Descriptive geometric mean

The results from the several studies that have been conducted to measure exposures associated with the use of chlorpyrifos are summarized in Tables 1 and 2. Table 1 summarizes results from mixer-loader and applicator studies reported by Honeycutt et al.1 Listed for each work description are the number of replicates, the arithmetic mean, and the geometric mean for the replicates from both the passive dosimetry measurements and the biomonitoring tech-... 

Description n Arithmetic mean Geometric mean n Arithmetic mean Geometric mean... 

In Eqs. (16) and (17), a and P refer to atoms of the solute and solvent, respectively q is the permittivity of free space, Qa and Qp are atomic charges, and Rap is the distance between atoms a and p. The parameters eap, aap, Aap, Bap and Cap can either be assigned by fitting to experimental data or can be the arithmetic or geometric means of literature values for the individual atom types.10,65,66 The atomic charges are commonly determined by requiring that they reproduce the calculated molecular electrostatic potentials.10 In order to provide better descriptions of the solvent s structure, Eqs. (16) and (17) are generally extended to include solvent-solvent intermolecular interactions. 

The biomonitoring data presented in each of the national exposure reports include descriptive statistics on the distribution of blood or urine concentrations of each chemical, including geometric means and percentiles with confidence intervals (CDC 2003). Each report also includes brief toxicity profiles and information relating the findings to biological exposure indices and European reference values or ranges, if available. Additionally, the raw data from the reports are publicly available and serve as a valuable resource. 

Within the USEPA s EPI Suite, descriptions of basic physical-chemical properties for 2,4-dichloroaniline (DCA CAS No. 554007), pentachlorophenol (PCP CAS No. 87865), nonylphenol (NP CAS No. 104405), and linear alkylbenzenesulfonate (C12 LAS CAS No. 25155300) were obtained using the EPI Suite (Table 3.1), whereas acute and chronic toxicity estimates for the median effective and chronic effect concentrations (the geometric mean between chronic lowest-observed-effect concentration [LOEC] and no-observed-effect concentration [NOEC]) and for fish, daphnids, and algae for each substance were estimated from ECOSAR (Table 3.2). These compounds were chosen based on their widespread use. 

Due to the small sample size, all variables were only presented descriptively for the different bioanalytical data and pharmacokinetic parameters calculated number of relevant observations, geometric mean, geometric standard deviation, arithmetic mean, standard deviation, coefficient of variation, median, minimum and maximum. 

Second, there is no possibility of separately determining either a+ or d. For, by definition, /r+ = (9G/9/i+)r,p, that is, the determination of requires addition of only positive ions to the solution, while holding the concentration of anions fixed. However, this step, which violates the Law of Electroneutrality, cannot be carried out operationally. Consequently, one cannot deal with individual ionic activities rather, as Eq. (4.1.3c) shows, the ionic activities occur in such a manner that only their geometric mean or appropriate logarithmic sum is involved. Third, since thermodynamic descriptions must be confined to measurable properties we may regard the quantity, /x -F i Tlna on the left of Eq. (4.1.3a) as an effective chemical potential that is to be used to represent the behavior of the electrolytes. On the other hand, one does not wish to ignore the ionic nature of the solution hence it is customary to write... 

This description may be quite adequate for two close sieves in a 2 series, but it can be absolutely meaningless for two sieves at extreme ends of the mesh range. Another simple average diameter is the geometric mean, defined by... 

TABLE 23.2 Geometric Mean PbB and Other Descriptive Statistics for Silver Valley, ID, Children in Areas 1—y 1974 ... 

Olaj and Zifferer [229] have performed Monte Carlo simulations in which they placed 300 polymer chains in a lattice, considering excluded volume effects. From the obtained configurations they evaluated the shielding factor which describes how severely the presence of a polymer coil retards the termination of its own radical compared to small (unshielded) radicals. For chains of unequal size Olaj and Zifferer showed that the geometric mean model provided a reasonable mathematical description of their results, although the harmonic mean model. 

These considerations and the existence of many solutions for whidi the excess functions g and have not the same sign indicate that presumably no theory limited to first order terms can at all give an adequate description of the thermodynamic properties of mixtmres. This may be taken as a consequence of the fact that the combining rule (2.7.8) seems to be a feiir ai roximation for mixtures of spherical nonpolar molecules. If the interaction i2 is somewhere between the arithmetical and geometrical mean of u and 22. i becomes a second order quantity (cf. 2.7.9) and the theory of conformal solutions would predict no deviations from the laws of perfect solutions. In order to force agreement using only first order terms, this approach has to overemphasize the deviations from the combining rules (2.7.8) or (2.7.10). This IS probably the reason of the discrepancy between (4.4.1) and (4,4.2). 

The properties characteristic to fractal objects were mentioned first by Leonardo da Vinci, but the term fractal dimension appeared in 1919 in a publication by Felix Hausdorff [197], a more poetic description of fractals was given by Lewis Richardson in 1922 [198] (cited by [199]), but the systematic study was performed by Benoit B. Mandelbrot [196], Mandelbrot transformed pathological monsters by Hausdorff into the scientific instrument, which is widely used in materials science and engineering [200-202]. Geometrical self-similarity means, for example, that it is not possible to discriminate between two photographs of the same object taken with two very different scales. 

A coordinate system is a mathematical device for naming positions in space. From the geometric standpoint, a coordinate system is an artificial construct, because the structure of an object is defined by the relationships between its component parts, independent of any external references. Nevertheless a description of geometric objects by means of a coordinate system is a convenient way to describe numerically the internal structure of an object and the relationships between different positions and orientations of an object in space. 

For each electron energy E, the interactions with tissue (frequently represented by liquid water) are simulated event by event, which means that one obtains a detailed description of the geometrical coordinates, energy locally deposited, and type of energy transfer (ionization or excitation) for each interaction point, primary or secondary. 

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