Global existence and time decay of the non-cutoff Boltzmann equation with hard potential

This paper is concerned with the Cauchy problem on the Boltzmann equation without angular cutoff assumption for hard potential in the whole space. When the initial data is a small perturbation of a global Maxwellian, the global existence of solution to this problem is proved in unweighted Sobolev spaces $H^N\left({\mathbb{R}}_{x,v}^6\right)$ with $N\ge 2$. But if we want to obtain the optimal temporal decay estimates, we need to add the velocity weight function, in this case the global existence and the optimal temporal decay estimate of the Boltzmann equation are all established. Meanwhile, we further gain a more accurate energy estimate, which can guarantee the validity of the assumption in Chen et al. (0000).     

Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions

In this paper, we consider a viscoelastic wave equation of variable coefficients in the presence of past history with nonlinear damping and delay in the internal feedback and dynamic boundary conditions. Under suitable assumptions, we establish an explicit and general decay rate result without imposing restrictive assumption on the behavior of the relaxation function at infinity by Riemannian geometry method and Lyapunov functional method.     

Capturing the experimental behaviour of a point-absorber WEC by simplified numerical models

The paper presents a wave basin experiment of a direct-driven point-absorber wave energy converter moving in six degrees of freedom. The goal of the work is to study the dynamics and energy absorption of the wave energy converter, and to verify under which conditions numerical models restricted to heave can capture the behaviour of a point-absorber moving in six degrees of freedom. Several regular and irregular long-crested waves and different damping values of the power take-off system have been tested. We collected data in terms of power output, device motion in six degrees of freedom and wave elevation at different points of the wave basin. A single-body numerical model in the frequency domain and a two-body model in the time domain are used in the study. Motion instabilities due to parametric resonance observed during the experiments are discussed and analysis of the buoy motion in terms of the Mathieu instability is also presented. Our results show that the simplified models can reproduce the body dynamics of the studied converter as long as the transverse non-linear instabilities are not excited, which typically is the case in irregular waves. The performance of the more complex time domain model is able to reproduce both the buoy and PTO dynamics, while the simpler frequency domain model can only reproduce the PTO dynamics for specific cases. Finally, we show that the two-body dynamics of the studied wave energy converter affects the power absorption significantly, and that common assumptions in the numerical models, such as stiff mooring line or that the float moves only in heave, may lead to incorrect predictions for certain sea states.     

Blow-up of solution for beam equation with delay and dynamic boundary conditions

In this paper, we consider an elastic beam equation with delay, source term, and boundary conditions together with some suitable initial data. Using the Faedo-Galerkin approximation and some estimates, we get the local existence of solution. Moreover, we obtain the finite time blow-up of solution by constructing suitable Lyapunov functionals.     

Periodic solutions of a periodic stochastic human immunodeficiency virus model with distributed delay and cytotoxic T lymphocytes immune response

In this paper, a periodic stochastic human immunodeficiency virus (HIV) model with distributed delay and cytotoxic T lymphocytes (CTL) immune response is investigated. First, by It $ô$'s formula, we show that the solution with any positive initial value is global and positive. Then, by the stochastic comparison theorem, we obtain the sufficient conditions guaranteeing the existence and global attractivity of infection-free periodic solution. Furthermore, we discuss the existence of the infective periodic solution by Has'minskii theory. Finally, numerical examples are given to illustrate the results.     

一种时间相关吸收光谱技术分析新方法

时间相关吸收光谱技术，如腔衰荡光谱技术(CRDS)和腔衰减相移光谱技术(CAPS)，是近三十几年发展起来的一类新型吸收光谱检测技术，它具有探测灵敏度高、响应速度快、不受光源强度起伏变化影响等优点。传统的吸收光谱技术都是基于Lambert-Beer定律，如直接吸收光谱技术(DAS)、波长调制光谱技术(WMS)和腔增强吸收光谱技术(CEAS)等，这类光谱技术在探测物质微弱吸收的时候一旦遇到较强的背景光信号就变得难以测量，而且光源的不稳定性也会对检测带来一定的限制。时间相关吸收光谱技术由于其不受光源强度起伏变化的特点，在很大程度上能够弥补传统吸收光谱技术所存在的缺陷，但其也有自身的局限性。首先在理论上，CRDS和CAPS这两种时间相关吸收光谱技术并不统一，而且在现有光谱理论下，Pulse-CRDS在应用时使用的脉冲光源的脉宽必须远小于谐振腔本身的时间常数，对于长脉宽的脉冲光或者反射率低(小于99.9%)的腔体，现有理论将不再适用；CAPS在应用时光源调制信号必须是周期性的正弦信号或者方波信号，对于其他类型的周期调制信号或者非周期性信号，现有理论并没有涉及。针对上述提到的时间相关吸收光谱技术的局限性，提出了一种新的分析时间相关吸收光谱技术的方法，即利用一阶传递函数，将谐振腔视为一阶传感系统，对时间相关吸收光谱技术理论进行统一解释，在公式推导上证明新方法下的推导结果和现有理论结果的一致性。针对Pulse-CRDS，以高斯脉冲光为例，给出一阶传感理论下的透射光强表达式，并对一系列不同的脉冲宽度γ、谐振腔时间常数τ_real以及从输出信号中拟合而得的时间常数τ_anal进行了模拟仿真。经过分析比较后发现，当γ<0.3τ_real时，τ_anal和τ_real的偏差小于1%；当γ>0.3τ_real时，τ_anal和τ_real的偏差渐渐变大，将不再满足实验条件。为了使Pulse-CRDS在长脉宽脉冲光下也能应用，本文给出了修正函数，使得在脉宽大于腔衰荡时间0.3倍的情况下，经过修正补偿后，衰荡时间的误差小于1%。对于CAPS系统，搭建相应实验平台，LED中心波长选用405 nm，使用方波调制信号，测量不同频率下的入射参考信号与探测信号的相位差和探测信号峰-峰值，通过由一阶传递函数推导而得的相频特性和幅频特性，拟合得到时间常数τ，结果分别为7.24和7.25 μs，残差范围分别为[-0.01, 0.02]和[-0.02, 0.025]，两者结果基本一致。实验结果验证了一阶传感系统理论完全适用于时间相关光谱的信号分析，并且一阶传感系统理论还使得时间相关光谱技术的理论得到了统一。     

Delay driven Hopf bifurcation and chaos in a diffusive toxin producing phytoplankton‐zooplankton model

This paper deals with a diffusive toxin producing phytoplankton‐zooplankton model with maturation delay. By analyzing eigenvalues of the characteristic equation associated with delay parameter, the stability of the positive equilibrium and the existence of Hopf bifurcation are studied. Explicit results are derived for the properties of bifurcating periodic solutions by means of the normal form theory and the center manifold reduction for partial functional differential equations. Numerical simulations not only agree with the theoretical analysis but also exhibit the complex behaviors such as the period‐3, 5, 6, 7, 8, 11, and 12 solutions, cascade of period‐doubling bifurcation in period‐2, 4, quasi‐periodic solutions, and chaos. The key observation is that time delay may control harmful algae blooms (HABs). Moreover, numerical simulations show that the chaotic states induced by the period‐doubling bifurcation are purely temporal, which is stationary in space and oscillatory in time. The investigations may provide some new insights on harmful phytoplankton blooms.     

Direct calculation of time varying Aharonov-Bohm effect

The Aharonov-Bohm effect (ABE) for steady magnetic fields is a well known phenomenon. However, if the current in the infinite solenoid that creates the magnetic field is time-dependent, that is in the presence of both magnetic and electric fields, there is no agreement whether the effect would be present. In this note, we try to investigate time varying ABE by a direct calculation in a set-up with a weak time dependent magnetic field. We find that the electric field arising out of the time-varying magnetic field in the path of the electrons does not enter the action integral but only changes the path of the electron from the source to the slits and then on to the detector. We find a frequency dependent AB phase shift. At low frequencies the result smoothly approaches the one for a constant field as the frequency tends towards zero. On the other hand, for high frequencies such that the AB-phase induced in the path of the wave packet oscillates rapidly, the net effect will be very small which is borne out by our results.