A Note on Log-Convexity of q-Catalan Numbers

The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q = 1, to the Catalan numbers: 1, 1, 2, 5, 14,…, a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted. Received January 2, 2007     

Catalytic asymmetric addition of alkylzinc and functionalized alkylzinc reagents to ketones

We describe our full report of the catalytic asymmetric addition of simple and functionalized dialkylzinc reagents to a broad range of saturated ketones and enones. The functionalized organozinc reagents contain esters, silyl ethers, alkyl chlorides, and alkyl bromides. In general, the resulting tertiary alcohol products are isolated with high ee's. With some substrates, yields are low as a result of the formation of aldol byproducts. Most substrates undergo additions with good yields reaching as high as 91%.     

Filling in the gaps: modelling incomplete CBL data using a graphing calculator

This article discusses the real-world problem-solving lesson that emerged when a high school math teacher used a motion detector with a CBL and graphing calculator to obtain the 'bounce' data of a ping-pong ball. While practising the 'bounce' data collection--a series of diminishing parabolas--the teacher accidentally pulled the motion detector away and then, realizing his mistake, pulled it back. The resulting data showed a series of parabolas, but one was missing. The teacher used this opportunity to create a lesson in which his students collect 'bad data' and then fill in the missing parabola using critical components of parabolas, such as the latus rectum and the vertex, and using matrices. The article provides all the necessary directions, formulas, and names of resources needed to replicate the lesson. The creation of this lesson demonstrates that a serendipitous error can create a genuine and authentic problem-solving activity for math students.     

Sensory Response and Two-Photon-Fluorescence Study of Regioregular Polythiophene Nanoparticles

Nanoparticles of a regioregular and soluble polythiophene were fabricated by mini-emulsion technique. The fabricated nanoparticles were characterized by optical spectroscopy and dynamic light scattering. The fluorescence quenching of these fabricated nanoparticles with 2,4-dinitrotolune (DNT) in aqueous and organic solutions was investigated. Significant fluorescence quenching was observed. The Stern-Volmer constants were determined to be higher than that of the bulk polymer in solution, indicating that the nanoparticles provide better sensitivity in DNT sensing. Strong two-photon-induced fluorescence was measured from these nanoparticles.     

Synthesis of Main‐Chain Liquid‐Crystalline Polyesters Containing Diphenyl Mesogens by Chemo‐Enzymatic Route †

Main‐chain thermotropic liquid‐crystalline polyesters containing rigid biphenyl mesogens and flexible spacers were synthesized by chemo‐enzymatic route. The enzyme‐catalyzed polymerization showed high regio‐ and chemo‐ selectivity, and is environmentally friendly. The resulting polyesters were characterized with ¹H‐NMR, ¹³C‐NMR, gel permeation chromatography (GPC), differential scanning calorimetry (DSC), and polarized light optical microscopy (POM).     

Fabrication of Gold Nano‐Structures with Azopolymer Templates

Fabrication of gold nano‐patterns has been demonstrated employing surface relief structures created on films of an azobenzene‐functionalized polymer as templates. The surface relief templates were photoinscribed on the azopolymer films in one‐step with two laser beams. Thin layers of gold were over‐coated on the polymer templates by thermal evaporation. Gold lines of a few hundred nanometer width were successfully fabricated by pyrolyzing the azobenzene polymer. Sub‐micron gold dots were also created. The resulting gold structures exhibited the same periodicity as the polymer templates.     

Enhancement of naringenin solution concentration by solid dispersion in cellulose derivative matrices

Naringenin (Nar) is an important bioactive flavonoid with poor organic solubility and oral bioavailability. It is highly promising for treatment of conditions including diabetes, hyperlipidemia, and hepatitis C infection. Amorphous solid dispersion (ASD) of Nar is an appealing way to enhance its solubility, and carboxylated cellulose esters are attractive polymers for this purpose because of their ability to stabilize drugs against crystallization in both solid and solution phases, while restricting drug release to the pH of the small intestine (ca. 6.8). We demonstrate that ASDs of Nar can be formed using such carboxylated cellulose derivatives as cellulose acetate adipate propionate (CAAdP), carboxymethylcellulose acetate butyrate (CMCAB) and hydroxypropylmethylcellulose acetate succinate (HPMCAS). We compare Nar solution concentrations and release profiles from these cellulosic ASDs to those from pure crystalline Nar, and to Nar ASD in poly(vinylpyrrolidinone) (PVP). We show that all polymers in this study form ASDs with Nar, that the PVP ASDs release Nar at both gastric (1.2) and small intestine (6.8) pH, and that the cellulosic polymers release Nar selectively at neutral pH. Solution concentrations of Nar are significantly enhanced from these ASDs. These preliminary studies indicate that HPMCAS, CAAdP, and CMCAB are practical ASD polymers for Nar due to their ability to generate and stabilize high solution concentrations, and their pH-triggered drug release.     

Zeolites with Continuously Tuneable Porosity

Zeolites are important materials whose utility in industry depends on the nature of their porous structure. Control over microporosity is therefore a vitally important target. Unfortunately, traditional methods for controlling porosity, in particular the use of organic structure‐directing agents, are relatively coarse and provide almost no opportunity to tune the porosity as required. Here we show how zeolites with a continuously tuneable surface area and micropore volume over a wide range can be prepared. This means that a particular surface area or micropore volume can be precisely tuned. The range of porosity we can target covers the whole range of useful zeolite porosity: from small pores consisting of 8‐rings all the way to extra‐large pores consisting of 14‐rings.