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1.
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 相似文献
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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%. 相似文献
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David A Swingle Lynne M Pachnowski 《International Journal of Mathematical Education in Science & Technology》2013,44(3):361-370
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. 相似文献
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Soumitra Satapathi Lian Li Robinson Anandakathir Lynne A. Samuelson Jayant Kumar 《高分子科学杂志,A辑:纯化学与应用化学》2013,50(12):1049-1054
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. 相似文献
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Langang Niu Ramaswamy Nagarajan Dingsong Feng Lynne A. Samuelson 《高分子科学杂志,A辑:纯化学与应用化学》2013,50(12):1983-1990
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). 相似文献
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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. 相似文献