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comment Scaling a svg image while keeping the offset position.
I see. :S Second question (since I'm not fluent in SVG): What do cx and cy represent? Are they coordinates of the center of the image with the origin $(0, 0)$ at upper left? That is, are bbox.x and bbox.y equal to $(0, 0)$ before the user does any dragging, and are they not $(0, 0)$ after the user drags the map? Or does bbox refer to an SVG element the user has clicked? (Or is their meaning something else...?)
2h
comment Scaling a svg image while keeping the offset position.
Quick preliminary question: If you replace cx-(scale*cx) by cx and cy-(scale*cy) by cy throughout (i.e., both before and after scaling), does the code do what you expect?
3h
comment “Basis extension theorem” for local smooth vector fields
(+1), but it seems to me that some care is needed regarding the domain of the extending sections: Where you write "$U$ is contractible", it would be more accurate to say "there exists an open neighborhood $V$ of $p$ such that $U \cap V$ is contractible". (If one cannot shrink $U$, there are counterexamples to the OP's question.)
3h
answered construction of linear independent local section in vector bundle
22h
answered Cylinder volume with curved base area
1d
answered How do I prove that for any two points in $\mathbb{C}$, there exists a $C^1$-curve adjoining them?
1d
comment Bizarre definition of convergent sequences
Related: Slightly changing the formal definition of continuity. Particularly, my answer discusses the four conditions obtained by permuting the order of the quantifiers "for every" and "there exists" and which variable ($\epsilon$ or $\delta$, cf. $N$) each quantifies.
1d
answered Differentiating the exponent power series
1d
answered How are arc components of a spherical system derived?
1d
comment Why are compact complex manifolds Liouville?
For what it's worth, the question in your title is one of the first remarks in Chern's Complex Manifolds Without Potential Theory. Your second question follows because if you fix a frame for a trivial bundle $E \to M$ at one point, there's an induced fibre projection $E \to \mathbf{C}^{n}$.
2d
comment Properties of $\mathbb{C}P^n$
@koe: You're aware that $\mathbf{CP}^{2}$ doesn't embed holomorphically in any complex Euclidean space? (That is, no matter how you map $\mathbf{CP}^{2}$ into a Euclidean space, some tangent space of the image is not preserved by the ambient complex structure.) I ask because the wording of your question suggests you're viewing the ambient complex structure as a complex structure on the image manifold....
2d
comment For which complex $a, b, c$ does $(a^b)^c=a^{bc}$ hold?
If $\log(-a) = \log a + i\pi$ for $a > 0$ real, the argument goes through with obvious modifications. :) (Trying to make sense of things for $a = 0$ isn't worth the hassle; there's no notion of signed infinities in $\mathbf{C}$, and $\exp$ has an "essential singularity" at $\infty$, so defining $\log 0 = \infty$ is asking for trouble.)
2d
answered For which complex $a, b, c$ does $(a^b)^c=a^{bc}$ hold?
2d
answered Restriction of a linear algebra to an affine subspace?
Jul
2
comment How do I visualize this quotient space?
The dunce hat is not exactly the same space, but may be close enough to help you visualize. The wikipedia animation may be helpful as well.
Jul
1
comment How to get the potential and the gradient of this function?
@Arthur: On a simply-connected set (such as all of $\mathbf{R}^{3}$, as is the case here), a vector field $f$ has a potential if and only if the curl of $f$ is zero. On a set that is not simply-connected, a vector field can have vanishing curl at every point yet still not have a potential. :)
Jun
30
comment How to get the potential and the gradient of this function?
Do you have reason to believe a potential exists for this particular vector field...?
Jun
30
comment Definitions of complxe singularity exponent
It looks to me that if $c$ is strictly smaller than the supremum, then not only is $r^{-2c} \mu_{U}(\{\varphi < \log r\})$ bounded, but you have a bit of "wiggle" in the exponent of $r$, enough to "absorb" a small amount from the $1/r$ in $dr/r$ and get a finite integral on the right.
Jun
30
answered Line integral: $u = ( \frac{-y}{x^2 + y^2}, \frac{x}{x^2 + y^2}, z)$
Jun
30
comment How to calculate the area of a region with a closed plane curve boundary?
As it turns out, yes: A bit of trig and algebra shows $t_{0}$ is a root of$$20\cos^{2} t + 18\cos t - 5 = 0;$$that is, $t_{0} = 2\pi - \arccos\frac{-9 + \sqrt{181}}{20}$. (Also, as you may have noticed, it suffices to integrate from $\pi$ to $t_{0}$; the contribution from the vertical segment is zero for all three integrands since $x \equiv 0$ on the $y$-axis.)