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1. A measure model for the spread of viral infections with mutations

   https://doi.org/10.3934/nhm.2022015  

   Gong, Xiaoqian ; Piccoli, Benedetto ( January 2022 , Networks and Heterogeneous Media)

   Genetic variations in the COVID-19 virus are one of the main causes of the COVID-19 pandemic outbreak in 2020 and 2021. In this article, we aim to introduce a new type of model, a system coupled with ordinary differential equations (ODEs) and measure differential equation (MDE), stemming from the classical SIR model for the variants distribution. Specifically, we model the evolution of susceptible \begin{document}$ S $\end{document} and removed \begin{document}$ R $\end{document} populations by ODEs and the infected \begin{document}$ I $\end{document} population by a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for \begin{document}$ S $\end{document} and \begin{document}$ R $\end{document} contains terms that are related to the measure \begin{document}$ I $\end{document}. We establish analytically the well-posedness of the coupled ODE-MDE system by using generalized Wasserstein distance. We give two examples to show that the proposed ODE-MDE model coincides with the classical SIR model in case of constant or time-dependent parameters as special cases.
2. On the rigorous derivation of the incompressible Euler equation from Newton’s second law

   https://doi.org/10.1007/s11005-023-01630-w  

   Rosenzweig, Matthew ( January 2023 , Letters in Mathematical Physics)

   A long-standing problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan and Iacobelli (Proc Am Math Soc 149:3045–3061, 2021) showed that in the monokinetic regime, one can directly obtain the Euler equation from a system ofNparticles interacting in$${\mathbb {T}}^d$$$T^d$,$$d\ge 2$$$d\ge 2$, via Newton’s second law through asupercritical mean-field limit. Namely, the coupling constant$$\lambda $$$\lambda$in front of the pair potential, which is Coulombic, scales like$$N^{-\theta }$$$N^{-\theta }$for some$$\theta \in (0,1)$$$\theta \in (0,1)$, in contrast to the usual mean-field scaling$$\lambda \sim N^{-1}$$$\lambda \sim N^{-1}$. Assuming$$\theta \in (1-\frac{2}{d(d+1)},1)$$$\theta \in (1-\frac{2}{d(d+1)},1)$, they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as$$N\rightarrow \infty $$$N\to \infty$. Han-Kwan and Iacobelli asked if their range for$$\theta $$$\theta$was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit$$N\rightarrow \infty $$$N\to \infty$for$$\theta \in (1-\frac{2}{d},1)$$$\theta \in (1-\frac{2}{d},1)$. Our proof is based on Serfaty’s modulated-energy method, but compared to that of Han-Kwan and Iacobelli, crucially uses an improved “renormalized commutator” estimate to obtain the larger range for$$\theta $$$\theta$. Additionally, we show that for$$\theta \le 1-\frac{2}{d}$$$\theta \le 1-\frac{2}{d}$, one cannot, in general, expect convergence in the modulated energy notion of distance.
3. Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains

   https://doi.org/10.3934/dcds.2021188  

   Le, Nam Q. ( January 2022 , Discrete & Continuous Dynamical Systems)

   By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as \begin{document}$ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $\end{document} with zero boundary data, have unexpected degenerate nature.
4. Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation

   https://doi.org/10.3934/krm.2022012  

   Nazarov, Fedor ; Zumbrun, Kevin ( January 2022 , Kinetic and Related Models)

   We establish an instantaneous smoothing property for decaying solutions on the half-line \begin{document}$ (0, +\infty) $\end{document} of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of \begin{document}$ H^1 $\end{document} stable manifolds of such equations, showing that \begin{document}$ L^2_{loc} $\end{document} solutions that remain sufficiently small in \begin{document}$ L^\infty $\end{document} (i) decay exponentially, and (ii) are \begin{document}$ C^\infty $\end{document} for \begin{document}$ t>0 $\end{document}, hence lie eventually in the \begin{document}$ H^1 $\end{document} stable manifold constructed by Pogan and Zumbrun.
5. Generative imaging and image processing via generative encoder

   https://doi.org/10.3934/ipi.2021060  

   Ong, Yong Zheng ; Yang, Haizhao ( January 2022 , Inverse Problems & Imaging)

   This paper introduces a novel generative encoder (GE) framework for generative imaging and image processing tasks like image reconstruction, compression, denoising, inpainting, deblurring, and super-resolution. GE unifies the generative capacity of GANs and the stability of AEs in an optimization framework instead of stacking GANs and AEs into a single network or combining their loss functions as in existing literature. GE provides a novel approach to visualizing relationships between latent spaces and the data space. The GE framework is made up of a pre-training phase and a solving phase. In the former, a GAN with generator \begin{document}$ G $\end{document} capturing the data distribution of a given image set, and an AE network with encoder \begin{document}$ E $\end{document} that compresses images following the estimated distribution by \begin{document}$ G $\end{document} are trained separately, resulting in two latent representations of the data, denoted as the generative and encoding latent space respectively. In the solving phase, given noisy image \begin{document}$ x = \mathcal{P}(x^*) $\end{document}, where \begin{document}$ x^* $\end{document} is the target unknown image, \begin{document}$ \mathcal{P} $\end{document} is an operator adding an addictive, or multiplicative, or convolutional noise, or equivalently given such an image \begin{document}$ x $\end{document}more »in the compressed domain, i.e., given \begin{document}$ m = E(x) $\end{document}, the two latent spaces are unified via solving the optimization problem

   \begin{document}$ z^* = \underset{z}{\mathrm{argmin}} \|E(G(z))-m\|_2^2+\lambda\|z\|_2^2 $\end{document}

   and the image \begin{document}$ x^* $\end{document} is recovered in a generative way via \begin{document}$ \hat{x}: = G(z^*)\approx x^* $\end{document}, where \begin{document}$ \lambda>0 $\end{document} is a hyperparameter. The unification of the two spaces allows improved performance against corresponding GAN and AE networks while visualizing interesting properties in each latent space.

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