Cycles containing 12 vertices in 3-connected cubic graphs

A necessary and sufficient condition is obtained for a set of 12 vertices in any 3-connected cubic graph to lie on a common cycle.     

Conjugacies for Tiling Dynamical Systems

We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being of finite type is invariant under topological conjugacy. For substitution tiling systems under rather general conditions, including the Penrose and pinwheel systems, we show that substitutions are invertible and that conjugacies are generalized sliding block codes.Research supported in part by NSF Vigre Grant DMS-0091946Research supported in part by NSF Grant DMS-0071643 and Texas ARP Grant 003658-158Acknowledgement The authors are grateful for the support of the Banff International Research Station, at which we formulated and proved Theorem 3.     

On the importance of mathematical play

A number of existing theories and proposals for the meaning and characteristics of ‘play’ are considered before the authors suggest six characteristics of mathematical play, including the idea that it is not confined to childhood. Previous studies provide evidence for relating play to cognitive gain while the place of mathematical play in research activities is illustrated by describing a mathematician's approach to a number investigation from the classroom-The Six Circles. The problem-solving process for the Six Circles and observations of students solving calculator and integration problems are analysed in relation to theories of play and cognitive gain and also considered from the perspective of the students' experience. Piaget's theory for the assimilation and accommodation of new information and Davis's view of play as ‘space to support learning’ are reflected in the authors' rationale for suggesting that open questions and mathematical play provide opportunities for students to develop their own conjectures, with no threat of failure, and provide a foundation for mathematical learning. Some difficulties of implementing a ‘play’ approach in the classroom are discussed and further research questions proposed.     

Scaffolding and metacognition

This paper proposes an expanded conception of scaffolding with four key elements: i. scaffolding agency – expert, reciprocal, and self-scaffolding;

ii. scaffolding domain – conceptual and heuristic scaffolding;

iii. the identification of self-scaffolding with metacognition; and

iv. the identification of six zones of scaffolding activity; each zone distinguished by the matter under construction and the relative positioning of the participant(s) in the act of scaffolding.

These key elements are illustrated with empirical examples drawn from a variety of research studies. Scaffolding, thus conceived, brings together several theoretical domains, and by situating metacognition within a framework derived from the social activity of scaffolding, a bridge is formed between the instructional support a teacher might provide and the learner's self-control of the learning process. With regard to instruction and the role of the teacher, it is the authors’ contention that a major object of instruction is the progressive relocation of scaffolding agency in the direction of the learner with the long-term goal of equipping the learner to take control of their own learning.     

Predicting the solubility of xenon in n-hexane and n-perfluorohexane: a simulation and theoretical study

The solubility of xenon in n-hexane and n-perfluorohexane has been studied using both molecular simulation and a version of the SAFT approach (SAFT-VR). The calculations were performed close to the saturation line of each solvent, between 200 K and 450 K, which exceeds the smaller temperature range where experimental data are available in the literature. Molecular dynamics simulations, associated with Widom's test particle insertion method, were used to calculate the residual chemical potential of xenon in n-hexane and n-perfluorohexane and the corresponding Henry's law coefficients. The simulation results overestimate the solubility of xenon in both solvents when simple geometric combining rules are used, but are in good agreement if a binary interaction parameter is included. With the SAFT-VR approach we are able to reproduce the experimental solubility for xenon in n-hexane, using simple Lorentz-Berthelot rules to describe the unlike interaction. In the case of n-perfluorohexane as a solvent, a binary interaction parameter was introduced, taken from previous work on (xe + C₂F₆) mixtures. Overall, good agreement is obtained between the simulation, theoretical and experimental data.     

A new regiospecific synthesis of aryl ketones from palladocycles

Aromatic palladocycles undergo facile reaction with acid halides to reglospecifically provide aryl ketones In high yield.     

Stability of cartesian products

We complete the work started by Holton and Grant concerning the semi-stability of non-trivial connected cartesian products and show that all such products are semi-stable. Further we show that except for certain (listed) restricted graphs, connected cartesian products are semi-stable at every vertex. Finally, we show that the cartesian product of any two graphs is not semi-stable if and only if one of them is totally disconnected and the other is not semi-stable.     

Cycles Through 23 Vertices in 3-Connected Cubic Planar Graphs

 We establish that if A is a set of at most 23 vertices in a 3-connected cubic planar graph G, then there is a cycle in G containing A. This result is sharp. Revised: September 1, 1997     

Palladium-catalyzed cascade cyclization of ynamides to azabicycles

Cascade reactions: A modular assembly of azabicycles by using a cascade cyclization/Suzuki coupling/6π-electrocyclization of bromoenynamides is reported. The reaction offers a wide substituent scope on the bicyclic aminodiene products, which can be selectively oxidized as a general approach to aromatic azabicycles.