When time is an implicit variable: An investigation of students’ ways of understanding graphing tasks

In this study, we investigate students’ ways of understanding graphing tasks involving quantitative relationships in which time functions as an implicit variable. Through task-based interviews of students ages 14–16 in a summer mathematics program, we observe a variety of ways of understanding, including thematic or visual association, pointwise thinking, and reasoning parametrically about changes in the two variables to be graphed. We argue that, rather than comprising a hierarchy, these ways of understanding complement one another in helping students discover an invariant relationship between two dynamically varying quantities, and develop a graph of the relationship that captures this invariance. From these ways of understanding, we conjecture several mathematical meanings for graphing that may account for students’ behavior when graphing quantitative relationships.     

Operator algebras and a theorem of Dieudonne

LetA be aC*-algebra with second dualA″. Let (φ _n)(n=1,...) be a sequence in the dual ofA such that limφ _n(a) exists for eacha εA. In general, this does not imply that limφ _n(x) exists for eachx εA″. But if limφ _n(p) exists whenever p is the range projection of a positive self-adjoint element of the unit ball ofA, then it is shown that limφ _n(x) does exist for eachx inA″. This is a non-commutative generalisation of a celebrated theorem of Dieudonné. A new proof of Dieudonné’s theorem, for positive measures, is given here. The proof of the main result makes use of Dieudonné’s original theorem.     

A Mössbauer effect study of structural ordering in rapidly quenched Fe–Ga alloys

Samples of Fe_100−xGa_x (x=8.3, 17.9, 20.5 and 23.3) were prepared by rapid solidification from the melt using a single Cu roller. X-ray diffraction studies of all samples showed them to be single phase with the disordered BCC structure. No evidence of superlattice reflections from D0₃ ordering was observed for any of the samples. Room-temperature ⁵⁷Fe Mössbauer effect spectra indicated that all samples were ferromagnetically ordered. Spectra were fit to distributions of hyperfine fields. The x=8.3 sample showed a hyperfine field distribution that was single peaked and indicated a reasonably random distribution of local Fe environments. The x=17.9 and 20.5 samples showed hyperfine field distributions that were bimodal and indicated two distinct local Fe environments. The x=23.3 sample showed three distinct field components. It is suggested that the x=8.3, 17.9 and 20.5 alloys are primarily a disordered BCC phase. The x=8.3 alloy shows a small amount of short-range Ga–Ga pairing, while this short-range pairing is significantly greater in the x=17.9 and 20.5 alloys. The three field components in the x=23.3 alloy correspond well to the two sites associated with the D0₃ phase and a third component corresponding to a remaining L1₂ phase suggesting the presence of at least short-range D0₃ clustering in this alloy.     

Study of molecular orientation in poly(ethylene terephthalate) fibers by fluorescence polarization

Molecular orientation in poly(ethylene terephthalate) (PET) fibers was studied by polarized fluorescence. The observed amorphous orientation of the spun as fiber was not random but uniaxial along the fiber axis. This orientation increased with draw ratio up to about 2 and then remained fairly constant. The amorphous regions of PET fibers were disoriented when the fibers were heated while unconstrained. The fluorescence data obtained were correlated with shrinkage measurements. Fluorescence data indicated that spin drawing had more effect upon orientation than subsequent drawing of the fiber, whereas birefringence data indicated the opposite. The reason for this behavior is discussed.     

Vibrational analysis of polyethylene terephthalate and its deuterated derivatives

The Vibrational analysis of polyethylene terephthalate, polyethylene-d₄ terephthalate, and polyethylene terephthalate-d₄ has been carried out using a valence force field calculated from the infrared and Raman spectra of a series of low molecular weight aromatic esters. The Raman spectra for polyethylene-d₄ terephthalate and polyethylene terephthalate-d₄ are presented and band assignments for these compounds and polyethylene terephthalate are discussed.