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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.

    

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2. Chemical Systems with Limit Cycles

   https://doi.org/10.1007/s11538-023-01170-3  

   Erban, Radek ; Kang, Hye-Won ( July 2023 , Bulletin of Mathematical Biology)

   The dynamics of a chemical reaction network (CRN) is often modeled under the assumption of mass action kinetics by a system of ordinary differential equations (ODEs) with polynomial right-hand sides that describe the time evolution of concentrations of chemical species involved. Given an arbitrarily large integer$$K \in {\mathbb N}$$$K\in N$, we show that there exists a CRN such that its ODE model has at leastKstable limit cycles. Such a CRN can be constructed with reactions of at most second-order provided that the number of chemical species grows linearly withK. Bounds on the minimal number of chemical species and the minimal number of chemical reactions are presented for CRNs withK stable limit cycles and at most second order or seventh-order kinetics. We also show that CRNs with only two chemical species can haveKstable limit cycles, when the order of chemical reactions grows linearly withK.

    

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3. 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 $\end{document} in the compressed domain, i.e., given \begin{document}$ 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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4. The mean-field limit of the Lieb-Liniger model

   https://doi.org/10.3934/dcds.2022006  

   Rosenzweig, Matthew ( January 2022 , Discrete and Continuous Dynamical Systems)

   We consider the well-known Lieb-Liniger (LL) model for \begin{document}$ N $\end{document} bosons interacting pairwise on the line via the \begin{document}$ \delta $\end{document} potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [3] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65,66,67] and Knowles and Pickl [44]. To overcome difficulties stemming from the singularity of the \begin{document}$ \delta $\end{document} potential, we introduce a new short-range approximation argument that exploits the Hölder continuity of the \begin{document}$ N $\end{document}-body wave function in a single particle variable. By further exploiting the \begin{document}$ L^2 $\end{document}-subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence when the limiting solution to the NLS has finite mass, but only for a very special class of \begin{document}$ N $\end{document}-body initial states.

    

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5. Algorithmic reconstruction of the fiber of persistent homology on cell complexes

   https://doi.org/10.1007/s41468-024-00165-w  

   Leygonie, Jacob ; Henselman-Petrusek, Gregory ( April 2024 , Journal of Applied and Computational Topology)

   Let Kbe a finite simplicial, cubical, delta or CW complex. The persistence map $$\textrm{PH}$$$\text{PH}$takes a filter $$f:K\rightarrow \mathbb {R}$$$f:K\to R$as input and returns the barcodes of the sublevel set persistent homology of fin each dimension. We address the inverse problem: given target barcodes D, computing the fiber $$\textrm{PH}^{-1}(D)$$${\text{PH}}^{-1}\left(D\right)$. For this, we use the fact that $$\textrm{PH}^{-1}(D)$$${\text{PH}}^{-1}\left(D\right)$decomposes as a polyhedral complex when Kis a simplicial complex, and we generalise this result to arbitrary based chain complexes. We then design and implement a depth-first search that recovers the polytopes forming the fiber $$\textrm{PH}^{-1}(D)$$${\text{PH}}^{-1}\left(D\right)$. As an application, we solve a corpus of 120 sample problems, providing a first insight into the statistical structure of these fibers, for general CW complexes.

    

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