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Abstract In this paper we study the degenerate parabolicp-Laplacian, ${\partial }_{t}u-v^{-1}\mathrm{div}\left({\left|\sqrt{Q}\nabla u\right|}^{p-2}Q\nabla u\right)=0${\partial_{t}u-v^{-1}\operatorname{div}(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)=0},where the degeneracy is controlled by a matrixQand a weightv.With mild integrability assumptions onQandv, we prove theexistence and uniqueness of solutions on any interval $[0,T]${[0,T]}. If we further assumethe existence of a degenerate Sobolev inequality with gain, thedegeneracy again controlled byvandQ, then we can prove bothfinite time extinction and ultracontractive bounds. Moreover, weshow that there is equivalence between the existence ofultracontractive bounds and the weighted Sobolev inequality. 

Abstract Variational implicit solvation models (VISMs) have gained extensive popularity in the molecular-level solvation analysis of biological systems due to their cost-effectiveness and satisfactory accuracy. Central in the construction of VISM is an interface separating the solute and the solvent. However, traditional sharp-interface VISMs fall short in adequately representing the inherent randomness of the solute–solvent interface, a consequence of thermodynamic fluctuations within the solute–solvent system. Given that experimentally observable quantities are ensemble averaged, the computation of the ensemble average solvation energy (EASE)–the averaged solvation energy across all thermodynamic microscopic states–emerges as a key metric for reflecting thermodynamic fluctuations during solvation processes. This study introduces a novel approach to calculating the EASE. We devise two diffuse-interface VISMs: one within the classic Poisson–Boltzmann (PB) framework and another within the framework of size-modified PB theory, accounting for the finite-size effects. The construction of these models relies on a new diffuse interface definition $u\left(x\right)$u\left(x), which represents the probability of a point $x$xfound in the solute phase among all microstates. Drawing upon principles of statistical mechanics and geometric measure theory, we rigorously demonstrate that the proposed models effectively capture EASE during the solvation process. Moreover, preliminary analyses indicate that the size-modified EASE functional surpasses its counterpart based on the classic PB theory across various analytic aspects. Our work is the first step toward calculating EASE through the utilization of diffuse-interface VISM. 

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. 

We revisit the theory of first-order quasilinear systems with diagonalizable principal part and only real eigenvalues, what is commonly referred to as strongly hyperbolic systems. We provide a self-contained and simple proof of local well-posedness, in the Hadamard sense, of the Cauchy problem. Our regularity assumptions are very minimal. As an application, we apply our results to systems of ideal and viscous relativistic fluids, where the theory of strongly hyperbolic equations has been systematically used to study several systems of physical interest. 

Tropomyosin kinase receptor B (TrkB) has been explored as a therapeutic target for neurological and psychiatric disorders. However, the development of TrkB agonists was hindered by our poor understanding of the TrkB agonist binding location and affinity (both affect the regulation of disorder types). This motivated us to develop a combined computational and experimental approach to study TrkB binders. First, we developed a docking method to simulate the binding affinity of TrkB and binders identified by our magnetic drug screening platform from Gotu kola extracts. The Fred Docking scores from the docking computation showed strong agreement with the experimental results. Subsequently, using this screening platform, we identified a list of compounds from the NIH clinical collection library and applied the same docking studies. From the Fred Docking scores, we selected two compounds for TrkB activation tests. Interestingly, the ability of the compounds to increase dendritic arborization in hippocampal neurons matched well with the computational results. Finally, we performed a detailed binding analysis of the top candidates and compared them with the best-characterized TrkB agonist, 7,8-dyhydroxyflavon. The screening platform directly identifies TrkB binders, and the computational approach allows for the quick selection of top candidates with potential biological activities based on the docking scores.