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If $f: [0,1] to mathbb{R}$ is differentiable, $g$ is continuous, and$f'(t) = g(f(t)),$ then $f$ is monotonic.Example 1: Suppose $g(x) = x Rightarrow f'...

Asked on 01/07/2022

1My work so far $${displaystyleint_0^1}dfrac{mathrm{e}^{12x}-mathrm{e}^{-12x}}{mathrm{e}^{12x}+mathrm{e}^{-12x}},mathrm{d}x$$ using substitution with $u=mathrm{e}^{12x}+mathrm{e}^{-12x}$ $$={displaystyleint}dfrac{1}{12u},mathrm{d}u$$ $$=class{steps-node}{cssId{steps-node-1}{dfrac{1}{12}}}{displaystyleint}dfrac{1}{u},mathrm{d}u$$ $${displaystyleint}dfrac{1}{u},mathrm{d}u$$ which is the standard integral: $$=lnleft(uright)$$ $$=dfrac{lnleft(uright)}{12}$$ and using substitution with...

Asked on 01/07/2022

4$G$ is a finite group. Let $H$ be a subgroup of $G$. Is there an example of $G$ and $H$ such that$${rm Card}({Hxhmid...

Asked on 01/07/2022

3First let $N = p^2$. I think that to show that any number less than $N$ is prime, you only have to check to show it has factors...

Asked on 01/07/2022 by Mondo Duke

2Here is part of Exercise 5.5.19 in Dummit & Foote's Abstract Algebra:Let $H$ be a group of order $n$, let $K=operatorname{Aut}(H)$ and $G=operatorname{Hol}(H)=Hrtimes K$ (where ...

Asked on 01/07/2022

1First question, do probability problems generally consisting of two categories always assume two events are dependent? Let's say there are two categories from a set of n people. They are...

Asked on 01/07/2022

0Let $Y$ be a connected normal Noetherian scheme, $f: Xto Y$ is an etale morphism of finite type. We assume $X$ is also connected. I have proved...

Asked on 01/07/2022

0For simplicity, I'll focus on the bivariate case. Let $(X_1,X_2)$ be a random vector that obeys bivariate Bernoulli. $X_i$ takes either zero or one. The associated pdf can...

Asked on 01/07/2022

1I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare"). One example would be "cancelling" the 6's in ...

Asked on 01/07/2022

41Let$B = { mathbb{R} } cup { (a,b) capmathbb {Q} , alt b , a,b inmathbb{Q}}$ Thus, a set $V in B$...

Asked on 01/07/2022 by roslavets

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