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## 1. chinaXiv:202002.00021 [pdf]

Subjects: Mathematics >> Mathematics （General）

 2019年12月，新型冠状病毒肺炎(NCP，又称2019-nCoV)疫情从武汉开始爆发,几天内迅速传播到全国乃至海外，对我国的工农业生产和人民生活产生了重要影响。科学有效掌控疫情发展对疫情防控至关重要。本文基于中国卫健委及湖北省卫健委每日公布的累计确诊数，采用逻辑斯蒂模型对数据进行了拟合，以期给该疾病的防控治提供科学依据。通过公布的疫情数据，我们反演了模型的参数，进而有效地模拟了目前疫情的发展，并预测了疫情未来的趋势。我们预测，湖北省疫情还要持续至少2周，而在全国其他地区，疫情可望1周左右达到顶峰。

submitted time 2020-02-18 Hits26952Downloads2888 Comment 0

## 2. chinaXiv:201809.00178 [pdf]

Subjects: Mathematics >> Mathematics （General）

 The aim of this paper is to study the heterogeneous optimization problem \begin{align*} \mathcal {J}(u)=\int_{\Omega}(G(|\nabla u|)+qF(u^+)+hu+\lambda_{+}\chi_{\{u>0\}} )\text{d}x\rightarrow\text{min}, \end{align*} in the class of functions $W^{1,G}(\Omega)$ with $u-\varphi\in W^{1,G}_{0}(\Omega)$, for a given function $\varphi$, where $W^{1,G}(\Omega)$ is the class of weakly differentiable functions with $\int_{\Omega}G(|\nabla u|)\text{d}x<\infty$. The functions $G$ and $F$ satisfy structural conditions of Lieberman's type that allow for a different behavior at $0$ and at $\infty$. Given functions $q,h$ and constant $\lambda_+\geq 0$, we address several regularities for minimizers of $\mathcal {J}(u)$, including local $C^{1,\alpha}-$, and local Log-Lipschitz continuities for minimizers of $\mathcal {J}(u)$ with $\lambda_+=0$, and $\lambda_+>0$ respectively. We also establish growth rate near the free boundary for each non-negative minimizer of $\mathcal {J}(u)$ with $\lambda_+=0$, and $\lambda_+>0$ respectively. Furthermore, under additional assumption that $F\in C^1([0,+\infty); [0,+\infty))$, local Lipschitz regularity is carried out for non-negative minimizers of $\mathcal {J}(u)$ with $\lambda_{+}>0$.

submitted time 2018-09-23 Hits17799Downloads1362 Comment 0

## 3. chinaXiv:201809.00180 [pdf]

Subjects: Mathematics >> Mathematics （General）

 In this paper, we prove several regularity results for the heterogeneous, two-phase free boundary problems $\mathcal {J}_{\gamma}(u)=\int_{\Omega}\big(f(x,\nabla u)+(\lambda_{+}(u^{+})^{\gamma}+\lambda_{-}(u^{-})^{\gamma})+gu\big)\text{d}x\rightarrow \text{min}$ under non-standard growth conditions. Included in such problems are heterogeneous jets and cavities of Prandtl-Batchelor type with $\gamma=0$, chemical reaction problems with $0<\gamma<1$, and obstacle type problems with $\gamma=1$. Our results hold not only in the degenerate case of $p> 2$ for $p-$Laplace equations, but also in the singular case of $1 submitted time 2018-09-22 Hits15151Downloads1213 Comment 0 ## 4. chinaXiv:201809.00179 [pdf] Subjects: Mathematics >> Mathematics （General）  We establish regularity of solutions to the$G$-Laplace equation$-\text{div}\ \bigg(\frac{g(|\nabla u|)}{|\nabla u|}\nabla u\bigg)=\mu$, where$\mu$is a nonnegative Radon measure satisfying$\mu (B_{r}(x_{0}))\leq Cr^{m}$for any ball$B_{r}(x_{0})\subset\subset \Omega$with$r\leq 1$and$m>n-1-\delta\geq 0$. The function$g(t)$is supposed to be nonnegative and$C^{1}$-continuous in$[0,+\infty)$, satisfying$g(0)=0$, and for some positive constants$\delta$and$g_{0}$,$\delta\leq \frac{tg'(t)}{g(t)}\leq g_{0}, \forall t>0$, that generalizes the structural conditions of Ladyzhenskaya-Ural'tseva for an elliptic operator. submitted time 2018-09-22 Hits857Downloads857 Comment 0 ## 5. chinaXiv:201611.00721 [pdf] Subjects: Mathematics >> Mathematics （General）  Explicit equations are obtained to convert Cartesian coordinates to elliptic coordinates, based on which an elliptic-coordinate function can be readily mapped on a uniform Cartesian mesh.Application to Kirchhoff vortex is provided. submitted time 2018-09-22 Hits36109Downloads1810 Comment 0 ## 6. chinaXiv:201809.00176 [pdf] 郭旭; 王浩帆; 郑军 Subjects: Mathematics >> Mathematics （General）  本文研究一类高阶非线性微分方程的Lyapunov 不等式，是对《Lyapunov-type inequalities for$\psi$-Laplacian equations》有关结论的进一步探讨和推广. submitted time 2018-09-18 Hits10801Downloads1273 Comment 0 ## 7. chinaXiv:201809.00116 [pdf] Subjects: Mathematics >> Mathematics （General）  The aim of this paper is to study the obstacle problem associated with an elliptic operator having degenerate coercivity, and with a low order term and$L^1-$data. We prove the existence of an entropy solution to the obstacle problem and show its continuous dependence on the$L^{1}-$data in$W^{1,q}(\Omega)$with some$q>1. submitted time 2018-09-13 Hits14346Downloads1083 Comment 0 ## 8. chinaXiv:201806.00016 [pdf] Subjects: Mathematics >> Mathematics （General）  In this work, we establish several Lyapunov-type inequalities for a class of nonlinear higher order differential equations having a form \begin{align*} (\psi(u^{(m)}(x)))'+\sum_{i=0}^nr_i(x)f_i(u^{(i)}(x))=0, %\ \ \ \ \text{or}\ \ \ \ (\psi(u^{(m)}))^{(m)}+r_i(x)f(u)=0, \end{align*} with anti-periodic boundary conditions, wherem> n\geq 0$are integers,$\psi$and$f_i (i=0,1,2,...,n)$satisfy certain structural conditions such that the considered equations have general nonlinearities. The obtained inequalities are extensions and complements of the existing results in the literature. submitted time 2018-06-04 Hits13192Downloads1340 Comment 0 ## 9. chinaXiv:201805.00171 [pdf] Subjects: Mathematics >> Mathematics （General）  In this work, we present several Lyapunov-type inequalities for a class of$\psi-Laplacian equations of the form \begin{align*} (\psi(u'(x)))'+r(x)f(u(x))=0, \end{align*} with Dirichlet boundary conditions, where\psi$and$f$satisfies certain structural conditions with general nonlinearities. We do not require any sub-multiplicative property of$\psi$, and any convexity of$\frac{1}{\psi(t)}$or$\psi (t)t$in the establishment of Lyapunov-type inequalities. The obtained inequalities can be seen as extensions and complements of the existing results in the literature. submitted time 2018-05-22 Hits8822Downloads1191 Comment 0 ## 10. chinaXiv:201712.02142 [pdf] Subjects: Mathematics >> Mathematics （General）  In this paper, we introduce the concept of Z$_1$-eigenvalue to infinite dimensional generalized Hilbert tensors (hypermatrix)$\mathcal{H}_\lambda^{\infty}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}})$, $$\mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots i_{m}+\lambda},\ \lambda\in \mathbb{R}\setminus\mathbb{Z}^-;\ i_{1},i_{2},\cdots,i_{m}=0,1,2,\cdots,n,\cdots,$$ and proved that its$Z_1$-spectral radius is not larger than$\pi$for$\lambda>\frac{1}{2}$, and is at most$\frac{\pi}{\sin{\lambda\pi}}$for$\frac{1}{2}\geq \lambda>0$. Besides, the upper bound of$Z_1$-spectral radius of an$m$th-order$n$-dimensional generalized Hilbert tensor$\mathcal{H}_\lambda^n$is obtained also, and such a bound only depends on$n$and$\lambda\$.

submitted time 2017-12-12 Hits6641Downloads1478 Comment 0

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