Generalizations of the distance and dependent function in extenics to 2D, 3D, and n--D.
1 IntroductionExtension Theory (or Extenics) was developed by Professor Cai Wen in 1983 by publishing a paper called Extension Set and Non-Compatible Problems. Its goal is to solve contradictory problems and also nonconventional, nontraditional ideas in many fields. Extenics is at the confluence of three disciplines: philosophy, mathematics, and engineering. A contradictory problem is converted by a transformation function into a non-contradictory one. The functions of transformation are: extension, decomposition, combination, etc. Extenics has many practical applications in Management, DecisionMaking, Strategic Planning, Methodology, Data Mining, Artificial Intelligence, Information Systems, Control Theory, etc. Extenics is based on matter-element, affair-element, and relation-element.
2 Extension Distance in 1 D-space
Let's use the notation <a, b> for any kind of closed, open, or half-closed interval [a, b], (a, b), (a, b], [a, b). Prof. Cai Wen has defined the extension distance between a point [x.sub.0] and a real interval X = <a, b>, by
[rho]([x.sub.0], X) = [absolute value of [x.sub.0] a + b/2] - b - a/2, (1)
where in general:
[rho]: (R,[R.sub.2]) [right arrow] (-[infinity], +[infinity]). (2)
Algebraically studying this extension distance, we find that actually the range of it is:
[rho]([x.sub.0],X) [member of] [- b - a/2+[infinity]] (3)
or its minimum range value - (b-a/2) depends on the interval A' extremities a and b, and it occurs when the point [x.sub.0] coincides with the midpoint of the interval X, i.e. [x.sub.0] = a+b/2, The closer is the interior point [x.sub.0] to the midpoint of the interval <a, b>, the negatively larger is [rho] ([x.sub.0], X).
In Fig. 1, for interior point [x.sub.0] between a and the extension distance [rho] ([x.sub.0], X) = a - [x.sub.0] is the negative length of the brown line segment [left side]. Whereas for interior point [x.sub.0] between a+b/2 and b, the extension distance p([x.sub.0],X) = [x.sub.0] - b is the negative length of the blue line segment [right side]. Similarly, the further is exterior point [x.sub.0] with respect to the closest extremity of the interval <a, b> to it (i.e. To either a or b), the positively larger is p ([x.sub.0], X).
In Fig. 2, for exterior point [x.sub.0]<a, the extension distance [rho] ([x.sub.0], X) = a - [x.sub.0] is the positive length of the brown line segment [left side]. Whereas for exterior point [x.sub.0]>b, the extension distance [rho] ([x.sub.0], X) = [x.sub.0] - b is the positive length of the blue line segment [right side].
3 Principle of the Extension 1 D-Distance
Geometrically studying this extension distance, we find the following principle that Prof. Cai Wen has used in 1983 defining it:
[rho] ([x.sub.0], X) is the geometric distance between the point [x.sub.0] and the closest extremity point of the interval <a, b > to it (going in the direction that connects [x.sub.0] with the optimal point), distance taken as negative if [x.sub.0] [member of] <a, b>, and as positive if [x.sub.0] [subset] <a, b >.
This principle is very important in order to generalize the extension distance from 1D to 2D (two-dimensional real space), 3D (three-dimensional real space), and n -D (n-dimensional real space).
The extremity points of interval < a, b> are the point a and b, which are also the boundary (frontier) of the interval < a, b>.
4 Dependent Function in 1 D-Space
Prof. Cai Wen defined in 1983 in 1D the Dependent Function K(y). If one considers two intervals [X.sub.0] and X, that have no common end point, and [X.sub.0] [subset] X, then:
K(y) = [rho](y,X)/ [rho](y,X) - [rho] (y, [X.sub.0])
Since K(y) was constructed in 1D in terms of the extension distance [rho] (.,.), we simply generalize it to higher dimensions by replacing [rho] (.,.) with the generalized in a higher dimension.
5 Extension Distance in 2D-Space
Instead of considering a segment of line AB representing the interval <a, b> in 1R, we consider a rectangle AMBN representing all points of its surface in 2D. Similarly as for 1Dspace, the rectangle in 2D-space may be closed (i.e. all points lying on its frontier belong to it), open (i.e. no point lying on its frontier belong to it), or partially closed (i.e. some points lying on its frontier belong to it, while other points lying on its frontier do not belong to it).
Let's consider two arbitrary points A([a.sub.1], [a.sub.2]) and B([b.sub.1], [b.sub.2]). Through the points A and B one draws parallels to the axes of the Cartesian system XY and one thus one forms a rectangle AMBN whose one of the diagonals is just AB.
Let's note by O the midpoint of the diagonal AB, but O is also the center of symmetry (intersection of the diagonals) of the rectangle AMBN. Then one computes the distance between a point P ([x.sub.0], [y.sub.0]) and the rectangle AMBN. One can do that following the same principle as Dr. Cai Wen did:
--compute the distance in 2D (two dimensions) between the point P and the center O of the rectangle (intersection of rectangle's diagonals);
--next compute the distance between the point P and the closest point (let's note it by P') to it on the frontier (the rectangle's four edges) of the rectangle AMBN.
This step can be done in the following way: considering P' as the intersection point between the line PO and the frontier of the rectangle, and taken among the intersection points that point P which is the closest to P; this case is entirely consistent with Dr. Cai's approach in the sense that when reducing from a 2D-space problem to two LD-space problems, one exactly gets his result.
The Extension 2D-Distance, for P [not equal to] O, will be:
[rho] (([x.sub.0],[y.sub.0]), AMBN) = d (point P, rectangle AMBN) =
= [absolute value of PO] - [absolute value of PO] = [+ or -][absolute value of PP'], (5)
i) which is equal to the negative length of the red segment [absolute value of PP] in Fig. 3, when P is interior to the rectangle
AMBN;
ii) or equal to zero, when P lies on the frontier of the rectangle AMBN (i.e. on edges AM, MB, BN, or NA) since P coincides with P';
iii) or equal to the positive length of the blue segment [absolute value of PP'] in Fig. 4, when P is exterior to the rectangle AMBN, where [absolute value of PO] means the classical 2D-distance between the point P and O, and similarly for [absolute value of P'O] and [absolute value of PP'].
The Extension 2D-Distance, for the optimal point, i.e. P = O, will be
[rho] (O, AMBN) = d(point O, rectangle AMBN) =
= - max d (point O, point M on the frontier of AMBN. (6)
The last step is to devise the Dependent Function in 2D-space similarly as Dr. Cai's defined the dependent function in 1D. The midpoint (or center of symmetry) O has the coordinates
O([a.sub.1] + [b.sub.1]/2, [a.sub.2] + [b.sub.2]/2) (7)
Let's compute the
[absolute value of PO] - [absolute value of P'O]. (8)
In this case, we extend the line OP to intersect the frontier of the rectangle AMBN. P' is closer to P than P", therefore we consider P'. The equation of the line PO, that of course passes through the points P ([x.sub.0], [y.sub.0]) and O ([a.sub.1] + [b.sub.1]/2, [a.sub.2] + [b.sub.2]/2), is:
y - [y.sub.0] = ([a.sub.2] + [b.sub.2]/2-[y.sub.0]/ [a.sub.1] + [b.sub.1]/2) - [x.sub.0] (x - [x.sub.0])(9)
Since the x-coordinate of point P' is [a.sub.1] because P' lies on the rectangle's edge AM, one gets the y-coordinate of point P' by a simple substitution of [x.sub.P], = [a.sub.1] into the above equality:
[y.sub.P'] = [y.sub.0] + [a.sub.2] + [b.sub.2] - 2[y.sub.0]/[a.sub.1] + [b.sub.1] - 2[x.sub.0]([a.sub.1] - [x.sub.0]) (10)
Therefore P' has the coordinates
P'[[x.sub.P'] = [a.sub.1], [y.sub.P], = [y.sub.0] +[a.sub.2] + [b.sub.2] - 2[y.sub.0]/[a.sub.1] + [b.sub.1] - 2[x.sub.0] ([a.sub.1] - [x.sub.0])]. (11)
The distance
d(PQ) = [absolute value of PQ] = [square root of [([x.sub.0] - [a.sub.1] + [b.sub.1]/2).sup.2] + [([y.sub.0] - [a.sub.2] + [b.sub.2]/2).sup.2] (12)
while the distance
d(P',Q) = [absolute value of P'Q] = [square root of [([a.sub.1] - [a.sub.1] + [b.sub.1]/2).sup.2] + [([y.sub.P'] - [a.sub.2] + [b.sub.2]/2).sup.2] [square root of [([a.sub.1] + [b.sub.1]/2).sup.2] + [([y.sub.P'] - [a.sub.2] + [b.sub.2]/2).sup.2]
Also, the distance
d(PP') = [absolute value PP'] = [square root of [([a.sub.1] - [x.sub.0]).sup.2] + [([y.sub.P'] - [y.sub.0]).sup.2]] (14)
Whence the Extension 2D-distance formula
[rho][([x.sub.0], [y.sub.0]), AMBN] = = d[P([x.sub.0], [y.sub.0]), A([a.sub.1], [a.sub.2])MB([b.sub.1], [b.sub.2])N] = = [absolute value PO] - [absolute value P'Q] (15)
= [square root of [([x.sub.0] - [a.sub.1] + [b.sub.1]/2).sup.2] + [([y.sub.0] - [a.sub.2] + [b.sub.2]/2).sup.2]] [square root of [([a.sub.1] + [b.sub.1]/2).sup.2] + [([y.sub.P'] - [a.sub.2] + [b.sub.2]/2).sup.2] (16)
= [+ or -][absolute value PP'] (17)
= [+ or -][square root of [([a.sub.1] - [x.sub.0]).sup.2] + [([y.sub.P'] - [y.sub.0]).sup.2], (18)
where
[y.sub.P'] = [y.sub.0] + [a.sub.2] + [b.sub.2] - 2[y.sub.0]/ [a.sub.2] + [b.sub.2] - 2[x.sub.0] ([a.sub.1] - [x.sub.0]) (19)
6 Properties
As for 1 D-distance, the following properties hold in 2D:
6.1 Property 1
a) (x, y) [member of] Int(AMBN) if [rho] [(x, y),AMBN] < 0, where Int (AMBN) means interior of AMBN;
b) (x, y) [member of] Fr(AMBN) if [rho] [(x, y), AMBN] = 0, where Fr (AMBN) means frontier of AMBN;
c) (x, y) [not member of] AMBN if [rho] [(x, y), AMBN] > 0.
6.2 Property 2
Let [A.sub.0][M.sub.0][B.sub.0][N.sub.0] and AMBN be two rectangles whose sides are parallel to the axes of the Cartesian system of coordinates, such that they have no common end points, and [A.sub.0][M.sub.0] [B.sub.0][N.sub.0] [subset] AMBN. We assume they have the same optimal points [O.sub.1] = [O.sub.2] = O located in the center of symmetry of the two rectangles. Then for any point (x, y) [subset] [R.sup.2] one has [rho] [(x, y),[A.sub.0][M.sub.0][B.sub.0][N.sub.0]] > [rho] [(x, y),AMBN]. See Fig. 5.
7 Dependent 2 D-Function
Let [A.sub.0][M.sub.0][B.sub.0][N.sub.0] and AMBN be two rectangles whose sides are parallel to the axes of the Cartesian system of coordinates, such that they have no common end points, and [A.sub.0][M.sub.0] [B.sub.0][N.sub.0] [subset] AMBN.
The Dependent 2D-Function formula is:
[K.sub.2D(x,y)] = [rho][(x,y),AMBN] [rho][(x,y),AMBN,]-[rho][{x,y),[A.sub.0][M.sub.0][B.sub.0][N.sub.0]] (20)
7.1 Property 3
Again, similarly to the Dependent Function in 1 D-space, one has:
a) If (x, y) [member of] Int([A.sub.0][M.sub.0] [B.sub.0][N.sub.0]), then [K.sub.2D(x,y)] > 1;
b) If (x,y) [member of] Fr ([A.sub.0][M.sub.0][B.sub.0][N.sub.0]), then [K.sub.2D(x,y)] = 1;
c) If (x, y) [member of] Int (AMBN-[A.sub.0][M.sub.0][B.sub.0][N.sub.0]), then 0 < [K.sub.2D(x,y)] < 1;
d) If (x, y) [member of] Fr (AMBN), then [K.sub.2D(x,y)] = 0;
e) If (x, y) [not member of] AMBN, then K2D(x, y) < 0.
8 General Case in 2D-Space
One can replace the rectangles by any finite surfaces, bounded by closed curves in 2D-space, and one can consider any optimal point O (not necessarily the symmetry center). Again, we assume the optimal points are the same for this nest of two surfaces. See Fig. 6.
9 Linear Attraction Point Principle
We introduce the Attraction Point Principle, which is the following:
Let S be a given set in the universe of discourse U, and the optimal point O [subset] S. Then each point P ([x.sub.1], [x.sub.2],..., [x.sub.n]) from the universe of discourse tends towards, or is attracted by, the optimal point O, because the optimal point O is an ideal of each point. That's why one computes the extension (n -D)-distance between the point P and the set S as [rho] [([x.sub.1], [x.sub.2],..., [x.sub.n]), S] on the direction determined by the point P and the optimal point O, or on the line PO, i.e.:
a) [rho] [([x.sub.1], [x.sub.2],..., [x.sub.n]), S] is the negative distance between P and the set frontier, if P is inside the set S;
b) [rho] [([x.sub.1], [x.sub.2],..., [x.sub.n]), S ] = 0, if P lies on the frontier of the set S;
c) [rho] [([x.sub.1], [x.sub.2],..., [x.sub.n]), S] is the positive distance between P and the set frontier, if P is outside the set.
It is a king of convergence/attraction of each point towards the optimal point. There are classes of examples where such attraction point principle works. If this principle is good in all cases, then there is no need to take into consideration the center of symmetry of the set S, since for example if we have a 2D piece which has heterogeneous material density, then its center of weight (barycenter) is different from the center of symmetry. Let's see below such example in the 2D-space: Fig. 7.
10 Remark 1
Another possible way, for computing the distance between the point P and the closest point P' to it on the frontier (the rectangle's four edges) of the rectangle AMBN, would be by drawing a perpendicular (or a geodesic) from P onto the closest rectangle's edge, and denoting by P' the intersection between the perpendicular (geodesic) and the rectangle's edge. And similarly if one has an arbitrary set S in the 2Dspace, bounded by a closed urve. One computes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
as in the classical mathematics.
11 Extension Distance in 3D-Space
We further generalize to 3D-space the Extension Set and the Dependent Function. Assume we have two points ([a.sub.1], [a.sub.2], [a.sub.3]) and ([b.sub.1], [b.sub.2], [b.sub.3]) in D. Drawing through A end B parallel planes to the planes' axes (XY,XZ, YZ) in the Cartesian system XYZ we get a prism A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3] (with eight vertices) whose one of the transversal diagonals is just the line segment AB. Let's note by O the midpoint of the transverse diagonal AB, but O is also the center of symmetry of the prism.
Therefore, from the line segment AB in iD-space, to a rectangle AMBN in 2D-space, and now to a prism A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3] in 3D-space. Similarly to 1D- and 2Dspace, the prism may be closed (i.e. all points lying on its frontier belong to it), open (i.e. no point lying on its frontier belong to it), or partially closed (i.e. some points lying on its frontier belong to it, while other points lying on its frontier do not belong to it).
Then one computes the distance between a point P ([x.sub.0], [y.sub.0], [z.sub.0]) and the prism A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2] [N.sub.3]. One can do that following the same principle as Dr. Cai's:
--compute the distance in 3D (two dimensions) between the point P and the center O of the prism (intersection of prism's transverse diagonals);
--next compute the distance between the point P and the closest point (let's note it by P') to it on the frontier of the prism A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3] (the prism's lateral surface); considering P' as the intersection point between the line OP and the frontier of the prism, and taken among the intersection points that point P which is the closest to P; this case is entirely consistent with Dr. Cai's approach in the sense that when reducing from 3D-space to 1D-space one gets exactly Dr. Cai's result;
--the Extension 3D-Distance d(P, A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3]) is d(P, A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3]) = [absolute value of PO] - [absolute value of PO] = [+ or -][absolute value of PP'[absolute value of, where |PO| means the classical distance in 3D-space between the point P and O, and similarly for [absolute value of P'O] and [absolute value of PP']. See Fig. 8.
12 Property 4
a) (x,y, z) [member of] Int(A[M.sub.1] [M.sub.2] [M.sub.3] B[N.sub.1] [N.sub.2] [N.sub.3]) if [rho] [(x,y, z), A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3]] < 0, where Int(A[M.sub.1] [M.sub.2][M.sub.3] B[N.sub.1][N.sub.2][N.sub.3]) means interior of A[M.sub.1] [M.sub.2][M.sub.3] B[N.sub.1][N.sub.2] [N.sub.3];
b) (x,y,z) [member of] Fr(A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3]) if [rho] [(x, y, z), A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3]] = 0 means frontier of A[M.sub.1] [M.sub.2][M.sub.3] B[N.sub.1] [N.sub.2][N.sub.3];
c) (x, y, z) [not member of] A[M.sub.1] [M.sub.2] [M.sub.3] B[N.sub.1] [N.sub.2] [N.sub.3] if [rho] [(x, y, z), A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3]] > 0.
13 Property 5
Let [A.sub.0][M.sub.01][M.sub.02][M.sub.03] [B.sub.0][N.sub.01][N.sub.02] [N.sub.03] and A[M.sub.1] [M.sub.2][M.sub.3] B[N.sub.1][N.sub.2][N.sub.3] be two prisms whose sides are parallel to the axes of the Cartesian system of coordinates, such that they have no common end points, and [A.sub.0][M.sub.01][M.sub.02][M.sub.03][B.sub.0][N.sub.01][N.sub.02][N.sub.03] [subset] A[M.sub.1] [M.sub.2][M.sub.3] B[N.sub.1][N.sub.2][N.sub.3]. We assume they have the same optimal points [O.sub.1] [equivalent to] [O.sub.2] [equivalent to] O located in the center of symmetry of the two prisms.
Then for any point (x, y, z) [member of] [R.sup.3] one has
[rho] [[(x, y, z), [A.sub.0][M.sub.01][M.sub.02][M.sub.03][B.sub.0][N.sub.01][N.sub.02]N].sub.03] [greater than or equal to] [ho] [(x, y,z)A[M.sub.1] [M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3]].
14 The Dependent 3D-Function
The last step is to devise the Dependent Function in 3D-space similarly to Dr. Cai's definition of the dependent function in 1D-space. Let the prisms [A.sub.0][M.sub.01][M.sub.02][M.sub.03][B.sub.0][N.sub.01][N.sub.02][N.sub.03] and A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3] be two prisms whose faces are parallel to the axes of the Cartesian system of coordinates XYZ, such that they have no common end points in such a way that [A.sub.0][M.sub.01][M.sub.02] [M.sub.03] [B.sub.0] [N.sub.01] [N.sub.02] [N.sub.03] [subset] A[M.sub.1] [M.sub.2] [M.sub.3] B[N.sub.1][N.sub.2][N.sub.3]. We assume they have the same optimal points [O.sub.1] [equivalent to] [O.sub.2] [equivalent to] O located in the center of symmetry of these two prisms.
The Dependent 3D-Function formula is:
[K.sub.3D(x,y,z)] = ([rho] [(x, y, z), A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3]]) X X ([rho] [(x, y,z),A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3],] - [rho] [(x, y,z), [[A.sub.0][M.sub.01][M.sub.02][M.sub.03]B[N.sub.01][N.sub.02][N.sub.03]] . (22)
15 Property 6
Again, similarly to the Dependent Function in 1D- and 2D-spaces, one has:
a) If (x, y, z) [member of] Int([A.sub.0] [M.sub.01] [M.sub.02] [M.sub.03] [B.sub.0] [N.sub.01] [N.sub.02] [N.sub.03]), then [K.sub.3D] (x, y, z) > 1;
b) If (x, y,z) [member of] Fr([A.sub.0] [M.sub.01] [M.sub.02] [M.sub.03] [B.sub.0] [N.sub.01] [N.sub.02] [N.sub.03]), then [K.sub.3D] (x, y, z) = 1;
c) If (x, y, z) [member of] Int(A[M.sub.1] [M.sub.2] [M.sub.3] B[N.sub.1][N.sub.2] [N.sub.3]- [A.sub.0] [M.sub.01] [M.sub.02] [M.sub.03] [B.sub.0] [N.sub.01] [N.sub.02] [N.sub.03]), then O < [K.sub.3D] (x, y, z) < 1;
d) If (x, y, z) [member of] Fr (A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3]), then [K.sub.3D] (x, y, z) = 0;
e) If (x, y,z) [not member of] A[M.sub.1][M.sub.2][M.sub.3]B[N.sub.1][N.sub.2][N.sub.3], then [K.sub.3D] (x, y, z) < 0.
16 General Case in 3D-Space
One can replace the prisms by any finite 3D-bodies, bounded by closed surfaces, and one considers any optimal point O (not necessarily the centers of surfaces' symmetry). Again, we assume the optimal points are the same for this nest of two 3D-bodies.
17 Remark 2
Another possible way, for computing the distance between the point P and the closest point P' to it on the frontier (lateral surface) of the prism AMX[M.sub.2][M.sub.3]BNj[N.sub.2][N.sub.3] is by drawing a perpendicular (or a geodesic) from P onto the closest prism's face, and denoting by P' the intersection between the perpendicular (geodesic) and the prism's face.
And similarly if one has an arbitrary finite body B in the 3D-space, bounded by surfaces. One computes as in classical mathematics:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
18 Linear Attraction Point Principle in 3D-Space
19 Non-Linear Attraction Point Principle in 3D-Space, and in (n -D)-Space
There might be spaces where the attraction phenomena undergo not linearly by upon some specific non-linear curves. Let's see below such example for points [P.sub.i] whose trajectories of attraction towards the optimal point follow some nonlinear 3D-curves.
20 (n - D)-Space
In general, in a universe of discourse U, let's have an (n -D)-set S and a point P. Then the Extension Linear (n -D)-Distance between point P and set S, is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
where O is the optimal point (or linearly attraction point); d(P, P') means the classical linearly (n -D)-distance between two points P and P'; Fr(S) means the frontier of set S; and [absolute value of OP'] means the line segment between the points O and P' (the extremity points O and P' included), therefore P [member of] [absolute value of OP'] means that P lies on the line OP', in between the points O and P'.
For P coinciding with O, one defined the distance between the optimal point O and the set S as the negatively maximum distance (to be in concordance with the 1D-definition).
And the Extension Non-Linear (n - D)-Distance between point P and set S, is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)
where means the extension distance as measured along the curve c; O is the optimal point (or non-linearly attraction point); the points are attracting by the optimal point on trajectories described by an injective curve c; dc (P, P) means the non-linearly (n -D)-distance between two points P and P', or the arc length of the curve c between the points P and P'; Fr (S) means the frontier of set S; and c (OP') means the curve segment between the points O and P (the extremity points O and P included), therefore P [member of] (OP') means that P lies on the curve c in between the points O and P'.
For P coinciding with O, one defined the distance between the optimal point O and the set S as the negatively maximum curvilinear distance (to be in concordance with the 1D-definition).
In general, in a universe of discourse U, let's have a nest of two (n - D)-sets, [S.sub.1] [subset] [S.sub.2], with no common end points, and a point P. Then the Extension Linear Dependent (n -D)Function referring to the point P ([x.sub.1], [x.sub.2],..., [x.sub.n]) is:
[K.sub.nD](P) = [rho](P,[S.sub.2])/[rho](P,[S.sub.2]) - [rho](P,[S.sub.1]), (26)
where is the previous extension linear (n -D)-distance between the point P and the (n - D)-set [S.sub.2].
And the Extension Non-Linear Dependent (n -D)-Function referring to point P ([x.sub.1], [x.sub.2],..., [x.sub.n]) along the curve c is:
[K.sub.nD](P) = [[rho].sub.c](P,[S.sub.2])/[[rho].sub.c](P,[S.sub.2]) - [[rho].sub.c](P,[S.sub.1]), (27)
where is the previous extension non-linear (n -D)-distance between the point P and the (n - D)-set [S.sub.2] along the curve c.
21 Remark 3
Particular cases of curves c could be interesting to studying, for example if c are parabolas, or have elliptic forms, or arcs of circle, etc. Especially considering the geodesics would be for many practical applications. Tremendous number of applications of Extenics could follow in all domains where attraction points would exist; these attraction points could be in physics (for example, the earth center is an attraction point), economics (attraction towards a specific product), sociology (for example attraction towards a specific life style), etc.
22 Conclusion
In this paper we introduced the Linear and Non-Linear Attraction Point Principle, which is the following:
Let S be an arbitrary set in the universe of discourse U of any dimension, and the optimal point O [member of] S. Then each point P ([x.sub.1], [x.sub.2],..., [x.sub.n]), n [greater than or equal to] 1, from the universe of discourse (linearly or non-linearly) tends towards, or is attracted by, the optimal point O, because the optimal point O is an ideal of each point.
It is a king of convergence/attraction of each point towards the optimal point. There are classes of examples and applications where such attraction point principle may apply.
If this principle is good in all cases, then there is no need to take into consideration the center of symmetry of the set S, since for example if we have a 2D factory piece which has heterogeneous material density, then its center of weight (barycenter) is different from the center of symmetry.
Then we generalized in the track of Cai Wen's idea to extend 1D-set to an extension (n -D)-set, and thus defined the Linear (or Non-Linear) Extension (n -D)-Distance between a point P ([x.sub.1], [x.sub.2],..., [x.sub.n]) and the (n -D)-set S as [rho] [([x.sub.1], [x.sub.2], ..., [x.sub.n]), S] on the linear (or non-linear) direction determined by the point P and the optimal point O (the line PO, or respectively the curvilinear PO) in the following way:
1) [rho] [([x.sub.1], [x.sub.2],..., [x.sub.n]), S] is the negative distance between P and the set frontier, if P is inside the set S;
2) [rho] [([x.sub.1], [x.sub.2],..., [x.sub.n]), S ] = 0, if P lies on the frontier of the set S;
3) [rho] [([x.sub.1], [x.sub.2],..., [x.sub.n]), S] is the positive distance between P and the set frontier, if P is outside the set.
We got the following properties:
4) It is obvious from the above definition of the extension (n - D)-distance between a point P in the universe of discourse and the extension (n -D)-set S that:
i) Point P ([x.sub.1], [x.sub.2],..., [x.sub.n]) [member of] Int (S) if [ho][([x.sub.1], [x.sub.2],..., [x.sub.n]),S] < 0;
ii) Point P ([x.sub.1], [x.sub.2],..., [x.sub.n]) [member of] Fr (S) if [rho] [([x.sub.1], [x.sub.2],..., [x.sub.n]), S ] = 0;
iii) Point P ([x.sub.1], [x.sub.2],..., [x.sub.n]) [not member of] S if [rho] [([x.sub.1], [x.sub.2],..., [x.sub.n]), S ] > 0.
5) Let [S.sub.1] and [S.sub.2] be two extension sets, in the universe of discourse U, such that they have no common end points, and [S.sub.1] [subset] [S.sub.2]. We assume they have the same optimal points [O.sub.1] [equivalent to] [O.sub.2] [equivalent to] O located in their center of symmetry. Then for any point P ([x.sub.1], [x.sub.2],..., [x.sub.n]) [member of] U one has:
[rho] [([x.sub.1], [x.sub.2],..., [x.sub.n]), [S.sub.2]] [greater than or equal to] [rho] [([x.sub.1], [x.sub.2],..., [x.sub.n]), [S.sub.1]] . (28)
Then we proceed to the generalization of the dependent function from 1D-space to Linear (or Non-Linear) (n -D)space Dependent Function, using the previous notations.
The Linear (or Non-Linear) Dependent (n -D)-Function of point P ([x.sub.1], [x.sub.2],..., [x.sub.n]) along the curve c, is:
[K.sub.nD]([x.sub.1], [x.sub.2],..., [x.sub.n]) = ([[rho].sub.c][([x.sub.1], [x.sub.2],..., [x.sub.n]), [S.sub.2]]) X
x ([[rho].sub.c][([x.sub.1], [x.sub.2],..., [x.sub.n]), [S.sub.2]] - [[rho].sub.c][([x.sub.1], [x.sub.2],..., [x.sub.n]), [S.sub.1]]) (29)
(where c may be a curve or even a line) which has the following property:
6) If point P ([x.sub.1], [x.sub.2],..., [x.sub.n]) [member of] Int ([S.sub.1]), then [K.sub.nD] ([x.sub.1], [x.sub.2],..., [x.sub.n]) > 1;
7) If point P ([x.sub.1], [x.sub.2],..., [x.sub.n]) [member of] Fr([S.sub.1]), then [K.sub.nD] ([x.sub.1], [x.sub.2],..., [x.sub.n]) = 1;
8) If point P ([x.sub.1], [x.sub.2],..., [x.sub.n]) [member of] Int([S.sub.2] - [S.sub.1]), then [K.sub.nD]([x.sub.1], [x.sub.2],..., [x.sub.n]) [member of] (0,1);
9) If point P ([x.sub.1], [x.sub.2],..., [x.sub.n]) [member of] Int ([S.sub.2]), then [K.sub.nD] ([x.sub.1], [x.sub.2],..., [x.sub.n]) = 0;
10) If point P ([x.sub.1], [x.sub.2],..., [x.sub.n]) [not equal to] Int([S.sub.2]), then [K.sub.nD]([x.sub.1], [x.sub.2],..., [x.sub.n]) < 0.
Submitted on July 15, 2012 / Accepted on July 18, 2012
Florentin Smarandache
University of New Mexico, Mathematics and Science Department, 705 Gurley Ave., Gallup, NM 87301, USA
E-mail: smarand@unm.edu
References
[1.] Cai Wen. Extension Set and Non-Compatible Problems. Journal of Scientific Exploration, 1983, no. 1, 83-97; Cai Wen. Extension Set and Non-Compatible Problems. In: Advances in Applied Mathematics and Mechanics in China. International Academic Publishers, Beijing, 1990, 1-21.
[2.] Cai Wen. Extension theory and its application. Chinese Science Bulletin, 1999, v. 44, no. 7, 673-682. Cai Wen. Extension theory and its application. Chinese Science Bulletin, 1999, v. 44, no. 17, 1538-1548.
[3.] Yang Chunyan and Cai Wen. Extension Engineering. Public Library of Science, Beijing, 2007.
[4.] Wu Wenjun et al. Research on Extension Theory and Its Application. Expert Opinion. 2004, http://web.gdut.edu.cn/extenics/jianding.htm
[5.] Xiangshan Science Conferences Office. Scientific Significance and Future Development of Extenics--No. 271 Academic Discussion of Xiangshan Science Conferences, Brief Report of Xiangshan Science Conferences, Period 260, 2006, 1.
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| Author: | Smarandache, Florentin |
|---|---|
| Publication: | Progress in Physics |
| Date: | Jul 1, 2012 |
| Words: | 5828 |
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