A refined convergence result in homogenization of second order parabolic systems

We derive the sharp $O(\epsilon )$ convergence rate in $L^2(0,T;L^{q_0}(\mathrm{\Omega })),q_0=2d/(d?1)$ in periodic homogenization of second order parabolic systems with bounded measurable coefficients in Lipschitz cylinders. This extends the corresponding result for elliptic systems established in [20] to parabolic systems and improves the corresponding result in $L^2$ settings derived in [7], [28] for second order parabolic systems with time-dependent coefficients.     

Numerical and experimental investigations on the band-gap characteristics of metamaterial multi-span beams

A novel metamaterial multi-span beam with periodic simple supports and local resonators is designed and investigated. The frequency responses of the proposed metamaterial multi-span beam are computed by the spectral element method (SEM). The accuracy and feasibility of the SEM are verified by the finite element method (FEM) and the vibration experiments. The results show that the metamaterial multi-span beam could generate both the local resonance band-gaps in the low-frequency ranges and the Bragg band-gaps in the medium and high frequency regions. By adjusting the natural frequencies of the local resonators, the thickness of the base beam and the length of the unit-cell, the local resonance and the Bragg band-gaps can be controlled, respectively. The coupling effects of these two kinds of band-gaps are investigated by the parametrical design, which broadens the band-gaps and consequently improves the vibration reduction performance.     

一阶非线性泛函微分方程的周期解

本文研究了一阶非线性泛函微分方程的周期解,得到了周期解存在唯一的一些充分条件,推广和改进了这方面的已有结果。     

一类带有治愈率的HIV感染的CD4＋T细胞模型的动力学性质

本文主要介绍一类带有治愈率的HIV感染的CD4 T细胞模型的动力学性质,同时证明了如果基本再生数R0＜1,HIV感染消失;如果R0＞1,HIV感染持续.然后进行数值模拟,给出了地方性平衡点E·全局稳定的参数域,得到了地方性平衡点E·不稳定时周期解存在.     

Harmonic approximation of difference operators

For a general class of difference operators Hε=Tε+Vε on ?²(d(εZ)), where Vε is a multi-well potential and ε is a small parameter, we analyze the asymptotic behavior as ε→0 of the (low-lying) eigenvalues and eigenfunctions. We show that the first n eigenvalues of Hε converge to the first n eigenvalues of the direct sum of harmonic oscillators on Rd located at the several wells. Our proof is microlocal.     

On the finiteness property for rational matrices

We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This “finiteness conjecture” is now known to be false but no explicit counterexample is available and in particular it is unclear if a counterexample is possible whose matrices have rational or binary entries. In this paper, we prove that all finite sets of nonnegative rational matrices have the finiteness property if and only if pairs of binary matrices do and we state a similar result when negative entries are allowed. We also show that all pairs of 2×2 binary matrices have the finiteness property. These results have direct implications for the stability problem for sets of matrices. Stability is algorithmically decidable for sets of matrices that have the finiteness property and so it follows from our results that if all pairs of binary matrices have the finiteness property then stability is decidable for nonnegative rational matrices. This would be in sharp contrast with the fact that the related problem of boundedness is known to be undecidable for sets of nonnegative rational matrices.     

Reducibility for wave equations of finitely smooth potential with periodic boundary conditions

In the present paper, the reducibility is derived for the wave equations with finitely smooth and time-quasi-periodic potential subject to periodic boundary conditions. More exactly, the linear wave equation $u_{tt}?u_{xx}+Mu+\epsilon (V_0(\omega t)u_{xx}+V(\omega t,x)u)=0,x\in \mathbb{R}/2\pi \mathbb{Z}$ can be reduced to a linear Hamiltonian system with a constant coefficient operator which is of pure imaginary point spectrum set, where V is finitely smooth in $(t,x)$, quasi-periodic in time t with Diophantine frequency $\omega \in {\mathbb{R}}^n$, and $V_0$ is finitely smooth and quasi-periodic in time t with Diophantine frequency $\omega \in {\mathbb{R}}^n$. Moreover, it is proved that the corresponding wave operator possesses the property of pure point spectra and zero Lyapunov exponent.     

New insights into pharmacologic inhibition of pyroptotic cell death by necrosulfonamide: A PDE model

Pyroptosis is a highly inflammatory form of cell death, which uses intracellularly generated pores to destroy electrolyte homeostasis and perform cell death. Gasdermin D, the pore-forming effector protein of pyroptosis, plays a critical role in coordinating membrane lysis and the release of highly inflammatory molecules. Recently, necrosulfonamide as a direct chemical inhibitor of gasdermin D has been confirmed to bind gasdermin D to inhibit pyroptotic cell death. To provide a more effective theoretical guidance for the influence of gasdermin D inhibitors on pyroptosis, we derive a novel PDE model from the genetic level, and study its threshold dynamics in terms of the basic reproduction number $R_0$. It turns out that threshold dynamics is determined by the sign of $R_0-1$. Under some suitable parameters, our numerical simulations show that environmental heterogeneity may increase transmission risk $R_0$ in time periodic environments. We may underestimate $R_0$ if the time average system is used. Based on some published experimental data, the administration of necrosulfonamide maybe strengthen the health condition of patients rapidly, which may become a new strategy to maintain CD4+ T cell counts at a safe level.     

Numerical analysis and computation of a multi-order laminated microstructure

The mesh transformation method is applied in a finite element approximation to a multi-well problem. It is proved that, compared with standard finite element methods, significantly higher convergence rate for the finite element approximations of multi-level microstructures can be obtained by combining the mesh transformation method with the periodic relaxation technique. Numerical examples are given to show the method can be efficiently implemented in computing multi-level microstructures.     

Gevrey class regularity for solutions of micropolar fluid equations

In this paper we consider the solutions of micropolar fluid equations in space dimension two with periodic boundary condition. We show that the strong solutions are analytic in time with values in an appropriate Gevrey class of function, provided that external forces and moments are time-independent and are in a Gevrey class.