This is a question about time dilation and different observers. Say you had two observers. One on earth and the other one traveling around 20% of the speed of light. The observer on earth will age faster than the observer traveling at 20% the speed of light.  The question is, will the observer traveling at 20% the speed of light travel into the person that's on earth's future? Say 30 years go by for the person on earth but only 5 years pass for the person traveling at 20% the speed of light.   Is the person traveling at 20% the speed of light just in another reference frame and the person on earth will still age and experience 30 years passing or will the person on earth and the one traveling at 20% of the speed of light both experience 5 years passing but the person traveling at 20% the speed of light will just end up in the person on earth's fututre?  If I understand this correctly, the person traveling at 20% the speed of light isn't traveling into the future of the person that's on earth, their just both in different reference frames relative to each other so the person on earth will have experienced 30 years and the person traveling at 20% of the speed of light will experience 5 years. Is this correct?  Also, does the rate entropy increases slow down as you move closer towards the speed of light?
Please will someone explain what time dilation really is and how it occurs? There are lots of questions and answers going into how to calculate time dilation, but none that give an intuitive feel for how it happens.
If virtual particles sometimes add more mass/energy to a system then was inputed or comes out in the output, how do they not violate conservation of mass/energy. Notice: Undefined variable: logged in C:\xampp\htdocs\Site\Site\view_category.php What is the best way(s) to fake function overloading in Javascript?   I know it is not possible to overload functions in Javascript as in other languages.  If I needed a function with two uses foo(x) and foo(x,y,z) which is the best / preferred way:   Using different names in the first place Using optional arguments like y = y || 'default' Using number of arguments Checking types of arguments Or how? Injection from $\mathbb{N}\to\{0,1\}$ to the set of all monotonic decreasing functions from $\mathbb{N}\to\mathbb{Z}$ Number Theory: Find all solutions of $\phi(n)=16$ and $\phi(n)=24$ Let $\mathcal{C}$ be a finite binary code such that for any two sequences $\{a_n\},\{b_n\}\in \mathcal{C}$ if $a_j\not=b_j$ for some $j$ then the concatenation binary sequence $a_1a_2\cdots a_n\not=b_1b_2\cdots b_n$, the Kraft-Mcmillan Inequality says that $$\sum_{i}\frac{N_i}{2^{i}}\le 1,$$where $N_i$ is the number of codes in $\mathcal{C}$ with length of $i$.  If we assume the code is prefix free, we will have an elegant probabilistic proof at  (consider the infinite binary sequence generated by i.i.d. binomial random variable with probability $1/2$, define $E_x$ be the event that the sequence has prefix $x\in \mathcal{C}$, then $\sum_i\frac{N_i}{2^i}\le \sum_i P[E_x]=  P[\cup_i E_x]=1$). But for the general case one will not necessarily have the events $E_i$s disjoint, I'm curious if similar argument exists for the general case. (The problem is an excercise in the probabilistic method of alon, I guess some probabilistic argument should exist)  (My attempt is that consider the product topology $T=\prod_i\mathcal C$ whose open sets generated by $E_x$s where $x$ is the concatenation codes of $\mathcal{C}$, we define a natural product measure $\mu$ over $T$ by $\mu(E_x)=2^{-|x|}$. Clearly $\mu(T)\le 1$ (as $T$ is a sub-topology of $\prod_i\{0,1\}$), I want to show $\mu(E_x\cap E_y)=0$ for $x\not=y\in \mathcal{C}$, if not we can find $E_z$ such that $E_z\subset E_x\cap E_y$ by compactness, if we can show $z$ can be write as concatenation code start with both $x,y$ we will done, but I get stuck at here.) Are Jedi bound by a code of honor or morals? In The Phantom Menace alone, Qui-Gon (and in part Obi-Wan) are quick to use Jedi mind tricks and not only gamble but also cheat at it. They seem happy with this fact.  Do Jedi have morals and ethics, or a code of honor? $\displaystyle\int_0^0\frac 1x\:dx$ This integral is defined ? If it define, what is the value ?  Please,step by step I've been having this issue for a while where I come to open the Terminal and I'm met with this:  Login incorrect login: _   I've already deleted the com.apple.terminal.plist to no avail. The really interesting thing is that sometimes it works, maybe 1 in 50 times. I literally have no idea what might be causing it and after searching the web for the better part of half a day it seems everyone else's issues are fixed when they delete the plist file.  How do I sort this out? I've tried running different shells (bash and zsh), but I still get this log in screen.  Also when I use Activity Monitor, I can see there is no shell running just the login. I have a char array buffer that I am using to store characters that the user will input one by one.  My code below works but has a few glitches that I can't figure out:   when I execute a printf to see what's in Buffer, it does fill up but I get garbage characters at the end it won't stop at 8 characters despite being declared as char Buffer[8];   Can somebody please explain to me what is going on and perhaps how I could fix this? Thanks.  char Buffer[8]; //holds the byte stream int i=0;  if (/* user input event has occurred */)  {         Buffer[i] = charInput;         i++;          // Display a response to input         printf("Buffer is %s!\n", Buffer);  }   Output:   tagBuffer is 1┬┬w! tagBuffer is 12┬w! tagBuffer is 123w! tagBuffer is 1234! tagBuffer is 12345! tagBuffer is 123456=! tagBuffer is 1234567! tagBuffer is 12345678!  tagBuffer is 123456789! Lack of Articles in Headlines Are there any rules as to an article before a noun in the singular in headlines please, i.e. lack of an article?       Description of Type of Investment Is there a single word for "moving unhinderedly"? Ubuntu Server 20.04 Nvidia Driver Dependencies I had a hard time installing Nvidia drivers in a fresh install of Ubuntu 20.04.  Installing via sudo apt install nvidia-driver-430 resulted in a list of 542 packages to install.  I ended up spending a bit of manual effort to get the Nvidia-provided driver installed without nearly so many dependencies (only had to install gcc and make).  Is this massive number of dependencies normal for Ubuntu Server?  I did a very vanilla install without the GUI.  Reddit suggested I ask here, as their opinion is that I should just install from the default repos, but I felt that so many packages was absurd just to install a driver.  Edit for clarity: installing the nvidia-driver-430 package appears to install the entire Gnome shell, which I feel is unnecessary on a server. In the double-slit experiment of electrons (observed by photons), is it correct to say the collapse is caused by the momentum of the photons? I'm working off the article, . I think it's well-written, but I'm not convinced about this part:     So what is causing the change in the particle’s behaviour?      In the example given above, when electrons are fired at a fluorescent screen, when we bombard the electrons with photons the interaction changes the state of the system catastrophically as, despite lacking mass, photons do carry momentum.   Here the author strongly implicates the momentum of photons as being responsible for the collapse. But is that correct to say?  I have read that the  verified that obscuring which-way information restores wave-like behavior. So it feels to me it's more about some "outer" state (perhaps well-characterized as "information") that we don't yet understand that collapses the system, not per se the photon-electron interaction.  Or am I wrong, and in fact for example, the process of obscuring in the quantum eraser experiments somehow "undoes" the photon-electron interaction or restores the quantum state through another interaction? Necromancy In Harry Potter How is Necromancy accomplished in the Harry Potter universe?  I thought that people cannot be raised from the dead in Harry Potter.  Were bodies just use as puppets that can only do simple tasks, for instance when Dumbledore and Harry were getting the locket, the  were there to guard the locket.  Also, besides Inferi were there any Zombies in Harry Potter? Associated elements in a ring Please help me to find elements $a,b$ in a ring $R$ such that $a\mid b$ and $b\mid a$, but there does not exist any unit $u$ in $R$ such that $a=ub$. Getting hyperref to work with \citeyear and \citeyearpar in biblatex I would like to open 4 Notepad++ windows at once, in the same pannel. Currently, I am only able to split it into two, and would like to know if there is a plugin that enables 4 views at at same time, or 3.   thanks When I watch the news I've noticed that the Gallup polls for things like presidential elections have [I assume random] sample sizes of well over 1,000.  From what I remember from college statistics was that a sample size of 30 was a "significantly large" sample.  It was made to seem that a sample size over 30 is pointless due to diminishing returns. I have been wanting to understand how to find the sum of this series.   $$1^p + 2^p + 3^p +{\dots} + n^p$$  I am familiar with Gauss' diagonalised adding trick for the sum of the first $n$ natural numbers.  I can prove the formulas for  $$\begin{align}   \sum_{1}^{n} k^2 &amp;= \frac{n(n+1)(2n+1)}{6}\\  \sum_{1}^{n} k^3 &amp;= \frac{n^2(n+1)^2}{4}\\ \sum_{1}^{n} k^4 &amp;= \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30} \\ \end{align}$$  With mathematical induction. But, beyond that even proofs with mathematical induction are difficult.  I'm interested in learning the theory and the proof behind Faulhaber's formula. What is the knowledge required to understand this proof ?