Extended conditional G-expectations and related stopping times

<p style='text-indent:20px;'>In this paper, we extend the definition of conditional <inline-formula> <tex-math id="M2">\begin{document}$ G{\text{-}}{\rm{expectation}} $\end{document}</tex-math> </inline-formula> to a larger space on which the dynamical consistency still holds. We can consistently define, by taking the limit, the conditional <inline-formula> <tex-math id="M3">\begin{document}$ G{\text{-}}{\rm{expectation}} $\end{document}</tex-math> </inline-formula> for each random variable <inline-formula> <tex-math id="M4">\begin{document}$ X $\end{document}</tex-math> </inline-formula>, which is the downward limit (respectively, upward limit) of a monotone sequence <inline-formula> <tex-math id="M5">\begin{document}$ \{X_{i}\} $\end{document}</tex-math> </inline-formula> in <inline-formula> <tex-math id="M6">\begin{document}$ L_{G}^{1}(\Omega) $\end{document}</tex-math> </inline-formula>. To accomplish this procedure, some careful analysis is needed. Moreover, we present a suitable definition of stopping times and obtain the optional stopping theorem. We also provide some basic and interesting properties for the extended conditional <inline-formula> <tex-math id="M7">\begin{document}$ G{\text{-}}{\rm{expectation}} $\end{document}</tex-math> </inline-formula>. </p>

Lightning Data Assimilation Scheme in a 4DVAR System and its Impact on Very-Short-Term Convective Forecasting

A proof-of-concept method for the assimilation of total lightning observations in the 4DVAR framework is proposed and implemented into the Variational Doppler Radar Analysis System (VDRAS). Its performance is evaluated for the very-short-term precipitation forecasts of a localized convective event over northeastern China. The lightning DA scheme assimilated pseudo observations for vertical velocity fields derived from observed total lightning rates and statistically computed vertical velocity profile from VDRAS analysis data. To reduce representative errors of the derived vertical velocity, a distance-weighted horizontal interpolation is applied to the input data prior to the DA. The case study reveals that although 0–2 hour precipitation nowcasts are improved by assimilating lightning data alone compared to CTRL (no radar or lightning) and RAD (radar only), better results are obtained when the lightning data are assimilated with radar data simultaneously. The assimilation of both data sources results in improved dynamical consistency with enhanced updraft and latent heat as well as improved moisture distributions. Additional experiments are conducted to evaluate the sensitivity of the combined DA scheme to varied vertical velocity profiles, radii of horizontal interpolation, binning time intervals, and relationships used to estimate the maximum vertical velocity from lightning flash rates. It is shown that the scheme is robust to these variations with both radar and lightning assimilated data.

Design of nonstandard computational method for stochastic susceptible–infected–treated–recovered dynamics of coronavirus model

The current effort is devoted to investigating and exploring the stochastic nonlinear mathematical pandemic model to describe the dynamics of the novel coronavirus. The model adopts the form of a nonlinear stochastic susceptible-infected-treated-recovered system, and we investigate the stochastic reproduction dynamics, both analytically and numerically. We applied different standard and nonstandard computational numerical methods for the solution of the stochastic system. The design of a nonstandard computation method for the stochastic system is innovative. Unfortunately, standard computation numerical methods are time-dependent and violate the structure properties of models, such as positivity, boundedness, and dynamical consistency of the stochastic system. To that end, convergence analysis of nonstandard computational methods and simulation with a comparison of standard computational methods are presented.

The Hamiltonian dynamics of Hořava gravity

We consider the Hamiltonian formulation of Hořava gravity in arbitrary dimensions, which has been proposed as a renormalizable gravity model for quantum gravity without the ghost problem. We study the full constraint analysis of the non-projectable Hořava gravity whose potential, $$\mathcal{V}(R)$$V(R), is an arbitrary function of the (intrinsic) Ricci scalar R but without the extension terms which depend on the proper acceleration $$a_i$$ai. We find that there exist generally three distinct cases of this theory, A, B, and C, depending on (i) whether the Hamiltonian constraint generates new (second-class) constraints or just fixes the associated Lagrange multipliers, or (ii) whether the IR Lorentz-deformation parameter $${\lambda }$$λ is at the conformal point or not. It is found that, for Cases A and C, the dynamical degrees of freedom are the same as in general relativity, while, for Case B, there is one additional phase-space degree of freedom, representing an extra (odd) scalar graviton mode. This would achieve the dynamical consistency of a restricted model at the fully non-linear level and be a positive result in resolving the long-standing debates about the extra graviton modes of the Hořava gravity. Several exact solutions are also studied as some explicit examples of the new constraints. The structure of the newly obtained, “extended” constraint algebra seems to be generic to Hořava gravity and its general proof would be a challenging problem. Some other challenging problems, which include the path integral quantization and the Dirac bracket quantization are discussed also.

Analysis and Nonstandard Numerical Design of a Discrete Three-Dimensional Hepatitis B Epidemic Model

In this work, we numerically investigate a three-dimensional nonlinear reaction-diffusion susceptible-infected-recovered hepatitis B epidemic model. To that end, the stability and bifurcation analyses of the mathematical model are rigorously discussed using the Routh–Hurwitz condition. Numerically, an efficient structure-preserving nonstandard finite-difference time-splitting method is proposed to approximate the solutions of the hepatitis B model. The dynamical consistency of the splitting method is verified mathematically and graphically. Moreover, we perform a mathematical study of the stability of the proposed scheme. The properties of consistency, stability and convergence of our technique are thoroughly analyzed in this work. Some comparisons are provided against existing standard techniques in order to validate the efficacy of our scheme. Our computational results show a superior performance of the present approach when compared against existing methods available in the literature.

Extending the Community Multiscale Air Quality (CMAQ) modeling system to hemispheric scales: overview of process considerations and initial applications

The Community Multiscale Air Quality (CMAQ) modeling system is extended to simulate ozone, particulate matter, and related precursor distributions throughout the Northern Hemisphere. Modeled processes were examined and enhanced to suitably represent the extended space and timescales for such applications. Hemispheric-scale simulations with CMAQ and the Weather Research and Forecasting (WRF) model are performed for multiple years. Model capabilities for a range of applications including episodic long-range pollutant transport, long-term trends in air pollution across the Northern Hemisphere, and air pollution–climate interactions are evaluated through detailed comparison with available surface, aloft, and remotely sensed observations. The expansion of CMAQ to simulate the hemispheric scales provides a framework to examine interactions between atmospheric processes occurring on various spatial and temporal scales with physical, chemical, and dynamical consistency.

Extending the Community Multiscale Air Quality (CMAQ) Modeling System to Hemispheric Scales: Overview of Process Considerations and Initial Applications

The Community Multiscale Air Quality (CMAQ) modeling system is extended to simulate ozone, particulate matter, and related precursor distributions throughout the Northern Hemisphere. Modelled processes were examined and enhanced to suitably represent the extended space and time scales for such applications. Hemispheric scale simulations with CMAQ and the Weather Research and Forecasting model are performed for multiple years. Model capabilities for a range of applications including episodic long-range pollutant transport, long-term trends in air pollution across the Northern Hemisphere, and air pollution-climate interactions are evaluated through detailed comparison with available surface, aloft, and remotely sensed observations. The expansion of CMAQ to simulate the hemispheric scales provides a framework to examine interactions between atmospheric processes occurring at various spatial and temporal scales with physical, chemical, and dynamical consistency.