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单词 CategoryOfMolecularSets
释义

category of molecular sets


0.1 Molecular sets as representations of chemical reactions

A uni-molecular chemical reaction is defined by the natural transformations

η:hAhB,

specified in the following commutative diagramMathworldPlanetmath representing molecular sets and their quantum transformationsPlanetmathPlanetmath:

\\xymatrix@M=0.1pc@=4pchA(A)=Hom(A,A)\\ar[r]ηA\\ar[d]hA(t)&hB(A)=Hom(B,A)\\ar[d]hB(t)hA(B)=Hom(A,B)\\ar[r]ηB&hB(B)=Hom(B,B),(0.1)

with the states of molecular sets Au=a1,,an andBu=b1,bn being defined as the endomorphismPlanetmathPlanetmathPlanetmath sets Hom(A,A) and Hom(B,B), respectively. In general, molecular sets MS are defined as finite setsMathworldPlanetmath whose elements are molecules; the molecules are mathematically defined in terms of their molecular observables as specified next. In order to define molecular observables one needs to define first the concept of a molecular class variable or m.c.v.

A molecular class variables is defined as a family of molecular sets [MS]iI,with I being either an indexing set, or a proper classMathworldPlanetmath, that defines the variation range of the m.c.v.Most physical, chemical or biochemical applications require that I is restricted to a finite set, (that is, without any sub-classes). A morphismMathworldPlanetmath, or molecular mapping, Mt:MSMS of molecular sets, with tT being real time values, is defined as a time-dependent mapping or function MS(t) also called a molecular transformation, Mt.

An m.c.v. observable of B, characterizing the productsPlanetmathPlanetmathPlanetmath of chemical type “B” of a chemical reaction is defined as a morphism:

γ:Hom(B,B),

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

\\xymatrix@M=0.1pc@=4pcHom(A,A)\\ar[r]f\\ar[d]e&Hom(B,B)\\ar[d]γHom(A,A)\\ar[r]δ&R,(0.2)

with c:Au*Bu*, and Au*, Bu* being, respectively,specially prepared fields of states of the molecular sets Au, and Bu within a measurement uncertainty range, Δ, which is determined by Heisenberg’s uncertainty relationMathworldPlanetmathPlanetmathPlanetmath, or the commutatorPlanetmathPlanetmath of the observable operators involved, such as [A*,B*], associated with the observable A of molecular set Au, and respectively, with the obssevable B of molecular set Bu, in the case of a molecular set Au interacting with molecular set Bu.

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

0.2 Category of molecular sets and their transformations

Definition 0.1.

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

Remark 0.1.

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

References

  • 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 MolecularBiology., in Proceedings of the SIAM Natl. Meet., Denver,CO.; Eprint at cogprints.org with No. 3675.
  • 5 Baianu, I.C.: 1984, A Molecular-Set-Variable Model of Structuraland Regulatory Activities in Metabolic and Genetic NetworksFASEB Proceedings 43, 917.
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