Mathematical viewpoint

From the information-theoretical point of view, calibration corresponds to the coding of the input quantity into the output quantity and, vice versa, the evaluation process corresponds to decoding of output data. From the mathematical viewpoint, qin is the independent quantity in the calibration step and qout the dependent one. In the evaluation step, the situation is reverse qout is the independent, and qin the dependent quantity. From the statistical standpoint, qout is a random variable both in calibration and evaluation whereas qin is a fixed variable in the calibration step and a random variable in the evaluation step. This rather complicated situation has some consequences on which will be returned in Sect. 6.1.2. 

The second major difference found in vapor-liquid extraction of polymeric solutions is related to the low values of the diffusion coefficients and the strong dependence of these coefficients on the concentration of solvent or monomer in a polymeric solution or melt. Figure 2, which illustrates how the diffusion coefficient can vary with concentration for a polymeric solution, shows a variation of more than three orders of magnitude in the diffusion coefficient when the concentration varies from about 10% to less than 1%. From a mathematical viewpoint the dependence of the diffusion coefficient on concentration can introduce complications in solving the diffusion equations to obtain concentration profiles, particularly when this dependence is nonlinear. On a physiced basis, the low diffiisivities result in low mass-transfer rates, which means larger extraction equipment. 

The mathematics we shall need is confined to the properties of vector spaces in which the scalar values are real numbers. From a mathematical viewpoint the whole discussion will take place in the context of two vector spaces, an S-dimensional space of chemical mechanisms and a Q-dimen-sional space of chemical reactions, which are related to each other by the fact that each mechanism m is associated with a unique reaction R(m) which it produces. The function R is a transformation of mechanisms to reactions which is linear by virtue of the fact that reactions are additive in a chemical system and that the reaction associated with combined mechanisms mt + m2 is R(m,) + R(m2). All mechanisms are combinations of a simplest kind of mechanism, called a step, which ideally consists of a one-step molecular interaction. Each step produces one of the elementary reactions which form a basis for the space of all reactions. 

To introduce recent advances without offending old friends who cherish the foundations of a successful past to keep pace with present tendencies toward the mathematical viewpoint without driving away students who are inadequately prepared and to sift out the permanent from the trivial are the privileges and responsibilities [of the author]. (2)... 

One special type of optimization problem involving restrictions or constraints has been solved quite successfully by a technique known as linear programming. From a mathematical viewpoint the basic form of the problem may be stated very briefly. Consider a linear response function of n variables ... 

Note that the employed definition of a system contains an important asymmetry between the system and the external world. The system s description includes the influence of the external forces on the system, but not the reverse, the influence of the system on the outside world. From a mathematical viewpoint the equations that describe the system s behavior contain the external forces, but the latter have to be taken from information outside the model. As will been shown, the specific hierarchy between system and outside world can often (but not always) be justified based on the respective strength of the interactions. Take the system of the earth. Solar radiation is a very strong driving force for the earth, but the back-radiation from the earth to the sun is so tiny that nobody would want to include it as a feedback mechanism in a radiation model of the sun. 

The second input scenario, simpler from a mathematical viewpoint but less probable to occur, is a generalization of the instantaneous (5-)input. It is assumed that the temporal variation of the input is Gaussian and leads to an initial longitudinal variation of the concentration cloud with standard deviation ... 

Before you do that, said A, I want to ask you about this binocular vision business. You were making a big fuss, a few moments ago, about the need to keep both the physical and the mathematical viewpoints in balance. Haven t you gone overboard with the mathematics ... 

From a mathematical viewpoint the necessary and sufficient condition for azeot-ropy is ... 

In any chemical or electrochemical process, the application of the conservation principles (specifically to the mass, energy or momentum) provides the outline for building phenomenological mathematical models. These procedures could be made over the entire system, or they could be applied to smaller portions of the system, and later integrated from these small portions to the whole system. In the former case, they give an overall description of the process (with few details but simpler from the mathematical viewpoint) while in the later case they result in a more detailed description (more equations, and consequently more features described). 

If the backbone torsion angles of a polypeptide are kept constant from one residue to the next, a regular repeating structure will result. While all possible structures of this type might be considered to be helices from a mathematical viewpoint, they are more commonly described by their appearance hence a helices and /3 pleated sheets. 

For the mathematical models based on transport phenomena as well as for the stochastic mathematical models, we can introduce new grouping criteria. When the basic process variables (species conversion, species concentration, temperature, pressure and some non-process parameters) modify their values, with the time and spatial position inside their evolution space, the models that describe the process are recognized as models with distributed parameters. From a mathematical viewpoint, these models are represented by an assembly of relations which contain partial differential equations The models, in which the basic process variables evolve either with time or in one particular spatial direction, are called models with concentrated parameters. 

It is important to note that, except for the heat transfer problems, which have not been considered here, the model contains, in a particular form, all the transport phenomena relationships given at the start of this chapter. From the mathematical viewpoint, we have an assembly of differential and partly differential equations, which show the complexity of this example. However, this relative mathematical complexity can be matched with the simplicity of the descriptive model. Indeed, it will be convenient to simplify general mathematical models in order to comply with the descriptive model. Two variants can be selected to simplify the flow characterization in the membrane filtration unit. 

From the mathematical viewpoint, it is important to assume that a very rapid heat transfer occurs at the extremities of the stick, and that a rapid cooling system is activated when the heating source is stopped. In addition, as far as we only take... 

An assembly of relations that contains the introduction of expressions for equality type constraints this assembly links some or all of the parameters of the model. From the mathematical viewpoint we can write these relations as follows ... 

From the mathematical viewpoint, this solution is quite different, from the solutions obtained with the open-open boundary conditions, although in the latter case we do not make use of the part of the column for which < 0. 

It is apparent that Eq. (121) contains the J function as a limiting case, at r = 1. This equation also has been shown to reduce to the constant-pattern result (Eq. 96) with r proportionate-pattern result (Eq. 75) with r 1, in work by Hiester and Vermeulen (H6) and Gilliland and Baddour (G2). Goldstein (G6) has reviewed this result from a mathematical viewpoint, and has presented limiting forms which give an accurate approximation in certain regions his variables u, s, and y correspond respectively to the present x, N, and ZN. 

We have shown from a mathematical viewpoint that similarity solutions of the boundary-layer equations exist whenever... 

This problem has also been solved numerically, and the function j(rf) is tabulated in [424]. We point out that in this case the solution differs from the corresponding Blasius solution. Thus, although physical consideration suggests that the inversion of flow is possible, the solution shows that it is impossible from the mathematical viewpoint. This is due to the fact that problems (1.7.5) and (1.7.11) are nonlinear. 

Obviously, from the mathematical viewpoint, problem (3.1.33), (3.1.34) describing the body-medium heat exchange is identical to problem (3.1.8)—(3.1.10) describing the flow-particle mass exchange in the case of a diffusion regime of reaction on the particle surface. 

From the mathematical viewpoint, the diffusion problem (4.3.1)-(4.3.3) is equivalent to the problem on the electric field of a charged conductive body in a homogeneous charge-free dielectric medium. Therefore, the mean Sherwood number in a stagnant fluid coincides with the dimensionless electrostatic capacitance of the body and can be calculated or measured by methods of electrostatics. 

In the following sections, relationships between three-way methods are outlined, facilitating comparisons between different models used on the same data set. An overview of hierarchical relationships between the major three-way models is given by Kiers [1991a], While a mathematical viewpoint is taken in this chapter, a more practical viewpoint for choosing between competing models is taken in Chapter 7. 

From a mathematical viewpoint this small difference is essential, although there is a strong similarity between the representation theories of the standard and g-deformed su(2)-algebras. In the case of the standard algebra (3) one can define the following operator ... 

From a mathematical viewpoint the word large is more difficult to explain, since there is no typical lifetime associated to a differential equation. Hence, in order to give the word stability a meaning in the sense above it is essential to consider the dependence of T on e (or on go). In this respect the continuity with respect to initial data does not help too much. For instance, if we consider the trivial example of the differential equation x = x one will immediately see that if a (0) = Xo > 0 is the initial point, then we have x(t) > 2xo for t > T = In 2 no matter how small is Xq] hence T may hardly be considered to be large , since it remains constant as Xq decreases to 0. 

The reactor design equations in this book can be applied to all components in the system, even inerts. When the reaction rates are formulated using Equation 2.8, the solutions automatically account for the stoichiometry of the reaction. This is the simplest and preferred approach, but it has not always been followed in this book. Several examples have ignored product concentrations when they do not affect reactions rates and when they are easily found from the amount of reactants consumed. Also, some of the analytical solutions have used stoichiometry to ease the algebra. The present section formalizes the use of stoichiometric constraints. We begin with a matrix formulation for the reaction rates of the components in multiple reactions. The presentation is rather elegant from a mathematical viewpoint and does have some practical utility. 

The assumption that, for a nonuniform surface, E increases linearly with increase of coverage is unrealistic from a physical viewpoint it is however, a convenient postulate from a mathematical viewpoint, particularly when it is realized that a surface comprising a small number of homogeneous patches, each patch having different E values on which there may or may not be induced effects, gives rise to an adsorption rate which subscribes well to an Elovich equation this model is an acceptable physical description for adsorbents in the form of powders or evaporated films. Similarly, models comprising uniform surfaces, but with site creation or exclusion, may be analyzed and extended to give conclusions of the same natures as those derived from a variation of E over a nonuniform surface mathematically, however, the extension to, e.g., interaction effects between two different adsorbates is more cumbersome. 

As it is clearly seen the catalyst effectivenes factors can be obtained for Langmuir-Hinshelwood kinetics by interpolating between 0th and 1st order kinetics. From the mathematical viewpoint, the Langmuir-Hinshelwood form of kinetics is similar to Michaelis-Menten kinetics. The influence of internal mass transfer on Michaelis-Menten kinetics will be discussed in the section 9.6. 

The mathematical complexity increases considerably from steady state to dynamic simulation, changing from algebraic to differential-algebraic equations. From mathematical viewpoint we may distinguish between two categories of models ... 

The origin of the unphysical behavior of the diffusion equation and the reaction-diffusion equation can be understood from three different viewpoints (i) the mathematical viewpoint, (ii) the macroscopic or phenomenological viewpoint, and (iii) the mesoscopic viewpoint. 

From a mathematical viewpoint, the origin of the infinitely fast spreading of local disturbances in the diffusion equation can be traced to its parabolic character. This can be addressed in an ad hoc manner by adding a small term rdffp to the diffusion equation or the reaction-diffusion equation to make it hyperbolic. From the diffusion equation (2.1) we obtain the telegraph equation, a damped wave equation. 

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