# All Questions

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### Help showing subadditivity of a map

I'm stuck with the following problem. Show that the map: $$r(x)=\inf\limits_{k\in\mathbb{N}}\limsup\limits_{m\to\infty}\frac{1}{k}\sum\limits_{j=0}^{k-1}S^j(x)(m)$$ is subadditive on ...
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### exp(ab) decomposition

How can one write $e^{a(x) \cdot b(x)} = c(x) e^{ b(x) }$ with $c(x)$ not implicitly depending on $b(x)$. I do not believe this is generally possible so alternatively one can use an infinite series or ...
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### Collapsing the boundary of a Möbius Strip to a point

I'm strunggling to prove that when we collapse the boundary of a Möbius strip we obtain the RP² thanks
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### Determining the probability density function from an equation

I have the following (for me quite interesting) densities for which I am completely stuck. I only hope that you can provide me some help. Let me introduce my problem. I have two probability ...
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### Definition of $\operatorname{arg}(z)$ choosing value of $\operatorname{Arg}(z)$

On what curve is $\arg (z)$ discontinuous if it is defined as the value of $Arg(z)$ satisfing the inequality: $$|z|-2\pi<\operatorname{Arg}(z)\leq|z|$$ would it be a ray from the origin with ...
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### Counting multidimensional structures (Chomp game states)

The game Chomp is described as follows on Wikipedia: Chomp is a 2-player game of strategy played on a rectangular "chocolate bar" made up of smaller square blocks (rectangular cells). The ...
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### Finding probabilities of $X_{(N)}$ (order statistics)

Draw N samples from unif(a,b) and consider the largest realization $X_{(N)}$. What's the probability that this value falls within a certain subset of (a, b)? What expression has to be integrated ...
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### Verifying a bijection

Let $V$ be vector space over $\mathbb{F}$, and $W\subseteq V$ a subspace. Let $p:V\rightarrow V/W$ be the canonical projection. Let $X$ be the set of all subspaces containing $W$ and $Y$ be the sets ...
For some additional excitement, I've been searching for primes $p \gg q = 104729$, where $q$ is of course the ten-thousandth prime. It seems that the best way to search for prime candidates $p$ is to ...
Result: Fix $a \in \mathbb{R}$. Then $(\mathbb{R} \backslash \{a\}, *)$ is a group, where our group operation is defined by $x*y = (x-a)(y-a) + a$. One consequence of this is the standard fact that ...