We give a “second generation” exposition of the slope filtration theorem for modules with Frobenius action over the Robba ring, providing a number of simplifications in the arguments. Some of these are inspired by parallel work of Hartl and Pink, which points out some analogies with the formalism of stable vector bundles.

Abstract Background Nitrogen (N) saturation theory proposes that an ecosystem might switch from N limitation to carbon (C), phosphorus (P), or other nutrient limitations if it receives continuous input. Yet, after is removed, which the most limited and whether topography modulates such change rarely tested at a microbial level. Here, we conducted two-year addition experiment under two different...

To explore the water flow characteristics under action of floating rigid vegetation groups, several laboratory tests are conducted in a flume North Minzu University, which is replaced by glass rods. In tests, two groups placed at different positions and length into changed. Experimental results show that both upstream level group surface slope along way grow with increasement root. When root be...

In this paper, we determine geometric information on slope lengths of a large class of knots in the 3–sphere, based only on dia-grammatical properties of the knots. In particular, we show such knots have meridian length strictly less than 4, and we find infinitely many families with meridian length approaching 4 from below. Finally, we present an example to show that, in contrast to the case of...

In [1], Berger and Colmez introduced a theory of families of étale (φ,Γ)-modules associated to families of p-adic Galois representations. Inspired by their works, we consider families of φ-modules over a reduced affinoid algebra where the base is fixed by Frobenius actions. We prove some results in this framework including the semicontinuity of variation of HN-polygons. These results will be ap...