Partition Lattice q-Analogs Related to q-Stirling Numbers

We construct a family of partially ordered sets (posets) that are q-analogs of the set partition lattice. They are different from the q-analogs proposed by Dowling [5]. One of the important features of these posets is that their Whitney numbers of the first and second kind are just the q-Stirling numbers of the first and second kind, respectively. One member of this family [4] can be constructed using an interpretation of Milne [9] for S[n, k] as sequences of lines in a vector space over the Galois field F _q. Another member is constructed so as to mirror the partial order in the subspace lattice.     

Crutch or Catalyst: Teachers' Beliefs and Practices Regarding Calculator use in Mathematics Instruction

This study investigated K‐12 teachers' beliefs and reported teaching practices regarding calculator use in their mathematics instruction. A survey was administered to more than 800 elementary, middle and high school teachers in a large metropolitan area to address the following questions: (a) what are the beliefs and practices of mathematics teachers regarding calculator use? and (b) how do these beliefs and practices differ among teachers in three grade bands? Factor analysis of 20 Likert scale items revealed four factors that accounted for 54% of the variance in the ratings. These factors were named Catalyst Beliefs, Teacher Knowledge, Crutch Beliefs, and Teacher Practices. Compared to elementary teachers, high school teachers were significantly higher in their perception of calculator use as a catalyst in mathematics instruction. However, the higher the grade level of the teacher, the higher the mean score on the perception that calculator use may be a way of getting answers without understanding mathematical processes. The mean scores for teachers in all three grade bands indicated agreement that students can learn mathematics through calculator use and using calculators in instruction will lead to better student understanding and make mathematics more interesting. The survey results shed light on teachers' self reported beliefs, knowledge, and practices in regard to consistency with elements of the National Council of Teachers of Mathematics Principles and Standards for School Mathematics (2000) technology principle and the NCTM use of technology position paper (2003). This study extended previous research on teachers' beliefs regarding calculator use in classrooms by examining and comparing the results of teacher surveys across three grade bands.     

Effects of pi-bonding on stability and reactivity in

Recent synthetic and theoretical investigations of organometallic compounds containing the CpCr(NO) fragment have shown an interdependence of the electronic nature of ancillary ligands and the stability of a given Cr oxidation state. Understanding the correlation between ligand pi-bonding properties and the nature of the metal-based frontier orbitals permits the rationalization of observed reactivity patterns, and the identification and preparation of new classes of target molecules.     

Fundamental role of the fostriecin unsaturated lactone and implications for selective protein phosphatase inhibition

Key derivatives and analogues of fostriecin were prepared and examined that revealed a fundamental role for the unsaturated lactone and confirmed the essential nature of the phosphate monoester. Thus, an identical 200-fold reduction in protein phosphatase 2A (PP2A) inhibition is observed with either the saturated lactone (7) or with an analogue that lacks the entire lactone (15). This 200-fold increase in PP2A inhibition attributable to the unsaturated lactone potentially may be due to reversible C269 alkylation within the PP beta12-beta13 active site loop accounting for PP2A/4 potency and selectivity.