function takes values that are less than $f(c) = f(d)$. Let $A$ be the greatest of the numbers $f(c-\delta), f(c+\delta), f(d-\delta)$. By the intermediate value theorem, there are $x\in [c-\delta, c), y\in (c, c+\delta]$ and $z\in [d-\delta, d)$ such that $f(x)=f(y)=f(z)=A$. Contradiction.   Well, why the last part? Why is it that I can apply the intermediate value theorem to these values? For example, $&lt;f(c-\delta)&lt;p&lt;f(c)$, then by the theorem I know that there exists $m\in [c-\delta, c)$ such that $f(m) = p$. Same for the other intervals. But what guarantees thhat the greatest of the values between  $x\in [c-\delta, c), y\in (c, c+\delta]$ will be inside the intervals $[c-\delta), c), (c,c+\delta), [d-\delta, d)$? $\gcd(b^x - 1, b^y - 1, b^ z- 1,\dots) = b^{\gcd(x, y, z,\dots)} -1$ Possible Duplicate:        $b$, $x$, $y$, $z$, $\ldots$ are integers greater than 1. How can we prove that  $$ \gcd (b ^ x - 1, b ^ y - 1, b ^ z - 1 ,\dots)= b ^ {\gcd (x, y, z, \dots)} - 1\ ? $$ Zorn's Lemma states that if every chain $C$ in a partially ordered set $X$ has an upper bound then there is at least one maximal element in $X$.  Let $n\in \mathbb{N}$, $C_n \subset \mathbb{N}$ be the set containing the natural numbers $\{1,\dots,n\}$. Then if $k\le n$, $C_k\subset C_n$, so $C_n$ is a chain. Also, $C_n$ has an upper bound, such as $n+1$. But what is the maximal element of $\mathbb{N}$?  There's something I don't get in Zorn's Lemma statement. Can a player character use the Misty Step spell to bring another character or creature with them when they teleport? I am an Eladrin elf and have the ability to use the misty step spell to teleport with my clothing and items. However, can I grapple a party member (or NPC or other creature) and use the misty step spell to bring them with me when I teleport? Would they have to be willing? Noetherian property for exact sequence Let $0 \to M' \xrightarrow{\alpha} M \xrightarrow{\beta} M'' \to 0$ be an exact sequence of $A$-modules. Then $M$ is Noetherian is equivalent to $M'$ and $M''$ are Noetherian.  For the ''$\Leftarrow$'' case:  I guess if we let $(L_n)_{n\geq 1}$ be an ascending chain of submodules of $M$, then $(\alpha ^{-1}(L_n))_{n\geq 1}$ is a chain in $M'$, and $(\beta(L_n))_{n\geq 1}$ is a chain in  $M''$. For large $n$ both these chains are stationary. Then why we know the chain $(L_n)$ is stationary?   Thanks! Making dual power supply from single for opamp? For the circuit of opamp 741. we have to supply the dual power for it. How can I make a dual power supply from single. For example, make a (+9V)-GND-(-9V) from single 9V battery ? How to simulate simple on/off, Single-Pole-Single-Throw switch? I am trying to simulate ON, OFF switch in this simple circuit:  V1 0 1 1.5v Rfan1 3 4 1 S1 1 4 SW ON Rwire1 6 5 0.00001 VV1 0 2 dc 0  VRfan1 3 5 dc 0  VRwire1 2 6 dc 0  .dc V1 1.5 1.5 1 .print dc i(VV1) i(VRfan1) i(VRwire1)  .end   but I got this error:  ngspice stopped due to error, no simulation run!   the circuit I am trying to simulate is like this one:    if I replace the switch with a resistor, the circuit will work normally.  any idea? EDIT  I have fixed my switch as the following:  S1 1 4 98 99 mySwitch ON .model mySwitch SW vt=0 vh=1 ron=1n roff=10k I block everything Google, including Google APIs, as I don't like to to have Google tracking all of my usage on the Internet. Is there any way to use Stack Exchange without Google APIs?  I usually am browsing with Windows 7 and Firefox, but of course not Firefox with StackExchange. How to make my own Dropbox / Ubuntu One server at home? Does anybody know of any resources that can show me how to make my own "Dropbox, Ubuntu One" server at home?  I really like the idea of these services, but I don't want to put my 'stuff' in the clouds. Ideally, it should have a client that runs on Linux and Windows.  I tried to setup iFolder on my Ubuntu 10.04, but without any success so far. How can I wake-up a PC using a command? (i.e. equivalent to moving a mouse, etc) It doesn't seem like this would be very difficult, but how can I wake-up a PC by doing the equivalent of moving a mouse or pressing keys on a keyboard?  My PC is running Windows 10 and I only have the monitors enter sleep mode after 15 minutes.  The rest of the PC is still awake and doesn't use hibernate, etc.  I'm trying to use EventGhost to receive a TCP/IP command that tells the PC to wake up the displays but it's proving quite difficult. How do I pipe terminal standard output (stdout) to the clipboard? For example,   Say I want to list the contents of a folder and directly paste them into a chat window for a friend to see.    I realize I could do ls &gt; filename.txt to create a file (filename.txt) with those contents; I'd then have to open or print the file and manually select and copy the text block (which can be annoying/tedious.)  I clearly could also select and copy the output of ls directly from within the terminal window.  It would be much faster/easier to simply pipe standard output to the clipboard.  What terminal command allows me to do this? Fix for Hibernate error "Use of the same entity name twice" How you fix the following Hibernate error:  What does "Use of the same entity name twice". 2016: a year in closing Youtube says "This video is currently unavailable" Similes and Metaphors - are similes a subset of metaphors? When I learned about the Minkowski Space and it's coordinates, it was explained such that the metric turns out to be $$ ds^{2} = -(c^{2}dx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2}   $$ where $ x^0,x^1,x^2,x^3 $ come from $ x^{\mu} : \mu = 0,1,2,3 $, and $ c $ is the speed of light. The first resource I had access to--I have to do a bit of digging for the exact paper--of course addressed that this invariant takes the tensor form: $$ ds^2 = g_{\mu \nu}dx^{\mu}dx^{\nu} $$ as well. This two things I've seen in all of my texts and online resources regarding resources. The element in question that varies between authors, is the time coordinate, $ x^0 $. When it was first explained to me, it was using the standard Cartesian representation for the spatial portion of the coordinates, and the time coordinate was labeled as $ x^0 = ict $. Squaring gives $ (x^0)^2 = -c^2t^2 $, and applying differential calculus, we get $ (dx^0)^2 = -c^2dt^2 $. Sensible and expected to have the tensor formula spit out the Minkowski Metric. The author then later explicitly states the coordinates are $ x^0 = ict, x^1=x,x^2=y,$ and $x^3 = z$.  My questions is then, why is it most authors on the subject omit the imaginary unit on the time coordinate? For example, .   The only reason I can fathom omission is if the author is using metric signature $ [+,-,-,-] $, where I started off learning the theory with signature $ [-,+,+,+] $ which may be the reason seeing the time coord with no imaginary unit seems dissonant to me. All help appreciated!  Edit: After reading the other answers, my questions is now one of why and how (mathematically) do we obtain the Minkowski Metric Signature. More specifically the one element with a different sign.