Ok so please excuse me if the following mental argument is completely ridiculous or obviously flawed :P  I was reading about how, even at 0 K (assuming we could experimentally reach such a temperature), there must be some slight particulate motion, because of the uncertainty principle. What about if we have some particles in a closed box that is in a dark vacuum (assuming a complete absence of any energy or light). If we then proceed to cool the particles to 0 K, then wouldn't we have succeeded at violating the uncertainty principle? We know that the speed of the particles is exactly 0 m/s, and we know with 100% certainty that they are in the box; we do not need to observe them to know so.
I got to read Volume I of Feynmann's lectures. It said that at absolute zero, molecular motion doesn't cease at all, because if that happens, we will be able to make precise determination of position and momentum of the atom. We do know that Heisenberg uncertainty principle holds for microscopic particles in motion. But what then is wrong to consider that all molecular motion ceases at absolute zero? In other words, does the uncertainty principle not hold when there is no motion?  Need some help!
Accidental chown under / as root Given two positive integers $a$ and $b$, prove that  $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$  I think there should be a good combinatorial proof for this, given the simplicity of the right-hand side... my hunch is to form all possible subsets of size $a+b+1$ by dividing the set into $k$ and $n-k$ elements, then choosing $a$ from the first, $b$ from the second, iterating over all values of $k$.  I can't quite make this work, so I've also been trying to rewrite ${n+1 \choose a+b+1}$ to make it clear.  Open to any proof method, all help appreciated. Elementary central binomial coefficient estimates Why doesn't my "Nikkor AF" lens autofocus on my Nikon D3300? Angle between 2 points Given the following image:   $\hskip{1.5 in}$   Supposing $A(100, 300)$ and $B(300, 100)$, how can I find the angle $\alpha$ between A and B?  On a side note, what's the main difference between a point and a vector? translating a point to a vector is as simple as Point = Vector? Sometimes I find articles where the terms are interchangable Novel about alien conquerers who conduct games to select servants I read this maybe 10-15 years ago.   Aliens have conquered earth. Possibly quite some time before the time then novel is set in. They have built their own cities. They conduct periodic athletic competitions for humans similar to the current olympics. The winners are take into the city to act as servants to the masters. The protagonist is part of a human resistance movement who is competing in the games. The plan is to infiltrate the city, gather intel and escape. He wins, get in, finally has to kill his master and escape down a waste disposal system which ends up outside the city.   Anybody knows which one I am talking about? Creating Workflow -> Error How do I select all the columns in a table that only contain NULL values for all the rows?  I'm using MS SQL Server 2005.  I'm trying to find out which columns are not used in the table so I can delete them. Why does lightning emit light? What exactly is causing the electric discharge coming from the clouds to emit light while traveling through the air. I've read and thought about it a little but with my current knowledge I cant really figure what is happening at a subatomic level. In other words, what causes the emission of photons during lightning. What I want to know is, is there some sort of collisions taking place that produces the photons we see, or is it some other interaction that is causing it?. I don't know too much about particle physics, I've only read a couple of pages in the Introduction to Elementary Particles by Griffihts. So if you could keep the answer simple then that would be great. I have a valid B1/B2 US visa expiring about a year from now. The passport it's in expired a few years ago, and I recently got a new one.   The passport number does not match, but the national ID number, mentioned in both, does (it used to be the same as your national ID, but not anymore)  Can I travel to the US carrying both passports? Or is it asking for trouble? (I read a few years ago that it was possible, but that might have changed)  Should I just get a new visa to avoid potential complications? Proof of $(a+b)^{n+1}$ I have to do proof of $(a+b)^n$ and $(a+b)^{n+1}$ with mathematical induction.  I finished the first one $(a+b)^n = a^n + na^{n-1}b+\dots+b^n$.  I however have trouble with the second one, I don't know what to start exactly. Any help/hints? What should I do to avoid ear pain when flying? I always feel a pain in my ears when flying. Specially when descending. The last time I had extreme pain, feeling it inclusive around my eye balls. It felt like someone was touching the eye and pulling it from the inside. I really had to focus, close my eyes and hold to avoid screaming. I am not sure but I think both things might be related. Such a severe situation only happened once.  I just heard that with a cold the ear pain gets worse. I am bit worried that I might go through that situation again. I have a cold and I am going to fly soon.  What can I do to improve this?  I see that some consider this a duplicate of the question "best way to make my ears pop" but after reading the answers I think my question goes a bit beyond that since my pain is more extreme and I think it goes beyond just popping the ears. I also want to know if other methods, like using earplugs, might be effective. How would you prove that the generating function of $\binom{2n}{n}$ is $\frac{1}{\sqrt{1-4x}}$?  More precisely, prove that( for $|x|&lt;\frac{1}{4}$ ):  $$\sum^{\infty}_{n=0}x^n\binom{2n}{n}=\frac{1}{\sqrt{1-4x}}$$  Background: I was trying to solve $$S=\sum^{\infty}_{n=0}\frac{(2n+1)!}{8^n(n!)^2}=\sum^{\infty}_{n=0}\frac{(2n+1)}{8^n}\binom{2n}{n}$$ Which if we let $f(x)$ be the generating function in question, would be simply $$f(x)+2xf'(x)$$ With $x=\frac{1}{8}$. Is there a simple proof of the first identity?  states it without a proper reference ( states it without proof). Is there an easier way of calculating $S$? (which is $\sqrt{8}$, by the way) Childrens fantasy, blind monk leading boy who can control the elements How to visualize Reverse Triangle Inequality $||\mathbf{a}| − |\mathbf{b}|| ≤ |\mathbf{a} − \mathbf{b}|$? pp  and ,   illustrates Triangle Inequality. But without referring to Triangle Inequality at all, how to picture  Reverse Triangle Inequality for all $\mathbf{a,b} \in C^n$? I don't want proofs or rigour. I plugged in my external harddrive (which was formatted on my Mac into HFS+ journaled) to my Ubuntu desktop 9.04 64bit.  I am not able to get the drive to mount with write capability, how do I do that?  Right now all I'm getting is read access, I tried  sudo mount -t hfsplus /dev/sdf2 /media/"Portable HD"   but that still gave me only read access...  ideas?? PSTricks error "Undefined control sequence  \c@lor@to@ps" Do plugin files have to follow a specific convention to be "picked up" by WordPress? Fluent and Query Expression — Is there any benefit(s) of one over other? How to parse hundred html source code files in shell?