This is the question that has been on my mind lately:  If an object A is moving away from an object B which is presumed stationary by A, then the time observed from the perspective of object B will flow (if you will) faster than the time on the object A, correct?   However in the same way according to my understanding, if   the view points are reversed, then the same effect is to take place from the perspective of the observer in the Object A towards object B. From perspective of observer A (granted there is no point of any observable reference besides object B, object B is moving away and object A is stationary. In which case the time dialation applicable to A and the same effect applicable to B effectively cancel each other out.    Would this be an accurate assumption?   And if later A or B or both A and B reversed their direction and met, would the time lapsed in each of their measurement be equal to the time either of them have spent in apart and therefore be of equal duration relative to both A and B?
Suppose we have two twins travelling away from each other, each twin moving at some speed $v$:    Twin $A$ observes twin $B$’s time to be dilated so his clock runs faster than twin $B$’s clock. But twin $B$ observes twin $A$’s time to be dilated so his clock runs faster than twin $A$’s clock. Each twin thinks their clock is running faster. How can this be? Isn’t this a paradox?
Can Superman see through his eyelids? Is it possible to tell a Valyrian steel sword from a very well crafted steel sword, by sight alone I'm sure in Westeros there are some incredibly well crafted steel swords, but could someone instantly spot a Valyrian steel sword over one of these?  Does Valyrian steel have any visual indications that it is Valyrian steel and not just a very well crafted steel sword?  I'm wondering, in canon, has anyone seen a sword and instantly recognised it as Valyrian steel.  For the purpose of this question, no one is allowed pick up or use the sword, so they cant feel it's weight, balance, sharpness, etc... I've noticed an interesting phenomenon in the user interfaces of many famous applications, they're moving away from the glossy complex to a more dull and bare minimum design.  Why the sudden change? It also appears that most of these companies have adopted this design around the same time, was that linked to some new study?  Examples: $H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$ Given the following definition of normal subgroup     A subgroup $H$ of a group $G$ is said to be normal if, for every $g\in G$: $$gH=Hg$$   I've tried to show that $H\mathrel{\unlhd} G$ if and only if we have $gHg^{-1}=H$ for every $g\in G$:  (if) Let $H\mathrel{\unlhd} G$, $g\in G$; if $l\in gHg^{-1}$, there exists $h\in H$ such that $l=ghg^{-1}$ and since $gh$ = $h'g$ for some $h'\in H$, then $l=ghg^{-1}=h'(gg^{-1}) = h'\in H$. Let $h\in H$; we can write $h$ as $hgg^{-1}$, and since $hg = gh'$ for some $h'\in H$, we obtain $h = hgg^{-1} = gh'g^{-1}\in ghg^{-1}$. So $gHg^{-1}=H$.  (only if) Let $gHg^{-1}=H$, $g\in G$; if we take an $l\in Hg$, $l = hg$ for some $h\in H$, we note immediately that $l=hg=(gh'g^{-1})g=gh'\in gH$. What should I do to show that $gH\subset Hg$?  I'm also wondering if this mess is correct, and if it is possible a more elegant proof of this fact. I used to say "sharp sign" to refer to the # sign.   Today a friend told me that the correct term is number sign or hash sign or even just hash.  What is the difference between these options and what's the correct usage of the names for this sign? Link to a note in Apple Notes, from another note In Apple Notes on the Mac, is it possible to reference a note from within another note (by way of hyperlink)? So, for example, if I had two notes:  Thoughts about pastries Pastries are great. Truly wonderful treats. But you know what my favorite types of pastries are? Donuts. I've even made a list of the types of [donuts I've eaten].  and  Donuts I've Eaten Chocolate, Glazed, Frosted, Encrusted…  How could I link the text [donuts I've eaten] from the first note to the second note? I made changes to iptables config file in /etc/iptables/filter in Ubuntu and want to reload them. I read the man page and also googled but couldn't find the information. Any help will be appreciated. Assumed True until proven False. The Curious Case of the Vacuous Truth Given two statements, $P$ and $Q$, and the logical connective, $\implies$, the truth table for $P \implies Q$ is:  $$\begin{array}{ c | c || c | }   P &amp; Q &amp; P\Rightarrow Q \\ \hline   \text T &amp; \text T &amp; \text T \\   \text T &amp; \text F &amp; \text F \\   \text F &amp; \text T &amp; \text T \\   \text F &amp; \text F &amp; \text T  \end{array}$$  Lines one and two are quite clear. What is ambiguous is lines three and four.  One explanation as to why $P \implies Q$ is true when $P$ is false, provided by Velleman goes:     Let $P(x)$ be the statement $x&gt;2$ and $Q(x)$, $x^2 &gt; 4$. When $x=3$, $P$ is true, and $Q(x) = 9$ thus $Q$ is true. If $P(x) = 1$ then $Q(x) = 1$ and they are both false. If $P(x) = -3$ then $Q(x) = 9$ and so the statement is true.   This explanation was quite unsatisfactory to me. Looking at Enderton, we have:     For example, we might translate the English sentence, ”If you're telling the truth then I'm a monkey's uncle,” by the formula ($V \implies M$). We assign this formula the value $T$ whenever you are fibbing. In assigning the value $T$, we are certainly not assigning any causal connection between your veracity and any simian features of my nephews or nieces. The sentence in question is a conditional statement. It makes an assertion about my relatives provided a certain condition — that you are telling the truth — is met. Whenever that condition fails, the statement is vacuously true.      Very roughly, we can think of a conditional formula ($p\implies q$) as expressing a promise that if a certain condition is met (viz., that $p$ is true), then q is true. If the condition $p$ turns out not to be met, then the promise stands unbroken, regardless of $q$.   Though a significant improvement over the Velleman explanation, I still feel uncomfortable with it.  Really, it seems we can conjure up as many silly counter-examples as we like, such as:     If pigs can fly, then I can walk on water.   Though following the above truth table, the implication is that my ability to walk on water is true.  After considering it, it seems to me that $\implies$ is only meaningful when $P$ is true, then we can look at it in relation to $Q$. However, if $P$ is false, then we actually know nothing about the relationship between $P$ and $Q$. This would give a truth table of:  $$\begin{array}{ c | c || c | }   P &amp; Q &amp; P\Rightarrow Q \\ \hline   \text T &amp; \text T &amp; \text T \\   \text T &amp; \text F &amp; \text F \\   \text F &amp; \text T &amp; \text ? \\   \text F &amp; \text F &amp; \text ?  \end{array}$$  where the $?$ denotes that given $P$ we actually don't know anything about $ \implies $.  Thus. one way to clear this up would be to assume that $? = T$. Thus the vacuous truth is a "definition of convenience" in a sense.  The above is my take on the matter.  Could someone provide some clarification on the conditional logical connective? Determining if a game is CPU- or GPU-limited What is a reliable, but not too time-consuming way to determine if a certain game is limited by the graphics card or by the processor speed?  I have some preconceptions about whether I'm mostly GPU or CPU limited, but I'd like to verify if I'm right about that. I have some ideas how I would go around and test that but performing accurate benchmarks is notoriously tricky and I don't have much experience with it.   So I'm wondering what would be an easy way to determine the bottleneck, having only one computer with a certain configuration available? What tools would I use for that purpose?  It would also be interesting to determine if the amount of VRAM on my graphics card is a limiting factor. Disallow closing of questions with a certain number of question / answer upvotes With reference to this currently open question   At the time of writing, it has 3 close votes yet has 7 question upvotes and the single answerer (11 upvotes) will have been awarded the enlightened badge (silver).  Therefore in the opinion of many (i) the question adds value and (ii) the answer adds value.  As a staring point for discussion, I suggest that 5 net question upvotes (and perhaps 5 answer net upvotes) should be enough to keep a question permanently open. Let's haggle over the numbers though. Thoughts? Why does an argument similiar to 0.999...=1 show 999...=-1? I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc. Can anyone point me to resources that would explain what the below argument that shows $999\ldots = -1$ is about? Here is the most usual proof I see that $0.999\ldots = 1$:  $x=0.999\ldots$ $10x=9.999\ldots$ $10x - x = 9$ $x=1$  Using this same argument template I can show $999\ldots=-1$:  $x= \ldots9999.0 $ $0.1x= \ldots9999.9$ $0.1x - x = 0.9$ $x=-1$  What might this mean? Edit from one of the comments: $$\sum_{k=0}^{\infty}{9 \cdot 10^k}=-1$$ I don't follow understand what this theorem is saying: "Every function $f: A \to{\cal P}(A)$ is not surjective."   Is this saying so for example let A = {1,2,3} then the power set of A is {},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3} Is this saying that there is no surjective function that takes every element in each of its power sets to atleast 1 element in A?  Is my understanding correct? Requirements for a clan war? How can I add certain mobs to a team and make them passive to me if they would normally attack me? I know how to do it in 1.8, but how can I do it now in 1.16.4? Can't learn the third Word of Power from Wulfgar GRUB2 does not show Windows 8 option to boot into after Dual Boot Yesterday, I tried to dualboot Ubuntu 16.04 onto my hard drive, creating a new partition for Ubuntu alongside my existing Windows 8. First, I disabled the fast-boot option in Windows 8, and still GRUB did not load, and it went directly to Windows boot-up. Then I did some research and found this article:     Basically, I issued the BCDEdit command in Windows, and entered in the following code:  bcdedit /set {bootmgr} path \EFI\ubuntu\grubx64.efi   When I reboot, I get a basic GRUB2 terminal, and after I type in the exit command, then a second Shell prompt appears, and after typing the exit command again, the GRUB2 menu to select the OS to boot finally appears... with only Ubuntu and Ubuntu memory test as options, not Windows 8.  After booting into Ubuntu, I tried to run the Boot-repair Utility, but that shows this following error:   How do I approach this problem? The Pastebin log is here:   Thanks for the help! In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? Can I extend a class using more than 1 class in PHP? If I have several classes with functions that I need but want to store separately for organisation, can I extend a class to have both?  i.e. class a extends b extends c  edit: I know how to extend classes one at a time, but I'm looking for a method to instantly extend a class using multiple base classes - AFAIK you can't do this in PHP but there should be ways around it without resorting to class c extends b, class b extends a How and why do you use an image histogram? I realize that an image histogram is a graphical display of an images tonal distribution (i.e. horizontal darks to lights, vertical pixel distribution), but how does one really use it and why? I mean, can't you determine everything you need just by looking at the image? for any ring $A$ the matrix ring $M_n(A)$ is simple if and only if $A$ is simple Let integer $n\geq 1$. I have obtained that for any field $k$, the matrix ring $M_n(k)$ is simple, i.e., $M_n(k)$ contains no nonzero proper two sided ideals. Now I want to prove that: for any ring $A$ (not necessarily commutative), the matrix ring $M_n(A)$ is simple if and only if $A$ is simple.   How to prove?