Model, mathematical weighting

Two procedures for improving precision in calibration curve-based-analysis are described. A multiple curve procedure is used to compensate for poor mathematical models. A weighted least squares procedure is used to compensate for non-constant variance. Confidence band statistics are used to choose between alternative calibration strategies and to measure precision and dynamic range. 

Keywords polymerization kinetics, polymerization reactors, mathematical modelling, molecular weight distribution (MWD), chemical composition distribution (CCD), Ziegler-Natta catalysts, metallocenes, microstructure, isotacticity distribution, mass transfer resistances, heat transfer resistances, effects of multiple site types. 

In the previous four sections, the sum of squared residuals that was minimized was that given by Eq. (7.160). This was the sum of squared residuals determined from fitting one equation to measurements of one variable. However, most mathematical models may involve simultaneous equations in multiple dependent variables. For such a case, when more than one equation is fitted to multiresponse data, where there are v dependent variables in the model, the weighted sum of squared residuals is given by... 

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

As already discussed, variations of a field unknown within a finite element is approximated by the shape functions. Therefore finite element discretization provides a nat ural method for the construction of piecewise approximations for the unknown functions in problems formulated in a global domain. This is readily ascertained considering the mathematical model represented by Equation (2.40). After the discretization of Q into a mesh of finite elements weighted residual statement of Equ tion (2.40), within the space of a finite element T3<, is written as... 

It cannot be stated generally how accurate the predicted results are. Due to the limitations of geometric, physical, and mathematical modeling, not all of the produced numbers (e.g., air velocity vectors) are at a high level of accuracy, and the results are therefore subjected to experienced weighting. In some cases, the values can be as accurate as within 5% of the real values in other cases, they are not as accurate as could be wished. But results can be still very strong and helpful in a comparative judgment, i.e., if a number of similar case.s are compared with observed tendencies. 

Hydrodynamic volume refers to the combined physical properties of size and shape. Molecules of larger volume have a limited ability to enter the pores and elute the fastest. A molecule larger than the stationary phase pore volume elutes first and defines the column s void volume (Vo). In contrast, intermediate and smaller volume molecules may enter the pores and therefore elute later. As a measure of hydrodynamic volume (size and shape), SE-HPLC provides an approximation of a molecule s apparent molecular weight. For further descriptions of theoretical models and mathematical equations relating to SE-HPLC, the reader is referred to Refs. 2-5. 

Mathematical models of the reaction system were developed which enabled prediction of the molecular weight distribution (MWD). Direct and indirect methods were used, but only distributions obtained from moments are described here. Due to the stiffness of the model equations an improved numerical integrator was developed, in order to solve the equations in a reasonable time scale. 

Mathematical models of the reaction system have been developed, enabling prediction of the molecular weight... 

As with most modeling efforts, the mathematical formulation is the easy part. Picking the right values from the literature or experiments is more work. An immediate task is to decide how to characterize the substrate and product concentrations. The balance equations for substrate and product apply to the carbon content. The glucose molecule contains 40% carbon by weight so S will be 0.4 times the glucose concentration, and 5q = 0.04. Similarly,... 

When one tries to fit a mathematical function to a set of data one has to choose a method or algorithm for achieving this end, and choose a weighting model with which to judge the goodnss of fit. 

The algebraic/iterative and the brute force methods are numerical respectively computational techniques that operate on the chosen mathematical model. Raw residuals r are weighted to reflect the relative reliabilities of the measurements. 

In implicit estimation rather than minimizing a weighted sum of squares of the residuals in the response variables, we minimize a suitable implicit function of the measured variables dictated by the model equations. Namely, if we substitute the actual measured variables in Equation 2.8, an error term arises always even if the mathematical model is exact. 

In parameter estimation we are occasionally faced with an additional complication. Besides the minimization of the objective function (a weighted sum of errors) the mathematical model of the physical process includes a set of constrains that must also be satisfied. In general these are either equality or inequality constraints. In order to avoid unnecessary complications in the presentation of the material, constrained parameter estimation is presented exclusively in Chapter 9. 

A mathematical model has been developed which allows the calculation of the degradation of polymeric drug delivery systems. The model has been shown to accurately simulate both the drug release and molecular weight changes in such systems. The concentration of anhydride levels affect the erosion characteristics of... 

Detailed studies on the lipase-catalyzed polymerization of divinyl adipate and 1,4-butanediol were performed [41-44]. Bulk polymerization increased the reaction rate and molecular weight of the polymer however, the hydrolysis of the terminal vinyl ester significantly limited the formation of the polyester with high molecular weight. A mathematical model describing the kinetics of this polymerization was proposed, which effectively predicts the composition (terminal structure) of the polyester. 

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