Mathematical model of constitutional

The mathematical model of constitutional chemistry. The programs EROS and IGOR... 

When our group started in 1972 to develop a computer program for the design of syntheses, we based it on a mathematical model of constitutional chemistry4,5. ... 

It has been demonstrated that a mathematical model of constitutional chemistry can serve as a basis for a computer program to automatically generate sequences of reactions and for planning organic syntheses. This gives credibility to other applications of this model. 

The mathematical model of constitutional chemistry which has been described in the preceeding sections can be used as a basis for a modular system of computer programs for the deductive solution of chemical problems. 

Ugi and co-workers have developed a mathematical model of constitutional chemistry and used it as a basis for deductive computer programs synthetic routes may be designed in this way/ but there is a problem in minimizing the number of routes so generated. Weise and Scharnow have critically discussed some existing computer programs for designing syntheses and provided their own approach to the subject. The topic has also been discussed briefly by Bruns in an article which includes comment on quantitative structure-activity relationships. 

The purpose of this work is to formulate such a mathematical model of constitutional chemistry that allows a simple description of the topology of molecular systems and their changes during chemical reactions. 

Ugi and Dugundji developed a mathematical model of constitutional chemistry. This model is based on the concept of isomerism of molecules which has been extended to ensembles of molecules. For example, a theoretical reaction A -I- B C -I- D can be seen as the conversion of an ensemble of molecules (A - - B) into an isomeric ensemble (C D). As an extension, the discovering of a synthesis Target = Precursor 1 Precursor 2 => Starting materials, may... 

An R-matrix has a series of interesting matheinatical properties that directly reflect chemical laws. Thus, the sum of all the entries in an R-matrix must be zero, as no electrons can be generated or annihilated in a chemical reaction. Furthermore, the sum of the entries in each row or column of an R-matrix must also he zero as long as there is not a change in formal charges on the corresponding atom. An elaborate mathematical model of the constitutional aspects of organic chemistry has been built on the basis of BE- and R-matriccs [17. ... 

As outlined in the previous section, there is a hierarchy of possible representations of metabolism and no unique definition what constitutes a true model of metabolism exists. Nonetheless, mathematical modeling of metabolism is usually closely associated with changes in compound concentrations that are described in terms of rates of biochemical reactions. In this section, we outline the nomenclature and the essential steps in constructing explicit kinetic models of metabolic networks. 

For the sake of brevity the reader is referred to Paper II for the details regarding the constitutive mathematical models of the method applied to measure the mass flow and stoichiometry of conversion gas as well as air factors for conversion and combustion system. Below is a condensed formulation of the mathematical models applied. Here a distinction is made between measurands and sought physical quantities of the method. 

A mathematical model of a plant or a section of a plant can be judged only by comparison with actual plant data. The model may be considered as good when the simulated variables can predict with some level of confidence the plant parameters which are important in determining the cost and quality of the finished product. Failures of the model are likely to be a result of (1) oversimplification of the equations that constitute the model, (2) inadequacy of the numerical solution of the equations. 

Interatomic distance is calculated by mathematical modelling of the electron exchange that constitutes a covalent bond. Such a calculation was first performed by Heitler and London using Is atomic wave functions to simulate the bonding in H2. To model the more general case of homonuclear diatomic molecules the interacting atoms in their valence states are described by monopositive atomic cores and two valence electrons with constant wave functions (3.36). 

The mathematical model of catalyst/polymer particle evolution consists of the set of differential-algebraic equations (61)-(64). The constitutive equations describing the force interactions, transport of monomer, phase equilibria at the interface between polymer and pore phase as well as the rules for connectivity of micro-elements have to be specified (Grof and Kosek, 2005 Grof et al., 2005a). 

From 1970 to 1984 Dugundji, Ugi et al. [9, 10] analyzed the logical structure of chemistry. They found that chemistry consists of two parts with distinct logical structures, constitutional chemistry and stereochemistry. This division is already implied by the nongeometric definition of stereoisomers [11] (see Sect. 1.3). Accordingly, Dugundji and Ugi formulated a qualitative mathematical model of the logical structure of chemistry that consists of two distinct parts. 

Mathematical Models of the Logical Structure of Constitutional Chemistry... 

The inclusion of bonds with fractional bond orders extends the scope of the mathematical model of the constitutional chemistry from classical organic chemistry also to modern inorganic and organometallic chemistry. 

The state equations with the associated state variables constitute the mathematical model of a process, which yields the dynamic or static behavior of the process. The application of the conservation principle... 

The state equations (4.4a) and (4.5b), with the state variables, the inputs, and the parameters, constitute the mathematical model of the stirred tank heater. We need only solve them in order to find the tank s dynamic or steady-state behavior. 

This article focuses on an approximate mathematical model of adsorption kinetics on rough (fractal) interfaces based on a constitutive equation of Riemann-Liouville type... 

Since the fluid velocity within the geothermal reservoirs created in the fractured rock can be relatively high and the characteristic sizes of the solid blocks of the rock, which constitute the fissured reservoir media, are relatively large, the high mean values of heat flow rates are typically attained within the stimulated region. In this situation the effect of thermal dispersion is an important factor that should be included into the mathematical model of the heat transport within the natural or artificially developed fissured geothermal reservoirs. Employing different... 

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