# All Questions

2k views

### Motivation behind standard deviation?

Let's take the numbers 0-10. Their mean is 5, and the individual deviations from 5 are -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 And so the average (magnitude of) ...
331 views

615 views

760 views

### Classification of prime ideals of $\mathbb{Z}[X]$

Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable. My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this? (1) ($0$) (2) ...
3k views

### Taking Seats on a Plane

This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless Imagine there are a 100 people in line to ...
837 views

### Show $\sum_{n=0}^\infty\frac{1}{a^2+n^2}=\frac{1+a\pi\coth a\pi}{2a^2}$

How to show the following equality? $$\sum_{n=0}^\infty\frac{1}{a^2+n^2}=\frac{1+a\pi\coth a\pi}{2a^2}$$
834 views

### Prove: $\int_0^1 \frac{\ln x }{x-1} d x=\sum_1^\infty \frac{1}{n^2}$

I'd like your help with proving that $$\int_0^1 \frac{\ln x }{x-1}d x=\sum_{n=1}^\infty \frac{1}{n^2}.$$ I tried to use Fourier series, or to use a power series and integrate it twice but it didn't ...
2k views

### How calculate the probability density function of $Z = X_1/X_2$

Let $X_1$ and $X_2$ be two continuous r.v., my question is: what is the p.d.f of $Z=X_1/X_2$?
462 views

571 views

### Which sets are removable for holomorphic functions?

[Note: I received a version of this question via email and decided to answer it on MSE, where it might be useful to others.] Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some ...
4k views

### Prove $2^{1/3}$ is irrational.

Please correct any mistakes in this proof and, if you're feeling inclined, please provide a better one where "better" is defined by whatever criteria you prefer. Assume $2^{1/2}$ is irrational. ...
1k views

### Comparing $\pi^{e}$ and $e^{\pi}$

How can I calculate without calculator or something like this the values of $\pi^{e}$ and $e^{\pi}$ in order to compare them ?
4k views

### What does closed form solution usually mean?

This is motivated by this question and the fact that I have no access to Timothy Chow's paper What Is a Closed-Form Number? indicated there by Qiaochu Yuan. If an equation $f(x)=0$ has no closed form ...
### How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?
How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not a clue as to ...