Recent Questions - MathOverflow

Let $A\in\{0,1\}^{n\times n}$ be a $n\times n $ matrix with entries in the discrete set $\{0,1\}$.

My question: What is the number of matrices in $\{0,1\}^{n\times n}$ that are normal, that is, that satisfy $AA^\top-A^\top A=0$?

If we restrict the attention to the subclass of symmetric matrices, my question becomes quite trivial. However, t...

Fix $n$ a natural number. Consider the set of all $2n \times 2n$ matrices with entries from {0,1}. This is clearly a finite set. I would like to count the number of such normal matrices for fixed $n$, call this number $N(n)$. What are the asymptotics for $N(n)$? It is easy to see that if the matrix is symmetric, then it is normal so we can get a lower bound on $N(n)$. Does an...

I am interested in exponential Diophantine equations. Recently, I have been investigating the following equation with mixed bases and exponents:

$$x^z + y^y = z^x$$

For positive integer solutions $(x, y, z) \in \mathbb{Z}^+$, it is easy to find some trivial families if we allow $x=1$ (for instance, $(1, y, 1+y^y)$ is a solution for any $y$).

However, look...

The following is perhaps a standard question, but i could not find a plain enough answer by simply searching online.

Q: Given a knot $K$ and its $(p,q)$-cable $K_{p,q}$ what is a relation between the Vassiliev invariants of $K$ and $K_{p,q}$?

In particular, I would be happy with a formula for the 2nd coefficient of the Conway polynomial. (One may attempt to s...

I propose a new approach to the Collatz (3n+1) conjecture based on a quantity I call Entropic Inertia:

λ(n) = v₂(n+1)

where v₂ denotes the 2-adic valuation. I build around it a framework I name "Arithmetic Thermodynamics." The core idea: each growth step (n → 3n+1) consumes exactly one unit of Entropic Inertia, and a "Depreciation Theorem" suggests this inertia ...