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olivier dot begassat dot cours at gmail dot com


Dec
15
comment Map from $n$-sphere to $n$ dimensional torus
@MikeEarnest No problem : )
Dec
15
comment If a linear transformation $T$ has $z^n$ as the minimal polynomial, there is a vector $v$ such that $v, Tv,\dots, T^{n-1}v$ are linearly independent
Hint: if such a $v$ exists, it can't be in in $\ker(T^{n-1})$...
Dec
15
comment Spectrum of left shift operator $L\in B(H)$
@student There is more to invertibility in infinite dimensions (i.e. in an infinite dimensional Banach space) than kernel=0. You also have to be surjective, which is a separate condition.
Dec
12
comment Showing that $F^*(L_{\mathbb{Y}}\omega)=L_{\mathbb{X}}(F^*\omega)$
Just plug in $k-1$ vector fields on both sides, and see for yourself.
Dec
12
comment Showing that $F^*(L_{\mathbb{Y}}\omega)=L_{\mathbb{X}}(F^*\omega)$
You could use Cartan's formula $$\mathcal{L}_X=\iota_X\circ d+d\circ \iota_X$$
Dec
10
comment Variation of geodesic and Jacobi field
You can get families of geodesics using the exponential, what precisely is your question?
Dec
9
comment Retracts of $\mathbb{Q}$
Not always, there is no such map when $A\neq \Bbb Q$ is dense in $\Bbb Q$ for instance.
Dec
9
comment Show that $|sin(x)+cos(x)|$ is continuous at $\pi$
How about as a composite of continuous function?
Dec
7
comment Embedding $S_n$ in $A_{2n}$
Hint: $\lbrace 1,\dots,2n\rbrace$ is two copies of $\lbrace 1,\dots,n\rbrace$.
Dec
4
comment action of $GL_3$ on $P^2$
Orbit(s) should be clear! Do you have a hunch?
Dec
3
comment A simple finite group $G$ with $n$ p-Sylows is isomorphic to a subgroup of $\mathbb A_n$
@user16924 Careful, one of the inclusions you wrote down doesn't make sense!
Dec
1
comment How to draw the picture of vector field
Missed it ^^, thanks @vadim123
Dec
1
comment How to show $g^{-1}\circ f:\partial N\longrightarrow \partial N$ extends to a diffeomorphism $h:N\longrightarrow N$?
$F$ won't be a diffeomorphism, do you mean isotopy?
Nov
30
comment Minimal and characteristic polynomials
This isn't worth a whole answer: it's just that the characteristic polynomial of an $n\times n$ matrix has degree $n$. This rules out option d).
Nov
28
comment Exponential of a 2-form
@Léo yes. ${}{}$
Nov
26
comment Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$
@JyrkiLahtonen I included your comment in my answer.
Nov
26
comment How to understand $d^2=0$ in differential form?
@Surb I think you can get away with twice differentiable at a point. That is, $f$ differentiable on some neighborhood of $p$, and its differential $Tf$ differentiable at $p$.
Nov
26
comment How to understand $d^2=0$ in differential form?
For functions, it expresses the fact that the second derivative is symmetric, i.e., in local coordinates, $$\frac{\partial^2f}{\partial x^i\partial x^j}=\frac{\partial^2f}{\partial x^j\partial x^i}$$ I would also like an interpretation for higher forms.
Nov
25
comment When is a vector field on a manifold restricted to a submanifold $X$ a vector field on $X$?
Your claim that the projection is independent of the Riemannian metric is false! Projection means projecting parallel to the orthogonal complement of $T_xX$ inside the euclidean space $(T_xM,g_x)$: this orthogonal complement may be different for different inner products.
Nov
25
comment Automorphisms of the field of real numbers
@Crostul What do you mean when you write "you cannot define the field of real numbers in a purely algebraic way"?