211 research outputs found
Analytic solutions and Singularity formation for the Peakon b--Family equations
Using the Abstract Cauchy-Kowalewski Theorem we prove that the -family
equation admits, locally in time, a unique analytic solution. Moreover, if the
initial data is real analytic and it belongs to with , and the
momentum density does not change sign, we prove that the
solution stays analytic globally in time, for . Using pseudospectral
numerical methods, we study, also, the singularity formation for the -family
equations with the singularity tracking method. This method allows us to follow
the process of the singularity formation in the complex plane as the
singularity approaches the real axis, estimating the rate of decay of the
Fourier spectrum
Quasivariational solutions for first order quasilinear equations with gradient constraint
We prove the existence of solutions for an evolution quasi-variational
inequality with a first order quasilinear operator and a variable convex set,
which is characterized by a constraint on the absolute value of the gradient
that depends on the solution itself. The only required assumption on the
nonlinearity of this constraint is its continuity and positivity. The method
relies on an appropriate parabolic regularization and suitable {\em a priori}
estimates. We obtain also the existence of stationary solutions, by studying
the asymptotic behaviour in time. In the variational case, corresponding to a
constraint independent of the solution, we also give uniqueness results
A rarefaction-tracking method for hyperbolic conservation laws
We present a numerical method for scalar conservation laws in one space
dimension. The solution is approximated by local similarity solutions. While
many commonly used approaches are based on shocks, the presented method uses
rarefaction and compression waves. The solution is represented by particles
that carry function values and move according to the method of characteristics.
Between two neighboring particles, an interpolation is defined by an analytical
similarity solution of the conservation law. An interaction of particles
represents a collision of characteristics. The resulting shock is resolved by
merging particles so that the total area under the function is conserved. The
method is variation diminishing, nevertheless, it has no numerical dissipation
away from shocks. Although shocks are not explicitly tracked, they can be
located accurately. We present numerical examples, and outline specific
applications and extensions of the approach.Comment: 21 pages, 7 figures. Similarity 2008 conference proceeding
Stable standing waves for a class of nonlinear Schroedinger-Poisson equations
We prove the existence of orbitally stable standing waves with prescribed
-norm for the following Schr\"odinger-Poisson type equation \label{intro}
%{%{ll} i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0
\text{in} \R^{3}, %-\Delta\phi= |\psi|^{2}& \text{in} \R^{3},%. when . In the case we prove the existence and
stability only for sufficiently large -norm. In case our approach
recovers the result of Sanchez and Soler \cite{SS} %concerning the existence
and stability for sufficiently small charges. The main point is the analysis of
the compactness of minimizing sequences for the related constrained
minimization problem. In a final section a further application to the
Schr\"odinger equation involving the biharmonic operator is given
A characteristic particle method for traffic flow simulations on highway networks
A characteristic particle method for the simulation of first order
macroscopic traffic models on road networks is presented. The approach is based
on the method "particleclaw", which solves scalar one dimensional hyperbolic
conservations laws exactly, except for a small error right around shocks. The
method is generalized to nonlinear network flows, where particle approximations
on the edges are suitably coupled together at the network nodes. It is
demonstrated in numerical examples that the resulting particle method can
approximate traffic jams accurately, while only devoting a few degrees of
freedom to each edge of the network.Comment: 15 pages, 5 figures. Accepted to the proceedings of the Sixth
International Workshop Meshfree Methods for PDE 201
Face Mask Use in the Community for Reducing the Spread of COVID-19: A Systematic Review.
Background: Evidence is needed on the effectiveness of wearing face masks in the community to prevent SARS-CoV-2 transmission. Methods: Systematic review and meta-analysis to investigate the efficacy and effectiveness of face mask use in a community setting and to predict the effectiveness of wearing a mask. We searched MEDLINE, EMBASE, SCISEARCH, The Cochrane Library, and pre-prints from inception to 22 April 2020 without restriction by language. We rated the certainty of evidence according to Cochrane and GRADE approach. Findings: Our search identified 35 studies, including three randomized controlled trials (RCTs) (4,017 patients), 10 comparative studies (18,984 patients), 13 predictive models, nine laboratory experimental studies. For reducing infection rates, the estimates of cluster-RCTs were in favor of wearing face masks vs. no mask, but not at statistically significant levels (adjusted OR 0.90, 95% CI 0.78–1.05). Similar findings were reported in observational studies. Mathematical models indicated an important decrease in mortality when the population mask coverage is near-universal, regardless of mask efficacy. In the best-case scenario, when the mask efficacy is at 95%, the R0 can fall to 0.99 from an initial value of 16.90. Levels of mask filtration efficiency were heterogeneous, depending on the materials used (surgical mask: 45–97%). One laboratory study suggested a viral load reduction of 0.25 (95% CI 0.09–0.67) in favor of mask vs. no mask. Interpretation: The findings of this systematic review and meta-analysis support the use of face masks in a community setting. Robust randomized trials on face mask effectiveness are needed to inform evidence-based policies. PROSPERO registration: CRD42020184963
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