6 problems

Stochastic embedding sufficiency conjecture. The bound

Embedding conjecture. It has been conjectured that, for a larger class of nonlinear discrete-time reservoir systems, the associated GS is an embedding; the same may also hold for c…

Let $d4a4^s(d6a9)$ denote the fractional Sobolev space on the domain $d6a9$, and let $d700_s(q)$ and $d4c0_s(q)$ be the two extremal functions defined in the source for $q>2$. Naza…

Consider an ESN driven by one-dimensional observations of a dynamical system on a manifold $M$, with random reservoir matrix $A$ and input matrix $W^{\text{in}}$. Let $f$ denote th…

Let $M$) be a compact $m$-manifold and let $n\in\mathbb{N}$ satisfy $n>2m$. Let $A$ be an $n\times n$ matrix with $\lVert A\rVert_2<\min(1/\lVert D\phi^{-1}\rVert_{\infty},1)$, let…

Let $r,k\in\mathbb{Z}_+$ with $r>k$, and let $1\leq p,q\leq\infty$. Define … where the minimum is taken over … An extremal is a function attaining this minimum. Symmetry conjecture…