The notion of*
distribution *from mathematics has been introduced as a
generalization of the *function *concept, in order to also allow
oprerations with the dependence between different variables, which
could not be considered as functions, strictly algebraically speaking
(for example, the discontinuous dependences). Few examples of this
kind of discontinuous distributions, with a wide application field
can be mentioned and these are: Dirac distribution
(also known as the *impulse function*), Heaviside distribution
(also known as the *step function*), etc.

For a better
understanding of the distributions, it must be first understood the
notion of *dependence *between the values of two *amounts.*

Comment
2.2.1: The meaning of the very general term of *amount*
(magnitude, size) used in the mathematics
field could be better understood by the reader after reading the
entire paper, especially chapter 9. The only explanation which is
made in advance is that an attribute (that is a property of a real or
abstract object) has two components, according to the objectual
philosophy: the *qualitative *component, represented by the
name or the symbol of that property (property’s semantic value)
and the *quantitative* component, which is also called
*existential attribute* in the present paper, a number (scalar)
which shows the measure (size, amount, degree) of existence of that
property. As we are about to see next, the two components are
conjunctively associated within the relationships in this paper,
which means that, in case of a certain object, they can exist only
together (a null value for the existential attribute implies the
non-existence of an associated qualitative property). In mathematics,
for providing the language universality, the associated qualitative
attribute is most of the times let aside, operating in most of the
cases only with existential attributes (numerical values), or with
literal or graphic symbols for displaying them. Because it cannot be
omitted the fact that the numerical values are however the attributes
of some properties, in case of real objects, this aspect shall be
mentioned anywhere necessary. Basically, this is not a mathematic
paper, but the mathematic is used as an universal language for
expressing the relations between various amounts.

Let us assume that there is a
qualitative attribute *X,* which belongs to a certain object
which is related to a quantitative attribute *x* at a certain
moment, whose possible numerical values make-up an ordered set *{x}*.
In mathematics, the quantitative value *x* which may belong to a
property *X* is called *amount x*, and because it can take
any value from *{x}*, it is also known as the *variable x*.
By assuming also that there is another qualitative property *Y*,
with the existential attribute *y*, whose values belong to set
*{y}*, another amount *y *(or variable) is therefore
made-up. If the value *y* is modified as a result of changing
the value *x* and it remains invariant if *x* is invariant
too, we may say that there is a *dependence relation* between
the two amounts. This relation may be univocal (in a single sense) or
biunivocal (interdependence). For the time being, we are only
interested in the univocal dependence relation, the univocal
character being presumed when we are talking about dependence. The
sets of numerical values *{x}* and *{y}* which were
above-mentioned, have *the singular numerical values, *as basic
(non-decomposable) elements.

**Definition
2.2.1:** The __invariant__ numerical value which is attributed
at a certain moment to a variable is called the** singular value**
of that variable (synonym - the** concrete value**).

Comment
2.2.2: The association of “invariant” property to a
variable seems to be quite bizarre in the first place, that is why an
explanation is needed. At a certain moment, when concrete numerical
values are given to a variable, both the independent and the
dependent variable have only a single value. It is true that the two
variables can take any kind of single value from their values domain,
but by means of __repeated__, __successive__ assignments for
each value. The variables values remain invariant between two
assignments. The reader will better understand this aspect after
reading the entire chapter, because the fact that a quantitative
attribute (numerical value of a variable) cannot have more than one
value at a certain moment, this will be considered as a consequence
of the specific approach of the objectual philosophy regarding the
distribution’s definition.

**Definition
2.2.2:** Amount *y* is* ***dependent **on amount *x*
if each singular value *x*_{k} from *{x}*
determines each singular value *y*_{k} from *{y}*,
by means of a relation *f*_{k}*.*

In other words, the
amount *y* can be modified only by means of amount *x*.
According to the literal mathematic syntax, one of the possible ways
of writing these dependence relations is:

* * (2.2.1)

where
is
the running number of the numerical values from the ordered set *{x}*.

**Definition
2.2.3:** Two variables *x *and *y* are **independent**
if there is no dependence relation between their singular values (the
dependence relations are null for all the values from {x}).

**Definition
2.2.4:** The ordered set of the singular values included between
other __two different__ singular values
and
,
which are accessible ()
for a variable *x* makes-up a **domain** of values (synonym –
**interval**, **range**) of this variable.

The amount, quantitative value of this interval is:

(2.2.2)

**Definition
2.2.5:** The singular values
and
,
are the domain **boundaries.**

The domain boundaries can be part of
this interval (*boundaries included* inside the interval), case
when the interval is considered to be closed, or to not be part of
this domain (to be only adjacent to the interval), case when we are
dealing with an open interval (with *asymptotic boundaries*).

The underlining made at the definition
2.2.4 has the role to draw the reader’s attention that the
condition required for the existence of a *domain of values*
must be the existence of its two boundaries, and their values to
comply with the relation 2.2.2. Expressions such as “null
interval” or “void interval” are not allowed
because, according to this paper, an object with a null
(quantitative) existential attribute means that __it does not exist__.

**Definition
2.2.6:** If a finite interval contains an infinity of singular
values, those values are called **absolute accurate values**
(AAV).

AAV case is
minutely presented in annex X.3, and for the time being we may assert
that these values make-up the so-called “set of real
numbers”{R}from mathematics, and each of these values comprise
an infinite quantitative information amount (with an infinite of
figures), therefore, they are actually *virtual numbers** ^{5}*.

Each of the amounts
implied in relation 2.2.1 can take single numerical values from a
certain domain (a certain values interval) known as the domain of the
independent variable, respectively, the domain of the dependent
variable (sets *{x}* and *{y}* which were above mentioned).
According to the most general case of dependence, each singular value
*x*_{k} is related to a certain relation *f*_{k}
and to a certain value *y*_{k}, as it is shown in
relation 2.2.1.

**Definition
2.2.7:** Set *{f}* of the assignment
relation between each singular value of set *{x}* (of the
independent amount) and the corresponding singular value belonging to
set *{y}* (of the dependent amount) is the **primary
distribution **(synonym - **distribution of the singular values**)
of the amount *y* along the domain of amount *x.*

**Definition
2.2.8:** The domain of the singular values of the independent
variable (set *{x}) *is the **support **of the primary
distribution.

Comment
2.2.3: After the reading of the following chapter, in which the
*strict set * shall be defined, we are about to notice that the
support set of a distribution is this kind of set (which does not
include identical objects, namely, identical singular numerical
values). The sets *{y}* and *{f}* can be strict sets, but
generally, they are not subjected to this kind of condition.

The qualitative attribute whose
quantitative values make-up the set *{y} *is also called in this
paper as *distributed attribute*, and the one whose values
make-up the set *{x}* is also known as *support attribute*.
Even if it is not necessary, we shall make once again the following
assertion: the three sets *{y}*, *{f}* and *{x}* have
__the same number of elements__ (they are equipotent sets).

**Definition
2.2.9:** If the attribute *y* is cumulative^{6},
the total amount of attribute y, distributed along the support domain
represents the **stock **of the primary distribution.

In case of the
virtual distributions with continuous support, this support is
usually represented by an interval from the set of the real numbers
{R}, and this set, as we have previously mentioned, contains an
infinite number of singular values in any interval, also resulting
that the number of the relations from a distribution on this kind of
range should also be infinite. The question is simplified if the
relation *f* is __invariant__^{7}
along the support domain or along its subdomains (independent from
the concrete, numerical values which are assigned to *x*). In
this case, that particular relation is the classic *continuous
function* stipulated in the mathematic analysis (where the term
“continuous” means both the support continuity, but
mostly the maintenance of the same dependence relations on the
support range), which may be applied in the domain where this
invariance is maintained. The continuous functions are therefore
particular cases of distributions.

The main advantage of the continuous functions is that they replace an infinity of individual relations (for each numerical support value) with a single one, which is valid on its support domain (continuity domain). Most of the virtual distributions (mathematic) consist of several invariant relations of this kind (functions), which are defined on continuous subranges of the support domain, the gathering of this subranges making-up the overall distribution support.

Comment
2.2.4: For example, the unit Heaviside distribution with the internal
reference *x*_{0} (an AAV from R)
is defined as it follows:

(2.2.3)

One may notice that two continuous functions are defined: , which is valid on the open support subdomain and , which is valid on the semi-open support subdomain . Heaviside distribution is therefore made-up from two invariant relations (continuous functions), each of them with its own support subdomain (it may be considered as a two-functions system).

According to the most general case of primary distribution, when the assignment relations are not invariant, we are dealing with a distinct relation for each singular value of the support amount (that is the case of the distributions made as lists, tables, matrix, images, etc.).

Comment
2.2.5: A simple example for showing this kind of distribution is
Dirac ***(x)*
distribution, which in case of the *unit impulse* may be defined
as it follows:

We
may notice the existence of two continuous functions, as in the case
of Heaviside distribution, defined along two open intervals, but also
the existence of a distinct assignment relation on a single *x*_{0}
value, internal distribution reference (the concept of *internal
reference* shall be approached in the following chapter).

Based on the issues which were presented so far, the result is that a primary distribution is decomposable up to its basic element - the individual assignment relation between a singular value of the distributed attribute (dependent) and a singular value of the support attribute (relation 2.2.1) - even in case of the continuous primary distributions (algebraic functions).

Comment 2.2.6: When the plot of a continuous, algebraic function is drawn-up on a computer, this shall use the assignment relation, by means of a repeated process, between the allotted value and the support value according to the number of the actual values (singular) which exist in the support range. The way of defining the distributions in this paper had to be consistent to SOP, which was presented within chapter 1, and due to this reason, a distribution must be considered as a system decomposable up to a basic element and composable up to the limit of the highest possible domain of the support attribute. SOP is one of the basic principles of this paper, so that, at each description of the new distributions-based objects, which are about to be presented in the following chapters, their (de)composition degree shall be minutely approached.

Therefore, the elements of a single *k
*element of primary distribution are *y*_{k},
*f*_{k} and *x*_{k}. The
dependence between the singular distributed value and the support
value shall be also written as a product:

(2.2.5.a)

or:

(2.2.5.b)

where **_{k}
(in case of a distribution element) is a simple numerical value.

**Definition
2.2.10:** The amount given by the __local__ assignment relation,
equal with the ratio between the actual distributed value and the
actual support value __within a distribution element__ is called
the **density** of that distribution element.

In other words, the density is an
attribute specific __only to the elements__ of a distribution, it
is therefore a local amount for a certain distribution. However, if
the assignment relation depends on the support value by means of a
relation
and this dependence is invariant on the support domain, then, we are
talking about a *density function *of a distribution (of a
primary distribution, in this case).

The assignment
relations may be either simple or more complex^{8}.
We have previously seen that a distribution is simple if the
assignment relation is invariant on the support domain (a continuous
function). In such a case, there are once again simple or complex
relations (functions); the most simple relation of this kind is a
numerical constant (a numerical invariant value), evenly assigned
along the entire support domain, which also gives the name of this
kind of distribution, that is **uniform distribution***^{9}*.

This type of
distribution with the most simple assignment function is a basic
distribution, it can be therefore used as an element within more
complex distributions. Right next to the uniform distributions, on a
higher rank from the point of view of the complexity of the
assignment relation, the * linear distributions* may be
found (synonym -

Comment 2.2.7: For instance, the distribution:

* * (2.2.6)

is
this kind of linear function (equation of a straight line which
crosses the axis y in *y*_{0}), where amount *m*
is invariant in case of a certain distribution and it is called the
distribution’s *angular coefficient*, equal to the tangent
of the angle between that straight line and the axis X (support
attribute). It may be noticed that if *m *is null, the linear
distribution becomes an uniform distribution.

So far, we have
discussed about the relations between the __singular__ values of
the two attributes involved in a primary distribution; it is the
moment to see what kind of relations are between the __variations__
of the numerical values of the two attributes, namely, between some
intervals (which include sets of singular values) of the two
variables. Therefore, we shall presume that the entire support domain
of
a primary distribution *{f}* is divided into elementary
intervals^{1}0
(variations) of the same amount
(amount
given by the relation 2.2.2 and imposed by the elementariness
condition), therefore, the support domain is made-up from an ordered
series of intervals (variations) with a constant amount
,
concatenated, in which each element of the series has a defined
position (within the series) by the singular values of its
boundaries. Thus, a support interval which has the lower boundary at
,
and
the other one at
,
we shall note it (provisional) with
(
pay attention, *m* is this time the running number of the
interval object
from
the arranged series of intervals, which is a number different from *k*
belonging to the primary distribution).

If the
elementariness condition of the support intervals is met, along the
interval
,
any kind of primary distribution *{f}* may be approximated by a
linear distribution (a continuous function on
),
namely *f*_{1} =* f*_{2} =*
f*_{m}, resulting the following variation for the
distributed amount:

(2.2.7)

where
is the running number of the interval
in the series of intervals
,
in which *{x}* support is divided.

Relation 2.2.7 is
similar with relation 2.2.1, but it defines a dependence between the
set
of
the elementary finite variations of amount *y* and the set
of
the elementary finite variations of the amount *x*, where *x*
and *y*, let’s not forget, are the amounts whose singular
values are linked by means of the primary distribution *{f}.*
Variations
which
comply with the data elementariness condition are also called as
first rank *finite differences* of the variable y, and the
variations
,
are first rank finite differences* *of the support variable *x*.

Comment
2.2.8: It is very important for the reader to notice that the way of
defining the elementary interval according to the objectual
philosophy does not contain any reference to the amount of this
interval, the only condition which must be fulfilled is to exist a
uniform variation of the distributed attribute along this interval
(or its real variation to be __considered__ as even, because the
support’s variation is uniform by definition). For this reason
we can utilize finite differences (whose size is not important in
this situation). As an example, see annex X.2.

**Definition
2.2.11:** The set
of
the dependence relations between each interval belonging to the
ordered set
of
the first rank finite differences (variations) of a support variable
*x *and each corresponding interval from set
of
the first rank finite variations of the distributed variable *y,*
where x and y are linked by means of a primary distribution *{f}*,
make-up the **first rank derived distribution** of the primary
distribution *{f}*.

In case of the
first rank derived distribution, set
of
the first rank finite variations of attribute *y* represents the
new distributed attribute, and the ordered set of the finite
variations
is
the support of this distribution. It is also clear that the sets
,
and
have
the same number of elements (but which is different from the number
of the primary distribution sets from which it derives, within ratio
,
and
,
being the number of the elements of set *{x}*, respectively, of
set
).

Comment
2.2.9: It is once again important to observe that the basic element
(non-decomposable) of the first rank derived distribution is an __even
variation__ (a set of singular values, an interval)
assigned
to an __even variation__
,
by means of relation
,
while the primary distribution (from which the derived distribution
arises) has as its basic element - as it was above mentioned - a
single value *y*_{k} assigned to a single value
*x*_{k} by means of a relation* f*_{k}.
The reader will better understand the difference between the
relations 2.2.1 and 2.2.7 after reading the following chapter in
which the term “object” shall be defined, what an inner
reference does represent, and moreover, after reading chapter 4,
where we are about to see what does it mean the variation of an
attribute’s value, namely a process. We shall therefore notice
that, although both relations display the same singular value *x*_{k},
in the relation 2.2.1, *x*_{k} is a *singular
value* type object, and according to the relations 2.2.7, *x*_{k}
is also a singular value type object but also an *inner reference*
of an interval type object.

Just as in the case
of the primary distributions, if the dependence relation
is
kept the same on the entire domain of the independent variable (in
our case, of the set
,
regardless of the concrete value of *m*) that particular
relation is a *continuous function* along that interval,
function which is called *the first rank derivative* of the
primary function *f.* Similarly with the primary distributions,
in case of the first rank derived distributions, the distribution
element may be written as a product:

(2.2.8.a)

or:

(2.2.8.b)

Where *
*
is also *the density* of the distribution element, but this
time, of the first rank derived distribution.

Comment 2.2.10: If in relations 2.2.8, the equation of a linear distribution given by the relation 2.2.6, is replaced, the result is:

* * (2.2.9)

hence, resulting that:

* (2.2.10)*

Namely,
the density of a derived distribution element (which is the density
of a linear primary distribution) is even the angular coefficient,
tangent of the angle between the primary linear distribution and the
axis of the independent variable (distribution support). The density
of the primary distributions does not have any practical utility (at
least for the time being), being introduced only for emphasizing the
generality of the abstract object model, that is the *density*,
valid __for any kind of distribution__, including the primary
ones. The density values of the derived distributions, as we shall
see in the following chapters, are abstract objects of major
importance in this paper for processes’ characterization, being
the substitutes of the local derivatives from the differential
calculus, also valid for the distributions with a discontinuous
support (see annex X.2.1).

For the reader, it is clear enough that
the derived distributions of a primary distribution *f* can also
have higher ranks, with the specification that the only primary
distribution remains *f*, all the other ones being derived
distributions
,
the elements of all these distributions having the same support (the
elementary interval
),
only the distributed amount (a finite difference of rank *n*)
and the number of distribution’s elements being different. For
all these distributions, the elementariness criterion is the same,
just like the definitions used for the distribution element and for
its density.

We cannot end this paragraph without making some observations concerning the differences between the mathematical models introduced, and the objects from the classic mathematics (differences which also exist in other mathematical fields and which are minutely approached in annex X.3). First of all, it is revealed the preoccupation of the objectual philosophy for the clear structure of each used object (abstract), moreover for the non-decomposable elements of this structure, which are the basic elements. Once with the definition of these elements and of the relations between them, a coherent structure of the whole mechanism is revealed (the object made-up from these elements), even if this object look (and it is otherwise denominated) in the official mathematics.

If there are not too many differences
between the primary distributions and the mathematic distributions
(besides the unusual concept on the density of an element or of a
function), the situation is not the same in case of the derived
distributions where there are major differences. The definition mode
of the derived distributions (and implicitly of the derived
functions) is much more different from the one used within the
differential calculus. An object similar with the classic *n*-ranked
local derivative of a function *f* is, according to the
objectual philosophy, *the density *of a *n*-ranked derived
distribution of that particular function.

5
The concepts of *information amount* and *virtual object*
are briefly defined in Annex X.3, but they will be minutely
presented in chapters 8 and 9. The fact that the numerical values
from set {R} are virtual, also gives the denomination of *virtual
distributions* for those distributions which have this kind of
support.

6 An attribute is cumulative if it allows the addition and subtraction operations. Attributes such as the frequency, color, temperature etc. are not cumulative, but the electric charge, mass, spatial dimensions, so on are.

7
The invariance of a relation means that, on a certain range of the
independent variable, also known as the function’s continuity
domain, the dependence relation *f* between the two amounts is
always kept the same (unchanged).

8 The relations are complex only in case of the invariant dependence relations (independent of the support actual value, that is the case of the continuous functions); the individual relations between an actual distributed value and one actual support relation (which define a distribution element) are always simple values (numerical or literal values).

9
An example of this kind of distribution was noticed at the Heaviside
distribution (given by the relations 2.2.3), which is made-up from
two concatenated uniform distributions **_{1}*(x)=0*
and **_{2}*(x)=1*.

10 Attention! We are talking about elementariness according to the objectual philosophy, that is, informational, which means that there is no more (by convention) inner differential information within the elementary domain, in case of a non-null variation this fact means that the variation is even, both for the support attribute and for the distributed one (see annex X.3).

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