The essence of the
objectual (systemic) approach consists in the organization of the
processing information into *objects* and* processes* by
using the notion of *distribution.* The identification of the
objects’ model attributes according to the generic object model
presented in chapter 3 is also taken into consideration, and if the
processes exist, the identification of the processual objects shall
be also undertaken. As for the set of objects which deploy external
relations (the case of the complex objects), we also know that these
relations are deployed __between the internal references__ of the
objects, their internal references being considered as objects’
substitutes within these relations, because quantitative relations
can exist only between the singular values.

According to the
facts described in chapter 2, the relation *y=f(x)* which was
mentioned at the beginning of section X.3.2.2. represents the
*distribution *of the attribute *y *on the support
attribute *x*, when the assignment relation is __invariant__
(a function) on the entire support range (the function’s
continuity range). In chapter 2, we have also noticed that in case of
a singular value considered to be invariant *x*_{k},
the value of the distributed attribute *y*_{k} is
also invariant, so that the values *y*_{k}
distributed on the values *x*_{k} are easily
recognized as abstract objects belonging to the class *S*_{0}.
A couple
which
is associated with the point *P*_{k} from the
plot from the figure X.3.2.3.1, with the position vector **r**_{k},
is an element from the set of assignment relations which make-up the
distribution (that is an element of the primary distribution).

If there are two symmetrical variations
of the support attribute ranging from the value
up to
against the singular value *x*_{k} (at values
which shall be assigned to the points
,
respectively
),
which are small enough so that the deployed distributions to be
considered as linear, then, they will be related to the attribute
variations:

(X.3.2.3.1)

and:

(X.3.2.3.2)

According to the aspects which were set in chapters 2, 3 and 4, the relation:

(X.3.2.3.3)

identical with the relation X.3.2.2.3,
means the even *density* of a linear distribution of the
attribute variation
on
a support interval
(that is the density of a P_{1}-type SEP) determined on the
left side of the reference *x*_{k}. In chapter 4,
we have seen that if the density of a SEP is invariant on its support
range representing an object, in this above-mentioned case, it is an
object belonging to the class *S*_{1}. This SEP
has the interval
as its support with an amount,
with the __internal reference__ at *x*_{k}
(right side reference, which means that the support interval is on
the left of this value. However, the same value *x*_{k}
may be also an internal reference for the support interval having the
same amount, but which is placed on the right side of the reference
*x*_{k},
,
that is a support interval of another PES with the following density:

(X.3.2.3.4)

*Fig. X.3.2.3.1*

One may note that according to the
objectual approach by means of distributions, the value *x*_{k}
(the same as the one from the above-mentioned classic approach)
becomes an internal reference for two interval-type objects (left and
right), but they are intervals which represent the support of two
even variations with amount
and
,
therefore, *x*_{k} will be an internal reference
both for these processes and for their density values (evenly
distributed on the two support intervals with the reference *x*_{k}).

Comment
X.3.2.3.1: The fact that two abstract objects have the same internal
reference does not always mean that the two objects are one and the
same. The abstract state object *S*_{0} with the
reference *x*_{k} is distributed on the
nondetermination interval of that DP, whereas the abstract object S_{1}
with the reference at *x*_{k} is distributed on a
finite interval which consists of more known singular values (so that
a non-zero process to be able to exist). In both cases, *x*_{k}
is identical, but the amount of the internal domains which are
referred to is different. If you read chapters 3 and 4, where the
constitutive elements of an object and of a process are mentioned, it
is clearly revealed that the objects belonging to classes S_{0}
and S_{1} cannot be mixed-up even if they have the same
internal reference. There was a clear specification in chapter 4 that
the objects S_{0} are states of some objects (with null
processes) and the objects S_{1} are states related to
specific even processes *P*_{1}).

The total variation density (which is
also considered as even) on the support interval
(by
subtracting and adding *f(x*_{k}*)* to the
numerator), results to be:

(X.3.2.3.5)

hence, it may be observed that this
density is equal to the mean value of the two left-right densities
with the reference point in *x*_{k}. In relation
to this mean density (which is equal to the tangent density in *P*_{k}),
considered as a *common component* of the two density values
(internal class reference), we shall have the two specific components
of the densities (given by the function D(), mentioned in chapter 3):

(X.3.2.3.6)

the specific component of the density
on the left side of the reference *x*_{k} and:

(X.3.2.3.7)

the specific component of the density
on the right side of the same reference. When we discussed in chapter
4 about the two concatenated SEP (such as the variations in
question), we noticed that in case of a non-linear dependence
relation such as *f(x)*, the two specific components of SEP must
exist (to be different from zero), otherwise, the relation *f(x)*
is a straight-line (tangent case).

As a result of this objectual analysis on the objects and processes involved in the definition of the first-rank derivative, the following aspects may be underlined:

The abstract object that is “

*the singular value of a dependent variable y*_{k}*assigned by means of a relation f to a single independent invariant x*_{k}*”*represents an object from the processual class*S*_{0}(class which is specific to the objects unfolding null processes).The abstract object “

*first rank finite variation of the attribute y evenly distributed on a support finite variation of the same rank*” (where*x*_{k}_{ }is the internal reference of the support interval), represents a SEP belonging to class*P*_{1}(a finite difference between two states*S*_{0}); the even density of this SEP is a process state belonging to class*S*_{1}.Two variations

*x*_{k}, shall represent two concatenated SEP where the final state of the first one is an initial state of the second one, this common state (*S*_{0}type) being the point

By considering this
latter amendment, we may keep the notation used for the infinitesimal
ranges from the differential calculus, the variations
and
becoming
*dy* and *dx*, which could have as an internal reference a
singular value, but __they could never be replaced by a singular
value__ (a single point). In such conditions, the relations
X.3.2.2.3 and X.3.2.2.4 remain valid in case of the
distributions-based mathematics, but the first rank derivative is no
longer the limit (asymptote) towards which the variations ratio tends
to, but it is the *density* of a first rank SEP. Another
relevant specification, that is the domain *dx* in case of the
realizable processes (of the numerical calculations) cannot be less
than,
the error interval used for representing the concrete numerical
values on the effector IPS. As for the concatenated SEP where the
support interval
cannot
be neglected (neglection imposed by the current formula of obtaining
the functions derivatives, which are valid for
),
thus, only the calculus with finite differences can be used for
computing the distributions’ density values.

Copyright © 2006-2011 Aurel Rusu. All rights reserved.