General Matrix Formulation

Consider an arbitrary three-dimensional enclosure of total volume V and surface area A which confines an absorbing-emitting medium (gas). Let the enclosure be subdivided (zoned) into M finite surface area and N finite volume elements, each small enough that all such zones are substantially isothermal. The mathematical development in this section is restricted by the following conditions and/or assumptions  

Radiative transfer with respect to the confined gas is either monochromatic or gray. The gray gas absorption coefficient is denoted here by fC(m-1). In subsequent sections the monochromatic absorption coefficient is denoted by KfX). 

All surface emissivities are assumed to be gray and thus independent of temperature. 

Noble (op. cit.) has extended the present matrix methodology to the case where the gaseous absorbing-emitting medium also scatters isotropically. 

Two arrays of direct exchange areas are now defined i.e., the matrix ss = IT, is the M X M array of direct surface-to-surface exchange areas, and the matrix sg= [Tg,] is the MxN array of direct gas-to-surface exchange areas. Here the scalar elements of ss and sg are computed from the integrals 

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In general matrix formulation the vibrational polar tensor of a molecule is given by... 

In the case of thicknesses larger than mentioned above the intensities must be calculated according to the more general many-beam theory. The calculation should include summation over different groups of crystals having a certain distributions of thickness and orientation. A method based on the matrix formulation of the many-beam theory was developed for partly-oriented thin films and have been successfully applied samples [2]. The main problem in using direct many-beam calculation is to find the distribution functions for size and orientation of the microcrystals. However, it is not always possible to refine these functions in the process of intensity adjustment. Additional investigation of the micro-structure by electron microscopy is very helpful in such case. 

Levy identified the unknown part of the exact universal D functional as the correlation energy Ed D] and investigated a number of properties of c[ D], including scaling, bounds, convexity, and asymptotic behavior [11]. He suggested approximate explicit forms for Ec[ D] for computational purposes as well. Redondo presented a density-matrix formulation of several ab initio methods [26]. His generalization of the HK theorem followed closely Levy s... 

It is now possible to see that the matrix formulation has the potential for describing the more-general Fredholm integral equation. This equation corresponds in spectroscopy to the situation where the functional form of s(x) varies across the spectral range of interest. In these circumstances, s is expressed as a function of two independent variables. Although we proceed with the present treatment formulated in terms of convolutions, the reader should bear the generalization in mind. 

Finally, MET was imparted the matrix form similar to that of IET. A newly developed original method based on the many-particle master equation led to an infinite hierarchy for vector correlation patterns (VCPs) that can be truncated in two different ways [43,44], The simplest one reproduces the conventional IET, while the other allows a general modification of the kernel, resulting in the matrix formulation of MET applicable to complex multistage reactions. 

Evaluation of the rate coefficients from the relaxation times of a complex system generally involves a matrix formulation of this type. Solution of eqn. (22) gives two relaxation times, T and rn, which are the roots of the characteristic equation... 

Let us now consider regression in general in terms of a matrix formulation of the fast wavelet transform. 

Many authors propose alternative mathematical treatments for kinetics equations. Some examples are a general approach based on a matrix formulation of the differential kinetic equations (Berberan-Santos Martinho, 1990) spreadsheets in which rate equations are integrated using the simple Euler approximation (Blickensderfer, 1990) a method for the accurate determination of the first-order rate constant (Borderie, Lavabre, Levy Micheau, 1990) a simplification of half-life methods that provides a fast way of determining reaction orders and rate constants (Eberhart Levin, 1991) a general approach to reversible processes, the special cases of which are shown to be equivalent to basic kinetic equations (Simonyi Mayer, 1985) an equation from which zero-, first- and higher order equations can be derived (Tan, Lindenbaum Meltzer, 1994). 

Farotimi, O., Dembo, A., and Kailath, T. 1991. A general weight matrix formulation using optimal control. IEEE Trans. Neural Networks, 2 378-394. 

The number of simultaneous equations provided by Equation 1.43 increases as the number of components increases. It then becomes more efficient to express the results in terms of matrices. Again, there are many equivalent matrix formulations that have been presented (Kirkwood and Buff 1951 O Connell 1971b Ben-Naim 2006 Nichols, Moore, and Wheeler 2009). Here, we present one of the simplest. A general formulation is easiest starting from the first expression in Equation 1.44. Writing the number fluctuations in matrix form for an component system where we also include the number densities (GD expression at constant T) in the first row provides. 

It should be pointed out that Greilf proposed that the thermostability of freeze-dried protein products could be a bell-shape funetion of the residual moisture eontent [26]. In other words, both overdrying and underdrying are detrimental. Hsu et al. also coneluded that the generally accepted concept the drier, the better is not neeessarily appropriate for their product [27]. It is noted that their data were generated from excipient-free proteins. In this study, however, our results did not demonstrate a bell-shape function for the crystalline matrix formulation. 

The best way to take into account also propagation during the iterations leading to self-consistency of a perturbation localized originally to one or a few cells is a general Green matrix formulation of the problem. " ... 

Elastic constants are directly related to the interchain and intrachain force field. A general matrix method for treating elastic constants was reported by Shiro (1968), Shiro and Miyazawa (1971), where the basic formulations of Bom and Huang (1954) were simplified with the use of matrix equations and symmetry considerations. 

A matrix formulation of these equations was developed in Section 14.1.2, using an orthonormal basis for the expansion of the perturbed wave functions (14.1.28). In this exercise, this matrix formulation is generalized to the expansion of the perturbed states ... 

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