非线性玻尔兹曼方程的傅里叶谱方法

玻尔兹曼方程作为空气动理学中最基本的方程之一,是连接微观牛顿力学和宏观连续介质力学的重要桥梁.该方程描述了一个由大量粒子组成的复杂系统的非平衡态时间演化:除了基本的输运项,其最重要的特性是粒子间的相互碰撞由一个高维,非局部且非线性的积分算子来描述,从而给玻尔兹曼方程的数值求解带来非常大的挑战.在过去的二十年间,基于傅里叶级数的谱方法成为了数值求解玻尔兹曼方程的一种很受欢迎且有效的确定性算法.这主要归功于谱方法的高精度及它可以被快速傅里叶变换加速的特质.本文将回顾玻尔兹曼方程的傅里叶谱方法,具体包括方法的导出,稳定性和收敛性分析,快速算法,以及在一大类基于碰撞的空气动理学方程中的推广.     

Stability analysis of fractional order fuzzy cellular neural networks with leakage delay and time varying delays

In this paper we investigated the stability of fractional order fuzzy cellular neural networks with leakage delay and time varying delays. Based on Lyapunov theory and applying bounded techniques of fractional calculation, sufficient criterion are established to guarantee the stability. Hybrid feedback control is applied to derive the proposed results. Finally, numerical examples with simulation results are given to illustrate the effectiveness of the proposed method.     

Linear independence of a finite set of time-frequency shifts

This paper introduces an open conjecture in time-frequency analysis on the linear independence of a finite set of time-frequency shifts of a given L² function. Firstly, background and motivation for the conjecture are provided. Secondly, the main progress of this linear independence in the past twenty five years is reviewed. Finally, the partial results of the high dimensional case and other cases for the conjecture are briefly presented.     

Enhancing signal-to-noise ratio and resolution in low-field NMR relaxation measurements using post-acquisition digital filters

The traditional way to enhance signal-to-noise ratio (SNR) of nuclear magnetic resonance (NMR) signals is to increase the number of scans. However, this procedure increases the measuring time that can be prohibitive for some applications. Therefore, we have tested the use of several post-acquisition digital filters to enhance SNR up to one order of magnitude in time domain NMR (TD-NMR) relaxation measurements. The procedures were studied using continuous wave free precession (CWFP-T₁) signals, acquired with very low flip angles that contain six times more noise than the Carr–Purcell–Meiboom–Gill (CPMG) signal of the same sample and experimental time. Linear (LI) and logarithmic (LO) data compression, low-pass infinity impulse response (LP), Savitzky–Golay (SG), and wavelet transform (WA) post-acquisition filters enhanced the SNR of the CWFP-T₁ signals by at least six times. The best filters were LO, SG, and WA that have high enhancement in SNR without significant distortions in the ILT relaxation distribution data. Therefore, it was demonstrated that these post-acquisition digital filters could be a useful way to denoise CWFP-T₁, as well as CPMG noisy signals, and consequently reducing the experimental time. It was also demonstrated that filtered CWFP-T₁ method has the potential to be a rapid and nondestructive method to measure fat content in beef and certainly in other meat samples.     

Convergence of solutions for the fractional Cahn–Hilliard system

This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called ?ojasiewicz–Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.     

Two weight commutators in the Dirichlet and Neumann Laplacian settings

In this paper we establish the characterization of the weighted BMO via two weight commutators in the settings of the Neumann Laplacian ${\mathrm{\Delta }}_{N_{+}}$ on the upper half space ${\mathbb{R}}_{+}^n$ and the reflection Neumann Laplacian ${\mathrm{\Delta }}_N$ on ${\mathbb{R}}^n$ with respect to the weights associated to ${\mathrm{\Delta }}_{N_{+}}$ and ${\mathrm{\Delta }}_N$ respectively. This in turn yields a weak factorization for the corresponding weighted Hardy spaces, where in particular, the weighted class associated to ${\mathrm{\Delta }}_N$ is strictly larger than the Muckenhoupt weighted class and contains non-doubling weights. In our study, we also make contributions to the classical Muckenhoupt–Wheeden weighted Hardy space (BMO space respectively) by showing that it can be characterized via the area function (Carleson measure respectively) involving the semigroup generated by the Laplacian on ${\mathbb{R}}^n$ and that the duality of these weighted Hardy and BMO spaces holds for Muckenhoupt $A^p$ weights with $p\in (1,2]$ while the previously known related results cover only $p\in (1,\frac{n+1}{n}]$. We also point out that this two weight commutator theorem might not be true in the setting of general operators L, and in particular we show that it is not true when L is the Dirichlet Laplacian ${\mathrm{\Delta }}_{D_{+}}$ on ${\mathbb{R}}_{+}^n$.