Remark on the Limit Case of Positive Mass Theorem for Manifolds with Inner Boundary

In [5] Herzlich proved a new positive mass theorem for Riemannian 3-manifolds $(N, g)$ whose mean curvature of the boundary allows some positivity. In this paper we study what happens to the limit case of the theorem when, at a point of the boundary, the smallest positive eigenvalue of the Dirac operator of the boundary is strictly larger than one-half of the mean curvature (in this case the mass $m(g)$ must be strictly positive). We prove that the mass is bounded from below by a positive constant $c(g), m(g) \geq c(g)$, and the equality $m(g) = c(g)$ holds only if, outside a compact set, $(N, g)$ is conformally flat and the scalar curvature vanishes. The constant $c(g)$ is uniquely determined by the metric $g$ via a Dirac-harmonic spinor.Comment: 12 pages, latex2

Some extensions of the Einstein-Dirac equation

We considered an extension of the standard functional for the Einstein-Dirac equation where the Dirac operator is replaced by the square of the Dirac operator and a real parameter controlling the length of spinors is introduced. For one distinguished value of the parameter, the resulting Euler-Lagrange equations provide a new type of Einstein-Dirac coupling. We establish a special method for constructing global smooth solutions of a newly derived Einstein-Dirac system called the {\it CL-Einstein-Dirac equation of type II} (see Definition 3.1).Comment: 21page

Switching voltage of micro-beam actuators - Analysis, fabrication and test

This thesis introduces an analysis, fabrication and testing of a //-cantilever. First, the theoretical method used to analyze the deflection of a micro-beam with two supports and switching voltage is introduced. The results are compared to an I-DEAS simulation. Second, the theoretical approach is modified in order to be suitable for the actual //- cantilever beam model because the material, geometry and the type of structure have been changed. The switching voltage of each beam size is predicted using the modified method. Third, a series of single //-cantilever beams are designed according to the MEMS design rules. Using the facilities at Semiconductor and Microsystems Fabrication Laboratory at RIT, the //-beams are fabricated and the switching voltages of each beam are measured. The detailed explanation of each fabrication step, measurement and testing method are described in this thesis. Finally, the results from the theoretical approach are compared to those from testing. Testing results show that for varying the beam dimensions, the trend in predicted switching voltage is similar to the experimental data. As the beam gets longer, a stronger sticking effect is present. From the testing of the micro-devices, it was found that the beam width does not influence the switching voltage very much. The two beams with the same length showed the almost same switching voltage even though the width is different from each other. So, the minimum width of //-beam can be an optimized design to minimize the use of wafer. Large //-beams (more than 100 //m width and 1000 //m length) are not recommended. Considering the etching process and testing convenience for the large //-beams, a much larger space pad for probing is required. The //-beam is deflected and contacted to the wafer by applying the switching voltage and recovered by removing the voltage applied in both theory and experiment. This single cantilever //-beam, therefore, can be used as an actuator

Film directors as hubs: how do the social networks of film crews interact with the spatial networks of physical locations?

Working relationships in Hollywood's film industry are some of the most tightly organized, and yet understudied, relationships in contemporary society. In this paper we study the linkages across the film industry by focusing on Hollywood directors as hubs in a network, with particular focus on the inhomogeneous networks that establish connectivity and dominance in a few particular cities (L.A., N.Y.). To date, no one has used a technique that directly connects individuals across space and empirically follows the social networks of film industry crews across major economic hubs. Overall, our analysis uses a dataset from the website imdb (including 177 movie titles, 25 film roles, and 11 directors) to study this social group and its network structures. In the end, our research makes clear that spatial proximity is crucial to cultivating creativity and innovation in the film industry, which further strengthens social and spatial networks. These findings prompt us to provide specific recommendations for the planning of urban spaces to further foster growth in this industry