Axiom of glueing: direct limit of sheaves in a noetherian topological space.
First axiom of sheaves: in noetherian topological spaces the direct limit presheaf is a sheaf.
Tachyons and Photons Is there a particle called a "tachyon" that can travel faster than light?  If so, would Einstein's relativity be wrong?  According to Einstein no particle can travel faster than light. Where do I ask where to ask? I would like to render an animation as a series of .png images with transparent backgrounds for use in a video. Is this possible? Which words in a title should be capitalized? Let $X$ be a set and consider the collection $\mathcal{A}(X)$ of countable or cocountable subsets of $X$, that is, $E \in \mathcal{A}(X)$ if $E$ is countable or $X-E$ is countable. If $X$ is countable, then $\mathcal{A}(X)$ coincides with the power set $\mathcal{P}(X)$ of $X$. Now suppose that $X$ is uncountable. Assuming the axiom of choice, we can conclude that $\mathcal{A}(X) \ne \mathcal{P}(X)$, since $|X| = |X| + |X|$. So the question is:     Can we prove in ZF that $\mathcal{A}(X) \ne \mathcal{P}(X)$ for every uncountable set $X$?   I'm assuming that a set $X$ is uncountable if there is no injective function $f : X \rightarrow \mathbb{N}$. I really like the 'ring chart' or 'wheel chart' given as an example on page 88 of the PGF manual. It is like a pie chart, but with a label in the centre. Unfortunately, the manual doesn't give the source for that chart. Does anyone have an example? Thanks. How do you get JavaScript/jQuery Intellisense Working in Visual Studio 2008? I thought jQuery Intellisense was supposed to be improved with SP1. I even downloaded an annotated version of jQuery 1.2.6, but intellisense will not work in a separate jscript file. I have the jQuery library referenced first on my web page in the &lt;head> tag. Am I doing anything wrong? How do I make Subversion (SVN) send email on checkins? I've always found checkin (commit) mails to be very useful for keeping track of what work other people are doing in the codebase / repository. How do I set up SVN to email a distribution list on each commit?  I'm running clients on Windows and the Apache Subversion server on Linux. The answers below for various platforms will likely be useful to other people though. $$\frac{1}{n+1}&lt; \log(1+ 1/n)$$  Any ideas? I tried estimating the difference between $1/n$ and the logarithm and comparing with $1/n-1/(n+1)$ but I miserably failed. I was trying to prove following inequality:  $$|\sin n\theta| \leq n\sin \theta \ \text{for all n=1,2,3... and } \  0&lt;\theta&lt;π $$  I succeeded in proving this via induction but I didn't get "feel" over the proof. Are there other proof for this inequality? How to implement Enums in Ruby? What's the best way to implement the enum idiom in Ruby? I'm looking for something which I can use (almost) like the Java/C# enums. What advantages does manual mode have over aperture priority mode? I'm wondering what are the most compelling reasons to switch to manual mode on my DSLR. I mostly just use aperture priority mode so I have good control over the depth of field combined with an awareness of the shutter speed that goes with it. My photography is mostly done wandering about with camera in hand, or occasionally on a tripod. I don't do studio photography and I just use the natural light.  I did learn how to take photos manually (with an Olympus OM-1) so I know how to do it, but I enjoy the extra convenience of aperture mode, particularly being able to take the photo quickly and capturing the moment rather than fiddling with dials and missing the shot.  So what would be some key advantages of using manual mode for my sort of photography that are worth the (small amount of) extra hassle involved?  And do you have any tips to minimise the downside (other than practicing to be quick)? $x^{x^{x^{\dots}}}=2$, what is the value of $x$? I came up with this little simple exercise, stating:  $x^{x^{x^{\dots}}}$ infinite times is equal to $2$, find $x$.   As we're dealing with infinity, we can just separate the first $x$ and get $x^2=2 \implies x=\sqrt2$  But then I thought what if we change $2$ to $4$? Now we get $x^4=4 \implies x=\sqrt[4]4=\sqrt2$  Something went wrong, can someone explain? Why do we use the Lagrangian and Hamiltonian instead of other related functions? There are 4 main functions in mechanics. $L(q,\dot{q},t)$, $H(p,q,t)$, $K(\dot{p},\dot{q},t)$, $G(p,\dot{p},t)$. First two are Lagrangian and Hamiltonian. Second two are some kind of analogical to first two, but we don't use them. Every where they write that using them can cause problems. But why? I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these two sets?  I have also learnt that there exists no set whose cardinality is strictly between the natural numbers and the real numbers. Is there a proof of this or at least some intuitiveness behind it? What would be the quickest way to construct a Python binding to a C or C++ library?  (I am using Windows if this matters.) I recall having heard somewhere that the only 1-manifolds (second countable, Hausdorff, connected spaces locally homeomorphic to $\mathbb R$) are $\mathbb R$ and $S^1$. Is this true? If so, is there a reasonably elementary proof? Why can't the indefinite integral $\int\frac{\sin(x)}{x}\mathrm dx$ be found? I came across a list of functions in my calculus textbook whose indefinite integral cannot be found. It was written that the integral $$\int \frac{\sin(x)}{x} dx$$ cannot be evaluated without any explanation as to why.  I did some research over the internet and found out that the definite integral $$\int_{0}^{\infty} \frac{\sin(x)}{x} dx$$ can be evaluated using Laplace transformation and is equal to $\pi /2$. But I still couldn't find answer to my original question. I read somewhere that the integral cannot be expressed using 'elementary functions'. A little help is appreciated, I'm in Calc 1 going advanced than my course but I am sorry if my post shows lack of research. Thank you! How are Star Wars blasters properly reloaded/charged/etc.? I've been looking into the inner workings of blaster weapons primarily using Wookieepedia. While most of it I understand, I still can't figure out how many rounds a gun would fire before needing the ammunition component replaced/reloaded. Is there any solid answer?  I'm someone who likes to design and draw his own weapons and such, but likes to keep to how the technology of an era is in mind. While Star Wars weapons don't really have magazines (and you don't really see them reloading during battle), other people's work still has magazines and treat ammunition consumption as though it's pretty much the same as our modern era.  I guess that I just find it hard to believe, with the technology of the Star Wars Universe, that characters would need to carry lots of ammunition (hence why clones/storm troopers/battle droids don't visually seem to carry very much).  Example: DC-15A     "The weapon's tibanna gas cartridge carried enough gas for up to 500 shots, depending on the power settings of the weapon, while the charge pack lasted about 50 shots"   Source:   I've read in several places that this can be seen in two different ways.  1. Which ever ran out first, had to be replaced before the weapon could fire again. 2. The weapon contained two different sources of ammunition (so I guess you could switch between the two) Does the faction of your armor/clothes affect character interactions in Skryim?