Model, mathematical scientific

Basak, S. C., and G. D. Grunwald, Use of Topological Space and Property Space in Selecting Structural Analogs, in Mathematical Modelling and Scientific Computing, in press. 1998. 

The implementation of mathematics into biology, physiology, pharmacology, and medicine is not new, but its use has grown in the last three decades as computer speeds have increased and scientists have begun to see the power of modeling to answer scientific questions. A conference was held in 1989 at the National Institutes of Health called Modeling in Biomedical Research An Assessment of Current and Potential Approaches. One conclusion from that conference was that biomedical research will be most effectively advanced by the continued application of a combination of models—mathematical, computer, physical, cell, tissue culture and animal—in a complementary and interactive manner. ... 

In a mathematical simulation, the behavior of a system is not examined on the system itself or in a physical model but with the help of a mathematical model. For this the behavior of the system is traced back to known logical, mathematical, scientific, and technical laws as well as to the known behavior of partial systems. The prerequisite in each case is that the system behavior of interest and the structure of the system including input and output quantities can be quantitatively described by means of mathematical equations. The value of mathematical models and simulations carried out with their help resides in the fact that the input data and the system parameters can be changed easily and that the influence of such modifications on the behavior can be studied. Mathematical simulation, in essence, can be of value for three types of projects ... 

Scientific models can take many shapes and forms, but they all seek to characterize response variables through relationships with appropriate factors. Traditional models can be divided into two main categories mathematical or theoretical models and statistical or empirical models. Mathematical models have the common characteristic that the response and predictor variables are assumed to be free of specification error and measurement uncertainty. Statistical models, on the other hand, are derived from data that are subject to various types of specification, observation, experimental, and/or measurement errors. In general terms, mathematical models can guide investigations, and statistical models are used to represent the results of these investigations. 

One of the most commonly used constructs is a model. A model is a simple way of describing and predicting scientific results, which is known to be an incorrect or incomplete description. Models might be simple mathematical descriptions or completely nonmathematical. Models are very useful because they allow us to predict and understand phenomena without the work of performing the complex mathematical manipulations dictated by a rigorous theory. Experienced researchers continue to use models that were taught to them in high school and freshmen chemistry courses. However, they also realize that there will always be exceptions to the rules of these models. 

In this chapter, we follow a typical scientific path. First, we collect experimental observations on the properties of gases and summarize these observations mathematically. We then formulate a simple qualitative molecular model of a gas suggested by these observations and go on to express it quantitatively. Finally, we use more detailed experimental observations to refine the model so that it accounts for the properties of real gases. 

Since electrochemical processes involve coupled complex phenomena, their behavior is complex. Mathematical modeling of such processes improves our scientific understanding of them and provides a basis for design scale-up and optimization. The validity and utility of such large-scale models is expected to improve as physically correct descriptions of elementary processes are used. 

Most of the constrained parameter estimation problems belong to this case. Based on scientific considerations, we arrive quite often at constraints that the parameters of the mathematical model should satisfy. Most of the time these are of the form,... 

Mathematical approaches used to describe micelle-facilitated dissolution include film equilibrium and reaction plane models. The film equilibrium model assumes simultaneous diffusive transport of the drug and micelle in equilibrium within a common stagnant film at the surface of the solid as shown in Figure 7. The reaction plane approach has also been applied to micelle-facilitated dissolution and has the advantage of including a convective component in the transport analysis. While both models adequately predict micelle-facilitated dissolution, the scientific community perceives the film equilibrium model to be more mathematically tractable, so this model has found greater use. 

There is no scientific reason for a soil model to be an unsaturated soil model only, and not to be an unsaturated (soil) and a saturated soil (groundwater) model. Only mathematical complexity mandates the differentiation, because such a model would have to be 3-dimensional (e.g., 7) and very difficult to operate. Most of the soil models account for vertical flows, groundwater models for horizontal flows. 

Mathematical groundwater modeling has been the least problematic in its scientific formulation, but has been the most problematic model category when dealing... 

The main tools used to provide global projections of future climate are general circulation models (GCMs). These are mathematical models based on fundamental physical laws and thus constitute dynamical representations of the climate system. Computational constraints impose a limitation on the resolution that it is possible to realise with such models, and so some unresolved processes are parameterised within the models. This includes many key processes that control climate sensitivity such as clouds, vegetation and oceanic convection [19] of which scientific understanding is still incomplete. 

Of all the mysteries of Nature time is the oldest and most daunting. It has been analyzed from many angles, mostly from a philosophical rather than a scientific point of view. These studies have produced a number of related descriptions, including definitions of psychological, biological, geological and mathematical time [28]. Despite the fact that time intervals can be measured with stupendous accuracy there is no physical model of time. This anomalous situation probably means that the real essence and origin of the concept time is not understood at all. 

Fourer, R. D. M. Gay and B. W. Kemighan. AMPL A Modeling Language for Mathematical Programming. Scientific Programming, San Francisco (1993). 

Models are an integral part of any kind of human activity. However, we are mostly unaware of this. Most models are qualitative in nature and are not formulated explicitly. Such models are not reproducible and cannot easily be verified or proven to be false. Models guide our activities, and throughout our entire life we are constantly modifying those models that affect our everyday behaviour. The most scientific and technically useful types of models are expressed in mathematical terms. This book focuses on the use of dynamic mathematical models in the field of chemical engineering. 

Francis Albarede has been a very active actor in this evolution towards quantitative science. His abundant scientific contributions published in the best international journals are all focussed on the goal of building a quantitative science. He is one of the leading scientists in this area and has now decided to broaden his approach by writing a book on geochemical modeling. This book has no equivalent in the present literature. It explains how we can build mathematical models to explain geochemical observations. 

This chapter is concerned with the design and improvement of chemically-active ship bottom paints known as antifouling paints. The aims have been to illustrate the challenges involved in working with such multi-component, functional products and to show which scientific and engineering tools are available. The research in this field includes both purely empirical formulation and test methods and advanced tools including mathematical modelling of paint behaviour. 

Despite the broad definition of chemometrics, the most important part of it is the application of multivariate data analysis to chemistry-relevant data. Chemistry deals with compounds, their properties, and their transformations into other compounds. Major tasks of chemists are the analysis of complex mixtures, the synthesis of compounds with desired properties, and the construction and operation of chemical technological plants. However, chemical/physical systems of practical interest are often very complicated and cannot be described sufficiently by theory. Actually, a typical chemometrics approach is not based on first principles—that means scientific laws and mles of nature—but is data driven. Multivariate statistical data analysis is a powerful tool for analyzing and structuring data sets that have been obtained from such systems, and for making empirical mathematical models that are for instance capable to predict the values of important properties not directly measurable (Figure 1.1). 

Typically extrapolations of many kinds are necessary to complete a risk assessment. The number and type of extrapolations will depend, as we have said, on the differences between condition A and condition B, and on how well these differences are understood. Once we have characterized these differences as well as we can, it becomes necessary to identify, if at all possible, a firm scientific basis for conducting each of the required extrapolations. Some, as just mentioned, might be susceptible to relatively simple statistical analysis, but in most cases we will find that statistical methods are inadequate. Often, we may find that all we can do is to apply an assumption of some sort, and then hope that most rational souls find the assumption likely to be close to the truth. Scientists like to be able to claim that the extrapolation can be described by some type of model. A model is usually a mathematical or verbal description of a natural process, which is developed through research, tested for accuracy with new and more refined research, adjusted as necessary to ensure agreement with the new research results, and then used to predict the behavior of future instances of the natural process. Models are refined as new knowledge is acquired. 

The success of any mathematical model, and in turn the computer code, depends completely on the clarity of the conceptual model (physical model). The authors have concluded from a comprehensive literature review on the subject of solid-fuel combustion, that there is a slight conceptual confusion in parts of this scientific domain. The first example of this is the lack of distinction between the thermochemical conversion of solid fuels and the actual gas-phase combustion process, which led these authors to the formulation of the three-step model. The thermochemical conversion of solid fuels is a two-phase phenomenon (fluid-solid phenomenon), whereas the gas-phase combustion is a one-phase phenomenon (fluid phenomenon). 

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