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4. Some classes of rational functions and related Banach spaces

Author

Dudley, R. M., Sidenko, Sergiy, Wang, Zuoqin, and Yang, Fangyun

Subjects
Mathematics - Functional Analysis, 46B99 (Primary), 46B22 (Secondary)
Abstract

For positive integers d, r, and M, we consider the class of rational functions on real d-dimensional space whose denominators are products of at most r functions of the form 1+Q(x) where each Q is a quadratic form with eigenvalues bounded above by M and below by 1/M. Each numerator is a monic monomial of the same degree as the corresponding denominator. Then we form the Banach space of countable linear combinations of such rational functions with absolutely summable coefficients, normed by the infimum of sums of absolute values of the coefficients. We show that for rational functions whose denominators are rth powers of a specific 1+Q, or differences of two such rational functions with the same numerator, the norm is achieved by and only by the obvious combination of one or two functions respectively. We also find bounds for coefficients in partial-fraction decompositions of some specific rational functions, which in some cases are quite sharp., Comment: LaTex, 18 pages, no figures. Replacement is just to insert a line acknowledging NSF grant support

Published
2007

11. Effects of Narrated Heuristic Feedback and Gender on College Students' Calculus Problem Solving Performance

Author

Xie, Ying, Coffland, David, and Yang, Fangyun

Abstract

Multimodal feedback could promote college students' conceptual, procedural and heuristic knowledge learning. This study used an experimental design to investigate the interaction effects of instructor's narrated feedback (with procedural feedback delivered visually and heuristic feedback auditorily) and gender on their post-test performance of a similar problem. Ninety-three first-year undergraduate students participated in the study and were randomly assigned into different feedback groups. Students in the control group received visual-only feedback containing procedural and heuristic information from the instructor. Those in the treatment group viewed a generic three-minute video with the same procedural message presented visually and heuristic knowledge in narration. A two-way ANOVA showed that students receiving the multi-modal feedback scored significantly higher than those receiving visual-only feedback on the post-test. This effect did not differ for female or male students. The result supports that multimodal feedback could be a valuable tool to improve college students' mathematical problem solving skills. (Contains 6 tables and 2 figures.)

Published
2012

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