4,819 research outputs found
The Alternative of Sensor Placement in Multi-Story Buildings through the Metric Dimension Approach: A Representation of Generalized Petersen Graphs
In a public facility or private office where many people can get together, a fire detection device is a mandatory tool as an emergency alarm in the facility. However, the expense of the installation of the device is a troublesome matter. So, optimization is needed to minimize the number of these devices. The way to implement is to select the appropriate position to place the devices in public facilities. The research discussed the placement of the sensors in multi-story buildings. The multi-story buildings could be represented as cube composition graphs with the number of rooms, and the connectivity between the floor and its rooms was equal. The concept of this multi-story building was modeled into a generalized Petersen graph where a vertex represented a room, and an edge was the connectivity of rooms. The basis obtained on that metric dimension was represented as a sensor placed on the building. Then, the optimization of device placement was seen as determining the metric dimensions of the Petersen graph. In the research, the alternative sensor placements were computed using the graph metric dimension approach implemented in Python. The research successfully implements the metric dimension of  to  using Python code to obtain the alternative of its basis. A basic alternative indicates the location of the device placement like fire detectors, network access points, or other sensors inside a building
Curvature based triangulation of metric measure spaces
We prove that a Ricci curvature based method of triangulation of compact
Riemannian manifolds, due to Grove and Petersen, extends to the context of
weighted Riemannian manifolds and more general metric measure spaces. In both
cases the role of the lower bound on Ricci curvature is replaced by the
curvature-dimension condition . We show also that for weighted
Riemannian manifolds the triangulation can be improved to become a thick one
and that, in consequence, such manifolds admit weight-sensitive
quasimeromorphic mappings. An application of this last result to information
manifolds is considered.
Further more, we extend to weak spaces the results of Kanai
regarding the discretization of manifolds, and show that the volume growth of
such a space is the same as that of any of its discretizations.Comment: 24 pages, submitted for publicatio
Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space
The rigidity of the Positive Mass Theorem states that the only complete
asymptotically flat manifold of nonnegative scalar curvature and zero mass is
Euclidean space. We study the stability of this statement for spaces that can
be realized as graphical hypersurfaces in Euclidean space. We prove (under
certain technical hypotheses) that if a sequence of complete asymptotically
flat graphs of nonnegative scalar curvature has mass approaching zero, then the
sequence must converge to Euclidean space in the pointed intrinsic flat sense.
The appendix includes a new Gromov-Hausdorff and intrinsic flat compactness
theorem for sequences of metric spaces with uniform Lipschitz bounds on their
metrics.Comment: 31 pages, 2 figures, v2: to appear in Crelle's Journal, many minor
changes, one new exampl
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