Dynamic Adjointness, Similarity and Analogous Systems.

Comparing Systems: Similarity Relations between Analogous or Adjoint Systems. Classification as a Dynamic Analogy, Categorical Adjointness or Functional Homology.

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

(0.1)

with and being isomorphic (that is, representing natural equivalences between adjoint functors of the same kind, either left or right), and as above in diagram (2.5), the two diagonals are, respectively, the state-space transition functions and 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, whereas the right-adjoint (or coadjoint) functor, such as G above, preserves colimits. (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).

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

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, whereas the right-adjoint (or coadjoint) functor, such as G above, preserves colimits. (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). Thus, dynamic attractors and genericity of states are preserved for 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 at all `mystical' in Husserl's sense, 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 (`mystical') Kantian-transcedental logic or by impossible, direct phenomenal observations at the Planck scale.