Bing and Moise proved, independently, that any Peano continuum admits a length metric d. We treat non-degenerate Peano continua with a length metric as evolution systems. For any compact length space ...(X,d) we consider a semiflow in the hyperspace 2X of all non-empty closed sets in X. This semiflow starts with a canonical copy of the Peano continuum (X,d) at t=0 and, at some time, collapses everything into a point. We study some properties of this semiflow for several classes of spaces, manifolds, graphs and finite polyhedra among them.
We prove that a power quasi-symmetric (or PQ-symmetric) homeomorphism between two complete metric spaces can be extended to a quasi-isometry between their hyperbolic approximations. This result can ...be used to prove that two visual Gromov hyperbolic spaces are quasi-isometric if and only if there is a PQ-symmetric homeomorphism between their boundaries with bounded visual metrics. Also, in the case of trees, we prove that two geodesically complete trees are quasi-isometric if and only if there is a PQ-symmetric homeomorphism between their boundaries with visual metrics based at infinity. We also give a characterization for a map to be PQ-symmetric based on the relative distortion of subsets.
In this paper we prove that if we consider the standard real metric on simplicial rooted trees then the category Tower-Set of inverse sequences can be described by means of the bounded coarse ...geometry of the naturally associated trees. Using this we give a geometrical characterization of Mittag-Leffler property in inverse sequences in terms of the metrically proper homotopy type of the corresponding tree and its maximal geodesically complete subtree. We also obtain some consequences in shape theory. In particular we describe some new representations of shape morphisms related to infinite branches in trees.
Parabolicity on Graphs Martínez-Pérez, Álvaro; Rodríguez, José M.
Resultate der Mathematik,
03/2024, Letnik:
79, Številka:
2
Journal Article
Recenzirano
Odprti dostop
Large scale properties of Riemannian manifolds, in particular, those properties preserved by quasi-isometries, can be studied using discrete structures which approximate the manifolds. In a sequence ...of papers, M. Kanai proved that, under mild conditions, many properties are preserved by a certain (quasi-isometric) graph approximation of a manifold. One of these properties is
p
-parabolicity. A manifold
M
(respectively, a graph
G
) is said to be
p
-parabolic if all positive
p
-superharmonic functions on
M
(resp.
G
) are constant. This is equivalent to not having
p
-Green’s function (i.e. a positive fundamental solution of the
p
-Laplace-Beltrami operator). Herein we study directly the
p
-parabolicity on graphs. We obtain some characterizations in terms of graph decompositions. Also, we give necessary and sufficient conditions for a uniform hyperbolic graph to be
p
-parabolic in terms of its boundary at infinity. Finally, we prove that if a uniform hyperbolic graph satisfies the (Cheeger) isoperimetric inequality, then it is non-
p
-parabolic for every
1
<
p
<
∞
.
Brazil is among the countries in South America where the COVID-19 pandemic has hit the general population hardest. Self-testing for SARS-CoV-2 infection is one of the community-based strategies that ...could help asymptomatic individuals at-risk of COVID-19, as well as those living in areas that are difficult for health personnel to reach, to know their infectious status and contribute to impeding further transmission of the virus. A population-based survey was conducted in November 2021, to assess the acceptability of rapid SARS-CoV-2 antigen self-testing among the population of São Paulo. Survey respondents were approached at more than 400 different street-points that were randomly selected using a five-stage randomization process. A 35-item structured questionnaire was used. Dependent variables for our analyses were the likelihood to use and willingness to pay for self-testing, and the likelihood of taking preventive measures to prevent onward transmission of SARS-CoV-2 following a reactive self-test result. Bivariate and multivariate regression analyses were performed. Overall, 417 respondents (44.12% female) participated; 19.66% had previously had COVID-19 disease. A minority (9.59%) felt at high-risk of COVID-19. The majority of both females and males (73.91% and 60.09%, respectively) were in favor of the idea of SARS-CoV-2 self-testing. Overall, if self-tests were available, almost half of the sample would be very likely (n = 54, 12.95%) or likely (n = 151, 36.21%) to use one if they felt they needed to. Upon receiving a positive self-test result, the majority of respondents would communicate it (88.49%), request facility-based post-test counseling (98.32%), self-isolate (97.60%), and warn their close contacts (96.64%). Rapid SARS-CoV-2 antigen self-testing could be an acceptable screening tool in São Paulo. The population would be empowered by having access to a technology that would allow them to test, even if asymptomatic, when traveling, or going to work or school. If there is a surge in the incidence of cases, self-testing could be a good approach for mass case detection by Brazil's already overstretched Unified Health System.
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