MathOverflow

Ref 1: https://nandacumar.blogspot.com/2014/12/filling-plane-with-non-congruent-pieces.html?m=1
Ref 2: To tile the plane with mutually non-congruent rational triangles of equal area
Question: Is it possible to tile the plane with mutually non-congruent triangles all of the same perimeter? No other constraints on the triangles - their areas could range from 0 to a finite upper bound. The answer seems "yes" but I have no clarity....

Let
$T(n,m,k)$ be a coefficients defined for an arbitrary $m,k$ (with single exception at $k=1$) with exponential generating function $$ \left(\frac{(1-(m-k)x)}{(1-(m-1)x)^k}\right)^{\frac{1}{k-1}}. $$
I conjecture that $T(n,m,k)$ is also the determinant of the $n \times n$ matrix, where the $(i,j)$-th entry is $$ \begin{cases}mi & \textrm{if } i = j \\kj & \textrm{if } j < i \\i & \textrm{otherwise}\end{cases} $$...

What mathematical structures are typically used to represent composition that depends on a mutable resource state?
In ordinary category theory, composition is defined unconditionally whenever domains and codomains match.
Consider a setting with a resource state $\Gamma$ (e.g., a budget, a counter, an availability set) that can change when morphisms are composed. Composition may only be admissible when $\Gamma$ satisfies certain conditions, and performing the composition...

Let $A$ be a discrete commutative ring, and take elements $f_1,\dots,f_n$.We define the derived quotient (as animated rings) by $A/^{\mathbb{L}}(f_1\dots f_n):= A\otimes^{\mathbb{L}}_{\mathbb{Z}[x_1,\dots,x_n]}\mathbb{Z}$. Then, we can compute the homology...