Existence Results for Semilinear Neutral Functional Differential Inclusions via Analytic Semigroups

In this paper we prove existence results for semilinear neutral functional differential inclusions with finite or infinite delay in Banach spaces. Our theory makes use of analytic semigroups and fractional powers of closed operators, integrated semigroups and cosine families.      

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     

Leggett-Williams norm-type theorems for coincidences

We study the existence of positive solutions to the operator equation Lx = Nx, where L is a linear Fredholm mapping of index zero and N is a nonlinear operator. Using the properties of cones in Banach spaces and Leray-Schauder degree for completely continuous operators, k-set contractions and condensing mappings, we obtain some refinements of the results established in [3] and [14]. Received: 9 July 2005; revised: 12 January 2006     

Predicting the performance of molecularly imprinted polymers: Selective extraction of caffeine by molecularly imprinted solid phase extraction

A rational design approach was taken to the planning and synthesis of a molecularly imprinted polymer capable of extracting caffeine (the template molecule) from a standard solution of caffeine and further from a food sample containing caffeine. Data from NMR titration experiments in conjunction with a molecular modelling approach was used in predicting the relative ratios of template to functional monomer and furthermore determined both the choice of solvent (porogen) and the amount used for the study. In addition the molecular modelling program yielded information regarding the thermodynamic stability of the pre-polymerisation complex. Post-polymerisation analysis of the polymer itself by analysis of the pore size distribution by BET yielded significant information regarding the nature of the size and distribution of the pores within the polymer matrix. Here is proposed a stepwise procedure for the development and testing of a molecularly imprinted polymer using a well-studied compound—caffeine as a model system. It is shown that both the physical characteristics of a molecularly imprinted polymer (MIP) and the analysis of the pre-polymerisation complex can yield vital information, which can predict how well a given MIP will perform.     

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.     

A robust spin-coated optical sensing device for rapid determination of predominant metal ions in wastewater streams

The potential of an optical sensing material for determination of metal ions in wastewater streams is presented. The initial results in the development of a chromo-ionophore-based optical sensor for metal ions are detailed. An azo dye species, 2-(5-bromo-2-pyridylazo)-5-diethylaminophenol (BrPADAP), together with an ionophore and plasticiser was immobilised in a film of poly(vinyl chloride) (PVC). Initial investigations using the dye-impregnated sensing film showed a reproducible and reversible response to Pb(II) ions in aqueous solution. The response of this sensing film to Pb(II) at pH 6 was found to be linear in the 0.1-10 mg/L range, with a %RSD of 1.55%. The film can be regenerated by immersion in a low-pH buffer solution. The authors demonstrate how this method shows potential for analysis using once-off sensing probes. The sensing film response for Pb(II) occurs within 30 s.