Of straight lines

C.I Linear Regression of Straight-Line Calibration Curves... 

This method of optimization is known as the generalized reduced-gradient (GRG) method. The objective function and the constraints are linearized ia a piecewise fashioa so that a series of straight-line segments are used to approximate them. Many computer codes are available for these methods. Two widely used ones are GRGA code (49) and GRG2 code (50). 

On the basis of straight-line depreciation, the average annual amount of depreciation Aq over a service life of s years is given by... 

FIG. 9-6 Effect of straight-line depreciation on rate of return for a project. Abd — annual depreciation allowance A c/ = annual net cash income after tax Awwp = annual net profit after payment of tax Cj = total capital cost. 

The annual rate of straight-line depreciation of the fixed-capital investment Cfc, from 1,000,000 at startup to a salvage value S, of zero at the end of a productive life s of 10 years, is given hy... 

Coefficient A and exponent a can be evaluated readily from data on Re and T. The dimensionless groups are presented on a single plot in Figure 15. The plot of the function = f (Re) is constructed from three separate sections. These sections of the curve correspond to the three regimes of flow. The laminar regime is expressed by a section of straight line having a slope P = 135 with respect to the x-axis. This section corresponds to the critical Reynolds number, Re < 0.2. This means that the exponent a in equation 53 is equal to 1. At this a value, the continuous-phase density term, p, in equation 46 vanishes. 

As the name suggests, the loss a feature causes is replaced by the loss due to a length of straight line, which gives the... 

Equation 13.50 gives the relation between liquid temperature, air enthalpy, and conditions at the interface, for any position in the tower, and is represented by a family of straight lines of slope —(ht/hop). The line for the bottom of the column passes through the point A( u,Hgi) and cuts the enthalpy-temperature curve for saturated air at the point C, representing conditions at the interface. The difference in ordinates of points A and C is the difference in the enthalpy of the air at the interface and that of the bulk air at the bottom of the column. 

H[ may be conveniently plotted against H G for a number of constant temperatures. If b and s are constant, a series of straight lines is obtained. The operating line AB given by equation 13.39 is drawn in Figure 13.21. Point A has coordinates (0L, //G1) corresponding to the bottom of the column. Point a has coordinates (0IA, H CA ), H[n being obtained from equation 13.77. 

In order to form in G o the grid with steps /ij = t /Ni and = I2/N2, where Ni and N2 are positive integers, we draw up two families of straight lines such as... 

Wald A (1940) The fitting of straight lines if both variables are subject to error. Ann Math Statist 11 284... 

Andrianov et al. (52) studied the rate of disappearance of Si—H as a function of time, temperature, and concentration of chloroplatinic acid in an equimolar solution of isoprene and dodecamethylhexasiloxane, H(Me2SiO)5SiMe2H. With a constant concentration of catalyst a plot of % conversion of SiH up to at least 60% vs. time at 20°, 50°, 70° and 110°C gave a family of straight lines. The slopes of the lines increased by a factor of 2.5-3 between 50° and 110°C. 

The latter discussion confirms the results of the potential dependence of the current in that the activation barrier for the hydrogen evolution reaction is, at least on copper and silver, not affected by the electrode potential. This behavior is, on the other hand, connected with the observation of straight lines in a Tafel plot. It would be premature to come up with a comprehensive model that would explain this behavior more experimental work is necessary to substantiate and quantify the effects for a larger variety of systems and reactions. A few aspects, however, should be pointed out. 

Forsythe, A. B. (1972). Robust estimation of straight line regression coefficients by minimizing p-th power deviations. Technometrics 14, 159-166. 

The validity of Johnston s interpretation of the experimental facts in terms of the simple unimolecular dissociation (1) has been questioned by Lindars and Hinshelwood120 and by Reuben and Linnett121. These workers maintain that isothermal plots of k versus p are not smooth curves, but consist of a number of straight lines linked by markedly curved portions. To explain such behaviour they incorporate into their mechanism a collision-induced crossover of vibrationally excited N20 (XS) to repulsive 3II and 3E states. While we incline towards the simpler view held by Johnston105 and others106-116, we feel that this feature of the decomposition kinetics merits further investigation. 

Figure 9. Sketch of the double spike Zn- Zn method. The surface is constructed by drawing an infinite number of straight-lines through the point representing the spike composition (supposed to be known with no error) and each point of the mass fractionation line going through the point representing the measured mixture. One of these straightlines, which is to be determined from the calculations, is the sample-spike mixing line (stippled line). Each determination of the Zn isotope composition of a sample involves only one run for the mixture of the sample with the spike. Since all natural samples plot on the same mass fractionation line, any reference composition will adequately determine isotope composition of the sample, note that, since the instrumental bias is not linear with mass, the mass discrimination lines are curved.

When data in the presence of an enzyme inhibitor are presented in the form of a Lineweaver-Burk plot, a series of straight lines should be obtained. The slopes of these hnes may or may not change, and the hnes may or may not intersect at a common point. The relationships between slopes, intersection points, and inhibitor mechanisms are outlined later. Further information regarding these mechanisms, including velocity equations describing data obtained in the presence of inhibitors with diverse mechanisms, can be found in (Segel, 1993). 

Plots devised by Dixon to determine K, for tight-binding inhibitors, (a) A primary plot of v versus total inhibitor present ([/Id yields a concave line. In this example, [S] = 3 x Km and thus v = 67% of Straight lines drawn from Vo (when [/It = 0) through points corresponding to Vq/2, Vq/3, etc. intersect with the x-axis at points separated by a distance /Cj app/ when inhibition is competitive. When inhibition is noncompetitive, intersection points are separated by a distance equivalent to K. The positions of lines for n = 1 and n = 0 can then be deduced and the total enzyme concentration, [EJt, can be determined from the distance between the origin and the intersection point of the n = 0 line on the x-axis. If inhibition is competitive, this experiment is repeated at several different substrate concentrations such that a value for K, app is obtained at each substrate concentration. (b) Values for app are replotted versus [S], and the y-intercept yields a value for /Cj. If inhibition is noncompetitive, this replot is not necessary (see text)... 

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