We combine the pointed Gromov-Hausdorff metric with the locally
C
0
-distance to obtain the pointed
C
0
-Gromov-Hausdorff distance between maps of possibly different non-compact pointed metric ...spaces. The latter is combined with Walters’s locally topological stability proposed by Lee–Nguyen–Yang, and
GH
-stability from Arbieto-Morales to obtain the notion of topologically
GH
-stable pointed homeomorphism. We give one example to show the difference between the distance when taking different base points in a pointed metric space.
Abstract
We provide an isoperimetric comparison theorem for small volumes in an $n$-dimensional Riemannian manifold $(M^n,g)$ with $C^3$ bounded geometry in a suitable sense involving the scalar ...curvature function. Under $C^3$ bounds of the geometry, if the supremum of scalar curvature function $S_g<n(n-1)k_0$ for some $k_0\in \mathbb{R}$, then for small volumes the isoperimetric profile of $(M^n,g)$ is less then or equal to the isoperimetric profile of the complete simply connected space form of constant sectional curvature $k_0$. This work generalizes Theorem $2$ of 12 in which the same result was proved in the case where $(M^n, g)$ is assumed to be compact. As a consequence of our result we give an asymptotic expansion in Puiseux series up to the 2nd nontrivial term of the isoperimetric profile function for small volumes, generalizing our earlier asymptotic expansion 29. Finally, as a corollary of our isoperimetric comparison result, it is shown that for small volumes the Aubin–Cartan–Hadamard’s conjecture is true in any dimension $n$ in the special case of manifolds with $C^3$ bounded geometry, and $S_g<n(n-1)k_0$. Two different intrinsic proofs of the fact that an isoperimetric region of small volume is of small diameter. The 1st under the assumption of mild bounded geometry, that is, positive injectivity radius and Ricci curvature bounded below. The 2nd assuming the existence of an upper bound of the sectional curvature, positive injectivity radius, and a lower bound of the Ricci curvature.
Objectives
At the end of 2019, SARS‐CoV‐2 was identified, the one responsible for the COVID‐19 disease. Between a 5.1% and a 98% of COVID‐19 patients present some form of alteration in their sense of ...smell. The objective of this study is to determine the diagnostic yield of the smell dysfunction as screening tool for COVID‐19.
Methods
Cross‐sectional, observational, and pro‐elective study was performed in a tertiary care hospital from May 25th to June 30th, 2020. One hundred and thirty‐nine patients were included in the study. Demographic characteristics were collected from anamnesis. A Self‐Perception Questionnaire and psychophysical olfactory test (POT) were applied to all participants. The presence of SARS‐CoV2, was detected by RT‐PCR methods.
Results
51.7% of patients were SARS‐CoV‐2 positive. A sensitivity of 50% was obtained for the self‐perception questionnaire as a screening tool for SARS‐CoV2, with a specificity of 80.59%. The positive predictive value (PPV) was of 73.46%, the negative predictive value (NPV) was of 60%. The POT as a screening tool had a PPV of 82.35%, a NPV of 52.45%, a LR+ of 4.34, a LR‐ 0.84. The combination of anosmia (according to the POT) plus cough and asthenia got an OR of 8.25 for the SARS CoV‐2 infection.
Conclusion
There is a strong association between olfactory dysfunction and COVID‐19. However, it is not really efficient in the screening of SARS‐CoV‐2 infection and thus, they should not be considered as a single diagnostic instrument.
Level of Evidence
4.
A cross‐sectional, observational and pro‐elective study, to determine the diagnostic yield of the questionnaire and the brief smell test as scrutiny instruments for COVID‐19.
We give a multiplicity result for solutions of the Van der Waals–Cahn–Hilliard two phase transition equation with volume constraints on a closed Riemannian manifold. Our proof employs some results ...from the classical Lusternik–Schnirelman and Morse theory, together with a technique, the so-called photography method, which allows us to obtain lower bounds on the number of solutions in terms of topological invariants of the underlying manifold. The setup for the photography method employs recent results from Riemannian isoperimetry for small volumes.
A aplicação de novas metodologias para o ensino de física no ensino médio, que facilitam a compreensão de conceitos e incentivam o interesse no estudo das ciências e da matemática, tornou-se um ...grande desafio. O objetivo deste trabalho é apresentar uma estratégia didática planejada para o estudo de fenômenos fundamentais do eletromagnetismo através do desenvolvimento de projetos e do uso de novas tecnologias para experimentação. Para isso, foi proposta a construção de um transmissor de FM, que permitiu motivar e integrar o conhecimento adquirido durante o processo.
We give a multiplicity result for solutions of the Van der Waals-Cahn-Hilliard two-phase transition equation with volume constraints on a closed Riemannian manifold. Our proof employs some results ...from the classical Lusternik--Schnirelman and Morse theory, together with a technique, the so-called \emph{photography method}, which allows us to obtain lower bounds on the number of solutions in terms of topological invariants of the underlying manifold. The setup for the photography method employs recent results from Riemannian isoperimetry for small volumes.
The statement and the proof of a technical lemma in Benci et al. (2022) turn out to be incorrect. Nonetheless, the main result of the paper remains valid, and in this Corrigendum we give an ...alternative approach which provides a correct proof of Benci et al. (2022, Theorem 2.1).
We combine the pointed Gromov-Hausdorff metric Ron10 with the locally \(C^0\) distance to obtain the pointed \(C^0\)-Gromov-Hausdorff distance between maps of possibly different non-compact pointed ...metric spaces. The latter is then combined with Walters's locally topological stability LNY18 and \(GH\)-stability from AMR17 to obtain the notion of topologically \(GH\)-stable pointed homeomorphism. We give one example to show the difference between the distance when take different base point in a pointed metric space.
We provide an isoperimetric comparison theorem for small volumes in an \(n\)-dimensional Riemannian manifold \((M^n,g)\) with strong bounded geometry, as in Definition \(2.3\), involving the scalar ...curvature function. Namely in strong bounded geometry, if the supremum of scalar curvature function \(S_g<n(n-1)k_0\) for some \(k_0\in\mathbb{R}\), then for small volumes the isoperimetric profile of \((M^n,g)\) is less then or equal to the isoperimetric profile of \(\mathbb{M}^n_{k_0}\) the complete simply connected space form of constant sectional curvature \(k_0\). This work generalizes Theorem \(2\) of Dru02b in which the same result was proved in the case where \((M^n, g)\) is assumed to be just compact. As a consequence of our result we give an asymptotic expansion in Puiseux's series up to the second nontrivial term of the isoperimetric profile function for small volumes. Finally, as a corollary of our isoperimetric comparison result, it is shown, in the special case of manifolds with strong bounded geometry, and \(S_g<n(n-1)k_0\) that for small volumes the Aubin-Cartan-Hadamard's Conjecture in any dimension \(n\) is true.
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