MathOverflow

I am working in or adjacent to machine learning these days and basically everything is a "function approximator." Given a set of data $X$ and observations $y$, I want to construct a function approximator $f(x)$ that 'approximates' the relationship between...

Consider the following system of equations:$$\sigma_{2k}(x_1,x_2,\dots,x_{2n+1}) = (-1)^k \binom{n}{k}, \quad 1 \le k \le n,$$where $\sigma_k(x_1,x_2,\dots,x_{2n+1})$ denotes the $k$-th elementary symmetric polynomial.
Question 1. How can one find integer solutions $(x_1,\dots,x_{2n+1})$ satisfying the following conditions:...

Is it true that any group with arbitrarily long finite chains of subgroups has an infinite chain of subgroups?

Let $\mathfrak T$ be a unital C*-algebra. For a canonical state $\tau$ let $(\pi_\tau,H_\tau,\xi_\tau)$ be its GNS triple, $\mathfrak{M}:=\pi_\tau(\mathfrak T)''$ a von Neumann algebra. For any nonempty convex set $\Delta\subset S(\mathfrak T)$ (i.e...