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We study an inverse problem for the time-dependent Maxwell system in an inhomogeneous and anisotropic medium. The objective is to recover the initial electric field $$\mathbf{E}_0$$ in a bounded domain $$\Omega \subset \mathbb{R}^3$$, using boundary measurements of the electric field and its normal derivative over a finite time interval. Informed by practical constraints, we adopt an under-determined formulation of Maxwell's equations that avoids the need for initial magnetic field data and charge density information. To address this inverse problem, we develop a time-dimension reduction approach by projecting the electric field onto a finite-dimensional Legendre polynomial-exponential basis in time. This reformulates the original space-time problem into a sequence of spatial systems for the projection coefficients. The reconstruction is carried out using the quasi-reversibility method within a minimum-norm framework, which accommodates the inherent non-uniqueness of the under-determined setting. We prove a convergence theorem that ensures the quasi-reversibility solution approximates the true solution as the noise and regularization parameters vanish. Numerical experiments in a fully three-dimensional setting validate the method's performance. The reconstructed initial electric field remains accurate even with $$10\%$$ noise in the data, demonstrating the robustness and applicability of the proposed approach to realistic inverse electromagnetic problems. 

We introduce a time-dimensional reduction method for the inverse source problem in linear elasticity, where the goal is to reconstruct the initial displacement and velocity fields from partial boundary measurements of elastic wave propagation. The key idea is to employ a novel spectral representation in time, using an orthonormal basis composed of Legendre polynomials weighted by exponential functions. This Legendre polynomial-exponential basis enables a stable and accurate decomposition in the time variable, effectively reducing the original space-time inverse problem to a sequence of coupled spatial elasticity systems that no longer depend on time. These resulting systems are solved using the quasi-reversibility method. On the theoretical side, we establish a convergence theorem ensuring the stability and consistency of the regularized solution obtained by the quasi-reversibility method as the noise level tends to zero. On the computational side, two-dimensional numerical experiments confirm the theory and demonstrate the method's ability to accurately reconstruct both the geometry and amplitude of the initial data, even under substantial measurement noise. The results highlight the effectiveness of the proposed framework as a robust and computationally efficient strategy for inverse elastic source problems. 

This paper studies the fusogenicity of cationic liposomes in relation to their surface distribution of cationic lipids and utilizes membrane phase separation to control this surface distribution. It is found that concentrating the cationic lipids into small surface patches on liposomes, through phase-separation, can enhance liposome’s fusogenicity. Further concentrating these lipids into smaller patches on the surface of liposomes led to an increased level of fusogenicity. These experimental findings are supported by numerical simulations using a mathematical model for phase-separated charged liposomes. Findings of this study may be used for design and development of highly fusogenic liposomes with minimal level of toxicity. 

"This paper generalizes the results obtained by the authors in Dang et al. (SIAM J. Appl. Math. 81(6):2547--2568, 2021) concerning the homogenization of a non-dilute suspension of magnetic particles in a viscous flow. More specifically, in this paper, a restrictive assumption on the coefficients of the coupled equation, made in Dang et al. (SIAM J. Appl. Math. 81(6):2547--2568, 2021), that significantly narrowed the applicability of the homogenization results obtained is relaxed and a new regularity of the solution of the fine-scale problem is proven. In particular, we obtain a global L∞-bound for the gradient of the solution of the scalar equation −divax∕$$\epsilon$$∇$$\phi$$\epsilon$$(x)=f(x){\$$}{\$$}- {\backslash}operatorname {\{}{\{}{\backslash}mathrm {\{}div{\}}{\}}{\}} {\backslash}left [ {\backslash}mathbf {\{}a{\}} {\backslash}left ( x/{\backslash}varepsilon {\backslash}right ){\backslash}nabla {\backslash}varphi ^{\{}{\backslash}varepsilon {\}}(x) {\backslash}right ] = f(x){\$$}{\$$}, uniform with respect to microstructure scale parameter $$\epsilon$${\thinspace}≪{\thinspace}1 in a small interval (0, $$\epsilon$$0), where the coefficient a is only piecewise H{\"o}lder continuous. Thenceforth, this regularity is used in the derivation of the effective response of the given suspension discussed in Dang et al. (SIAM J. Appl. Math. 81(6):2547--2568, 2021)." 

Water plays a critical role in our living and manufacturing activities. The continuously growing exploitation of water over the aquifer poses a risk for over-extraction and pollution, leading to many negative effects on land irrigation. Therefore, predicting aquifer water levels accurately is urgently important, which can help us prepare water demands ahead of time. In this study, we employ the Long-Short Term Memory (LSTM) model to predict the saturated thickness of an aquifer in the Southern High Plains Aquifer System in Texas and exploit TensorBoard as a guide for model configurations. The Root Mean Squared Error of this study shows that the LSTM model can provide a good prediction capability using multiple data sources, and provides a good visualization tool to help us understand and evaluate the model configuration.