Deterministic models definition

The mass balance of the retained solid is described by Eq. (4.280). The Mint deterministic model results from the coupling of this relation vfith the definition of the filtration coefficient. The result is written in Eq. (4.281) for the start time of the filtration and in Eq. (4.282) for the remaining filtration time. Here a is the detachment coefficient of the retained particle its dimension is T h... 

Expressions for Growth Rate. At this stage, a model must be introduced if further progress is to be made, and we choose a deterministic model. All deterministic models which have been introduced (with two exceptions) have been autonomous they assume that the growth rates are explicit functions only of the state of the system, and not of time. But state in an unstructured model can only refer to population density or to concentration of protoplasm, since, by definition, a model is unstructured if only one state variable appears. Thus, the model is... 

Deterministic model Mathematical model in which each variable and parameter can be assigned a definite fixed number. 

By definition, a deterministic model assumes quantitative, physical-based knowledge of the causal laws that govern corrosion, which is a very complex phenomenon. Accordingly, strong hypotheses simplifying the problem have had to be introduced to allow causal natural laws to be applied. Some of the factors required, and thus research opportunities presented, to perform realistic deterministic modeling are the following ... 

Within these two classes of models, there exist numerous subclasses. For example, within the empirical class, there are functional models, in which (discrete) data are represented by continuous mathematical functions or by approximations that sometimes follow a natural law. Within the broad class of deterministic models there can exist definite models that yield a single output for a given set of input values and probabilistic models, in which the inputs are distributed, resulting in a distributed output from which the probability of an event occurring can be estimated. Also, as mentioned above, there are other possible ways to classify models ... 

Deterministic models or elements of models are those in which each variable and parameter can be assigned a definite fixed number, or a series of numbers, for any given set of conditions, i.e. the model has no components that are inherently uncertain. In contrast, the principle of uncertainty is introduced in stochastic or probabilistic models. The variables or parameters used to describe the input-output relationships and the structure of the elements (and the constraints) are not precisely known. A stochastic model involves parameters characterized by probability distributions. Due to this the stochastic model will produce different results in each reahzation. 

In this chapter, we explain the technique of sequential bifurcation and add some new results for random (as opposed to deterministic) simulations. In a detailed case study, we apply the resulting method to a simulation model developed for Ericsson in Sweden. In Sections 1.1 to 1.3, we give our definition of screening, discuss our view of simulation versus real-world experiments, and give a brief indication of various screening procedures. 

Often, the strict definition of ergodicity must be relaxed to accommodate the realistic modelling setting for example only some portion of the eno gy surface may be accessible by trajectories in the time-interval available, due to the cost of computing steps with a numerical method. For a more detailed discussion of these and other ergodicity issues in the context of deterministic molecular dynamics, see the articles of Tupper [379, 381]. 

In quite a few cases, chaos can be predicted based on appropriate mathematical model (Lorenz or Rosselor) (Chapter 12). The strange attractor obtained in such cases signifies Deterministic Chaos . Such models exhibit cross-catalysis and inhibition. Experimentally observed deterministic chaos can be easily ascertained using definite criteria, although prediction of complex time order is quite difficult in many cases. In terms of causality principle, the complexity in the system arises due to complex interdependence of a number of causes and effects. 

New methodology for exact reliability quantification of highly reliable systems with maintenance was introduced in (Bris 2008a). It assumes that the system structure is mathematically represented by the use of directed acyclic graph (AG), see more details in (Bris 2008b). Terminal nodes of the AG that represent system components are established by the definition of deterministic or stochastic process, to which they are subordinate. From them we can compute a time dependent unavailability function, of individual terminal nodes. Finally a correspondent time dependent imavailability function U(x,t) of the highest node (SS node or top event in classic PRA model) which represents rehabdity behaviour of the whole system may be found. It is clear that U(x,t) < Us(x). 

When the stirring rate is decreased, the flow branch is favoured. One could speak of a "forced nucleation" but this term is somehow misleading since the process is purely deterministic and the local perturbation due to the feed is permanent. The validity of the conclusions implies that the input flow is large enough to define the small volume above. This is certainly the case close to ks, where the flow becomes dominant compared to the reaction processes on a macroscopic scale. The transition is shifted to lower values of the flow rate. For the reverse transition at k-j, where the flow is often very small and where the reaction process becomes dominant no definite conclusion can be drawn. We shall now develop these ideas on simple models and show that localized injection of reactants induces effects in qualitative agreement with the experiments. 

A stochastic system is one whose behavior is not purely deterministic and predictable, but rather has (assumed inherent) randomness. The theory of probability provides a mathematical fi amework for understanding and modeling such systems. Rather than provide abstract definitions, we introduce the subject through an example modeling the distribution of polymer chain lengths in condensation polymerization. 

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