similarity and analogous systems: dynamic adjointness and topological equivalence
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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 $({G}^{\prime},{F}^{\prime})$ 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 nonhyperbolic class of twodimensional (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 statespaces that commute with their transition (state) function, or dynamic laws. However, the extension^{} of this concept to either complex or supercomplex systems^{} has not yet been investigated, and may be similar in importance to the introduction of the LorentzPoincaré 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 $\mathcal{A}$ and ${\mathcal{A}}^{*}$, are dynamically adjoint if they can be represented naturally by the following (functorial) diagram:
$$\text{xymatrix}\mathrm{@}M=0.1pc\mathrm{@}=4pc\mathcal{A}\text{ar}{[r]}^{F}\text{ar}{[d]}_{{F}^{\prime}}\mathrm{\&}{\mathcal{A}}^{*}\text{ar}{[d]}^{G}{\mathcal{A}}^{*}\text{ar}{[r]}_{{G}^{\prime}}\mathrm{\&}\mathcal{A}$$  (0.1) 
with $F\approx {F}^{\prime}$ and $G\approx {G}^{\prime}$ being isomorphic (that is, $\approx $ 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 statespace transition functions^{} $\mathrm{\Delta}:\mathcal{A}\to \mathcal{A}$ and ${\mathrm{\Delta}}^{*}:{\mathcal{A}}^{*}\to {\mathcal{A}}^{*}$ 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 leftadjoint 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 $\mathcal{A}$ of any functor’), whereas the rightadjoint (or coadjoint) functor, such as G above, preserves colimits (or commutes with the projective limit in ${\mathcal{A}}^{*}$ 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, 19861987a; 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 nanoautomata (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 LMlogic 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, wellbeyond 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  20130322 18:12:56 
Last modified on  20130322 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 1800 
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 