Undergraduate mathematics majors’ writing performance producing proofs and counterexamples about continuous functions

In advanced mathematical thinking, proving and refuting are crucial abilities to demonstrate whether and why a proposition is true or false. Learning proofs and counterexamples within the domain of continuous functions is important because students encounter continuous functions in many mathematics courses. Recently, a growing number of studies have provided evidence that students have difficulty with mathematical proofs. Few of these research studies, however, have focused on undergraduates’ abilities to produce proofs and counterexamples in the domain of continuous functions. The goal of this study is to contribute to research on student productions of proofs and counterexamples and to identify their abilities and mathematical understandings. The findings suggest more attention should be paid to teaching and learning proofs and counterexamples, as participants showed difficulty in writing these statements. More importantly, the analysis provides insight into the design of curriculum and instruction that may improve undergraduates’ learning in advanced mathematics courses.     

A mass and time-of-flight spectroscopy study of the formation of clusters in free-jet expansions of normal D2

The mass spectra in the range of 2(D+)-38(D19+) amu of clusters formed in a supersonic free-jet expansion of normal D2 are investigated as functions of source temperature in the range of 95-220 K and of source pressure in the range of 10-120 bars. For some of the small ion fragments, time-of-flight distributions are also measured. For large clusters (n > 200) the intensities of the odd-numbered ion fragments exhibit magic numbers at D9+ and D15+ in accordance with previous experiments and calculations. The even-numbered ion fragments have much smaller intensities and exhibit new magic numbers at D10+ and D14+. For source conditions such that large clusters are formed, the intensities of the various different ion fragments are observed to saturate beyond a certain source pressure. At lower source pressures, where only small clusters are formed, the terminal mole fractions of the neutral dimers are analyzed in the light of available theories which take into account both the thermodynamics and the kinetics of the expansion. At higher source pressures and lower temperatures, where larger clusters are formed, the sizes of the neutral clusters are estimated using scaling laws and are found to be consistent with the mass spectra and measured time-of-flight distributions. By using a variety of techniques it has been possible to obtain reliable conclusions about the neutral cluster sizes for the present free-jet expansion conditions.     

Efficient representation of perm groups

This note presents an elementary version of Sims's algorithm for computing strong generators of a given perm group, together with a proof of correctness and some notes about appropriate lowlevel data structures. Upper and lower bounds on the running time are also obtained. (Following a suggestion of Vaughan Pratt, we adopt the convention that perm=permutation, perhaps thereby saving millions of syllables in future research.)Dedicated to the memory of Marshall HallThis research was supported in part by the National Science Foundation under grant CCR-86-10181, and by Office of Naval Research contract N00014-87-K-0502.     

Untersuchungen an Stichstoff–Jod-Verbindungen. VI. Präparative und infrarotspektroskopische Untersuchungen an N-Jodmethylaminen

Studies on nitrogen iodine compounds. VI. Preparation and infrared studies on N-iodomethylamines N-diiodomethylamine and N-iododimethylamine can both be prepared under special conditions only. N-diiodomethylamine forms solid adducts with methylamine, trimethylamine and pyridine. The diodomethylamines show IR spectra similar to those shown by (NI₃ · NH₃)_n and demonstrating an analogous polymeric structure; iododimethylamine according to its spectrum is thought to be a monomeric substance.     

A Learning Progression for Elementary Students’ Functional Thinking

In this article we advance characterizations of and supports for elementary students’ progress in generalizing and representing functional relationships as part of a comprehensive approach to early algebra. Our learning progressions approach to early algebra research involves the coordination of a curricular framework and progression, an instructional sequence, written assessments, and levels of sophistication describing students’ algebraic thinking. After detailing this approach, we focus on what we have learned about the development of students’ abilities to generalize and represent functional relationships in a grades 3–5 early algebra intervention by sharing the levels of responses we observed in students’ written work over time. We found that the sophistication of students’ responses increased over the course of the intervention from recursive patterning to correspondence and in some cases covariation relationships between variables. Students’ responses at times differed by the particular tasks that were posed. We discuss implications for research and practice.     

Accuracy of the multipole expansion applied to a sphere in a creeping flow parallel to a wall

An example system is studied to discuss precision of the multipoleexpansion, applied to determine forces exerted on particlesby a viscous low-Reynolds-number fluid flow. A single spherein an ambient flow (pure shear, quadratic, and modulated shear)parallel to a close plane wall is considered. Forces and torquesexerted by the ambient flow on a motionless sphere are evaluated.Their precision is determined and related to a multipole orderof the truncation. Similar analysis is performed for a movingsphere with no ambient flow and for a freely moving sphere.Relative motion of the sphere with respect to the wall givesrise to strong lubrication interactions. It is analysed howthese interactions affect accuracy of the pure multipole expansion,and what are the smallest distances where it becomes insufficient.An alternative precise method is applied, in which lubricationexpressions are subtracted from the hydrodynamic forces andtorques, and the residue is evaluated as a fast-convergent seriesof inverse powers of the distance between the sphere centreand the wall. The accuracy of this procedure is carefully analysed.     

A Longitudinal Examination of Middle School Students' Understanding of the Equal Sign and Equivalent Equations

This longitudinal study investigated (a) middle school students' understanding of the equal sign, (b) students' performance solving equivalent equations problems, and (c) changes in students' understanding and performance over time. Written assessment data were collected from 81 students at four time points over a 3-year period. At the group level, understanding and performance improved over the middle school years. However, such improvements were gradual, with many students still showing weak understanding and poor performance at the end of grade 8. More sophisticated understanding of the equal sign was associated with better performance on equivalent equations problems. At the individual level, students displayed a variety of trajectories over the middle school years in their understanding of the equal sign and in their performance on equivalent equations problems. Further, students' performance on the equivalent equations problems varied as a function of when they acquired a sophisticated understanding of the equal sign. Those who acquired a relational understanding earlier were more successful at solving the equivalent equations problems at the end of grade 8.     

Construction of a random sequence

A completely equidistributed sequence of real numbers is constructed explicitly.