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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
1 answerMy 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 answer$G$ 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
3 answerFirst 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
2 answerHere 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
1 answerFirst 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
0 answerLet $Y$ 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
0 answerFor 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
1 answerI'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
41 answerLet$B = { mathbb{R} } cup { (a,b) capmathbb {Q} , alt b , a,b inmathbb{Q}}$ Thus, a set $V in B$...
Asked on 01/07/2022 by roslavets
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