Uncertainty fraction

The method of the uncertainty dimension was introduced in 1985 by Grebogi et al. Suppose we are given a fractal that can be covered by the interval [0,1]. Also, suppose that we are provided with a mapping function / that generates the fractal and determines the orbits of seeds xo in [0,1]. Choose e > 0 and N = 1/e random points xj in [0,1]. Now, for every one of the N points determine its lifetime Ij. The lifetime is the number of times the seed xj can be iterated forward with / without ionizing it. Then, for every Xj determine the lifetimes of Xj + e and Xj — e. Call them Ij and lj respectively. For Ij = Ij = lj the point Xj is called e certain. Otherwise, the point Xj is called e uncertain. This procedure measures the fraction of gaps in the fractal on a scale e. Define /(e) as the ratio of e uncertain points and the total number of points N. Then, /(e) is also called the uncertainty fraction. 

Unfortunately, many commonly used methods for parameter estimation give only estimates for the parameters and no measures of their uncertainty. This is usually accomplished by calculation of the dependent variable at each experimental point, summation of the squared differences between the calculated and measured values, and adjustment of parameters to minimize this sum. Such methods routinely ignore errors in the measured independent variables. For example, in vapor-liquid equilibrium data reduction, errors in the liquid-phase mole fraction and temperature measurements are often assumed to be absent. The total pressure is calculated as a function of the estimated parameters, the measured temperature, and the measured liquid-phase mole fraction. 

Experimentally, tire hard-sphere phase transition was observed using non-aqueous polymer lattices [79, 80]. Samples are prepared, brought into the fluid state by tumbling and tlien left to stand. Depending on particle size and concentration, colloidal crystals tlien fonn on a time scale from minutes to days. Experimentally, tliere is always some uncertainty in the actual volume fraction. Often tire concentrations are tlierefore rescaled so freezing occurs at ( )p = 0.49. The widtli of tire coexistence region agrees well witli simulations [Jd, 80]. 

The fraction 0.1% is chosen to be so low that individuals living near a nuclear plant should have no special concern because of the closeness. Uncertainties in the analysis of risk are not caused by the "quantitative methodology" but are highlighted by it. Uncertainty reduction will be achieved by methodological improvements mean values should be calculated. As a guideline for rcinilatory implementation, the following is recommended ... 

It is conventional to take as the activation volume the value of AV when P = 0, namely —bRT. (This is essentially equal to the value at atmospheric pressure.) Pressure has usually been measured in kilobars (kbar), or 10 dyn cm 1 kbar = 986.92 atm. The currently preferred unit is the pascal (Pa), which is 1 N m 1 kbar = 0.1 GPa. Measurements of AV usually require pressures in the range 0-10 kbar. The units of AV are cubic centimeters per mole most AV values are in the range —30 to +30 cm moP, and the typical uncertainty is 1 cm moP. Rate constant measurements should be in pressure-independent units (mole fraction or molality), not molarity. ... 

The conclusions on the occurrence of ion-molecule reaction in the radiolysis of ethylene are not seriously affected by the uncertainties in the neutralization mechanism. It must be assumed that neutralization results in the complex species which constitute the ionic polymer, — i.e., the fraction of the ethylene disappearance which cannot be accounted for by the lower molecular weight products containing up to six carbon atoms. 

Uncertainties in Photochemical Models. The ability of photochemical models to accurately predict HO concentrations is undoubtedly more reliable in clean vs. polluted air, since the number of processes that affect [HO ] and [H02 ] is much greater in the presence of NMHC. Logan et al (58) have obtained simplified equations for [HO ] and [HO2 ] for conditions where NMHC chemistry can be ignored. The equation for HO concentration is given in Equation E6. The first term in the numerator refers to the fraction of excited oxygen atoms formed in R1 that react to form HO J refers to the photodissociation of hydrogen peroxide to form 2 HO molecules other rate constants refer to numbered reactions above. 

In this Section we aim to make the CRM user aware of the uncertainty budgets that need to be considered with the use of CRMs. Certified values in CRMs are the property values (mass fraction, concentration, or amount of substance) and their uncertainty, the uncertainty being in many instances a specified confidence interval for the certified property. As we discussed before, this uncertainty value is not always a complete uncertainty budget for an analytical process from sampling to production of data. But even when disregarding the subtle differences in the certificates, the way a CRM is used has serious consequences on the uncertainty budget that has to be applied to a user s result. This is summarized in Table 7.2. These uses may affect accuracy claims as well as traceability claims. It is the user s obligation to establish com-... 

The uncertainty in the predicted CHF of rod bundles depends on the combined performance of the subchannel code and the CHF correlation. Their sensitivities to various physical parameters or models, such as void fraction, turbulent mixing, etc., are complementary to each other. Therefore, in a comparison of the accuracy of the predictions from various rod bundle CHF correlations, they should be calculated by using their respective, accompanied computer codes.The word accompanied here means the particular code used in developing the particular CHF correlation of the rod bundle. To determine the individual uncertainties of the code or the correlation, both the subchannel code and the CHF correlation should be validated separately by experiments. For example, the subchannel code THINC II was validated in rod bundles (Weismanet al., 1968), while the W-3 CHF correlation was validated in round tubes (Tong, 1967a). 

In this study, 4.4 mg of lead equivalent was applied to the skin under a covered wax/plastic patch on the forearms of human subjects of the applied dose, 1.3 mg of lead was not recovered from skin washings. The amount that actually remained in (or on) the skin and the mass balance of the fate of this lead was not determined it may have been absorbed or eliminated from the skin by exfoliation of epidermal cells. Thus, while this study provides evidence for dermal absorption of lead, it did not quantity the fraction of applied dose that was absorbed. The quantitative significance of the dermal absorption pathway as a contributor to lead body burden remains an uncertainty. The wax/plastic patch provided a means by which the lead compounds could permeate or adhere to the skin. The effect of concentration in aqueous solution may cause skin abrasion through enhanced acidity since the lead ion is acidic. Abraded skin is known to promote subsequent higher lead penetration. 

A fifth success concerns carbon monoxide, the dominant interstellar molecule from an observer s point of view. Despite all the uncertainties and problems with the model calculations, which will be amply brought out in this review, the predicted fractional abundance of CO is large and in the range of 10"5 to 10-4, in excellent agreement with observation. 

In the previous section we indicated how various mathematical models may be used to simulate the performance of a reactor in which the flow patterns do not fit the ideal CSTR or PFR conditions. The models treated represent only a small fraction of the large number that have been proposed by various authors. However, they are among the simplest and most widely used models, and they permit one to bracket the expected performance of an isothermal reactor. However, small variations in temperature can lead to much more significant changes in the reactor performance than do reasonably large deviations inflow patterns from idealized conditions. Because the rate constant depends exponentially on temperature, uncertainties in this parameter can lead to design uncertainties that will make any quantitative analysis of performance in terms of the residence time distribution function little more than an academic exercise. Nonetheless, there are many situations where such analyses are useful. 

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