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Motivated by the polynuclear growth model, we consider a Brownian bridge b(t) with b(±T) = 0 conditioned to stay above the semicircle$c_{T}(t)=\\sqrt{T^{2}-t^{2}}$. In the limit of large T, the fluctuation scale of b(t)-cT(t) is T1/3and its time-correlation scale is T2/3. We prove that, in the sense of weak convergence of path measures, the conditioned Brownian bridge, when properly rescaled, converges to a stationary diffusion process with a drift explicitly given in terms of Airy functions. The dependence on the reference point t = τT, τ ∈ (-1, 1), is only through the second derivative of cT(t) at t = τT. We also prove a corresponding result where instead of the semicircle the barrier is a parabola of height Tγ, γ > 1/2. The fluctuation scale is then T(2-γ)/3. More general conditioning shapes are briefly discussed.

Gelation through the liquid-liquid contact between a polymer solution and a gelator solution has been attempted with various combinations of gelator and polymer solutions. In many combinations, the gel growth dynamics is expressed as X∼t, where X is the gel thickness and t is the elapsed time, and the scaling law holds for the relationship between X and t. In the blood plasma gelation, however, the crossover of the growth behavior from X∼t in the early stage to X∼t in the late stage was observed. It was found that the crossover behavior is caused by a change in the rate-limiting process of growth from the free-energy-limited process to the diffusion-limited process. How, then, would the crossover phenomenon be described in terms of the scaling law? We found that the scaling law does not hold in the early stage owing to the characteristic length attributable to the free energy difference between the sol-gel phases, but it does in the late stage. We also discussed the analysis method for the crossover in terms of the scaling law.

Anodic oxidation is a promising method for removing organic pollutants from water due to its high nonselectivity and effectiveness. Nevertheless, its widespread application is limited due to its low current efficiency, high energy consumption and low treatment rates. These problems may be overcome by the optimization of the process parameters, reactor design and electrode geometry, by coupling the experimental investigations with mathematical modeling. Here we review the modeling of anodic oxidation with focus on basics of this process, the competition phenomenon in real wastewater, flow cells and batch cells, historical aspects, general modeling equations, modeling with plate electrodes, modeling with porous 3-dimension electrodes and the density functional theory. Mathematical modeling can provide current, voltage and concentration distributions in the system. Mathematical modeling can also determine the effects on the performance of parameters such as diffusion layer thickness, flow velocity, applied current density, solution treatment time, initial concentration and diffusion coefficients of organic pollutants, electrode surface area, and oxidation reaction rate constant. Mathematical models allow to determine whether the limiting factor of the process is kinetics or diffusion, and to study the impact of competition of phenomena. The density functional theory provides information on probable reaction pathways and by-products.

A novel dense diffusion barrier material (Y x Sr 1− x Ti 0.9 In 0.1 O 3− δ ( x = 0.03, 0.05, 0.07)) was prepared by using a sol-gel method. The crystal structure, microstructures, electrical conductivity and ionic conductivity of barrier material were characterized. The results show that the samples exhibit the formation of cubic perovskite structure phase. The increase of Y-doping amount on A-site improved electrical conductivity and sinterability of materials. A limiting current oxygen sensor based on Y 0.07 Sr 0.97 Ti 0.9 In 0.1 O 3− δ as a dense diffusion barrier shows excellent sensing performance. The linear relationship between limiting current log I L and 1000/ T can described log I L = 4.6038 − 3.8475·1000/ T . At 750 °C, 0.25% ≤ x (O 2 ) ≤ 5.0%, the linear relationship between limiting current ( I L ) and oxygen amount ( x (O 2 )) can described as I L = 7.0476 + 3.8751 x (O 2 ).

Due to the instability of FeO at temperatures below 843 K, the fluidization reduction pathway of iron ore powder changes with the reduction temperature. Thus, the effect of temperature and reaction pathway interaction on the kinetics of fluidization reduction of iron ore powder under low-temperature conditions ranging from 783 to 903 K was investigated to describe the fluidization reduction rate of iron ore powder from three aspects: microstructure change, reaction limiting link, and apparent activation energy of the reaction, exploring their internal correlation. The experimental results revealed that in a temperature range of 783–813 K, the formation of a dense iron layer hindered the internal diffusion of reducing gas, resulting in relatively high gas diffusion resistance. In addition, due to the differences in limiting links and reaction pathways in the intermediate stage of reduction, the apparent activation energy of the reaction varied. The apparent activation energy of the reaction ranged from 23.36 to 89.13 kJ/mol at temperature ranging from 783 to 813 K, while it ranged from 14.30 to 68.34 kJ/mol at temperature ranging from 873 to 903 K.

The North Pacific is known with iron limitation for phytoplankton growth in the subarctic region and nitrogen limitation in the subtropical gyre. Although the growth rate of phytoplankton is determined by the concentration of limiting nutrient, the supply ratio of iron to nitrogen is suggested to be essential to this biogeographic pattern. However, the underlying dynamics determining the ratio remain largely unknown. We investigated mechanisms of dissolved iron (dFe) and nitrate (NO3−) transport to the euphotic zone of the North Pacific using an eddy‐resolvable biogeochemical model. We show that lateral advection and atmospheric deposition are dominant drivers for dFe transport, resulting in high Fe:N supply ratio in both subarctic and subtropical regions. Conversely, significant vertical supplies of NO3− through upwelling and diffusion processes markedly reduce the supply ratio in the subarctic region. These dynamics combined lead to high Fe:N supply ratio in the gyre and low ratio in the subarctic, ultimately driving high nitrogen fixation condition in the gyre and the iron‐limited phytoplankton growth condition in the subarctic region. Plain Language Summary Iron is a critical trace element for the photosynthesis and nitrogen fixation of phytoplankton in the ocean. In North Pacific Subarctic region, although there is plenty of nitrate, the growth of phytoplankton is limited due to the lack of iron. In the North Pacific Subtropical Gyre (NPSG), nutrient supply to the surface is restricted due to ocean stratification, but diazotrophs can fix nitrogen from the atmosphere. However, their growth is also constrained by iron availability. Understanding how nutrients like iron reach the ocean's surface is vital for predicting the productivity of marine life. Our research employed advanced computer models to explore how dissolved iron is transported in the North Pacific. We discovered that lateral transport by ocean currents, followed by atmospheric deposition, is the primary pathway for iron delivery to the sunlit layer of the NPSG. In the Subarctic Gyre, atmospheric deposition and vertical advection are the major sources of iron. However, we found different transport patterns for nitrate, revealing that physical process‐controlled supply ratio of iron to nitrate may determine where different types of phytoplankton thrive in the surface ocean. This research helps understand the complex processes that supply nutrients to ocean surface. Key Points Lateral transport and atmospheric deposition dominate supplies of dissolved iron (dFe) to the euphotic zone of the North Pacific Upwelling and vertical diffusion control nitrate (NO3−) supply in subarctic region Lateral dFe and vertical NO3− transports determine the stoichiometric supply ratio and shape the biogeographic pattern

The entropy production in the polarization phenomena occurring in the underlimiting regime, when an electric current circulates through a single cation-exchange membrane system, has been investigated in the 3–40 °C temperature range. From the analysis of the current–voltage curves and considering the electro-membrane system as a unidimensional heterogeneous system, the total entropy generation in the system has been estimated from the contribution of each part of the system. Classical polarization theory and the irreversible thermodynamics approach have been used to determine the total electric potential drop and the entropy generation, respectively, associated with the different transport mechanisms in each part of the system. The results show that part of the electric power input is dissipated as heat due to both electric migration and diffusion ion transports, while another part is converted into chemical energy stored in the saline concentration gradient. Considering the electro-membrane process as an energy conversion process, an efficiency has been defined as the ratio between stored power and electric power input. This efficiency increases as both applied electric current and temperature increase.

In this paper, we focus on the following nearly nonsationary autoregressive model: yt=qnyt-1+ut, t=1,…,n, where qn=1+c/kn with c a non-zero constant and {kn,n⩾1} a sequence of positive constants increasing to ∞ such that kn=o(n) as n→∞, and {ut,t⩾1} is a sequence of independent and identically distributed random variables which are in the domain of attraction of the normal law with zero mean and possibly infinity variance. The weighted composite quantile estimate of qn is examined, and the corresponding limiting distributions under the cases of c>0 and c<0 are established. Monte Carlo simulations are conducted to illustrate the theoretical results on finite-sample performance. The simulation results show that the weighted composite quantile estimate method is more robust and efficient than the composite quantile estimate method in terms of bias and accuracy, and we employ this estimator to analyze a real-world data set.

We build the iDLA cluster using drifted random walks, and study the limiting shapes they exhibit, with the help of sandpile models. For constant drift, the normalised cluster converges to a canonical shape S , which can be termed a true heat ball, in that it gives rise to a mean value property for caloric functions. The existence and boundedness of such a shape answers the natural yet open question of the existence and boundedness of a shape that satisfies a mean value property for caloric functions.

In this paper we prove a weak necessary and sufficient maximum principle for Markovian regime switching stochastic optimal control problems. Instead of insisting on the maximum condition of the Hamiltonian, we show that 0 belongs to the sum of Clarke’s generalized gradient of the Hamiltonian and Clarke’s normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle.