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8h
comment Algebraic methods to compute the cohomology ring of the complex topology of a variety?
@QiaochuYuan Induction method is concerned with the embedding $\mathbb P_\mathbb C^{n-1}\to\mathbb P_\mathbb C^n$, and the spectral sequence associates to $S^1\to S^{2n+1}\to\mathbb P_\mathbb C^n$. Right, $\mathbb P_\mathbb C^n=e^0\cup e^2\cup\dotsb\cup e^{2n}$, but I don't know whether the cup product is easily computed.
8h
comment Algebraic methods to compute the cohomology ring of the complex topology of a variety?
@MarianoSuárez-Alvarez Thanks. If it's a complex manifold, there's a CW decomposition of $V(\mathbb C)$ with only even dimensional cells (cf. Milnor's Morse Theory) and therefore the cohomology is torsion-free, and one can recover $\mathbb Z$-cohomology from $\mathbb R$-cohomology? Thanks for the reference.
9h
asked Algebraic methods to compute the cohomology ring of the complex topology of a variety?
Jul
30
comment Cohomology with Coefficients in the sheaf of distributions
@Exterior The fine resolution is similar to $0\to\mathbb R\to\Omega^0\to\Omega^1\to\Omega^2\to\dotsb$, where $\Omega^p$ is the sheaf of $p$-forms, replacing $\Omega^p$'s with corresponding sheaves of distributions of cross-sections.
Jul
30
comment Cohomology with Coefficients in the sheaf of distributions
@Exterior Suppose $\xi\colon E\to M$ is a smooth vector bundle, then we can define sheaf of distribution cross-sections of $\xi$, see Gunning's Lectures on Riemann Surfaces, section 6 (some terms are archaic). Especially, consider exterior products of the cotangent bundle, we obtain a sequence of vector bundles, and we can consider the corresponding sheaves.
Jul
18
comment How can I prove Dini's theorem using the Baire Category theorem?
Fix $\epsilon>0$, and set $E_n=\{f(x)\ge\epsilon\}$, which is decreasing and $\bigcap_nE_n=\emptyset$, then there exists $N$ s.t. $E_N=\emptyset$.
Jul
18
comment Approximation of irrational numbers?
The original problem from the interview (finished but maybe inappropriate to propagate) is that when $\theta$ is the unique root of $P\in\mathbb Z[X]$ such that $\lvert\theta\rvert\ge1$, where $P$ is monic and $d=\operatorname{deg}P\ge2$, then there's a constant $C>1$ independent of $d,P$ such that $\lvert\theta\rvert\ge C$. They gave me a hint that assuming $P(X)/(X^dP(1/X))=\sum_jb_jX^j$ and the identity $1+\theta^2=b_0^2+\sum_j(b_j-\theta b_{j-1})^2$. I have no clear idea to do next. Now your post tells me that my conjectures are all wrong. Do you have any idea? Thanks!
Jul
18
accepted Approximation of irrational numbers?
Jul
17
revised Approximation of irrational numbers?
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Jul
17
revised Approximation of irrational numbers?
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Jul
17
revised Approximation of irrational numbers?
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Jul
17
revised Approximation of irrational numbers?
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Jul
17
comment Approximation of irrational numbers?
@Théophile I found number theorists usually call algebraic integers as integers, thus to disambiguate, I added the adjective rational.
Jul
17
asked Approximation of irrational numbers?
Jun
7
comment Quotient manifold theorem provides a fibration?
Okay, the submersion, in fact, furnishes local sections, and therefore bundle structure. I would rather refer to the bundle structure theorem in section 7.4 of Steenrod's Topology of fiber bundles.
Jun
7
accepted Quotient manifold theorem provides a fibration?
Jun
7
revised Quotient manifold theorem provides a fibration?
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Jun
7
asked Quotient manifold theorem provides a fibration?
May
9
comment Cohomology with Coefficients in the sheaf of distributions
@ZhenLin Ugh, but why can't we restrict a Schwartz distribution to an open set? Even more, it constitutes a fine sheaf, so are sheaves of currents, which forms a fine resolution of the constant sheaf $\mathbb R$, say.
May
9
comment Why does the Residue Theorem still hold, when I let my contour get infinitely large?
You cannot make the contour infinitely large. Instead, you choose a family of contours, deriving an equation from each contour, then take limit.