Szeg\H{o} kernel asymptotic expansion on CR manifolds with S1S^1 action

Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n \ge 1$ with a transversal CR $S^1$ action on $X$. We establish an asymptotic expansion for the $m$-th Fourier component of the Szeg\H{o} kernel function as $m\rightarrow\infty$, where the expansion involves a contribution in terms of a distance function from lower dimensional strata of the $S^1$ action. We also obtain explicit formulas for the first three coefficients of the expansion.Comment: 27pages, final version, to appear in the International journal of mathematic

On the stability of equivariant embedding of compact CR manifolds with circle action

We prove the stability of the equivariant embedding of compact strictly pseudoconvex CR manifolds with transversal CR circle action under circle invariant perturbations of the CR structures.Comment: 21 pages, final versio

Large time behavior and energy relaxation time limit of the solutions to an energy transport model in semiconductors

AbstractIn this paper, the global existence and the large time behavior of smooth solutions to the initial boundary value problem for the multi-dimensional energy transport model are studied. It is also proved that the solutions of the problem converge to an isothermal drift–diffusion model as energy relaxation time τ goes to 0 by compactness argument with the help of energy estimates and entropy inequality

Crossing the borders: an eco-cinecritical analysis of "Plant Wars" and "Bird Without Borders"

Este artículo trata sobre dos películas orientadas a la naturaleza desde la perspectiva de la cinecrítica ecológica, una crítica cinematográfica con conciencia ecológica. "Plant Wars", producida por PTS Taiwan, cuestiona la barrera que existe entre plantas nativas y plantas exóticas, así como entre plantas y humanos, para poder reclasificar las especies que se encuentran entre estas demarcaciones. El método narrativo que utiliza consiste en presentar 14 plantas nativas como protagonistas, y pone de manifiesto su relación con los nativos, los granjeros, los colmeneros y con otras especies de insectos. "Bird Without Borders", también producido por PTS, cuenta la historia del viaje migratorio de un grupo de espátulas menores ("Platalea minor") ("black-faced spoonbills") que vuelan 16.000 kilómetros desde Tainan, Taiwán, pasando por Japón, a lo largo de la costa de Corea del Sur y Corea del Norte, hasta una remota isla de China para buscar alimento, y su vuelta a Taiwán para pasar el invierno. Bird muestra diferentes niveles de cruce de fronteras. Estas dos producciones PTS son similares en cuanto a la búsqueda de cruces de frontera para redefinir lo que significa frontera o límite: "Plant" cuestiona la categorización biológica de las plantas nativas y las exóticas, mientras que "Bird" utiliza el ejemplo de la tripulación que sigue físicamente la ruta del vuelo de los pájaros, que cruza fronteras nacionales, para dar una visión de reclasificación del territorio natural mediante el cuestionamiento de la idea de "lugar" fijo. Ambas películas buscan modos de cruzar las fronteras hechas por los hombres: "Plant" lo hace a través de la imaginación medioambiental y "Bird" por medio de movimientos físicos de diferentes tipos. "Plant Wars" parece expresar su posición de modo más fuerte, por medio de "guerras", mientras que "Bird Without Borders" trata de alcanzar una visión sin fronteras.This essay reads two Asian nature-oriented films from the perspective of eco-cinecriticism, an ecologically minded film criticism. “Plant Wars”, produced by PTS Taiwan, questions the demarcation between native plants and exotic plants as well as between plants and human beings so as to remap the in-between border area. The storytelling method it applies is to feature 14 currently native plants as the main characters, and illustrates their relationship to aboriginal people, farmers, beekeepers and to other insect species. “Bird Without Borders”, also produced by PTS, tells the story of the migration journey of a group of black-faced spoonbills flying 16,000 kilometers all the way from Tainan, Taiwan, to Japan, along the border of South and North Korea, to a remote island of China to look for feeding places and then back to Taiwan to spend the winter. Different layers of border-crossing are shown in “Bird”. One similarity of these two PTS productions is the search for border-crossing so as to redefine or revise what the border means. “Plant” questions the biological categorization of the native plants and the exotic plants; “Bird” illustrates how the crew physically follows the route of the birds’ flight which crosses the national borders to provide a vision of remapping the natural territory through challenging the idea of a fixed “place.” Both are seeking for ways to trespass the man-made borders: “Plant” via environmental imagination; “Bird” via some physical movement. While the title of “Plant Wars” seemingly expresses its position more strongly through “wars,” “Bird Without Borders” explicitly asserts a dream of no borders

Improving the Quality of Mathematics Education:Two Teaching Modes and Taiwanese Student Learning

Students’ and teachers’ long-term (i.e. three years) experiences in three classes of the traditional direct instruction and constructivist class discussion approach to the mathematics teaching at a Taiwanese junior high school and at an experimental school in Taiwan were discussed in this study. This research utilized qualitative methods. The study adopted content analysis approaches from a qualitative perspective. This was combined with the perspectives of social constructivism and situated learning theories to interpret students’ learning and growth. The research findings of this study revealed differences in the group of students exposed to the constructivists teaching environment. These differences were evident in their mathematical competencies and richer students’ autonomy. However, when compared to the traditional teaching environment there were several challenges such as time use, understanding all classmates’ dialogue, mathematical writing ability in explaining and communicating their thinking and more teacher work. Constructivist class discussion classrooms in this study appeared open, relaxed, lively, friendly, and supportive of each other in building new knowledge. This was apparent in School E where the environment provided more opportunities for students to develop their own mathematical ideas. This environment also produced a more social/collective/adaptive form of mathematical knowledge, with ongoing assessment of information provided by the teachers, to inform instructional practices. The data presented here show that students exposed to the constructivist discussion approach had richer learning experiences which may be viewed as a result of their active participation during instruction. Compared to the their peers in School T, the traditional direct instructional group, School E students had more learning roles - (knowledge explorers, knowledge producers, and knowledge adventurers). Student in School T acted mainly as knowledge receivers; they mostly received and followed the teacher’s instruction and explanations of mathematical concepts, and then applied the received procedures to solve given mathematical problems. The findings of the sequential relationship between teachers’ perceptions of mathematics/learning, teaching practice, and students’ knowledge/perceptions sheds new light on the social relationships between teaching and learning and the situated influences among classroom practices and students’ knowledge/competencies/perceptions. This investigation revealed that the constructivist approach seems to be an excellent medium to provide quality education. It is recommended that educators should re-introduce the use of a constructivist approach to teaching Mathematics because of its potential to enhance the quality of Mathematics education, which in turn augments students’ competency as future Mathematicians

Crossing the borders: an eco-cinecritical analysis of "Plant Wars" and "Bird Without Borders"

Este artículo trata sobre dos películas orientadas a la naturaleza desde la perspectiva de la cinecrítica ecológica, una crítica cinematográfica con conciencia ecológica. "Plant Wars", producida por PTS Taiwan, cuestiona la barrera que existe entre plantas nativas y plantas exóticas, así como entre plantas y humanos, para poder reclasificar las especies que se encuentran entre estas demarcaciones. El método narrativo que utiliza consiste en presentar 14 plantas nativas como protagonistas, y pone de manifiesto su relación con los nativos, los granjeros, los colmeneros y con otras especies de insectos. "Bird Without Borders", también producido por PTS, cuenta la historia del viaje migratorio de un grupo de espátulas menores ("Platalea minor") ("black-faced spoonbills") que vuelan 16.000 kilómetros desde Tainan, Taiwán, pasando por Japón, a lo largo de la costa de Corea del Sur y Corea del Norte, hasta una remota isla de China para buscar alimento, y su vuelta a Taiwán para pasar el invierno. Bird muestra diferentes niveles de cruce de fronteras. Estas dos producciones PTS son similares en cuanto a la búsqueda de cruces de frontera para redefinir lo que significa frontera o límite: "Plant" cuestiona la categorización biológica de las plantas nativas y las exóticas, mientras que "Bird" utiliza el ejemplo de la tripulación que sigue físicamente la ruta del vuelo de los pájaros, que cruza fronteras nacionales, para dar una visión de reclasificación del territorio natural mediante el cuestionamiento de la idea de "lugar" fijo. Ambas películas buscan modos de cruzar las fronteras hechas por los hombres: "Plant" lo hace a través de la imaginación medioambiental y "Bird" por medio de movimientos físicos de diferentes tipos. "Plant Wars" parece expresar su posición de modo más fuerte, por medio de "guerras", mientras que "Bird Without Borders" trata de alcanzar una visión sin fronteras.This essay reads two Asian nature-oriented films from the perspective of eco-cinecriticism, an ecologically minded film criticism. “Plant Wars”, produced by PTS Taiwan, questions the demarcation between native plants and exotic plants as well as between plants and human beings so as to remap the in-between border area. The storytelling method it applies is to feature 14 currently native plants as the main characters, and illustrates their relationship to aboriginal people, farmers, beekeepers and to other insect species. “Bird Without Borders”, also produced by PTS, tells the story of the migration journey of a group of black-faced spoonbills flying 16,000 kilometers all the way from Tainan, Taiwan, to Japan, along the border of South and North Korea, to a remote island of China to look for feeding places and then back to Taiwan to spend the winter. Different layers of border-crossing are shown in “Bird”. One similarity of these two PTS productions is the search for border-crossing so as to redefine or revise what the border means. “Plant” questions the biological categorization of the native plants and the exotic plants; “Bird” illustrates how the crew physically follows the route of the birds’ flight which crosses the national borders to provide a vision of remapping the natural territory through challenging the idea of a fixed “place.” Both are seeking for ways to trespass the man-made borders: “Plant” via environmental imagination; “Bird” via some physical movement. While the title of “Plant Wars” seemingly expresses its position more strongly through “wars,” “Bird Without Borders” explicitly asserts a dream of no borders