Sigmoiditis

The Langmuir equation (Eq. XI-4) applies to many systems where adsorption occurs from dilute solution, but some interesting cases of sigmoid isotherms have been reported [54-56]. In several of these studies [54,55] the isotherms... 

Ways to circumvent the above-mentioned problems have been to simply increase the cutoff distance to larger values, to use more than one cutoff value with different update frequencies, or to define more sophisticated cutoff schemes. In the last case, a truncation of the non-bonded interactions was replaced by shifting the interaction energies to zero or by additionally applying a switched sigmoidal func-... 

The net signal is then modified by a so-called transfer function and sent as output to other neurons. The most widely used transfer function is sigmoidal it has two plateau areas having the values zero and one. and between these an area in which it is increasing nonlinearly. Figure 9-15 shows an example of a sigmoidal transfer function. 

Figure 9-15. Sigmoidal transfer function, in which the area between the plateaus does not increase linearly.

Fi. 4.30 A sigmoidal dielectric model smoothly varies the effective permittivity from SO to 1 as shown. 

The titration curve in Figure 9.1 is not unique to an acid-base titration. Any titration curve that follows the change in concentration of a species in the titration reaction (plotted logarithmically) as a function of the volume of titrant has the same general sigmoidal shape. Several additional examples are shown in Figure 9.2. 

For those pesticides which are utilized as microbial growth substrates, sigmoidal rates of biodegradation are frequentiy observed (see Fig. 2). Sigmoidal data are more difficult to summarize than exponential (first-order) data because of their inherent nonlinearity. Sigmoidal rates of pesticide metabohsm can be described using microbial growth kinetics (Monod) however, four kinetics constants are required. Consequentiy, it is more difficult to predict the persistence of these pesticides in the environment. 

Fig. 4. Typical sigmoid curve for the response of a biological system to chemical injury.

It states that the rate is proportional to the fraction x that has decomposed (which is dominant early in the reaction) and to the fraction not decomposed (which is dominant in latter stages of reaction). The decomposition of potassium permanganate and some other solids is in accordance with this equation. The shape of the plot of x against t is sigmoid in many cases, with slow reactions at the oeginning and end, but no theory has been proposed that explains everything. 

As for the dielectric constant, when explicit solvent molecules are included in the calculations, a value of 1, as in vacuum, should be used because the solvent molecules themselves will perform the charge screening. The omission of explicit solvent molecules can be partially accounted for by the use of an / -dependent dielectric, where the dielectric constant increases as the distance between the atoms, increases (e.g., at a separation of 1 A the dielectric constant equals 1 at a 3 A separation the dielectric equals 3 and so on). Alternatives include sigmoidal dielectrics [80] however, their use has not been widespread. In any case, it is important that the dielectric constant used for a computation correspond to that for which the force field being used was designed use of alternative dielectric constants will lead to improper weighting of the different electrostatic interactions, which may lead to significant errors in the computations. 

The sigmoid aetivation funetion is popular for neural network applieations sinee it is differentiable and monotonie, both of whieh are a requirement for the baek-propagation algorithm. The equation for a sigmoid funetion is... 

If the activation function is the sigmoid function given in equation (10.56), then its derivative is... 

A neural network has a strueture as shown in Figure 10.43. Assuming that all the aetivation funetions are sigmoids, ealeulate the values of yn and >22 when the inputs are... 

The heat removal line is unehanged irrespeetive of the kineties, and the fraetional eonversion Xa has a qualitatively similar sigmoidal shape for seeond-order kineties. Numerieal values as in Example 6-11 have been put into both the mass balanee and heat balanee equations. Fraetional eonversion Xa from both the mass balanee and heat balanee equations at effluent temperatures of 300, 325, 350, 375, 400, 425, 450, and 475 K, respeetively, were determined using the Mierosoft Exeel Spreadsheet (Example6-12.xls). Table 6-8 gives the results of... 

Enzymatic reactions frequently undergo a phenomenon referred to as substrate inhibition. Here, the reaction rate reaches a maximum and subsequently falls as shown in Eigure 11-lb. Enzymatic reactions can also exhibit substrate activation as depicted by the sigmoidal type rate dependence in Eigure 11-lc. Biochemical reactions are limited by mass transfer where a substrate has to cross cell walls. Enzymatic reactions that depend on temperature are modeled with the Arrhenius equation. Most enzymes deactivate rapidly at temperatures of 50°C-100°C, and deactivation is an irreversible process. 

The ANN model had four neurones in the input layer one for each operating variable and one for the bias. The output was selected to be cumulative mass distribution thirteen neurones were used to represent it. A sigmoid functional... 

The concentration Cz is a sigmoid function ( growth curve ) of time. Presumably some of product Z must be present at r = 0 in order to initiate the reaction. 

Figure 6-9 shows Fhs and Fs plotted against pH, according to Eqs. (6-61) and (6-62), for a weak acid of p/c = 4.0. Because of their appearance such curves are called S-shaped or sigmoid curves. 

If k is much larger than k", Eq. (6-64) takes the form of Eq. (6-61) for the fraction Fhs thus we may expect the experimental rate constant to be a sigmoid function of pH. If k" is larger than k, the / -pH plot should resemble the Fs-pH plot. Equation (6-64) is a very important relationship for the description of pH effects on reaction rates. Most sigmoid pH-rate profiles can be quantitatively accounted for with its use. Relatively minor modifications [such as the addition of rate terms first-order in H or OH to Eq. (6-63)] can often extend the description over the entire pH range. 

This property of the sigmoid curve permits to be easily estimated. This is an advantage of the k-pH plot. If the inflection point cannot be accurately located, the dissociation constant may still be estimated. Let [H j = Ka in Eq. (6-64) then Eq. (6-66) results. 

The kinetic analysis of the sigmoid pH-rate profile will yield numerical estimates of the pH-independent parameters K, k, and k". With these estimates the apparent constant k is calculated using the theoretical equation over the pH range that was explored experimentally. Quantitative agreement between the calculated line and the experimental points indicates that the model is a good one. A further easy, and very pertinent, test is a comparison of the kinetically determined value with the value obtained by conventional methods under the same conditions. 

Many sigmoid rate curves have been reported. A typical example is provided by the hydrolysis of phthalamic acid. ... 

The hydrolysis of aspirin [Eq. (6-70), R = CH3] is a classic example demonstrating a sigmoid pH-rate effect. Figure 6-13 shows this curve for trimethyl-... 

A frequently encountered pH-rate profile exhibits a bell-like shape or hump, with two inflection points. This graphical feature is essentially two sigmoid curves back-to-back. By analogy with the earlier analysis of the sigmoid pH-rate curve, where the shape was ascribed to an acid-base equilibrium of the substrate, we find that the bell-shaped curve can usually be accounted for in terms of two acid-base dissociations of the substrate. The substrate can be regarded, for this analysis, as a dibasic acid H2S, where the charge type is irrelevant we take the neutral molecule as an example. The acid dissociation constants are... 

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