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Sep
4
comment If $(a^{n}+n ) \mid (b^{n}+n)$ for all $n$, then $ a=b$
@Chandru1: You have some imbalanced braces after "rearranging," so it's hard to see where you're going.
Sep
4
comment How to solve DE that relate values of derivatives at different points?
Differential-difference equation. (Google it.)
Sep
4
comment Is there a (deep) relationship between these various applications of the exponential function?
(4) and (5) are related to combinatorial limits and the Central Limit Theorem, whereas (1) - (3) are mutually connected in obvious ways. So the challenge comes down to finding a deep link between any of {1,2,3} and {4,5}. One possible avenue of investigation is via the "characteristic function" (Fourier transform) of the Gaussian, which is a key tool in most proofs of the CLT. Another promising route is via the Heat equation, diffusions, and random walks.
Sep
4
comment Evaluating a convergent improper triple integral over the unit sphere
Additional response to Américo's comment: Integrals like these arise in the solution of the Schrodinger equation for central forces (e.g., one-electron atoms). This is a quantum mechanical system with a large group of symmetries. (The full group is SO(4), but that's a subtle result; in any event the group SO(3) acts in the obvious way on the coordinates.) A deep way to solve and explore such highly-symmetric systems is to study the representations of the symmetry group. That provides an algebraic--and extremely practical--perspective on the solutions of PDEs and on computing related integrals.
Sep
4
comment Evaluating a convergent improper triple integral over the unit sphere
Interesting question. Most of Calculus amounts to algebraic manipulation of expressions according to rules--sum rule, product rule, chain rule for derivatives; integration by parts, substitution, etc., etc. The rules themselves of course obtain their meaning from analytical considerations, traditionally encapsulated in a few basic theorems about derivatives of polynomials and elementary functions, the Fundamental Theorem of Calculus, and some limit theorems. Thus, what is new to you here most likely is the notation, which highlights the algebraic patterns in this problem.
Sep
4
answered If $(a^{n}+n ) \mid (b^{n}+n)$ for all $n$, then $ a=b$
Sep
3
comment Evaluating a convergent improper triple integral over the unit sphere
Integration by parts can reduce [n,0,0;a] ultimately to [0,0,0;-1], which gives the surface area $4 \pi$. That makes the solution entirely algebraic, showing directly why we should expect integrals of the form [i,j,k;a] to be rational multiples of $4 \pi$ whenever $a$ is rational. This approach generalizes to other dimensions, too.
Sep
3
comment Evaluating a convergent improper triple integral over the unit sphere
@Américo, @Moron: you guys are kind. I'm sorry to make you wait; I thought it would be fun to post a solution of a different nature.
Sep
3
answered Evaluating a convergent improper triple integral over the unit sphere
Sep
3
comment Proving that this sum $\sum\limits_{0 < k < \frac{2p}{3}} { p \choose k}$ is divisible by $p^{2}$
This is identical to a problem that appeared on the Putnam exam a couple decades ago.
Sep
2
comment What is your favorite proof that $e^{ix}$ has a period of $2\pi$?
Well, yes, but then it's a tautology. Now you have to prove that the lcm of the periods of sine and cosine equals $2 \pi$!
Sep
2
comment Evaluating a convergent improper triple integral over the unit sphere
The spherical coordinate conversion for z is not correct.
Sep
2
comment Find polynomials such that $(x-16)p(2x)=16(x-1)p(x)$
Where do you use the value of 16 in the denominator of the right hand side? If you replace that value by anything else, the solution is the empty set (as is readily seen by letting $x = 0$).
Sep
2
comment Find the Frequency Components of a Time Series Graph
@Qiaochu: this "acceleration history" would have to be sampled to yield a discrete approximation to what was recorded. Different sampling methods would yield different approximations and also be accompanied by measurement error. Thus it represents an approximation of one realization of a random process which itself can be identified only up to some statistical error. Coming up with a "mathematical" description, such as an interpolating polynomial or finite trigonometric series, misses these important statistical points. I imagine that's the spirit in which "cannot be expressed" was said.
Sep
2
answered Find polynomials such that $(x-16)p(2x)=16(x-1)p(x)$
Sep
2
comment What is your favorite proof that $e^{ix}$ has a period of $2\pi$?
That pushes the question back: how do you define cosine and sine, and then--based on that definition--how do prove that their common period is 2 pi? One would like a definition that makes it relatively easy to prove the important properties of exp, especially that it is an entire analytic function of the complex plane.
Sep
2
answered What is your favorite proof that $e^{ix}$ has a period of $2\pi$?
Sep
1
comment Why does “convex function” mean “concave *up*”?
"Epi" means "above."
Sep
1
comment Why does “convex function” mean “concave *up*”?
@Srikant: "Convex" does have a standard meaning in English. To strengthen his point, GottfriedLeibniz might also appeal to the widespread consistent use of "convex" in many branches of mathematics. In effect, this views the question as probing for connections between an idea in one area of mathematics (single variable calculus) and possibly related ideas.
Sep
1
comment Why does “convex function” mean “concave *up*”?
@Srikant: agreed, but that seems like the less significant part of the question to me. Mnemonics are extremely useful, whence their popularity, but in general they do nothing to help one's understanding. Answers like that of GottfriedLeibniz are much deeper and satisfying, even if (in some opinions) they might turn out to be incomplete or even wrong.