BIO-INFORMATICS IN THE LIGHT OF THE MAXIMUM
ORDINALITY PRINCIPLE
The Case of Duchenne Muscular Dystrophy
Corrado Giannantoni
ENEA’s Researcher and Consultant of Duchenne Parent Project Onlus, Via Aurelia 1299, 00166 Rome, Italy
Keywords: Intractable Problems, High Performance Computing (HPC), Traditional Differential Calculus (TDC),
Incipient Differential Calculus (IDC), Molecular Docking, Drug Design.
Abstract: In a previous paper (presented at the Third International Conference on Bioinformatics) we have shown that
Protein Folding, although considered as being an “intractable” problem that would require thousands of
years to be solved, in reality can be solved in less than 10 minutes when modeled in terms of Incipient
Differential Calculus (IDC). Such an evaluation was specifically made with reference to Dystrophin,
precisely because, being made up of about 100,000 atoms, it represents the largest protein in a human being.
Consequently it can be considered as being the most significant ostensive example in the context of such
Informatics problems. The present paper aims to show that the folding of Dystrophin can also be run on a
simple PC in less than two hours, as a consequence of very profound “symmetry” properties of the Ordinal
Matrices that characterize the mathematical model adopted. The same happens in the case of dynamic
interactions, such as Molecular Docking and computer-aided Drug Design, which can be obtained in
absolutely comparable computation time. This is also why, by keeping the original reference to Dystrophin,
we assumed Duchenne Muscular Dystrophy as the pertinent corresponding example. The paper will also
point out that such advantages are strictly referable to a different gnoseological (and mathematical)
approach based on the Maximum Ordinality Principle, which can be considered as being the most advanced
Ordinal Self-organization Principle for living (and also non-living) Systems.
1 INTRODUCTION
The paper presents some Informatics advances with
respect to the results shown at the Third
International Conference on Bioinformatics
(Giannantoni 2010b).
On that occasion we dealt with the well-known
Problem of Protein Folding, which is usually
considered as being one of the most important
“intractable” problems. This is because, although the
problem is thought of as being theoretically solvable
in principle, the time required in practice to be
solved may range from hundreds to some thousands
of years, even when run on the most updated
computers.
In this respect we have shown that, by
introducing a new concept of derivative (the
“incipient” derivative) (ib.), the Maximum Emergy
Power Principle (Odum 1994a,b,c) can be
reformulated in a more general form, by replacing
both Emergy and Transformity by the concept of
Ordinality. The principle can be thus renamed as the
Maximum Ordinality Principle (Giannantoni 2010a).
Its corresponding enunciation then becomes: “Every
System tends to Maximize its own Ordinality,
including that of the surrounding habitat”. In formal
terms
:
0}{)/(
)/(
=
∼∼∼
∼∼
rtdd
nm
Maxnm →
∼∼
)/(
(1)
where:
)/(
∼∼
nm
is the Ordinality of the System, which
represents the Structural Organization of the same in
terms of Co-Productions, Inter-Actions, Feed-Backs
(see also Appendix), while
}{
∼
r
= the proper Ordinal Space of the System.
At this stage, by modeling Protein Folding as a
Self-organizing System which evolves in adherence
to the Maximum Ordinality Principle, the problem
becomes soluble in explicit terms. This enabled us to
244
Giannantoni C..
BIO-INFORMATICS IN THE LIGHT OF THE MAXIMUM ORDINALITY PRINCIPLE - The Case of Duchenne Muscular Dystrophy.
DOI: 10.5220/0003273002440250
In Proceedings of the International Conference on Bioinformatics Models, Methods and Algorithms (BIOINFORMATICS-2011), pages 244-250
ISBN: 978-989-8425-36-2
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

assert that the simulation of Protein Folding, even in
the case of a macroscopic protein, such as
Dystrophin (made up of about 100,000 atoms), can
be obtained in a few minutes, when run on the next
generation computers (1 Petaflop).
A fortiori, any Protein Folding becomes a
“tractable” problem, with a corresponding solution
obtainable in even much lower computation time.
This led us to think of developing a computer code,
first finalized to analyze biological systems made up
of a limited number of atoms (e.g. sugars), before
modeling the smallest proteins (about 2.000 atoms).
In reality, during the development of the code,
we discovered some additional properties of the
mathematical model adopted, which enabled us to
further improve the solution in terms of Informatics.
2 INFORMATICS ADVANCES
As just anticipated, the improvements here
considered are directly related to some formal
properties which are intrinsic to the mathematical
model. Such mathematical properties (that will be
dealt with in the next section) “emerged” more
clearly during the development of the code and led
us to recognize the possibility of some decisive
advantages with respect to the previous stage
presented in (Giannantoni 2010b). Namely: i) a
reduced number of computations; ii) a reduced need
of a high computation power; iii) a reduced
incidence of special numerical methods to be
adopted.
In order to adequately point out the relevance of
the improvements obtained, it is worth recalling the
present state of High Performance Computing
(HPC) and its foreseeable perspectives.
The most powerful computers at present
available (IBM, June 2010) have a computation
power of about 1 Petaflop. Their power supply,
however, is of about 500 MW. This value represents
a sort of a “threshold”, which seems to prevent
further developments, at least in terms of the same
technology. On the other hand, even in the case of a
very rapid change of technology, an increase of the
corresponding power of 10
6
Flops would require
(according to Moore’s Law) not less than 30 years.
In such a general context, the afore-mentioned
advantages enable us to largely overcome the same
improvements pertaining to the previous solution.
This in fact, although obtainable in a few minutes,
always had to be run on very advanced computers,
characterized by a computation power of at least 1
Petaflof. Such an aspect would certainly limit the
application (and the diffusion) of the new
methodology to special cases only, as a consequence
of the very high costs associated to the use of such
powerful computers.
Vice versa, the possibility of obtaining the same
solution by means of less powerful computers
should increase the diffusion of the methodology
proposed and the number of problems which can
adequately be solved.
3 ORDINAL PROPERTIES
OF THE MATHEMATICAL
MODEL
Let us then consider the intrinsic Ordinal properties
of the Model which facilitate the research for a
solution, in faithful adherence to the Maximum
Ordinality Principle.
These properties are related to the fact that, when
a Self-organizing System, which persistently tends
toward the Maximum Ordinality conditions,
effectively reaches such very special conditions, it
presents itself as being self-structured in a radically
different way with respect to its initial Ordinality.
This is because the latter has evolved according to
the following Trans-formation
→
∼∼
)/( nm
∼∼∼∼
↑↑↑ N}}2{}2/2{{
(2)
where:
}2/2{
∼∼
represents a “binary-duet” coupling
(see Appendix); the Ordinal power
}2{ ↑
∼
indicates
the “perfect specularity” of the previous “binary-
duet” structure; while
∼
↑ N
indicates the Ordinal
Over-structure of the
∼
N
elements of the System
considered as a Whole (this is the reason for the
“tilde” notation).
Under such conditions, the solution to Eq. (1)
assumes the form of an exponential Ordinal Matrix
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎬
⎫
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
∼
∼∼∼
∼∼∼
∼∼∼
=
)(...)()(
............
)(...)()(
)(...)()(
21
22221
11211
}{
ttt
ttt
ttt
NNNN
N
N
er
ααα
ααα
ααα
(3)
in which any element
ij
∼
α
is characterized by the
Ordinality
}2{}2/2{ ↑↑
∼∼∼
. Ordinal Matrix (3) in
fact, as already shown in (Giannantoni 2010b),
reflects the fact that the relationships between the
different parts of the System cannot be reduced to
BIO-INFORMATICS IN THE LIGHT OF THE MAXIMUM ORDINALITY PRINCIPLE - The Case of Duchenne
Muscular Dystrophy
245

mere “functional” relationships between the
corresponding cardinal quantities. This is because
such quantities always “vehicle” something else,
which leads us to term those relationships as
“Ordinal” relationships. The term “Ordinal” would
thus explicitly remind us that each part of the
System is related to the others essentially because,
prior to any other aspect, it is related to the Whole
or, even better, it is “ordered” to the Whole. This is
also the basic reason why the most important terms,
when understood in such an Ordinal sense, are
usually capitalized to expressly point out such a
fundamental concept.
Given these conditions, each element of the
Ordinal Matrix can be interpreted as being Inter-
Acting (in Ordinal terms) with all the other elements
of the System. This leads to a first simplification,
because in this case the elements of the main
diagonal result as being equal to zero. In addition,
the afore-mentioned perfect specularity reveals itself
as being a property which also characterizes the
Ordinal Matrix as a Whole. This in turn suggests we
give an equivalent representation of the System by
choosing, as a preferential reference perspective, any
of the
N elements of the Ordinal Matrix.
Such a preferential choice introduces a further
simplification, due to the fact that any preferential
description adopted is “perfectly specular” to any
other perspective specifically associated to each one
of the remaining
1−N
elements of the System. This
evidently reduces the description to
2/)1)(1( −− NN
distinct elements, which are
coupled between them in the form of “binary-duet”
structures.
However, it is also possible to show that all these
distinct basic elements are so strictly related to each
other (in Ordinal terms) that the description can
equivalently be given by means of one sole element
(assumed as a preferential reference perspective) and
only
)1( −N
correlating factors
i
λ
.
Clearly, all these properties are exclusively
related to the concept of Ordinal Matrix. These
intrinsic properties, in fact, express a much more
profound concept of “symmetry” (with respect to the
traditional one), which can more properly be termed
as “specularity”. That very aspect which offers such
relevant advantages when developing a computer
code based on an Ordinal Model.
More specifically, by considering the Folding of
Dystrophin, the above-mentioned properties allow
us to reduce the corresponding computation power
of about 10
6
Flops. This means that the same
Ordinal Model can also be run in less than 2 hours
on a common PC, usually characterized by a
computation power of about 1 Gigaflop (such as, for
instance, a traditional Pentium IV processor).
4 BIO-INFORMATICS
IN THE LIGHT
OF THE M. O. PRINCIPLE
Mathematical Models and Ordinal Methods of
solution illustrated with reference to Protein Folding
are also applicable to the majority of Biological
Problems usually dealt with through Informatics
Methods.
Protein Folding, in fact, is only an ostensive
example. In such a context the Dystrophin Folding,
in addition, represents the most significant case,
precisely because Dystrophin is the largest protein in
a human body.
The same Ordinal Methods can also be applied to
the research for the best therapy in the case of a
Protein mis-Folding. This can be obtained by
considering the dynamic Ordinal Inter-Action
between the considered protein and any given
chemical compound.
As an example, the exon skipping method, at
present adopted in Duchenne Muscular Dystrophy
(Aartsma-Rus et al. 2006, Wilton 2007) could surely
be improved by selecting the most appropriate
AONs (Antisense Oligo-Nucleotides). Their
potential Ordinal Inter-Actions with the mis-folded
Dystrophin, in fact, could be analyzed in advance,
by modeling their potential Inter-Actions on the
basis of an appropriate Ordinal Model. This would
lead to a significant acceleration in such a research,
by also saving time and costs associated to a reduced
number of experimental tests (in vitro and in vivo).
The Ordinal Methodology here proposed could
also become even more effective when finalized to
Drug Design. That is, when thought of as a
supporting method in the research for new
compounds, precisely because exon skipping
method cannot be considered, at present, as being
the definitive solution to DMD (ib.).
This is even truer when considering that, as a
consequence of the reduced computation power
need, such a fundamental research could be diffused
to all those research laboratories involved in the
field and, in this way, increasing the probability of a
more rapid success.
This latter aspect can be considered as being the
fundamental advantage of the present Methodology
with respect to that presented in (Giannantoni
BIOINFORMATICS 2011 - International Conference on Bioinformatics Models, Methods and Algorithms
246

2010b). In that case, in fact, the Methodology still
required very high computation powers.
Consequently, in that preliminary version it could
only be adopted by very important Research Groups
and/or big Pharmaceutical Companies.
From a more general point of view, the main aim
of this paper is thus to show that, in the light of the
Maximum Ordinality Principle, it is possible to
realize Ordinal Models of several biological
Systems, with very significantly related Informatics
advantages.
In fact the adoption of the Maximum Ordinality
Principle as reference criterion leads to minimize all
the associated cardinalities, in all their various
forms: i) a reduced incidence of computation,
because the “sequence” of traditional “functional
operations” (e.g. successive derivatives) is replaced
by “co-instantaneous” derivatives (Giannantoni
2010b); ii) a reduction of computation power, due to
always explicit solutions; iii) a substantial absence
of sophisticated numerical methods for getting a
solution; iv) with the additional advantage of
eliminating the correlative solution “drifts” (ib.).
The relevance of these advantages could also be
pointed out by considering that the Clay
Mathematics Institute announced in the year 2000
that they would pay a US$ 1,000,000 prize for the
first person to prove a solution to the famous P vs
NP problem.
In this respect, the solution to Protein Folding
previously shown seems to indicate that there are no
NP problems, in any case. Obviously, this is true
only when the considered Processes are analyzed in
adherence to the Maximum Ordinality Principle, and
consequently modeled in terms of IDC. Such an
Ordinal Approach in fact does not limit its validity
exclusively to the case of Protein Folding but, as
already said, is also applicable to Molecular
Docking and Drug Design. This is precisely because
the ever-present specularity of the pertinent Ordinal
Matrices always leads to solutions which can be run
in absolutely comparable computation time.
However, beside the afore-mentioned
advantages, it is worth emphasizing that Methods of
solution based on IDC operate in terms of
Ordinality, whose various forms are always
represented through associated cardinalities, which
are consequently always understood as “cipher-
values”.
Such an extremely important aspect will be gone
into more in depth in the next section.
5 CONCLUSIONS
The adoption of the Maximum Ordinality Principle
as reference criterion in modeling “intractable”
problems in Biology (e.g. Protein Folding) led us to
a preliminary result concerning their solubility in
explicit terms, with a consequential significant
reduction of the computation time (from thousand of
years to some minutes), always evaluated, in both
cases, with specific reference to the most updated
computers.
Afterwards, a much deeper analysis of the
emergent properties of the Ordinal Model adopted
led us to recognize some profound forms of
“symmetry” (thus termed as “specularity”). This
enabled us to further improve the Informatics
methods of solution, especially because of the
extremely reduced influence of the cardinal values to
be calculated. This led to: i) a reduced number of
computations required to obtain the solution; ii)
much lower computation powers; iii) a reduced need
of electrical power supply, iv) without mentioning
the higher precision achieved, because IDC always
adopts “co-instantaneous” derivatives, which
eliminate the “drift” associated to a “step by step”
derivation, which characterizes TDC.
The generality of the approach based on the
Maximum Ordinality Principle (and associated IDC)
also enables us to assert that the achieved results are
not limited to the specific case analyzed, but can
easily be extended to other aspects, such as, for
instance, Molecular Docking and Drug Design. In
actual fact the results previously shown indicate that
the dynamic evolutions of the above-mentioned
Processes can be described in a similar way. They
can consequently be run in comparable computation
time and, above all, on computers of much lower
computation power (1 Gigaflop).
This clearly means that, in the near future, any
researcher would be able to analyze the dynamic
behavior of any biological Process, by means of
his/her own personal computer, simply sitting at
his/her own desk. This is because the optimization of
any Process could then be obtained by means of
successive attempts, each one lasting a few minutes.
However, as already anticipated, all these
advantages should not lead to underestimate the
unavoidable “training” on behalf of the User,
because the computer code operates in terms of both
Ordinality and associated cardinality. This means
that the code interprets cardinal inputs in terms of
“cipher” values and, correspondently, furnishes
cardinal outputs that must analogously be interpreted
as “cipher” of their corresponding Ordinality. The
BIO-INFORMATICS IN THE LIGHT OF THE MAXIMUM ORDINALITY PRINCIPLE - The Case of Duchenne
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outputs of any Ordinal Model, in fact, cannot be
understood as mere cardinalities, because this would
alter the corresponding proper meaning, by making
the solution vanish.
Such a “training” period is the essentially
finalized to get familiar with such a “com-possible”
Ordinal Approach, whose basic characteristics have
already been presented in (Giannantoni 2010a,b).
In this respect, it is worth recalling what was
already said on that occasion. Any Mathematical
Model (and associated code) based on the Maximum
Ordinality Principle should not be considered as
being reducible to a mere mathematical “tool”, that
is as simply being able to solve “intractable”
problems in a more efficient way. This is because it
is the reflex of a radically new methodology,
precisely because based on IDC. This new
differential calculus, in fact, “translates” into an
adherent formal language a gnoseological approach
which is completely different from the traditional
one. This difference resides on the three new basic
presuppositions: Generative Causality, Adherent
Logic, Ordinal Relationships (ib.).
From an even more general point of view, all the
afore-mentioned results are substantially due to the
independence of the Maximum Ordinality Principle
from the “rigidity” of Classical Thermodynamics.
The latter in fact prevents us from getting the correct
solutions (Giannantoni 2010a), especially because of
the Energy conservation Principle, which represents
“a limitation imposed to freedom of complex
systems” (Poincaré 1952, p. 133) and excludes the
“emergent properties that arise through the multiple
relations existing between individual components of
the System” (Van Regenmortel 2001).
REFERENCES
Aartsma-Rus A. et al., 2006. Exploring the Frontiers of
Therapeutic Exon Skipping for Duchenne Muscular
Dystrophy by Double Targeting within One or
Multiple Exons. Molecular Therapy, Vol. 14, 401-407.
Giannantoni C., 2001. The Problem of the Initial
Conditions and Their Physical Meaning in Linear
Differential Equations of Fractional Order. Applied
Mathematics and Computation 141 (2003) 87-102.
Giannantoni C., 2002. The Maximum Em-Power Principle
as the basis for Thermodynamics of Quality. Ed.
S.G.E., Padua, ISBN 88-86281-76-5.
Giannantoni C., 2004. Mathematics for Generative
Processes: Living and Non-Living Systems. 11
th
International Congress on Computational and Applied
Mathematics, Leuven, July 26-30, 2004. Applied
Mathematics and Computation 189 (2006) 324-340.
Giannantoni C., 2006. Emergy Analysis as the First
Ordinal Theory of Complex Systems. Proceedings of
the Fourth Emergy Conference 2006. Gainesville,
Florida, USA, January 17-22, pp. 15.1-15.14.
Giannantoni C., 2008. From Transformity to Ordinality, or
better: from Generative Transformity to Ordinal
Generativity. Proceedings of the Fifth Emergy
Conference. Gainesville, Florida, USA, January 31-
February 2, 2008.
Giannantoni C., 2009. Ordinal Benefits vs Economic
Benefits as a Reference Guide for Policy Decision
Making. The Case of Hydrogen Technologies. Energy
n. 34 (2009), pp. 2230–2239.
Giannantoni C., 2010a. The Maximum Ordinality
Principle. A Harmonious Dissonance. Proceedings of
the Sixth Emergy Conference. Gainesville, Florida,
USA, January 14-16, 2010.
Giannantoni C., 2010b. Protein Folding, Molecular
Docking, Drug Design. The Role of the Derivative
“Drift” in Complex Systems Dynamics. Proceedings
of Third International Conference on Bioinformatics,
Valencia, Spain, January 20-24, 2010.
Odum H. T., 1994a. Ecological and General Systems. An
Introduction to Systems Ecology. Revised Edition.
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Odum H. T., 1994b. Environmental Accounting. Environ.
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Odum H. T., 1994c. Self Organization and Maximum
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Poincaré H., 1952. Science and Hypothesis. New York:
Dover.
Van Regenmortel M.H.V., 2001. Pitfalls of Reductionism
in the Design of Peptide-based Vaccines. Vaccine 19
(2001) 2369-2374.
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Exon Skipping Across the Human Dystrophin Gene
Transcript. Molecular Therapy, Vol. 15, 1288-96.
APPENDIX
The basic concepts of IDC and, in particular, the
concept of “incipient” derivative, have already been
presented in (Giannantoni 2010b). This appendix is
then finalized to recall that the three Fundamental
Processes introduced by H.T. Odum (1994a,b,c),
that is Co-Production, Inter-Action and Feed-Back,
when formalized in terms of IDC, and analyzed
under Maximum Ordinality conditions, appear as
being one sole Generative Process.
For the sake of simplicity, we can always refer to
Ordinal relationships represented by exponential
functions (in the most general form
)(t
e
α
) because,
as is well know, any function
)(tf
can always be
written as
)()(ln
)(
ttf
eetf
α
==
(4) (Giannantoni
(ib.)).
BIOINFORMATICS 2011 - International Conference on Bioinformatics Models, Methods and Algorithms
248

Co-Production Process
This Process, schematically graphed in Fig. 1, can
formally be represented by means of the “incipient”
derivative of order 1/2. This derivative, in fact, gives
rise to a “binary” function, that is: an output made
up of two distinct entities, which however form one
sole thing. This is equivalent to say that the two “by-
products”, precisely because generated by the same
unique (Generative) Process, keep memory of their
common and indivisible origin, even if they may
have, later on, completely different topological
locations in time:
)(t
e
α
)(
)(
)(
t
e
t
t
α
α
α
⋅
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
+
D
D
Figure 1: Representation of a Co-Production Process.
The genesis of “binary” functions (from a
Co-Production Process) can formally be represented
as:
)()(
2
1
)(
)(
)(
tt
e
t
t
e
td
d
αα
α
α
⋅
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
+
=
∼
∼
D
D
(5)
where
)(t
D
α
represents the first order incipient
derivative (ib.), while the derivative 1/2 explicitly
reminds us that the output generated is “1” sole
entity, although made up of “2” parts. In other terms
the output, when understood as a whole, is much
more than the simple sum of its single parts. Said
differently, the uniqueness of the Generative
Process, recognized as being a specific property of a
Co-Production Process, remains as being in-
divisible, and thus also ir-reducible to the
component parts. This is why the correspondent
Ordinality should better be represented as
∼
2/1
.
However, in order to simplify the various formulae,
the “tilde” notation indicating the Ordinality can also
be omitted (even if always understood).
Inter-Action Process
This Generative Process can easily be illustrated by
considering first a single input Process (see Fig. 2).
In such a case the Process, modeled through the
incipient derivative of Order 2, represents a
reinforcement of the same input, so giving rise to a
new entity which, however, is much more than the
simple (cardinal) product of the
original input by itself
considered, and it can be thus represented as
)(t
e
α
)(
)](),([
t
ett
α
αα ⋅
DD
Figure 2: Representation of a “duet” Process.
This Process can be termed as “Generative”
precisely because the two contributions not only
reinforce each other, but are also unified in a new
one sole entity. In other terms, they not only increase
the cardinality of their joint action, but also generate
an exceeding Quality, represented by the uniqueness
and irreducibility of their co-operating activity,
because solidly and indissolubly orientated in the
same “direction”. This is why the corresponding
output can be termed as a “duet” function, while the
Process can formally be represented as follows
)()(2
)](),([)/(
tt
ettetdd
αα
αα ⋅=
∼∼ DD
(6)
It is then easy to recognize that Eq. (7) can never
be represented in terms of TDC because, in this case,
we have
)(2)(2
)]()([)/(
tt
ettedtd
αα
αα ⋅+=

(7)
Accordingly, even if the term
)(t
α

(due to the
“step by step” derivation) equals zero, the Process is
always understood in mere cardinal terms. The
output reduces in fact to the traditional scalar
product between the two quantities
)(t
α

. This is
because TDC, as already shown in (Giannantoni
2006, 2008, 2010b), aprioristically “filters” any
form of Ordinality.
The True Inter-Action Process
The Inter-Action Process, in its proper definition,
manifests its true essence when associated to a Co-
Production Process. In such a case we can also speak
of an Inter-Action Process characterized by a
“subjacent” Co-Production Process (with its
associated “binary” function). The Process can be
then characterized by a derivative of Order 2/2 and
thus represented as in Fig. 3
)(
)(
2
1
t
t
e
e
α
α
Figure 3: Representation of a true Inter-Action Process.
2
)/( tdd
∼∼
2/2
)/( tdd
∼∼
Second hand of
Eq. (8)
2
1
)/( tdd
∼∼
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249

In such a case the two inputs not only contribute to a
reciprocal reinforcement, but are also reciprocally
coupled in the form of a “binary” function. The
Process thus gives rise to a “duet-binary” function:
=
∼
∼
][)(
)()(
2
2
21
tt
ee
td
d
αα
)()(
1
2
2
1
21
)(
)(
,
)(
)(
tt
ee
t
t
t
t
αα
α
α
α
α
⋅
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
+
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
+
D
D
D
D
(8)
In addition, such a coupling is further enhanced
by the inter-exchange (and successive coupling) of
the specific “genetic” properties of the input Ordinal
functions (see
)(
1
t
D
α
and
)(
2
t
D
α
, respectively).
A significant example of this Generative Process
can be represented by the generation of a living
being. The formal expression (8), in fact, would be a
preliminary representation of the re-composition of a
completely new couple of chromosomes by starting
from one chromosome pertaining to the father and
the other pertaining to the mother. Evidently, the
Process is here extremely simplified. In fact, in the
human case (for instance) we should have to
consider 23 couples of chromosomes deriving from
the father and 23 from the mother, respectively,
which give rise to a completely new human being,
characterize by 46 new couples of chromosomes.
Ordinal Feed-back
This Process can easily be illustrated on the basis of
the Inter-Action Process, by assuming that the
Ordinal output of the Process contributes, together
with the input, to its same genesis (see Fig. 4).
)(t
e
α
Figure 4: Representation of an Ordinal Feed-Back Process.
In such a case the output represents a reproduction
of the input, although at a higher Ordinality level.
This is why the derivative of Order (2/2) is
specifically represented in brackets: to expressly
point out such a harmonic consonance between the
input and the output of the Ordinal Feed-Back
Process, which can be represented in formal terms as
follows
=
∼∼
)()2/2(
)/(
t
etdd
α
)(
)(
)(
,
)(
)(
t
e
t
t
t
t
α
α
α
α
α
⋅
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
+
−
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
+
=
D
D
D
D
(9)
We are now able to formalize the fact that, under
Maximum Ordinality conditions, the three above-
mentioned Processes constitute one sole Process,
which can be represented by the Ordinality
}2/2{
,
because characterized by a perfect internal
specularity.
Under the same conditions the Ordinal Inter-
Action between two distinct Processes generates a
new entity of Ordinality
}2/2{
}2/2{
(also
represented as {2/2}↑{2↑}), where the bracket
notation {2↑} remind us the perfect internal
“specularity” between the two basic Processes.
Such an internal “specularity” is due to interior
harmony relationships of the Process, whose number
and typology are defined by Transformation (2).
These in fact express particular “coupling
conditions” between different order fractional
derivatives (Giannantoni 2001, 2004). For example:
()
()
()
() =
∼∼∼∼
tftddtftdd
2/12/1
)/()/( D
() ()
tftddtf
)2/2(
)/(
∼∼
= D
(10)
which is always valid for any function
)(tf
, also
under steady state conditions.
Additional mathematical details about such
harmony conditions can be found in (Giannantoni
2009).
)2/2(
)/( tdd
∼∼
Second hand of
E
q
.
(
9
)
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