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 Yearling
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16m
comment what are the principal applications of symetric matrix in physics giving examples of it's applications?
See inertia matrix (en.wikipedia.org/wiki/…)
1d
comment Show that $\mathbb{R}^m$ is not homeomorphic to $\mathbb{R}^n$
If two (locally compact Hausdorff) spaces are homeomorphic, their one point compactifications are too (via an extension of the original homeomorphism). The one point compactification of $R^n$ ($n\geq 1$) is $S^n$.
1d
comment Show that $\mathbb{R}^m$ is not homeomorphic to $\mathbb{R}^n$
Have you heard about one point compactification?
1d
comment Does the product functor preserve quotient maps?
This math.stackexchange.com/questions/833042/… is related, and talks a little about the categorical part ($-\times Y$ having an adjoint)
2d
comment Redundance of the Smoothness of the Inversion Map in the Definiton of a Lie Group.
Thanks : ) ${}$
2d
comment Redundance of the Smoothness of the Inversion Map in the Definiton of a Lie Group.
You don't need smoothness of the inverse map, you only need smoothness of the maps "left multiplication by some fixed $g$" and "right multiplication by some fixed $h$" (of course you'll take $h=g^{-1}$), which follow from the assumption that multiplication is a smooth map.
2d
revised Redundance of the Smoothness of the Inversion Map in the Definiton of a Lie Group.
added 290 characters in body
2d
answered Redundance of the Smoothness of the Inversion Map in the Definiton of a Lie Group.
2d
comment There is no smooth submersion from $S^2$ to $S^1$.
$\pi_1(S^2)=0$.
2d
comment There is no smooth submersion from $S^2$ to $S^1$.
Another approach to showing that there is no such submersion $f:S^2\to S^1$ would be to endow $S^2$ with its standard riemannian metric, and to pull the vector field $\frac\partial{\partial t}$ back to $S^2$ along the submersion, by selecting the only tangent vector at $m\in S^2$ that projects onto $\frac\partial{\partial t}$ while being orthogonal to $ker(d_mf)$. This contradicts the hairy ball theorem.
2d
comment There is no smooth submersion from $S^2$ to $S^1$.
The immediate proof that comes to mind lifts such a map to the real numbers, and shows that submersivity is impossible in the presence of a maximum. It's short, did you have this in mind or something else?
2d
comment Groups with finite automorphism groups.
@ClémentGuérin Yes, it's infinite.
May
25
comment How to prove $\sinh^{-1}x=\ln\left(x+\sqrt{x^2+1}\right)$
Fix some real number $x\in\Bbb R$, you are looking for a formula for $y\in\Bbb R$ such that $\sinh(y)=x$. If you write $Y=e^y$, then you are trying to solve the equation $$\frac{Y-\frac1Y}2=x$$ i.e. $$Y^2-2Yx-1=0$$ and you can apply the quadratic formula to get the (right) $Y$, thus $y$.
May
23
comment Are closed simple curves with this property necessarily circles?
Many curves have this property, for instance ellpises, or squares, or regular polygons drawn from the $(2n)$-th roots of unity...
May
23
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May
19
comment Which other “exotic” permutation-related things exist?
I apologize, this is very basic, and I don't doubt you are aware of it, but it relates to $S_4$. There are 3 ways to cut a 4 element set in two equal halves, hence there exists a (clearly) surjective morphism $S_4\to S_3$.
May
17
revised prove $p (\bigcup A_i) \leq \sum p (A_i) $
added 2 characters in body; edited title
May
15
comment compact operators surjection
@r.pomegranate finite rank continuous linear maps are always compact.
May
15
comment compact operators surjection
@r.pomegranate $A(B_X)$ is open by the open mapping theorem, and contains zero, so there is some positive real number $c$ such that $cB_X$ is contained in $A(B_X)$.
May
15
comment compact convergence for a series in complex space
Compact convergence means that the series $\sum f_n\big|_K$ converges uniformly for every compact set $K\subset D$. But the series already converges normally on $D$, hence uniformly on $D$ and thus uniformly on every compact subset of $D$.