On the Relationship between the One-Corner Problem and the M−M-Corner Problem for the Vortex Filament Equation

In this paper, we give evidence that the evolution of the vortex filament equation (VFE) for a regular M-corner polygon as initial datum can be explained at infinitesimal times as the superposition of M one-corner initial data. This fact is mainly sustained with the calculation of the speed of the center of mass; in particular, we show that several conjectures made at the numerical level are in agreement with the theoretical expectations. Moreover, due to the spatial periodicity, the evolution of VFE at later times can be understood as the nonlinear interaction of infinitely many filaments, one for each corner; and this interaction turns out to be some kind of nonlinear Talbot effect. We also give very strong numerical evidence of the transfer of energy and linear momentum for the M-corner case; and the numerical experiments carried out provide new arguments that support the multifractal character of the trajectory defined by one of the corners of the initial polygon

Vortex filament equation for a regular polygon

In this paper, we study the evolution of the vortex filament equation,$$X_t = X_s \wedge X_{ss},$$with $X(s, 0)$ being a regular planar polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that $X(s, t)$ is also a polygon at any rational time; moreover, it can be fully characterized, up to a rigid movement, by a generalized quadratic Gau$\beta$ sum. We also study the fractal behaviour of $X(0, t)$, relating it with the so-called Riemann's non-differentiable function, that was proved by Jaffard to be a multifractal

Vortex Filament Equation for a regular polygon in the hyperbolic plane

The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow exponentially, which makes the problem more challenging from a numerical point of view. However, with fixed boundary conditions, a finite difference scheme and a fourth-order Runge--Kutta method in time, we show that the numerical solution is in complete agreement with the one obtained from algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with that in the Euclidean case

On the Evolution of the Vortex Filament Equation for regular M-polygons with nonzero torsion

In this paper, we consider the evolution of the Vortex Filament equa- tion (VFE): Xt = Xs ∧ Xss, taking M-sided regular polygons with nonzero torsion as initial data. Us- ing algebraic techniques, backed by numerical simulations, we show that the solutions are polygons at rational times, as in the zero-torsion case. However, unlike in that case, the evolution is not periodic in time; more- over, the multifractal trajectory of the point X(0,t) is not planar, and appears to be a helix for large times. These new solutions of VFE can be used to illustrate numerically that the smooth solutions of VFE given by helices and straight lines share the same instability as the one already established for circles. This is accomplished by showing the existence of variants of the so-called Rie- mann’s non-differentiable function that are as close to smooth curves as desired, when measured in the right topology. This topology is motivated by some recent results on the well-posedness of VFE, which prove that the selfsimilar solutions of VFE have finite renormalized energy

A pseudospectral method for the one-dimensional fractional Laplacian on R

In this paper, we propose a novel pseudospectral method to approximate accurately and efficiently the fractional Laplacian without using truncation. More precisely, given a bounded regular function defined over R, we map the unbounded domain into a finite one, and represent the resulting function as a trigonometric series. Therefore, the central point of this paper is the computation of the fractional Laplacian of an elementary trigonometric function. As an application of the method, we also do the simulation of Fisher's equation with fractional Laplacian in the monostable case

Global compliance with hepatitis b vaccine birth dose and factors related to timely schedule. A literature review

Objectives: Identify global barriers for delivery of hepatitis B vaccine birth dose. Methods: A search for cross sectional studies published between January 2001 and December 2017 was conducted using the following Mesh terms: "Vaccination"[Mesh], "Mass Vaccination"[Mesh], "Hepatitis B"[Mesh], "Hepatitis B virus"[Mesh], "Hepatitis B Surface Antigens"[Mesh]. Databases consulted included: PUBMED, SCIELO, EMBASE and BIREME. To evaluate the quality of studies, we used an adapted version of the Newcastle-Ottawa Quality Assessment Scale for cross sectional studies. Results: An initial list of 6,789 articles were generated by the combination of search terms. After reviewing titles and abstracts, they were reduced to 134 for full reading, and 22 studies were included in the barriers analysis. The region with more references was Western Pacific while eastern Mediterranean had the lowest. Being born outside of a health facility and weakness of outreach vaccination service seems to be the most important an cited factors related to underperformance of birth dose delivery. In developed countries, hospital policies on birth dose vaccination was the main factor associated to no vaccintion with the birth dose. Conclusions: New ways to deliver hepatitis B vaccines to neonates being born at home or outside health facilities should be envisaged and applied, if the goal of eliminating perinatal transmission of hepatitis B is to be achieved

Educational inequality trends in mortality due to pneumonia in Colombia, 1998-2015

We aimed to explore the existence and trends of social inequalities related to pneumonia mortality in Colombian adults using education as a proxy of socioeconomic status. We obtained death certificates due to pneumonia and population data from Departamento Administrativo Nacional de Estadística for 1998-2015. Educational level data were gathered from microdata of the Colombian Demography Health Surveys. Annual trends in Age Standardized Mortality Rates by sex and educational level were quantified by calculating the Estimated Annual Percentage Change (EAPC). We estimated Rate Ratios (RR) by using Poisson regressionmodels, comparing mortality of educational groups with mortality in the highest education group. We estimated the Relative Index of Inequality (RII) to assess changes in disparities, regressing mortality on the mid-point of the cumulative distribution of education, thereby considering the size of each educational group. All analyses were conducted in SAS®V.9.2. For adults 25+ years, the risk of dying was significantly higher among lower educated. The RRs depict increased risks of dying comparing lower and highest education level, and this tendency was stronger in woman than in men [RRprimary=2.34 (2.32-2.36), RRsecondary=1.77 (1.75-1.78) vs. RRprimary=1.83(1.81-1.85), RR secondary=1.51 (1.50-1.53)]. According to age groups, young adults (25-44 years) showed the largest inequality in terms of educational level; RRs for pneumonia mortality regarding to the tertiary educated groups show increased mortality in the lower and secondary educated, and these differences decreased with ages. RII in pneumonia mortality among adult men was 2.01 (95%CI 2.00-2.03) and in women 2.46 (95%CI 2.43-2.48). The RII was greatest at young ages, for both sexes. Time trends showed steadily significant increases for RII in both men and women (EAPCmen=3.8; EAPCwomen=2.6). Pneumonia mortality rates in adults evidenced a clear age-dependency, with lowest rates for young and much higher rates for senior adults. All estimated mortality rates were higher in men than in women