At the beginning of
this chapter we have established that the mathematical model of a
process is also a distribution, more precisely, a *derived
distribution* of a *primary distribution*, and if a single
attribute *Y* is distributed, then, the process is *specific*.

Both the primary
and derived distributions (as we have shown in chapter 2), have __the
same type of qualitative attribute__ *Y*, __distributed on
the same type of qualitative support attribute__ *X*, but,
within a primary distribution, the distribution elements are made-up
from quantitative* *__singular values__ *y*_{k}
(virtual or normal), assigned by means of relations *f*_{k}
or
on singular values *x*_{k} of the support
attribute *X*, whereas in case of a derived distribution (such
as the first rank distribution), the distribution element is made-up
from an elementary
quantitative __variation__ assigned through a relation
or
to an elementary variation
of the support attribute (the notations are those from chapter 2, but
the difference is that the lower index *m1* represents the
running number of the support element from a first rank derived
distribution, whereas, in case of a second rank distribution, it
would be *m2* etc. (see also the example from annex X.2). As
objects, the support elementary intervals are limited by the two
boundaries (*x*_{1}_{ }and *x*_{2}
from relation 2.2.2) with values settled in relation to the inner
reference *x*_{k}, corresponding with the values
*
*and
,
which were mentioned in section 2.2.

Otherwise speaking,
a support elementary variation starts from the singular value *x*_{1}_{
}and it ends with the singular value *x*_{2},
the two values being related to the singular dependent values *y*_{1}
and *y*_{2}, which are the interval’s
boundaries of the distributed variation
.
But this fact also means that a derived distribution element has two
primary distribution elements as boundaries. These two primary
distribution elements which “frame” a derived
distribution element have a special name in this paper.

**Definition
4.2.1:** The total amount of the attribute’s invariant values
of an object Ob, at a singular value *x*_{k} of
the support attribute, makes-up the** state** abstract object of
object Ob, at a value *x*_{k}_{ }.

Comment
4.2.1: Definition 4.2.1 is a much more general definition than the
case approached in the definition’s preamble, that is why few
explanations are required. When we have defined the object’s
general model in chapter 3, we have noticed that it is an union of
invariant distributions on a common support. Let us assume that we
have a real object, such as a piece of wood, whose properties are
distributed on a spatial support, with the inner domain defined
through its boundary - the object’s outer surface. This piece
of wood is decomposable in its elements (elements of space volume
which are selected so that the properties to be evenly distributed on
their inner domain), elements which have an invariant distribution of
their spatial positions. This spatial distribution which is
determined against the inner reference system, mostly for the
elements which make-up the object’s surface represents the
*shape* of this object. Specific colors, hardness, temperature,
etc. shall be assigned to a certain position from the body surface,
all of these being properties which are specific to that position.
The union of these properties within the defined spatial position
makes-up a *state* of the body element from that position. In
this case, the support attribute is the spatial one, but the state’s
definition is the same if the support attribute is another kind of
attribute, such as the temporal or frequential one. If we move that
piece of wood, the variable attribute is the spatial position of the
object against an external reference. The object’s spatial
position together with all the other inner properties at the initial
moment *t*_{1} of the motion is a *state* of
the object at that moment, and the position and all the other
properties existing at other moment *t*_{2}
represent a state of the object at another moment. The following
paragraphs will reveal that there are various sorts of *state*
objects and as a result, we could identify to what class the
above-mentioned state (definition 4.2.1) belongs to.

As we have noticed
in chapter 2, within the support interval
of the first rank derived distribution element, the distribution of
the dependent attribute
is a linear distribution, with constant density and specific to the
inner reference value *x*_{k} of the support
interval:

(4.2.1)

**Definition
4.2.2:** The **specific elementary process **(SEP) is a __linear__
distribution (with constant, even density), with finite support, of a
finite state variation of __an attribute__.

In other words, a
derived distribution element of a single attribute *Y* is a SEP
of this attribute. As we have also noticed in chapter 2, the
distribution elementariness (and of the process at the same time) is
given by the fact that the variation density (the same with the
density of the linear primary distribution) is evenly distributed,
which means that there is no inner differential information
(contrast) between the two values of the density from this interval.

This invariant
density of a SEP has various denominations, depending on the type of
the support attribute; if this (support) is for example, a temporal
attribute, the density given by the relation 4.2.1 is also called
*velocity *(or variation *rate*) of the distributed
attribute.

**Definition
4.2.3:** The even density module of a SEP (a scalar) is called the**
intensity** of the specific elementary process.

The presentation had a general nature in the previous mentioned paragraph, valid both for the virtual distributions (ideal, mathematic) and also for the realizable ones. As we have seen in chapter 2, the differences consist only in the type of the singular values (AAV or normal) and in the amount of the support intervals.

Let us assume that
we are dealing with a specific *realizable* process (first rank,
too) of the attribute *Y* (that is a derived distribution of a
primary *realizable* distribution of this attribute). As for the
primary distribution, we shall have therefore a discrete realizable
support made-up from an ordered set of normal singular values
(made-up from DP range quanta, of
even amount), each with *x*_{k} __inner
reference__. According to this distribution version, the *state*
objects of an object *Ob* shall consist in normal invariant
values of the object’s attributes, related by means of
assignment relations to some normal support values. The assembly
made-up from the distributed attribute, support attribute,
distribution type, inner range and the reference system form an
*object*, as we have previously seen. Within a realizable
process, the main invariance condition imposed to the properties of
*state *object, in contradiction with the variability condition
required to the process, lead to a compromise (an equilibrium) as
regards the support DP amount, so that the attribute variation in
this interval to be __under the perception threshold__ of IPS
which analyse this process.

Comment
4.2.2: A snapshot taken by a photo camera (at the end of a running
contest, for example) represents a state of the *spatial position*
attribute of the objects taken into consideration at a certain moment
(that is the camera tripping moment). The exposure duration is
settled so that the motion of the objects from the set to not be
perceivable (to not impair the picture definition). This kind of time
interval is in this case the temporal DP. Obviously, the motion
processes have also continued during the exposure duration, but this
motion is non-perceivable for the human IPS which analyzes the
picture and it does not exist any more for the material support of
the state information (picture), which must be invariant
(information) once it has been recorded (stored). The faster the
observing process, the lower must be the state’s support
interval (such as the exposure duration in this example). The same
method - sampling - is being used in the case of electronic data
acquisition taken from the real processes, characterized by two time
intervals: sample duration and the duration between two successive
samples. The sample duration is selected so that the variation
process to be insignificant (null) during its unfolding, and the
interval between the samples is selected so that a process to arise
during its unfolding, but it must be considered even in case of the
most uneven stage of the real process. These conditions are met if
the sampling frequency
and
the maximum frequency of the sampling process
comply
with the Niquist criterion
.

The state of an
attribute is therefore a stationary abstract object (deliberate
pleonasm), namely, invariant, that is an object with *a null
process*, as its characteristic process deployed during its
existence.

Comment 4.2.3: According to the natural language, the processes are represented by verbs (with all their flexional forms). For displaying the invariance of an object or of a property, the static verbs are also used, for example “to remain”, “to stay” etc. These are examples of null processes which are specific to the objects during their invariance period.

We have discussed so far about first rank SEP, the ones represented by a first rank derived distribution element. If that specific process is more complex, there will be two different successive (concatenated) SEP, which means that there will also be a second rank derivative for the primary distribution. The element of such a second rank distribution shall be a second rank SEP, namely, a second rank finite difference assigned to a support element by means of that relation ( or ). Here, a special remark is required: because all the support elements are equal one another, there will be no finite difference of higher rank (all of them being null), so that, all the elements of the derived distributions, regardless of their rank, shall have a support element of the same amount , but integer exponents of shall interfere in the expressions of the various ranks densities.

As a conclusion, any specific process, no matter how complex it is, can be modeled by some derived distributions of a primary distribution belonging to the variable attribute. The primary distribution is exclusively made-up from state-type elementary objects (we shall see next what kind of state), and the derived distributions are made-up from SEP-type elementary processes of various ranks..

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