Is everything relative? Length and time?
If everything in the universe doubled in size overnight, would it be noticeable?
Why didn't James' love and sacrifice for Lily protect her? We know that Harry survived Voldemort's killing curse because Lily loved him so much that she sacrificed herself in an attempt to protect him. This resulted in a very powerful charm that pretty much rendered Harry immune to Voldemort (until Voldemort figured out the work-around). Then there's this passage describing that night:  [Voldemort] was over the threshold as James came sprinting down the hall. It was easy, too easy, he had not even picked up his wand ... &quot;Lily, take Harry and go! It's him! Go! Run! I'll hold him off -&quot; Hold him off, without a wand in his hand! ... He laughed before casting the curse ... &quot;Avada Kedavra!&quot; Harry Potter and the Deathly Hallows Chapter 17: Bathilda's Secret  So James Potter, fully aware that he was sacrificing himself (he said &quot;I'll hold him off,&quot; not &quot;I'll get rid of him&quot;) ran to face Voldemort because of his love for Lily and Harry. This seems to be very similar to how Lily tried to protect Harry from Voldemort even though she knew she would die. It seems to me that the same charm should have been cast upon Lily when James died, but just moments later Voldemort killed her too. Apparently James' sacrifice did not result in a charm that protected Lily, even though he did it from love for her and Harry. Why is this? Was something different in the circumstances? In Volo's Guide to Monsters, there is a race in the bestiary called Grungs. Were these created for 5e specifically?  I'm pretty sure they were created for 5e, as they have no deities listed which is always a bit suspicious, but I'm just making sure. Alternative methods to prove that $Z_m$ is a field under addition and multiplication $\bmod\ m$ iff $m$ is a prime I am looking for ways to prove that $\mathbb{Z}_m=\{0,1,2,\dots,m-1\}$ is a field under addition and multiplication $\bmod\ m$ iff $m$ is a prime.   I tried it this way: If $m$ is a prime $p$,it is obvious that most importantly, we need to prove that every non-zero element in $\mathbb{Z_p}$ has an inverse in $\mathbb{Z_p}$.We note that if $a,b\in \mathbb{Z_p}$, then $a\cdot b\in \mathbb{Z_p} \dots \boxed{*}$. Next, $a^p\equiv a \bmod p$ for prime $p$(the proof follows by induction on $a$).As  $\displaystyle 0\leq a\leq p-1$,$a\not =p$. Thus $$a^{p-1}\equiv 1 \bmod p$$ for prime $p$ which,together with $\boxed{*}$ guarantees the existence of an inverse in $\mathbb{Z_p}$.Proving the other properties is simple.  Conversely,if $\mathbb{Z_m}$ is a field,then every non-zero element of $\mathbb{Z_m}$ has an inverse.Note that inverse of an element is unique.For if $a\cdot b\equiv a\cdot c\equiv 1 \bmod m$,then $(a\cdot b)\cdot c\equiv c\equiv (a\cdot c)\cdot b\equiv b \bmod m$ i.e $m|(b-c)\implies b-c=0(0\leq a\leq m-1,a\in \mathbb{Z_m})$.So, inverses must be unique and every element is either an inverse of itself or has in inverse in the field.That allows us to conclude that $[(m-1)!]^2\equiv 1 \bmod m$ which can only mean that either  $(m-1)!\equiv 1 \bmod m$ or $(m-1)!\equiv -1 \bmod m$. If $m$ is not a prime, it has a prime factor, say $p$. $ 0\leq p\leq m-1\implies p|(m-1)!\implies p|1$or $-1$ which is absurd, a contradiction.Thus $m$ is prime.  I feel that this method is too involved and complicated. Can anyone please suggest some other method?(Some context: This is a problem from Finite Dimensional Vector Spaces by Paul Halmos. I am learning linear algebra for the first time.It seems some of the comments have been posted on the assumption that I know rings and fields; I don't. ) A spaceship crew stuck in a damaged spaceship that's losing power Which Marvel TV episodes are related to the MCU movies? I've kept up to date with the movies from the MCU, but I am grossly out of date with the TV series, of which I think there is only Agent Carter (not as relevant for the movies I think, besides that scene in Captain America 2), and S.H.I.E.L.D. (which from what I've seen seems more relevant, with  coming in phase 3).   I've gained what I can from . But I can't tell which of the episodes listed are relevant to the movies (clearly, it seems that none are essential), and which movies are relevant to the episodes.  So which episodes are related? Suppose we're tying to differentiate the function $f(x)=x^x$. Now the textbook method would be to notice that $f(x)=e^{x \log{x}}$ and use the chain rule to find $$f'(x)=(1+\log{x})\ e^{x \log{x}}=(1+\log{x})\ x^x.$$  But suppose that I didn't make this observation and instead tried to apply the following differentiation rules:  $$\frac{d}{dx}x^c=cx^{c-1} \qquad (1)\\ \frac{d}{dx}c^x = \log{c}\ \cdot c^x \quad (2)$$  which are valid for any constant $c$. Obviously neither rules are applicable to the form $x^x$ because in this case neither the base nor the exponent are constant. But if I pretend that the exponent is constant and apply rule $(1)$, I would get $f'(x)\stackrel{?}{=}x\cdot x^{x-1}=x^x.$ Likewise, if I pretend that the base is constant and apply rule $(2)$ I obtain $f'(x)\stackrel{?}{=}\log{x}\cdot x^x$.  It isn't hard to see that neither of the derivatives are correct. But here's where the magic happens: if we sum the two “derivatives” we end up with $$x^x+ \log{x}\cdot x^x=(1+\log{x})\ x^x$$ which is the correct expression for $f'(x)$.  This same trick yields correct results in other contexts as well. In fact, in some cases it turns out to be a more efficient way of taking derivatives. For example, consider $$g(x)=x^2 = \color{blue} x\cdot \color{red} x.$$ If we pretend the blue $\color{blue} x$ is a constant we would get $g'(x)\stackrel{?}{=}\color{blue}x\cdot 1=x$. Now if we pretend the red $\color{red}x$ is constant we get $g'(x)\stackrel{?}{=}1\cdot \color{red} x=x$. Summing both expressions we end up with $2x$ which is of course a correct expression for the derivative.  These observations have led me to the following conjecture:     Let $f(x,y)$ be a differentiable function mapping $\mathbb{R}^2$ to $\mathbb{R}.$ Let $f'_1 (x,y)=\frac{\partial}{\partial x} f(x,y)$ and $f'_2 (x,y)=\frac{\partial}{\partial y} f(x,y)$. Then for any $t$ we have: $$\frac{d}{dt}f(t,t)=f'_1 (t,t) + f'_2 (t,t).$$   (I apologise for the somewhat awkward notation which I could not seem to get around without causing undue ambiguity.)  This formulation also seems to lend itself to following generalisation:     Let $f:\mathbb{R}^N \to \mathbb{R}$ be a function differentiable in each of its variables $x_1,x_2,\ldots,x_N$. For $n=1,2,\ldots,N$ define $f'_n(x_1,x_2,\ldots,x_N)=\frac{\partial}{\partial x_n}f(x_1,x_2,\ldots,x_N)$. Let $t$ be any real number and define the $N$-tuple $T=(t,t,\ldots,t)$. Then one has: $$\frac{d}{dt} f(T)=\sum_n f'_n(T).$$   Thus my question is:   Is this true? How can it be proven? (Specifically in the case $N=2$ but also in the general case.) Is it possible to change the colors in the command prompt for the user@computer, as well as the the current directory and command parts of the prompt display?   I've already seen something like this done by OSX users, but I don't know how to do the same thing in gnome terminal (I can only change foreground and background colors).   It'd be very useful when, for example, trying to compile programs that have errors, since long, unformatted messages make it hard to distinguish which lines are commands and which are output. 123.SOGOU.COM - trojan horse in ukui-screensaver What is the expected minimum euclidean distance between $N$ points uniformly and independently chosen on the shell(sphere) of 3-D ball of radius $R$?   Note that the expected minimum distance might be difficult to compute, so a good lower bound is also fine.   My approach  I take the approach similar to this   .   Suppose we put $N$ circles of radius $r$ uniformly on the surface of the ball. Therefore we could compute the expected minimum distance by the following expression \begin{align} E[D]=\int_0^\infty P(D&gt;t) dt. \end{align}  So, it remains to compute  $P(D&gt;t)$. Actually, since we are after a lower bound on  $E[D]$ it is enought to give a lower bound on $P(D&gt;t)$.  Let $S_i$ be the event that the pair of circles does not intersect which should be given by \begin{align} P(S_i) =  \frac{ Surf(R)-Surf(r)}{Surf(R)}= 1-\frac{  \pi r^2}{ 4\pi R^2}=1-\frac{ r^2}{ 4 R^2} \end{align}  Now the probability that euclidean minimum distance $D$ between $N$ balls is bigger than $t$ should be similar to  \begin{align} P[ D \ge t] \stackrel{?}{=}  P[ \cap S_i] \end{align} However, I think we somehow need to converte from the geodesic distance to euclidean distance but I am not very sure how to do that? What is the proper name for a "river plot" visualisation In a famous plot, Charles Minard visualised the losses of the French Army in the Russian campaign of Napoleon:    (another nice example is  xkcd plot)  Is there a canonical name for this type of visualisation? I'm actually looking for an R package to create such plots, but I don't even know how to look for it.  EDIT:  As I could not find a good package in R do do this type of plots, I have created my own, called "riverplot" -- you can download it . Here is a simplified version of the above diagram:    And an example of what other diagrams can be created with the package: Why do the $n \times n$ non-singular matrices form an "open" set? Why is the set of $n\times n$ real, non-singular matrices an  open subset of the set of all $n\times n$ real matrices? I don't quite understand what "open" means in this context.  Thank you. Securely erase hard drive using the Disk Utility I’m currently in the process of reconsidering how I do bibliography management. It seems to be most sensible to use biblatex with biber. I’d like to know what tools are available on which I can build a solution that works for me. I’m interested in everything from full-blown bibliography management suites to pretty-printers for .bib files; both as traditional programs and online services.  Tools in question should fulfill the following criteria:   Unicode support. Support for the additional entry types and fields that biblatex supports. Support for biber’s improved cross-referencing. Equation of motion of a free particle We know that the equation of motion of particle can be derived from the respective action. But in the book I am reading, the author is saying:     ... timelike worldline of a massive particle is parametrised by proper time, $x^a=x^a(\tau)$. The velocity (tangent) vector is $u^a \equiv \dot{x}^a(\tau)$. This is normalised as $u^au_a =-c^2$.   How can one derive e.o.m? Difference between staticmethod and classmethod What is the difference between a function decorated with  and one decorated with ? Why do we need both the Suffrage and Vox Populi badges? The Suffrage badge is awarded for using 30 votes in the day, and the Vox Populi for using 40.  I understand from  that the idea of Vox Populi is to incentivise voting on questions, but is there any evidence that having two badges does so better than just having Vox Populi? I have been working on a Java project for a class for a while now. It is an implementation of a linked list (here called AddressList, containing simple nodes called ListNode). The catch is that everything would have to be done with recursive algorithms. I was able to do everything fine sans one method: public AddressList reverse()      ListNode:    public class ListNode{   public String data;   public ListNode next; }   Right now my reverse function just calls a helper function that takes an argument to allow recursion.  public AddressList reverse(){   return new AddressList(this.reverse(this.head)); }   With my helper function having the signature of private ListNode reverse(ListNode current).  At the moment, I have it working iteratively using a stack, but this is not what the specification requires. I had found an algorithm in C that recursively reversed and converted it to Java code by hand, and it worked, but I had no understanding of it.  Edit: Nevermind, I figured it out in the meantime.  private AddressList reverse(ListNode current, AddressList reversedList){   if(current == null)        return reversedList;   reversedList.addToFront(current.getData());   return this.reverse(current.getNext(), reversedList); }   While I'm here, does anyone see any problems with this route? I see a lot of questions comparing full frame (e.g EF for Canon, FX for Nikon) and crop lenses (e.g. EF-S for Canon, DX for Nikon) but I can't find the answer to what would be the differences between the photos taken with say a 50mm f/1.4 EF and EF-S lenses both used with   same APS-C sensor sized camera same ISO same f/1.4 f-stop same focal length same shutter speed   What would be the resulting differences in the photos? Which one would   be brighter than the other due the EF-S lens concentrating the light more? no that's wrong. Both put the same amount of light from the same FoV on the sensor. have greater depth-of-field? neither less problems towards the edges? generally the full frame more lens flare? generally the full frame lens unless there is, or you provide more shading anything else? not really, unless you start down the road of which one is typically of better build quality.   Edit: Added the answers from what seems to be the general consensus from the multiple answers and comments.  Results: My desire to buy a fast medium-high quality lens compatible with both full frame and crop camera's doesn't have to be hampered by the APS-C sensor size of my current camera.  This means I'm willing to part with more money to get a higher performance/quality lens as the performance/quality will be useful now and when I upgrade my camera.  Though reading reviews of some full frame lenses with short focal lengths, look out, as some are really only made to perform well on crop cameras even though they fit on a full frame camera and will give you a full frame picture, the edge quality is very bad. How do I parameterize a query containing an IN clause with a variable number of arguments, like this one?  SELECT * FROM Tags  WHERE Name IN ('ruby','rails','scruffy','rubyonrails') ORDER BY Count DESC   In this query, the number of arguments could be anywhere from 1 to 5.  I would prefer not to use a dedicated stored procedure for this (or XML), but if there is some elegant way specific to , I am open to that. Word or phrase for making something seem better by comparison What is a word or phrase that expresses the idea of purposely making something seem better by comparison? For example, lemon Starburst was created to make the cherry flavor seem that much tastier.