Impurity-stimulated heterogeneous nucleation of supercooled H2 clusters

The sizes and mass spectra of large (N=1900-13,700 molecules) cold (approximately 3.1 K) H2 clusters have been measured after scattering from CO molecules. Cluster-size measurements after between 2 and 8 collisions indicate that 7% of the H2 molecules are evaporated. This loss agrees with calculations for the number of H2 molecules evaporated by the heat released in the transition from an initial liquid state to a final solid state. Even though heterogeneous nucleation is initiated after only a few collisions with CO molecules, the mass spectra show that additional captured CO molecules coagulate to form large CO clusters with up to n=11 molecules, suggesting that the outer layer is sufficiently liquidlike to facilitate mobility of the CO molecules. Since the calculated H2 cluster temperature (approximately 3.1 K) is below the superfluid transition temperature predicted for pH2 with density between 40% and 80% of the triple-point density, a shell-like region of low density near the cluster surface can be expected to be superfluid.     

Untersuchungen an Stickstoff-Jod-Verbindungen. IX. Präparative und infrarotspektroskopische Untersuchungen an Jod-Komplexen von N-Jodmethylaminen

Studies of Nitrogen Iodine Compounds. IX. Preparation and I. R. Investigations of Iodine Complexes of N-Iodine Methylamines [CH₃NI₂]₂ · I₂, [(CH₃)₂NI]₂ · I₂ and (CH₃)₂NI · I₂, have been prepared. The IR-spectra have been obtained in the region 33–4000 cm^?1. The spectra can be coordinated on the ground of a molecule model with 5 atoms Z₂XYW of the symmetry C_s and are compared to that of CH₃NI₂. The structures of the compounds are proposed to be monomolecular.     

A one-way,stackless quicksort algorithm

This note describes a sorting technique that is similar to the well-known quicksort method, but it is undirectional and avoids recursion. The new approach, which assumes that the keys to be sorted are positive numbers, leads to a much shorter program.The preparation of this paper was supported in part by National Science Foundation grant DCR-83-00984. The authors thank the referees for several important suggestions.     

Persistent radicals of trivalent tin and lead

In this report we present synthetic, crystallographic, and new electron paramagnetic resonance (EPR) spectroscopic work that shows that the synthetic route leading to the recently reported, first persistent plumbyl radical *PbEbt3 (Ebt = ethylbis(trimethylsilyl)silyl), that is, the oxidation of the related PbEbt3-anion, was easily extended to the synthesis of other persistent molecular mononuclear radicals of lead and tin. At first, various novel solvates of homoleptic potassium metallates KSnHyp3 (4a), KPbHyp3 (3a), KSnEbt3 (4b), KPbIbt3 (3c), and KSnIbt3 (4c) (Hyp = tris(trimethylsilyl)silyl, Ibt = isopropylbis(trimethylsilyl)silyl), as well as some heteroleptic metallates, such as [Li(OEt2)2][Sn(n)BuHyp2] (3d), [Li(OEt2)2][Pb(n)BuHyp2] (4d), [Li(thf)4][PbPhHyp2] (3e), and [K(thf)7][PbHyp2{N(SiMe3)2}] (3f), were synthesized and crystallographically characterized. Through oxidation by tin(II) and lead(II) bis(trimethylsilyl)amides or the related 2,6-di-tert-butylphenoxides, they had been oxidized to yield in most cases the corresponding radicals. Five novel persistent homoleptically substituted radicals, that is, *SnHyp3 (2a), *PbHyp3 (1a), *SnEbt3 (2b), *SnIbt3 (2c), and *PbIbt3 (1c), had been characterized by EPR spectroscopy. The stannyl radicals 2a and 2c as well as the plumbyl radical 1c were isolated as intensely colored crystalline compounds and had been characterized by X-ray diffraction. Persistent heteroleptically substituted radicals such as *PbHyp2Ph (1e) or *PbHyp2Et (1g) had also been generated, and some selected EPR data are given for comparison. The plumbyl radicals *PbR3 exhibit a clean monomolecular decay leading to the release of a temperature-dependent stationary concentration of branched silyl radicals. They may thus serve as tunable sources of these reactive species that may be utilized as reagents for mild radical silylations and/or as initiators for radical polymerizations. We present EPR-spectroscopic investigations for the new tin- and lead-containing compounds giving detailed insights into their electronic and geometric structure in solution, as well as structural studies on the crystalline state of the radicals, some of their anionic precursors, and some side-products.     

The distribution of continued fraction approximations

The continued fraction convergents to a random real number are shown to approximate that number with a limiting probability distribution that had been conjectured by H. W. Lenstra.     

Teachers’ gestures and speech in mathematics lessons: forging common ground by resolving trouble spots

This research focused on how teachers establish and maintain shared understanding with students during classroom mathematics instruction. We studied the micro-level interventions that teachers implement spontaneously as a lesson unfolds, which we call micro-interventions. In particular, we focused on teachers’ micro-interventions around trouble spots, defined as points during the lesson when students display lack of understanding. We investigated how teachers use gestures along with speech in responding to such trouble spots in a corpus of six middle-school mathematics lessons. Trouble spots were a regular occurrence in the lessons (M = 10.2 per lesson). We hypothesized that, in the face of trouble spots, teachers might increase their use of gestures in an effort to re-establish shared understanding with students. Thus, we predicted that teachers would gesture more in turns immediately following trouble spots than in turns immediately preceding trouble spots. This hypothesis was supported with quantitative analyses of teachers’ gesture frequency and gesture rates, and with qualitative analyses of representative cases. Thus, teachers use gestures adaptively in micro-interventions in order to foster common ground when instructional communication breaks down.     

Middle School Mathematics Teachers' Knowledge of Students' Understanding of Core Algebraic Concepts: Equal Sign and Variable

This article reports results from a study focused on teachers' knowledge of students' understanding of core algebraic concepts. In particular, the study examined middle school mathematics teachers' knowledge of students' understanding of the equal sign and variable, and students' success applying their understanding of these concepts. Interview data were collected from 20 middle school teachers regarding their predictions of student responses to written assessment items focusing on the equal sign and variable. Teachers' predictions of students' understanding of variable aligned to a large extent with students' actual responses to corresponding items. In contrast, teachers' predictions of students' understanding of the equal sign did not correspond with actual student responses. Further, teachers rarely identified misconceptions about either variable or the equal sign as an obstacle to solving problems that required application of these concepts. Implications for teacher professional development are discussed.     

Mathematics in the home: Homework practices and mother-child interactions doing mathematics

Parents are a largely untapped resource for improving the mathematics performance of American children, which lags behind the performance of children from other nations. The purpose of the research reported here was to assess homework practices in the home, and to examine interactions between mothers and their 5th grade children as they worked challenging mathematics problems (pre-algebra equivalence problems). Results indicated that children spent on average 23 min per day on mathematics homework, with an average of 8 min of help from parents. Videotapes of mother-child interactions indicated that mothers varied considerably in the quality of the mathematics content that they conveyed while teaching, and in the quality of their scaffolding of the material for the child. As expected, mothers who themselves had more mathematics preparation performed better in conveying mathematical content and in scaffolding. Mothers with more mathematics self-confidence also performed better. The results suggest that children face inequities in the parental resources available to them for math learning; these inequities might be remedied by school-family partnership programs.