[2] **viXra:1206.0043 [pdf]**
*replaced on 2012-06-11 04:21:17*

**Authors:** Florentin Smarandache, Victor Vladareanu

**Comments:** 12 Pages.

In this article one proposes several numerical examples for applying the extension set to 2D- and 3D-spaces. While rectangular and prism geometrical figures can easily be decomposed from 2D and 3D into 1D linear problems, similarly for the circle and the sphere, it is not possible in general to do the same for other geometrical figures.

**Category:** Artificial Intelligence

[1] **viXra:1206.0014 [pdf]**
*submitted on 2012-06-04 23:37:54*

**Authors:** Florentin Smarandache

**Comments:** 17 Pages.

Dr. Cai Wen defined in his 1983 paper:
- the distance formula between a point x0 and a one-dimensional (1D) interval ;
- and the dependence function which gives the degree of dependence of a point with respect to
a pair of included 1D-intervals.
His paper inspired us to generalize the Extension Set to two-dimensions, i.e. in plane of real
numbers R2 where one has a rectangle (instead of a segment of line), determined by two
arbitrary points A(a1, a2) and B(b1, b2). And similarly in R3, where one has a prism determined by
two arbitrary points A(a1, a2, a3) and B(b1, b2, b3). We geometrically define the linear and nonlinear
distance between a point and the 2D- and 3D-extension set and the dependent function
for a nest of two included 2D- and 3D-extension sets. Linearly and non-linearly attraction point
principles towards the optimal point are presented as well.
The same procedure can be then used considering, instead of a rectangle, any bounded 2Dsurface
and similarly any bounded 3D-solid, and any bounded n-D-body in Rn.
These generalizations are very important since the Extension Set is generalized from onedimension
to 2, 3 and even n-dimensions, therefore more classes of applications will result in
consequence.
Introduction.

**Category:** Artificial Intelligence