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Consider a measure space $(S, Sigma, mu)$ and the normed vector space $mathcal{L}^2(mu)$. Then for any measurable function $f: S to mathbb{R}$ with $f in mathcal{L}^2(mu)$ the norm is...
Asked on 12/29/2021 by bcf
3 answerGiven a smooth real function $f$, we can approximate it as a sum of polynomials as$$f(x+h)=f(x)+h f'(x) + frac{h^2}{2!} f''(x)+ dotsb = sum_{k=0}^n frac{h^k}{k!} f^{(k)}(x) + h^n...
Asked on 12/29/2021
7 answerExample 1Rational tangle dance mentioned here with operations:T(tangle)R(rotate)For example, sequence of operations $TTRTT$ is considered as $T^2 cdot R cdot T^2$ but not $2T...
Asked on 12/29/2021 by user1787812
1 answerLet $U sim Unif(S^{d-1}).$ I was wondering if it's true that, and if yes, how could we prove that: $U = frac{Z}{|Z|}$ where $Z sim mathcal{N}(0, I_d),...
Asked on 12/29/2021
1 answerI have measured the parameters for a hyperbola and an ellipse, let us call them$$begin{cases} a^2x^2 + b^2y^2 = 1 \ c^2x^2 + d^2y^2 = 1 end{cases}...
Asked on 12/29/2021 by a20
1 answerLet $X$ and $Y$ be compact and closed subset of a metric spaces $M$ respectively, then prove that $inf {d(x,y)|xin X, yin Y}=d(x,y)$ for some ...
Asked on 12/29/2021
1 answerHow to show that the distance of the points of tangency along a tangent line on two tangent circles with radius $a$ and $b$ is equal to ...
Asked on 12/29/2021
2 answerSolve in $mathbb{R}$ $begin{cases}&x+y+frac{x^2}{y^2}=7 ~cdots (text{I})\&frac{(x-y)x^2}{y^2}=12~~~~~ cdots (text{II})\end{cases}$ A friend's attempt:$ I.(x-1)= x^2 - 7x + 12= y^2 - 7y rightarrow x...
Asked on 12/29/2021 by peta arantes
3 answerI have a double integral: $$iint (x+y),dx, dy$$ with circle constraint:$$x^{2}+y^{2}=x+y$$ I tried to calculate it with transition to polar coordinates: $$x^{2}+y^{2}=x+y$$$$left(x-frac{1}{2}right)^{2}+left(y-frac{1}{2}right)^{2}=frac{1}{2}$$ In polar...
Asked on 12/29/2021
3 answerLet $I=Atimes B,$ where $A,Bsubset mathbb{R}$ are closed sets of positive Lebesgue measure, and $Esubset mathbb{R}^2,$ be a set of zero Lebesgue measure. Is it true that...
Asked on 12/29/2021 by Prof.Hijibiji
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