On some notions of good reduction for endomorphisms of the projective line

Let $\Phi$ be an endomorphism of \SR(\bar{\Q}), the projective line over the algebraic closure of \Q, of degree $\geq2$ defined over a number field $K$. Let $v$ be a non-archimedean valuation of $K$. We say that $\Phi$ has critically good reduction at $v$ if any pair of distinct ramification points of $\Phi$ do not collide under reduction modulo $v$ and the same holds for any pair of branch points. We say that $\Phi$ has simple good reduction at $v$ if the map $\Phi_v$, the reduction of $\Phi$ modulo $v$, has the same degree of $\Phi$. We prove that if $\Phi$ has critically good reduction at $v$ and the reduction map $\Phi_v$ is separable, then $\Phi$ has simple good reduction at $v$.Comment: 15 page

Limiting and sill-controlled adverse-slope hydraulic jump

This note analyzes, from a theoretical and experimental point of view, hydraulic jumps on adverse slopes in rectangular prismatic channels. The analysis is carried out for the classical adverse-slope hydraulic jump and the jump forced by the presence of a sill. Data collected in two series of experiments involving different equipment were added to available results to obtain a general relationship for the sequent depth ratio as a function of the upstream Froude number and adverse-slope angle. The presence of a sill stabilizes the jump