Error complement

Here erfc( ) is the error function complement, a mathematical function of the argument u given by 1 - erf( ), where erf( ) is the error or Euler-Laplace integral. This integral in turn is defined by the expression... 

The error function complement erfcy is defined by the relationship erfcy = 1 — erf y. The concentration at the reference plane c is... 

Great simplification is achieved by introducing the hypothesis of independent reaction times (IRT) that the pairwise reaction times evolve independendy of any other reactions. While the fundamental justification of IRT may not be immediately obvious, one notices its similarity with the molecular pair model of homogeneous diffusion-mediated reactions (Noyes, 1961 Green, 1984). The usefulness of the IRT model depends on the availability of a suitable reaction probability function W(r, a t). For a pair of neutral particles undergoing fully diffusion-con-trolled reactions, Wis given by (a/r) erfc[(r - a)/2(D t)1/2] where If is the mutual diffusion coefficient and erfc is the complement of the error function. 

Here, erfcx is the error function complement of x and ierfc is its inverse. The physical properties are represented by a, the thermal diffusivity, which is equal to K/pCp, where k is the thermal conductivity, p is the density and Cp, the specific heat capacity at constant pressure. The surface temperature during this irradiation, Ts, at x = 0, is therefore... 

The possible drying times for the main drying are estimated in Section 1.2.1 and complemented by examples. The decisive qualities are the heat transfer coefficient from the shelf to the sublimation front of the ice (Ktot). The heat conductivity in the product does normally not play an important part (see Fig. 1.67), except that a granulated product is dried from the surface to the center (see Fig. 1.68). The shortest possible main drying time can be estimated with 5 or 10 % error, if the dimensions of the product and the maximum tolerable TKe (e. g. -10 °C) are given (Eq. (12), (12 a-c) in Section 1.2.1). 

The integral of the error function complement ierfc is defined as... 

As the interpretation of data in some cases is quite definite, while, in other eases, a wider range of assessments seems possible, the qualitative assessment seheme is complemented by providing a subjective confidenee interval for each indicator. This subjective confidence interval is the result of critical discussion among the authors with respect to the possible margins of error of the assessment made. 

A+B equal to total volume, and erfc( l ) is the complement error function of... 

As would be expected, in order to be able to have at least 95% confidence that the true CV p does not exceed its target level, we must suffer the penalty of sometimes falsely accepting a "bad" method (i.e. one whose true CV p is unsatisfactory). Such decision errors, referred to as "type-1 errors", occur randomly but have a controlled long-term frequency of less than 5% of the cases. (The 5% probability of type-1 error is by definition the complement of the confidence level.) The upper confidence limit on CV p is below the target level when the method is judged acceptable... 

The quantitative treatment for i as a function of a varying T f was first solved analytically by Sevdk in 1948. The solution involves Laplace transformation and the error function complement expressions applied in Vol. I, Section (4.2.11). It is better to quote here the rather simpler equations that can be found if one takes the entire surface as available for the exchange of electrons, i.e., the easy case of 0 = 0. Then (Gileadi, 1993),22 with this assumption, the peak potential is related to the rate constant (Ay) for the interfacial reaction, to the Tafel constant b, and to the sweep rate s, by the equation ... 

The error function, erf (z) and its complement, erfc (z) — 1 — erf (z) are frequently encountered in the solution of diffusion problems. They can be handled numerically using their tabulated values [42], or graphical representations, or preferably by means of computer algorithms as, for example, recently published by Oldham [43]. 

Figure 3. Gel permeation data for linear randomly coiled polypeptides on various agarose resins, plotted according to the method of Ackers (9). M0 555 is plotted vs. the inverse error function complement of Kd (erfc 1 Kd). Lines drawn through the data points represent best fits obtained from linear least-squares analysis of the data. Numerical designation of each curve represents the percent agarose composition for the resin used. Filled triangles on the curve for the 6% resin, and the filled squares on the curve for the 10% resin are points determined using fluorescent proteins. Data for the labeled polypeptides were not included in the least-squares analysis.

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