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olivier dot begassat dot cours at gmail dot com


Jan
16
reviewed Leave Open Metric $p := p(x,y)= \min(|x-y|, 1- |x-y|)$ $x,y \in [0,1)^2$. Prove metric space is compact.
Jan
15
comment pronunciation of sinh x, cosh x, tanh x for short
The position of "hyperbolic" and the name of the trigonometric function are reversed in french. I'm tempted to call the hyperbolic analogues 'ship', 'chip' and 'tip' from now on. Square chips are one better than square ships.
Jan
15
comment Variation on Stokes Theorem for Manifolds (2)
I don't understand what you want.
Jan
15
comment Variation on Stokes Theorem for Manifolds (2)
You won't find one, because your own hypothesis demands $d\omega=0$...
Jan
15
comment Variation on Stokes Theorem for Manifolds (2)
A closed $0$ form on a connected manifold is just a constant function. If you want to apply Stokes' theorem, you will want to have $M$, which is a finite number of points, you will want $M=\partial N$ for some connected compact submanifold $N$ of (orientability isn't a problem since $N$ is one dimensional). such an $N$ exists iff the number $M$ has even cardinality.
Jan
14
comment number of nilpotent $n\times n$ matrices over $\mathbb F_2$
Every nilpotent matrix is conugate to a Jordan form. For a given Jordan form you could (in theory) compute the cardinal of its stabilizer under conjugation, thus deduce the size of the orbit, and sum over all conjugacy classes of nilpotent matrices. But that's far from explicit.
Jan
14
comment constant-curvature Riemannian metric for Bring's surface
I doubt this will be of any help, but if you think of the $[z_0:\cdots:z_4]$ as roots of polynomials, then the equations defining Bing's surface (modulo the $S_5$-symmetry) and Newton's identities for symmetric polynomials tell you that $$\prod_{i=0}^4(X-z_i)=X^5+\lambda X+\mu$$ with at least one of $\lambda,\mu$ nonzero, and thus you get a ramified covering $B/S^5\to\mathbb{CP}^1$ by the formula $$[z_0:\cdots:z_4]\mapsto\left[\left(\sum_{i=0}^4\prod_{j\neq i}z_j\right)^5:\left(\prod_{i=0}^4z_i\right)^4\right]$$ The ramification points are related to the vertices of type A and B you describe.
Jan
14
comment constant-curvature Riemannian metric for Bring's surface
I guess you don't have an explicit description of the cover $p:\mathbf H\to B$, perhaps as a concrete map $\mathbf H\to\Bbb{CP}^4$?
Jan
14
comment Retraction and intersection
$U\subset X$ doesn't imply that $\pi_1(U)$ is a subgroup of $\pi_1(X)$: the canonical morphism $i_*:\pi_1(U)\to\pi_1(X)$ need not be injective.
Jan
14
comment Retraction and intersection
What is $f|_{\cap_I U_i}$ supposed to represent? Do you mean that, what you call $f$, is actually $f_*^I$ for some continuous map $f^I:X\to\cap_{i\in I}U_i$ (one for each finite set $I$ of indices), which furthermore is a retraction?
Jan
14
comment Retraction and intersection
Not in this generality, for instance $X,U,V$ might be connected while the intersection $U\cap V$ isn't.
Jan
14
comment Every 1- or 2-dimensional compact, connected Lie group is abelian
For the unimodular part, is it correct to say that $e^{\mathrm{ad}(x)}\in\mathrm{Ad}(G)\subset\mathrm{GL}(\mathfrak g)$ [hence its determinant $e^{\mathrm{Tr}(\mathrm{ad}(x))}$ is in $\lbrace 1\rbrace$, the only connected compact subgroup of $\mathbb R^*$, and $\mathrm{ad}(x)$ has $0$ trace]?
Jan
13
revised Every 1- or 2-dimensional compact, connected Lie group is abelian
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Jan
13
reviewed Leave Open closure of a convex set in a normed linear space is convex ?
Jan
13
reviewed Close Limit of $x^2+3\sin x$ as $x$ goes to negative infinity
Jan
13
reviewed Leave Open $T^n = 0$ but $T^{n-1} \neq 0$: find basis such that $Ta_j = a_{j+1}$ for $j < n$ and $Ta_n = 0$
Jan
13
reviewed Approve functional-equations tag wiki excerpt
Jan
13
reviewed Reject functional-equations tag wiki
Jan
12
revised Every 1- or 2-dimensional compact, connected Lie group is abelian
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Jan
12
revised Every 1- or 2-dimensional compact, connected Lie group is abelian
added 2011 characters in body