Rank and k-nullity of contact manifolds

We prove that the dimension of the 1-nullity distribution N(1) on a closed Sasakian manifold M of rankl is at least equal to 2l−1 provided that M has an isolated closed characteristic. The result is then used to provide some examples of k-contact manifolds which are not Sasakian. On a closed, 2n+1-dimensional Sasakian manifold of positive bisectional curvature, we show that either the dimension of N(1) is less than or equal to n+1 or N(1) is the entire tangent bundle TM. In the latter case, the Sasakian manifold Mis isometric to a quotient of the Euclidean sphere under a finite group of isometries. We also point out some interactions between k-nullity, Weinstein conjecture, and minimal unit vector fields.

The impact of eliminating extraneous sound and light on students' achievement: An empirical study

The impact of eliminating extraneous sound and light on students’ achievement was investigated under four conditions: Light and Sound controlled, Sound Only controlled, Light Only controlled and neither Light nor Sound controlled. Group, age and gender were the control variables. Four randomly selected groups of high school freshmen students with different backgrounds were the participants in this study. Academic achievement was the dependent variable measured on a pretest, a posttest and a post-posttest, each separated by an interval of 15 days. ANOVA was used to test the various hypotheses related to the impact of eliminating sound and light on student learning. Independent sample T tests on the effect of gender indicated a significant effect while age was non- significant. Follow up analysis indicated that sound and light are not potential sources of extraneous load when tested individually. However, the combined effect of sound and light seems to be a potential source of extrinsic load. The findings revealed that the performance of the Sound and Light controlled group was greater during the posttest and post-posttest. The overall performance of boys was greater than that of girls. Results indicated a significant interaction effect between group and gender on treatment subjects. However gender alone was non-significant. Performance of group by age had no significant interaction and age alone was non-significant in the posttest and post-posttest. Based on the results obtained sound and light combined seemed to be the potential sources of extraneous load in this type of learning environment. This finding supports previous research on the effect of sound and light on learning. The findings of this study show that extraneous sound and light have an impact on learning. These findings can be used to design better learning environments. Such environments can be achieved with different electric lighting and sound systems that provide optimal color rendering, low glare, low flicker, low noise and reverberation. These environments will help people avoid unwanted distraction, drowsiness, and photosensitive behavior.