The method of moments

The remaining part of the collision integral can be calculated analytically by identifying the different cases, thereby reducing the discretized KE to a form similar to Eq. (7.52). A detailed description can be found in Aristov (2001). 

The basic idea behind the MOM is to solve the evolution equations for the moments of the NDF instead of determining the evolution of the NDE by solving the actual GPBE. The general evolution equation for the kth moment (k = (A i. m) is the moment exponent vector containing the order of the moments with respect to each of the components of ) for a homogeneous system can easily be obtained by applying the moment transform to Eq. (7.1)  

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The method of moments reduees the eomputational problem to solution of a set of ordinary differential equations and thus solves for the average properties of the distribution. 

Estimation of model parameters is frequently accomplished by the method of moments. For example, for the uniform distribution, the mean is... 

Bulman RA (1978) Chemistry of Plutonium and the Transuranics in the Biosphere. 34 39-77 Bulman RA (1987) The Chemistry of Chelating Agents in Medical Sciences. 67 91-141 Burdett JK (1987) Some Structural Problems Examined Using the Method of Moments. 65 29-90... 

The present section analyzes the above concepts in detail. There are many different mathematical methods for analyzing molecular weight distributions. The method of moments is particularly easy when applied to a living pol5mer polymerization. Equation (13.30) shows the propagation reaction, each step of which consumes one monomer molecule. Assume equal reactivity. Then for a batch polymerization. 

Example 13.8 Apply the method of moments to an anionic polymerization in a CSTR. 

Application of the Method of Mcanents. In order to apply the method of moments (6,7), the pseudo-kinetic rate constant for the crosslinking reaction should be defined as follows. 

The effects of various pore-size distributions, including Gaussian, rectangular distributions, and continuous power-law, coupled with an assumption of cylindrical pores and mass transfer resistance on chromatographic behavior, have been developed by Goto and McCoy [139]. This study utilized the method of moments to determine the effects of the various distributions on mean retention and band spreading in size exclusion chromatography. 

Burden, J. K. Some Structural Problems Examined Using the Method of Moments. Vol. 65, pp. 29-90. 

APPENDIX. CALCULATION OF THE DENSITY OF ELECTRONIC STATES WITHIN THE TIGHT BINDING THEORY BY THE METHOD OF MOMENTS... 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

The method of moments of coupled-cluster equations (MMCC) is extended to potential energy surfaces involving multiple bond breaking by developing the quasi-variational (QV) and quadratic (Q) variants of the MMCC theory. The QVMMCC and QMMCC methods are related to the extended CC (ECC) theory, in which products involving cluster operators and their deexcitation counterparts mimic the effects of higher-order clusters. The test calculations for N2 show that the QMMCC and ECC methods can provide spectacular improvements in the description of multiple bond breaking by the standard CC approaches. 

The Method of Moments of Coupled-Cluster Equations An Overview of the Ground-State Formalism... 

In conclusion, the method of moments can be used to obtain a state space model for the dynamics of the moments of the CSD. The method is limited to MSMPR crystallizers with size-independent growth or size-dependent growth described by... 

Example. Consider an evaporative, isothermal. Class II crystallizer with fines removal. It is assumed that the growth rate is size-independent and that there is no growth rate dispersion. Because we assume fines removal, the method of moments can not be applied. 

The first method, the method of moments, is restricted to MSMPR crystallizers with size-independent growth or very simple size-dependent growth rate kinetics. Depending on the control demand, several process outputs can be chosen for the control algorithm. When population densities, or the number or mass of crystals in a size range are to be controlled, the method of moments can not be used because it reveals information on the dynamics of the moments of the crystal size distribution only. 

The most familiar estimation procedure is to assume that the population mean and variance are equal to the sample mean and variance. More generally, the method of moments (MOM) approach is to equate sample moments (mean, variance, skewness, and kurtosis) to the corresponding population. Software such as Crystal Ball (Oracle Corporation, Redwood Shores, CA) uses MOM to fit the gamma and beta distributions (see also Johnson et al. 1994). Use of higher moments is exemplified by fitting of the... 

This expression includes the curvature corrections to the Gaussian function, which play an important role in the averaging procedure they ensure the smoothed spectrum g(E) to be approximated, through the above definition, by its own truncated Taylor expansion. This smoothing procedure of the one-electron spectrum is an application of the method of moments, also used in other systems [23]. 

Generalization of the method of moments of coupled-cluster equations to excited electronic states Exact formalism... 

Detonation, Free Volume Theory of Multi-component Fluid Mixtures. The free volume theory of the liq state is extended to multi-component fluid mixts by using the method of moments in the treatment of the order-disorder problem. The results of this extension are given in the article by Z.W. 

Hiickel calculations have been employed extensively in other approaches such as the angular overlap model and the method of moments developed by Burdett and coworkers. Stabilities of crystal structures, pressure- and temperature-induced transitions, dynamical pathways in reactions and other phenomena have been analysed using angular overlap models. Thus, the electronic control of rutile structures and the stability of the defect structure of NbO have been examined (Burdett, 1985 Burdett Mitchell, 1993). In the case of NbO, the structure is stable at involving the formation... 

Theoretical analysis has always been an important part of MCD spectroscopy. The parameters Aj, Bj, and Cj can be extracted from an experimental spectrum by a fit to a suitable set of functions or through the method of moments (27-28). The interpretation of these parameters is generally not a trivial task. For smaller, symmetrical molecules group theory has been used to good effect to extract information from an MCD spectrum (11). In recent years, quantum chemical calculation has proven a very useful aid in the interpretation of the often-complicated spectra of larger, nonsymmetric molecules. 

Saidel and Katz (97) have used an extension of the method of moments to calculate MJMn for a batch polymerization with transfer to polymer, and with termination exclusively by combination, but with restriction to mono-radicals. As the equations for the second and higher moments contain terms involving still higher moments it is necessary to approximate them in terms of lower moments, but this requires assumptions about the form of the distribution that may not be justified. 

Whereas there is little doubt that the method of moments, as the procedure is called, is basically sound, it is obvious that for reliable results high-quality experimental data over a broad range of frequencies and temperatures are desirable. As importantly, reliable models of the interaction potential must be known. Since these requirements have rarely been met, ambiguous dipole models have sometimes been reported, especially if for the determination of the spectral moments substantial extrapolations to high or to low frequencies were involved. Furthermore, since for most works of the kind only two moments have been determined, refined dipole models that attempt to combine overlap and dispersion contributions cannot be obtained, because more than two parameters need to be determined in such case. As a consequence, empirical dipole models based on moments do not attempt to specify a dispersion component, or test theoretical values of the dispersion coefficient B(7) (Hunt 1985). 

Using the results in Example 18.7, estimate the asymptotic covariance matrix of the method of moments estimators of P and 7. based on m[ and tn 2. [Note You will need to use the data in Example C.l to estimate V.]... 

Using only the nonlimit observations, repeat the Exercise 2 in the context of the truncated regression model. Estimate u and a by using the method of moments estimator outlined in Example 20.4. Compare your results to those in the previous problems. 

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