Students' strategies and difficulties: the case of algebraic inequalities

This article describes a study related to secondary school students' ways of thinking about linear, quadratic, rational and square-root inequalities. Findings show that using graphic representations of parabolas when solving rational and quadratic inequalities usually yielded correct solutions. Difficulties arose when students failed to reject the excluded values or chose inappropriate, logical connectives. The most prevalent source of difficulties was inappropriate analogies between equations and inequalities. The article concludes with some suggested educational implications.     

Effective communication,critical aspects and compositionality in algebra

This paper contains a discussion of how the concept of critical aspects and the principle of compositionality can provide a powerful tool to analyse and understand the communications that occur in the classroom. It is grounded in data collected in a longitudinal study. The content chosen is algebra. It is argued that the critical aspects and the principle of compositionality should be considered as a methodological principle that describes how communication in the classroom should be designed. Here, I present the power of using variation theory whose main purpose is to generate an understanding of critical aspects and compositionality in practice.     

Factorial sums

Two sums involving squares of factorials are evaluated using Taylor series and numerical integration.     

Numerical solution of a multidimensional two-phase stefan problem

This paper treats a multidimensional two-phase Stefan problem with variable coefficients and mixed type boundary conditions. A numerical method for solving the problem is of fixed domain type, based on a variational inequality formulation of the problem. Numerical solutions are obtained by using piecewise linear finite elements in space and finite difference in time, and by solving a strictly convex minimization problem at each time step. Some computational results are presented.     

Intensity distribution for the broadband aberrated partially coherent flat-topped beam propagating through turbulent atmosphere

In this paper, taking the partially coherent flat-topped (PCFT) beam as an example, we investigate the intensity distribution of broadband beams with astigmatic aberration propagating through a turbulent atmosphere. The analytic expressions for the intensity of the beams propagating through the turbulent atmosphere are derived. It is shown that the intensity distribution of the broadband aberrated PCFT beams in the turbulent atmosphere are closely related with the astigmatism aberration, the strength of atmospheric turbulence, the order of flatness, the correlation of source and the bandwidth of PCFT beams. It is important to note that the position of the focal plane of the broadband aberrated (or not) PCFT beams can be modulated by changing the aberration, the order of flatness, the correlation and the bandwidth of PCFT beams.     

ESR and optical study of Cu2+-doped bis-(5,5′-diethylbarbiturato)bis picoline Zn(II)

ESR studies were conducted on Cu²⁺-doped bis-(5,5′-diethylbarbiturato)bis picoline Zn(II). Two Cu²⁺ lattice sites, Cu²⁺(I) and Cu²⁺(II), were identified. These sites exhibit two sets of four hyperfine lines in all directions. The g factor and hyperfine splitting were calculated from ESR absorption spectra: g_x ?=?2.0201?±?0.002, g_y ?=?2.0900?±?0.002, g_z ?=?2.1634?±?0.002, A_x ?=?(30?±?2)?×?10^?4?cm^?1, A_y ?=?(40?±?2)?×?10^?4?cm^?1 and A_z ?=?(154?±?2)?×?10^?4?cm^?1. It was found that Cu²⁺ enters the lattice substitutionally. The ground-state wavefunction of the Cu²⁺ ion in this lattice was determined from the spin Hamiltonian constants obtained from the ESR studies. With the help of an optical absorption study, the nature of the bonding in the complex is also discussed.     

基于非均匀螺旋线数据和布雷格曼迭代的快速磁共振成像方法

数据采集时间长是制约磁共振成像技术发展的重要瓶颈.为了解决这一问题, 本文基于压缩感知成像理论, 提出了一种结合非均匀螺旋线磁共振数据采集序列和布雷格曼迭代重建的快速磁共振成像方法, 通过欠采样缩短数据采集时间.欠采样引起混迭伪影则通过非均匀螺旋线欠采样特性和布雷格曼迭代重建去除.水模磁共振成像实验和在体磁共振成像实验结果表明: 欠采样情况下, 所提出的方法能有效去除欠采样导致的混迭伪影, 获得的图像结构信息完整的成像结果, 在缩短采样时间的同时, 具有较高的准确度. 关键词： 磁共振成像 非均匀螺旋线 全变分 布雷格曼迭代