Sharp Inequalities for BMO Functions

The purpose of the paper is to study sharp weak-type bounds for functions of bounded mean oscillation. Let 0 p ∞ be a fixed number and let I be an interval contained in R. The author shows that for any φ : I → R and any subset E I of positive measure,For each p, the constant on the right-hand side is the best possible. The proof rests on the explicit evaluation of the associated Bellman function. The result is a complement of the earlier works of Slavin, Vasyunin and Volberg concerning weak-type, L ~p and exponential bounds for the BMO class.     

Combined effects of singular and critical nonlinearities in elliptic problems

In this paper we consider the existence and multiplicity of positive weak solutions for a class of elliptic problems with the nonlinearity containing both singular and critical terms. By means of the concentration compactness principle due to Lions and Ekeland’s variational principle, two positive weak solutions are obtained.     

Effect of long-period stacking ordered phase on thermal stability of refined grains in Mg-RE-based alloys

The refined grains in both as-cast Mg-10Y-1Zn-2Zr (WZ101?K, wt.%) and Mg-10Y-1Al (WA101, wt.%) alloys have been found to have superior thermal stability after solution treatment. Plate-shaped, but different polytypes of long-period stacking ordered (LPSO) phases distributed along grain boundaries are considered as the predominant inhibitors of grain growth. Based on the plate-shaped features of the LPSO phases, a modified Zener’s model is established and the predicted limiting grain sizes are in a good agreement with the experimental results. Therefore, this model has strong potential for designing advanced Mg-RE-based alloys with high strength and high creep resistance at high temperatures.     

Continuous dependence estimates for viscosity solutions of integro-PDEs

We present a general framework for deriving continuous dependence estimates for, possibly polynomially growing, viscosity solutions of fully nonlinear degenerate parabolic integro-PDEs. We use this framework to provide explicit estimates for the continuous dependence on the coefficients and the “Lévy measure” in the Bellman/Isaacs integro-PDEs arising in stochastic control/differential games. Moreover, these explicit estimates are used to prove regularity results and rates of convergence for some singular perturbation problems. Finally, we illustrate our results on some integro-PDEs arising when attempting to price European/American options in an incomplete stock market driven by a geometric Lévy process. Many of the results obtained herein are new even in the convex case where stochastic control theory provides an alternative to our pure PDE methods.     

二维轴对称图象高阶胡氏不变矩快速算法的改进

提出了基于二维轴对称性目标图象矩快速算法的实用化改进方案.算法实用化改进是通过两个命题的证明实现的.一是通过轴对称目标图象质心在对称轴上的命题证明,实现了将二维轴对称图象矩快速算法用于轴对称性图象中心矩的计算;二是通过目标图象以坐标系中任意两点为圆心,旋转相同角度,结果图象具有平移性的命题证明,实现了算法用于对旋转不同角度的对称目标图象的中心矩的计算改进.在此基础上,完成了二维轴对称目标图象高阶胡氏不变矩实用化快速算法的实现.实验证明,该算法具有较好的实时性能,且具有较小的引入误差.     

两等同双能级原子与q模腔场任意N∑光子共振相互作用辐射谱研究

本文建立了“q模腔场-两原子”系统的任意N^∑光子相互作用模型,利用这一模型研究了两个偶极一偶极力关联的等同双能级原子与q模腔场任意N^∑光子.共振相互作用的辐射谱.对q模腔场分别处于不同数态:即q模腔场均为真空场、q模腔场均为强场、1模为真空场(q-1)模为强场、q₀模为真空场(qq₀)模为强场、以及(q-1)模为真空场1模为强场时辐射谱的结构持征及物理特性进行了详细分析,揭示出“q模腔场-两原子”系统任意N^∑光子共振相互作用辐射谱的一般特征.现有文献报道,仅仅是本文的普遍性结果在各种不同情况下的特例.     

Secondary-school teachers’ noticing of aspects of mathematics teaching talk in the context of one-day workshops

In a research project with one-day teacher education workshops for secondary-school mathematics teachers, our study explores the potential of tool-supported discussions in helping them to notice important and critical aspects of mathematics teaching talk. Mathematical practices of naming and explaining in teaching talk, students’ content learning challenges, and noticing processes of identifying, interpreting and deciding are the components of our framework and the tools that guided the design and implementation of three workshops on linear equations, fractions and plane isometries. The data was collected during the discussions with the seven teachers and the teacher educator throughout these workshops. The coding of the discussions allowed us to see discourse moves that reveal the teachers’ noticing of: (i) challenges in the identification of mathematical naming, (ii) mathematical explaining that voices the students’ learning, (iii) classroom practice in relation to mathematical naming and explaining.     

Surface Acoustic Wave Characterization of Equivalent Young's Moduli for Patterned Films北大核心CSCD

Based on the equivalent elasticity theory for layered materials, the micro-mechanics equivalent models for single and dual damascene structures were established. The equivalent elastic constant of the patterned structure was introduced, to establish the propagation model for the surface acoustic waves propagating in the layered structure of the patterned film/ substrate, and the theoretical dispersion curves of the surface acoustic waves were calculated with Green’s function and the matrix method. The finite element method was used to calculate 24 numerical examples of damascene structures with different volume ratios, and the results were compared with those of the strain energy method. The results show that, the average relative errors of the equivalent Young’s moduli of the 300 nm-thick dual damascene film and the 100 nm-thick single damascene film are 2.06% and 2.27%, respectively. The research verifies the correctness of the equivalent patterned structure model and the feasibility of the surface acoustic wave method to characterize the mechanical properties of patterned films, and provides a reference for the development of suitable chemico-mechanical polishing technologies for patterned films under low pressure. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

A generalized Duffing equation with the Coulomb’s friction law and Signorini–type contact conditions

This work provides mathematical and numerical analyses for a spring–mass system, in which Signorini–type contact conditions and Coulomb’s friction law with thermal effects are taken into consideration. The motion of a mass attached to a viscoelastic (Kelvin–Voigt type) nonlinear spring is described by a generalized Duffing equation. Signorini contact conditions are understood as extended complementarity conditions (CCs), where convolution is incorporated, allowing to consider thermal aspects of an obstacle. We prove the existence of global weak solutions for the highly nonlinear differential equation system with all the conditions, based on the regularized differential equation and the normal compliance condition with the standard mollifier. In addition, we investigate what side effects produce higher singularities of contact forces in dynamic contact problems, which is also supported by numerical evidences. Numerical schemes are proposed and then several groups of data are selected for the display of our numerical simulations.