Summing correction, mathematical

These network equations differ from the Kirchhoff equations used in electrical circuit theory where the sum in the node Equation 10.5 is zero. O Keeffe (Struct. Bonding 1989, 71, 161-190) has shown that a correct mathematical correspondence requires that Kirchhoff s loop law be equivalenced with Equation 10.5 and the junction law with Equation 10.6. This requires replacing the nodes of the bond network with the loops of the equivalent Kirchhoff network and vice versa. For practical purposes it is simpler to stay with Equations 10.5 and 10.6... 

To sum up, in some instances the proposed tangent method and procedure of systematic error correction allows excluding the necessity of mathematical or chemical resolution of overlapped peak-shaped analytical signals. 

Dalton s law Dalton s law states that in a mixture of gases (A + B + C. . . ) the total pressure is simply the sum of the partial pressures (the pressures associated with each individual gas), decomposition reactions Decomposition reactions are reactions in which a compound breaks down into two or more simpler substances, diamagnetism Diamagnetism is the repulsion of a molecule from a magnetic field due to the presence of all electrons in pairs, dilute Dilute is a qualitative term that refers to a solution that has a relatively small amount of solute in comparison to the amount of solvent, dimensional analysis Dimensional analysis, sometimes called the factor label method, is a method for generating a correct setup for a mathematical problem. 

Although mathematically not quite correct, this enhances readability. Just always keep in mind that it represents the sum over all species using their corresponding stoichiometric factors. 

Mathematically these are radically different functions. Du Di, and D3 are all double exponential decays, but their preexponential factors deviate radically and the lifetimes differ noticeably. The ratio of preexponentials for the fast and slow components vary by a factor of 16 D has comparable amplitudes, while D2has a ratio of short to long of 4, and D3 has a ratio of short to long of 1/4. D4 is a sum of three exponentials. All five functions vary from a peak of about 104 to 25, and all four functions, if overlaid, are virtually indistinguishable. To amplify these differences, we assume that the Gaussiandistribution, Da, is the correct decay function and then show the deviations of the other functions from Do. These results are shown in Figure 4.10. The double exponential D fits the distribution decay essentially perfectly. Even Dj and Ds are a very crediblefit. >4 matches Do so well that the differences are invisible on this scale, and it is not even plotted. 

Mathematically, all this iirformation is used to calculate the best fit of the model to the experimental data. Two techniques are currently used, least squares and maximum hkelihood. Least-squares refinement is the same mathematical approach that is used to fit the best line through a number of points, so that the sum of the squares of the deviations from the line is at a minimum. Maximum likelihood is a more general approach that is the more common approach currently used. This method is based on the probability function that a certain model is correct for a given set of observations. This is done for each reflection, and the probabilities are then combined into a joint probability for the entire set of reflections. Both these approaches are performed over a number of cycles until the changes in the parameters become small. The refinement has then converged to a final set of parameters. 

If the width of the filter is approximately equal to the narrowest feature of the absorption spectra, the resultant mathematical function will be different, it will resemble a sum of one or more Gaussian functions and it will not correctly represent the original absorption function. Individual features will be suppressed relative to the overall function. This is the situation normally seen in the vision literature where a filter of ten to twenty nanometers has been used. 

It is clear from Eq. 9.3 and Fig. 9.11 that the net magnetization, that is, the vector sum Ma + M, never has a component in the x direction at any time. Because the component along y oscillates according to the cosine relation, at time t =1/2. J (and at subsequent times 3/2J, 5/2J, etc.) the total net magnetization is zero. Although mathematically correct, this result obscures the physical reality that each component retains its absolute value M at all times. 

Some points are noteworthy. According to Fourier, formally, the series must be summed over all integral frequencies from —oo to +00 to be mathematically exact. In practice of course, this is never possible. As the number of terms increases, however, as higher frequency terms are included, the approximation to the exact resultant wave function becomes more nearly correct. As shown in Figure 4.9, it often doesn t require all that many terms before a quite acceptable result is obtained. The difference between the exact waveform and the one we obtain from summing a limited series of Fourier terms is known as series termination error. As illustrated by the two-dimensional case in Figure 4.11, the phases of the component waves in the synthesis play a crucial role in determining the form of the resultant wave. 

Time and lack of data often prevent such a procedure from being adopted, and addition is therefore often assumed in practice. In general terms such summation of effects of individual compounds in a mixture can only be expected to predict the effects following exposure to a mixture if these compounds exert the same type of toxic effect(s) in the same organ(s), and if their mechanism of action is the same. Under these circumstances, additivity of effects may be assumed to be an acceptable estimate of the upper-bound risk level, and the sum of the individual (or corrected ) concentrations can be used to calculate a total level of exposure. However, for the exposure estimate, addition of mass (concentration) may not be the mathematical procedure which best reflects the biological principles involved in the comfort and health effects. Better models may be established in the future. 

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