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关键词： 分数阶随机时滞微分方程   波形松弛方法   分裂函数   均方收敛  

摘要： 研究了分数阶随机时滞微分方程的波形松弛方法。在分裂函数满足Lipschitz条件下,给出波形松弛方法的误差估计,该误差估计说明此方法在均方意义上是收敛的。最后通过几个算例证明了波形松弛方法求解分数阶随机时滞微分方程的有效性,验证了收敛理论的正确性。

关键词： 键极化加和模型   超拉曼光谱   微观极化率张量元微分   C_(2v)对称性  

摘要： 相干反斯托克斯拉曼光谱(CARS)和相干反斯托克斯超拉曼光谱(CAHRS)等高阶非线性光谱技术已经应用在动力学过程、基因表达谱筛选、高分辨率光谱分析等诸多领域。但因其涉及到的高阶微观极化率张量元数量众多,对其光谱信号进行定量分析非常困难。文献通过理论分析将CARS和CAHRS的微观极化率张量元分解为拉曼微观极化率张量元微分α′i′j′和超拉曼微观极化率张量元微分β′i′j′k′的乘积,将高阶光谱定量分析的困难简化为对低阶光谱的分析。该研究处理了具有C_(2v)对称性分子基团,提出一种利用键极化加和模型及其实验修正的方法简化超拉曼微观极化率张量元微分β′i′j′k′比值的方案。首先利用键极化加和模型方法,在对C_(2v)分子基团局域模式假设的基础上,进行单键的局域模式假设,通过对单键伸缩振动进行简正振动模式的分析,并与单键的偶极矩表达式进行比较,得到单键的超拉曼微观极化率张量元微分β′i′j′k′。在单键β′i′j′k′结果的基础上利用键极化加和模型,对C_(2v)分子基团进行对称性分析,将基团中两个单键伸缩振动的β′i′j′k′耦合,得到C_(2v)分子基团对称振动和反对称振动两种简正振动模式的超拉曼微观极化率张量元微分β′i′j′k′的表达式。在此基础上对C_(2v)分子基团对称振动和反对称振动的β′i′j′k′表达式进行理论推导,结合对应振动模式下不同偏振组合的超拉曼退偏率ρHR-SS和ρHR-AS及HV偏振组合时实验测量的超拉曼光谱强度比,得到超拉曼实验修正的C_(2v)分子基团β′i′j′k′之间的比值。再结合文献中拉曼实验修正的拉曼微观极化率张量元微分α′i′j′之间的比值,即可得到C_(2v)分子基团CARS和CAHRS的微观极化率张量元之间的比例关系,为定量分析高阶非线性光谱信息提供理论基础。

关键词： 导数   圆锥曲线   应用  

摘要： 在高中数学课本中,导数是核心知识点之一,并在求圆锥曲线参数方程中得到了很好的运用.导数加入高中数学体系后,使高中数学的知识体系得到了极大的延展,也为一些比较难的数学问题提供了一种新的解题思路.基于此,本文将通过具体例题来说明导数在圆锥曲线参数方程问题中的一些应用策略.

关键词： 高中数学   函数与导数   教学方略探索  

摘要： 高中数学是高中基础课程的重要组成部分,数学教学可以全面提高学生对数学知识的学习兴趣,并促使学生表述能力得以提高,学习积极性也获得提升.通过构建高质量的高中阶段数学教学,可以促进学生思维能力的有效发展,并提高学生对数学内涵的理解能力,以此全方位地促进学生学习能力的提高.最终促使学生的核心素养得以发展.

关键词： 数学竞赛   普及读物   数学教育家   数学爱好者   辅导讲座   数学传播   博士生导师   前沿领域  

摘要： 单缚,数学家,数学教育家,教授,博士生导师,我国著名数学传播、普及和数学竞赛专家。1964年扬州师范学院数学系毕业,从事数论及数学课程与教学论研究,在数学普及与数学竞赛方面,倾注了大量的时间和精力,培养了一批活跃在我国数学教育前沿领域的专家学者。精彩的辅导讲座、高水平的普及读物,受到数学爱好者的喜爱。

关键词： Stochastic systems   Finite-time   Annular domain stability   Wiener noises   Poisson jumps  

摘要： This paper investigates the finite-time annular domain stability and stabilization of stochastic systems, described by an Ito-type differential equation, in which the systems are driven by both Wiener noises and Poisson jumps. First, a new inequality called reverse differential Gronwall inequality is established to achieve less conservative conditions for finite-time annular domain stability. Second, some sufficient conditions are derived to guarantee that the closed-loop system is finite-time annular domain stable by constructing a state feedback controller and an observer-based controller, respectively. All related conditions can be expressed in terms of matrix inequalities and corresponding algorithms are given. Finally, two numerical examples are presented to verify the validity of the derived results, and the effect of Poisson jump intensity on the boundary of the system states is illustrated. (c) 2021 Elsevier Inc. All rights reserved.

关键词： 高阶导数   课程思政   数学文化   数学思维  

摘要： 数学分析课程蕴含了数学思维培养、人生情感引领和学习兴趣引导等方面的思政元素.以高阶导数教学为例,从高阶导数概念的导入、高阶导数的计算以及蕴含在Leibniz公式中的数学文化3个方面开展课程思政,引导学生形成良好的数学思维品质和健康的人生观、价值观、世界观,提升学生对数学分析的学习兴趣.

关键词： PBL   常微分方程   课堂教学   团队协作能力  

摘要： 通过认识PBL的基本运行模式，了解PBL的丰富内涵，掌握PBL的基本理念和核心要素，可以看到PBL教学模式在培养学生具有较强适应能力?团队协作能力和创新精神方面所展示的突出优势?本报告主要结合常微分方程课程的课程特点和教学实际情况，以及笔者在奥尔堡大学6个月以来的学习实践和PBL具体执行情况，探讨基于问题的(Problem-Based Learning，简称PBL)的教学方法在“常微分方程”课程教学中的应用?此方法强调在“常微分方程”的课堂教学中，教师通过一些实际问题让学生了解微分方程?学习微分方程并掌握

关键词： Reduced basis method   Parametrized nonlinear PDEs   Time-dependent PDEs   Hyperreduction   Empirical quadrature   Compressible flow  

摘要： We present a projection-based model reduction formulation for parametrized time-dependent nonlinear partial differential equations (PDEs). Our approach builds on the following ingredients: reduced bases (RB), which provide rapidly convergent approximations of the parameter-temporal solution manifold;reduced quadrature (RQ) rules, which provide hyperreduction of the nonlinear residual;and the dual-weighted residual (DWR) method, which provides an error representation formula for the quantity of interest. To find the RQ rules, we develop an empirical quadrature procedure (EQP) for time-dependent problems;we analyze the output error due to hyperreduction using a space-time DWR framework and identify appropriate constraints so that the output error due to hyperreduction is controlled. We in addition equip our reduced model with an online-efficient DWR a posteriori error estimate for the output;we again analyze the error in the hyperreduced dual problem and DWR expression to find appropriate constraints for the EQP so that the error in the error estimate is controlled. In the offline stage, the RBs and RQs, as well as the finite element mesh, are simultaneously constructed using a POD-greedy algorithm that leverages the online-efficient output error estimate. We demonstrate the framework for parametrized unsteady flows in a lid-driven cavity and over a NACA0012 airfoil. Reduced models achieve over two orders of magnitude reduction in the number of degrees of freedom, number of quadrature points, and wall-clock computational time, while achieving less than 0.5% output error and providing efficient error estimates in predictive settings. (C) 2021 Elsevier B.V. All rights reserved.

关键词： Open channel flow   Unsteady flow   Suspended sediment concentration   Mixing length   Hindered settling   Nonlinear partial differential equation (PDE)  

摘要： A model on one-dimensional unsteady suspended sediment transport has been developed in this study by including the effect of hindered settling and from mixing length point of view. The sediment diffusion term has been related to mixing length, which has been taken as a function of concentration. The mixing length and settling velocity are reduced due to the presence of particles in the flow. By considering these effects in the governing equation, the resulting partial differential equation (PDE), which becomes highly nonlinear, has been solved numerically using the most generalized boundary conditions available in the literature. For the purpose of validation, the derived model is compared with similar existing works under certain specified conditions. Apart from that, the obtained solution has also been compared with available laboratory data for steady and uniform flow because over a large span of time, the model behaves like a steady one. Furthermore, effects of damping function and hindered settling are explained both graphically and physically.