74
B. Steinbach and C. Posthoff
belong to the class
C N 3
g
re
3 (x), g
id
3 (x)
with the mark function of the single maximum of the class C N with regard to x i
g
re
3 (x) = max
x i
f
re (x),
(3.24)
and the independence function g id
3 (x) associated with
IDM(g 3 ) = UM(IDM(f
re (x)), x i ).
The three independence functions are equal to each other:
g
id
1 (x) = g
id
2 (x) = g
id
3 (x) = g
id (x),
(3.25)
with
IDM(g) = UM(IDM(f ), x i )
and
rank(IDM(g)) = rank(IDM(f )) + 1.
If Condition (3.21) is not satisfied, we have
s min = MIDC(IDM(f ), x i ) = 0
and all functions of the given class C N do not depend on the change of x i . In
this case the mark functions of the single derivative operations of the given class
C N
f re (x), f id (x)
are
g
re
1 (x) = 0,
IDM(g 1 ) = I n ,
(3.26)
g
re
2 (x) = f
re (x), IDM(g 2 ) = IDM(f ),
(3.27)
g
re
3 (x) = f
re (x), IDM(g 3 ) = IDM(f ),
(3.28)
where I n is the identity matrix of the size n.
From (3.26) follows that the single derivatives with regard to x i of all functions
f (x) of the given class C N
f re (x), f id (x)
, which do not depend on the change of
x i , i.e., s min = 0, are equal to the constant function f (x) = 0(x).