Scalar and vector line integrals: A conceptual analysis and an initial investigation of student understanding

This paper adds to the growing body of research happening in multivariable calculus by examining scalar and vector line integrals. This paper contributes in two ways. First, this paper provides a conceptual analysis for both types of line integrals in terms of how theoretical ways of thinking about definite integrals summarized from the research literature might be applied to understanding line integrals specifically. Second, this paper provides an initial investigation of students’ understandings of line integral expressions, and connects these understanding to the theoretical ways of thinking drawn from the literature. One key finding from the empirical part is that several students appeared to understand individual pieces of the integral expression based on one way of thinking, such as adding up pieces or anti-derivatives, while trying to understand the overall integral expression through a different way of thinking, such as area under a curve.     

基于傅里叶级数的定积分计算技巧

对数三角函数可以在长度为π的区间上展开为傅里叶级数.通过交换积分和求和的次序,对数三角函数的定积分就转化为无穷求和形式,后者可以通过基本的求和方法求出.     

Definite Condition of the Evolutionary $\vec{p}(x)$−Laplacian Equation

For the nonlinear degenerate parabolic equations, how to find an appropriate boundary value condition to ensure the well-posedness of weak solution has been an interesting and challenging problem. In this paper, we develop the general characteristic function method to study the stability of weak solutions based on a partial boundary value condition.     

Ramanujan's Master Theorem applied to the evaluation of Feynman diagrams

Ramanujan's Master Theorem is a technique developed by S. Ramanujan to evaluate a class of definite integrals. This technique is used here to produce the values of integrals associated with Feynman diagrams.     

Students’ understandings of multivariate integrals and how they may be generalized from single integral conceptions

Prior research has documented several conceptions students have regarding the definite integral, but these largely describe how students understand single-variable integral expressions. There is little research that has focused specifically on investigating students’ personal understandings of advanced types of integrals, including multiple integrals, nor how these might be generalized from the students’ understanding of single-variable definite integrals. Through interview data from 10 students and survey data from 42 students, we present a variety of student understandings of multiple integrals. We also investigate the relationship between these understandings and students’ conceptions of single-variable integrals.     

超短激光脉冲在共振介质中传播的时空特性

 用有限差分方法求解Maxwell和 Bloch 的联立方程，计算得到共振介质中超短光传播的时空特性。结果表明，在考虑空间效应时，峰面积为2π 的超短光脉冲不再以孤子的形式传播，光场的形状将发生变化，变形的程度和光与物质相互作用的强度有关：强度较小时，光场包络在时间和空间两方面都展宽；强度较大时，则发生分裂。光与物质相互作用的强度与光场大小、光场频率、偶极矩、失谐以及共振粒子浓度等因素有关。     

关于实方阵的几个问题

本文讨论如下内容：１．把有关对称正定（半正定）的一些性质推广到广义正定（半正定）。２．给定ｘ∈Ｒｍ×ｍ，∧为对角阵，求ＡＸ＝ｘ∧在对称半正定矩阵类中解存在的充要条件及一般形式，并讨论了对任意给定的对称正定（半正定）矩阵Ａ，在上述解的集合中求得Ａ，使得     

Integrals, partitions, and cellular automata

We prove that

where is the decreasing function that satisfies , for . When is an integer and we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.

     

Some Spectral Properties of Metapositive Definite Matrice

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小波算子矩阵法求定积分的近似值

结合Haar小波和算子矩阵的思想,给出一种新的Haar小波积分算子矩阵.利用所得小波积分算子矩阵来求定积分的近似值,将求定积分的问题转化为算子矩阵相乘,从而更容易计算机求解.特别是对于无法求得原函数的定积分,采用本文方法可以有效的求其近似值.最后数值算例验证了方法的可行性和有效性.