I'm a little bit confused about the nature of symmetrical reference frames and how they relate to time dilation. To clarify my question, I've written a scenario.  Bob stands on a platform as a train moving at some high velocity goes by. He observes that while his own watch measures 10 seconds elapsed, the train's clock measures 15 seconds elapsed. The train stops, its clock 5 seconds ahead of Bob's.  However, since from the train's point of view, Bob was moving at the same velocity, the passengers measure his watch to have measured 15 seconds while their clock measures 10. When the train stops, Bob's watch is ahead of the train's clock by 5 seconds.  How are these reconciled?
There is a problem with my logic and I cannot seem to point out where. There's a rocket ship travelling at close-to-c speed v without any acceleration (hypothetically), and there is an observer AA with a clock A on Earth, and there's another observer on the rocket BB with a clock B and these two clocks were initially in sync when the rocket was at rest with a FoR (frame of reference) attached to Earth. Now, this rocket's moving and AA tells us that B is running slower than A by a factor of$$ \gamma = 1 /(1-v^2/c^2)^{1/2} $$ $$ t_A/t_B = \gamma $$ where v is the relative 1d- velocity between the two i.e the earth and the rocket! That would mean the time elapsed on A is greater than that on B but this will happen only in the FoR of AA? So $t_B$ in this equation must be the time on B as observed by AA? Is this correct? What do the terms mean in the equations? If the symmetry holds and BB doesn't accelerate, then BB could say that $$ t_B/t_A = \gamma $$ right? where $t_B$ and $t_A$ are the times on B and A with respect to FoR of BB? But I was solving this problem and I took the earth FoR of A, but the prof took the rocket FoR of B? Like how will I know which FoR to solve the problem from? It'd greatly help if the terms in all the above equations were laid down neatly! DO we even need these FoRs?? Because in all the solved problems the prof is't specifying any and is using random ones! Please help!!!! This is the question where I messed up. The first rocket bound for Alpha Centauri leaves Earth at a velocity (3/5)c. To commemorate the ten year anniversary of the launch, the nations of Earth hold a grand celebration in which they shoot a powerful laser, shaped like a peace sign, toward the ship.  According to Earth clocks, how long after the launch (of the rocket) does the rocket crew first see the celebratory laser light?  This must be 25 years. My reasoning is: If v = 3c/5 10v + vt = ct where t is time taken by the light to reach the rocket from Earth as calculated from Earth.. and I solved that for t. and added 10 years to that because the time starts at the launch of the rocket!  According to clocks on the rocket, how long after the launch does the rocket crew first see the celebratory laser light?  This is 20 years. Here, I say: If it takes 25 years as observed by clocks on earth for the laser to reach the rocket, what should be the corresponding time as seen on a clock on the rocket? Using the formula: 25 = $\gamma$t where $\gamma$= 5/4 solved for t!  According to the rocket crew, how many years had elapsed on the rocket's clocks when the nations of Earth held the celebration? That is, based on the rocket crews' post-processing to determine when the events responsible for their observations took place, how many years have passed on the rocket's clocks when the nations of Earth hold the celebration?  For this, I did the following: 10 years on earth = T years on rocket ship where T must be lesser than 10 as observed from Earth FoR! Therefore, T= 4(10)/5 years = 8 years! But, prof says, 10 years on Earth = T years in rocket ship where T must be GREATER than 10 as observed from the rocket FoR??? Therefore, T = 10(5/4) years = 12.5 years!! What does this question actually want?
As most of you know, in the Microsoft Store, there are . This means that on Windows I can emulate Ubuntu and have the Ubuntu command line directly on Windows.  The question is very easy: what can’t I do from the Ubuntu command line emulated in Windows that I can do on a proper Linux-based Ubuntu? Is it useful to download this Ubuntu application, install it, and work just with it instead of the real OS? Is it possible to install all the development libraries? Can I write (or not) device drivers? In other words: what are the limits?  What features of Ubuntu Linux-based are missing from Ubuntu-on-Windows?  New answers are welcome: I know that every software is always changing by improving characteristics and features! Zero visibility while editing a message in a chat room Uniform convergence with two sequences of functions For, $n\ge 1$ let, $g_n(x)=\sin^2(x+1/n)$ , $x\in [0,\infty)$ and $f_n(x)=\int_0^xg_n(t)\,dt.$  Then, which is(/are) correct ?  (A) $\{f_n(x)\} $ converges pointwise to a function $f$ on $[0,\infty)$ but does not converge uniformly on $[0,\infty)$.  (B) $\{f_n(x)\} $ does not converge pointwise to any function on $[0,\infty)$.  (C) $\{f_n(x)\} $ converges uniformly on $[0,1]$.  (D)  $\{f_n(x)\} $ converges uniformly on $[0,\infty)$.  I found that, $$f_n(x)=\frac{1}{2}\left[x-\frac{1}{2}\sin(x+1/n)+\frac{1}{2}\sin(1/n)\right].$$  So, $f(x)=\lim_{n}f_n(x)=\frac{x}{2}-\frac{1}{4}\sin x$.  Now, $|f_n(x)-f(x)|\le 3/4$. So, by M-test, $f_n(x)$ is not uniformly convergent, but point-wise convergent. So option (A) is correct..Am I right?? Is an automorphism of the field of real numbers the identity map? Between you and ("me" or "I")? Is it correct to say "between you and me" or "between you and I"?  I am not a native English speaker, so please bear with me. How do I "replay" the "Caveats" section from a homebrew recipe I put 4 different layers into my ArcMap document, but only one of them will show up on the map! I should be seeing the other 3 layers as well. When I do a Zoom to layer, I can see one layer, but I cannot figure out how to make the other 3 show up at once on top of each other.  Before I opened ArcMap, I used ArcCatalog where I right-clicked 3/4 of the shapefiles and  clicked Properties &rarr; Coordinate system &rarr; Import and then clicked on my Trees layer, which has the coordinate system I want the other 3 to have. But when I opened ArcMap the 4 layers did not overlap, nor could I move them above or below each other. Google Play: Error downloading app - insufficient space How can I get a reference to a method? Number of ways in which $5$ boys and $5$ girls are ordered in such a way that exactly $4$ girls stand consecutively in the queue Let $n$ be the number of ways in which $5$ boys and $5$ girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let $m$ be the number of ways in which $5$ boys and $5$ girls in such a way that exactly $4$ girls stand consecutively in the queue. Then find $\frac{m}{n}$.    I could find $n$ correctly as $6!\times 5!$, but I am finding $m$ as $7!\times 4!\times \binom{5}{1}$, but the book says $m$ is $6!\times 4!\times 5\times \binom{5}{4}$. Where have I gone wrong? What is the proper logic to find $m$? Mathoverflow - why so different Curious why  (se site) doesn't have hot questions list on the right.  It also uses the old stackoverflow design.  Why has this site's design and functionality been abandoned (or at least look abandoned?) The origin of my question arose from a problem: Let $q: X \to Y$ and $r: Y \to Z$ be covering maps, let $p= r \circ q$. Show that if $r^{-1}(z)$ is finite for each $z \in Z$, then $p$ is a covering map.  I am just wondering is there an easy counter example to show the necessary of "inverse image of every point is finite" condition?  I have already checked Hatcher's book (the wedged circle example) but I wonder is there a less complex and more obvious example?  I tried to construct a counter example by composition of a covering map from $R_+$ to $S^1$ with $R$ in the middle, but I failed due to I can't find a covering map from $R^+$ to $R$ which does the job. Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of continuous X Let $X$ be a random variable with a continuous and strictly increasing c.d.f. function $F$ (so that the quantile function $F^{−1}$ is well-deﬁned). Deﬁne a new random variable $Y$ by $Y = F(X)$. Show that $Y$ has a uniform distribution on the interval $[0, 1]$.  My initial thought is that $Y$ is distributed on the interval $[0,1]$ because this is the range of $F$.  But how do you show that it is uniform? Color change of string delimiter in listings with macro $f$ holomorphic on $D\setminus \{0\}$ and takes no values in $(-\infty,0],$ then $0$ removable If $f$ is holomorphic on $D\setminus \{0\}$ and takes no values in $(-\infty,0]$ then $0$ is a removable singularity.   I thought to prove this by elimination, but I can't really tell anything about the behavior of $f$ around $0$. How can one translate the information about the definition of $f$ in semi-open interval $(-\infty,0]$. After trying to delete apache and tomcat, and now trying to reinstall apache 2, I am missing the init.d/apache2 file to start/stop my server.  I am not sure if the new install actualy worked, and my apt-get purge apache2 didn't remove everything in the first place!  How can I get the init.d/apache2 file to test if server starts / is installed properly (I had removed files and directories manually)? Low Voltage AC Smoothing Capacitor Calculation I have a 9V AC source which will have a max load of 100mA.  I'm trying to figure the value of a smoothing capacitor with less than a 10% ripple.  I don't have a scope or else I'd just try a bunch out.  I'm just a hobbyist, but I'm trying to understand this.  Since I'm not sure if I'm doing this right, I'll just step through what I've done so far.  First I found this formula: $$V_{ripple}={V_p \over R_LC} \times \Delta T$$  I'm using 60Hz 9VAC that is full wave rectified and I want a 10% ripple so:  $$(9V \times .1)={9V \over R_LC} \times {1 \over 60Hz}$$  I'm calculating my resistive load with Ohm's law. $$R = {9V \over .1A} \quad R = 90 \Omega$$  So plugging that in I get: $$.9V={9V \over 90\Omega \; C} \times {1 \over 60Hz}$$  If I solve for C (admittedly using wolfram alpha), I get: $$C={1\over 540}F \quad C=1851 \mu f$$  1850µf seems pretty high to me...  And if I lower the ripple percent it get crazy high.  Is this the right way to find this?  Or did I screw up somewhere?  Also, my circuit has the capacitor between the full bridge rectifier's positive and negative outputs.  I've only seen this type of smoothing circuit, but maybe my circuit needs to change? Run command whenever someone joins a Minecraft server Is there any way to run a command automatically when a player joins the server? I need to do this without the use of a Bukkit plugin. I think it's possible with MCP but I want to know how! Game about placing pennies on table This problem is from The Art and Craft of problem solving book:  Consider the following two player game. Each player takes turns placing a penny on the surface of a rectangular table. No penny can touch a penny which is already on the table. The table starts out with no pennies. The last player who makes a legal move wins. Does the first player have a winning strategy?  Now its not hard to see that the first player will always win if you place the coin in the center. But I have been thinking about what would happen If I also allow Quarter, Loonie and Twoonies. Does the first player have a strategy to always win? Could someone provide an explanation? How to best respond to a request to send a passport scan by unsecured email? It is not uncommon for travel or holiday accommodation companies to ask for a scan of your passport by email. Sometimes local laws require them to keep a record of the passport numbers of all visitors. I'm happy to provide that information but you usually have to send it in an unsecured email, which may have consequences .  Could these companies be more responsible, and provide a public cryptographic key at least?  Are there other steps you should take to protect yourself, if you do send by unsecured email?