Arithmetic-geometric

Arithmetic-Geometric Inequality Let and denote respectively the arithmetic and the geometric means of a set of positive numbers ai, a2, The i.e.,... 

One such summarizing parameter expresses the central tendency of the distribution. A number of choices are available for this measure, including the median, mode, and various averages, such as the arithmetic, geometric, and harmonic means. Each may be most appropriate for different distributions. The arithmetic mean is usually used with synthetic polymers. This is because it was very much easier, until recently, to measure the arithmetic mean directly than to characterize the whole distribution and then compute its central tendency. The distribution must be known to derive the mode or any simple average except the arithmetic mean. (Some methods like those based on measurement of sedimentation and diffusion coefficients measure more complicated averages directly. They are not used much with synthetic polymers, however, and will not be discussed in this text.)... 

Example 4. The number of particles that fall between different size ranges are counted using a microscope and shown in Table 2. The arithmetic, geometric, and harmonic mean diameters along with their arithmetic and geometric standard deviations can be calculated using Equations (24)-<33) and is shown in Table 2. 

TABLE 2 Method of Determining the Arithmetic, Geometric and Harmonic Mean Diameter with Standard Deviation... 

In which Atfe 1/2 is the thermal conductivity at temperature f 1//2- This requires a suitable mean value to be chosen, the arithmetic, geometric or harmonic mean of the thermal conductivities at the known temperatures and tA+1 or nd The type of mean value formation does not play a decisive role if A is only weakly dependent on d or if the step size Ax is chosen to be very small. D. Marsal [2.53] recommends the use of the harmonic mean, so... 

Area and samolina method Number of subiects Year Arithmetic mean PCB level ng/mL (ppb) Arithmetic Geometric standard mean deviation 95% Confidence interval Ranae Reference... 

Population Number of subjects Year Arithmetic Geometric mean mean Arithmetic standard deviation Range Reference... 

If a population of particles is to be represented by a single number, there are many different measures of central tendency or mean sizes. Those include the median, the mode and many different means arithmetic, geometric, quadratic, cubic, bi-quadratic, harmonic (ref. 1) to name just a few. As to which is to be chosen to represent the population, once again this depends on what property is of importance the real system is in effect to be represented by an artificial mono-sized system of particle size equal to the mean. Thus, for example, in precipitation of fine particles due to turbulence or in total recovery predictions in gas cleaning, a simple analysis may be used to show that the most relevant mean size is the arithmetic mean of the mass distribution (this is the same as the bi-quadratic mean of the number distribution). In flow through packed beds (relevant to powder aeration or de-aeration), it is the arithmetic mean of the surface distribution, which is identical to the harmonic mean of the mass distribution. 

N is the number of particles in the system. According to the arithmetic-geometric-harmonic mean inequalities, (1/r) is always greater than 1 jf and, flius, the right-hand side of the above equation is negative. This shows that the size distribution is always narrowed when the growth rate is proportional to 1 jr and Q and are constants, flinf. use the relation cr = r — (r). )... 

Several standard expressions of a population that are used include mode, median, and mean differences (arithmetic, geometric, square, harmonic, etc.), as shown in Figure 9.10. 

The empirical AN and DN indexes obtained by our approach also lend themselves easily for calculations of pair interaction numbers. Tliere is no formal theoretical guideline on how best to combine individual AN and DN numbers. Arithmetic, geometric and harmonic mean averaging may be used, with a decision as to preferred approach left to an empirical examination of results. One pair interaction number, I p, which has proven to be useful, is defined as follows ... 

The average degree of polymerization, denoted X, were k represent the type of average (i.e., arithmetic, geometric), represents the dimensionless number of monomer units or residues R constitutive of a macromolecular chain. It can be defined as the ratio of the relative molar mass of the macromolecule to the relative molar mass of the monomer unit ... 

Calculate the arithmetic, geometric, and harmonic mean sizes. 

In most practical applications, we require to describe the particle size of a population of particles (millions of them) by a single number. There are many options available the mode, the median, and several different means including arithmetic, geometric, quadratic, harmonic, etc. Whichever expression of central tendency of the particle size of the population we use must reflect the property or properties of the population of importance to us. We are, in fact, modelling the real population with an artificial population of mono-sized particles. This section deals with calculation of the different expressions of central tendency and selection of the appropriate expression for a particular application. 

Arithmetic-Geometric Series Another common scries is... 

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