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.     

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.     

An alternative preparation of the 2-dimethylaminomethylene-1,3-bis(dimethylimmonio)propane salt from phosphonoacetic acids and some applications in heterocyclic synthesis

An alternative preparation of a 2-iminovinamidinium salt from phosphonoacetic acids is described along with its application to the synthesis of 5-formylpyrimidines and masked 4-formylpyrazoles.     

A simple and efficient one step synthesis of benzoxazoles and benzimidazoles from carboxylic acids

Benzoxazoles or benzimidazoles can be rapidly and efficiently synthesized from a variety of carboxylic acids with 2-aminophenols or 1,2-phenylenediamines in one simple step, respectively. The use of commercially available PS-PPh₃ resin combined with microwave heating delivered a variety of benzoxazoles and benzimidazoles in high yields and purities.     

Metal-catalyzed di-tert-butylsilylene transfer: synthesis and reactivity of silacyclopropanes

Metal-catalyzed di-tert-butylsilylene transfer was developed as a mild, operationally simple, functional-group-tolerant method for silacyclopropane formation. Di-tert-butylsilylene was transferred from cyclohexene silacyclopropane 1 to an alkene through the use of a metal salt. Silacyclopropanation occurred at temperatures as low as -27 degrees C when AgOTf or AgOC(O)CF(3) were used as catalysts. Complex silacyclopropanes were formed stereospecifically and diastereoselectively from functionalized alkenes. Silacyclopropanes reacted with various carbonyl compounds, including aldehydes, ketones, formate esters, and formamides, in an overall process that efficiently converts alkenes into oxasilacyclopentanes with defined stereochemistry.     

Syntheses and Crystal Structures of Some CCo3 Clusters Containing PPh3 or dppm Ligands

Molecular structural determinations are reported for six Co₃C carbonyl cluster complexes containing tertiary phosphines, which have been isolated as by-products from a variety of reactions. Structural features are similar to those of related complexes already reported. Some discussion of apparent orientational preferences of the CH₂ group of dppm ligands, which appear to enter into H-bonding interactions with amido or carboxylate substituents, is given. Appropriate comparisons are made with unsubstituted analogues.     

Mechanism of silver-mediated di-tert-butylsilylene transfer from a silacyclopropane to an alkene

Kinetic studies of the reactions of cyclohexene silacyclopropane 1 and monosubstituted alkenes in the presence of 5 mol % of (Ph3P)2AgOTf suggested a possible mechanism for silver-mediated di-tert-butylsilylene transfer. The kinetic order in cyclohexene silacyclopropane 1 was determined to be one. Inverse kinetic saturation behavior (rate inhibition) was observed in monosubstituted alkene and cyclohexene concentrations. Saturation kinetic behavior in catalyst concentration was observed. A reactive intermediate, a silylsilver complex, was observed using low temperature 29Si NMR spectroscopy. Competition experiments between substituted styrenes and a deficient amount of 1 correlated well with the Hammett equation and provided a rho value of -0.62 +/- 0.02 using sigmap constants. These data support a mechanism involving reversible silver-promoted di-tert-butylsilylene extrusion from 1 followed by irreversible concerted electrophilic attack of the silylsilver intermediate on the alkene.     

Convergence of Padé approximants for aq-hypergeometric series (Wynn's Power Series I)

We investigate the convergence of sequences of Padé approximants for power series