# All Questions

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### Pear Orchard (word problem)

A farmer has $70$ acres on which to plant a pear orchard. three neighboring farms with similar soil conditions already established orchards. one of these orchards has $250$ tress planted per acre ...
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### If $0< 2\varepsilon < \sigma^2 < 1$ then $\prod\limits_{i = 1}^n (1 + \varepsilon + \sigma \xi _i )$ converges almost surely to $0$

I posted this question a few days ago and there were some errors in my post. I have fixed them and it should be all right now. Hope someone can help with my confusion. Let $(X_n)_{n\ge0}$ denote an ...
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### Nilpotent differential operators of degree $2$?

I've reflecting about global solvability of a class of (pseudo-)differential operators. In the middle of my researchs a condition like $P^2=0$ appeared where $P$ is a differential operator. However, ...
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### Size of group of roots of unity

Let $p$ be an odd prime and $G=(\mathbb{Z}/p^N\mathbb{Z})^{\times}$. Let $a=p^{N-1}$, $b=p-1$, $A=\{g\in G: g^{a}=1\}$, $B=\{g\in G: g^{b}=1\}$. Prove that $A,B$ are subgroups of size $a,b$ ...
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### Showing Markov kernel properties

For $a\in\mathbb R$ and $\sigma^2>0$ let $\phi_{a,\sigma^2}$ be the Gauss distribution on $(\mathbb R, \mathcal B)$ with the expectation $a$ and variance $\sigma^2$. Show that ...
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### Change of the sequence of differentation in physics?

Assume to have a quantity A which is calculated from the formula $A=\frac{dB}{dC}$. dC can be written as dC=dEdF so $A=\frac{dB}{dEdF}$. I assume that the differential of A is also ...
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### Ordinal Numbers within Simple Type Theory

A little preamble: I recently came across the system of higher-order logic known as simple type theory, and I was immediately intrigued by it, as it seemed to be exactly what I was looking for as an ...
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### Verbiage in book, …uniformly in $\theta$ belonging to compact subsets…

I am reading Statistical Estimation and Asymptotic Theory by Ibragimov and Has'minskii and was confused by the following passage: ... and that we have uniformly in $\theta$ belonging to the compact ...
### Prove that $\mathbb{Z}$-module is flat.
Let $p$ be prime and let $\mathbb{Z}_{(p)} \subset \mathbb{Q}$ denote the set of all fractions $n/q$ for which $p \nmid q$. Is it true that $\mathbb{Z}_{(p)}$ is flat as a $\mathbb{Z}$-module? ...
Suppose $\Omega=(0,1)$, $\mathcal{B}$ is the Borel $\sigma$-algebra on $(0,1)$ and $P$ the Lebesgue measure. Let $\Phi:\mathbb{R}\to(0,1)$ denote the standard normal CDF. Then, we can generate \$X\sim ...