The proof to both of these relies on INDUCTION.
The basic idea behind induction is the following:
1) Prove that a statement P(n) is true for n=1.
2) Prove that if P(n) is true, then P(n+1) is true, so long as n is at least 1.
If we can prove the above two statements, then we have...
Let's compile a list of theorems we think every mathematician ought to know!
I'll start:
Stoke's Theorem: If M is a smooth n-dimensional manifold, and \omega is a compactly supported (n-1) form on M, then \int_{M} d\omega = \int_{\partial M} \omega
I'm currently registered for:
Complex Variables II
Differential Geometry II
Basic Probability
Functional Analysis
Philosophy of Mind
Advanced Logic
I'm actually really excited for all my courses this semester!
It's going to be so much work, though. I'll have to be sure to devote...
You know, I agree with you that take-home finals better test your ability than in-class finals.
I have a much better understanding of real analysis than I do of differential geometry. Differential Geometry had a take-home final, and I got an A. Real Variables had an in-class final, and I got a...
OK, I'll give you a few questions, and let's see if you can answer it. If you can't answer it without looking at a book, I'll just tell you the answer, and ask you a similar question.
Question 1: Is it possible for |x| to be less than 1/n for any natural number n, but for x to not equal zero...
The probability is actually (3/5)^10, which is about 0.006, which is 0.6%
3/5 is the probability that any given ticket will NOT be given on a Monday or a Friday, and there are 10 independent tickets being given.
The probability of something just as surprising happening is actually 10 times...
If P(A) is small, then the probability of [receiving 10 tickets overall but 0 tickets on Monday or Friday] is small.
If the probability of not receiving any tickets on Monday or Friday is very low under the assumption that tickets are given out uniformly across all days, then this gives us...
How do you define "equidistant"?
Here's the definition I assume you are using:
If C is a curve, and E is an ellipse, then C is "equidistant from E" if there is some positive real number r such that for all x in C, inf{|x-y| : y in E} = r.
This is actually an interesting question.
There is a...
FAIL.
From wikipedia: http://en.wikipedia.org/wiki/Outer_measure#Formal_definitions
Defining properties of an outer measure:
* The empty set has measure 0.
* Monotonicity: If A is a subset of B, then the measure of A is at most the measure of B.
* Countable Subadditivity: The measure of a...
Because in a CRT, there is no diffraction taking place.
The narrowness of the slit through which the electrons pass through determines the wideness of the diffraction pattern. If the slit were very narrow, the electrons would leave a very wide diffraction pattern.
The slit itself is what...
Ah, I read the paper, and the Glan-Thompson prism is actually just there to separate the entangled photons exiting the BBO. The BBO creates entagled photons with different polarizations (but not in different directions), and then the Glan-Thompson prism directs the two entangled photons in...
No, that text you are quoting is referring to the prism PS, not the Glan-Thompson prism. The Glan-Thompson prism somehow affects polarization, and I'm trying to figure out why it's necessary (it's not mentioned in the wikipedia description).
Yes, but the light is coherent to begin with! The...
Wait a second, how can this be true?
After "PS", the "red" photons and "blue" photons are traveling on entirely different paths. Detectors D3 and D4 could only possibly detect photons that traveled through the "blue" and "red" slits, respectively.
I don't see where the phase information...