Light at the Speed of Light
Does a photon in vacuum have a rest frame?
How much money did Harry Potter have? So, we know Harry . By descriptions from the books, it looks like he had enough money to spend during his school years with plenty to spare for later life.  Is there an estimate on how much money he had? How much would that be in muggle money? Was he extremely rich?  I'm thinking here that he could have received more than just an inheritance. For example, anonymous supporters could have given him money through the bank while he was a baby, or when he was about to go to Hogwarts, and he wouldn't have known (not even Dumbledore would have known). Automate keystrokes in Windows 7 64-bit I need to automate the pressing of the  key then pressing 5 digits then press  in Windows 7 64-bit.  I have an app that comes up on the screen and I need the "script" to press the keys.  I don't even need mouse support.  What can I use to do this?  Any examples or links? Find the value of : $\cos x \cos 2x...\cos 999x$ given that $x=\frac {2\pi}{1999}$ Given $x=\frac {2\pi}{1999}$  Find the value of $$\cos x \cos 2x \cos 3x ...\cos 999x$$  So I tried expanding $\sin {2000x}=2\sin 1000x \cos 1000x$  Then rewriting  $\cos 1000x= \cos {(999x+x)}$   No luck so far. Where can I find free elevation data for Europe?   The available resolutions and terms of use are important details that I also seek. I have been given the task to rewrite a function as a sum of an even and an odd function. But when I went to analyze the parts of the original function I noticed that one part was neither even nor odd.   The original function was: $g(x) = (x + 1) / (x^2 - 3x + 4$)  After analyzing I realized that when using f(-x) to determine symmetry it produced a 'neither' result. How would I go about writing a function that is neither even nor odd as an even function. (It would have to be an even function as the (x+1) part of the original function is odd and since I need to write it as a sum of even and odd, the denominator would need to be even. When and how did "momentarily" come to mean "in a moment", rather than "for a moment"? "Momentarily" used to mean "for a moment" only, and not "in a moment". Thus, newscasters could be divided into two clear groups: those who would say "we'll be back momentarily," and those who would not.  This restriction made sense to me, because having both definitions would promote ambiguity if a unique interpretation could not always be derived from the context. But in recent years it seems "momentarily" is regularly, maybe even more often, used to mean "in a moment" by newscasters of every caliber, and in fact this is even shown to be the definition when looked up in most dictionaries.  When did this word's meaning change? How did it come about? How can I use the instant messenger program Kik - most commonly installed on smartphones and other mobile devices - on a laptop or desktop running Win7Pro, without a phone being involved and without needing any kind of Google account?  The two ways of using Kik from my laptop that I have so far found are both inadequate, or at least I do not how to tweak them to make them meet the requirements.  The first is to install on my laptop the Bluestacks emulator of Google's Android operating system, and then to run Kik on it. Unfortunately to run Kik on Bluestacks requires (or at least I have not found a way to avoid) having a Google Play account.  The second is to run Bluestacks on Manymo, the online emulator of Google's Android OS. Unfortunately, using Kik on Manymo similarly requires a Google Play account.  I realise Kik was made for use on a mobile OS, but it is supposed to be cross-platform where mobile OSes are concerned, and therefore not to be tied into Google's mobile OS - or, as far as I know, into any other software supplied by Google or any procedure prescribed by that company, such as giving it personal information - so I am optimistic that someone will be able to advise on what I am asking for here.  I notice that Microsoft have made a , so there might be a way to get Kik to work on this without dealing with Google. I couldn't find anything online about this, though. Can anyone help me identify a story about kids with silver blood? Proving a relation between inradius ,circumradius and exradii in a triangle Prove that in a triangle $$r^2+r_1^2+r_2^2+r_3^2=16R^2-(a^2+b^2+c^2)$$ where the symbols have their usual meanings. I am looking for a smaller or elegant proof using trigonometry. A geometric proof would be super cool. I did prove it but it took me ages to do the calculation. Thanks. What does **Ens** stand for? Earlier someone was asking about the category "Ens" described in Categories for the Working Mathematician. My question is more basic: What does Ens stand for?  Most of the categories have names that are self explanatory, and maybe this one does too.  But for the life of me I can't think of it, and a Google search was no help. Under TeX, when I write a line too long, it prints a fat black box at the end of the line.  That is probably a feature, but I dislike it.  How to turn that OFF? Is there a command-line option or something else?   I usually use LaTeX, but I have a complicated file which happens to be plain TeX. What USB drive filesystem works out of the box on Windows, Linux, and OS X? Well, I'm trying to find the Thevenin equivalent of this circuit:    As you can see, the only current source is a dependent source. However, the control variable is in a wire where, if nothing is connected between A and B, is \$i_\Delta = 0\$ A, and that makes the source 0 A. However, if I connect an impedance \$Z\$ to the circuit, I will have current, and a Thevenin equivalent. I don't know how to solve this problem. I've bee searching here for a while, and I found this:   . I can see how is solved that case. However, I still don't know how to apply that method to my circuit.  Any help about finding the Thevenin equivalent is appreciated n.n Present or Past tense to describe a past condition which is still present? Scenario: My friend John has a personality disorder, intensely paranoid for example.  Can I say:     There were three events last year that told me John had a problem.   or can I say:     There were threee events last year that told me John has a problem. How do I show that if there are functions $f,g$ such that$$ f(g(x)+g(y))=bx+cy $$holds for all $b,c\in\mathbb{R}$, then we necessarily have $b=c$? Is this even true? It seems so, but I'm just not sure how to explain it in detail and using theory from functional equations. Any ideas?  Extra: In addition, I'd like to prove that there are no pair of functions $f,g$ such that$$ y^2+z^2=f(x,g(y-x)+g(z-x)). $$ Any ideas on how to prove this? Plus, does anyone know some literature regarding this type of problems? how to bring up the menu at the bottom left hello everyone I have a little problem, how do I bring up the menu in the circle? when I select another object the menu becomes non-existent What determines speciation at a molecular level?  At what point does a scientist determine two lineages are different enough to be considered separate species?  Does it have a margin of error? How to prepare files for rsync on a case insensitive filesystem? I am transferring a large number of files on a HFS+ filesystem.  The files are currently on ext2 partitions.  I have conflicts due to case insensitivity of the target partition (HFS+).  I want to identify the files that have duplicates filenames once they are in lower case, and delete them if they are actually duplicates.  I also found that I will have duplicate folder names if I convert everyhing to lower case. Basically these hard drives contain years of unsorted data, and I happen to have this problem with folder names too.   Does this seem reasonable:  find . -type f | while read f; do echo $f:l; done | sort | uniq -d    $f:l is ZSH for convert to lower case.  Now I want to keep only one instance of each file that have duplicates. How to do this efficiently ?  I do not want to find files with duplicate content, unless they have the same lower case filename. I will deal with duplicates later. How can I find the possible values that $\gcd(a+b,a^2+b^2)$ can take, if $\gcd(a,b)=1$ I have come across numerous questions where I am given the picture such as the above one been asked "how many rectangles are there?". I have even come across some slightly different images that instead of rectangles you are supposed to find the number of triangles. Well, I was thinking whether there is any formula or strategy that is used to solve these problems without having to manually count every shape.  Help would be appreciated.  Thank you :)