Computational mathematical models capturing

At the time of this writing, it must be conceded that there have been no fundamental principles-based mathematical model for Nafion that has predicted significantly new phenomena or caused property improvements in a significant way. Models that capture the essence of percolation behavior ignore chemical identity. The more ab initio methods that do embrace chemical structure are limited by the number of molecular fragments that the computer can accommodate. Other models are semiempirical in nature, which limits their predictive flexibility. Nonetheless, the diversity of these interesting approaches offers structural perspectives that can serve as guides toward further experimental inquiry. 

There is an increased use of flammability tests, which measure fundamental properties as opposed to tests that simulate a specific fire scenario. The former can be used in conjunction with mathematical models to predict the performance of a material in a range of fire scenarios. This approach has become feasible due to the significant progress that has been made in the past few decades in our understanding of the physics and chemistry of fire, mathematical modeling of fire phenomena and measurement techniques. However, there will always be materials that exhibit a behavior that cannot be captured in bench-scale tests and computer models. The fire performance of those materials can only be determined in full-scale tests. 

In the preceding chapter we saw how the chemostat could be modified to account for a new phenomenon - the presence of an inhibitor. In this chapter we extend the idea behind the simple chemostat to a new apparatus in order to model a property of ecological systems that it is not possible to model in the simple chemostat. The idea is to capture the essentials of the new phenomenon without destroying the tractability of the chemostat either as a mathematical model or as an experimental one. A very simple situation will be described here a more complicated one -with a less explicit (in the sense of less computable) analysis - will be discussed in the next chapter. Just as the chemostat is a basic model for competition in the simplest situation, the apparatus here shows promise of being the model for competition along a nutrient gradient. 

In fact, the success of any optimization technique critically depends on the degree to which the model represents and accurately predicts the investigated system. For this reason, the model must capture the complex dynamics in the system and predict with acceptable accuracy the proper elements of reality. Moreover, it is important to be able to recognize the characteristics of a problem and identify appropriate solution techniques within each class of problems there are different optimization methods which vary in computational requirements and convergence properties. These problems are generally classified according to the mathematical characteristics of the objective function, the constraints, and the controllable decision variables. 

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