Frank Science
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 Mar 1 comment On the Banach–Alaoglu theorem: is the unit ball of an equivalent norm also weak-* compact? @G.Sassatelli Thanks for clarification. Yes. Mar 1 asked On the Banach–Alaoglu theorem: is the unit ball of an equivalent norm also weak-* compact? Feb 19 revised A function of which $n(x)$-times derivative vanashes for all $x$ is a polynomial? added 9 characters in body Feb 19 revised A function of which $n(x)$-times derivative vanashes for all $x$ is a polynomial? added 458 characters in body Feb 19 asked A function of which $n(x)$-times derivative vanashes for all $x$ is a polynomial? Feb 18 comment Closed Graph Theorem Application Suppose $X,Y,Z$ are Banach spaces and $B\colon X\times Y\to Z$ is bilinear, $B(x,\bullet)\colon Y\to Z$ is continuous for all $x\in X$ and $B(\bullet,y)\colon X\to Z$ is continuous for all $y\in Y$, then $B$ is continuous. (Hint: apply Banach-Steinhaus theorem) Feb 17 comment Is a strongly holomorphic function automatically continuous? More succinctly, $\lim_{w\to z}(w-z)\cdot(w-z)^{-1}(f(w)-f(z))=0$ since scalar multiplication is continuous for TVSs. Feb 17 comment Comparing the U.S. undergraduate math education to the French “classes préparatoires” @Potato Here are reports (and subjects) of international selections, which are easier than those of concours (only math major is considered), but with English editions and still valuable. Feb 17 comment Comparing the U.S. undergraduate math education to the French “classes préparatoires” @Potato In short, it's an exam based on linear algebra and calculus. Maybe some abstract algebra is also assumed, but no more than rudiments of group theory and field theory. Samples: this one is representative. It demands students to prove Liouville's theorem testing whether the primitive of a specific kind of function is elementary. This one is also interesting. It's a special case of Mordell's theorem for elliptic curves. Feb 17 comment Comparing the U.S. undergraduate math education to the French “classes préparatoires” As far as I know, the most brilliant students in France, at the stage of the first year of graduate school, learn much less than those in the United States because they need to practice VERY MUCH to be skilled against concours. I should stress again that the comparison is held between the most brilliant ones, since the variance of levels of U.S. students is very huge. It takes 6 or 4 hours to finish a math exam of ENS concours, and very rare French students can finish them all, so don't be worry if you cannot finish them in 2 hours, say. These concours don't demand you a lot of knowledge. Feb 17 revised Studying mathematics abroad - specifically France added 7 characters in body Feb 17 comment Studying mathematics abroad - specifically France And for the answer: international students are not usually selected through «concour», but a specific selection procedure is organized by ENS Ulm and X, respectively, as far as I know. Feb 17 comment Studying mathematics abroad - specifically France @emka I don't know whether you're still interested, but ENS Paris has a program called sélection internationale which is suitable for international students. No French is required before the examination. Feb 14 awarded Autobiographer Feb 12 revised Let $d \in \mathbb{Z}$, $d > 1$. Determine all the ideals of $\mathbb{Z}/d\mathbb{Z}$ which are prime or maximal edited body; edited title Feb 12 revised What is the mathematical nature of $i$? edited tags Feb 12 comment $ax=0$ if and only if $a=0$ or $x=0$ Hint: if $a\neq0$, then $x=a^{-1}ax$. The theorem fails to be true for Abelian groups. Consider $M=\mathbb Z/n\mathbb Z$, and find a characterization of $ax=0$. Feb 11 comment Can an infinite sum of irrational numbers be rational? @JackM See here or here Feb 3 comment Support of a module with extended scalars Except the obvious one indicating something about the radical ideal. Feb 3 comment Support of a module with extended scalars I'm pretty curious whether we can obtain some relation between $\operatorname{Ann}_A(M)$ and $\operatorname{Ann}_B(M\otimes_AB)$?