On a Nonlinear Mixed Problem for a Parabolic Equation with a Nonlocal Condition

The aim of this work is to prove the well-posedness of some linear and nonlinear mixed problems with integral conditions defined only on two parts of the considered boundary. First, we establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense using a functional analysis method. Then by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem.

Well posedness of a nonlinear mixed problem for a parabolic equation with integral condition

The aim of this work is to prove the well posedness of some posed linear and nonlinear mixed problems with integral conditions. First, an a priori estimate is established for the associated linear problem and the density of the operator range generated by the considered problem is proved by using the functional analysis method. Subsequently, by applying an iterative process based on the obtained results for the linear problem, the existence, uniqueness of the weak solution of the nonlinear problems is established.

Bayesian updating: increasing sample size during the course of a study

Background A priori sample size calculation requires an a priori estimate of the size of the effect. An incorrect estimate may result in a sample size that is too low to detect effects or that is unnecessarily high. An alternative to a priori sample size calculation is Bayesian updating, a procedure that allows increasing sample size during the course of a study until sufficient support for a hypothesis is achieved. This procedure does not require and a priori estimate of the effect size. This paper introduces Bayesian updating to researchers in the biomedical field and presents a simulation study that gives insight in sample sizes that may be expected for two-group comparisons. Methods Bayesian updating uses the Bayes factor, which quantifies the degree of support for a hypothesis versus another one given the data. It can be re-calculated each time new subjects are added, without the need to correct for multiple interim analyses. A simulation study was conducted to study what sample size may be expected and how large the error rate is, that is, how often the Bayes factor shows most support for the hypothesis that was not used to generate the data. Results The results of the simulation study are presented in a Shiny app and summarized in this paper. Lower sample size is expected when the effect size is larger and the required degree of support is lower. However, larger error rates may be observed when a low degree of support is required and/or when the sample size at the start of the study is small. Furthermore, it may occur sufficient support for neither hypothesis is achieved when the sample size is bounded by a maximum. Conclusions Bayesian updating is a useful alternative to a priori sample size calculation, especially so in studies where additional subjects can be recruited easily and data become available in a limited amount of time. The results of the simulation study show how large a sample size can be expected and how large the error rate is.

Expansions for distributional solutions of the elliptic equation in two dimensions

Assume [Formula: see text] is a planar domain, and [Formula: see text] is a locally bounded distributional solution to the elliptic equation [Formula: see text] where [Formula: see text] is a constant, [Formula: see text] and [Formula: see text] are real analytic functions defined on [Formula: see text] and the real line [Formula: see text], respectively. We establish asymptotic expansions of [Formula: see text] to arbitrary orders near [Formula: see text], which complements the recent results of Han–Li–Li on the Yamabe equation, Guo–Li–Wanon the weighted Yamabe equation, and partly extends that of Guo–Wan–Yang on the Liouville equation in a punctured disc. Our method is a combination of a priori estimate and mathematical induction.

Satellite retrieval of aerosol combined with assimilated forecast

We developed a new aerosol satellite retrieval algorithm combining a numerical aerosol forecast. In the retrieval algorithm, the short-term forecast from an aerosol data assimilation system was used as an a priori estimate instead of spatially and temporally constant values. This method was demonstrated using observation of the Advanced Himawari Imager onboard the Japan Meteorological Agency's geostationary satellite Himawari-8. Overall, the retrieval results incorporated strengths of the observation and the model and complemented their respective weaknesses, showing spatially finer distributions than the model forecast and less noisy distributions than the original algorithm. We validated the new algorithm using ground observation data and found that the aerosol parameters detectable by satellite sensors were retrieved more accurately than an a priori model forecast by adding satellite information. Further, the satellite retrieval accuracy was improved by introducing the model forecast instead of the constant a priori estimates. By using the assimilated forecast for an a priori estimate, information from previous observations can be propagated to future retrievals, leading to better retrieval accuracy. Observational information from the satellite and aerosol transport by the model are incorporated cyclically to effectively estimate the optimum field of aerosol.

Bayesian Updating: Increasing Sample Size During the Course of A Study

Background: A priori sample size calculation requires an a priori estimate of the size of the effect. An incorrect estimate may result in a sample size that is too low to detect effects or that is unnecessarily high. An alternative to a priori sample size calculation is Bayesian updating, a procedure that allows increasing sample size during the course of a study until sufficient support for a hypothesis is achieved. This procedure does not require and a priori estimate of the effect size. This paper introduces Bayesian updating to researchers in the biomedical field and presents a simulation study that gives insight in sample sizes that may be expected for two-group comparisons. Methods: Bayesian updating uses the Bayes factor, which quantifies the degree of support for a hypothesis versus another one given the data. It can be re-calculated each time new subjects are added, without the need to correct for multiple interim analyses. A simulation study was conducted to study what sample size may be expected and how large the error rate is, that is, how often the Bayes factor shows most support for the hypothesis that was not used to generate the data. Results: The results of the simulation study are presented in a Shiny app and summarized in this paper. Lower sample size is expected when the effect size is larger and the required degree of support is lower. However, larger error rates may be observed when a low degree of support is required and/or when the sample size at the start of the study is small. Furthermore, it may occur sufficient support for neither hypothesis is achieved when the sample size is bounded by a maximum. Conclusions: Bayesian updating is a useful alternative to a priori sample size calculation, especially so in studies where additional subjects can be recruited easily and data become available in a limited amount of time. The results of the simulation study show how large a sample size can be expected and how large the error rate is.

A locally one-dimensional scheme for a general parabolic equation describing microphysical processes in convective clouds

A locally one-dimensional difference scheme for a general parabolic equation in a p-dimensional parallelepiped is considered. To describe microphysical processes in convective clouds, non-local (nonlinear) integral sources of a special type are included in the equation under consideration. An a priori estimate for the solution of a locally one-dimensional scheme is obtained and its convergence is proved.

The Regularity Criteria and the A Priori Estimate on the 3D Incompressible Navier-Stokes Equations in Orthogonal Curvilinear Coordinate Systems

The paper considers the regularity problem on three-dimensional incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems. We establish one regularity criteria of the weak solutions involving only in a vorticity component ω 3 and one a priori estimate on the solution that H 3 u 3 L ∞ 0 , T ; L p ℝ 3 is bounded for 1 ≤ p ≤ ∞ to three-dimensional incompressible Navier-Stokes equations in orthogonal curvilinear coordinate systems. These extent greatly the corresponding results on axisymmetric cylindrical flow.

Frequently cocited publications: Features and kinetics

Cocitation measurements can reveal the extent to which a concept representing a novel combination of existing ideas evolves towards a specialty. The strength of cocitation is represented by its frequency, which accumulates over time. Of interest is whether underlying features associated with the strength of cocitation can be identified. We use the proximal citation network for a given pair of articles ( x, y) to compute θ, an a priori estimate of the probability of cocitation between x and y, prior to their first cocitation. Thus, low values for θ reflect pairs of articles for which cocitation is presumed less likely. We observe that cocitation frequencies are a composite of power-law and lognormal distributions, and that very high cocitation frequencies are more likely to be composed of pairs with low values of θ, reflecting the impact of a novel combination of ideas. Furthermore, we note that the occurrence of a direct citation between two members of a cocited pair increases with cocitation frequency. Finally, we identify cases of frequently cocited publications that accumulate cocitations after an extended period of dormancy.