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关键词： Existence   Uniqueness   Caputo   Caputo-Fabrizio   Atangana-Baleanu-Caputo   Numerical simulations  

摘要： An epidemiological model is proposed for three different types of novel fractional-order derivative operators known as the Caputo, the Caputo-Fabrizio, and the Atangana-Baleanu-Caputo operators. The classical model is fractionalized while taking care of the dimensional consistency for each ordinary differential equation of the model. The true field data, obtained through some authentic sources, for the dengue fever outbreak in Cape Verde islands in 2009 is the major motivation behind analysis of the present research study. The proposed model, being nonlinear, possesses the possibility of having no closed form solution. It is, therefore, existence and uniqueness for the solutions of the models are investigated via fixed point theory. The residuals computed via least-squares approach for all types of cases under consideration reveal the better performance of the models under fractional-order derivative operators proving that the dynamics of the disease (dengue virus) can be well understood if non-local effects are taken into consideration within the model. Based upon the results obtained, the efficiency rates of the fractional-order operators under the Caputo, Caputo-Fabrizio and the Atangana-Baleanu-Caputo are higher than that of the existing classical model. (C) 2019 Elsevier B.V. All rights reserved.

关键词： 数学分析   极限理论   一题多解   变式教学  

摘要： 充分利用教材,注重例题的结论推广、思维延伸、一题多解与变式教学,能更有效地改善数学概念、定理、公式的教学质量.本文将华东师范大学数学系编著的《数学分析》教材中采用的一道数列极限例题的结论推广到一般情形,对该结论给出了六种证明方法,并举例展示了推广所得的结论的广泛应用性与应用方法.

关键词： p-V图像   数学分析   汞柱注入   汞柱溢出   模型建构   例题分析  

摘要： 在同一单位制中,压强与体积可以具有相同的量纲。在此条件下,对一定质量理想气体的等温过程,依托p-V图像中的等温线进行分析,可得压强微分量与体积微分量之间的关系,以此建构汞柱注入与溢出模型。

关键词： 表征能力   数学分析   小学生   培养   结构关系   中小学教育   发展中心   连云港市  

摘要： 江苏省连云港市赣南区教师发展中心教师朱贵玺在《现代中小学教育》2019年第4期上撰文认为,数学分析表征能力是指学生透过数学现象发现数学本质和结构关系的能力水平,是学生数学学习的核心能力之一。培养小学生数学分析表征能力,教师首先要引导学生观察直观形象的主题图,找出图中蕴含的数学信息和知识之间存在的结构关系,进而提出数学问题并尝试解决问题,从而训练学生的图示分析表征能力.

关键词： 实数集完备性定理   极限   一元函数  

摘要： 文章首先介绍六大实数集完备性定理以作为求取极限的基础,然后介绍一元函数求极限的方法,包括极限定义法、极限的四则运算、函数迫敛性、两个重要极限、单调有界原理、洛必达法则、泰勒公式,重点在于对它们互相进行对比,找出它们各自的优点,指出它们各自所适用的情况,力求在遇到极限问题时能灵活运用各种方法,找到最简易的解法。

关键词： Sampling strategies   Optimal sampling design   Information theory   Entropy and conditional entropy   Uncertainty reduction   Multiple-point simulations   Channelized facies models   Geostatistics  

摘要： The task of optimal sampling for the statistical simulation of a discrete random field is addressed from the perspective of minimizing the posterior uncertainty of non-sensed positions given the information of the sensed positions. In particular, information theoretic measures are adopted to formalize the problem of optimal sampling design for field characterization, where concepts such as information of the measurements, average posterior uncertainty, and the resolvability of the field are introduced. The use of the entropy and related information measures are justified by connecting the task of simulation with a source coding problem, where it is well known that entropy offers a fundamental performance limit. On the application, a one-dimensional Markov chain model is explored where the statistics of the random object are known, and then the more relevant case of multiple-point simulations of channelized facies fields is studied, adopting in this case a training image to infer the statistics of a non-parametric model. In both contexts, the superiority of information-driven sampling strategies is proved in different settings and conditions, with respect to random or regular sampling.

关键词： basic model   dynamic   mathematical analysis   New classes   binary mixtures   surface diffusion   least action   comprehensive analysis   CAHN-HILLIARD EQUATION   Energy dissipation   solid walls   Boundary conditions   uniqueness   derivation  

摘要： The Cahn-Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In recent years, several types of dynamic boundary conditions have been proposed in order to account for possible short-range interactions of the material with the solid wall. Our first aim in this paper is to propose a new class of dynamic boundary conditions for the Cahn-Hilliard equation in a rather general setting. The derivation is based on an energetic variational approach that combines the least action principle and Onsager's principle of maximum energy dissipation. One feature of our model is that it naturally fulfills three important physical constraints: conservation of mass, dissipation of energy and force balance relations. Next, we provide a comprehensive analysis of the resulting system of partial differential equations. Under suitable assumptions, we prove the existence and uniqueness of global weak/strong solutions to the initial boundary value problem with or without surface diffusion. Furthermore, we establish the uniqueness of the asymptotic limit as t -> +infinity and characterize the stability of local energy minimizers for the system.

关键词： 问题式教学   问题式导向   数学分析  

摘要： 本文通过问题式引导教学模式参与到数学分析课程的课前预习、课堂教学、思维导图构建、复习巩固课后习题总结分类等环节,在节省课堂教学时间、提升课堂教学效率、重视数学知识体系的渗透等方面,进行了深入的思考和新的探索尝试。