Algebraic Enumerations

In the algebraic enumerations of classes of polyhexes the achievements of Harary and Read [9] exhibit a peak of professional expertise. The work has been quoted several times [14,30,58]. Herein the two mathematicians [9] developed a generating function with the explicit form... 

Enumerating contacts, Proceedings of Symposia in pure Mathematics of the AMS, Volume 46 part 2, Algebraic Geometry 1985, 401-418. 

When we have reduced the representing of a problem by an equation to be no more than an algebraic expression, the first point we have to examine is the following How many distinct quantities are there in the equations of the problem And the examination of this point is immediately followed by the study of this other point Among these distinct quantities, how many independent relations does algebra furnish In making this double enumeration for the problems of chemical mechanics, J. Willard Gibbs was led to the propositions whose ensemble constitutes the phase rule. 

Exact results of combinatorial optimization of PWCs structure are presented in the Table. The data were obtained by different methods. For small and middle-size clusters (N < 28), we used a complete enumeration method. For large clusters (N = 36, 60), the data were obtained using symbolic calculations and max-plus algebra. Characteristics of two lowest energy levels for the SWEB model are presented. For water fullerene N = 60) on the base of SWEB model, the data obtained only for ground state level without isomorphism analysis. Values in italics correspond to ground sates of the SWB model. For example, the... 

The object of all arithmetical operations is to save direct enumeration. Having done a sum once, we seek to preserve the answer foi future use so too the purpose of algebra, which, by substituting relations for values, symbolizes and. definitely fixes all numerical operations which follow the same rule. —E. Mach. 

The calculations for the nonconstant-density case may be greatly simplified by using a differential-algebraic equation (DAE) solver, All three cases enumerated above can be handled by modifying the residual equations provided to the DAE solver. We do not have to differentiate the equation of state or perform other algebraic manipulations that are required if one uses an ordinary differential equation (ODE) solver. 

We will enumerate several examples of the above-mentioned algebras, namely those which are of interest from the point of view of multidimensional analogues of the equations of motion of the various mechanical systems. 

Enumerations of coronoid systems is a substantial part of the work. Algebraic methods involving combinatorics and generating functions are employed on one hand, and computer programming on the other. The whole book is supposed to demonstrate a piece of mathematical chemistry, which can be characterized as lying on the "interfaces between mathematics, chemistry and computer science", a formulation used for the MATH/CHEM/COMP Conferences cf. Cyvin SJ, BrunvoU and Cyvin (1989d) in BibUography. 

The computation of the characteri.stic polynomial for a molecular graph is a difficult numerical problem, and a wide range of methods have been developed for the efficient and accurate determination of its coefficients subgraph enumeration methods, recurrence relationships, graph decomposition, and linear algebra numerical methods. 

Table 10.1 enumerates the MATLAB operators. Most of these are the same as for Excel and VBA. A notable difference is the backslash operator (called left division), which is used primarily when solving sets of linear algebraic equations. Examples using this operator appear in the sequel. 

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