Uncertain quantities

The lifetime of a spin adduct is thus seen to be dependent on both its rate of formation and its decay, meaning that the half-life of a particular spin adduct under the conditions of its generation is a somewhat uncertain quantity, as frequently noticed in the literature. The intrinsic stability of a particular spin adduct should preferably be determined on isolated specimens, as for example reported for the trichloromethyl adduct of PBN, which was shown to survive almost unchanged in benzene solution after 90 days in the dark (Janzen et al., 1993). 

Perhaps the simplest classification of flow regimes is on the basis of the superficial Reynolds number of each phase. Such a Reynolds number is expressed on the basis of the tube diameter (or an apparent hydraulic radius for noncircular channels), the gas or liquid superficial mass-velocity, and the gas or liquid viscosity. At least four types of flow are then possible, namely liquid in apparent viscous or turbulent flow combined with gas in apparent viscous or turbulent flow. The critical Reynolds number would seem to be a rather uncertain quantity with this definition. In usage, a value of 2000 has been suggested (L6) and widely adopted for this purpose. Other workers (N4, S5) have found that superficial liquid Reynolds numbers of 8000 are required to give turbulent behavior in horizontal or vertical bubble, plug, slug or froth flow. Therefore, although this classification based on superficial Reynolds number is widely used... 

We see that this trouble gets worse in the next step because the object estimate o2 is given by the magnified small difference between the uncertain quantity o1s0 and, itself uncertain. The difficulties increase as we proceed, and it is clear that uncertainty in the observations plays a fundamental role in our ability to restore the true object. The naive approach with which we started exemplifies the way we may be misled when we do not provide explicitly for the effects of noise. 

Ibrekk H, Morgan MG (1987) Graphical communication of uncertain quantities to nontechnical people. Risk Analysis, 7(4) 519-529. 

The most uncertain quantity here is Iq+, for which one obtains the lowest threshold P0+ > -P0- + Pps + 47rP2 —4.3 eV, which occasionally coincides with P. It implies that the above-mentioned relationship Vq P underestimates Vq+ due to neglect of the e+ zero-point kinetic energy contribution. 

C = sum of the mean specificlieats of the products at the temp of expln. It is a vety uncertain quantity Tj = the absolute temperature of the expl before it is fired... 

Supernovae, planetary nebulae (PNe) and, to a minor extent, stellar winds are the means to restore the nuclearly enriched material into the ISM, thus giving rise to the process of chemical evolution. There are two main Types of SNe (II, I) then divided in subclasses SNe IIL, IIP and SNe la, lb, Ic. As already mentioned before, SNe II, which are believed to be the end state of stars more massive than 1OM0 exploding after a Fe core is formed (core-collapse SNe), produce mainly a-elements (O, Ne, Mg, Si, S, Ca) plus some Fe. The amount of Fe produced by type II SNe is one of the most uncertain quantities since it depends upon the so-called mass cut (how much Fe remains in the collapsing core and how much is ejected) and on explosive nucleosynthesis. 

While more detailed descriptions of DEMOS are available elsewhere (8,9.10.11) it seems appropriate to provide here a very simple Illustration of the system. Suppose one Is concerned with estimating the mortality Impact of a chemical pollutant vdilch for convenience I ll call XYZ. In the problem of interest, there are two sources of XYZ exposure. The strength of these source terms, call them SI and S2, Is uncertain. The total population at risk Is P, also a somewhat uncertain quantity. The exposure process Is such that exposure Is proportional to source strength through a constant of proportionality, C, which is known only approximately. Finally, while a linear damage function with no threshold Is known to be appropriate the slope of the damage function, D, Is uncertain. A simple quantitative model of the potential health Inqpact can thus be written ... 

Even when the production profiles are uncertain, it is possible to apply optimization methods similar to those used in the deterministic case. A very simple approach to this problem is to simply replace the uncertain quantities in the model by point estimates, and then solve the optimization problem deterministically using these estimates. If the imcertainty is not too substantial, this approach may produce reasonably good results. On the other hand if the optimal solution varies a lot even for small changes in the parameter values, this approach may not produce a robust solution. 

The most uncertain quantity is the parameter v for the fission width distribution. The values one, two, and three have been recommended by different authors. Garrison (54) finds that a maximum likelihood test of the combined fission width distribution of four fissile isotopes yields V = 2.0, whereas Fischer 32) finds v = 2.2 for and v = 1.2 for... 

For a spherical particle, the diameter is taken as the size. However, the size of an irregularly shaped particle is a rather uncertain quantity. We therefore need to define what the particle size represents. One simple definition of the size of an irregularly shaped particle is the diameter of the sphere having the same volume as the particle. This is not much help because in many cases the volume of the particle is ill-defined or difficult to measure. Usually, the particle size is defined in a fairly arbitrary manner in terms of a number generated by one of the measuring techniques described later. A particle size measured by one technique may therefore be quite different from that measured by another technique, even when the measuring instruments are operating properly. 

The reliability assessment is carried out by using advanced reliability theory most research in this field has dealt with the issues of the probabilistic characterization of the uncertain quantities, the computational efficiency in case of large dimension problems, and the lack of analytical solutions that occurs, for example, in case of structures that exhibit a strongly nonlinear dynamic behaviour. 

In addition to the qualitative uncertainty assessment, the risk description by Flage and Aven (2009) also considers sensitivity S, which relates to the change in the relevant calculated risk measures conditional to changes in the uncertain quantities. This leads to a risk description ... 

Two types of errors usually happen at this stage (1) systematic error, which influences all measurements within one microarray chip with similar effect — this error may be corrected by estimation and (2) random error that cannot be explained or corrected, which is typically laiown as noise. Such errors are totally stochastic and have different influence on different probes. (Tibshirani et al., 2005) Typically, the pre-processing stage contains three steps background correction, normalization and summarization. For the widely used Affymetrix chips, many Bioconductor routines are available in R for pre-processing. These require creation of an AffyBatch object based on raw Affymetrix data (in a. cel file). The first step is the background adjustment. In this step, one tends to subtract the control intensity from the treatment, to denoise the intensity. However, direct subtraction of uncertain quantities can increase the level of noise and possibly result in negative intensity values for certain spots. Various methods to circumvent these problems are available as metitod parameters in the bg, correct function in R ... 

Input data for technical calculations can be exact quantities or uncertain quantities. Normally, exact quantities specify a niunber of units (e.g. 12 protons, 60 seconds), or they express a physical or mathematical definition (e.g. 101325 Pa/atm or the value of ir). Exact quantities are not encumbered with uncertainty. In contrast, uncertain quantities are measured physical quantities or rounded numerical values encumbered with more or less uncertainty (e.g. R = 8.31441 0.00026 J/molK, n 3.14). 

The uncertainty on the equilibrium constant Ka is then calculated by the procedure (f) it should be remembered that there are only two uncertain quantities in the expression ... 

Many of the terms used here involve uncertain quantities, often called random variables. Uncertain is used here because it applies to quantities that change unpredictably (e.g., whether a tossed coin will land heads or tails side up on the next toss) and to quantities that do not vary but that are not known with certainty. For example, a particular building s capacity to resist collapse in an earthquake may not vary much over time, but one does not know that capacity before the building collapses, so it is uncertain. In this entry, uncertain variables are denoted by capital letters, e.g., D particular values are denoted by lower case, e.g., d probability is denoted by P[] and conditional probability is denoted by P[AIB], that is, probability that statement A is tme given that statement B is true. 

Both have in common that the numerical effort increases with growing numbers of uncertain parameters. This means that reducing the number of uncertain quantities improves the numerical efficiency. This necessary categorization of uncertain quantities into relevant and nomelevant is possible by analyzing the sensitivity for each input quantity according to the output quantity of interest (see, e.g., Saltelli et al. 2008). Metamodels are versatile, e.g., they are applicable for pattern classification, functitMi approximation, or computing sensitivity measures. The possibility to approximate functional data, a mapping of input quantities to output quantities... 

The robustness measure for a structure is analyzed considering the uncertain quantities with n uncertain input parameters x. hi contrast, I load and failure scenarios, represented by the uncertain load input variables, are ctmsidered separately. There are m uncertain result variables z, which have to be taken into account All uncertain variables y 6 xJJ, z have in common that the measures describing the uncertainty... 

Information reducing measures M map uncertain quantities to deterministic values ... 

In general, the measures for uncertain quantities can be classified into representative values 971 and values to quantify the amount of uncertainty 11. For fuzzy quantities, the reduction of information is called defuzzification. This contribution shows information reducing measures for random and fuzzy variables. To avoid confusion, random variables Y G PfO.M) are defined by the CDF... 

The optimization objective function under consideration of uncertain quantities is written as... 

Numerical Optimization Algorithm for Consideration of Uncertain Quantities... 

The active approach is also denoted as here-and-now strategy (Tintner 1960). The generalized formulation for polymorphic uncertain quantities is... 

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