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关键词： 数学分析   混合式教学   教学模式  

摘要： 《数学分析》是数学专业最重要的基础课，学生初学时普遍感觉学习困难较多。文章讨论了如何将线上线下相结合的混合式教学引入《数学分析》教学实践，尝试探索一种符合实际的、有效的《数学分析》混合式教学模式。

关键词： 函数的奇偶性   数形结合   函数图象   数学分析   函数奇偶性   定义域   判定方法   定义法  

摘要： 函数的奇偶性反映了函数图象的对称性,充分体现了"数"与"形"可以互相转化的思维.数形结合是进行数学分析和数学研究的有力工具,对函数部分的知识体系和综合应用具有纽带的作用,而正确快速判断函数的奇偶性是解决问题的关键所在.1定义法利用函数奇偶性的定义进行判别时,要注意以下两点:1)在整个函数的定义域内考虑;2)具有奇偶性的函数,其定义域一定关于原点对称.

关键词： 初中   数学教学   解题能力  

摘要： 随着初中数学教学改革深入开展,在初中数学教学课堂,教师要加强学生数学分析和解题能力,通过进一步教学实践研究,本文从多方面进行了实践探索,总结了提高学生分析和解题能力的具体途径,希望分析能够全面提高数学教学有效性,从而为学生日后发展奠定良好基础。

关键词： 培养策略   解决问题   小学数学分析  

摘要： 为了全面提升小学生在学习数学时,分析和解决问题的能力,需要教师全面分析在教学过程中存在的不足。并充分重视分析和解决问题能力对于学生学习数学的重要性,制定完善的教学方案,有效提升课堂教学质量。

关键词： 数学分析   高斯光束   光斑半径   测量  

摘要： 采用光阑法和光点法测量高斯光束光斑半径时,未对光斑图像进行增强处理,导致测量精准度较低,为了解决该问题,提出了基于数学分析的高斯光束光斑半径测量方法研究。根据高斯光束瞬时辐射照度示意图,求解电场振幅和辐照度时域平均值,从而得到高斯光束光强分布情况。依据分析结果采用数学分析法建立光斑几何模型,重构多维度数据,将重构数据块映射到一个新的坐标系之中,使数据差异性达到最大,并增强处理光斑图像。参考光斑图像增强结果获取光斑边缘特征中的最佳特征点,使用最小二乘法进行边缘拟合,以此来提高拟合精准度,最终完成对高斯光束光斑半径的测量。通过实验结果表明,该方法测量误差最大为230μm,最小为20μm,相比于其他两种测量方法来说测量结果较为精准,充分验证了该方法的可行性与优势性。

关键词： 极值问题   中学物理   内部规则   终极状态   物理现象   法国物理学家   解析理论   数学分析  

摘要： 物理中求最大值或最小值,以及对物理现象的终极状态进行判断都属于极值问题。分析极值是物理中常见的问题,中学物理中对极值的处理往往针对具体的模型有固定的套路,但这些套路因为和模型结合过于紧密而显得局限性较强,这是导致学生解决极值问题显得困难的原因之一。法国物理学家傅立叶在《热的解析理论》一书中指出:“数学分析已经超过观察,它弥补了我们感官的不足,在某种意义上,它使我们亲眼看到物体内部规则和谐的振动。”

关键词： 数学分析   数学史   数学思维   教学方法  

摘要： 数学分析是数学专业学生的重要基础课程之一,是学习数学专业后继课程的基础,也是其他非数学专业学生和科研工作者用于打好数学基本功的重要课程之一。该文结合个人从事数学方向学术研究的感悟和数学分析的教学实践经历,分别从数学思维的培养、学生学习兴趣的培养、教师的模范作用、概念的讲解、反例的重要性以及作业、习题课和章节测试等六个方面,通过具体的教学实例,给出了个人教授数学分析这门课程的几点体会和建议。

关键词： Markov chains  

摘要： We study the learning of an external signal by a neural network and the time to forget it when this network is submitted to noise. The presentation of an external stimulus to the recurrent network of binary neurons may change the state of the synapses. Multiple presentations of a unique signal lead to its learning. Then, during the forgetting time, the presentation of other signals (noise) may also modify the synaptic weights. We construct an estimator of the initial signal using the synaptic currents and in this way define a probability of error. In our model, these synaptic currents evolve as Markov chains. We study the dynamics of these Markov chains and obtain a lower bound on the number of external stimuli that the network can receive before the initial signal is considered forgotten (probability of error above a given threshold). Our results are based on a finite-time analysis rather than large-time asymptotic. We finally present numerical illustrations of our results.

关键词： HBV infection   Fractional-order model   Global stability   Simulation  

摘要： Fractional differential mathematical model describing the dynamics of hepatitis B viral infection with DNA-containing capsids, the liver hepatocytes and the humoral immune response is presented and investigated in this paper. The humoral immunity is represented by antibodies, the principal role of those antibodies is to attack the free viruses. In order to describe the time needed for the interaction between biological liver cells and viral particles and also the time needed for the activation of the humoral immune response, a memory term represented by a fractional derivative will be added in each equation of our suggested model. The positivity and boundedness of all solutions with non negative initial condition will be proved which is consistent biologically. Moreover, the disease-free equilibrium, the infection steady state without humoral immunity and the infection steady state with humoral immunity are given. By constructing some suitable Lyapunov functionals, the global stability of all equilibria are proven depending on the basic reproduction number and on the antibody immune response reproduction number. Finally, different numerical simulations using the multistage generalized differential method are established in order to illustrate our theoretical findings. It was revealed from both the theoretical and the numerical results that the order of the fractional derivative has no effect on the three equilibria stability. However, for increased values of the fractional derivative order, which describes the long memory behavior, each solution converge more rapidly to its stationary state.

关键词： Stability analysis   Population model   Numerical simulation   Delivery protocol  

摘要： In this paper we study a system of partial differential equations that couple nonfickian diffusion of one of two species with the fickian diffusion of a chemical or biological agent. This system can be used to describe the evolution a population (biological species, cells) that switch between two phenotypes: migratory and proliferative. This switching process is very common in certain types of cancer or even viral diseases. It is assumed that the population/cell migration occurs in a viscoelastic extracellular matrix (ECM). This has an important role in the transport inducing to a nonfickian motion. A chemical agent, which can represent, for instance, a treatment drug in a cancer disease scenario, acts on the population according to Michaelis-Menten or linear reactions, eliminating individuals from the population. Under suitable regularity conditions we establish upper bounds for an energy functional, with respect to theL(2)norm, leading to the stability of the model. Agent administration protocols based on stronger assumptions on the agent/population kinetics are also proposed with the purpose of controlling the total amount of individuals of the population. These protocols are based on suitable estimates for the total mass of individuals in the system after a certain number of agent administration sessions. A numerical method based on finite differences and finite elements is introduced and its stability properties are analysed. The qualitative behaviour of the solutions of the system is explored and discussed. Several protocols are defined within the context of chemotherapy in brain tumours and their effectiveness is numerically studied considering Michaelis-Menten or linear chemical effects.