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22

$e$ appears in the number of derangements The formula for the number of derangements of length $n$ turns out to be $$n! \cdot \sum_{j=0}^n \frac{(-1)^j}{j!}$$ Since the second part is just the standard series for $e^{-1}$ this can also be written as $$\bigl[ \frac{n!}{e} \bigr]$$ where $[ . ]$ denotes the closest integer. This also implies that the ...


16

My favourite, also in the area of probability, is the secretary problem. Copied (with editing) from the Wikipedia site: The task is to hire the best of $n$ applicants for a position. The applicants are interviewed one by one in random order. A decision about each one must be made immediately after the interview. Once rejected, an applicant cannot be ...


16

We have $$e=\lim_{n\to\infty}\sqrt[\large^n]{\text{LCM}[1,2,3,\ldots,n]},$$ where LCM stands for least common multiple.


11

An almost magical appearance of $e$ comes from Pascal's Triangle. Let $s_n$ be the product of the terms on the $n$-th row of the Pascal's Triangle, that is: $$s_n=\prod_{k=0}^n\binom{n}{k}$$ Then $$\lim_{n\to \infty}\frac{s_{n-1}s_{n+1}}{s_n^2}=e$$ A proof of this fact can be found here. I think it's one of the things that struck me the most about ...


8

Here's a nice (longish) one. A sequence of numbers $x_1,x_2,...$ is generated randomly from $[0,1]$. This process is continued so long as the sequence is monotonically increasing or monotonically decreasing. Q: What is the the expected length of the monotonic sequence? The probability that the length $L$ of the monotonic sequence is greater than $k$ is ...


8

I kind of think this is cheating, but Euler's Identity comes to mind: $$e^{i\pi}+1=0$$ This is a specific case of $e^{ix}=\cos x+i\sin x$ when $x=\pi$. Deriving the formula requires only a knowledge of the Taylor expansions of $e^x$, $\sin x$, and $\cos x$. I suppose this is not particularly surprising, but it reveals a very deep connection between ...


6

I am not sure if the result is "most wanted", but we disproved a well-known conjecture of John Milnor, that every connected solvable Lie group admits a left-invariant affine structure, by a very hard computation concerning a refinement of Ado's theorem for Lie algebras. We proved the following result, yielding a counterexample to Milnor's conjecture: ...


5

$e$ finds itself in formulas involving $\pi$. Ramanujan's constant $$e^{\pi \sqrt{163}} = 262537412640768743.99999999999925\ldots \approx 640320^3+744$$ is related to Heegner numbers and has deep connections to number theory. The Gaussian integral $$\int_{-\infty}^{+\infty} e^{-x^2}\,dx = \sqrt{\pi}$$ is related to polar coordinates and thus Euler's ...


4

Pick non-overlapping pairs $(x,x+1)$ of integers from $[1,n]$ until no more pairs can be picked (i.e., until no more consecutive integers remain), and let $p$ be the number of integers that were picked (i.e. $w$ is twice the number of picked intervals). Then you have for the expected value $\mathbb{E}\ p$ of picked integers that $$ \lim_{n\to\infty} ...


4

Not really. Trigonometry in its limited, familiar form is not an area of active research. You will however find trigonometry all over various different fields of mathematical research which are active, just masked under different names and forms. At this point it is mostly applications or generalizations of what you and I would call trigonometry. ...


4

A Lagrange point relative to the positions of two massive objects is a third point with the property that a small object moving under the gravitational influence of the others remains in relative equilibrium. The following image (taken from Wikipedia and licensed under CC) shows the five Lagrange points in a Sun-Earth like system, together with contours of ...


3

The definition of a closed form solution, in particular what people agree to call an 'elementary' function, is culturally determined. A certain solution to a problem might be expressed as a series, or an integral, or some other limit, until its frequent use makes it worthwile having a standard name. That is how natural logarithms became accepted as ...


3

I don't know if this is relevant, but the fact that both $e$ and $\pi$ appear in the Gaussian function defining the normal distribution (which is very important in probability and in statistics) is something I find beautiful : $$f(x) = \frac{1}{\sqrt{2π}}e^{-\frac{x^2}{2}}$$ Another example : $e$ appears in the definition of the moment-generating function ...


3

The graph of $$y=x^x, x>0$$ has minimum value $$y=\left(\frac 1e\right)^{\frac 1e}$$ when $x=\frac 1e$


3

The arithmetic mean of the first $N$ positive integers is about $N/2$. This is easy to justify without any computation. Less obviously, the geometric mean of the first $N$ positive integers is about $N/e$.


3

In section 1.3 of Mathematical Constants by S.R. Finch we find this connection to prime number theory \begin{align*} \lim_{n\rightarrow \infty}\left(\prod_{{p\leq n}\atop{p \text{ prime}}}p\right)^{\frac{1}{n}}=e \end{align*} and also some Wallis-like infinite products \begin{align*} e=&\frac{2}{1}\cdot\left(\frac{4}{3}\right)^{\frac{1}{2}}\cdot ...


2

What is the radius of convergence of the power series $$ f(x)=\sum_{n=0}^\infty \frac{n! x^n}{n^n}? $$ Answer. Exactly $\mathrm{e}$.


2

I would add to the list Dobiński's formula for the $n^{\rm th}$ Bell's number (the number of partitions of an $n$-element set) which is given by $$B_n = {1 \over e}\sum_{k=0}^\infty \frac{k^n}{k!}.$$ When I first saw this formula I was amazed by an appearance of $e$ in a formula for a very concrete natural number.


2

Throw $N$ balls into $N$ bins at random. The probability that any given bin is empty is $e^{-1}$, and thus with high probability the fraction of empty bins is $e^{-1}$.


2

You could look up some interesting topics such as Bernoulli trials, which use Euler's number to approximate probabilities involving large numbers, and Stirling's approximation which provides an approximation for factorials. I always liked the inequality rule that e is the only real number for which the following is true: $$\left(1+\frac{1}{x}\right)^x < ...


2

Closed-form solutions are the exception rather than the rule. They are droplets in an ocean of intractable computations. In the frame of algebra/calculus, we have Linear equations and systems of equations: yes, always (fortunately). Algebraic equations and systems of equations: no (with a few exceptions, like second, third and fourth degree). ...


2

All such functions can be written in the form $$f(z) = \exp(g(z))$$ where $g$ is holomorphic. Such functions certainly satisfy your condition, and the fact that all such functions can be written in this form follows from the existence of the logarithm of non-zero functions.


1

Perhaps you're looking for something like this. In elementary probability/statistics courses one deals with random variables that are either "discrete" or "continuous". The discrete ones have countably many possible values with positive probabilities; the continuous ones have a density (piecewise continuous in all examples, but easily generalized to ...


1

Although you can witness complex (or chaotic) behavior via a simulation with some of the bodies fixed, it usually pays to run a complete numeric solution to the general problem, since any non-fixed bodies can always run chaotically fairly easily. If, on the other hand, you wish to fix some of the bodies, you can tweak the general solution by giving them ...


1

I know I'm a bit late, but I recently came across the lovely book Treks into Intuitive Geometry, which consists of a series of conversations between Gen and Kyu, as can be seen in the image below. The whole book appears to be written in this format.


1

Lawvere and Schanuel, Conceptual Mathematics: A first introduction to categories has many such passages, reflecting the classroom atmosphere from when the material was first taught.


1

You cannot say it is dead. Much of algorithm are available in software package. You might conisder there is no new breakrthough in theory or algorithms.


1

For a few examples in recent TCS (Theory of Computer Science): http://arxiv.org/abs/1111.0837 (STOC 2012) http://arxiv.org/abs/1311.2369 (STOC 2014) and a blog post about these. This is more about negative results -- i.e., trying to rule out approaches that would show $\sf P=NP$, by (roughly speaking) proving that any linear programming formulation of ...


1

I've got quite a few: $$f(x)=\frac{1}{2}e^{-|x|}$$ $$f(x)=\frac{1}{\sqrt \pi} e^{{-x}^2}$$ And on and on. The point I am trying to prove it that any function $\phi$ such that $$\int_{-\infty}^\infty \phi(x)=B \text{ s.t. }$$ $$B\in\Bbb{R}$$ Can give yet another solution to your integral: $$f(x)=\frac{1}{B}\phi(x)$$


1

Here is Euler's Other Proof by Gerald Kimble \begin{align*} \frac{\pi^2}{6}&=\frac{4}{3}\frac{(\arcsin 1)^2}{2}\\ &=\frac{4}{3}\int_0^1\frac{\arcsin x}{\sqrt{1-x^2}}\,dx\\ &=\frac{4}{3}\int_0^1\frac{x+\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!}\frac{x^{2n+1}}{2n+1}}{\sqrt{1-x^2}}\,dx\\ &=\frac{4}{3}\int_0^1\frac{x}{\sqrt{1-x^2}}\,dx ...



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