Narrow Results

In this paper, we suggest a mathematical representation to the holographic principle through the theory of topological retracts. We found that the topological retraction is the mathematical analog of the hologram idea in modern quantum gravity and it can be used to explore the geometry of the hologram boundary. An example has been given on the five-dimensional (5D) wormhole spacetime $W$ which we found can retract to lower-dimensional circles $S_i\subset W$. In terms of the holographic principle, the description of this volume of spacetime $W$ is encoded on the lower-dimensional circle which is the region boundary.

This study aims to investigate $L^B$-valued GFA from algebraic and topological perspectives, where $L$ stands for residuated lattice and B is a set of propositions about the general fuzzy automata, in which its underlying structure is a complete infinitely distributive lattice. Further, the concepts of $L^B$-valued general fuzzy automata (or simply $L^B$-valued GFA) contractible spaces, $L^B$-valued GFA path homotopy, $L^B$-valued GFA retraction, $L^B$-valued GFA deformation retraction, $L^B$-valued GFA path connected space and $L^B$-valued GFA homotopy equivalent space are introduced and explicated. In addition, $L^B$-valued GFA fundamental groups are proposed and studied. Regarding these issues, some properties are also established and explained.

A method to extract the boundary between the regions in a d-dimensional regular grid is presented. The grid may be a discrete representation of a configuration space in motion planning, and the regions may be the nearest neighborhoods of convex regions into which the obstacle space of the configuration space is partitioned. Then the boundary represents a retraction space of the free space which can be investigated for reachability more efficiently than the whole free space. On the cell complex representation of the boundary which is deduced here, skeletons can be defined which may further reduce the portion of the free space to be searched. From the cell complex, analogous concepts can be derived on the original grid.