To confirm no differences preceding entrance into the intersection, we used a repeated-measures analysis of variance (ANOVA) with two factors: Trial Set (two levels: 1 10, 11 20), and Time Bin (10 levels: 1-s increments preceding and following intersection entrance).
In the meanwhile, Ha [3] recently established a minimax theorem ([3], Theorem 3.1), which uses the following whole intersection theorem.
The following whole intersection property for the map-values of a KKM map is a standard form of the KKM type theorems.
above-D-quasi-convex on W if for each (w in W), the set ({u in U: F u) subset w -D}) is a convex subset of U. By definition, the above-D-convex mapping is also an above-naturally D-quasi-convex on U. The following whole intersection theorem is a variant form of Ha [8].
Shop c computes the following set intersections begin{aligned} mathsf{Items}_c &= {mathtt{MP}}_c cap mathsf{EWL}_u mathsf{Items}_mathsf{dir} &= {mathtt{MP}}_c^mathsf{dir}cap mathsf{EWL}_u, end{aligned}for (mathsf{dir}in {N, NE, E, SE, S, SW, W, NW}).
One uses here the following trick: the intersection of a projective variety X with a hypersurface is equal to the intersection of X with a hyperplane, but for a different embedding of X; this trick is used several times, starting with (X = {mathbb {P}}^N).
We have considered the following crossover operators, called intersection operators (denoted CrossRegionCrossRegion, hereafter), which take in input two individuals and produce in output two new individuals (see Algorithm 2).
Given any two approximation sets X i and X j in the approximation space A ij =(U ij,R), we observe the following properties for the intersection of their approximations [7]: A inf(X i )∩A inf(X j ) =A inf(X i ∩X j ) and A sup(X i )∩A sup(X j )⊇A sup(X i ∩X j ).
