arXiv research

New R-equivalence classes found for torus knot diagrams.

Let $A \subset \mathbb{R}^d$, $d\ge 2$, be a compact convex set and let $μ= \varrho_0 dx$ be a probability measure on $A$ equivalent to the restriction of Lebesgue measure. Let $ν= \varrho_1 dx$ be a probability measure on $B_r := \{x\colon |x| \le r\}$ equivalent to the restriction of Lebesgue measure. We prove that t…

Let $Y\subset{\mathbb R}^n$ be a triangulable set and let $r$ be either a positive integer or $r=\infty$. We say that $Y$ is a $\mathscr{C}^r$-approximation target space, or a $\mathscr{C}^r\text{-}\mathtt{ats}$ for short, if it has the following universal approximation property: For each $m\in{\mathbb N}$ and each loc…

In the first part of the paper, comprising section 1 through 6, we introduce a sequence of functions in the tangent bundle TM of any smooth two-dimensional manifold M with smooth Riemannian metric g that correspond to the higher order Schwarzians of the linearized geodesic flow. With these functions and a classical the…