similarity and analogous systems: dynamic adjointness and topological equivalence

similarity and analogous systems: dynamic adjointness and topological equivalence

\xyoption

curve

0.1

Dynamic Adjointness, Similarity and Analogous Systems.

1.

Comparing Systems: Similarity Relations

;

2.

Analogous and

Adjoint

Systems

;

3.

Classification as a Dynamic

Analogy

;

4.

Categorical

Adjointness and

Functional

Homology

;

5.

Topological Dynamics:

Topological Conjugation

/Conjugacy and Dynamic

Equivalence

;

6.

Applications of Adjointness in

Quantum Field Theories

(QFT)

;

7.

Applications of Categorical Adjointness in Mathematical Biology (

http://planetmath.org/MathematicalBiology

);

Categorical comparisons of different types of systems in

diagrams

provide useful means for their classification and understanding the

relations

between them. Thus, two dynamic systems whose

state spaces

are

isomorphic

such that their dynamics commute with the

isomorphism

of their state spaces are defined to be

analogous systems

. A related,
dynamic similarity between systems is recalled in the next

section

.

From a global viewpoint, comparing

categories

of such different systems does reveal useful
analogies, or similarities, between systems and also their

universal properties

. According to Rashevsky (1969), general
relations between sets of biological organisms can be compared with those between societies, thus leading to more general principles pertaining to both. This can be considered as a further, practically useful
elaboration of Spencer’s philosophical principle ideas in biology and sociology.

0.1.1

Topological Dynamics: Topological Conjugation/Conjugacy and Dynamic Equivalence.

When two dynamic systems have state spaces with defined

topologies

one can naturally
define their dynamic equivalence in terms of topological conjugation (

http://planetmath.org/TopologicalConjugation

) as a form of
dynamic,

topological equivalence

. Thus, topological conjugation can be considered as a particular case of the
commutative square diagram (0.1) in the next subsection where the four corners of diagram (0.1) are replaced sequentially by

topological spaces

X

,

Y

,

X

and

Y

, respectively, and the two pairs of

adjoint functors

(

F

,

G

)

and

(

G

′

,

F

′

)

are then naturally replaced by two corresponding pairs of homeomorphisms between

X

and

Y

.

Remark:

One notes that topological equivalence is considered to be a weaker equivalence by comparison with topological conjugacy or

conjugation

because–unlike topological conjugacy–a topological equivalence of

dynamical systems

does not map the time variable along with the orbits and their orientation. An often cited example of topologically equivalent, but not

topologically conjugate

systems, is that of the non-hyperbolic class of two-dimensional (2D)

solutions of systems

of

differential equations

which have closed orbits.

0.1.2

Diagrams Linking Super– and Ultra– Complex/Meta–Levels.

When viewed from a formal perspective of Poli’s theory of levels (Baianu and Poli, 2008), the two levels of super– and ultra–

complex systems

are quite

distinct

in many of their defining properties, and therefore, categorical diagrams that ‘mix’ such distinct levels

do not commute

.
Considering dynamic similarity, Rosen (1968) introduced the

concept

of

‘analogous’

(classical) dynamical systems in terms of categorical, dynamic isomorphisms between their isomorphic state-spaces that commute with their transition (state)

function

, or dynamic laws. However, the

extension

of this concept to either complex or

super-complex systems

has not yet been investigated, and may be

similar

in importance to the

introduction

of the Lorentz-Poincaré group of

transformations

for reference frames in Relativity theory. On the other hand, one is often looking for

relational invariance

or

similarity in functionality

between different organisms or between different stages of

development

during ontogeny–the development of an organism from a fertilized egg. In this context,
the categorical concept of ‘

dynamically adjoint systems

’ was introduced in relation to the data obtained through nuclear transplant experiments (Baianu and Scripcariu, 1974). Thus, extending the latter concept to super– and ultra– complex systems , one has in general, that two complex or supercomplex systems with ‘state spaces’ being defined respectively as

𝒜

and

𝒜

*

, are dynamically adjoint if they can be represented naturally by the following (functorial) diagram:

\xymatrix

⁢

@

⁢

M

=

0.1

⁢

p

⁢

c

⁢

@

=

4

⁢

p

⁢

c

⁢

𝒜

⁢

\ar

⁢

[

r

]

F

⁢

\ar

⁢

[

d

]

F

′

⁢

&

⁢

𝒜

*

⁢

\ar

⁢

[

d

]

G

⁢

𝒜

*

⁢

\ar

⁢

[

r

]

G

′

⁢

&

⁢

𝒜

(0.1)

with

F

≈

F

′

and

G

≈

G

′

being isomorphic (that is,

≈

representing

natural equivalences

between adjoint functors of the same kind, either left or right), and as above in diagram (0.1), the two diagonals are, respectively, the state-space

transition functions

Δ

:

𝒜

→

𝒜

and

Δ

*

:

𝒜

*

→

𝒜

*

of the two adjoint dynamical systems. (It would also be interesting to investigate dynamic adjointness in the context of quantum dynamical systems and

quantum automata

, as defined in Baianu, 1971a).

A

left-adjoint

functor

, such as the functor F in the above

commutative diagram

between categories representing state spaces of

equivalent

cell nuclei

preserves limits

(or ‘commutes with the

inductive limit

in

𝒜

of any functor’), whereas the

right-adjoint

(or coadjoint) functor, such as G above,

preserves

colimits

(or commutes with the

projective limit

in

𝒜

*

of any functor). (For precise definitions of adjoint functors the reader is
referred to Brown, Galzebrook and Baianu, 2007, as well as to Popescu, 1973,
Baianu and Scripcariu, 1974, and the initial paper by Kan, 1958).

0.1.3

Nuclear Transplant Experiments, Cloning, Embryogenesis, Development and Other Biological applications

Consider

dynamic

attractors

and genericity of states

as in the above diagram that are preserved in differentiating cells up to the blastula stage of organismic development. Subsequent stages of ontogenetic development can be considered only ‘weekly adjoint’ or partially analogous. Similar dynamic controls may operate for controlling division cycles in the cells of different organisms; therefore, such

instances

are also good example of the dynamic adjointness relation between cells of different organisms that may be very far apart phylogenetically, even on different ‘branches of the tree of life.’ A more elaborate dynamic concept of ‘homology’ between the genomes of different species during evolution was also proposed (Baianu, 1971a), suggesting that an entire phylogenetic series can be characterized by a topologically–rather than biologically–

homologous

sequence

of genomes which preserves certain genes

encoding

the essential

biological functions. A striking example was recently suggested involving the differentiation of the nervous system in the fruit fly and mice (and perhaps also man) which leads to the formation of the back, middle and front parts of the neural tube. A related, topological

generalization

of such a dynamic similarity between systems was previously introduced as

topological conjugacy

(Baianu, 1986-1987a; Baianu and Lin, 2004), which replaces

recursive

, digital

simulation

with symbolic, topological modelling for both super– and ultra– complex systems (Baianu and Lin., 2004; Baianu, 2004c; Baianu et al., 2004, 2006b). This approach stems logically from the introduction of topological/

symbolic computation

and topological computers Baianu, 1971b), as well as their natural extensions to

quantum nano-automata

(Baianu, 2004a), quantum automata and

quantum computers

(Baianu, 1971a, and 1971b, respectively); the latter may allow us to make a ‘quantum leap’ in our understanding Life and the higher

complexity

levels in general. Such is also the relevance of

Quantum Logics

and LM-logic

algebra

to
understand the

immanent

operational logics of the human brain and the associated mind meta–level. Quantum Logics concepts are introduced next that are also relevant to the fundamental, or ‘ultimate’, concept of spacetime, well-beyond our phenomenal reach, and thus in this specific sense,

transcedental

to our physical experience (perhaps vindicating the need for a Kantian–like

transcedental logic

, but from a quite different standpoint than that originally advanced by Kant in his critique of ‘pure’ reason; instead of being ‘mystical’- as Husserl might have said–the transcedental logic of quantized spacetime is very different from the Boolean logic of digital computers, as it is

quantum

, and thus non–

commutative

). A

Transcedental

Ontology

, whereas with a definite Kantian ‘flavor’, would not be as unacceptable as it was to Husserl, but would rely on ‘verifiable’ many–valued, non–commutative logics, and thus contrary to Kant’s original presupposition, as well as untouchable by Husserl’s critique. The fundamental nature of spacetime would be ‘provable’ and ‘verifiable’, but only to the extent allowed by Quantum Logics, not by an arbitrary Kantian–‘transcedental’ logic or by impossible, direct phenomenal observations at the Planck scale.

Title

similarity and analogous systems: dynamic adjointness and topological equivalence

Canonical name

SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence

Date of creation

2013-03-22 18:12:56

Last modified on

2013-03-22 18:12:56

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

45

Author

bci1 (20947)

Entry type

Topic

Classification

msc 03C98

Classification

msc 18-00

Classification

msc 37F99

Classification

msc 37F05

Classification

msc 03D80

Classification

msc 03D15

Classification

msc 03C52

Classification

msc 18A40

Synonym

analogy

Synonym

isomorphic dynamic systems

Synonym

dynamic similarity

Synonym

adjoint functor

Synonym

adjointness

Synonym

conjugacy

Synonym

isomorphic dynamical system

Related topic

SystemDefinitions

Related topic

TopologicalConjugation

Related topic

MathematicalBiology

Related topic

CommutativeVsNonCommutativeDynamicModelingDiagrams

Related topic

GroupoidCDynamicalSystem

Related topic

AdjointFunctor

Defines

analogous systems

Defines

topological conjugacy

Defines

dynamic adjointness

Defines

adjoint dynamical systems

Defines

equivalent dynamic systems

Defines

analogous systems

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