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We here investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the potential of a neuron increases when some of the others fire: precisely, the kick it receives is proportional to the instantaneous proportion of firing neurons at the same time. From a mathematical point of view, the coefficient of proportionality, denoted by α, is of great importance as the resulting system is known to blow-up for large values of α. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when α is small enough.

This paper addresses a general nonhomogeneous incompressible cell-fluid Navier-Stokes model incorporating chemotaxis in a two or three-dimensional bounded domain. This model comprises two mass balance equations and two general momentum balance equations, specifically for the cell and fluid phases, combined with a convection-diffusion-reaction equation for oxygen. We establish the existence and uniqueness of a local strong solution under initial data that satisfy natural compatibility conditions. Additionally, we present a blow-up criterion for the strong solution.

This paper presents a probabilistic–statistical approach to the analysis of diffusion processes in randomly nonhomogeneous multilayered bodies under conditions of incomplete experimental information on the boundary. The boundary condition is reconstructed from experimental data using linear regression, while the solution of the corresponding contact initial-boundary value problem is obtained in the form of a Neumann series and averaged over an ensemble of phase configurations. A system of statistical estimates for the solution is developed, including confidence intervals and two-sided critical regions, which provide complementary characteristics of uncertainty. Numerical experiments are performed for six representative samples differing in sample size, variance, and observation interval. It is shown that, despite significant differences in the statistical properties of the input data, the averaged concentration field preserves a qualitatively stable spatio-temporal structure. The results of the article address gaps in existing research by applying a probabilistic-statistical approach that consistently integrates two key elements for the analysis of diffusion processes in multilayer media. The first of these is the reconstruction of boundary conditions using linear regression to recover the conditions at the body boundary based on incomplete experimental data. The second key point is the analysis of uncertainty propagation by combining the regression model with a probabilistic analysis of the corresponding contact initial-boundary value problem, which allows us to quantitatively assess how the errors in the experimental data affect the final solution. From the point of view of mathematical modeling methods, the novelty of the approach lies in the creation of a structural-hierarchical scheme that synthesizes the approaches of mathematical statistics and the theory of random fields. The developed method is a theoretical and computational innovative basis for the analysis of specific physical and technological processes.

The Gompertz nonlinear growth (GNG) model with independently and identically distributed (i.i.d.) errors is often employed for describing growth data. However, the corresponding stochastic differential equation (SDE) variant is more realistic for modeling growth data, as it is capable of taking into account the effect of randomly fluctuating parameters, such as birth and death rates. However, one limitation of this prescription is that the diffusion term is assumed to be time independent. The purpose of this article is to generalize the Gompertz SDE model by taking the diffusion coefficient as timevarying. The resultant model is solved analytically and methodology for estimation of parameters, based on the method of maximum likelihood, is developed. Formulas for optimal predictors and prediction error variances and the linear Gompertz SDE (LGSDE) model and modified Gompertz SDE (MGSDE) model are also derived. Superiority of the proposed MGSDE model is shown over the LGSDE and GNG models for pig growth data.

In this paper, we develop a theoretical framework for a research into spatial patterns in a three-species Holling II and Leslie-Gower type food chain model with cross-diffusion, the results of which show that the cross-diffusion induces the spatial patterns. When biological pattern formation has been concerned with the method of reaction-diffusion theory, in most of the previous works, as a precondition, the assumption of the existence of nonhomogeneous steady state is presented essentially. We give a rigorous proof to the assumption that the model has at least a nonhomogeneous stationary solution by the Leray-Schauder degree theory. Moreover, the numerical simulations for spatial pattern is also carried out, we propose a method to estimate the wavenumber of the spatial patterns.

To investigate the characteristics of water droplets on the gas diffusion layer from both top-view and side-view of the flow channel, a rig test apparatus was designed and fabricated with prism attached plate. This experimental device was used to simulate the growth of a single liquid water droplet and its transport process with various air flow velocity and channel height. Not only dry condition but also fully humidified condition was also simulated by using a water absorbing sponge. The detachment height of the water droplet with dry and wet conditions was measured and analyzed. It was found that the droplet tends towards becoming unstable by decreased channel height, increased flow velocity or making a gas diffusion layer (GDL) dryer. Also, peculiar behavior of the water droplet in the channel was presented like attachment to hydrophilic wall or sudden breaking of droplet in case of fully hydrated condition. The simplified force balance model matches with experimental data as well.

Shell shaped carbon nanoparticles were synthesized from acetylene flow injected inside an oxygen-hydrogen diffusion flame when it was irradiated with a focused beam of a CW CO 2 laser above a certain laser power and at a certain position [1]. In the present study, the evolution of carbon nanoparticles generated under laser irradiation has been investigated together with a study on the visualization of particle generating flames. The size distributions of carbon shell nanoparticles and soots have been determined by examining the images of TEM grid on which particles were captured by local thermophoretic sampling. The variations of radii of gyration and fractal dimensions of soot and shell shaped particulate aggregates are obtained at different laser powers.

We consider a suspension of particles in a fluid settling under the influence of gravity and dispersing by Brownian motion. A mathematical description is provided by the Stokes equations and a Fokker-Planck equation for the one-particle phase space density. This is a nonlinear system that depends on a number of parametric functions of the spatial concentration of the particles. These functions are known only empirically or for dilute suspensions. We analyze the system, its stability, its asymptotic behavior under different scalings and its validity from more microscopic description. We summarize our conclusions at the end.