On the boomerang uniformity of a class of permutation quadrinomials over finite fields

Let ${\mathbb{F}}_{2^n}$ be a finite field with $2^n$ elements and $f_{\underset{\_}{c}}(x)=c_0{x}^{2^m(2^k+1)}+c_1{x}^{2^{m+k}+1}+c_2{x}^{2^m+2^k}+c_3{x}^{2^k+1}\in {\mathbb{F}}_{2^n}[x]$, where n, m and k are positive integers with $n=2m$ and $\gcd ?(m,k)=e$. In this paper, motivated by a recent work of Li, Xiong and Zeng (Li et al. (2021) [12]), we further study the boomerang uniformity of $f_{\underset{\_}{c}}(x)$ by using similar ideas and carrying out particular techniques in solving equations over finite fields. As a consequence, we generalize Li, Xiong and Zeng's result from the case of m being odd and $e=1$ to that of both $m/e$ and $k/e$ being odd.     

初始微观结构缺陷和断裂的温度依赖及相关性

纳米线(NW)结构内的微观结构缺陷对NW的机械性能存在一定的影响。NW断裂位置的预测关系着纳米器件应用的寿命,进而引起了人们的广泛关注。在本工作中,基于统计分析,分别研究了单晶铜纳米线(Cu NW)拉伸过程中出现的断裂位置以及在应力屈服点处产生的初始微观结构缺陷(初始缺陷)的位置对温度的依赖性,进一步探究了两者之间的联系。利用分子动力学(MD)模拟了单晶Cu NW在20~300 K的温度范围内的拉伸状态,共包含6个体系,各温度体系包含300个独立的样本。基于机器学习,采用density-based spatial clustering of applications with noise (DBSCAN)算法,将hexagonal close-packed (hcp)原子划分为各个初始缺陷以进一步确定其位置。统计结果显示,当温度低于50 K时,初始缺陷的位置集中在NW的两端。随着模拟温度的上升,MD模拟结果展现了单晶Cu NW的拉伸过程中的杨氏模量、平均屈服应力、平均势能等机械性能对温度的依赖性。温度的升高进一步促使了更多初始缺陷的产生,并使得初始缺陷的位置由统计分布的两端向中间平均化。与初始缺陷相比,各温度下的断裂位置集中在两端。统计结果表明,模拟的温度范围对NW的断裂位置无明显影响,但对初始缺陷的产生具有明显影响。当温度低于100 K时,初始缺陷的位置分布与断裂位置分布呈现了一致性。由于两者具有不同的温度依赖,其差异随着温度的上升逐渐显现。对不同温度下的微观结构形变行为观察发现,断裂失效明显受到NW两端的表面效应和阻挡效应的影响。最终的断裂位置受塑性形变中后期的影响,与应力屈服区产生的初始缺陷无直接联系。     

Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

We consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold M isometrically immersed into another Riemannian manifold $\overline{M}$. We first assume the pull back Weitzenböck operator of $\overline{M}$ bounded from below, and obtain an extrinsic lower bound for the first eigenvalue of Hodge-Laplacian. As applications, we obtain some rigidity results. Second, when the pull back Weitzenböck operator of $\overline{M}$ bounded from both sides, we give a lower bound of the first eigenvalue by the Ricci curvature of M and some extrinsic geometry. As a consequence, we prove a weak Ejiri type theorem, that is, if the Ricci curvature bounded from below pointwisely by a function of the norm square of the mean curvature vector, then M is a homology sphere. In the end, we give an example to show that all the eigenvalue estimates are optimal when $\overline{M}$ is the space form.     

水岩化学作用对黑色页岩的化学损伤及力学劣化试验研究

为模拟黑色页岩化学风化中酸性水-页岩化学作用过程,本文对其进行氧化条件下的不同pH值H2SO4溶液的非平衡流动态腐蚀性试验,获得了黑色页岩化学腐蚀前后矿物变化、相对质量损失、次生孔隙率、纵波波速变化及微观结构特征的变化。通过单轴压缩试验,获得黑色页岩在不同浸泡时段的变形和强度特性规律,探讨了酸性水对黑色页岩化学作用的化学损伤和力学劣化的腐蚀效应及机制。研究表明,页岩试件在化学腐蚀后,易溶性矿物成分减小,黏土矿物增加,同时矿物胶结变得松散,矿物边缘变得模糊;页岩试件的相对质量损失与次生孔隙率随pH值减小和浸泡时间的增长而增大,而纵波波速则减小;其力学特性有从脆性破坏向延性破坏转化的趋势,单轴抗压强度和弹性模量有随pH值减小和浸泡时间的增长而减小的趋势。基于次生孔隙率,构建化学损伤变量来描述试件化学-力学损伤演化过程。分析酸性水-页岩化学作用的机理主要为:溶解作用、氧化作用、水解作用及离子交换吸附作用。     

周期结构后屈曲新算法及其应用

提出了周期结构后屈曲分析的一种新算法。在屈曲点附近,通过加载模型和诱导后屈曲边值问题之间的相互切换,避开屈曲点附近刚度矩阵的奇异性,并诱导结构产生预期的后屈曲变形,避免了以往后屈曲算法中引入几何初始缺陷后对系统带来的可能影响。通过对三种由超弹性材料所构成的周期孔隙结构的后屈曲分析,验证了本文所提出的后屈曲算法的有效性和灵活性。分析了周期孔隙材料多向加载对屈曲模式转换的影响,以及后屈曲变形对弹性波传播带隙的影响,为周期结构中弹性波传播的调控提供良好的基础。     

Biaxial tensile behavior and strength of architectural fabric membranes

The utilization of composite fabric membrane materials for large-span membrane structures has attracted considerable attention in recent decades due to enhanced material properties. Biaxial mechanical properties with respect to real engineering applications are essential and indispensable in comparison with uniaxial ones. This study focuses on true biaxial characteristics of a typical polyvinylidene fluoride (PVDF)-coated polyester membrane material in terms of stress-strain characteristics and breaking criteria.The true stress-strain curves obtained from an experimental study, i.e. seven loading ratios on the basis of symmetry and typical conditions, are investigated with digital image correlation method. The interpolation of these curves in combination of least square method achieves a three-dimensional strain surface as a function of warp and weft strains, which is useful to assess reasonable structural behavior. A new breaking criteria intended for architectural fabric membrane is proposed in analogy to Tsai-Hill, Yeh-Stratton and Norris failure criteria. The basic constants in the criteria are determined using experimental results. A comparative analysis between available uniaxial and biaxial criteria shows that the new criteria can cover all criteria due to the fact that biaxial mechanical properties are larger than uniaxial ones. Furthermore, a similar but glued specimen is employed to compare welded specimens. It is obtained that observations, values and curve tendency are similar, demonstrating the suitability of using new specimens to identify true biaxial properties.     

An explicit series solution of the squeezing flow between two infinite plates by means of the homotopy analysis method

In this paper, we investigated an axisymmetric Newtonian fluid squeezed between two parallel plates. The steady nonlinear governing equations are reduced to a single differential equation using integrability condition. Homotopy analysis method (HAM) is used to solve the nonlinear differential equation analytically. Numerical solutions indicate this method is satisfactory.     

Stress-driven morphological instability and catastrophic failure of microdevices

In microdevices, the competition between surface energy and elastic energy could lead at the phenomenon known as stress-driven morphological instability (MI), causing an increase of surface roughness with time. Several different mass transport mechanisms can trigger such a morphological alteration and operate simultaneously: surface and bulk diffusion, evaporation and condensation, chemical reactions. Unstable solids could eventually evolve towards crack-like surfaces thus altering mechanical, electrical and optical properties of the devices or even leading to catastrophic failures by supercritical crack propagation. In this work, a more general kinetic law is employed to estimate the onset of MI, considering the effect of the stress field on the atomic mobility. A more intuitive and straightforward approach is used to determine the stability conditions, where the rate of atomic mass motion is introduced as a stability parameter. The critical loads and wavelengths for the onset of MI, determined as a function of material parameters α and β, are compared with the limiting conditions for the supercritical crack propagation (SC) of a crack-like surface in order to asses if and under which situations catastrophic failures by SC can be observed. Two practical cases are investigated: fixed wavelength (Case I) and arbitrary rough surface with a fixed remote load (Case II). In Case I, absolute and relative threshold loads are found below which MI could never occur and a transitional wavelength over which MI would always lead to SC is introduced. In Case II, it is shown that dominant perturbation for MI would always lead to SC given enough time for the surface to evolve towards a crack-like profile. The influence of the material properties α and β on the critical parameters is also addressed.