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molecular set and molecular class variables
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Definition 0.1 Molecular sets  are defined as finite sets of molecules that are being represented as elements of such sets.
An alternative definition is available in terms of natural transformations of organismic structures or quantum functorial morphisms, as further specified next.
Definition 0.4 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:
![$\displaystyle \xymatrix@M=0.1pc @=4pc{Hom(A,A) \ar[r]^{f} \ar[d]_{e} & Hom(B,B)\ar[d]^{\gamma} \\ {Hom(A,A)} \ar[r]_{\delta} & {R},}$](http://images.planetmath.org:8080/cache/objects/10867/l2h/img18.png "$\displaystyle \xymatrix@M=0.1pc @=4pc{Hom(A,A) \ar[r]^{f} \ar[d]_{e} & Hom(B,B)\ar[d]^{\gamma} \\ {Hom(A,A)} \ar[r]_{\delta} & {R},}$") |
(0.1) |
with
 , and  ,  being specially prepared fields of states, within a measurement uncertainty range,  , specified by the observable operator commutation relation as generally defined by the Heisenberg Uncertainty Principle in Quantum Mechanics.
Remark: The family
![$ ([M]_t)_{t \in T}$](http://images.planetmath.org:8080/cache/objects/10867/l2h/img23.png "$ ([M]_t)_{t \in T}$") and the associated class of its molecular transformations can be thus employed to define a category of molecular sets, with composition defined by the concatenation of sequential molecular transformations.
- 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
- Baianu, I. C.: 1983, Natural Transformation Models in Molecular Biology., in Proceedings of the SIAM Natl. Meet., Denver, CO.; Eprint No. 3675 at cogprints.org/3675/01 as ``Naturaltransfmolbionu6.pdf''.
- 4
- Baianu, I.C.: 1984, A Molecular-Set-Variable Model of Structural and Regulatory Activities in Metabolic and Genetic Networks FASEB Proceedings 43, 917.
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"molecular set and molecular class variables" is owned by bci1.
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See Also: category of molecular sets, molecular set theory, categories and supercategories in relational biology, complex systems biology, abstract relational biology, category of molecular sets, supercategory of variable molecular sets
| Other names: |
msv's, variable molecular sets |
| Also defines: |
molecular set, molecular class variable, mcv observable |
| Keywords: |
category of molecular sets, molecular set variable, msv, variable molecular set, molecular set theories, natural transformations of molecular structures |
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Cross-references: concatenation, composition, category of molecular sets, uncertainty principle, relation, operator, commutativity, field, type, products, functorial morphisms, natural transformations of organismic structures, molecular transformation, function, mapping, real, morphism, applications, variation, range, indexing set, class, terms, mathematical representations, molecular set variables, finite sets
There are 2 references to this entry.
This is version 26 of molecular set and molecular class variables, born on 2008-07-25, modified 2008-10-16.
Object id is 10867, canonical name is MolecularSetVariable.
Accessed 717 times total.
Classification:
| AMS MSC: | 18-00 (Category theory; homological algebra :: General reference works ) | | | 18E05 (Category theory; homological algebra :: Abelian categories :: Preadditive, additive categories) |
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Pending Errata and Addenda
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![$ ([M]_t)_{t \in T}$](http://images.planetmath.org:8080/cache/objects/10867/l2h/img3.png "$ ([M]_t)_{t \in T}$"){kind=small}
![$ [M_S]_{i \in I}$](http://images.planetmath.org:8080/cache/objects/10867/l2h/img6.png "$ [M_S]_{i \in I}$"){kind=small}
