# similarity and analogous systems: dynamic adjointness and topological equivalence

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## 0.1 Dynamic Adjointness, Similarity and Analogous Systems.

1. 1.

Comparing Systems: Similarity Relations;

2. 2.

3. 3.

Classification as a Dynamic Analogy;

4. 4.

5. 5.

Topological Dynamics: Topological Conjugation/Conjugacy and Dynamic Equivalence;

6. 6.

Applications of Adjointness in Quantum Field Theories (QFT);

7. 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 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 $\mathcal{A}$ and $\mathcal{A}^{*}$, are dynamically adjoint if they can be represented naturally by the following (functorial) diagram:

 $\xymatrix@M=0.1pc@=4pc{\mathcal{A}\ar[r]^{F}\ar[d]_{F^{\prime}}&\mathcal{A}^{*% }\ar[d]^{G}\\ {\mathcal{A}^{*}}\ar[r]_{G^{\prime}}&{\mathcal{A}}}$ (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 state-space transition functions $\Delta:\mathcal{A}\rightarrow\mathcal{A}$ and $\Delta^{*}:\mathcal{A}^{*}\rightarrow\mathcal{A}^{*}$ 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 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 $\mathcal{A}$ of any functor’), whereas the right-adjoint (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, 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.