Jun 4, 2012 · A remark: regardless of whether it is true that an infinite union or intersection of open sets is open, when you have a property that holds for every finite collection of sets (in …

I was playing with my calculator when I tried $1.5!$. It came out to be $1.32934038817$. Now my question is that isn't factorial for natural numbers only? Like $2!$ is $2\\times1$, but how do …

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Nov 11, 2018 · The Integration by Parts formula may be stated as: $$\\int uv' = uv - \\int u'v.$$ I wonder if anyone has a clever mnemonic for the above formula. What I often do is to derive it …

Nov 12, 2015 · J. P. Aubin, Un théorème de compacité, C.R. Acad. Sc. Paris, 256 (1963), pp. 5042–5044. It seems this paper is the origin of the "famous" Aubin–Lions lemma. This lemma …

When can we say a multiplicative group of integers modulo $n$, i.e., $U_n$ is cyclic? $$U_n=\\{a \\in\\mathbb Z_n \\mid \\gcd(a,n)=1 \\}$$ I searched the internet but ...

Mar 27, 2019 · I know the proof using binomial expansion and then by monotone convergence theorem. But i want to collect some other proofs without using the binomial expansion. *if you …

Apr 28, 2017 · You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation …

(If you know about ring theory.) Since $\mathbb Z_n$ is an abelian group, we can consider its endomorphism ring (where addition is component-wise and multiplication is given by …

Oct 2, 2011 · I need to graph a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? I can't find it anywhere.