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Jul
15
comment An equivalent definition of the rotation number of a circle homeomorphism
I took the liberty of rewriting the question (which I like a lot). I hope you don't mind.
Jul
15
revised An equivalent definition of the rotation number of a circle homeomorphism
rewording of the question
Jul
15
answered Smooth isometric embeddings of Riemannian manifolds
Jul
9
answered Curvature flow for convex planes curves
Jul
9
revised Curvature flow for convex planes curves
deleted 80 characters in body
Jul
9
comment Understanding definition of Riemann Integral
E may come before I in your name, but it's the other way around in Bernhard's :-)
Jul
9
comment Curvature flow for convex planes curves
Je ne vois pas comment un phénomène ou un exemple réel pourrait expliquer un résultat mathématique. Cherches-tu une application de cette théorie ou s'agit-il d'autre chose ? [Je veux bien traduire la question en anglais, mais il faudrait déjà que je sois sûr de bien la comprendre...]
Jun
25
comment Find whats's wrong with this proof about the reflexive vector space
@BrianM.Scott: I guess some people use this word to talk about the purely algebraic fact that the canonical morphism $X \to X^{**}$ is an isomorphism if $X$ is a finite-dimensional vector space (Bourbaki does in the module setting, for an example), but I agree the vocabulary comes from the topological, infinite-dimensional case.
Jun
25
comment Algorithm to find non-zero matrix $N$ such that $N \times M = 0$
Note that if $M$ is in echelon form, the question is easy. So Gauss's algorithm (en.wikipedia.org/wiki/Gaussian_elimination) will allow you to construct such an algorithm.
Jun
25
comment Find whats's wrong with this proof about the reflexive vector space
@BrianM.Scott: your first comment seems off-topic to me. The original question deals with "naked" vector spaces, without topology.
Jun
25
comment Find whats's wrong with this proof about the reflexive vector space
Of course, since $X^*$ is in a sense always strictly larger than $X$ when $\dim X$ is infinite, there's no hope for preserving a meaningful notion of reflexivity (at least purely algebraically, with topological vector space, the notion is highly interesting).
Jun
25
comment Find whats's wrong with this proof about the reflexive vector space
If $X$ is the set of real sequences with only finitely many terms, it is quite easy to identify $X^*$ to the set of all real sequences, which is strictly larger (and it's a general property of infinite-dimensional spaces, cf. for instance math.stackexchange.com/questions/1297845/…).
Jun
19
awarded  Nice Answer
Jun
16
answered Does there exist a double cover with trivial deck transformation group?
Jun
10
comment Coordinate systems on manifolds
@Will: you can always put some Coordinates (they're useful for computations), but you cannot ask them to preserve the metric (=to be isometries). If you have a chart which is an isometry, then your manifold is locally Euclidean, which is a very exceptional case. In Riemannian Geometry, it is sometimes important to use charts with extra properties (so that the problem you're studying has a nice expression), but these "extra properties" are always weaker than being an isometry because the manifold is not flat.
Jun
10
comment Coordinate systems on manifolds
It depends on what you ask for your coordinate systems. If you want only them to reflect the topology of the manifold, then you're right, it's always locally possible and then a collection of charts will cover your whole manifold. The thing you've read, on the other hand, deals with more that mere topology. If you want your chart to reflect geometric properties of your space (angles, lengths, areas...) then it is in general not possible, even locally. It's a thing cartographers learned a long time ago: you can represent angles nicely, areas nicely, but not both at the same time.
Jun
4
revised Request for a comparison between these 3 (advanced?) functional analysis books?
better formatting (explanation of links)
Jun
4
comment Does pushing point along a loop on a surface induce a homotopy from identity to a homeomorphism of the surface?
Is your loop a closed simple curve? If it is, what you're defining is a Dehn twist, and you can read about them in the same Primer on Mapping Class Groups (chapter 3) [but I don't think I agree with your computation of $f_*$]. If not, I'm not sure that you're really defining a homeomorphism, even up to homotopy.
Jun
4
comment Are normal subgroups of $S_4$ transitive?
A group of order $4$ is either cyclic or the product of two cyclic groups of order $2$. Because you know very well the order of the elements of $S_4$, you can use this remark to classify completely the 4-element subgroups of $S_4$ and check that $V_4$ is indeed the only normal one. That's really easy, but not very conceptual.
Jun
4
comment Solutions to the matrix equation $\mathbf{AB-BA=I}$ over general fields
I'm pretty late, but it's never too late to praise so great an answer. Do you have references for the non-simple finite-dimensional quotients?