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In this paper, we explore pattern formation in a four-species variational Gary-Scott model, which includes all reverse reactions and introduces a virtual species to describe the birth–death process in the classical Gray-Scott model. This modification transforms the classical Gray-Scott model into a thermodynamically consistent closed system. The classical two-species Gray-Scott model can be viewed as a subsystem of the variational model in the limiting case when the small parameter ε, related to the reaction rate of the reverse reactions, approaches zero. We numerically explore pattern formation in this physically more complete Gray-Scott model in one spatial dimension, using non-uniform steady states of the classical model as initial conditions. By decreasing ε, we observed that the stationary patterns in the classical Gray-Scott model can be stabilized as the transient states in the variational model for a significantly small ε. Additionally, the variational model admits oscillating and traveling wave-like patterns for small ε. The persistent time of these patterns is on the order of O(1/ε). We also analyze the energy stability of two uniform steady states in the variational Gary-Scott model for fixed. Although both states are stable in a certain sense, the gradient flow type dynamics of the variational model exhibit a selection effect based on the initial conditions, with pattern formation occurring only if the initial condition does not converge to the boundary steady state, which corresponds to the trivial uniform steady state in the classical Gray-Scott model. 

The state-of-the-art determines the remaining useful lifetime (RUL) through a steady-state, fixed power cycling tests (PCT) without considering the impact of dynamically changing environmental conditions. It has resulted in considerable RUL prediction errors in the real world. However, the dynamic changing conditions (e.g., large temperature swings) may affect the degradation evolution of SiC MOSFET, which could eventually result in RUL changes. Thus, it must be integrated to make accurate predictions. To precisely understand the RUL variation complexity, the junction temperature (Tj) has been measured with a Negative Thermal Coefficient (NTC) thermistor, Temperature Sensitive Electrical Parameter (TSEP), and these profiles have been modeled through the thermal model RC foster network using Extended Kalman Filter (EKF). Then, the on-state resistance (Rds,on) variations and Degradation Acceleration Factor (DAF) under the dynamic environment conditions are integrated into a lifetime prediction model to accurately predict the RUL through the Long Short-Term Memory (LSTM) machine learning algorithm. 

In this paper, we provide a detailed theoretical analysis of the numerical scheme introduced in [C. Liu, C. Wang, and Y. Wang, J. Comput. Phys., 436:110253, 2021] for the reaction kinetics of a class of chemical reaction networks that satisfies detailed balance condition. In contrast to conventional numerical approximations, which are typically constructed based on ordinary differential equations (ODEs) for the concentrations of all involved species, the scheme is developed using the equations of reaction trajectories, which can be viewed as a generalized gradient flow of a physically relevant free energy. The unique solvability, positivity-preserving, and energy-stable properties are proved for the general case involving multiple reactions, under a mild condition on the stoichiometric matrix. 

Abstract Assouad–Nagata dimension addresses both large‐ and small‐scale behaviors of metric spaces and is a refinement of Gromov's asymptotic dimension. A metric space is a minor‐closed metric if there exists an (edge‐)weighted graph satisfying a fixed minor‐closed property such that the underlying space of is the vertex‐set of , and the metric of is the distance function in . Minor‐closed metrics naturally arise when removing redundant edges of the underlying graphs by using edge‐deletion and edge‐contraction. In this paper, we determine the Assouad–Nagata dimension of every minor‐closed metric. Our main theorem simultaneously generalizes known results about the asymptotic dimension of ‐minor free unweighted graphs and about the Assouad–Nagata dimension of complete Riemannian surfaces with finite Euler genus (Bonamy et al., Asymptotic dimension of minor‐closed families and Assouad–Nagata dimension of surfaces,JEMS(2024)). 

In this article, we introduce a new method for discretizing micro-macro models of dilute polymeric fluids by integrating a finite element method (FEM) discretization for the macroscopic fluid dynamics equation with a deterministic variational particle scheme for the microscopic Fokker-Planck equation. To address challenges arising from micro-macro coupling, we employ a discrete energetic variational approach to derive a coarse-grained micro-macro model with a particle approximation first and then develop a particle-FEM discretization for the coarse-grained model. The accuracy of the proposed method is evaluated for a Hookean dumbbell model in a Couette flow by comparing the computed velocity field with existing analytical solutions. We also use our method to study nonlinear FENE dumbbell models in different scenarios, such as extensional flow, pure shear flow, and lid-driven cavity flow. Numerical examples demonstrate that the proposed deterministic particle approach can accurately capture the various key rheological phenomena in the original FENE model, including hysteresis and δ-function-like spike behavior in extensional flows, velocity overshoot phenomenon in pure shear flows, symmetries breaking, vortex center shifting, and vortices weakening in lid-driven cavity flows, with a small number of particles.