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## Properties of $\mathcal{C}^1$-diffeomorphisms which keep invariant the uniform distribution on the n-cube?

Let us consider the n-cube (n-dimensional hypercube) $H_P={]0,\,1[}^P$ and let $\psi:\,H_P\rightarrow{}H_P$ be a $\mathcal{C}^1$-diffeomorphism which keep the uniform distribution (with respect to the Lesbegue measure) on $H_P$ invariant. Is $\psi$ an isometry (with respect to the usual euclidean distance)? Is it possible to express $\psi$ as the composition of rotations and reflections? Thank you!

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## Doubt in bayesian theorem

Is P(A|B) a function of A or B? my doubt is P(A) is function of event A outcome. What about P(A|B)? is it funtion of A outcome? i think no right? P(A|B) is what fraction of P(A,B) in P(B)?? Please elaborate. I am a beginer. If i am making any mistake in terminology or thinking, […]

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## Zero set of the Fourier transform of measures

Let $\mu$ be a probability measure on $\mathbb R$ and the Fourier transform of $\mu$ is defined by $$\widehat{\mu}(\xi) = \int_{\mathbb{R}} e^{-2\pi i \xi x} ~ d \mu(x).$$ Let $\mathcal{Z}(\widehat{\mu})=\{ \xi \in \mathbb R: \widehat{\mu}(\xi)=0 \}$. Question: how about the set $\mathcal{Z}(\widehat{\mu})$? Can we show that the set $\mathcal{Z}(\widehat{\mu})$ is at most countable? […]

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## How do I get bounding lines of planes

I have a problem where I have the origin and normal vector of several planes describing the walls of a building(inner and outer walls). What I don’t have, is the boundaries of those planes. Those boundaries would be on the intersection lines of those planes, or the corners of the rooms/house. My question is, how […]

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