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%%% Primary Title: highly complex systems
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%%% Filename: HighlyComplexSystems.tex
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\begin{document}
\subsection{Highly-complex systems}
\subsubsection{Introduction}
Modeling the emergence of the ultra-complex system of the
human mind--based on the super-complex human organism-- one needs to consider an associated
progression towards \htmladdnormallink{higher dimensional algebras}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html} from the lower dimensions of human neural network \htmladdnormallink{dynamics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html} and the simple algebra of physical dynamics, as shown in the following, essentially \emph{\htmladdnormallink{non-commutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}} \htmladdnormallink{categorical diagram}{http://planetphysics.us/encyclopedia/CategoricalDiagramsDefinedByFunctors.html}.
One can represent by \htmladdnormallink{square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} categorical diagrams the emergence of ultra-complex
dynamics from the super-complex dynamics of human organisms coupled {\em via} social interactions
in characteristic patterns represented by Rosetta biogroupoids, together with the complex--albeit inanimate--systems with `chaos' as discussed next.
\subsection{Diagrams of simple and highly-complex systems}
\begin{definition}
An \emph{ultra-complex system, $U_{CS}$} is defined as an \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} in the following \htmladdnormallink{non-commutative diagram}{http://planetphysics.us/encyclopedia/NonCommutativeDynamicModelingDiagrams.html} of \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} and \htmladdnormallink{dynamic system}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html} \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} or `dynamic transformations':
$$ \xymatrix@C=5pc{[SUPER-COMPLEX] \ar [r] ^{(\textbf{Higher
Dim})} \ar[d] _{\Lambda}& [[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]])(U_{CS}= ULTRA-COMPLEX) \ar [d]^{onto}\\ COMPLEX&
\ar [l] ^{(\textbf{Generic Map})}[SIMPLE]} $$
\end{definition}
\subsection{Remarks}
Note that the above \htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} is indeed not `natural' (i.e. it is not commutative) for reasons
related to the emergence of the higher dimensions of the super--complex
(biological/organismic) and/or ultra--complex (psychological/neural network dynamic) levels in comparison with
the low dimensions of either simple (physical/classical) or complex (chaotic) dynamic systems.
An \emph{ultra-complex system} represents the human mind and consciousness from the standpoint
of a \htmladdnormallink{categorical ontology}{http://planetphysics.us/encyclopedia/CategoricalOntology.html} theory of levels as the highest level of \htmladdnormallink{complexity}{http://planetphysics.us/encyclopedia/Complexity.html} that
emerged through biological and social coevolution over the last $2.2$ million years on Earth.
\end{document}
