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Axiomatic Panbiogeography

offers an application of incidence geometry to historical biogeography by defining collection localities as points, tracks as lines and generalized tracks as planes.
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Incidence Geometry
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Quaternion Algebraic Geom
Primate Vicariances
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Martitrack Panbiogeograph
Processing Parallels
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Axiomatic Panbiogeography and Graph Theory

Panbiogeography is a discipline of historical biogeography that utilizes track graphs to present geographic distribution data.

Graph databases offer a means to demonstrate biogeographic relationships that enables simple visual  discrimination of different representations within panbiogeography.  It is especially handy when many tracks are being considered worthwhile.

Additionally, graph databases offer a means to realize the Panbiogeographic Atlas proposed by Craw, Grehan and Heads  (1999) and supports rigorous definitions of different kinds of nodes being utilized in the discipline. Graph databases offer a new database exploratory vehicle useful in the construction of generalized tracks.

Graph traversals offer new query possibilities. Above is a panbiogeographic structure one might traverse. Nodes, masses and baselines along with track widths explain the multicolored parallelism uniting polyphletic groups geographically.  So while it may not be that organisms have a paricular space species do when graphed relative to others.

One needs a way to relate the number C(added units) of a to unit decomposable Cgrossdigits record/symbolization through a grossone or larger numeral system.  We can make a graph database with the nodes equal to the record/symbol and the edges being the number to another node/record such that the entire graph is the numeral system itself.

Different graphs express different numeral systems of grossone depending on the grossdigits entered. Other edges can express addition, mulitiplication subtraction and division. The entire graph is a new model domain in which to optimize social selections.

A sufficiently large enough grossone graph database should enable a new framing for social evolution on two tiers (behavioral/developmental - evolutionary). The different social selection of salamanders via pheromones may be modeled and use a grossone representation into their biogeography.

This use of grossone in social evolution permits a reevaluation of Price and Smith  1973 s' individual category to intensity distinction in terms of infinite series of plays (individual lifetime fitness sums) where different C and D tactics can be continuously smoothed in intensity and number.  Thus we may be able to assert postively why a snake is genetically more able through social selection to use C to D  with differing intensity probabilites in terms of the evolution of the payoff matrix. Thus the notion of Evolutionary Stable Strategy  ESS can become enlarged and redefined in terms of graph traversals that measure social infrastructure selected as creatures optimize a disparate (infintesimal to infinite) stable equilibrium spaces of their NCE and NBS respectively.

1C(1) 2C(1) 1C(1) 2D(1 ) 3C(1) 1C(1) 1D(1) 2C(1)...n times

1C(1) 2C(1) 1C(1) 1C(1) 2C(1) 1C(1)  1D(1) 2 C(1)..k  times

Where (1) is grossone.

Thus the arguments for the advantage of grossone to divergence as applied to  zeta and eta functions advances the notion of evolutionary stable strategy within Social Selection Sensu Roughgarden similarly! A probe or provacation  (D after C) as defined by Price and Smith is precisely thought by Sergeyev's cut grossone. The category to intensity distinction of this difference in step is measureable by some NBS over an NCE!!  The notion of blinkingfractals (two infinite processes (behavorial to evolutionary tier in one population across populations/demes)) can help explore this amelioration of strategy and preference in the new larger theoretical space opened by creating this alternative to sexual selection.  Thank you Roughgarden!!!  Creating a calculus for grossone computations is all that remains needed. This will allow for instance the use of unconfoundation of physiological mechanisms on the molecular bond level per game through a somatic program in the individual to coalition transition per population to materialize Mayr's suggestion applied to Price and Smith's notion of a snake genetical fang "program" in its full information metaphor thanks to graph traversal languages' linguistics.

A probe or provOcation is a numeral system starting at finite number of steps beyond what display or tournament it is intended to probe or provocate.  It ends maximally grossone wise numerically this finite number of steps beyond where the other
tournament or display ends.  Thus an ending if not needed may be reflected in the evolutionary history as a recapitulation needed to be gone through.

Here are two other ways to use graph databses in panbiogeography.  Rather than have the species tracks be the verticies and the panbiog concepts the edges consider the geographic lat longs as the verticies (which they are commonly represented) and the min spans between them the edges.  Now scale across increasing area size or any kind of cladogram (molecular and morphological) and then compute the most central verticies with page rank or some other kind of traversal.  This would enable a demonstration of a generalized track in form-making or in space. It could also be used in time if geographic places of the various geological horizons were plotted in the same plane.  The generalized track thus pathed may help to geologists with issues of terrane accrection.

Thus one could use the geographic scaled genearlized track to compare the different monophyletic cladograms (polyphyletic track to Linnean hierarchy of different content cladograms).

Most Gremlin pipelines are serial in that one step feeds to the next, so on and so forth. There are situations where it is desirable to split a pipeline and thus, have n-parallel steps that are later merged back into a serial flow. To support this type of traversal, there are split and merge steps.

  • Split
    • copySplit: copy the incoming object to each pipeline
  • Merge
    • fairMerge: merge the parallel traversals in a round-robin fashion
    • exhaustMerge: merge the parallel traversals by exhaustively getting the objects of the first, then the second, etc.

This would be useful where one wishes to keep Eurycea and Plethodon glutinosis complex in parallel that are serial with respect to Desmognathus and Other Plethodon while reamaing open on the left and right to work with other taxa (those in Texas). Use of grossone numeral systems on different computers could facilitate this integration of the numbers involved as one works to determine where to split and where to merge to get the best database relative to the overall phylogeny.

Panbiogeography with Taxonomy can provide graphdatabase programmers a meta-graph for doing rich breadth first traversals by using a common rooting scheme. Translating these nodes, masses and baseline in the roughly approximate graph below

into  a container root  shows how track^node^mass^baseline panbiogeography with underlying variance is a breadth first walk.

 If the taxonomic tress of common node edges are included into a heterogenous graph then the track^node^mass^baseline can be the index for the multiple taxonomic graph tress which provide monophylogenetic "breadths" that can be explored before moving along the index.  the panbiogeographic atlas thus can provide a new semantics for graph database traversal pattern matching.  One would thus be able to ask of a general nonbiological (or biological) graph if it matches the  NODE3SALAMANDER2t^n^m^b pattern for instance.

Here are various ways to layout such a graphdatabase.