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Oct
29
revised How to show that a measurable function on $R^d$ can be approximated by step functions?
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Oct
25
comment Proper and free action of a discrete group
Thanks a lot! It seems that Lee's book on smooth manifolds also takes this as definition.
Oct
25
accepted Proper and free action of a discrete group
Oct
25
asked Proper and free action of a discrete group
Oct
10
awarded  Nice Question
Sep
30
awarded  Explainer
Sep
11
comment Is every local ring a valuation ring?
But, $F$ isn't the field of fractions of $\{0,1\}\subseteq F$?
Sep
11
revised Is every local ring a valuation ring?
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Sep
11
revised Difference between two concepts of homotopy for simplicial maps?
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Sep
11
comment Difference between two concepts of homotopy for simplicial maps?
@ZhenLin It seems that you are right, if the definition of the product is essentially same as the one in Goerss & Jardine. It might be a mistranlation from Russian.
Sep
11
comment Difference between two concepts of homotopy for simplicial maps?
@QiaochuYuan Simply homotopy of $f,g\colon X\to Y$ here, informally speaking, is just a simplicial map $F\colon\Delta[1]\times X\to Y$ from a cylinder $\Delta[1]\times X$ (triangulated canonically, where $\Delta[1]$ is just the simplicial set associated with $I=[0,1]$, i.e., $1$-simplex) to $Y$, both of which are simplicial sets. Sorry for my ignorance. I don't know what you're referring to. Thanks, anyway.
Sep
11
revised Difference between two concepts of homotopy for simplicial maps?
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Sep
11
revised Finite intersection property in any metric space
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Sep
11
asked Difference between two concepts of homotopy for simplicial maps?
Aug
31
revised Differentiable manifolds $\mathscr C^k$ vs. $\mathscr C^\infty$
edited body
Aug
27
comment 6.17 Theorem : Show that $f \ \ \in \mathfrak R(\alpha)$ if and only if $ f\alpha' \ \ \in \mathfrak R$ ( walter rudin)
It follows from Rudin's argument that the upper and lower (Darboux) integrals of $fd\alpha$ and $f\alpha'dx$ are the same.
Aug
27
comment Differential identity and wedge products
$f\omega$ is just $f\wedge\omega$ where $f$ is considered as a $0$-form, and $d(\omega_1\wedge\omega_2)=d\omega_1\wedge\omega_2+(-1)^p\omega_1\wedge d\omega_2$ where $\omega_1$ is a $p$-form.
Aug
27
comment $f\in C(\mathbb{R})$. What does it mean?
But it's usually denoted as $C^n(\mathbb R)$ or $\mathcal C^n(\mathbb R)$ as well as $\mathcal C^0(\mathbb R)$ without parentheses.
Aug
27
comment A question on short exact sequences.
I found learning diagram chasing isn't that easy for a beginner from books, but there're two materials which seems more accessible: Hatcher's Algebraic Topology, pp115, subsection Relative Homology Group, and Eisenbud's Commmutative Algebra, pp637, subsection A3.7.
Aug
27
answered Prove $a^3+b^3+c^3\ge a^2+b^2+c^2$ if $ab+bc+ca\le 3abc$