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1. Impulsive stochastic functional differential equations with Markovian switching: study of exponential stability from a numerical solution point of view

   https://doi.org/10.1093/imamci/dnaf008  

   Tran, Ky_Q; Yin, George (May 2025, IMA Journal of Mathematical Control and Information)

   Abstract This paper is devoted to the study of stochastic functional differential systems with Markov switching. It focuses on the stability of numerical solutions. To gain insight on how impulses, functional past dependence, random switching and stochastic disturbances can have impact on dynamic systems, this paper addresses exponential stability of the Euler–Maruyama approximations of stochastic functional differential equations with impulsive perturbations and Markovian switching. It begins with a presentation of the definitions of exponential stability in mean square and in the almost sure sense for stochastic functional differential equations with impulsive perturbations and Markovian switching. Then, it is devoted to showing that if the underlying system is stable in the aforementioned sense then the Euler–Maruyama approximation method faithfully reproduces exponential stability in the mean square and almost sure sense for sufficiently small step sizes and large iteration number. Two examples are provided to demonstrate the effectiveness of our results. 

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2. Alternative stable states and hydrological regime shifts in a large intermittent river

   https://doi.org/10.1088/1748-9326/ac7539  

   Zipper, Sam; Popescu, Ilinca; Compare, Kyle; Zhang, Chi; Seybold, Erin C (June 2022, Environmental Research Letters)

   Abstract Non-perennial rivers and streams make up over half the global river network and are becoming more widespread. Transitions from perennial to non-perennial flow are a threshold-type change that can lead to alternative stable states in aquatic ecosystems, but it is unknown whether streamflow itself is stable in either wet (flowing) or dry (no-flow) conditions. Here, we investigated drivers and feedbacks associated with regime shifts between wet and dry conditions in an intermittent reach of the Arkansas River (USA) over the past 23 years. Multiple lines of evidence suggested that these regimes represent alternative stable states, including (a) significant jumps in discharge time series that were not accompanied by jumps in flow drivers such as precipitation and groundwater pumping; (b) a multi-modal state distribution with 92% of months experiencing no-flow conditions for <10% or >90% of days, despite unimodal distributions of precipitation and pumping; and (c) a hysteretic relationship between climate and flow state. Groundwater levels appear to be the primary control over the hydrological regime, as groundwater levels in the alluvial aquifer were higher than the stream stage during wet regimes and lower than the streambed during dry regimes. Groundwater level variation, in turn, was driven by processes occurring at both the regional scale (surface water inflows from upstream, groundwater pumping) and the reach scale (stream–aquifer exchange, diffuse recharge through the soil column). Historical regime shifts were associated with diverse pressures including network disconnection caused by upstream water use, increased flow stability potentially associated with reservoir operations, and anomalous wet and dry climate conditions. In sum, stabilizing feedbacks among upstream inflows, stream–aquifer interactions, climate, vegetation, and pumping appear to create alternative wet and dry stable states at this site. These stabilizing feedbacks suggest that widespread observed shifts from perennial to non-perennial flow will be difficult to reverse. 

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3. Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective

   https://doi.org/10.46298/mna.7203  

   Liu, Jian-Guo; Wang, Ziheng; Xie, Yantong; Zhang, Yuan; Zhou, Zhennan (November 2021, Mathematical Neuroscience and Applications)

   In the mean field integrate-and-fire model, the dynamics of a typical neuronwithin a large network is modeled as a diffusion-jump stochastic process whosejump takes place once the voltage reaches a threshold. In this work, the maingoal is to establish the convergence relationship between the regularizedprocess and the original one where in the regularized process, the jumpmechanism is replaced by a Poisson dynamic, and jump intensity within theclassically forbidden domain goes to infinity as the regularization parametervanishes. On the macroscopic level, the Fokker-Planck equation for the processwith random discharges (i.e. Poisson jumps) are defined on the whole space,while the equation for the limit process is on the half space. However, withthe iteration scheme, the difficulty due to the domain differences has beengreatly mitigated and the convergence for the stochastic process and the firingrates can be established. Moreover, we find a polynomial-order convergence forthe distribution by a re-normalization argument in probability theory. Finally,by numerical experiments, we quantitatively explore the rate and the asymptoticbehavior of the convergence for both linear and nonlinear models. 

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4. Effects of Abdominal Rotation on Jump Performance in the Ant Gigantiops destructor (Hymenoptera, Formicidae)

   https://doi.org/10.1093/iob/obz033  

   Ye, Dajia; Gibson, Joshua C; Suarez, Andrew V (January 2020, Integrative Organismal Biology)

   Synopsis Jumping is an important form of locomotion, and animals employ a variety of mechanisms to increase jump performance. While jumping is common in insects generally, the ability to jump is rare among ants. An exception is the Neotropical ant Gigantiops destructor (Fabricius 1804) which is well known for jumping to capture prey or escape threats. Notably, this ant begins a jump by rotating its abdomen forward as it takes off from the ground. We tested the hypotheses that abdominal rotation is used to either provide thrust during takeoff or to stabilize rotational momentum during the initial airborne phase of the jump. We used high speed videography to characterize jumping performance of G. destructor workers jumping between two platforms. We then anesthetized the ants and used glue to prevent their abdomens from rotating during subsequent jumps, again characterizing jump performance after restraining the abdomen in this manner. Our results support the hypothesis that abdominal rotation provides additional thrust as the maximum distance, maximum height, and takeoff velocity of jumps were reduced by restricting the movement of the abdomen compared with the jumps of unmanipulated and control treatment ants. In contrast, the rotational stability of the ants while airborne did not appear to be affected. Changes in leg movements of restrained ants while airborne suggest that stability may be retained by using the legs to compensate for changes in the distribution of mass during jumps. This hypothesis warrants investigation in future studies on the jump kinematics of ants or other insects. 

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5. A High-Order Hybrid Approach Integrating Neural Networks and Fast Poisson Solvers for Elliptic Interface Problems

   https://doi.org/10.3390/computation13040083  

   Ren, Yiming; Zhao, Shan (April 2025, Computation)

   A new high-order hybrid method integrating neural networks and corrected finite differences is developed for solving elliptic equations with irregular interfaces and discontinuous solutions. Standard fourth-order finite difference discretization becomes invalid near such interfaces due to the discontinuities and requires corrections based on Cartesian derivative jumps. In traditional numerical methods, such as the augmented matched interface and boundary (AMIB) method, these derivative jumps can be reconstructed via additional approximations and are solved together with the unknown solution in an iterative procedure. Nontrivial developments have been carried out in the AMIB method in treating sharply curved interfaces, which, however, may not work for interfaces with geometric singularities. In this work, machine learning techniques are utilized to directly predict these Cartesian derivative jumps without involving the unknown solution. To this end, physics-informed neural networks (PINNs) are trained to satisfy the jump conditions for both closed and open interfaces with possible geometric singularities. The predicted Cartesian derivative jumps can then be integrated in the corrected finite differences. The resulting discrete Laplacian can be efficiently solved by fast Poisson solvers, such as fast Fourier transform (FFT) and geometric multigrid methods, over a rectangular domain with Dirichlet boundary conditions. This hybrid method is both easy to implement and efficient. Numerical experiments in two and three dimensions demonstrate that the method achieves fourth-order accuracy for the solution and its derivatives. 

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