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"Arithmetic is the hardest of the sciences, this is well-known." -V. Touraev

Comathematician: A device for turning cotheorems into ffee.


2d
comment Proving limits using epsilon definition
It's often useful to look at the ratio of the dominant terms in the numerator and denominator. In this case, that ratio looks like 1/x, so you are led to consider proving it for 1/x and then adapting that to the original problem.
Dec
18
comment Proving the Cone is Contractible: Is my Proof correct?
Many upvotes and no correcting answer do not necessarily mean that you are clear and correct. Even if you were completely wrong, I would still have heartily upvoted for investing effort into the problem and showing your work.
Dec
17
awarded  Good Answer
Dec
15
comment Any finite set is compact; what exactly is a finite set?
+1 for a question which demonstrates a thought process and outlines previous attempts at an answer.
Dec
15
comment What level of math is needed to learn fractional calculus?
Ok, I see more clearly what you mean.
Dec
15
comment What level of math is needed to learn fractional calculus?
You suggest jumping into Hormander, Vol 1, after just two semesters of baby Rudin? Or is there an easier reference for microlocal analysis you have in mind?
Dec
5
answered Is there a non-simply-connected space with trivial first homology group?
Dec
3
comment Riemann-Lebesgue Lemma for Spherical Harmonics expansion
This is because on a closed Riemannian manifold, eigenfunctions of the Laplace operaror form an orthonormal basis of $L^2$.
Dec
3
comment Riemann-Lebesgue Lemma for Spherical Harmonics expansion
The spherical harmonic expansion is the fourier expansion on the round sphere. Spherical harmonics : sphere :: sines and cosine : circle.
Dec
3
comment Riemann-Lebesgue Lemma for Spherical Harmonics expansion
In $L^2$, at least, the fourier coefficients must decay to zero at infinity because the $L^2$ norm of a function is the sum of the squares of its fourier coefficients.
Dec
1
comment Does $\lim_{n\to\infty}\frac{1}{n}\sum_{t=1}^na_t=L$ imply that $\{a_t\}$ is bounded?
Hmmm. I have removed the $\log\log\log(t)$ example. I'll see if I can cook up an explicit subsequence example.
Dec
1
revised Does $\lim_{n\to\infty}\frac{1}{n}\sum_{t=1}^na_t=L$ imply that $\{a_t\}$ is bounded?
deleted 66 characters in body
Dec
1
comment Is this some kind of trick question? Submanifold “proof”
@jip Precompose with a local coordinate map.
Dec
1
answered Does $\lim_{n\to\infty}\frac{1}{n}\sum_{t=1}^na_t=L$ imply that $\{a_t\}$ is bounded?
Dec
1
comment Does $\lim_{n\to\infty}\frac{1}{n}\sum_{t=1}^na_t=L$ imply that $\{a_t\}$ is bounded?
@Aryabhata Sure, why not?
Dec
1
comment Does $\lim_{n\to\infty}\frac{1}{n}\sum_{t=1}^na_t=L$ imply that $\{a_t\}$ is bounded?
What if $a_t$ is something like $\log\log(t)$, whose partial sums grow more slowly than $1/n$?
Dec
1
comment Is this some kind of trick question? Submanifold “proof”
@jip It is straightforward. You just have to parse the statement of the theorem and apply it.
Dec
1
comment How Many Triangles are Created by n Lines in the Plane?
The trick is showing that adding another line creates at least one more triangle.
Dec
1
comment Is this some kind of trick question? Submanifold “proof”
The relevant theorem is not the rank theorem! The rank theorem shows up in the proof of the relevant theorem, but is about linear operators on finite-dimensional vector spaces, not smooth functions on manifolds.
Dec
1
answered Is this some kind of trick question? Submanifold “proof”