Talk:PlanetPhysics/Category of Molecular Sets 3

Original TeX Content from PlanetPhysics Archive

edit
%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: category of molecular reactions
%%% Primary Category Code: 00.
%%% Filename: CategoryOfMolecularSets3.tex
%%% Version: 4
%%% Owner: bci1
%%% Author(s): bci1
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.        
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}

\usepackage{html}

% this is the default PlanetPhysics preamble. 

% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}

% define commands here
\usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym}
\usepackage{xypic}
\usepackage[mathscr]{eucal}
\theoremstyle{plain}
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{example}{Example}[section]
%\theoremstyle{remark}
\newtheorem{remark}{Remark}[section]
\newtheorem*{notation}{Notation}
\newtheorem*{claim}{Claim}

\renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@
}}}
\numberwithin{equation}{section}

\newcommand{\Ad}{{\rm Ad}}
\newcommand{\Aut}{{\rm Aut}}
\newcommand{\Cl}{{\rm Cl}}
\newcommand{\Co}{{\rm Co}}
\newcommand{\DES}{{\rm DES}}
\newcommand{\Diff}{{\rm Diff}}
\newcommand{\Dom}{{\rm Dom}}
\newcommand{\Hol}{{\rm Hol}}
\newcommand{\Mon}{{\rm Mon}}
\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Ker}{{\rm Ker}}
\newcommand{\Ind}{{\rm Ind}}
\newcommand{\IM}{{\rm Im}}
\newcommand{\Is}{{\rm Is}}
\newcommand{\ID}{{\rm id}}
\newcommand{\GL}{{\rm GL}}
\newcommand{\Iso}{{\rm Iso}}
\newcommand{\Sem}{{\rm Sem}}
\newcommand{\St}{{\rm St}}
\newcommand{\Sym}{{\rm Sym}}
\newcommand{\SU}{{\rm SU}}
\newcommand{\Tor}{{\rm Tor}}
\newcommand{\U}{{\rm U}}
\newcommand{\A}{\mathcal A}
\newcommand{\Ce}{\mathcal C}
\newcommand{\D}{\mathcal D}
\newcommand{\E}{\mathcal E}
\newcommand{\F}{\mathcal F}
\newcommand{\G}{\mathcal G}
\newcommand{\Q}{\mathcal Q}
\newcommand{\R}{\mathcal R}
\newcommand{\cS}{\mathcal S}
\newcommand{\cU}{\mathcal U}
\newcommand{\W}{\mathcal W}
\newcommand{\bA}{\mathbb{A}}
\newcommand{\bB}{\mathbb{B}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\bD}{\mathbb{D}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bG}{\mathbb{G}}
\newcommand{\bK}{\mathbb{K}}
\newcommand{\bM}{\mathbb{M}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bO}{\mathbb{O}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bV}{\mathbb{V}}
\newcommand{\bZ}{\mathbb{Z}}
\newcommand{\bfE}{\mathbf{E}}
\newcommand{\bfX}{\mathbf{X}}
\newcommand{\bfY}{\mathbf{Y}}
\newcommand{\bfZ}{\mathbf{Z}}
\renewcommand{\O}{\Omega}
\renewcommand{\o}{\omega}
\newcommand{\vp}{\varphi}
\newcommand{\vep}{\varepsilon}
\newcommand{\diag}{{\rm diag}}
\newcommand{\grp}{{\mathbb G}}
\newcommand{\dgrp}{{\mathbb D}}
\newcommand{\desp}{{\mathbb D^{\rm{es}}}}
\newcommand{\Geod}{{\rm Geod}}
\newcommand{\geod}{{\rm geod}}
\newcommand{\hgr}{{\mathbb H}}
\newcommand{\mgr}{{\mathbb M}}
\newcommand{\ob}{{\rm Ob}}
\newcommand{\obg}{{\rm Ob(\mathbb G)}}
\newcommand{\obgp}{{\rm Ob(\mathbb G')}}
\newcommand{\obh}{{\rm Ob(\mathbb H)}}
\newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}}
\newcommand{\ghomotop}{{\rho_2^{\square}}}
\newcommand{\gcalp}{{\mathbb G(\mathcal P)}}
\newcommand{\rf}{{R_{\mathcal F}}}
\newcommand{\glob}{{\rm glob}}
\newcommand{\loc}{{\rm loc}}
\newcommand{\TOP}{{\rm TOP}}
\newcommand{\wti}{\widetilde}
\newcommand{\what}{\widehat}
\renewcommand{\a}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\de}{\delta}
\newcommand{\del}{\partial}
\newcommand{\ka}{\kappa}
\newcommand{\si}{\sigma}
\newcommand{\ta}{\tau}
\newcommand{\lra}{{\longrightarrow}}
\newcommand{\ra}{{\rightarrow}}
\newcommand{\rat}{{\rightarrowtail}}
\newcommand{\oset}[1]{\overset {#1}{\ra}}
\newcommand{\osetl}[1]{\overset {#1}{\lra}}
\newcommand{\hr}{{\hookrightarrow}}

\begin{document}

 \subsection{Molecular sets and representations of chemical reactions}

A \emph{uni-molecular chemical reaction} is represented by the \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html}
$\eta: h^A\longrightarrow h^B$, through the following \htmladdnormallink{commutative diagram}{http://planetphysics.us/encyclopedia/Commutativity.html}:
\begin{equation}
\def\labelstyle{\textstyle}
\xymatrix@M=0.1pc @=4pc{h^A(A) = Hom(A,A) \ar[r]^{\eta_{A}}
\ar[d]_{h^A(t)} & h^B (A) = Hom(B,A)\ar[d]^{h^B (t)} \\ {h^A (B) =
Hom(A,B)} \ar[r]_{\eta_{B}} & {h^B (B) = Hom(B,B)}}
\end{equation}

with the \emph{states of molecular sets} $A_u = a_1, \ldots, a_n$ and
$B_u = b_1, \ldots b_n$ being represented by certain endomorphisms
in $Hom(A,A)$ and $Hom(B,B)$, respectively. In general, \emph{molecular sets} $M_S$ are defined as finite sets whose elements are \emph{molecules} defined in terms of their molecular \htmladdnormallink{observables}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} that are specified next. One need to define first the \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of a molecular class variable. A \emph{molecular class variable}, or $m.c.v$ is defined as a family of molecular sets $[M_S]_{i \in I},$ with $I$ being an indexing set, or class, defining the \emph{molecular range of variation} of the $m.c.v$. Most applications in Physics, Chemistry or Biochemistry require that $I$ is a finite set, (that is, without any sub-classes). A \emph{homomorphism of molecular sets} $M_t: M_S \to M_S$, with $t \in T$ being real time values, is defined as a time-dependent mapping or \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} $M_S (t)$ also called a $M_t$ \emph{molecular transformation}.

An \emph{$m.c.v.$ observable} of $B$, characterizing the products of chemical \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} ``B'' of a chemical reaction is defined as a \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}:

$$\gamma : Hom(B,B) \longrightarrow \Re ,$$
where $\Re$ is the set or \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} of real numbers. This \emph{mcv-observable} is subject
to the following \htmladdnormallink{commutativity}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} conditions:
\begin{equation}
\def\labelstyle{\textstyle}
\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},}
\end{equation}~
with $c: A^*_u \longrightarrow B^*_u$, and $A^*_u$, $B^*_u$ being, respectively,
specially prepared \emph{fields of states} of the molecular sets $A_u$, and $B_u$ within a measurement uncertainty range, $\Delta$, which is determined by Heisenberg's uncertainty \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html}, or the \htmladdnormallink{commutator}{http://planetphysics.us/encyclopedia/Commutator.html} of the observable \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} involved, such as $[A^*, B^*]$, associated with the observable $A$ of molecular set $A_u$, and respectively, with the obssevable $B$ of molecular set $B_u$, in the case of a molecular set $A_u$ interacting with molecular set $B_u$.

With these concepts and preliminary data one can now define the \htmladdnormallink{category of molecular sets}{http://planetphysics.us/encyclopedia/ChemicalTransformations.html} and their transformations
as follows.

\subsection{Category of molecular sets and their transformations}
\begin{definition}
The \emph{category of molecular sets} is defined as the \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} $C_M$ whose \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are molecular sets $M_S$ and whose morphisms are molecular transformations $M_t$.
\end{definition}

\begin{remark}
This is a mathematical \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of chemical reaction \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} in terms of molecular sets that vary with time
(or $msv$'s), and their transformations as a result of diffusion, \htmladdnormallink{collisions}{http://planetphysics.us/encyclopedia/Collision.html}, and chemical reactions.
\end{remark}

\begin{thebibliography}{9}

\bibitem{BAF60}
Bartholomay, A. F.: 1960. Molecular Set Theory. A mathematical representation for chemical reaction mechanisms. \emph{Bull. Math. Biophys.}, \textbf{22}: 285-307.

\bibitem{BAF65}
Bartholomay, A. F.: 1965. Molecular Set Theory: II. An aspect of biomathematical theory of sets., \emph{Bull. Math. Biophys.} \textbf{27}: 235-251.

\bibitem{BAF71}
Bartholomay, A.: 1971. Molecular Set Theory: III. The Wide-Sense Kinetics of Molecular Sets ., \emph{Bulletin of Mathematical Biophysics}, \textbf{33}: 355-372.

\bibitem{ICB2}
Baianu, I. C.: 1983, Natural Transformation Models in Molecular
Biology., in \emph{Proceedings of the SIAM Natl. Meet}., Denver,
CO.; Eprint at cogprints.org with No. 3675.

\bibitem{ICB2}
Baianu, I.C.: 1984, A Molecular-Set-Variable Model of Structural
and Regulatory Activities in Metabolic and Genetic Networks
\emph{FASEB Proceedings} \textbf{43}, 917.

\end{thebibliography} 

\end{document}
Return to "PlanetPhysics/Category of Molecular Sets 3" page.