Mixing solutions for the Muskat problem with variable speed

We provide a quick proof of the existence of mixing weak solutions for the Muskat problem with variable mixing speed. Our proof is considerably shorter and extends previous results in Castro et al. (Mixing solutions for the Muskat problem, 2016, arXiv:1605.04822) and Förster and Székelyhidi (Comm Math Phys 363(3):1051–1080, 2018).

Circles, Epistemic and Benign

If we reject level-crossing principles, some apparently circular reasoning becomes licensed. This looks like a problem for normative externalism. This chapter responds to the problem. The response takes some time, because it turns out there are many different ways to understand what it is for reasoning to be circular. The ultimate argument is that for every such way, either it is not problematic, or normative externalism does not license it. But there is no quick proof of this; each way to understand circular reasoning has to be treated separately. The chapter ends with a discussion of the Problem of Easy Knowledge, and of the norms for proper testing of measuring devices.

On the Blow-Up of Solutions to Liouville-Type Equations

We estimate some complex structures related to perturbed Liouville equations defined on a compact Riemannian 2-manifold. As a byproduct, we obtain a quick proof of the mass quantization and we locate the blow-up points.

Graph Properties and Stratified Presentations of Partially Ordered Sets

In this paper, we introduce some new definitions such as the U* L* condition to describe the zero-divisor graph G=Γ(P) of a poset P, and give a new and quick proof to a main result in [2, 4]. By deleting a typical vertex with least degree, we provide an algorithm for finding a maximum clique of a finite graph G. We study some properties of the zero-divisor graphs of posets concerning diameters and girths. We also provide stratified presentations of posets.