Small-group searches for mathematical proofs and individual reconstructions of mathematical concepts

This is a study of mathematics students working in small groups. Our research methodology allows us to examine how individual ideas develop in a social context. The research perspective used in this study is based on a co-constructive view of learning. Groups of three or four undergraduate mathematics majors, with prior experience writing mathematical proofs together, were asked to prove three statements. Computer software, such as Geometers Sketchpad, was available. Group work sessions were videotaped. Later, individuals viewed segments of the group video and were asked to reflect on group activities. Students in some groups did not share a common conception of proof, which seemed to hamper their collaboration. We observed interactions that fit with the co-constructive theory, with bidirectional interactions that shaped both group and individual conceptions of the tasks. These changes in understanding may result from parallel and successive internalization and externalization of ideas by individuals in a social context.     

The Hilbert–Carathéodory Form for Parametric Multiple Integral Problems in the Calculus of Variations

We define, and discuss some properties of, a form which bears a relation to the Carathéodory form for multiple integral problems in the calculus of variations similar to the relation that the Hilbert 1-form of Finsler geometry bears to the Poincaré–Cartan form for one-dimensional variational problems.     

The complexity of plane hyperbolic incidence geometry is ∀∃∀∃

We show that plane hyperbolic geometry, expressed in terms of points and the ternary relation of collinearity alone, cannot be expressed by means of axioms of complexity at most ???, but that there is an axiom system, all of whose axioms are ???? sentences. This remains true for Klingenberg's generalized hyperbolic planes, with arbitrary ordered fields as coordinate fields. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalization of a Circular Property

We generalize a property of a circle concerning the power of a point.     

Validation of Solutions of Construction Problems in Dynamic Geometry Environments

This paper discusses issues concerning the validation of solutions of construction problems in Dynamic Geometry Environments (DGEs) as compared to classic paper-and-pencil Euclidean geometry settings. We begin by comparing the validation criteria usually associated with solutions of construction problems in the two geometry worlds – the ‘drag test’ in DGEs and the use of only straightedge and compass in classic Euclidean geometry. We then demonstrate that the drag test criterion may permit constructions created using measurement tools to be considered valid; however, these constructions prove inconsistent with classical geometry. This inconsistency raises the question of whether dragging is an adequate test of validity, and the issue of measurement versus straightedge-and-compass. Without claiming that the inconsistency between what counts as valid solution of a construction problem in the two geometry worlds is necessarily problematic, we examine what would constitute the analogue of the straightedge-and-compass criterion in the domain of DGEs. Discovery of this analogue would enrich our understanding of DGEs with a mathematical idea that has been the distinguishing feature of Euclidean geometry since its genesis. To advance our goal, we introduce the compatibility criterion, a new but not necessarily superior criterion to the drag test criterion of validation of solutions of construction problems in DGEs. The discussion of the two criteria anatomizes the complexity characteristic of the relationship between DGEs and the paper-and-pencil Euclidean geometry environment, advances our understanding of the notion of geometrical constructions in DGEs, and raises the issue of validation practice maintaining the pace of ever-changing software.     

C60富勒烯及其非经典衍生物:C58,C59和C62的第一性原理计算

利用密度泛涵理论对具有四元环、五元环、六元环、七元环、八元环和九元环等的C60富勒烯及其非经典衍生物:C62,C60,C59和C58共8个笼状分子进行了结构优化和电子结构计算,得到平衡构型下的结合能、HOMO-LUMO能隙、电离能、电子亲和能.对具有缺陷环的富勒烯,还研究了H2分子的贯穿行为,讨论了贯穿势垒与缺陷环尺寸的关系.     

A <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebraic Model for Locally Noncommutative Spacetimes

Locally noncommutative spacetimes provide a refined notion of noncommutative spacetimes where the noncommutativity is present only for small distances. Here we discuss a non-perturbative approach based on Rieffel’s strict deformation quantization. To this end, we extend the usual C ^*-algebraic results to a pro-C ^*-algebraic framework.