PlanetPhysics/Category of Molecular Sets 2

Molecular Sets and Representations of Chemical ReactionsEdit

The uni-molecular chemical reaction is represented by the natural transformations as specified by the following commutative diagram:

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with the states of the molecular sets and being represented by certain endomorphisms in and , respectively. In general, molecular sets are defined as finite sets whose elements are `molecules' defined in terms of their molecular observables that are specified below. molecular class variables, or 's are defined as families of molecular sets , with being an indexing set, or class, defining the range of molecular variation of the  ; most applications require that is a proper, finite set, (i.e., without any sub-classes). A morphism of molecular sets, with being real time values, is defined as a time-dependent mapping or function also called a molecular transformation, .

An observable of , characterizing the products of chemical type "B" of a chemical reaction is defined as a morphism:

where is the set or field of real numbers. This mcv-observable is subject to the following commutativity conditions:

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~

with , and , being, respectively, specially prepared fields of states of the molecular sets , and within a measurement uncertainty range, , which is determined by Heisenberg's uncertainty relation, or the commutator of the observable operators involved, such as , associated with the observable of molecular set , and respectively, with the obssevable of molecular set , in the case of a molecular set interacting with molecular set .

With these concepts and preliminary data one can now define the category of molecular sets and their transformations as follows.

Category of molecular sets and their transformationsEdit

The category of molecular sets is defined as the category whose objects are molecular sets and whose morphisms are molecular transformations .

This is a mathematical representation of chemical reaction systems in terms of molecular sets that vary with time (or 's), and their transformations as a result of diffusion, collisions, and chemical reactions.

All SourcesEdit

[1][2][3][4][4]

ReferencesEdit

  1. Bartholomay, A. F.: 1960. Molecular Set Theory. A mathematical representation for chemical reaction mechanisms. Bull. Math. Biophys. , 22 : 285-307.
  2. Bartholomay, A. F.: 1965. Molecular Set Theory: II. An aspect of biomathematical theory of sets., Bull. Math. Biophys. 27 : 235-251.
  3. Bartholomay, A.: 1971. Molecular Set Theory: III. The Wide-Sense Kinetics of Molecular Sets ., Bulletin of Mathematical Biophysics , 33 : 355-372.
  4. 4.0 4.1 Baianu, I. C.: 1983, Natural Transformation Models in Molecular Biology., in Proceedings of the SIAM Natl. Meet ., Denver, CO.; Eprint at cogprints.org with No. 3675. Cite error: Invalid <ref> tag; name "ICB2" defined multiple times with different content