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My profile pic is currently Sir Isaac's Newton coat of arms. It comes from wikipedia.


Jan
16
revised For which $n, k$ is $S_{n,k}$ a basis? Fun algebra problem
added 1808 characters in body
Jan
16
answered For which $n, k$ is $S_{n,k}$ a basis? Fun algebra problem
Jan
16
comment Connectedness arguments in elementary mathematics?
This is in my opinion a gorgeous proof. It shares one of the aspect I'm interested in: reducting a theorem to a trivial case, but the reduction is not the topological one I'm looking for. There is no a priori good reason that this equal-area property does not change when you move the triangle continuously (it would for example be obvious if we knew that all areas involved must be integers, but that's false). So my dream would be e.g. a property of triangles that you could prove saying: it's obvious for the equilateral triangle, it stays true when you deform a bit your triangle, so it's true!
Jan
16
comment Connectedness arguments in elementary mathematics?
Let me give an example of something that is not exactly what I'm looking for. I claim that when you draw the three medians of a triangle, the 6 triangles that are thus created (and which share the centroid as a vertex) have the same area. Proof: it's obvious for an equilateral triangle. I can pass from any triangle to an equilateral one via an affine transformation. Such a transformation does not change the property I want to prove (because an affine transformation multiply all the areas by the same factor – its determinant).
Jan
16
comment Connectedness arguments in elementary mathematics?
I understand that the vagueness of my question is frustrating (and I'm ready to apologise for that), but I am in any case unable to define "elementary mathematics". I guess it's one of those "I know it when I see it"-type things that are really hard to define...
Jan
16
awarded  Benefactor
Jan
16
comment Connectedness arguments in elementary mathematics?
but it's a nice spin on a classical proof. In any case, thank you for the time you spent on this question. I know it's frustratingly vague, and I sincerly appreciate the effort.
Jan
16
comment Connectedness arguments in elementary mathematics?
I don't think I misrepresented the first step. In the degree-genus formula, it is clear that if the polynomial varies in the space of nonsingular polynomials (the space $\mathfrak M$), the corresponding (real) surfaces are deformations of each other and therefore they have the same genus. On the other hand, I agree that I was quite vague on the connected/pathwise connected stuff. To be honest, I hoped for examples where the topological part would be so obvious that this kind of distinction wouldn't matter. +1 for the second paragraph: it's not really what I am looking for...
Jan
9
revised Binomial coefficients identity: $\sum i \binom{n-i}{k-1}=\binom{n+1}{k+1}$
edited body
Jan
9
comment Binomial coefficients identity: $\sum i \binom{n-i}{k-1}=\binom{n+1}{k+1}$
Yes, of course. My bad.
Jan
8
answered Binomial coefficients identity: $\sum i \binom{n-i}{k-1}=\binom{n+1}{k+1}$
Jan
7
asked Connectedness arguments in elementary mathematics?
Dec
18
comment Does there exist a continous function $f(t)$ on $[0,1]$ for which $\int_0^1 t^3 f(t) dt = 0$?
There must be: take two random (continuous) functions $f$ and $g$ and look at all possible combinations $\lambda f + \mu g$. In more abstract terms, $f \mapsto \int_0^1 t^3\, f(t)\, dt$ is a linear form, so its kernel is huge.
Nov
3
comment The set of functions which map convergent series to convergent series
You're absolutely right, of course. I fixed it.
Nov
3
revised The set of functions which map convergent series to convergent series
added 4 characters in body
Nov
2
awarded  Necromancer
Oct
29
comment Convergent complex series
Do you know Abel transform / sommation by parts? Or the alternating series test ?
Oct
5
awarded  Good Answer
Oct
1
comment $\mathbb{R}P^2$ and its fundamental group by identification of edges of unity square
Only the opposite corners are identified, aren't they?
Sep
30
awarded  Explainer