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Feb
11
comment Difference between Real Analysis and Probability Theory?
Nice little glitch in the SE technology: I have been able to upvote this answer twice--once on CV and once again here :-).
Feb
11
comment Difference between Real Analysis and Probability Theory?
+1 This answer gets to the heart of the matter. Analysis and probability have different interests: although they use similar sets of tools, they ask different questions and pursue almost completely different avenues of investigation. The two disciplines will nevertheless remain closely intertwined because insights from one can lead to progress in the other, much as (say) investigations of general relativity and quantum mechanics in physics have inspired advances in low-dimensional topology and operator theory, respectively.
Jan
30
comment What's a proof that the angles of a triangle add up to 180°?
@JoeZ. Or $-2\pi$: the complex arithmetic keeps track of angle orientation, too.
Dec
29
comment Interesting and unexpected applications of $\pi$
This is David H's answer in disguise: the ARE is algebraically related to (a) $e^0=1$ and (b) $\int_\mathbb{R}e^{-x^2}dx=\Gamma(1/2)=\sqrt{\pi}$. The appearance of $\pi$ is due to the duplication formula for the Gamma function, $\Gamma(z)\Gamma(1-z)=\pi\csc(\pi z)$, exhibiting $\Gamma$ as a kind of "square root" of a trig function. This is because $\Gamma(z)=\Gamma(z+1)/z$ implies $\Gamma$ has (simple) poles at $0,-1,-2,\ldots$, whence $\Gamma(z)\Gamma(1-z)$ has poles at $\mathbb Z$, strongly suggesting periodicity--which is why $\pi$ should show up!
Dec
28
revised Prove without induction $2^n \mid (b+\sqrt{b^2-4c})^n + (b-\sqrt{b^2-4c})^n $
Appended some more discussion.
Dec
28
revised Prove without induction $2^n \mid (b+\sqrt{b^2-4c})^n + (b-\sqrt{b^2-4c})^n $
[Edit removed during grace period]
Dec
28
revised Prove without induction $2^n \mid (b+\sqrt{b^2-4c})^n + (b-\sqrt{b^2-4c})^n $
deleted 4 characters in body
Dec
28
comment Prove without induction $2^n \mid (b+\sqrt{b^2-4c})^n + (b-\sqrt{b^2-4c})^n $
@Thank you for noticing that, Thomas. That is easily fixed by replacing one of the $c$'s by $1$; I will make the change.
Dec
24
comment Probability that there is sub-sequence of exact length
There is a lot of literature on this subject. The asymptotic distribution is known. I don't believe exact formulas for the case of finite $N$ are available.
Dec
24
revised Urn problem with balls
added 12 characters in body
Dec
24
comment Urn problem with balls
@Brian Thank you for that observation. It's very much in the spirit of this answer, which is an attempt to find as simple a solution as possible.
Dec
24
revised Urn problem with balls
added 2 characters in body
Dec
24
comment The point of contact of between two circles and common tangent at this point.
+1 for the simplified elementary approach. But note that the solution is incomplete: it has not (yet) addressed whether the circles are internally or externally tangent.
Dec
24
comment Urn problem with balls
It might be even simpler just to count the ways of choosing one ball in one urn and the other in the other: $(n+1)^2$. Divided by $\binom{2n+2}{2}$, this gives the answer directly.
Dec
24
answered Urn problem with balls
Dec
24
answered Game to maintain distinct number of balls in glasses
Dec
24
comment Prove without induction $2^n \mid (b+\sqrt{b^2-4c})^n + (b-\sqrt{b^2-4c})^n $
@Brian You're right; there are many inductions lurking here. The one you point to comes down to asserting that the integers are closed under addition and multiplication. I'll accept that as immediately obvious. The crux of the matter, though, is that scalars commute with matrices, whence $(2\mathbb P)^n = 2^n \mathbb P^n$: that shows where the powers of $2$ come from.
Dec
24
answered Prove without induction $2^n \mid (b+\sqrt{b^2-4c})^n + (b-\sqrt{b^2-4c})^n $
Dec
18
reviewed Approve Proof of the description of a set
Dec
15
comment Show that $\lim_{n\rightarrow\infty}{\displaystyle\sum_{i=1}^{n}{\frac{F_n}{2^n}}}=2$
The limit is $0$, not $2$. Use $(1 +\sqrt{5})/2$ instead of $2$ in the denominator to obtain a finite limit.