{\L}ojasiewicz exponent and pluricomplex Green function on algebraic sets

We study pluricomplex Green functions on algebraic sets. Let $f$ be a proper holomorphic mapping between two algebraic sets. Given a compact set $K$ in the range of $f$, we show how to estimate the pluricomplex Green functions of $K$ and of $f^{-1}(K)$ in terms of each other, the {\L}ojasiewicz exponent of $f$ and the growth exponent of $f$. This result leads to explicit examples of pluricomplex Green functions on algebraic sets. We also present an enhanced version of the Bernstein-Walsh polynomial inequality specific to algebraic sets. This article provides a theoretical framework for future investigations of the rate of polynomial approximation of holomorphic functions on algebraic sets in the style of Bernstein-Walsh-Siciak theorem

Evaluating Lebesgue constants by Chebyshev polynomial meshes on cube, simplex and ball

We show that product Chebyshev polynomial meshes can be used, in a fully discrete way, to evaluate with rigorous error bounds the Lebesgue constant, i.e. the maximum of the Lebesgue function, for a class of polynomial projectors on cube, simplex and ball, including interpolation, hyperinterpolation and weighted least-squares. Several examples are presented and possible generalizations outlined. A numerical software package implementing the method is freely available online

Charting a course to creativity in developmental education

textA central problem in community colleges' developmental education programs concerns the over-emphasis on basic skills instruction to the possible exclusion of higher order thinking. Although the ability to read, write, and compute establishes an indispensable foundation for future academic success, basic skills instruction alone does not teach students how to analyze, synthesize, and evaluate ideas--all of which are imperative in the global, knowledge-based economy where creative thinking constitutes the primary form of capital. The purpose of this study, therefore, was to synthesize creativity research and developmental education by investigating the significance of creative thinking in developmental courses taught at Florida Community College at Jacksonville's Kent Campus. To fulfill the study's purpose, the researcher employed a qualitative research design and methodology through which she explored the perspectives and practices of twelve participants selected through stratified purposeful sampling. Representing different disciplines, the participants varied in their instructional classification (full-time versus part-time) and developmental teaching experience. Having designed a basic interpretive qualitative study, the researcher, as a human instrument, sought to understand the participants' perceptions regarding the importance of promoting creativity in developmental courses; the characteristics of classroom environments that facilitate creative thinking; as well as the instructional approaches and methods that foster such thinking. By triangulating the data collection through interviews, observations, and document analyses and by obtaining member checks of the interviews from the participants, the researcher endeavored to enhance the trustworthiness of the findings. Presented in the rich, thick description distinctive of qualitative analysis, the study revealed that the enthusiastic, caring, and learner-centered participants possessed the personality characteristics necessary for the cultivation of creative thinking among students. Despite being intended to promote the acquisition of basic skills, many of the participants' approaches and methods, particularly the use of personalized instruction, verbal praise, cooperative learning, and figurative language, could also be employed to establish learning environments that facilitate creative thinking. Upon reviewing the data, the researcher made recommendations designed to contribute to the limited body of knowledge about the synthesis of creativity research and developmental education.Educational Administratio