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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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Center of mass of a Space blunder by two parametrische curves

Is there a way to calculate the centre of mass of a space bound by two general parametric curves. To be more specific, if you have an outer curve defined by: $X_o(t)$ and $Y_o(t)$, and an inner curve defined by: $X_i(t)$ and $Y_i(t)$. How would you solve the integral: $${\displaystyle \mathbf {R} =\iint\limits _{Q}(\mathbf {r} […]

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input output function difference in cryptography

Observations: Let 𝑋,𝑋prime be 𝑙-bit values, and Δ𝑋 =𝑋⊕𝑋prime. There are two difference properties of AND and OR operations: (𝑋∧𝐾) ⊕ (𝑋prime∧𝐾) = Δ𝑋∧𝐾, (𝑋∨𝐾) ⊕ (𝑋prime∨𝐾) = Δ𝑋⊕ (Δ𝑋∧𝐾) . Given output difference Δ𝑌 = Δ𝑌𝑙 ‖ Δ𝑌𝑟 (‖ is concatenate operator) and the key value 𝐾𝐿 = (𝐾𝐿1, 𝐾𝐿2) of F function, the […]

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Evaluate $\lim_{(x,y) \to (0,0)}\frac{y\sin(x^2y)}{x^2+y^2}$

Evaluate (if exists) $\lim_{(x,y) \to (0,0)}\frac{y\sin(x^2y)}{x^2+y^2}$. $$\left|\frac{y\sin(x^2y)}{x^2+y^2}\right|\leq\left|\frac{x^2y^2}{x^2+y^2}\right|\leq\left|\frac{x^2y^2}{2xy}\right|=\left|\frac{xy}{2}\right|\to0.$$ But wolframalpha says: no. The limit doesn’t exist. But is it really true? It would mean that $\sin(x^2y)$ cannot really be bounded by it’s argument.

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A fantasy game on the Angels&Demons novel

CONTEXT Recently, I have been reading the novel “Angels & Demons” by Dan Brown and I was kind of fascinated by the plot. This problem is inspired by the novel. PROBLEM Assume that the Vatican is a $2019\times 2019$ square. Cameras are placed at the vertices of unit squares. Each camera can cover four unit […]

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$h(0)=0$ and $h \in C^k$ imply $\frac{h(x)}{x} \in C^{k-1}$

This is a self-answered question. I post it here since it wasn’t obvious to me. (although I have seen similar questions-is it an exact duplicate?). Let $h:\mathbb{R}\to\mathbb{R}$ be $C^k$ and suppose that $h(0)=0$. Define $$ F(x) = \begin{cases} \frac{h(x)}{x} & \text{if $x\neq 0$} \\ h'(0) & \text{if $x=0$}\end{cases} $$ Then $F$ is $C^{k-1}$.

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How do I integrate exponential functions including $e^{x^2}$?

I am trying to prove the following equality: $$\lambda\int^\infty_{-\infty}\frac{\mathrm df}{\sqrt{2\pi}}\exp\left(i2\pi f(x-a)\right)\exp\left(\frac{-i\pi\lambda tf^2}{n}\right)=\frac{1}{\sqrt{\frac{2\pi it}{n\lambda}}}\exp\left(\frac{i\pi n(x-a)^2}{\lambda t}\right)$$ but am struggling to work out which method to use. I assume somewhere in this I need to use $$\int^\infty_{-\infty}e^{-f^2}\,\mathrm df=\sqrt{\pi}$$ but I haven’t currently got much further than that, can anyone advise on this?