Bounds on the critical times for the Fisher-KPP equation

The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented. doi:10.1017/S1446181121000365

Nonstandard Modeling of a Possible Blue Straggler Star, KIC 11145123

Nonstandard modeling of KIC 11145123, a possible blue straggler star, has been asteroseismically carried out based on a scheme to compute stellar models with the chemical compositions in their envelopes arbitrarily modified, mimicking the effects of some interactions with other stars through which blue straggler stars are thought to be born. We have constructed a nonstandard model of the star with the following parameters: M = 1.36 M ⊙, Y init = 0.26, Z init = 0.002, and f ovs = 0.027, where f ovs is the extent of overshooting described as an exponentially decaying diffusive process. The modification is down to the depth of r/R ∼ 0.6 and the extent ΔX, which is a difference in surface hydrogen abundance between the envelope-modified and unmodified models, is 0.06. The residuals between the model and the observed frequencies are comparable with those for the previous model computed assuming standard single-star evolution, suggesting that it is possible that the star was born with a relatively ordinary initial helium abundance of ∼0.26 compared with that of the previous models (∼0.30–0.40), then experienced some modification of the chemical compositions and gained helium in the envelope. Detailed analyses of the nonstandard model have implied that the elemental diffusion in the deep radiative region of the star might be much weaker than that assumed in current stellar evolutionary calculations; we need some extra mechanisms inside the star, rendering the star a much more intriguing target to be further investigated.

On The Dynamics of Interstate Diffusion of Firearm Violence and Impact of Firearm Regulations

Objectives : Our purpose was to test the impact of firearm regulations on the firearm violence flow across US state borders. Further we assessed the spatial variations in these impacts across different regions with the goal of identifying state groups that are especially vulnerable to cross-border firearm violence. Methods : Incidence of firearm violence (2000 to 2017) has been modelled as an inhomogeneous diffusion process whose parameters depend on state firearm regulations. Firearm regulations measurement for a state accounted for all 14 law categories across 54 states since 1991 as per State Firearm Law Database. The effects of regulations and other covariates were estimated across all states. Results : Six clusters of states were identified based on the variations of effects within and across those clusters. For 3 of these clusters the diffusive flow parameters were statistically significant. In all of these clusters the deterring effect of regulations on incidence and flow of crime was statistically significant. Conclusion : The clusters can be assigned to 5 descriptive categories based on their roles in the flow of firearm violence : Source states, Transitive states, Destination states, Isolated issue states and Stable . It was found that flow of firearm violence indeed does follow a diffusive process for most categories of states. It has also been recommended that while in-state regulations are important to curb firearm violence flowing into Destination states, they are not adequate unless regulatory stringency is also applied to neighbouring Source and Transitive States.

On The Dynamics of Interstate Diffusion of Firearm Violence and Impact of Firearm Regulations

Objectives – Our purpose was to test the impact of firearm regulations on the firearm violence flow across US state borders. Further we assessed the spatial variations in these impacts across different regions with the goal of identifying state-groups that are especially vulnerable to cross-border firearm violence.Methods – Incidence of firearm violence (2000-2017) has been modelled as an inhomogeneous diffusion process whose parameters depend on state firearm regulations. Firearm regulations measurement for a state accounted for all 14 law categories across 54 states since 1991 as per State Firearm Law Database. The effects of regulations and other covariates were estimated across all states. Results – Six clusters of states were identified based on the variations of effects within and across those clusters. For 3 of these clusters the diffusive flow parameters were statistically significant. In all of these clusters the deterring effect of regulations on incidence and flow of crime was statistically significant.Conclusion – The clusters can be assigned to 5 descriptive categories based on their roles in the flow of firearm violence – Source states, Transitive states, Destination states, Isolated issue states and Stable states. It was found that flow of firearm violence indeed does follow a diffusive process for most categories of states. It has also been recommended that while in-state regulations are important to curb firearm violence flowing into Destination states, they are not adequate unless regulatory stringency is also applied to neighboring Source and Transitive States.

BOUNDS ON THE CRITICAL TIMES FOR THE GENERAL FISHER–KPP EQUATION

The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented.

Study on Wear Dynamic Reliability of Gear System Based on Markov Diffusive Process

Wear is one of the most common failure forms of gear transmission system. In this paper, a dynamic wear reliability evaluation method for gear system, which subjects to stochastic external load, is proposed on the basis of Markov diffusive process. The stochastic load is considered as the combination of constant load and random noise. The failure is defined as the wear depth of the maximum wear point of gear under the constant load exceeds a specific threshold. The maximum wear point and relative sliding velocity are obtained by deterministic wear analysis. The stochastic noise is assumed as Gaussian white noise; hence the wear depth can be described as Markov diffusive process, and the transition probability is governed by Fokker-Planck-Kolmogorov (FPK) equation. With the transition probability function, the wear life and dynamic reliability of gear system with different noise spectral density are predicted. The results revel that the wear depth obeys normal distribution and becomes more and more scattered with noise spectral density increases. The wear life does not obey normal distribution, the effect of noise spectral density on mean life is neglectable while on standard deviation of life is considerable.

Water Absorption and Transport in Clay Bricks

Moisture and water transport in clay bricks are essential contributing factors towards brick decay in buildings. Experimental work is reported comparing water absorption and porosity of handmade and solid fired clay bricks from Paraguay. Their respective porosities were estimated by gravimetric test and, their pore distribution, degree of anisotropy and morphology were examined by SEM. Although it was found that both samples have a common composition (XRD), a higher degree of porosity was found in handmade bricks, which also absorb water at a higher rate accelerating the decay process. Experimental data is reported on the exponential kinetics behavior of water absorption similar to the one observed in capillary tubes, but with an additional diffusive process. A phenomenological model is proposed for water absorption in both types of bricks which compared with the standard model of water absorption available in the literature, produces a better representation of the experimental data.

Loss of Information Associated with the Perceptomotor Cognitive Domain: Praxis

We have proposed a random walk model to model perceptomotor cognitive domain damage in patients with neurocognitive disorders (NDs). The random walk model with truncated Lorentz memory profile. We have reported that memory damage has an impact on random walkers' praxis and, consequently, on the global diffusive process, fractal dimension and information loss, and it is the major impact on information loss. We have found two diffusive regimes: the ordinary and the superdiffusive. We have observed two superdiffusion regions separated by a region with ordinary diffusion regime: one in the anti-persistence region and one in the persistence region. These regions are characterized by diffusion level curves, invariant curves of scale variations in the Lorentz distribution. In the anti-persistence region, we found a greater variation in entropy, for example, greater loss of praxis-related information than in the persistence region. Therefore, when memory impairment is accompanied by observation of anti-persistent behavior, there are greater losses of information related to the perceptomotor domain of random walkers.