Akiva Weinberger
Discussion on question by Md Asfaque:…
To explain my question I need to discuss two notions from differential geometry (the Riemannian curvature tensor and sectional curvature), but this is really a linear algebra / Euclidean geometry question, so I will provide background material.
The Riemann curvature tensor is a certain linear fun...
A directed tree is called an in-tree if all of the edges points towards a root, and an out-tree if all edges point away from a root. (Equivalently, an in-tree is any connected directed graph for which every vertex has out-degree $1$ except for one, called the root, which has out-degree $0$; an ou...
Recall from model theory that two structures are called elementarily equivalent if they satisfy the same first-order sentences. In other words, two structures $\frak A$ and $\frak B$ of the language $\cal L$ are elementarily equivalent (written $\frak A\equiv\frak B$) if $\operatorname{Th}(\frak ...
I was messing around with Wolfram Alpha and eventually found the following surprising fact:
\begin{align*}\frac{\tanh^{-1}(\frac23\sqrt{1+x})-\tanh^{-1}(\frac23)}{\sqrt{1+x}}={}&\frac35x-\frac{21}{100}x^2+\frac{303}{1000}x^3-\frac{7\,533}{40\,000}x^4\\&+\frac{437\,289}{2\,000\,000}x^5-\frac{1\,34...
A fisher is somewhere on the edge of a rectangle-shaped lake, and a fish is somewhere inside of it. The lake measures $a\times b$. What is the probability that the fisher is closer to the center than the fish is? (Assume that the probability distributions of the fisher and fish are both uniform.)...
$\quad$What is the value of the integral
$$\int_0^1\int_0^1\frac{x^a(1-y)^b}{(1-xy)^c}\,{\rm d}x\,{\rm d}y?$$
for nonnegative integers $a,b,c$?
$\quad$For example, setting $b=4$, $c=3$, and letting $a$ vary, it seems that $\displaystyle\int_0^1\int_0^1\frac{x^a(1-y)^4}{(1-xy)^3}\,{\rm d}x\,{\rm d...