Mathematical model deterministic

H. I. Freedan, Deterministic Mathematical Models in Population Ecology (1980)... 

Any analysis of risk should recognize these distinctions in all of their essential features. A typical approach to acute risk separates the stochastic nature of discrete causal events from the deterministic consequences which are treated using engineering methods such as mathematical models. Another tool if risk analysis is a risk profile that graphs the probability of occurrence versus the severity of the consequences (e.g., probability, of a fish dying or probability of a person contracting liver cancer either as a result of exposure to a specified environmental contaminant). In a way, this profile shows the functional relationship between the probabilistic and the deterministic parts of the problem by showing probability versus consequences. 

Theoretical investigations of the problem were carried out on the base of the mathematical model, combining both deterministic and stochastic approaches to turbulent combustion of organic dust-air mixtures modeling. To simulate the gas-phase flow, the k-e model is used with account of mass, momentum, and energy fluxes from the particles phase. The equations of motion for particles take into account random turbulent pulsations in the gas flow. The mean characteristics of those pulsations and the probability distribution functions are determined with the help of solutions obtained within the frame of the k-e model. 

Mathematical optimization always requires a deterministic process model to predict the future behavior of a process. However, as previously mentioned, it is difficult to construct mathematical models that can cover the entire range of fermentation due to the complexity of intracellular metabolic reactions. As an alternative to the deterministic mathematical models, Kishimoto et al. proposed a statistical procedirre that uses linear multiple regression models [7], as shown below, instead of a deterministic mathematical model such as a Monod equation. 

Kousa et al. [20] classified exposure models as statistical, mathematical and mathematical-stochastic models. Statistical models are based on the historical data and capture the past statistical trend of pollutants [21]. The mathematical modelling, also called deterministic modelling, involves application of emission inventories, combined with air quality and population activity modelling. The stochastic approach attempts to include a treatment of the inherent uncertainties of the model [22],... 

This approach, taken to its logical conclusion, makes it possible to achieve two main goals first, to optimize the process so as to provide the maximum possible throughput at minimum cost, and, second, to ensure the required quality level of the final products. Development of a deterministic mathematical model of a process requires an adequate representation of its main constituent stages. This always involves an internal contradiction on the one hand, it is desirable to describe all... 

There are two possible approaches to estimating the human safe dose for chemicals that cause deterministic effects the use of safety and uncertainty factors and mathematical modeling. The former constitutes the traditional approach to dose-response assessment for chemicals that induce deterministic effects. Biologically-based mathematical modeling approaches that more realistically predict the responses to such chemicals, while newer and not used as widely, hold promise to provide better extrapolations of the dose-response relationship below the lowest dose tested. 

Mathematical models that describe and predict the inanimate world quite well are actually of little value in the system of deterministic chaos that governs biology. The answers one can expect from mathematical approaches to evolution (in contrast to my earlier perception) cannot be narrowed to less than the surface of the chaotic attractor of the system which is a little like watching evolution on earth from a satellite.1 The limits of the attractor surface are given by the initial conditions which are not knowable in sufficient detail.2 Empiricism can help, after all our laws of science by and large are the results of repeated observations. 

There are various ways to classify mathematical models (5). First, according to the nature of the process, they can be classified as deterministic or stochastic. The former refers to a process in which each variable or parameter acquires a certain specific value or sets of values according to the operating conditions. In the latter, an element of uncertainty enters we cannot specify a certain value to a variable, but only a most probable one. Transport-based models are deterministic residence time distribution models in well-stirred tanks are stochastic. 

In polymer processing, the mathematical models are by and large deterministic (as are the processes), generally transport based, either steady (continuous process, except when dynamic models for control purposes are needed) or unsteady (cyclic process), linear generally only to a first approximation, and distributed parameter (although when the process is broken into small, finite elements, locally lumped-parameter models are used). 

P. Erdi and J. Toth, Mathematical Models of Chemical Reactions Theory and Applications of Deterministic and Stochastic Models, Princeton University Press, Princeton, 1989. 

Since this monograph is devoted only to the conception of mathematical models, the inverse problem of estimation is not fully detailed. Nevertheless, estimating parameters of the models is crucial for verification and applications. Any parameter in a deterministic model can be sensibly estimated from time-series data only by embedding the model in a statistical framework. It is usually performed by assuming that instead of exact measurements on concentration, we have these values blurred by observation errors that are independent and normally distributed. The parameters in the deterministic formulation are estimated by nonlinear least-squares or maximum likelihood methods. 

Mathematical models are widely applied in biosciences and different modeling routes can be taken to describe biological systems. The type of model to use depends completely on the objective of the study. Models can be dynamic or static, deterministic or stochastic. Kinetic models are commonly used to study transient states of the cell such as the cell cycle [101] or signal transduction pathways [102], whereas stoichiometric models are generally used when kinetics parameters are unknown and steady state systems is assumed [48, 103]. 

Deterministic (point estimates) model A mathematical model in which the parameters and variables are not subject to random fluctuations, so that the system is at any time entirely defined by the initial conditions chosen - contrast with a stochastic model (Swinton, 1999). 

Stochastic model A mathematical model which takes into consideration the presence of some randomness in one or more of its parameters or variables. The predictions of the model therefore do not give a single point-estimate but a probability distribution of possible estimates (contrast with deterministic) (Swinton, 1999). 

Deterministic mathematical model given by analytical functions that are differentiable respect to each parameter... 

Various methods 1. Deterministic process mathematical model especially given by differential equations 2. With or without inequality type constraints... 

Using stochastic differential equations can also represent the stochastic models. A stochastic differential equation keeps the deterministic mathematical model but accepts a random behaviour for the model coefficients. In these cases, the problems of integration are the main difficulties encountered. The integration of stochastic differential equations is known to be carried out through working methods that are completely different from those used for the normal differential equations... 

Some domain ontologies have been developed, which so far cover only a small portion of the chemical engineering domain They enable the representation of material properties, experiments and process recipes, as well as the structural description of mathematical models. In parallel, an ontology for the modeling of work processes and decisions has been built. Yet the representation of work processes is confined to deterministic guidelines, the input/output information of activities cannot be modeled, and the acting persons are not... 

A deterministic mathematical model for simulating the biotechnological synthesis of acrylic acid was developed in a previous study [3] to explore an alternative process. The proposed... 

Mathematical models may be classified into deterministic and stochastic models. For deterministic models, knowledge of the relationship between dependent and independent variables is necessary. Consequently, the complex nature of polymer-based heterogeneous materials is rather incompatible with such requirements. Hence, stochastic models become necessary either when the existing knowledge about the stimulus-response behavior of a system is not enough as to ascertain its behavior or when it is not possible to build an efficient deterministic model able to score the system response. 

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