Geometric convergence results for closed minimal surfaces via bubbling analysis

We present some geometric applications, of global character, of the bubbling analysis developed by Buzano and Sharp for closed minimal surfaces, obtaining smooth multiplicity one convergence results under upper bounds on the Morse index and suitable lower bounds on either the genus or the area. For instance, we show that given any Riemannian metric of positive scalar curvature on the three-dimensional sphere the class of embedded minimal surfaces of index one and genus $$\gamma $$ γ is sequentially compact for any $$\gamma \ge 1$$ γ ≥ 1 . Furthemore, we give a quantitative description of how the genus drops as a sequence of minimal surfaces converges smoothly, with mutiplicity $$m\ge 1$$ m ≥ 1 , away from finitely many points where curvature concentration may happen. This result exploits a sharp estimate on the multiplicity of convergence in terms of the number of ends of the bubbles that appear in the process.

On the Morse index of branched Willmore spheres in 3-space

We develop a general method to compute the Morse index of branched Willmore spheres and show that the Morse index is equal to the index of certain matrix whose dimension is equal to the number of ends of the dual minimal surface (when the latter exists). As a corollary, we find that for all immersed Willmore spheres $$\vec {\Phi }:S^2\rightarrow \mathbb {R}^3$$ Φ → : S 2 → R 3 such that $$W(\vec {\Phi })=4\pi n$$ W ( Φ → ) = 4 π n , we have $$\mathrm {Ind}_{W}(\vec {\Phi })\le n-1$$ Ind W ( Φ → ) ≤ n - 1 .

Schreier graphs of spinal groups

We study Schreier dynamical systems associated with a vast family of groups that hosts many known examples of groups of intermediate growth. We are interested in the orbital graphs for the actions of these groups on [Formula: see text]-regular rooted trees and on their boundaries, viewed as topological spaces or as spaces with measure. They form interesting families of finitely ramified graphs, and we study their combinatorics, their isomorphism classes and their geometric properties, such as growth and the number of ends.

Coarse cohomology with twisted coefficients

To a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

Positive solutions to Schrödinger equations and geometric applications

A variant of Li–Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schrödinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant Sobolev inequality. Consequently, a finiteness result for the number of ends follows. In the case when the Sobolev inequality is of particular type, the finiteness result is proven directly. As an application, an estimate on the number of ends for shrinking gradient Ricci solitons and submanifolds of Euclidean space is obtained.

Structure theorems for singular minimal laminations

We apply the local removable singularity theorem for minimal laminations [W. H. Meeks III, J. Pérez and A. Ros, Local removable singularity theorems for minimal laminations, J. Differential Geom. 103 (2016), no. 2, 319–362] and the local picture theorem on the scale of topology [W. H. Meeks III, J. Pérez and A. Ros, The local picture theorem on the scale of topology, J. Differential Geom. 109 (2018), no. 3, 509–565] to obtain two descriptive results for certain possibly singular minimal laminations of {\mathbb{R}^{3}}. These two global structure theorems will be applied in [W. H. Meeks III, J. Pérez and A. Ros, Bounds on the topology and index of classical minimal surfaces, preprint 2016] to obtain bounds on the index and the number of ends of complete, embedded minimal surfaces of fixed genus and finite topology in {\mathbb{R}^{3}}, and in [W. H. Meeks III, J. Pérez and A. Ros, The embedded Calabi–Yau conjectures for finite genus, preprint 2018] to prove that a complete, embedded minimal surface in {\mathbb{R}^{3}} with finite genus and a countable number of ends is proper.

Content Analysis of Electronic Nicotine Delivery System Publications in Core Clinical Journals from 2012 to 2018

There is no consensus if electronic nicotine delivery systems (ENDS) should be used to reduce harm among the smoking population. Physicians, who represent a trusted source of health communication, are exposed to a range of often conflicting ENDS information and this information may be relayed to patients looking to quit smoking. Previous studies have examined ENDS content of various sources of media but there is a lack of knowledge about ENDS content in medical journals. We analyzed 421 ENDS publications printed between 2012 and 2018 from PubMed’s Core Clinical Journal list. Publications were analyzed based on publication type, journal type, study design, geographic focus, theme, terminology, outcomes, and positive/negative statements. The number of ENDS publications grew since 2012, and peaked in 2015. Across all years, negative statements about ENDS outnumbered positive statements, though this difference decreased over time. Over time, negative statements about “toxins and carcinogens” were made less frequently, while negative statements about “gateway effects” and “youth appeal” became more prevalent. UK journals had fewer negative statements relative to US journals. Only 12.6% of publications included guidance for healthcare professionals about ENDS. As published ENDS topics change over time, physicians’ communications with patients may be impacted.

Rational discrete first degree cohomology for totally disconnected locally compact groups

It is well known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group G can be detected on the cohomology group H1(G,R[G]), where R is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group G the same information about the number of ends of G in the sense of H. Abels can be provided by dH1(G, Bi(G)), where Bi(G) is the rational discrete standard bimodule of G, and dH•(G, _) denotes rational discrete cohomology as introduced in [6].As a consequence one has that the class of fundamental groups of a finite graph of profinite groups coincides with the class of compactly presented totally disconnected locally compact groups of rational discrete cohomological dimension at most 1 (cf. Theorem B).

Intelligent Tutoring Systems

The importance of intelligent tutoring systems has rapidly increased in past decades. There has been an exponential growth in the number of ends users that can be addressed as well as in technological development of the environments, which makes it more sophisticated and easily implementable. In the introduction, the paper offers a brief overview of intelligent tutoring systems. It then focuses on two types that have been designed for education of students in the tertiary sector. The systems use elements of adaptivity in order to accommodate as many users as possible. They serve both as a support of presence lessons and, primarily, as the main educational environment for students in the distance form of studies – e-learning. The systems are described from the point of view of their functionalities and typical features that show their differences. The authors conclude with an attempt to choose the best features of each system, which would lead to creation of an even more sophisticated intelligent tutoring system for e-learning.