如何在高等数学教学中培养学生的创新思维

创新是当今的时代精神.创新能力的培养是实施素质教育的重要目标之一.高等数学作为高等教育的重点基础课程,在训练和培养学生创新能力方面具有重要地位.如何在高等数学教学过程中培养学生的创新思维,提高创新能力是我们高等数学教学改革的重要任务.文章通过对当前教育形势的分析以及创新思维的特点的思考,从教学理念、教学模式以及教学内容三个方面讨论了在高等数学教学过程中学生的创新思维的培养问题.     

工科数学教学中非逻辑思维能力训练初探

为在高等数学教学中培养学生的创造性思维能力,本文强调了在传统数学教学中容易忽视的“组合法”、“相似性法”、“列举奇想法”、“一题多解”等发散性思维能力的训练方法．     

试论实分析教学实践中构建思维的培养

本文阐述了培养数学构建思维是培养创造性思维的一种体现。根据教学实践论述了构建函数是利于解证问题的心智活动；构建特定图形支持逻辑论证，是行之有效的方法。     

The liapunov dimension of strange attractors

Many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange. In this paper the question of estimating the dimension of the attractor is addressed. While more general conjectures are made here, particular attention is paid to the idea that if the Jacobian determinant of a map is greater than one and a ball is mapped into itself, then generically, the attractor will have positive two-dimensional measure, and most of this paper is devoted to presenting cases with such Jacobians for which the attractors are proved to have non-empty interior.     

半线性抛物方程在L2p空间中全局吸引子的存在性

本文研究了半线性抛物方程所生成的半群{S（t）}t≥0的吸引子的存在性.利用文献[1]中证明吸引子正则性的思想,分别得到半群{S（t）}t≥0在L^2p（Ω）空间中具有一个有界吸收集和一个全局吸引子.     

线性代数是蓝色的——大学非数学专业《线性代数》的课程设计

线性代数不仅是处理多元问题的有力工具,而且具有强烈的思辨性.大学本科教育应该是"泛专业的高等素质教育",作为公共基础课之一的《线性代数》,也应着眼于学生综合科学素质的培养,注意对学生进行思维训练.本文提出线性代数的教学理念:"提出处理多元问题的新要求,沿着多元整合的集成化思路推进,逐步把学生引上线性变换和线性空间的思维平台."为此,本文进行了"五模块、两阶段、三层次"的课程设计,希望对讲授该课程的教师有所裨益.     

A necessary and sufficient condition for the existence of an exponential attractor

We give a necessary and sufficient condition for the existence of an exponential attractor. The condition is formulated in the context of metric spaces. It also captures the quantitative properties of the attractor, i.e., the dimension and the rate of attraction. As an application, we show that the evolution operator for the wave equation with nonlinear damping has an exponential attractor.     

Partial differential equations with discrete and distributed state-dependent delays

This work is an attempt to treat partial differential equations with discrete (concentrated) state-dependent delay. The main idea is to approximate the discrete delay term by a sequence of distributed delay terms (all with state-dependent delays). We study local existence and long-time asymptotic behavior of solutions and prove that the model with distributed delay has a global attractor while the one with discrete delay possesses the trajectory attractor.     

ON THE EXISTENCE OF GLOBAL ATTRACTOR FOR A CLASS OF INFINITE DIMENSIONAL DISSIPATIVE NONLINEAR DYNAMICAL SYSTEMS

By means of a nonstandard estimation about the energy functional, the authors prove the existence of a global attractor for an abstract nonlinear evolution equation. As an application, the existence of a global attractor for some nonlinear reaction-diffusion equations with some distribution derivatives in inhomogeneous terms is obtained.     

Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

The nonlinear reaction‐diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor ?? in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ?‐entropy. Upper and lower bounds of this entropy are obtained. Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ?ⁿ. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional “time” if n > 1) acting on a phase space ??. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy. In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz‐continuous) embedding of this system into the spatial shifts on the attractor. Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.     

A four-wing chaotic attractor generated from a new 3-D quadratic autonomous system

This paper introduces a new 3-D quadratic autonomous system, which can generate two coexisting single-wing chaotic attractors and a pair of diagonal double-wing chaotic attractors. More importantly, the system can generate a four-wing chaotic attractor with very complicated topological structures over a large range of parameters. Some basic dynamical behaviors and the compound structure of the new 3-D system are investigated. Detailed bifurcation analysis illustrates the evolution processes of the system among two coexisting sinks, two coexisting periodic orbits, two coexisting single-wing chaotic attractors, major and minor diagonal double-wing chaotic attractors, and a four-wing chaotic attractor. Poincaré-map analysis shows that the system has extremely rich dynamics. The physical existence of the four-wing chaotic attractor is verified by an electronic circuit. Finally, spectral analysis shows that the system has an extremely broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.     

Finite dimensional global and exponential attractors for a class of coupled time-dependent Ginzburg-Landau equations

We study a coupled nonlinear evolution system arising from the Ginzburg-Landau theory for atomic Fermi gases near the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover.First,we prove that the initial boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses a global attractor.Then we establish the existence of an exponential attractor.As a consequence,we show that the global attractor is of finite fractal dimension.     

CABRI AS A COGNITIVE TOOL FOR THE EVOLUTION OF A SENSE OF PROVING

Cognitive technologies have been described in the literature as reorganisers of thinking processes, especially where problem solving is concerned. This paper aims to analyse the possible use of Cabri-Géomètre as a cognitive tool in the elaboration of mathematical justifications in the context of problem-based mathematics. Some empirical examples are given to illustrate the significance of the specific learning situation. The complexity of learning environments incorporating computer-based activities is stressed as a condition for them to be effective in the introduction of the idea of mathematical justification and its evolution towards a sense of proving.     

数学教学中思维的拓展与延伸

结合五个教学案例说明教师在教学中如何依据教学内容进行思维的拓展与延伸,从而改善学生的思维品质与思维习惯.     

Defining and demonstrating an equivalence way of thinking in enumerative combinatorics

Counting problems offer opportunities for rich mathematical thinking, and yet there is evidence that students struggle to solve counting problems correctly. There is a need to identify useful approaches and thought processes that can help students be successful in their combinatorial activity. In this paper, we propose a characterization of an equivalence way of thinking, we discuss examples of how it arises mathematically in a variety of combinatorial concepts, and we offer episodes from a paired teaching experiment with undergraduate students that demonstrate useful ways in which students developed and leverage this way of thinking. Ultimately, we argue that this way of thinking can apply to a variety of combinatorial situations, and we make the case that it is a valuable way of thinking that should be prioritized for students learning combinatorics.     

DYNAMIC BIFURCATION OF NONLINEAR EVOLUTION EQUATIONS

§1. IntroductionA key problem in the study of problems in mathematical physics and mechanics is tounderstand and predict patterns and their transitions/evolutions. In ?uid mechanics, forinstance, it is important to study the periodic, quasi-periodic, ape…     

某些多维Ginzberg-Landau方程混合初边值问题的整体吸引子

Ginzburg-Landau方程具有十分丰富的物理应用背景,它准确地描述了湍流行为、超导理论及相位跃迁等演化现象.本文研究一类多维Ginzburg-Landau方程的混合初边值,证明它存在弱的整体吸引子.     

Functions via everyday actions: Support or obstacle?

The general context of this paper is the power of intuitive thinking, and how it can help or hinder analytical thinking. The research literature in cognitive psychology teems with tasks where intuitive thinking leads subjects to “non-normative” answers, including tasks for which they have all the knowledge necessary for the normative answer. The best explanation to date for such phenomena is dual-process theory, which stipulates the activation of a quick automatic intuitive process (System 1), together with the failure of the heavy, lazy, and computationally expensive analytical process (System 2) to intervene and correct the intuitive response.In an earlier paper, we have documented a clash between intuitive and analytical thinking concerning functions, which we have termed the changing-the-input phenomenon. The discovery of the changing-the-input phenomenon, however, left us with a puzzle: Why has this phenomenon concerning functions – a purely mathematical concept – been observed in computer science classes but not in mathematics ones? The purpose of the present paper is to address this puzzle. More generally we ask, under what conditions the changing-the-input phenomenon will or will not be manifested? Still more generally, in learning about functions, when is the intuitive scaffolding of functions via actions-on-tangible-objects helpful, and when does it get in the way of deeper understanding?