Mathematical measures

The left-hand side of equation (9.124) is called the 2-norm of the input signal v (t) squared . Norms are mathematical measures that enable objects belonging to the... 

Mathematical measures of performance What is the required precision Why ... 

If the valley did not touch the baseline, it would have a length (A-C) that would be less than the distance to the baseline. If we divide the length of A-C by the length of A-B, we have a mathematical measure of the baseline resolution for the two peaks. Summing these ratios of baseline resolution for every peak pair gives us a resolution sum for the whole chromatogram. 

The evolution of this determinant first yields the eigenvalues. The solution of the whole eigenvalue problem provides pairs of eigenvalues and eigenvectors. The mathematical algorithm is described in detail in [MALINOWSKI, 1991]. A simple example, discussed in Section 5.4.2, will demonstrate the calculation. The following properties of these abstract mathematical measures are essential ... 

The communality is introduced as a mathematical measure of this common feature variance. The communality is the part of the variance of one feature which is described by the common factor solution in the factor analysis. High communalities, hj, mean that this feature variance is highly explained by the factor solution. Low communalities for one feature detect either a specific feature variance or high random error. 

Determination of Linearity and Range Determine the linearity of an analytical method by mathematically treating test results obtained from analysis of samples with analyte concentrations across the claimed range of the method. The treatment is normally a calculation of a regression line by the method of least squares of test results versus analyte concentrations. In some cases, to obtain proportionality between assays and sample concentrations, the test data may have to be subjected to a mathematical transformation before the regression analysis. The slope of the regression line and its variance (correlation coefficient) provide a mathematical measure of linearity the y-intercept is a measure of the potential assay bias. 

Mechanical advantage— A mathematical measure of the amount by which a machine magnifies the force put into the machine. 

In drawing a histogram for a set of data, one is representing the distribution of the data. Different sets of data will vary in relation to one another and, consequently, their histograms will be different. Basically, there are three characteristics that will distinguish the distributions of different sets of data. These are central location, dispersion, and skewness. These are characterized in Figure 132. Curves A and B have the same central location, but B is more dispersed. However, both A and B are symmetrical and are, therefore, not skewed. Curve C is skewed to the right and has a different central location than A and B. Mathematical measures of central location and dispersion are discussed later in this problem set. 

A recently often used practical method is that of proposed by Pipek and Mezey [26], Their intrinsic localization is based on a special mathematical measure of localization. It uses no external criteria to. define a priori orbitals. The method is similar to the Edmiston-Ruedenberg s localization method in the a-n separation of the orbitals while it works as economically as the Boys procedure. For the application of their localization algorithm, the knowledge of atomic overlap integrals is sufficient. This feature allows the adoption of their algorithm for both ab initio and semiempirical methods. The implementation of die procedure in existing program systems is easy, and this property makes the Pipek-Mezey s method very attractive for practical use. 

Acidity constant (K ) A mathematical measure of acid strenglli, determined from tlie equilibrium constant in reaction with water. 

Acid dissociation constant (Ka) A mathematical measure of acid strength, determined from the equilib-... 

Entropy is a mathematical measure of disorder. Living systems are highly ordered, if nothing else. Hence, living systems will have a negative entropy. When the living thing dies, and its elements scatter to the environment, its entropy increases. 

SCHEME 8.10 (a) The classification of acids as strong and weak is based on the concentration of H30 ions (represented simply as H+) in solutions of these acids, (b) The pH scale (as defined mathematically) measures an acid s strength the pH of pure water is 7, while acidic solutions have pH values smaller than 7. 

Variance A mathematical measure of the variation in the observed values of a sample population. 

Stevens SS (1951) Mathematics, measurement, and psychophysics. In Stevens SS (ed) Handbook of experimental psychology. John WUey, New York... 

The rate at which the attenuation increases is known as the slope of the filter - or rolloff. The rolloff is usually expressed as attenuation per unit interval, such as 6 dB per octave. In the stopband of an LPF with a 6 dB/octave slope, for example, every time the frequency doubles, the amount of attenuation increases by 6 dB. The slope of attenuation is determined by the order of the filter. Order is a mathematical measure of the complexity of a filter in a digital filter, it is proportional to the number of calculations performed on each sample. 

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