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seen Jan 28 at 15:19

Jan
28
comment When are two direct products of groups isomorphic?
For finitely generated Abelian groups, it seems right, as a result of the structure theorem. In general, maybe Jordan-Hölder theorem works. I don't know.
Jan
28
comment When are two direct products of groups isomorphic?
In infinite case, $H_1\cong H_2$ isn't necessary. Note that $\mathbb Z^\omega\times\mathbb Z^\omega\cong\mathbb Z^\omega$. That's simply to say, $\omega+\omega=\omega$, where $\omega$ is the countable cardinal.
Jan
28
comment Homotopically equivalent to Čech nerve?
@anomaly Well, I need time to read it. Please post an answer here so that I can accept it. That's exactly what I want.
Jan
28
comment When are two direct products of groups isomorphic?
Are you only interested in the finite case?
Jan
28
comment Homotopically equivalent to Čech nerve?
@anomaly I skimmed the proof and maybe the paracompactness is only used to give a partition of unity? It seems to me that the local finiteness of $\mathfrak U$ gives a partition of unity. I haven't gone through homotopy theory. I only want a reference to see what's the idea involved. Thanks!
Jan
28
comment Is every finite dimensional linear space a banach space
Well, we can specify arbitrary topology to a space, even for $\mathbb R^n$, say. Maybe A more interesting question is whether every finite dimensional topological vector space is complete with respect to the uniform structure.
Jan
19
comment Conditions on ideal b for fields or integral domains
It might be worthy to explicate the reason for me (and maybe others) to vote to close the question, and it might be the probable reason for these downvotes. That is to say, you have to write down your efforts when you post questions. It's a convention and a formal etiquette. It may not be that reasonable, though.
Jan
19
comment If $f$ is a continuous function in $[a,b]$ whose derivative $f~'$ exists at every point $c \in [a,b]$. Then, is $f~'$continuous?
Related: math.stackexchange.com/q/112067/23875
Jan
18
comment Convergence of a sequence with assumption that exponential subsequences converge?
@Mizar Thanks for the comment. Early I heard from Kai Lai Chung's A Course in Probability Theory that Etemadi proved SLLN for pairwise independent identically distributed random variables. That's it! But this problem doesn't help since one needs a countable version, but with more assumptions to the original sequence (Cesàro sum of nonnegative numbers).
Jan
17
comment Convergence of a sequence with assumption that exponential subsequences converge?
@tomasz It should have been $\lvert a_{\lfloor\alpha^l\rfloor}\rvert<\epsilon$. Still checking.
Jan
17
comment Convergence of a sequence with assumption that exponential subsequences converge?
@tomasz Thanks.
Jan
16
comment Show that function has removable singularity
Let $F=\phi\circ f$, then what about $\lim_{z\to a}(z-a)F(z)$? What can you conclude? Then use $f=\phi^{-1}\circ F$ to obtain what you want.
Jan
16
comment Show that function has removable singularity
Consider the biholomorphic map $\phi$ from the half plane $\Re w>0$ to the unit disc $\lvert\zeta\rvert<1$. Study the composite $\phi\circ f$.
Jan
16
comment What is the most appropriate book for teaching, not the content but skills of mathematics
I'd suggest Pólya's How to Solve It, Hilbert's Geometry and the Imagination, Klein's Development of Mathematics in the 19th Century, and his Lectures on Mathematics.
Jan
15
comment Intuitive understanding of the $BAB^{-1}$ formula for changing basis in linear transformations.
There's another way to look into such a transformation rule. In fact, the linear maps are (1,1)-tensors: $\operatorname{End}_k(V)\cong V^*\otimes_kV$. In tensor analysis and differential geometry, we usually write $y^j=\sum_ka_k^jx^k$ for linear maps.
Jan
7
comment a.e. convergence of a piecewise constant function $f_h(t)=\left\lfloor \frac{t}{h} \right\rfloor \cdot h$
Note that $(t/h-1)h<f_h(t)\le(t/h)h$.
Jan
7
comment Limit of $ \int_{0}^{\frac{\pi}{2}}\sin^n xdx$ and probability
For the first one, dominated convergence theorem also works.
Jan
2
comment Completeness implies geodesic completeness, a more conceptual way?
@studiosus I checked my original post and found that I wrongly owed my feeling of unnaturality to Riemannian structure. In fact, I think that it's a property of vector fields and manifolds, rather than specifically geodesic sprays which is somewhat related to the tangent bundle. In ODE terms, it's just a global existence theorem on the equation and the ambient space (here, the phase space $UM$), no matter whether it's originated from second order geodesic equations.
Jan
2
comment Completeness implies geodesic completeness, a more conceptual way?
@mollyerin It's impossible to formulate completeness without distance (or generally uniform) structure. Em, yes, we need some properties to characterize the boundedness property of the vector field $V$.
Jan
2
comment Can a function be continuous but not Hölder on a compact set?
@PedroTamaroff It appears in Fikhtengolts's A Course of Differential and Integral Calculus. I don't know whether there are English translations.