Schwarz’s model

Schwarz s model is a multiradical extension of the Ganguly-Magee model with some additional improvements, to be described later. Schwarz assumes that initially—that is, 10 11 s after the act of energy deposition in water—there appear five species, namely eh, H, OH, H30+, and H2. Their initial yields, indicated by superscript zero, are related by charge conservation and material balance. Thus, there are three independent initial yields, taken to be those of eh, H, and Hr The initial yield of H2 is identified with the unscavengable molecular hydrogen yield. No mechanism of its production is speculated, except that it is not formed by radical recombination. For the gaussian distribution of the radicals, two initial... 

Discovery of the hydrated electron and pulse-radiolytic measurement of specific rates (giving generally different values for different reactions) necessitated consideration of multiradical diffusion models, for which the pioneering efforts were made by Kuppermann (1967) and by Schwarz (1969). In Kuppermann s model, there are seven reactive species. The four primary radicals are eh, H, H30+, and OH. Two secondary species, OH- and H202, are products of primary reactions while these themselves undergo various secondary reactions. The seventh species, the O atom was included for material balance as suggested by Allen (1964). However, since its initial yield is taken to be only 4% of the ionization yield, its involvement is not evident in the calculation. 

TABLE 7.1 Reaction Scheme in Schwarz s Diffusion Model... 

While fitting five adjustable parameters to four sets of experimental data may not seem surprising, the strength of the diffusion model lies in predicting a much wider body of experimental results. Of these, the most important are the variations of molecular yields with LET and solute concentration. Since these calculated variations agree quite well with experiment, no further comment is necessary except to note that calculations often require normalization, so that only relative yields can be compared with experiment. One main reason is that the absolute yields often differ from laboratory to another for the same experiment. Thus, Schwarz s theoretical predictions have reasonable normalization constants, which, however, are not considered as new parameters. In the next subsection, we will consider some experimental features that could possibly be in disagreement with the diffusion model. 

Four observation were thought to be in disagreement with the diffusion model (1) the lack of a proportional relationship between the electron scavenging product and the decrease of H2 yield (2) the lack of significant acid effect on the molecular yield of H2 (3) the relative independence from pH of the isotope separation factor for H2 yield and (4) the fact that with certain solutes the scavenging curves for H2 are about the same for neutral and acid solutions. Schwarz s reconciliation follows. 

Although both models could be valid theoretically, Model 18 does not constitute a significant benefit over Model 7 in terms of AIC (Akaike Criteria) and SBC (Schwarz s Bayesian Criteria). In principle, any model should be practical and as simple as possible. UM-203 has its target within the central compartment, i.e., the platelets. Therefore, one could visualize the compound distributing between two compartments within the blood, i.e., plasma and platelets. For this reason, Model 7 was selected for further analyses. The final parameters from Model 7 are listed in Table 3. 

Kowalski and Hutmacher (17) have proposed using the Wald approximation to the likelihood ratio test in conjunction with Schwarz s Bayesian criterion (SBC) to determine the covariates for inclusion in a population PM model. In this approach all possible models (with or without each of the covariate parameters in the model) are tested. The process proceeds as follows ... 

Legend , model number OFV, NONMEM objective function value LRT, likelihood ratio test SBC, Schwarz s Bayesian criterion based on NONMEM OFV (larger is better) LRT, likelihood ratio test based on Wald s approximation SBC, Schwarz s Bayesian criterion based on Wald s approximation (larger is better) p, total number of estimable parameter values plus all estimable covariance terms in reference model q, number of estimable parameter values and estimable covariance terms in reduced model. SBC and SBC ranking based on the largest value. 

Sorensen, P.W. Stacey, N.E. 1990. Identified hormonal pheromones in the goldfish The basis for a model of sex pheromone function in teleost fish. In Chemical Signals in Vertebrates 5 (Ed. by D.W. MacDonald, D. Muller-Schwarze S.E. Natynczuk), pp 302—311. Oxford Oxford. 

Schwarz, S., Borovinskaya, E.S., and Reschetilowski, W. (2013) Base catalyzed ethanolysis of soybean oil in microreactors experiments and kinetic modeling. 

Mende M, Schwarz S, Petzold G, Jaeger W (2007) Destabilization of model silica dispersions by polyelectrolyte complex particles with differoit charge excess, hydrophobicity, and particle size. J Appl Polym Sci 103 3776... 

Structural identification, i.e. selection of the model type and structure, is always an arbitrary research decision. What is helpful is autocorrelation and spectrum analysis (detection of the intervals). Generally, the simplest possible model is chosen. A series of information criteria (algorithms) exist that may help in this process, usually defined as a combination of the model error and the number of model parameters, such as the AIC criterion (Akaike s information criterion), the criterion of the final error of the prediction, Ravelli Vulpiani criterion or Schwarz s BIC criterion (Bayesian information criterion comparison of log likelihood of specific models corrected by the number of estimated parameters and the number of observations). 

The second assumption has been effectively invalidated by the discovery of the hydrated electron. However, the effects of LET and solute concentration on molecular yields indicate that some kind of radical diffusion model is indeed required. Kuppermann (1967) and Schwarz (1969) have demonstrated that the hydrated electron can be included in such a model. Schwarz (1964) remarked that Magee s estimate of the distance traveled by the electron at thermalization (on the order of a few nanometers) was correct, but his conjecture about its fate was wrong. On the other hand, Platzman was correct about its fate—namely, solvation—but wrong about the distance traveled (tens of nanometers). 

In this instance, we may believe that for once Arturo Schwarz was right on the mark indeed, as he chose to practice it, chess provides the perfect metaphorical model for Duchamp s life and works. ... 

Hertel O, Skj0th CA, Frohn LM, Vignati E, Frydendall J, de Leeuw G, Schwarz U, Reis S (2002) Assessment of the atmospheric nitrogen and sulphur inputs into the North Sea using a Lagrangian model. Phys Chem Earth 27 1507-1515... 

Noh and Schwarz proposed a modified Huang s and Stumm method for the calculation of surface hydroxyl group ionization constants, based on Gouy-Chapman model [118]. In this method the reactions of the surface complex formations are neglected ... 

Drusch, S., Kaeding, J., and Kopka, S., and Schwarz. 2004. Stability of patulin in a juice-like aqueous model system in the presence of ascorbic acid. Food Chem. Submitted. 

Coyle, J.T., and Schwarz, R. Lesion of striatal neurons with kainic acid provides a model for Huntington s Chorea. Nature 263 244-246, 1976. 

A common form of model selection is to maximize the likelihood that the data arose under the model. For non-Bayesian analysis this is the basis of the likelihood ratio test, where the difference of two -2LL (where LL denotes the log-Ukelihood) for nested models is assumed to be approximately asymptotically chi-squared distributed. A Bayesian approach— see also the Schwarz criterion (36)—is based on computation of the Bayesian information criterion (BIC), which minimizes the KuUback-Leibler KL) information (37). The KL information relates to the ratio of the distribution of the data given the model and parameters to the underlying true distribution of the data. The similarity of the KL information expression (Eq. (5.24)) and Bayes s formula (Eq. (5.1)) is easily seen ... 

Luebeck, E. G., Buchmann, A., Stinchcombe, S., Moolgavkar, S. H., and Schwarz, M. (2000). Effects of 2,3,7,8-tetrachlorodibenzo-/j-dioxin on initiation and promotion of GST-P-positive foci in rat Uver A quantitative analysis of experimental data using a stochastic model. Toxicol Appl Pharmacol 167, 63-73. 

Schroder D, Schwarz H, Shaik S (2000) Characterization, Orbital Description, and Reactivity Patterns of Transtition-Metal Oxo Species in the Gas Phase. 97 91-124 Schubert K (1977) The Two-Correlations Model, a Valence Model for Metallic Phases. 33 ... 

Ludden, T.M., Beal, S.L., and Sheiner, L.B. Comparison of the Akaike Information Criterion, the Schwarz criterion, and the F-test as guides to model selection. Journal of Pharmacokinetics and Biopharmaceutics 1994 22 431-445. 

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