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Some of the most used distributions are the ones whose support is made-up from ordered and continuous intervals of the integers set {Z}, or natural numbers set {N}. For example, we shall consider a function-type distribution, such as:
(X.2.1.1)
for having non-null and non-even second-rank derived distributions. We are now reminding to the reader that according to the specific notation used in this paper for the calculus with finite differences (where the notation with the reversed rank sign is being used for preventing it to be mistakenly considered as an exponent), the posterior finite difference (which may be also named on the right) between the posterior element yk+1 and the local reference element from series yk is:
(X.2.1.2)
and the anterior finite difference (or on the left against the same reference element yk) is:
(X.2.1.3)
the same for
or
,
and:
(X.2.1.4)
respectively:
(X.2.1.5)
so on.
By looking at the Table 1, one may observe that in the first
four columns we are dealing with a primary distribution
,
where
,
in the columns 5,6,7 and 8, we may find a first rank derived
distribution of the primary distribution, and in the following
columns, the derived distributions of upper rank (II and III). If in
case of this type of discontinuous distributions, a derivative under
an algebraic meaning cannot be conceived, we may use instead, with no
restrictions, the more general term (introduced by the objectual
philosophy) known as density of distributions, which is a term
applicable also in the algebraic case (but only by some compulsory
conditions, see annex X.3). The table also shows that the two series
of support objects (series of the singular values for the primary
distribution and series of the elementary variations for the derived
distributions) do not have the same number of elements (index values
k, m1, m2,
m3), but both types of series are using the
same value xk as a local reference for the
determination of posterior or anterior finite difference (the
equivalent of the variations towards the right or left in case of the
differential calculus).
It may be also noticed that the elementary support intervals x have all the same size and there is no finite difference (of any rank) between these intervals and consequently, all the elements of the derived distributions, regardless their rank, shall have the same support, that is x. As for the first rank derived distribution, the posterior density (the only one which is computed in Table 1) against the same reference element xk which was above mentioned, shall be:
(X.2.1.6)
where, because x is the same, regardless the value of k, xk was no longer written.
In the actual case of the primary distribution given by the relation X.2.1.1, and by replacing the values given by X.2.1.2 and X.2.1.6 we shall get:
(X.2.1.7)
relation which complies perfectly with the values written in the table below.
Table 1
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
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|
|
|
|
|
|
|
|
1 |
0 |
0 |
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|
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|
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|
|
|
|
|
|
|
|
|
1 |
1 |
1 |
1 |
|
|
|
|
|
|
|
2 |
1 |
1 |
1 |
|
|
|
|
1 |
6 |
6 |
|
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|
|
|
|
|
2 |
1 |
7 |
7 |
|
|
|
1 |
6 |
6 |
|
3 |
2 |
8 |
4 |
|
|
|
|
2 |
12 |
12 |
|
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|
|
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|
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3 |
1 |
19 |
19 |
|
|
|
2 |
6 |
6 |
|
4 |
3 |
27 |
9 |
|
|
|
|
3 |
18 |
18 |
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|
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|
|
|
4 |
1 |
37 |
37 |
|
|
|
3 |
6 |
6 |
|
5 |
4 |
64 |
16 |
|
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|
|
4 |
24 |
24 |
|
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|
|
|
|
|
|
5 |
1 |
61 |
61 |
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4 |
6 |
6 |
|
6 |
5 |
125 |
25 |
|
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|
|
5 |
30 |
30 |
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|
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6 |
1 |
91 |
91 |
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5 |
6 |
6 |
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7 |
6 |
216 |
36 |
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|
6 |
36 |
36 |
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7 |
1 |
127 |
127 |
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8 |
7 |
343 |
49 |
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The values yk are (as we have mentioned in
chapter 4) objects which belong to the processual class S0,
and the values
are objects which belong to the processual class S1.
If the difference between two successive states S0
distributed (assigned) on an elementary support interval Δx
is an element of the first rank derived distribution, the difference
between two successive states S1 distributed
on the same interval Δx shall be the element of the
second rank derived distribution, whose density is:
(X.2.1.8)
By also making replacements in this case, the actual values given by the relations X.2.1.1, X.2.1.2 and X.2.1.6 within X.2.1.8, we shall finally get:
(X.2.1.9)
which once again, complies perfectly with the values from Table 1.
This example of a concrete distribution
with a support made-up from integers aimed to clearly emphasize the
accuracy of the general relations X.2.1.6 and X.2.1.8, as well as the
concrete relations X.2.1.7 and X.2.1.9, accurate and valid
relations for any kind of amount Δx of the
elementary support interval. Those who would neglect in some
circumstances the terms which contain Δx (according to
the classic differential calculus) have this right, but they will
reach to an approximate result, with the error in proportion to the
amount Δx. The amounts
and
are
the equivalent of the first and second rank local derivatives (right)
from the classic differential calculus.
Copyright © 2006-2011 Aurel Rusu. All rights reserved.
