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I hereby authorize whoever it may concern to post my comments as answers if they wish, whether as community wiki or otherwise if you like imaginary internet points.


1d
comment Why is that *any* union of open sets is open but only *finitely many* intersections of open sets is open?
Previously: Why can the intersection of infinite open sets be closed?
1d
comment Why is that *any* union of open sets is open but only *finitely many* intersections of open sets is open?
If you allow the intersection of infinitely many open sets to be open, then you have to accept, say, $\{x\}=\bigcup_{n=1}^\infty \bigl(x-\frac1n, x+\frac1n\bigr)$ as open for any $x\in\mathbb R$, and from there it's a hop, skip, and jump to having every possible subset of $\mathbb R$ be open.
1d
comment From 2 to 3 dimensions: integrating a force along a contour/surface.
In that case, the integral becomes $\oint(\mathbf x-\mathbf q)\times\mathrm d\mathbf x$ where the integral is taken along the contour of tangential points. One can see that the integral is independent of $\mathbf q$ because $\oint\mathrm d\mathbf x=0$, so you might as well take it to be the origin, yielding $\oint\mathbf x\times\mathrm d\mathbf x$.
1d
revised Understanding last step of a proof that $\text{two trajectories cannot cross at a finite value of } t$ (Phase trajectories/nodes)
why are you forcing text into math mode? it's weird.
Aug
17
comment Angular momentum of an accretion disk
The angular momentum of a system can change if momentum enters or leaves the system, for example if new material falls into the accretion disk from outside, or material in the disk disappears into the black hole.
Aug
16
comment Finding triangulations on 2D space by projecting lower hull of 3D
Just to clarify, a triangulation is given and the problem is to find the $z$-coordinates so that the lifted triangulation is part of the convex hull?
Aug
16
answered Intepolate from linear to step function, and one application for shading colors
Aug
14
comment Sphere degenerates to point in discrete space?
By discrete space do you mean a set equipped with the discrete metric $d(x,y)=\begin{cases}0&\text{if $x=y$,}\\1&\text{otherwise}\end{cases}$, or something else?
Aug
12
awarded  Good Answer
Aug
10
comment How is the curve with equation $1/x^4 + 1/y^4 = 1$ called?
en.wikipedia.org/wiki/Superellipse en.wikipedia.org/wiki/Squircle
Aug
10
comment If the function $f$ satisfies the equation $f(xf(y)+x)=xy+f(x)$, find $f$
I don't mean to disparage the question at all, but who comes up with such problems??
Aug
8
comment Ellipsoid-Sphere Intersection
achille: Good catch. The algebraic-geometry tag was added by @5xum for some reason. The original tags, optimization and nonlinear-optimization, were more appropriate.
Aug
8
comment From 2 to 3 dimensions: integrating a force along a contour/surface.
The presence of $\mathbf p$ is a red herring; you're really asking about the integral of the unit normal field on an open contour or surface. And I'm pretty sure Stokes' theorem implies that the integral only depends on the boundary of your integration region, i.e. the points $\mathbf a$ and $\mathbf b$ in 2D, and the boundary contour in 3D. So if you just want to compute the integral, you can replace the surface with one having the same boundary on which the integral is easier to compute, such as a cone joining an arbitrary point $\mathbf q$ to the boundary.
Aug
7
comment A sigma notation but with multiplication instead of addition?
This may already be obvious to you, but I'll leave it here anyway: S is for sum, P is for product. Sigma and pi are the Greek letters for S and P.
Aug
7
comment How come $\sum\limits_{n=1}^{\infty}\frac{1}{2^n}=\sum\limits_{n=1}^{\infty}\frac{1}{2^n\ln(2^n)}$?
Do you also consider $\sum_{n=1}^\infty1/2^n=\sum_{n=1}^\infty2/3^n$ a "miracle"? Sometimes an equality is just an equality.
Aug
7
comment project a point onto the intersection of surfaces
It's a sum of squares, so the minimum is attained when all the terms are zero, i.e. $g_1(\vec x)-c_1=0,$ $\ldots,$ $g_m(\vec x)-c_m=0$.
Aug
7
comment project a point onto the intersection of surfaces
Anyway, you can try following the gradient of $\big(g_1(\vec x)-c_1\big)^2+\cdots+\big(g_m(\vec x)-c_m\big)^2$.
Aug
7
comment project a point onto the intersection of surfaces
Do you want the point on $S$ closest to $P$ (which is usually what is meant by projection) or just any point on $S$? Because the "follow the gradient" procedure you mentioned doesn't necessarily give you the closest point.
Aug
7
awarded  Nice Question
Aug
6
comment What is this semicircle-like shape called?
Half a stadium?