Monday, December 31, 2007
Here are three tetrahedron overlaps. The right two differ from the disjoint one by single migrations. The middle one is a point migration since it is a point jumping across a boundary. The rightmost one is a segment migration since it is a segment jumping across another segment.
Saturday, December 8, 2007
Linear Space Equivalence
Two linear spaces are equivalent iff they are isomorphic to the same finite set of nonparallel noncoincidental linear surfaces. Two finite sets of nonparallel noncoincidental linear surfaces are equivalent iff they are isomorphic to the same linear space. A finite set of nonparallel noncoincidental linear surfaces are said to be equivalent to a linear space iff they are isomorphic.
Example Migration Theorem
Without loss of generality, assume
and its opposite is not empty in
There exists continuous
As a set of linear surfaces changes continuously, its regions cannot suddenly become empty. Before becoming empty, a region must become infinitesimally small. As a region becomes infinitesimally small, so does its opposite. Thus, as a region becomes empty, its opposite becomes nonempty.
A simplex overlap is two disjoint sets of boundaries, each of which is a simplex. The inside of a simplex overlap