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I think the problem that I was given by my school is wrong because $ \forall v \in N(T)$ , $ Tv=0$ , so we have $ STv=S(0)=0$ . Thus, $ v \in N(T)$ implies $ v \in N(ST)$ . Hence, $ N(T) \subseteq N(ST)$ . Am I right? Here, $ N(T), N(ST)$ means nullRead more

Using xcopy with the code below. I have read you can put double quotes around the source and destination folder so that the folders with spaces can be treated correctly as a directory. However, when I use quotes around the path it shows an error: Invalid number of parameters For /D %%K IN (“C:\This Folder”)Read more

so you may have guessed from the title that this is Exercise 1-21 of K&R (2nd edition). I’ve seen someone else’s post about it but it doesn’t solve the problem the way my code does, and I wasn’t sure whether I was allowed to post my code as an answer and ask for feedback (itRead more

I am looking for a standard name (if it exists) for the following property of a Schauder basis $ (e_i)_{i=1}^\infty$ in a Banach space $ X$ : $ $ \|\sum_{i\in F}x_ie_i\|\le\|x\|$ $ for any $ x=\sum_{i=1}^\infty x_ie_i\in X$ and any finite subset $ F\subset\mathbb N$ . This condiion implies that the Schauder basis is unconditional.Read more

There are several ways to construct good Cech covers of a smooth complex analytic space, by either using some little input from Riemannian geometry or complex analysis, for instance. What is a proof of existence of good Cech covers, that uses mostly the fact that a smooth complex analytic space $ X$ has an openRead more

If $ X$ is a proper smooth complex analytic space, one can define Chow groups of analytic cycles on $ X$ the usual way. We have a cycle map $ $ c^p_X: \text{CH}^p(X) \to \text{H}^{2p}_{D}(X,\mathbf{Z}(p))$ $ to Deligne cohomology of $ X$ . Is $ c^p_X$ an isomorphism? Is $ c^p_X\otimes\mathbf{Q}$ an isomorphism?Read more

I would like to know where about I can find the most updated results on the unstable groups $ \pi_{2k+1}P_{k+1}$ and $ \pi_{2k}P_k$ . I think there would be definitely computations Mahowald’s AMS memoir, but I presume there has been some work after this masterpiece! I also wonder how is known about the stable homotopyRead more

What is the CW-complex of Eilenberg-MacLane space $ K(\mathbb{Z}_2,2)$ ? What is the CW-complex of Eilenberg-MacLane space $ K(\mathbb{Z}_n,d)$ ? What is the CW-complex of Eilenberg-MacLane space $ K(\mathbb{Z}_n\times \mathbb{Z}_m,d)$ ? For example, I like to know the number of cells in each dimensions, and the chain complex formed by those cells, so that oneRead more

Problem. Does every compact metric space of finite packing dimension admit a bi-Lipschitz embedding into the Hilbert space $ \ell_2$ ? The packing dimension of a compact metric space $ (X,d)$ in the (finite or infinite) number $ $ Dim(X)=\limsup_{\varepsilon\to 0}\frac{\ln N_\varepsilon(X)}{\ln(1/\varepsilon)},$ $ where $ N_\varepsilon(X)$ is the cardinality of the smallest cover of $Read more

If $ X$ is a topological space, one can naturally view the set $ \pi_0(X)$ of path-components of $ X$ as a quotient space of $ X$ by collapsing each path-component to a point by a quotient map $ q:X\to \pi_0(X)$ . Of course, $ q$ is a homeomorphism if and only if $ X$Read more