Series arithmetic-geometric

Arithmetic-Geometric Series Another common scries is... 

Note that the logs of the numbers in a geometric series will form an arithmetic series (e.g. 0, 1, 2, 3, 4,... in the above case). Thus, if a quantity y varies with a quantity x such that the rate of change in y is proportional to the value of y (i.e. it varies in an exponential maimer), a semi-log plot of such data will form a straight line. This form of relationship is relevant for chemical kinetics and radioactive decay (p. 236). 

Once the range of substrate concentrations to be used is chosen, the intermediate levels must be decided upon. If substrate concentrations are spaced in an arithmetic series, too many of the points will be in the high concentration range. A geometric series is better, but the best procedure is probably to space the points evenly on a reciprocal plot. 

Richter, with his mathematical outlook on chemistry, had the idea that the combining proportions formed arithmetical or geometrical series. This error permeates the whole of his Anfangsgriinde der Stochiometrie, and in finding such numerical relations he corrected his experimental results in a very arbitrary way. The second volume of this book, published in 1793, contains a detailed account of the matter. In the preface he states that ... 

The point of change is determined for every series of dilution evaluated. It is defined as the geometric mean of the dilution of the last negative and the first positive answer. The arithmetical mean and its standard deviation are calculated from the logarithms of the points of change. 

In rubber testing, the surface finish of metals is of importance, for example on mould surfaces and compression set plates. There are a number of standards in the ISO Geometric Product Specification series but the most relevant is ISO 428729 which covers terms, definitions and surface texture parameters relating to the profile method of measuring surface finish. There are apparently over 1000 different parameters to characterize surface finish30 but only a few are generally encountered. The most commonly found is Ra (previously called CLA) which is the mean deviation of the surface profile above and below the center line, followed by Rz, a measure of the peak to valley height. For example, the arithmetic mean deviation (Ra) of the compression plates for compression set tests must be better than 0.2 m. 

Several models have been proposed to estimate the thermal conductivity of hydrate/gas/water or hydrate/gas/water/sediment systems. The most common are the classical mixing law models, which assume that the effective properties of multicomponent systems can be determined as the average value of the properties of the components and their saturation (volumetric fraction) of the bulk sample composition. The parallel (arithmetic), series (harmonic), or random (geometric) mixing law models (Beck and Mesiner, 1960) that can be used to calculate the composite thermal conductivity (kg) of a sample are given in Equations 2.1 through 2.3. 

Some number series can be categorized as arithmetic or geometric. Other number series are neither arithmetic or geometric and, thus, must be analyzed in search of a pattern. 

Table II lists slope and intercept values for the linear-regression equations and Gaussian statistics, both for the full data set and for subsets categorized by various sampling, meteorological, or oceanographic conditions. The overall statistics for the cruise (Case I in Table II) indicate that the geometric mean salt aerosol concentration was 11.5 3.0 jug/SCM, and the arithmetic mean wind speed was 9.8 3.9 m/s. Generally, data are included in the table to show that condensation processes, hysteresis effects, and advection impose difficulties when trying to match the time series of local wind speed and salt aerosol concentrations.

Electrical conductivity. Calculation of electrical conductivity of a two-component material (solid, fluid) as function of porosity. The following equations are used Voigt model (parallel, upper bound), Reuss model (series, lower bound), arithmetic mean, geometric mean, Krischer and Esdorn model with parameter a, generalized Lichtenecker-Rother model with parameter a. 

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