Genetic variations in the COVID19 virus are one of the main causes of the COVID19 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
Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is nonconservative even when restricted to SumofSquares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose
 Award ID(s):
 1931270
 Publication Date:
 NSFPAR ID:
 10354588
 Journal Name:
 Journal of Computational Dynamics
 Volume:
 0
 Issue:
 0
 Page Range or eLocationID:
 0
 ISSN:
 21582491
 Sponsoring Org:
 National Science Foundation
More Like this

and removed\begin{document}$ S $\end{document} populations by ODEs and the infected\begin{document}$ R $\end{document} population by a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for\begin{document}$ I $\end{document} and\begin{document}$ S $\end{document} contains terms that are related to the measure\begin{document}$ R $\end{document} . We establish analytically the wellposedness of the coupled ODEMDE system by using generalized Wasserstein distance. We give two examples to show that the proposed ODEMDE model coincides with the classical SIR model in case of constant or timedependent parameters as special cases.\begin{document}$ I $\end{document} 
This paper introduces a novel generative encoder (GE) framework for generative imaging and image processing tasks like image reconstruction, compression, denoising, inpainting, deblurring, and superresolution. GE unifies the generative capacity of GANs and the stability of AEs in an optimization framework instead of stacking GANs and AEs into a single network or combining their loss functions as in existing literature. GE provides a novel approach to visualizing relationships between latent spaces and the data space. The GE framework is made up of a pretraining phase and a solving phase. In the former, a GAN with generator
capturing the data distribution of a given image set, and an AE network with encoder\begin{document}$ G $\end{document} that compresses images following the estimated distribution by\begin{document}$ E $\end{document} are trained separately, resulting in two latent representations of the data, denoted as the generative and encoding latent space respectively. In the solving phase, given noisy image\begin{document}$ G $\end{document} , where\begin{document}$ x = \mathcal{P}(x^*) $\end{document} is the target unknown image,\begin{document}$ x^* $\end{document} is an operator adding an addictive, or multiplicative, or convolutional noise, or equivalently given such an image\begin{document}$ \mathcal{P} $\end{document} more »\begin{document}$ x $\end{document} and the image
is recovered in a generative way via\begin{document}$ x^* $\end{document} , where\begin{document}$ \hat{x}: = G(z^*)\approx x^* $\end{document} is a hyperparameter. The unification of the two spaces allows improved performance against corresponding GAN and AE networks while visualizing interesting properties in each latent space.\begin{document}$ \lambda>0 $\end{document} 
We consider the wellknown LiebLiniger (LL) model for
bosons interacting pairwise on the line via the\begin{document}$ N $\end{document} potential in the meanfield scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the timedependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the onedimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [\begin{document}$ \delta $\end{document} 3 ] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65 ,66 ,67 ] and Knowles and Pickl [44 ]. To overcome difficulties stemming from the singularity of the potential, we introduce a new shortrange approximation argument that exploits the Hölder continuity of the\begin{document}$ \delta $\end{document} body wave function in a single particle variable. By further exploiting the\begin{document}$ N $\end{document} subcritical wellposedness theory for the 1D cubic NLS, we can prove meanfield convergence when the limiting solution to the NLS has finitemore »\begin{document}$ L^2 $\end{document} 
Abstract We consider the problem of covering multiple submodular constraints. Given a finite ground set
N , a weight function ,$$w: N \rightarrow \mathbb {R}_+$$ $w:N\to {R}_{+}$r monotone submodular functions over$$f_1,f_2,\ldots ,f_r$$ ${f}_{1},{f}_{2},\dots ,{f}_{r}$N and requirements the goal is to find a minimum weight subset$$k_1,k_2,\ldots ,k_r$$ ${k}_{1},{k}_{2},\dots ,{k}_{r}$ such that$$S \subseteq N$$ $S\subseteq N$ for$$f_i(S) \ge k_i$$ ${f}_{i}\left(S\right)\ge {k}_{i}$ . We refer to this problem as$$1 \le i \le r$$ $1\le i\le r$MultiSubmodCover and it was recently considered by HarPeled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260 HarPeled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$ $r=1$MultiSubmodCover generalizes the wellknown Submodular Set Cover problem (SubmodSC ), and it can also be easily reduced toSubmodSC . A simple greedy algorithm gives an approximation where$$O(\log (kr))$$ $O(log(kr\left)\right)$ and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm for$$k = \sum _i k_i$$ $k={\sum}_{i}{k}_{i}$MultiSubmodCover that covers each constraint to within a factor of while incurring an approximation of$$(11/e\varepsilon )$$ $(11/e\epsilon )$ in the cost. Second, we consider the special case when each$$O(\frac{1}{\epsilon }\log r)$$ $O(\frac{1}{\u03f5}logr)$ is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover ($$f_i$$ ${f}_{i}$PartialSC ), covering integer programs (CIPs ) and multiple vertex cover constraintsmore » 
In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian
for\begin{document}$ ( \Delta)^\frac{{ \alpha}}{{2}} $\end{document} . One main advantage is that our method can easily increase numerical accuracy by using highdegree Lagrange basis functions, but remain its scheme structure and computer implementation unchanged. Moreover, it results in a symmetric (multilevel) Toeplitz differentiation matrix, enabling efficient computation via the fast Fourier transforms. If constant or linear basis functions are used, our method has an accuracy of\begin{document}$ \alpha \in (0, 2) $\end{document} , while\begin{document}$ {\mathcal O}(h^2) $\end{document} for quadratic basis functions with\begin{document}$ {\mathcal O}(h^4) $\end{document} a small mesh size. This accuracy can be achieved for any\begin{document}$ h $\end{document} and can be further increased if higherdegree basis functions are chosen. Numerical experiments are provided to approximate the fractional Laplacian and solve the fractional Poisson problems. It shows that if the solution of fractional Poisson problem satisfies\begin{document}$ \alpha \in (0, 2) $\end{document} for\begin{document}$ u \in C^{m, l}(\bar{ \Omega}) $\end{document} and\begin{document}$ m \in {\mathbb N} $\end{document} , our method has an accuracy of\begin{document}$ 0 < l < 1 $\end{document} for constant and linear basis functions, while\begin{document}$more » for quadratic basis functions. Additionally, our method can be readily applied to approximate the generalized fractional Laplacians with symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.\begin{document}$ {\mathcal O}(h^{\min\{m+l, \, 4\}}) $\end{document}