Experimental data mathematical analysis

A model of the dynamics of phytoplankton populations based on the principle of conservation of mass has been presented. The growth and death kinetic formulations of the phytoplankton and zooplankton have been empirically determined by an analysis of existing experimental data. Mathematical expressions which are approximations to the biological mechanisms controlling the population are added to the mass transport terms of the conservation equation for phytoplankton, zooplankton, and nutrient mass in order to obtain the equations for the phytoplankton model. The resulting equations are compared with two years data from the tidal portion of the San Joaquin River, California. Similar comparisons have been made for the lower portion of Delta and are reported elsewhere (62). 

Many problems in physical chemistry are analyzed with approximations or idealizations that make the mathematics of the analysis less complicated or that offer a more discernible physical picture. Experimental data and analysis offer a validation or a rejection of the approximation. 

Both the need to reduce experimental costs and increasing reHabiHty of mathematical modeling have led to growing acceptance of computer-aided process analysis and simulation, although modeling should not be considered a substitute for either practical experience or reHable experimental data. 

Over the years there have been many attempts to simulate the behaviour of viscoelastic materials. This has been aimed at (i) facilitating analysis of the behaviour of plastic products, (ii) assisting with extrapolation and interpolation of experimental data and (iii) reducing the need for extensive, time-consuming creep tests. The most successful of the mathematical models have been based on spring and dashpot elements to represent, respectively, the elastic and viscous responses of plastic materials. Although there are no discrete molecular structures which behave like the individual elements of the models, nevertheless... 

Thus, Tis a linear function of the new independent variables, X, X2,. Linear regression analysis is used to ht linear models to experimental data. The case of three independent variables will be used for illustrative purposes, although there can be any number of independent variables provided the model remains linear. The dependent variable Y can be directly measured or it can be a mathematical transformation of a directly measured variable. If transformed variables are used, the htting procedure minimizes the sum-of-squares for the differences... 

Because earlier experimental results and data analyses (3-10) had led us to anticipate the inadequacy of the simple approach considered above, we also planned and carried out (2) a second order factorial design of experiments and related data analysis. Mathematical analysis (of the results of 11 experiments) based on the second order model showed that all of these results could be represented satisfactorily by an equation of the form... 

Perhaps the most discouraging type of deviation from linearity is random scatter of the data points. Such results indicate that something is seriously wrong with the experiment. The method of analysis may be at fault or the reaction may not be following the expected stoichiometry. Side reactions may be interfering with the analytical procedures used to follow the progress of the reaction, or they may render the mathematical analysis employed invalid. When such plots are obtained, it is wise to reevaluate the entire experimental procedure and the method used to evaluate the data before carrying out additional experiments in the laboratory. 

The studies described in the preceding two sections have identified several processes that affect the dynamic behavior of three-way catalysts. Further studies are required to identify all of the chemical and physical processes that influence the behavior of these catalysts under cycled air-fuel ratio conditions. The approaches used in future studies should include (1) direct measurement of dynamic responses, (2) mathematical analysis of experimental data, and (3) formulation and validation of mathematical models of dynamic converter operation. 

Ideal reactors can be classified in various ways, but for our purposes the most convenient method uses the mathematical description of the reactor, as listed in Table 14.1. Each of the reactor types in Table 14.1 can be expressed in terms of integral equations, differential equations, or difference equations. Not all real reactors can fit neatly into the classification in Table 14.1, however. The accuracy and precision of the mathematical description rest not only on the character of the mixing and the heat and mass transfer coefficients in the reactor, but also on the validity and analysis of the experimental data used to model the chemical reactions involved. 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

Beeckmans, J. M. The depoution of aerosols in the respiratory tract. I. Mathematical analysis and comparison with experimental data. Can. J. Physiol. Pharmacol. 43 157-172, 1%5. 

Any statistical procedure that defines a set of experimental data in terms of a mathematical function. The most common curve-fitting protocol is the least squares method (also known as analysis of covariance). 

A number of useful reviews of small molecule-protein interaction have appeared (D2, E4, S26, W7), and these contain detailed treatments of the mathematical analysis of binding data as well as information on experimental methods. 

The remaining errors in the data are usually described as random, their properties ultimately attributable to the nature of our physical world. Random errors do not lend themselves easily to quantitative correction. However, certain aspects of random error exhibit a consistency of behavior in repeated trials under the same experimental conditions, which allows more probable values of the data elements to be obtained by averaging processes. The behavior of random phenomena is common to all experimental data and has given rise to the well-known branch of mathematical analysis known as statistics. Statistical quantities, unfortunately, cannot be assigned definite values. They can only be discussed in terms of probabilities. Because (random) uncertainties exist in all experimentally measured quantities, a restoration with all the possible constraints applied cannot yield an exact solution. The best that may be obtained in practice is the solution that is most probable. Actually, whether an error is classified as systematic or random depends on the extent of our knowledge of the data and the influences on them. All unaccounted errors are generally classified as part of the random component. Further knowledge determines many errors to be systematic that were previously classified as random. 

A mathematical analysis of such mechanisms shows that simple kinetics is only found when the equilibrium is set up rapidly and maintained throughout the reaction. In all other circumstances, complex kinetics would be expected, and computer analysis of the experimental data would be required. Fortunately, for most inorganic mechanisms the equilibrium is set up rapidly and maintained, and simple kinetics is observed. 

A reverse kinetic problem consists in identifying the type of kinetic models and their parameters according to experimental (steady-state and unsteady-state) data. So far no universal method to solve reverse problems has been suggested. The solutions are most often obtained by selecting a series of direct problems. Mathematical treatment is preceded by a qualitative analysis of experimental data whose purpose is to reduce drastically the number of hypotheses under consideration [31]. 

Different data interpretation models have been applied simple dissociation constants (Langford and Khan, 1975), discrete multi-component models (Lavigne et al., 1987 Plankey and Patterson, 1987 Sojo and de Haan, 1991 Langford and Gutzman, 1992), discrete kinetic spectra (Cabaniss, 1990), continuous kinetic spectra (Olson and Shuman, 1983 Nederlof et al., 1994) and log normal distribution (Rate et al., 1992 1993). It should be noted that for heterogeneous systems, analysis of rate constant distributions is a mathematically ill-posed problem and slight perturbations in the input experimental data can yield artefactual information (Stanley et al., 1994). 

Once the best estimates of the adjustable parameters have been computed, an analysis of the results allows one to evaluate the quality of the correspondence between experimental data and mathematical model and to identify the best model among the available alternatives. This analysis consists of different steps, mainly based on... 

The parameters of the models, which include the equilibrium constants, the rate constants of elementary processes, and the interaction potentials, differ when the same experimental data are described by using different approximations. This is why the description of complicated processes requires that all the subsystems be considered with one level of accuracy, i.e., with the employment of the same approximation. For this purpose, it is essential that the physical and chemical fundamentals introduced into a mathematical model correspond to the real conditions within a broad range of variations of the external conditions. Now the parameters of models found from an analysis of individual subsystems can also be used when describing a process as a whole. 

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