Does the positron prove that Dirac's electron sea must exist?
What was missing in Dirac's argument to come up with the modern interpretation of the positron?
How to show the solution of $\dot{x} = Ax $ is an invariant subspace? Consider the linear dynamical system $\dot{x} = Ax $ in $V$ a finite dimensional vector space.  The definition of an invariant subspace $U$ is as follows:  For all $x_0 \in U$, (initial condition), the solution $x(t)$ with initial condition $x(0)=x_0$ lies in $U$, i.e. $x(t) \in U$ for all $t \in \mathbb{R}$. Intuitively, it means if we start in a subspace we never leave that subspace.  Prove that a subspace $U \subset V$ is an invariant subspace if and only if $AU \subset U$. I was just wondering why Fresnel Lenses are not widely used in the production of solar electricity. Their use there would mean that you could produce heat within a fraction of a second, up to a few minutes and run a turbine to produce electricity.  Though it is used in welding, I am not sure what are the problems in producing electricity, as stated by :     New applications have appeared in solar energy, where Fresnel lenses can concentrate sunlight (with a ratio of almost 500:1) onto solar cells. Roman numerals with over- and underline This question led to a new package:      (using a variation of egreg's code, and stating this in the manual; the package isn't really new now, sorry for the delayed announcement)  I have read about "How to get Roman numerals" but asked myself how to get Roman numerals formatted as in     i.e. with fitted over- and underline (without manually using boxes and such)? I'm curious for your answer(s)! What happens to spirits/souls of other creatues than Elves and Men after their death? What are the entire functions $f$ such that $|f'(z)| \leq |f(z)| \ ? $ Could someone advise me on how to determine all entire functions $f$ such that $|f'(z)| \leq |f(z)|, \forall z\ ?$  Hints will suffice, thank you. I am considering buying my first DSLR.  How do I know whether I need a 18-200 lens, 17-85, or 18-55? Or something else? Print total free memory from `free -m` If the original question is "What time is it?" and I want to communicate it to another person, are these Embedded Questions OK?        I don't know what time is it.       or        I didn't know what time was it. (if communicated later) How do I disable "you need to format this disk" message in Windows Vista/7? I've started using TrueCrypt to encrypt a partition on an external USB hard drive. I've opted for using a raw partition as opposed to a file container.  But here's the annoyance. Whenever I connect this drive to a computer running Vista or Windows 7, I get the message popup "You need to format the disk in drive J: before you can use it".   The default option is "Format disk", and I'm afraid one of these days I'll accidentally trigger it.  Is there any way to disable this message? Non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i,$ for two consecutive integers. Given an example of a non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i, \forall a, b \in G$ for two consecutive integers.   This is question 5 from Herstein Page 35. I have proved that conclusion holds for three positive integers.Thanks. Ok, so at my school, everybody was supplied with a Macbook Pro. I set up the Messages app and it was all great until the county decided to block it. Now, instead of receiving text messages on my phone they go to my Mac. I have no idea how to fix this because I can't open the Messages app. Please help?! Why avoid increment ("++") and decrement ("--") operators in JavaScript? Calculating the number of possible paths through some squares I'm prepping for the GRE.  Would appreciate if someone could explain the right way to solve this problem. It seems simple to me but the site where I found this problem says I'm wrong but doesn't explain their answer. So here is the problem verbatim:    Find the number of paths from x to y moving only right (R) or down (D).  My answer is 6. What am I missing??  Thanks for any help. What potions actually grant fame or glory, or put a stopper in death? Are there any? ‘You are here to learn the subtle science and exact art of potionmaking,’ [Snape] began. He spoke in barely more than a whisper, but they caught every word – like Professor McGonagall, Snape had the gift of keeping a class silent without effort. ‘As there is little foolish wand-waving here, many of you will hardly believe this is magic. I don’t expect you will really understand the beauty of the softly simmering cauldron with its shimmering fumes, the delicate power of liquids that creep through human veins, bewitching the mind, ensnaring the senses ... I can teach you how to bottle fame, brew glory, even stopper death – if you aren’t as big a bunch of dunderheads as I usually have to teach.’&#185;   Philosopher's Stone - page 102 - Bloomsbury - chapter 8, The Potions Master   Are there any canon potions that actually grant fame or glory, or stopper death? I'm looking for specific canon-based answers from sources such as the Harry Potter books, interviews with J.K. Rowling, or information from Pottermore. Maybe because it's early I'm overlooking the obvious, but offhand I can't think of any canon potions that fit this criteria exactly.   &#185;I edited in the entire paragraph that I can teach you how to bottle fame, brew glory, even stopper death comes from for clarity. Hopefully this will be helpful and provide a better context for the comment in question. I was running out of space for Windows 10, so I deleted my Ubuntu partition and then combined the free space and added it to the Windows 10 partition into a single partition (the boot partition was untouched). Now all I get when I turn on my computer is:  Error no such partition. &gt;grub rescue   I have tried rebooting, but it did not work. Is there any way to fix it? My idea is that if i reinstalled Ubuntu it may fix it, but I'm not sure. I don't have a Windows repair disk, and I can't make one, but I do have a 4GB USB flash drive. Any way to tell apart a CPU defect from a mainboard defect? What's the difference between Normalization and Standardization? Once it is compiled, is there a difference between:  delegate { x = 0; }   and  () =&gt; { x = 0 }   ? I use this program in Windows that's one-box and searches all of my drives. By default it lists all files, everywhere, in a single list. Is there a Linux graphical utility that does this? Showing diffeomorphism between $S^1 \subset \mathbb{R}^2$ and $\mathbb{RP}^1$ I am trying to construct a diffeomorphism between $S^1 = \{x^2 + y^2 = 1; x,y \in \mathbb{R}\}$ with subspace topology and $\mathbb{R P}^1 = \{[x,y]: x,y \in \mathbb{R}; x \vee y \not = 0  \}$ with quotient topology and I am a little stuck.  I have shown that both are smooth manifolds, and I used stereographic projection for $S^1$, but now I am runing into trouble when I give the homeomorphism between $S^1$ and $\mathbb{RP}^1$ as the map that takes a line in $\mathbb{RP}^1$ to the point in $S^1$ that you get when letting the parallel line go through the respective pole used in the stereographic projection.   If I use the south and north poles I get a potential homeomorphism, but I cannot capture the horizontal line in my image, but when I pick say north and east then my map is not well defined as I get different lines for the same point in $S^1$. Can somebody give me a hint how to make this construction work, or is it better to move to a different representation of $S^1$ ?