a b s t r a c tThe notions of Lipschitz conditions with L average are introduced to the study of convergence analysis of Gauss-Newton's method for singular systems of equations. Unified convergence criteria ensuring the convergence of Gauss-Newton's method for one kind of singular systems of equations with constant rank derivatives are established and unified estimates of radii of convergence balls are also obtained. Applications to some special cases such as the Kantorovich type conditions, γ -conditions and the Smale point estimate theory are provided and some important known results are extended and/or improved.

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The present paper is concerned with the convergence problem of Newton's method to solve singular systems of equations with constant rank derivatives. Under the hypothesis that the derivatives satisfy a type of weak Lipschitz condition, a convergence criterion based on the information around the initial point is established for Newton's method for singular systems of equations with constant rank derivatives. Applications to two special and important cases: the classical Lipschitz condition and the Smale's assumption, are provided; the latter, in particular, extends and improves the corresponding result due to Dedieu and Kim in [J.P. Dedieu, M. Kim, Newton's method for analytic systems of equations with constant rank derivatives, J. Complexity 18 (2002) 187-209].

Coronavirus disease 2019 (COVID-19) was declared a pandemic at the beginning of 2020, causing millions of deaths worldwide. Millions of vaccine doses have been administered worldwide; however, outbreaks continue. Probiotics are known to restore a stable gut microbiota by regulating innate and adaptive immunity within the gut, demonstrating the possibility that they may be used to combat COVID-19 because of several pieces of evidence suggesting that COVID-19 has an adverse impact on gut microbiota dysbiosis. Thus, probiotics and their metabolites with known antiviral properties may be used as an adjunctive treatment to combat COVID-19. Several clinical trials have revealed the efficacy of probiotics and their metabolites in treating patients with SARS-CoV-2. However, its molecular mechanism has not been unraveled. The availability of abundant data resources and computational methods has significantly changed research finding molecular insights between probiotics and COVID-19. This review highlights computational approaches involving microbiome-based approaches and ensemble-driven docking approaches, as well as a case study proving the effects of probiotic metabolites on SARS-CoV-2.

The famous Newton-Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton's method to a solution of an equation. Here we present a "Kantorovich type" convergence analysis for the Gauss-Newton's method which improves the result in [W.M. Häußler, A Kantorovich-type convergence analysis for the Gauss-Newton-method, Numer. Math. 48 (1986) 119-125.] and extends the main theorem in [I.K. Argyros, On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math. 169 (2004) 315-332]. Furthermore, the radius of convergence ball is also obtained.

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Novel severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the etiologic agent of the ongoing coronavirus disease 2019 (COVID-19) pandemic, which has reached 28 million cases worldwide in 1 year. The serological detection of antibodies against the virus will play a pivotal role in complementing molecular tests to improve diagnostic accuracy, contact tracing, vaccine efficacy testing, and seroprevalence surveillance. Here, we aimed first to evaluate a lateral flow assay's ability to identify specific IgM and IgG antibodies against SARS-CoV-2 and second, to report the seroprevalence estimates of these antibodies among health care workers and healthy volunteer blood donors in Panama. We recruited study participants between April 30th and July 7th, 2020. For the test validation and performance evaluation, we analyzed serum samples from participants with clinical symptoms and confirmed positive RT-PCR for SARS-CoV-2, and a set of pre-pandemic serum samples. We used two by two table analysis to determine the test positive and negative percentage agreement as well as the Kappa agreement value with a 95% confidence interval. Then, we used the lateral flow assay to determine seroprevalence among serum samples from COVID-19 patients, potentially exposed health care workers, and healthy volunteer donors. Our results show this assay reached a positive percent agreement of 97.2% (95% CI 84.2–100.0%) for detecting both IgM and IgG. The assay showed a Kappa of 0.898 (95%CI 0.811–0.985) and 0.918 (95% CI 0.839–0.997) for IgM and IgG, respectively. The evaluation of serum samples from hospitalized COVID-19 patients indicates a correlation between test sensitivity and the number of days since symptom onset; the highest positive percent agreement [87% (95% CI 67.0–96.3%)] was observed at ≥15 days post-symptom onset (PSO). We found an overall antibody seroprevalence of 11.6% (95% CI 8.5–15.8%) among both health care workers and healthy blood donors. Our findings suggest this lateral flow assay could contribute significantly to implementing seroprevalence testing in locations with active community transmission of SARS-CoV-2.