A manuscript with the title above is now on preprint server (https://www.biorxiv.org/content/10.1101/780882v1). We have already developed a new complex metric “small s” as an indicator of species existence and resolving fractal dimension accompanied with species dynamics (https://arxiv.org/abs/1603.00959). This calculation only requires the information for species density along some time development. One of the interesting problems here is that apparently single-dimensional information elucidates time development model as 3 + 1 dimensions. Inspired by Bethe ansatz, we tried here to construct mathematically rigorous model for the induction of additional dimensions together with time. We need ideas such as homology/cohomology and hierarchies of the system.
To this aim, three and only distinct topologies isomorphic and holomorphic to simply connected region of Riemann sphere, i.e. open unit disk, complex plane and Riemann sphere, are considered. For clarification of the model, data from liquid-chromatography mass spectrometry of proteins, and species density data of Dictyostelia community in wild are examined. With Clifford algebra, congruent zeta function and Weierstraß p-function with a help from type VI Painlevé equation, we are able to demonstrate introduction of hierarchy and time through one-dimensional probability space with those topologies. Furthermore, we can also grasp information of interaction in the model. Previously developed “small s” metric thus fulfills the necessity for characterizing dynamical hierarchy and interactions of the system of interest, only with abundance data along time development.
In future, we would like to generalize this type of model to any system of interest in condensed matter physics, by treating dimensional analyses carefully. This work (https://www.biorxiv.org/content/10.1101/780882v1) is the second work of a trilogy begun with https://arxiv.org/abs/1603.00959, and the final work would be much more interesting, representing finalization of my researches before my career break.
I have found three typos; p. 1, right column in middle: and\hat{\mathbb{C}}, “and” should be deleted; p. 6, right column in middle: 1.001511 -> 1.001571; p. 9, left column: “$P_0, P_1, P_2$ correspond to $\Re(v), \Re(v'), \Re(v'')$. For $\Re(v), \Re(v'')$, values close to zero represent large contributions, and for $\Re(v')$, large values represent large contributions. The inverses of $\Re(v'), \Re(v'')$ scale for $\Re(v')$” should be “$P_0, P_1, P_2$ correspond to $\Re(v'), \Im(v'), \Re(v'')$. For $\Re(v'), \Re(v'')$, values close to zero represent large contributions, and for $\Im(v')$, large values represent large contributions. The inverses of $\Re(v'), \Re(v'')$ scale for $\Im(v')$”. These will be corrected at submission to a journal.
2019.10.05 update:
All the above errors are corrected in the current version of the preprint.
https://www.biorxiv.org/content/10.1101/780882v2
