General mathematical formulation

A sublattice phase can be envisaged as being composed of interlocking sublattices (Fig. 5.3) on which the various components can mix. It is usually crystalline in nature but the model can also be extended to consider ionic liquids where mixing on particular ionic sublattices is considered. The model is phenomenological in nature and does not define any crystal structure within its general mathematical formulation. It is possible to define internal parameter relationships which reflect structure with respect to different crystal types, but such conditions must be externally formulated and imposed on the model. Equally special relationships apply if the model is to be used to simulate order-disorder transformations. 

The starting point for the receptor model is the source model. Though the source model may not deliver accurate results under many conditions, its limitations are primarily due to its inability to include every environmentally relevant variable and inadequate measurements for the variables it does include. The general mathematical formulations, however, are representative of the way in which particulate matter travels from source to receptor. 

In this section, general mathematical formulations and graphic interpretations are presented for several resilience analysis problems (1) feasibility test, (2) resilience (flexibility) test, (3) flexibility index, and (4) resilience index. 

A general mathematical formulation and a detailed analysis of the dynamic behavior of this mass-transport induced N-NDR oscillations were given by Koper and Sluyters [8, 65]. The concentration of the electroactive species at the electrode decreases owing to the electron-transfer reaction and increases due to diffusion. For the mathematical description of diffusion, Koper and Sluyters [65] invoke a linear diffusion layer approximation, that is, it is assumed that there is a diffusion layer of constant thickness, and the concentration profile across the diffusion layer adjusts instantaneously to a linear profile. Thus, they arrive at the following dimensionless set of equations for the double layer potential, [Pg.117]

The general mathematical formulation of the equilibrium statistical mechanics based on the generalized statistical entropy for the first and second thermodynamic potentials was given. The Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles were investigated as an example. It was shown that the statistical mechanics based on the Tsallis statistical entropy satisfies the requirements of equilibrium thermodynamics in the thermodynamic limit if the entropic index z=l/(q-l) is an extensive variable of state of the system. 

A general mathematical formulation that is applicable to most models described in the earlier sections begins with the volume reaction model described by Equations (11.69) through (11.72). Then, for a reaction first order in the gaseous component, we recast these equations as... 

Objective functions define the aim of an optimization process that is by convention stated as a minimization problem. A general mathematical formulation is, for example ... 

For many polymers, crystal growth can take place either from the melt or from dilute solution to yield single crystals. Crystal formations in polymers were studied intensively almost from the time of recognition of their existence in macromolecules. As a result, certain basic principles were established (1) The melt crystallization process is a first-order phase transition (see Section 1.4.5). (2) Crystallization from a molten polymer follows the general mathematical formulation for the kinetics of a phase change.Equilibrium conditions, however, are seldom if ever attained and complete crystallinity is not reached. 

In general mathematical formulation, the y-distribution comprises the gamma function instead of the factorial. For integers k, it reads i ( ) = ( - 1) . Since in the context of discussion only integers of quantity k make sense, we stay with the factorial. 

Certain basic information was established about the crystallization from the melt [5] The process is a first-order phase transition and follows the general mathematical formulation for the kinetics of a... 

In this chapter, we will study the strategic SCM level and introduce a general mathematical formulation for optimizing plant location decisions considering different information. We will first formally define the problem of interest and then describe the mathematical formulation in detail. We end the chapter with some notes and recommendations for further readings. 

In this work, we will consider the case where the process flowsheet is given, as well as the process specifications (including set-points for controlled variables). The objective is to find the equipment static design variables, the operating conditions and the controller s parameters which optimise the plant economics and, simultaneously, a measure of the plant controllability, subject to a set of constraints which ensure appropriate dynamic behaviour and process specifications. The general mathematical formulation of the problem is ... 

The pursuit of operations research consists of (a) the judgment phase (what are the problems ), (b) the research phase (how to solve these problems), and (c) the decision phase (how to act on the finding and eliminate the problems). These phases require the evaluation of objectives, analysis of an operation and the collection of evidence and resources to be committed to the study, the (mathematical) formulation of problems, the construction of theoretical models and selection of measures of effectiveness to test the models in practice, the making and testing of hypotheses as to how well a model represents the problem, prediction, refinement of the model, and the interpretation of results (usually as possible alternatives) with their respective values (payoff). The decision-maker generally combines the findings of the... 

Invariance Properties.—Before delving into the mathematical formulation of the invariance properties of quantum electrodynamics, let us briefly state what is meant by an invariance principle in general. As we shall be primarily concerned with the formulation of invariance principles in the Heisenberg picture, it is useful to introduce the concept of the complete description of a physical system. By this is meant at the classical level a specification of the trajectories of all particles together with a full description of all fields at all points of space for all time. The equations of motion then allow one to determine whether the system could, in fact, have evolved in the way... 

Such a choice of pore shape and distribution leads to a relatively simple mathematical formulation without loss of generality. 

In this chapter, the mathematical tools and fundamental concepts utilized in the development and application of modem estimation theory are considered. This includes the mathematical formulation of the problem and the important concepts of redundancy and estimability in particular, their usefulness in the decomposition of the general optimal estimation problem. A brief discussion of the structural aspects of these concepts is included. 

The preceding section discusses the mathematical formulation of the problem under consideration and the general conditions for redundancy and estimability. Now, we are ready to analyze the decomposition of the general estimation problem. The division of linear dynamic systems into their observable and unobservable parts was first suggested by Kalman (1960). The same type of arguments can be extended here to decompose a system considered to be at steady-state conditions. 

A variety of specific mathematical formulations of the CTRW approach have been considered to date, and network models have also been applied (Bijeljic and Blunt 2006). A key result in development of the CTRW approach is a transport equation that represents a strong generalization of the advection-dispersion equation. As shown by Berkowitz et al. (2006), an extremely broad range of transport patterns can be described with the (ensemble-averaged) equation... 

In general the mathematical formulation of the quantitation problem is shown in Table 2 for n-components. The matrix equation, H=AW G, can be solved for the total polymer Gp and... 

Dimensional analysis is based on the recognition that a mathematical formulation of a physicotechnological problem can be of general validity only when the process equation is dimensionally homogenous, which means that it must be valid in any system of dimensions. 

In this section we show that a tensor product of state spaces is the proper mathematical formulation for a system with independent measurements. Each measurement has a corresponding complex scalar product space whose pro-jectivization is the state space for that measurement the projectivization of the tensor product of the complex scalar product spaces is the state space for all of the measurements together. We work with one example, leaving generalization to the reader. 

The mathematical formulation of the theory becomes drastically more complicated however, the physical conclusions in the part of the curve relating to the pressure change during the reaction, the selection principle, and the calculation of the detonation velocity and the effect of external losses on the detonation velocity remain practically unchanged. As was to be expected, a theory of pressure and velocity of a detonation wave based on the general conservation laws proves not very sensitive to the mechanism of chemical reaction. 

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