Old mathematical challenges : precedents to the millennium problems

The millennium problems set out by the Clay Mathematics Institute became a stimulus for mathematical research. The aim of this article is to highlight some previous challenges that were also a stimulus to finding proof for some interesting results. With this pretext, we present three moments in the history of mathematics that were important for the development of new lines of research. We briefly analyse the Tartaglia challenge, which brought about the discovery of a formula for third degree equations; Johan Bernoulli?s problem of the curve of fastest descent, which originated the calculus of variations; and the incidence of the problems posed by David Hilbert in 1900, focusing on the first problem in the list: the continuum hypothesis

Gelfand-type problems involving the 1-Laplacian operator

In this paper, the theory of Gelfand problems is adapted to the 1-Laplacian setting. Concretely, we deal with the following problem: −∆1u = λf(u) in Ω,u = 0 on ∂Ω, where Ω ⊂ RN (N ≥ 1) is a domain, λ ≥ 0, and f : [0, +∞[ → ]0, +∞[ is any continuous increasing and unbounded function with f(0) > 0. We prove the existence of a threshold λ∗ = h(Ω) f(0) (h(Ω) being the Cheeger constant of Ω) such that there exists no solution when λ > λ∗ and the trivial function is always a solution when λ ≤ λ∗. The radial case is analyzed in more detail, showing the existence of multiple (even singular) solutions as well as the behavior of solutions to problems involving the p-Laplacian as p tends to 1, which allows us to identify proper solutions through an extra condition

Elliptic equations having a singular quadratic gradient term and a changing sign datum

In this paper we study a singular elliptic problem whose model is \begin{eqnarray*} - \Delta u= \frac{|\nabla u|^2}{|u|^\theta}+f(x), in \Omega\\ u = 0, on \partial \Omega; \end{eqnarray*} where $\theta\in (0,1)$ and $f \in L^m (\Omega)$, with $m\geq \frac{N}{2}$. We do not assume any sign condition on the lower order term, nor assume the datum $f$ has a constant sign. We carefully define the meaning of solution to this problem giving sense to the gradient term where $u=0$, and prove the existence of such a solution. We also discuss related questions as the existence of solutions when the datum $f$ is less regular or the boundedness of the solutions when the datum $f \in L^m (\Omega)$ with $m> \frac{N}{2}$

The Dirichlet problem for the 11-Laplacian with a general singular term and L1L^1-data

We study the Dirichlet problem for an elliptic equation involving the $1$-Laplace operator and a reaction term, namely: $$\left\{\begin{array}{ll} \displaystyle -\Delta_1 u =h(u)f(x)&\hbox{in }\Omega\,,\\ u=0&\hbox{on }\partial\Omega\,, \end{array}\right.$$ where $\Omega \subset \mathbb{R}^N$ is an open bounded set having Lipschitz boundary, $f\in L^1(\Omega)$ is nonnegative, and $h$ is a continuous real function that may possibly blow up at zero. We investigate optimal ranges for the data in order to obtain existence, nonexistence and (whenever expected) uniqueness of nonnegative solutions

Enfermedades crónicas

Adherencia al tratamiento farmacológico y relación con el control metabólico en pacientes con DM2Aluminio en pacientes con terapia de reemplazo renal crónico con hemodiálisis en Bogotá, ColombiaAmputación de extremidades inferiores: ¿están aumentando las tasas?Consumo de edulcorantes artificiales en jóvenes universitariosCómo crecen niños normales de 2 años que son sobrepeso a los 7 añosDiagnóstico con enfoque territorial de salud cardiovascular en la Región MetropolitanaEfecto a corto plazo de una intervención con ejercicio físico, en niños con sobrepesoEfectos de la cirugía bariátrica en pacientes con síndrome metabólico e IMC < 35 KG/M2Encuesta mundial de tabaquismo en estudiantes de profesiones de saludEnfermedades crónicas no transmisibles: Consecuencias sociales-sanitarias de comunidades rurales en ChileEpidemiología de las muertes hospitalarias por patologías relacionadas a muerte encefálica, Chile 2003-2007Estado nutricional y conductas alimentarias en adolescentes de 4º medio de la Región de CoquimboEstudio de calidad de vida en una muestra del plan piloto para hepatitis CEvaluación del proceso asistencial y de resultados de salud del GES de diabetes mellitus 2Factores de riesgo cardiovascular en población universitaria de la Facsal, universidad de TarapacáImplicancias psicosociales en la génesis, evolución y tratamiento de pacientes con hipertensión arterial esencialInfarto agudo al miocardio (IAM): Realidad en el Hospital de Puerto Natales, 2009-2010Introducción de nuevas TIC y mejoría de la asistencia a un programa de saludNiños obesos atendidos en el Cesfam de Puerto Natales y su entorno familiarPerfil de la mortalidad por cáncer de cuello uterino en Río de JaneiroPerfil del paciente primo-consultante del Programa de Salud Cardiovascular, Consultorio Cordillera Andina, Los AndesPrevalencia de automedicación en mujeres beneficiarias del Hospital Comunitario de Til-TiPrevalencia de caries en población preescolar y su relación con malnutrición por excesoPrevalencia de retinopatía diabética en comunas dependientes del Servicio de Salud Metropolitano Occidente (SSMOC)Problemas de adherencia farmacológica antihipertensiva en población mapuche: Un estudio cualitativoRol biológico de los antioxidantes innatos en pacientes portadores de VIH/SidaSobrepeso en empleados de un restaurante de una universidad pública del estado de São Paul

The evolution of the ventilatory ratio is a prognostic factor in mechanically ventilated COVID-19 ARDS patients

Background: Mortality due to COVID-19 is high, especially in patients requiring mechanical ventilation. The purpose of the study is to investigate associations between mortality and variables measured during the first three days of mechanical ventilation in patients with COVID-19 intubated at ICU admission. Methods: Multicenter, observational, cohort study includes consecutive patients with COVID-19 admitted to 44 Spanish ICUs between February 25 and July 31, 2020, who required intubation at ICU admission and mechanical ventilation for more than three days. We collected demographic and clinical data prior to admission; information about clinical evolution at days 1 and 3 of mechanical ventilation; and outcomes. Results: Of the 2,095 patients with COVID-19 admitted to the ICU, 1,118 (53.3%) were intubated at day 1 and remained under mechanical ventilation at day three. From days 1 to 3, PaO2/FiO2 increased from 115.6 [80.0-171.2] to 180.0 [135.4-227.9] mmHg and the ventilatory ratio from 1.73 [1.33-2.25] to 1.96 [1.61-2.40]. In-hospital mortality was 38.7%. A higher increase between ICU admission and day 3 in the ventilatory ratio (OR 1.04 [CI 1.01-1.07], p = 0.030) and creatinine levels (OR 1.05 [CI 1.01-1.09], p = 0.005) and a lower increase in platelet counts (OR 0.96 [CI 0.93-1.00], p = 0.037) were independently associated with a higher risk of death. No association between mortality and the PaO2/FiO2 variation was observed (OR 0.99 [CI 0.95 to 1.02], p = 0.47). Conclusions: Higher ventilatory ratio and its increase at day 3 is associated with mortality in patients with COVID-19 receiving mechanical ventilation at ICU admission. No association was found in the PaO2/FiO2 variation