Display equations numbering

Number displayed equations by using any consistent system of sequencing. 

Chapter 11 Numbers, Mathematics,and Units of Measure >- 221 > Do not use any closing punctuation on the line with displayed equations. 

Various workers in the field of NIR have attempted to address this (unavoidable) part of generating an equation for a NIR analysis. A comprehensive review of published papers and conference proceedings from the recent past displays a number of NIR methods validated under ICH guidelines or application of these guidelines in general. 

Short chemical reactions may be run into text or they may be displayed and numbered, if numbering is needed. Long chemical reactions should be displayed separately from the text. The sequential numbering system used may integrate both chemical and mathematical equations, or separate sequences using different notations may be used for different types of equations (e. g., eqs 1 3 could be used for a set of chemical reactions and eqs I—III could be used for a set of mathematical equations). The use of lettering,... 

Numbers in parentheses refer to the equation number in which the symbol first appears or is first defined. When the first appearance of the symbol occurs within the text, rather than in a displayed equation, this is so indicated by giving the equation numbers lying on either side of that portion of the text. For example, (75)-(76) indicates that the first appearance of the symbol occurs somewhere between Eqs. (75) and (76),... 

For e en the simplest system, where there are only two possible positiorts for molecules (n = 2), most catciiators cannot display a number as large as the result of Equation 18.1 for even as few as 500 molecules—much less for the enormous number of molecules present in a maoDscopic sample. (If your calculator is like most, you can calculate the number of microstalBs ferx s 332 molecules.)... 

Equation (18) displays the relationship between the column efficiency defined in theoretical plates and the column efficiency given in effective plates. It is clear that the number of effective plates in a column is not aii arbitrary measure of the column performance, but is directly related to the column efficiency as derived from the plate theory. Equation (18) clearly demonstrates that, as the capacity ratio (k ) becomes large, (n) and (Ne) will converge to the same value. 

In Spite of the existence of numerous experimental and theoretical investigations, a number of principal problems related to micro-fluid hydrodynamics are not well-studied. There are contradictory data on the drag in micro-channels, transition from laminar to turbulent flow, etc. That leads to difficulties in understanding the essence of this phenomenon and is a basis for questionable discoveries of special microeffects (Duncan and Peterson 1994 Ho and Tai 1998 Plam 2000 Herwig 2000 Herwig and Hausner 2003 Gad-el-Hak 2003). The latter were revealed by comparison of experimental data with predictions of a conventional theory based on the Navier-Stokes equations. The discrepancy between these data was interpreted as a display of new effects of flow in micro-channels. It should be noted that actual conditions of several experiments were often not identical to conditions that were used in the theoretical models. For this reason, the analysis of sources of disparity between the theory and experiment is of significance. 

In summary, then, polymerization of ATP-actin under conditions of sonication displays two characteristic deviations from the simple law described by equation (4), which is only valid for reversible polymerization. These are (a) overshoot polymerization kinetics, and (b) the steady-state amount of polymer formed decreases, or the steady-state monomer concentration increases, with the number of filaments. These two features are the direct consequence of ATP hydrolysis accompanying the polymerization of ATP-actin, as will be explained now. 

Comment-. All models yield the same l-value, but differ in the number of degrees of freedom to be used. The difference between the means is barely significant in two cases. Suggestion acquire more data to settle the case. Program TTEST automatically picks the appropriate equation(s) and displays the result(s). Equation (1.21) is used to scan the parameter space (Xmean Sx, n) in the vicinity of the true values to determine whether a small change in experimental protocol (n) or measurement noise could have changed the interpretation from Hq to H or vice versa. 

Note that a number of complicating factors have been left out for clarity For instance, in the EMF equation, activities instead of concentrations should be used. Activities are related to concentrations by a multiplicative activity coefficient that itself is sensitive to the concentrations of all ions in the solution. The reference electrode necessary to close the circuit also generates a (diffusion) potential that is a complex function of activities and ion mobilities. Furthermore, the slope S of the electrode function is an experimentally determined parameter subject to error. The essential point, though, is that the DVM-clipped voltages appear in the exponent and that cheap equipment extracts a heavy price in terms of accuracy and precision (viz. quantization noise such an instrument typically displays the result in a 1 mV, 0.1 mV, 0.01 mV, or 0.001 mV format a two-decimal instrument clips a 345.678. .. mV result to 345.67 mV, that is it does not round up ... 78 to ... 8 ). 

Equations) gives these equations in a common format because not all r-test procedures return tbe same number of degrees of freedom or the same r-value, the interpretations can slightly differ. For each of the four combinations of N] (=, 7 ) Nj and Vi (=, 7 ) V2 all equations that apply are used and the corresponding results and interpretations are displayed. 

The low-pressure region displays the electroneutrality equation approximation [e ] = 2[Vx ]. Electrons predominate so that the material is an n-type semiconductor in this regime. In addition, the conductivity will increase as the partial pressure of the gaseous X2 component decreases. The number of nonmetal vacancies will increase as the partial pressure of the gaseous X2 component decreases, and the phase will display a metal-rich nonstoichiometry opposite to that in the high-pressure domain. Because there is a high concentration of anion vacancies, easy diffusion of anions is to be expected. 

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