Mathematical characterization

Despite being notoriously difficult to analyze formally, the behavior of general CA rules is nonetheless often amenable to an almost complete mathematical characterization. In this section we look at a simple method that exploits the properties of certain implicit deterministic structures of elementary one-dimensional rules to help determine the existence of periodic temporal sequences, rule inverses and homogeneous states. Additional details appear in [jen86a] and[jen86b]. 

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

Mathematical Characterization of Simple Constant Volume Reaction Systems... 

The notion of chaos is interwoven with the discussion of time evolution, which we do not pursue in this volume. It is worthwhile, however, to note that it is, by now, well understood that a quantum-mechanical system with a finite Hamiltonian matrix cannot satisfy many of the purely mathematical characterizations of chaos. Equally, however, over long periods of time such systems can manifest many of the qualitative features that one associates with classically chaotic systems. It is not our intention to follow this most interesting theme. Instead we seek a more modest aim, namely, to forge a link between the elementary notions of classical nonlinear dynamics and the algebraic approach. This turns out to be possible using the action-angle variables of classical mechanics. In this section we consider only the nonlinear dynamics aspects. We complete the bridge in Chapter 7. 

Before discussing methods for optimization of stationary points it is appropriate to state briefly the mathematical characterization of such points and lay out the basic strategies for their determination. 

In response to this we need to return to the mathematical characterization of extension that I have provided. In characterizing extension, I appealed to the notion of a set of points structured in some way. There are two components to such an idea - the set of points and the relations defined on them. The resulting structure, I said, is extension. Now, were one to abstract away the relations from extension, one would arrive at the set of points. Or, so one might think. But in fact, once the relations are removed, what one arrives at is a structureless set of objects with no intrinsic properties. And such objects could not be points. For, necessarily any point bears some spatial relation to something. Hence, without any relations defined on them, the resulting objects can hardly be called points. So, let us just call them objects. 

We will leave the proof of the conjecture to mathematically inclined readers interested in this problem and will focus attention on consequences of the novel definition of Clar structures. If the two definitions of Clar structure are equivalent (as we conjecture) there should be identical consequences and the new definition is not to make a difference. However, the new definition does offer a mathematical characterization of Kekule valence structures involved in construction of Clar structure, something that has been hitherto missing. 

Mathematical Characterization of the Compressive Stress-Strain Relationship of Polymeric Foams, Breads and Cakes... 

Peleg, M., Roy, I., Campanella, O.H., and Normand, M.D. (1989). Mathematical characterization of the compressive stress-strain relationships of spongy baked goods. J. Food Sci., 54, 947-949. 

In the second variant, the plug flow model is considered as a series of tanks with perfect mixing flow [3.22, 3.23]. In this case, the real filter will be supposedly replaced by a series of some small filters (three in this analysis) with perfect mixing flow. Figure 3.9 shows the scheme, relations and notations used. The filtrate transfer equation has been used for the mathematical characterization of each small filter for the total material balance equation and non-steady-state solid balance equation ... 

For the mathematical characterization of polystochastic chains, we often use the theory of systems with complete connections. According to the definition given in... 

The ELF is a scalar function of three variables, and in order to obtain more information from it, it is necessary to use a mathematical approach called differential topology analysis. This was first done by Silvi and Savin,11 and later on extended by them and co-workers.45,46 Unfortunately, one cannot visualize in a global way a three-dimensional function. Usually, one resorts to isosurfaces like the ones in Figure 1, or to contour maps. A three-dimensional function has a richer structure than a one-dimensional function, and their mathematical characterization introduces some new words which are necessary to understand in order to go further. It is the purpose of this section to explain this new terminology in a manner as simpler as possible. Let us begin with a one-dimensional (ID) example, a function f(x) like the one in Figure 3. The function has three maxima and two minima characterized by the sign of the second derivative. In three dimensions (3D) there are more possibilities, for there are nine second derivatives. Hence, one does not talk about maxima but about attractors. In ID, the attractors are points, in... 

Burnham and Braun [99] have provided a valuable review of the approaches used in the kinetic analysis of decompositions of complex materials such as polymers, minerals, fossil fuels and biochemicals. Mathematical characterization of these reaction systems, recognized as being too complex to be characterized in any fundamental way, is termed global kinetic analysis. One of its main objectives is to predict the reactivities of such materials at temperatures different from those for which kinetic measurements were made (see Section 5.5.13.). 

In order to reliably calculate microbial behavior, predictive microbiology requires a reliable combination of mathematical and statistical considerations (Roberts, 1995). It is, however, often inappropriate to extrapolate mathematical models used in different applications. In Table 10.1 the differences between mathematical characterization of bacterial growth in food microbiology and mathematical modeling techniques used in biotechnology are stipulated. 

Table 10.1 Differences between Mathematical Characterization of Bacterial Growth and Mathematical Modeling Techniques Used in Biotechnology...

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