A perturbed elementary operator and range-kernel orthogonality

Let denote the algebra of operators on a Hilbert . If and are commuting normal operators, and and are commuting quasi-nilpotents such that , then define and by , , and . It is proved that and , where is some scalar and is the quasi-nilpotent part of the operator .

     

独立学院大学物理教学研究

分析了独立学院大学物理教学工作面临的主要问题和困难,提出了通过加强微元法、物理模型教学,重视物理图景的建立、物理学史教育和学习方法的引导,理论联系实际等教学措施来提升大学物理的教学质量.     

Beyond abstract elementary classes: On the model theory of geometric lattices

Based on Crapo's theory of one point extensions of combinatorial geometries, we find various classes of geometric lattices that behave very well from the point of view of stability theory. One of them, $({\mathbf{K}}^3,?)$, is ω-stable, it has a monster model and an independence calculus that satisfies all the usual properties of non-forking. On the other hand, these classes are rather unusual, e.g. in $({\mathbf{K}}^3,?)$ the Smoothness Axiom fails, and so $({\mathbf{K}}^3,?)$ is not an AEC.     

Conceptualizations of slope in Mexican intended curriculum

Slope is a fundamental mathematics concept in middle and high school that transcends to the university level. An understanding of slope is needed at the university level since slope plays an important role in understanding problems involving variation and change. In this study Mexican curricula documents were examined to determine which conceptualizations of slope are addressed in the intended mathematics curriculum. To explain the results, we use conceptualizations of slope identified in previous research. Our findings reveal that, to a certain extent, the conceptualizations proposed in the Mexican intended mathematics curriculum differ slightly in terms of the emphasis and timing of instruction from what others have identified in the U.S., with slope as a geometric ratio receiving less emphasis in the Mexican curriculum. There was also noted discontinuity within the Mexican curriculum in introducing slope in grade 9 and subsequently introducing of linear functions in grade 10 without explicit mention of slope. Suggestions are made for future studies, both to consider the conceptualizations of slope promoted in the Mexican textbooks and the impact they have in classroom instruction and student learning of slope.     

Analytic and arithmetic methods in Liouville's identities

The history of Liouville's identities is of remarkable interest especially as far as the methods used to prove them are concerned. In Liouville's opinion, the identities had to be proved by merely arithmetic considerations. Some mathematicians, such as Uspensky, developed Liouville's approach and used elementary methods. However, other mathematicians (among them Nazimoff and Bell) followed Hermite's suggestions of deriving Liouville's identities from the theory of elliptic functions: most of them thought that analytic methods were preferable since they allowed one to discover new identities (and not only to prove known ones).     

Counter-examples for refinement of conjectures and proofs in primary school mathematics

The purpose of this study is to explore how primary school students reexamine their conjectures and proofs when they confront counter-examples to the conjectures they have proved. In the case study, a pair of Japanese fifth graders thought that they had proved their primitive conjecture with manipulative objects (that is, they constructed an action proof), and then the author presented a counter-example to them. Confronting the counter-example functioned as a driving force for them to refine their conjectures and proofs. They understood the reason why their conjecture was false through their analysis of its proof and therefore could modify their primitive conjecture. They also identified the part of the proof which was applicable to the counter-example. This identification and their action proof were essential for their invention of a more comprehensive conjecture.     

A task that elicits reasoning: A dual analysis

This paper reports on the forms of reasoning elicited as fourth grade students in a suburban district and sixth grade students in an urban district worked on similar tasks involving reasoning with the use of Cuisenaire rods. Analysis of the two data sets shows similarities in the reasoning used by both groups of students on specific tasks, and the tendency of a particular task to elicit numerous forms of reasoning in both groups of students. Attributes of that task and ways that those attributes can be replicated in other domains may have implications in the teaching of early reasoning.     

一类二维单个守恒律方程的Riemann问题

本文研究一类二维单个守恒律方程的Riemann问题.用广义特征分析方法研究了这类方程,给出了基本波的分类,解决了初值为两片常数的二维Riemann问题,给出了Riemann解.     

Consistencies and inconsistencies in students' solutions to algebraic ‘single-value’ inequalities

This paper describes students’ solutions to a commonly taught and not commonly taught inequality. The findings showed students’ difficulties. Participants implicitly and explicitly exhibited two intuitive beliefs: inequalities must result in inequalities and solving inequalities and equations are the same process. Following the analysis of students’ written solutions, individual interviews were conducted that gave a better insight into their reasoning and provided some ideas for teaching. The concluding section of the paper offers relevant educational implications.     

On the construction of involutory rhotrices

An involutory matrix is a matrix that is its own inverse. Such matrices are of great importance in matrix theory and algebraic cryptography. In this note, we extend this involution to rhotrices and present their properties. We have also provided a method of constructing involutory rhotrices.