How does Einstein got  equivalent mass- energy equation E=mc2 ?
Most people know the famous equation:    What were his steps of thinking for this equation that helped us discover so much about our world?
How do I make exponential equation with points?  Here are my points:  x = 0, y = 0 x = 1, y = 4.75000 x = 2, y = 9.26250 x = 3, y = 13.54937 x = 4, y = 17.62191   this is same as   x = 0, y = 0 x = 1, y = (0 + 5) * 0.95  x = 2, y = ((5 * 0.95) + 5) * 0.95 x = 3, y = (((5 * 0.95) + 5) * 0.95 + 5)) * 0.95 x = 4, y = ((((5 * 0.95) + 5) * 0.95 + 5)) * 0.95 + 5) ) 0.95   as x value increase by 1, y value is added by 5 and multiplied by 0.95 i tried to make an equation for this but i couldn't  also i tried to use natural exponential but the initial value is 0 so it didn't work  please help me Do you need to take the Hide action when you are Invisible, make no noise and leave no traces? The Situation: Lets say you are under the effect of Invisibility.  An invisible creature is impossible to see without the aid of magic or a special sense. For the purpose of hiding, the creature is heavily obscured. The creature's location can be detected by any noise it makes or any tracks it leaves.  And you are wearing the Boots of Elvenkind  While you wear these boots, your steps make no sound, regardless of the surface you are moving across. You also have advantage on Dexterity (Stealth) checks that rely on moving silently.  Now your groups ranger or druid casts Pass Without Trace on you.  [...] can’t be tracked except by magical means. A creature that receives this bonus leaves behind no tracks or other traces of its passage.  My Question(s): I know that when you are Invisible and you did not hide, you are still detectable because of the noise and the track you leave behind. But the Boots of Elvenkind and Pass Without Trace take care of that. So:  Are you still required to take the hide action in this situation if you don't want enemies to know your location/presence? If so, why? What gives away your location? If not and you are under the effect of greater Invisibility do you get the benefits of &quot;Unseen Attackers and Targets&quot; for every attack you make when you move between attacks? (You would give away your location, but are not detectable, so completely hidden again when you move) How to make sure an application keeps running on Linux I'm trying to ensure a script remains running on a development server. It collates stats and provides a web service so it's supposed to persist, yet a few times a day, it dies off for unknown reasons. When we notice we just launch it again, but it's a pain in the rear and some users don't have permission (or the knowhow) to launch it up.  The programmer in me wants to spend a few hours getting to the bottom of the problem but the busy person in me thinks there must be an easy way to detect if an app is not running, and launch it again.  I know I could cron-script ps through grep:  ps -A | grep appname   But again, that's another hour of my life wasted on doing something that must already exist... Is there not a pre-made app that I can pass an executable (optionally with arguments) and that will keep a process running indefinitely?  In case it makes any difference, it's Ubuntu. Trying to show that $\ln(x) = \lim_{n\to\infty} n(x^{1/n} -1)$ How can you test what color different people perceive? Why doesn't Java allow generic subclasses of Throwable? According to the , 3rd edition:        I wish to understand why this decision has been made. What's wrong with generic exceptions?  (As far as I know, generics are simply compile-time syntactic sugar, and they will be translated to Object anyway in the .class files, so effectively declaring a generic class is as if everything in it was an Object. Please correct me if I'm wrong.) Existence of a group isomorphism between $(\mathbb K,+)$ and $(\mathbb K^\times,\cdot)$ Let $(\mathbb K,+,\cdot)$ be a field. Is there a group isomorphism between $(\mathbb K,+)$ and $(\mathbb K^\times,\cdot) $ ?  The answer should clearly be negative. I tried to proceed via contradiction, but this has not led me very far. Since it's some kind of &quot;trick problem&quot;, I'm merely looking for hints. I was wondering why binary crossentropy can be used as the loss function in autoencoders trained on (normalized) images, e.g.  or ? I know that binary crossentropy can be used in binray classification problems where the ground-truth labels (i.e. $y$) are either 0 or 1 and therefore when predictions (i.e. $p$) are correct, in both cases, the loss value would be zero:  $$ BCE(y,p) = -y.log(p) - (1-y).\log{(1-p)} $$  $$ BCE(0,0) = 0, BCE(1,1) = 0 $$  However, binary crossentropy does not have a value of zero when neither of its arguments are both zero or one, which is the case for an autoencoder with ground-truth labels in range $[0,1]$ (i.e. assuming the input data has been normalized in this range). I thought a regression loss function such as mean squared error or mean absolute error must be used instead, which have a value of zero when labels and predictions are the same. What am I missing here? I am studying General Relativity and some basic QFTs. It bothers me a lot that different books use different metric signatures, i.e. $(-+++)$ and $(+---).$ Can anyone tell me the advantages of using both of them, or why some people prefer the former while some prefer the latter? I would like to know which one should I while tackling different problems. Does mathematics become circular at the bottom? What is at the bottom of mathematics? I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign), functions and relations.  Now is the "=" taken as undefined? I have seen it been defined in terms of the identity relation.  But in order to talk about functions and relations you need set theory.  However,  seems to be a part of mathematical logic.  Does this mean that (naive) set theory comes before sentential and predicate logic? Is (naive)set-theory at the absolute bottom, where we can define relations and functions and the eqality relation. And then comes sentential logic, and then predicate logic?  I am a little confused because when I took an introductory course, we had a little logic before set-theory. But now I see in another book on introduction to proofs that set-theory is in a chapter before logic. So what is at the bottom/start of mathematics, logic or set theory?, or is it circular at the bottom?  Can this be how it is at the bottom?  naive set-theory $\rightarrow$ sentential logic $\rightarrow $ predicate logic $\rightarrow$ axiomatic set-theory(ZFC) $\rightarrow$ mathematics  (But the problem with this explanation is that it seems that some naive-set theory proofs use logic...)  (The arrows are of course not "logical" arrows.)  simple explanation of the problem:  a book on logic uses at the start: functions, relations, sets, ordered pairs, "="  a book on set theory uses at the start: logical deductions like this: "$B \subseteq A$", means every element in B is in A, so if $C \subseteq B, B \subseteq A$, a proof can be "since every element in C is in B, and every element in B is in A, every element of C is in A: $C \subseteq A$". But this is first order logic? ($(c \rightarrow b \wedge b \rightarrow a)\rightarrow (c\rightarrow a)$).  Hence, both started from each other? Finitely generated projective module Hi is it possible to get images from go daddy  to visual force pages.if it possible can you explain clearly how the process is. How to find the shortest path between opposite vertices of a cube, traveling on its surface? I am stuck with the following problem that says:      Let $A,B$ be the ends of the longest diagonal of the unit cube . The length of    the shortest path from $A$ to $B$ along the surface is :     $\sqrt{3}\,\,$  2.$\,\,1+\sqrt{2}\,\,$  3.$\,\,\sqrt{5}\,\,$ 4.$\,\,3$    My Try:   So, the length of the longest diagonal $AB=\sqrt{3}$. If I reach from $A$ to $B$ along the surface line $AC+CD+BD$, then it gives $3$ units. But the answer is given to be option 3.   Can someone explain? Thanks in advance for your time. Why doesn't an Environment texture appear when the world has a Volume Scatter shader? I'm working on lighting a scene in which I need to have both an Environment texture for the background and Volume Scatter for light cones, but when I check the volume section in World setting and choose the Volume Scatter, the environment becomes black background (but the lighting works as I want).  I'm using Blender 2.72  How can this be fixed? How can I find the difference between two angles? Why does bash say No such file or directory when the file is on the path Problem 20-9 in Lee's Introduction to Smooth Manifolds (2nd Ed.) asks to prove that for a Lie group $G$, and for $X, Y\in \text{Lie}(G)=\mathfrak g$, there is $\epsilon&gt;0$ and a smooth map $Z:(-\epsilon, \epsilon)\to \mathfrak g$ such that   $$(\exp tX)(\exp tY) = \exp\left(t(X+Y) + \frac{1}{2}t^2[X, Y] + t^3 Z(t)\right)$$   What we want to show is that the first and second term in the Taylor expansion of $\exp^{-1}\left((\exp tX)(\exp tY)\right)$ are $t(X+Y)$ and $\frac{1}{2}t^2[X, Y]$ respectively.  Calculating the first term is easy since we just need to differentiate $\exp^{-1}((\exp tX)(\exp tY))$ with respect to $t$ at $t=0$ once, and this gives the desired result.  But for the second term we need to take the second derivative at which I am stuck. What if the universe is rotating as a whole? Suppose in the milliseconds after the big bang the cosmic egg had aquired some large angular momentum. As it expanded, keeping the momentum constant (not external forces) the rate of rotation would have slowed down, but it would never reach zero.  What implications would that have to measurments to distant supernova and CMB radiation? Do we have any experimental data that definitely rules out such as scenario? And to what confidence level?  Edit A  suggests that the universe might indeed be spinning as a whole. Anyone care to poke holes at it?  Edit 2 Even more  places limits on possible rotation. "No such file or directory" when trying to remove a file, but the file exists? Why do we use microwaves in microwave oven? We know that any object above absolute zero emits electromagnetic radiation. So hotter the object shorter the wavelengths. In the electromagnetic radiation spectrum radio waves has the longest then microwave, then infrared, then visible light, then ultraviolet, then x-ray, then gamma rays. Why do we use microwaves in microwave oven when infrared and visible light are much hotter and how do microwaves cook food when they are cooler than visible light and others.