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Consultant (environmental stats a specialty) and teacher.


Aug
29
comment A naive approach for nonlinear optimization on several real variables and one natural variable. Give examples of (potential) failure
It appears that this is an integral program, not a real program. The optimum of a function of a set of integers is being found by obtaining it over a real extension and then rounding to nearby integral coordinates. The question concerns why this might be a bad approach.
Aug
29
answered A naive approach for nonlinear optimization on several real variables and one natural variable. Give examples of (potential) failure
Aug
29
comment Intuition for the definition of the Gamma function?
@Qiaochu: The final paragraph was put there precisely to avoid being accused of using a trick! ;-) The reference to statistics is that the study of variances leads one to look at the distributions of sums of squares of Gaussian variables (called chi-squared distributions). Obtaining volumes of hyperspheres drops out almost by accident as a byproduct of this calculation.
Aug
29
comment Intuition for the definition of the Gamma function?
@Qiaochu: The method of stationary phase (Tao's notes), btw, is motivated by the central limit theorem. The gamma distribution, as you now know from other comments and another answer, is a sum of independent exponentials. The CLT implies it should be close to Gaussian for large n. This means the integrand should be well approximated by a parabolic fit to its logarithm. A good choice is the Taylor series around the maximum. This lets us approximate the integral by the Normal integral (equivalently, erf) with parameters depending on n, whence we obtain asymptotic formulae.
Aug
29
comment Intuition for the definition of the Gamma function?
The first method is motivated by the hope of finding integrable real functions f and g for which f(sqrt(x^2 + y^2 + ... + z^2)) = g(x)g(y)...g(z): the lhs will be the integrand in polar coordinates and the rhs is the factorization that makes everything work. Assuming f differentiable, equating partial derivatives on both sides shows that f'(r)/(r f(r)) must be a constant, whence f(r) = A*exp(b*r^2). We need b < 0 for integrability. This determines the form of g; a simple choice is g(x)=exp(-x^2). (I didn't go into this originally because the first method is now standard and well documented.)
Aug
29
comment How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?
@Qiaochu: How does one stop the animation (without blocking all images from downloading)? It can be distracting...
Aug
29
comment How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?
@Moron: the principal part of Qiaochu's answer--namely, that Euler's formula can be understood as the exponential map from R to SO(2)--is a modern way of expressing the elementary construction described in my answer. An advantage of the modern approach is its far reach, as hinted at by the application to SU(20; a possible disadvantage is that to the uninitiated it appears to obscure the basic geometric idea. But there's no "circular logic."
Aug
29
comment How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?
Thank you for the graphic!
Aug
29
answered How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?
Aug
28
awarded  Editor
Aug
28
revised Intuition for the definition of the Gamma function?
Added a summary paragraph.
Aug
28
answered Intuition for the definition of the Gamma function?
Aug
28
comment Intuition for the definition of the Gamma function?
Thanks for writing this up. I have been musing over the same ideas, so I think I can make the final connection with the symmetric group: recall how the Poisson distribution is computed as a limit of Binomials. In tracing through the derivation, you see that the 1/k! terms come from Binomial coefficients; those, originally, count ways of selecting subsets from a set. The denominator (k!) counts the ways of permuting elements in a k-element subset. This completes the train of thought: counting subsets --> symmetric group --> Binomial distribution --> Poisson --> Gamma.
Aug
27
comment (PQ)²=RSP, where P,Q,R,S are distinct single digit natural numbers, then R=?
Nice conjecture. But when M > 81 you can check that 93^2 = (9M+3)^2 = 81M^2 + 54M + 9 = (81)(54)9; i.e., (M,P,Q,R,S) = (M,9,3,81,54) is a solution for all M >= 82.
Aug
27
comment Intuition for the definition of the Gamma function?
I understand. Another train of thought: the gamma function shows up naturally in computing volumes of hyperspheres, whence one obtains that the volume of the 2n-dimensional unit ball is Pi^n/n!. Interestingly, Pi^n is the volume of the product of n unit disks (a solid torus), on which the symmetric group naturally acts by permuting coordinates. So: is there some natural geometric way to show that the volume of a fundamental domain of this action must equal the volume of the unit ball?
Aug
27
comment Which one is bigger $2^{n!}$ or $(2^{n})!$?
This appears to be a round-about way of reproducing yjj's earlier demonstration.
Aug
27
comment Which one is bigger $2^{n!}$ or $(2^{n})!$?
You seem to have independently reproduced yjj's earlier response.
Aug
27
comment Intuition for the definition of the Gamma function?
It would help to understand better what kind of "intuition" you are looking for. For example, it's a standard Calculus exercise in integration by parts to show that this sequence of integrals satisfies the recursion, so for some people that would be good enough. I take it you're looking for a deeper connection with the Symmetric groups, but (i) why should there be one that goes beyond the mere counting of their elements and (ii) what would constitute a deeper connection for you?
Aug
27
answered Intuition for the definition of the Gamma function?
Aug
27
comment Which one is bigger $2^{n!}$ or $(2^{n})!$?
Very nice! The "very simple effort" establishes the inequality for n >= 6. The cases n = 0, ..., 5 are quick checks (and for some of them the inequality is reversed, as Tobias Kienzler notes). That completely addresses the problem.