Functional test mathematical description

Identification of the material properties as an estimation of transfer function (TF) for the black box model. In this case the problem of identification is solving according to the results of the input (IN) and output (OUT) actions. There is a transfer of notion of mathematical description of TF on characterization of the material. This logical substitution gives us an opportunity to formalize testing procedure and describe the material as a set of formulae, which can be used for quantitative and qualitative characterization of the materials. 

The main concept addressed in this new multi-part series is the idea of correlation. Correlation may be referred to as the apparent degree of relationship between variables. The term apparent is used because there is no true inference of cause-and-effect when two variables are highly correlated. One may assume that cause-and-effect exists, but this assumption cannot be validated using correlation alone as the test criteria. Correlation has often been referred to as a statistical parameter seeking to define how well a linear or other fitting function describes the relationship between variables however, two variables may be highly correlated under a specific set of test conditions, and not correlated under a different set of experimental conditions. In this case the correlation is conditional and so also is the cause-and-effect phenomenon. If two variables are always perfectly correlated under a variety of conditions, one may have a basis for cause-and-effect, and such a basic relationship permits a well-defined mathematical description. 

A second objective of this study was to develop a convenient, but not necessarily rigorous, mathematical description of the octane number of blend (ONE) as a function of the base fuel and the volume percent methanol. For this reason, tests were planned for methanol percentages greater than 50% even though these mixtures were not considered viable fuels. Complete information for research and motor ratings of all base fuels with 0-100% methanol was desired. 

Through this step and based on experimental evidence we try to develop the appropriate model to describe the test chamber kinetics. As was anticipated in the introduction of this Chapter, from a conceptual point of view, two broad categories of models can be developed empirical-statistical and physical-based mass transfer models. It should be emphasized that, in several cases, even the fundamentally based mass transfer models are indistinguishable from the empirical ones. This happens because the mass transfer models are generally very complex in both the physical concept involved and the mathematical treatment required. This often leads the modelers to introduce approximations, making the mass transfer models not completely distinguishable from some empirical models in terms of both functional formulations and descriptive capabilities. Considering the current status of models which have been developed to describe VOC emissions (and/or sink processes), we could define the mass transfer models as hybrid-empirical models. 

Both selection and design of a fibre for a given end-use must take into account a wide range of properties. However, satisfactory mechanical performance is always essential. Various aspects of the mechanical properties of polymers are discussed in detail in other chapters of this work and elsewhere." " For fibres with cylindrical symmetry, the fundamental approach requires determination of five independent viscoelastic functions," which is prohibitively time-consuming. Besides, in practice, the fibres assembled in an end-product are often deformed in a complex way. The magnitudes and directions of the applied forces vary with time in a manner which precludes an exact mathematical description. At the same time, the fibres may be exposed to a changing environment. Consequently, it is often necessary to use various end-use oriented tests for a practical evaluation of a fibre. 

The Delta system contains many of the same tests contained in the APTS system, including those that monitor motor function, reaction time, attention and working memory, learning and memory, spatial perception and reasoning, mathematical calculation, and language (Table 6.1). More complete descriptions of the psychometric and validity studies supporting the utility of this testing system are available elsewhere.61,62... 

David and Augsburger [146] were the first to try this method of analysis. Further tests, for example, determination of complex functions based on methods of numerical mathematics, were performed by the research group of Muller [147, 148], Although the results were helpful, for exact description the models became rather complicated and the derived parameters were complex. According to Bauer [149], these models have to be three dimensional for a reasonable description. Another similar approach was used by the research group of Rippie [12,13]. They described the structure evolution in the tablet during decompression by the aid of vectorial 3D models and concluded that fracture and stress contribute to the final structure of the tablet. 

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