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3

The maximum of a large number of independent, identically distributed random variables -- e.g., the highest flood observed over a long period -- has an extreme value distribution. One common case is the Gumbel distribution, whose cdf has the double-exponential form $e^{-e^{-x}}$.


4

Gompertz model was created in 1825 to study human mortality curves. From the 1920's, it was used in economic fields, and from there it was also used in Biology to study cells and microorganisms, such as microbes, growth of tumours, and survival of cancer patients. The model is described by the differential equation $$\dfrac{dN}{dt} = rN \ln \left(\dfrac{K}{N}...


1

Let n be given and consider the set of logical formulas with $n$ variables. Two formulas are equivalent if they are both true, or both false, for any assignment of truth values to the variables. For example, the formulas $\lnot a\land b$ and $\lnot(a\lor\lnot b)$ are equivalent, because both are true only when $a$ is false and $b$ is true. This ...


8

The number of undirected graphs of order $n$ is $2^{n \choose 2} = 2^{(n^2-n)/2}$ (e.g. consider the adjacency vector as a binary-encoded number). You can describe this in terms of the number of different arrangements in which $n$ classmates can be friends. This also has the benefit of being easy to enumerate for small numbers.


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What a nice collection of answers! I am inclined to use $\lfloor A^{3^n} \rfloor$ where $A$ is Mill's constant, $$A \approx 1.3063778838630806904686144926 \;, $$ just because it is astounding that this evaluates to a prime for every $n \in \mathbb{N}$: $$ 2, 11, 1361, 2521008887, \ldots \;. $$ See A051254. But really, @JoelReyesNoche's answer is best.


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I think there are really two different ways that this can occur. (1) You can have people writing math in real-world contexts where it makes sense to use a notation that is this type of tower of exponents. (2) You can have a function of $x$ that blows up like $a^{b^x}$ or $a^{x^b}$ as $x\rightarrow\infty$. I'm a physicist, and the contexts where I see #1 are ...


3

It is easy to square a number. So instead of computing $\exp(z)$ you can compute $\exp(z/2^k)^{2^k}$ for some suitably large positive integer $k$. This is a very simple way of accelerating the convergence of the Taylor series for $\exp$, also known as "argument reduction". The double-exponential is important here because it allows you to get the $2^...


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If $S$ represents the set of $n$ students of a school, then $\mathscr P(S)$ is the set of rosters for all possible clubs that that school could host (if we assume that any two clubs with exactly the same set of students merged their interests into a single club charter). In that same scenario, the yearbook that printed pictures of all of the clubs would be ...


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Count all possible functions mapping $i$ input bits to $o$ output bits: For each of the $2^i$ input combinations, each output has two possible outputs ($0$ and $1$), i.e. you have $2^{2^i}$, leading to a total of $2^{o\cdot 2^i}$ binary functions. And course if you're working with $n$ different values per input/output instead of binary you end up with $n^{o\...


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If your students have learned some statistics, then you could point out that the normal distribution's probability density function uses this double exponential. $$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right)$$


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Inspired by your example, I like the number of matrices, tensors, binary or n-ary relations over a specified set (where the domain and range of such a relation need not match), such as asking about colorings using $k$ colors of a $n \times n$ board. Continuous or "non-combinatorial" applications feel hard to come up with, perhaps due to ...


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