Mathematical modeling hierarchic structure

Computer generation, storage and retrieval of substructures is greatly facilitated, when the mathematical model of structural chemistry is employed to generate a hierarchically organized substructure file in a systematic fashion. Such hierarchical ordering is not only a prerequisite in avoiding duplication of... 

Feature diagrams are a formal representation of product data with a precise semantics. Consequently, they define a unique mathematical model of the product data that consists of the hierarchical structure and additional logical formulas that specify the properties and links to aU product parts. The analyzing procedures presented here operate on purely logical representations of the product data. As a consequence, the structure and content of a feature diagram have to be transformed into an equivalent representation in logic. 

Of course, the micromechanical relations (bounds, approximations or fit models) presented in this chapter for the effective elastic properties are by no means restricted to the alumina-zirconia system but can be applied to many types of ceramies and eeramie composites. On the other hand they cannot be expected to be automatically applicable to matrix-inclusion type composites in cases where the matrix consists of nonlinearly elastic materials (polymers), viscoelastic materials (glasses or porcelain at high temperature) or elastoplastic materials (metals). In particular, they cannot be a priori expected to be justified for materials of biological origin, although their application to many of these materials, e g. bone, might be seductive and dictated by practical needs. With respect to the inherent anisotropy and the hierarchical microstructure of these materials [Ontanon et al. 2000], however, any mathematical modeling or description of their composition-structure-property relationships has to be performed with due caution. 

In this study, the life cycle assessment of microwave hot air systems were developed using the analytic hierarchy process and the fuzzy comprehensive evaluation. The hierarchical structure consists of assessment aspect and assessment objective were established. The fuzzy assessment matrices of the assessment mathematical model were calculated using eigenvalue method and Gauss-Seidel iterative matrix. The life cycle assessment results show that the microwave hot air systems have a good green degree. 

Some comparisons of a hierarchical data model with a relational data model are of interest here. The structures in the hierarchical model represent the information that is contained in the fields of the relational model. In a hierarchical model, certain records must exist before other records can exist. The hierarchical model is generally required to have only one key field. In a hierarchical data model, it is necessary to repeat some data in a descendant record that need be stored only once in a relational database regardless of the number of relations. This is so because it is not possible for one record to be a descendant of more than one parent record. There are some unfortunate consequences of the mathematics involved in creating a hierarchical tree, as contrasted with relations among records. Descendants cannot be added without a root leading to them, for example. This leads to a number of undesirable characteristic properties of hierarchical models that may affect our ability to easily add, delete, and update or edit records. 

The concept of ultrametrics has emerged as an important mathematical concept to describe complex, hierarchical dynamic structures, as spin glasses, computing devices, memory. It would be interesting to see, how to derive hierarchical chemical kinetic models by switching the microscopic and mesocopic level of the description. 

Can more elaborate VB calculations (those that are above the resonance theory of Herndon in the diagram shown in Figure 50, which represents the hierarchical relationship between different VB models according to Klein et al. ) be cast in an alternative but mathematically equivalent formalism based on the decomposition of Kekule valence structures in conjugated circuits, rather than using Kekule valence structures ... 

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