Linear convolution relationship

Equations (11) and (12) enable the generation of the total isotopic transient responses of a product species given (a) the transient response that characterises hypothesized catalyst-surface behaviour and (b) an inert-tracer transient response that characterises the gas-phase behaviour of the reactor system. Use of the linear-convolution relationships has been suggested as an iterative means to verify a model of the catalyst surface reaction pathway and kinetics. I This is attractive since the direct determination of the catalyst-surface transient response is especially problematic for non-ideal PFRs, since a method of complete gas-phase behaviour correction to obtain the catalyst-surface transient response is presently unavailable for such reactor systems.1 1 Unfortunately, there are also no corresponding analytical relationships to Eqs. (11) and (12) which permit explicit determination of the catalyst-surface transient response from the measured isotopic and inert-tracer transient responses, and hence, a model has to be assumed and tested. The better the model of the surface reaction pathway, the better the fit of the generated transient to the measured transient. 

The Laplace transform, which is a linear operator, is frequently used as a mathematical tool when dealing with linear systems. The Laplace transform is often useful in dealing with more complex convolution relationships. For example, consider the following property of the Laplace transform operation L ... 

The response, R, may be expressed as a concentration or an amount and is not limited to the parent drug but can be any biotransformed form of the drug. If the drug and metabolites are linearly related to the input (Theorem 16.5), then the sum of the drug and the metabolite input-response (concentration or amounts) is also linearly related to the input through a simple convolution relationship. This follows from the following derivation ... 

Thus, if the response measured is radioactivity from a labeling of the drug, then the total radioactivity resulting from the drug and metabolites is linearly related to input through a convolution relationship if both the metabolites and the drug are linearly related to the input. 

The relationship between R2 and R is unique and independent of the input. The relationship is independent of the input in the sense that it does not depend on the rate or extent of the input. However, as for any input-response system, the relationship depends on the site of the input. Thus, as long as the drug disposition does not change (time-invariant disposition) and the drug enters the system through the same input site, the relationship between the responses remains unchanged. The 7 i, R2 relationship is a linear operational relationship. In its simplest form it can be a simple convolution-type relationship otherwise it involves additional linear operations (see examples below). The relationship applies to any two PK responses in a multivariate PK system with a linear disposition and, as such, is an alternative to the traditional linear compartmental multivariate analysis. Perhaps the biggest power of the... 

We deemed it necessary to confirm the CV results by the alternate method using convolutive potential sweep voltammetry, which requires no assumptions as to the form of the free energy relationship and is ideally suited for an independent analysis of curvature revealed in Figure 7. In convolutive linear sweep voltammetry, the heterogeneous rate constant ke is obtained from the cur-... 

In cyclic voltammetry, simple relationships similar to equations (1.15) may also be derived from the current-potential curves thanks to convolutive manipulations of the raw data using the function 1 /s/nt, which is characteristic of transient linear and semi-infinite diffusion.24,25 Indeed, as... 

The convolution treatment of the linear and semi-infinite diffusion reactant transport (Section 1.3.2) leads to the following relationship between the concentrations at the electrode surface and the current ... 

Since the value of bounds has come to be widely accepted, numerous other effective bounded methods have appeared. Linear programming has provided the basis for a method presented by Mammone and Eichmann (1982a, 1982b). In a method loosely related to linear programming, MacAdam (1970) exploited the relationship between polynomial multiplication and convolution. His method is particularly suited to human interactive adjustment of constraints. 

A basic principle of protein chemistry is the central relationship between three-dimensional structure and activity. Unless the linear polypeptide chain folds into a particular three-dimensional configuration, the protein is inactive. As Fig. 2 illustrates, the active form of a protein is typically a highly convoluted, globular structure in which a particular small domain is the precise locus of interaction with reactant or binding ligand. 

LSA is a modeling approach in pharmacokinetics/pharmacodynamics (PK/PD) that applies general linear principles such as convolution and deconvolution to simplify and generalize linear PK/PD relationships. 

Both responses are related to the input through a convolution-type linear relationship. Accordingly, a unique inverse relationship to the input exists for the two responses ... 

This may be done according to LSA principles by convolution assuming a linear input-response (concentration) relationship where the required UlR is determined in step 1 from the reference administration. Accordingly, the IVIVR model of the dual-step method may be summarized as follows ... 

NIR analysts often use a statistical methodology called chemometrics to calibrate an NIR analysis. Chemometrics is a specialized branch of mathematical analysis that uses statistical algorithms to predict the identity and concentration of materials. Chemometrics is heavily used in NIR spectral analysis to provide quantitative and qualitative information about a variety of pure substances and mixtures. NIR spectra are often the result of complex, convoluted, and even unknown interactions of the different molecules and their environment. As a result, it is difficult to pick out a spectral peak or set of peaks that behave linearly with concentration or are definitive identifiable markers of particular chemical structures. Chemometrics uses statistical algorithms to pick out complex relationships between a set of spectra and the material s composition and then uses the relationship to predict the composition of new materials. Essentially, the NIR system, computer, and associated software are trained to relate spectral variation to identity and then apply that training to new examples of the material. 

In particular, the LS-Dyna finite viscoelastic relationship [175] takes into accotmt rate effects through linear viscoelasticity by a convolution integral. The model corresponds to a Maxwell fluid consisting of dampers and springs in series. The Abaqus FEA model is reminiscent of, and similar to, a well-established model of finite viscoelasticity, namely the Pipkin-Rogers model [161]. This model, with an appropriate choice of the constitutive parameters, reduces to the Fung (QLV) model [173, 177]. 

In 1990, Shao and Girault started a series of investigations based on the kinetic study of the transfer of acetylcholine Ac = CH3C02CH2CH2N (CH3)3 in which the physical properties of one of the solvents were varied. The experimental approach for the measurement of the kinetic parameters was chronocoulometry, a technique which, like convolution linear sweep voltammetry, does not impose any prerequisites on the potential dependence of the rate constants. To verify the suitability of the experimental method, they studied the potential dependence of the rate constant for Ac transfer from water to oil and from oil to water. As illustrated in Fig. 7, the results obtained show that the apparent rate constants obey the Butler-Volmer relationship, expressed by Eq. 10. Note that Fig. 7 has been obtained from two independent experiments. In the first experiment, acetylcholine was only present in the aqueous phase as a chloride salt and forced to cross to the organic phase, whereas in the second, acetylcholine was only present in the organic phase as a tetraphenylborate salt and forced to transfer to the aqueous phase. 

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