In chapter 2, we have seen that a
primary realizable distribution is a discrete distribution, whose
constitutive element is an equipartition of the attribute distributed
on a DP support. In case of the spatial distribution of the amount *M*
from the paragraph 5.2.1, the support of an element is a 3D DP with
an
volume (a volume quantum) on which a specific value of the amount *M
*is evenly distributed. If the amount *M* is cumulative, a
volume quantum with the position vector^{2}3
shall
contain a stockpile quantum:

(5.2.2.2.1)

where
is
the amount of attribute *M* associated (assigned) to a support
quantum (a 3D DP located at that position, that is the equivalent
from the objectual philosophy of the material point from the classic
physics), and
is
the distribution density, but also the assignment relation.

Comment
5.2.2.2.1: **Attention!** In this paper, the term “quantity”
has a quite different meaning than the one used in the common
language, denoting the value of the quantitative attribute associated
to a qualitative property. According to the common language, the word
“quantity” refers to the cumulative attributes
(countable, integrable, extensive) on a certain domain, resulting a
total quantity. According to the objectual philosophy, the word
“*stockpile*” defines this total quantity. In case
of the non-cumulative attributes, (intensive, such as the
temperature, pressure, etc.) the word “value” or “amount”
is being used, a quantitative quanta *ε* being also
related to it, but it does not represent a stockpile, but only a
normal numerical value. As for the intensive attributes, there is
also a difference regarding the meaning of the word “density”
of the distribution element. While for the extensive attributes, the
density was a ratio between an elementary evenly distributed
__stockpile__ and its elementary support interval, in case of the
intensive attributes, the density is only a ratio between an absolute
(in case of the primary distributions) or relative (in case of the
derived distributions) __value__, evenly distributed and the
elementary support interval. The reader is invited to accept the fact
that the concept of *quantity* in the objectual philosophy
refers to the existential, quantitative attribute of a qualitative
property, regardless if that property is cumulative or not. However,
the attention must be maintained on the integration reasoning; it may
exist a volume distribution of the temperature, but its integration
does make sense only if this attribute is converted into a cumulative
amount, such as the thermal energy quantity.

Because within the achievable
distribution 5.2.2.2.1, the variables *x*, *y*, *z*
vary gradually rather than continuously, with the amount
,
any interval of these variables shall contain a finite and integer
number of values such as *N*_{x}, *N*_{y},
*N*_{z} (see annex X.3.8), resulting also a
finite and integer number of volume quanta for any spatial domain of
the distribution of amount *M*. As it was mentioned in annex
X.3, the number of the elements belonging to the realizable primary
distribution 5.2.2.2.1 can be very high (but not infinite as in the
case of the virtual distributions), because
is usually very small (is
the number of bytes used for the display of a normal numerical
value). For reducing the information amount which must be processed
and for cutting-down the time required for this processing, in case
of the distributions realizable on AIPS, the volume which includes
the spatial distribution of amount *M* is divided not in 3D DP,
but into volume elements *dV* with the sizes *dx*, *dy*,
*dz*, which were selected so that the inner distribution of the
amount *M* with the density
can
be considered as even on this elementary unit.

Comment
5.2.2.2.2: Besides the amount of the inner domain, another major
difference between the volume quanta and the volume element is
represented by the composition of the inner reference system of the
two elementary objects. In case of the volume element *dV*, we
have seen that it has both a T reference and a R reference made-up
from the directions **X**,**Y**,**Z** of the three segments
*dx*, *dy*, *dz*. In case of the volume quanta *q*_{V}_{,
}the inner reference is made-up only from T reference, the
rotations being totally non-determined for this kind of object, just
as in case of the dimensionless point.

Under these circumstances, we may write (only for a cumulative attribute):

(5.2.2.2.2)

Where
is the stockpile from the amount *M* assigned to the elementary
volume *dV,* placed on that spatial position.

**Attention!** In this paper, both
the elementary volume and the volume quanta are *objects *with
an inner RS, and T reference (origin) of this inner RS is the one who
has assigned the position
,
defined against the inner RS of the distribution (as compound
object), and inner distribution’s RS position is defined
against an outer RS (an absolute one). According to the relations
5.2.2.2.1 and 5.2.2.2.2, the position vector
of each element is defined against this inner RS of the distribution
(with its components T and R). When the distribution starts to move
against an outer reference (for instance, an absolute reference), the
variable attribute shall be
,
the position vector of the inner T reference of the distribution,
against the outer T reference, and once with it, the positions of
all the distribution elements shall become variable. If there are
S-type relations (which shall be defined in the following chapter)
deployed between the distribution elements, an overall motion of the
inner R reference against the outer R reference may also occur.

The relation
5.2.2.2.2 is identical __from the point of view of its form__ with
the relations from the current scientific papers depicting spatial
distributions of some amounts, (and in which the expression with
“surrounded” point by the volume element is being used),
but its meaning is the above-mentioned one. The distribution
5.2.2.2.2 is also a primary spatial distribution of the amount *M*,
but with a higher approximation degree and requiring a less calculus
volume as compared to the distribution 5.2.2.2.1.

At the moment when
the distribution of the amount *M *which was initially static,
begins to move as a whole, the spatial position
of the inner *T *reference of the distribution becomes variable
and once with it, the positions of each distribution element become
variable as well. All of these variations are *time *dependent.

Comment
5.2.2.2.3: This
additional variable - that is the time - is approached in the present
paper just as any other realizable variable, which means that a
temporal quantum(a
temporal DP) shall be assigned to it and also an elementary interval
*dt* (see annex X.3 for the relation between the two elements
types). In case of the temporal attribute, the difference between
these two elements consist not only in their amount, but also in each
specific utilization. The temporal DP quantum of amount
with the inner *t* reference is the support of an element of
temporal primary distribution (a state *S*_{0}*(t)*),
while *dt* element (with an inner asymptotic reference at *t
*moment) is the support of a temporal derived distribution (a
state *S*_{x}*(t)*, *x=1…n*).

23 Attention! We are talking about the inner position vector determined in relation to the distribution’s inner reference

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