The Emergence of Mathematical Modeling Competencies: An Investigation of Prospective Secondary Mathematics Teachers

A study with prospective teachers without prior mathematical modeling experience sheds light on how their newly developed conceptual understanding of modeling manifested itself in their work on the final task of a modeling module within a pedagogy course in secondary mathematics curriculum and assessment. The main purpose of the module was to provide opportunity for the prospective teachers to experience the Common Core Mathematical Practice Model with Mathematics and begin to develop competency in modeling. Their work and reflections displayed a range of proficiency in several competencies associated with the modeling process. Examples of their work illustrating these ranges are provided. The prospective teachers expressed both struggle and rewards during the process, and reflected on challenges for teaching modeling. The results suggest that infusing modules in existing courses can be an effective way to elevate prospective teachers from unfamiliarity with modeling to noticeable levels of proficiency in various modeling sub-competencies.     

Non-symmetric Cauchy kernels for the classical groups

We give non-symmetric versions of the Cauchy kernel and Littlewood's kernels, corresponding to the types A, B, C and D, of the classical groups. Defining two families of key polynomials (one of them being the Demazure characters), we show that these new kernels are diagonal in the basis of key polynomials. We define scalar products such that the two families of key polynomials are adjoint to each other.     

Rhodium-Catalyzed Addition of Aryl Boronic Acids to 2,2-Disubstituted Malononitriles

β-Ketonitriles bearing a quaternary carbon at the 2-position were prepared through Rh-catalyzed addition of aryl boronic acids to 2,2-disubstituted malononitriles. In contrast to the previously described transnitrilative cyanation of aryl boronic acids with dialkylmalononitriles, the present reaction avoids retro-Thorpe collapse of the intermediate addition product through the use of a milder base. The reaction was amenable to a variety of aryl boronic acids and disubstituted malononitriles, providing a diverse array of β-ketonitriles. The products could be further derivatized to valuable chiral α,α-disubstituted-β-aminonitriles through addition reactions to the corresponding N-tert-butanesulfinyl imines.     

Jet impingement cooling of a discretely heated portion of a protruding pedestal with a single round air jet

The influence of a protruding pedestal on impinging jet heat transfer is investigated. A discretely heated portion of a protruding pedestal is exposed to a single circular impinging air jet with Re=10,000–30,000. Jet exit diameters of 3.5, 9.5 and 21 mm are positioned at jet exit-to-surface distances of 2–5 diameters. The nondimensional heat transfer over the discretely heated portion of the pedestal is compared to a flat plate design to gauge the effects of Reynolds number, jet diameter and jet exit-surface spacing. In all cases, the presence of the protruding pedestal downstream is found to increase heat transfer.     

Pricing and volume discounting for a dominant retailer with uncertain manufacturing cost information

A product costs the manufacturer c/unit to produce; the retailer sells it at p/unit to the consumers. The retail-market demand volume V varies with p according to a given demand curve D_p. How would or should the “players” (i.e., the manufacturer and the retailer) set their prices? In contrast to many studies that assume a dominant manufacturer implementing the “manufacturer-Stackelberg” (“[mS]”) game, this paper examines how a dominant retailer should operate when his knowledge of c is imperfect. We first derive optimal decisions (some of them counter-intuitive) for the dominant retailer when he is restricted to choosing between [rS] (retailer-Stackelberg) and [mS]. Second, we propose a “reverse quantity discount” scheme that the dominant retailer (i.e., the downstream player) can offer to the manufacturer (note that the standard discount scheme is offered by the upstream player). We show that this discounting scheme is quite effective compared to the considerably more complicated though nevertheless theoretically optimal “menu of contracts.” We also reveal a largely overlooked function of discounting; i.e., discounting enables an “ignorant” but dominant player to usurp the earnings attributable to the knowledge of the dominated player. Finally, we also show that discounting works well when the demand curve is linear, but becomes ineffective when the demand curve is iso-elastic – a result echoing the conclusions of some earlier related works.     

Characterization of cell lysis events on a microfluidic device for high-throughput single cell analysis

A microfluidic device capable of rapidly analyzing cells in a high-throughput manner using electrical cell lysis is further characterized. In the experiments performed, cell lysis events were studied using an electron multiplying charge coupled device camera with high frame rate (>100 fps) data collection. It was found that, with this microfluidic design, the path that a cell follows through the electric field affects the amount of lysate injected into the analysis channel. Elimination of variable flow paths through the electric field was achieved by coating the analysis channel with a polyamine compound to reverse the electroosmotic flow (EOF). EOF reversal forced the cells to take the same path through the electric field. The improved control of the cell trajectory will reduce device-imposed bias on the analysis and maximizes the amount of lysate injected into the analysis channel for each cell, resulting in improved analyte detection capabilities.