Slope of function

The parameter m in Eq. (5.6) is called the slope. It measures how steeply y rises or falls with x. Differential calculus, which we begin in the next chapter, is at its most rudimentary level, the computation of slopes of functions at different points. The slope tells how many units of y you go up or down when you travel along one unit of x. Symbolically,... 

The inverse susceptibility follows a linear dependence on temperature in both the paraelectric and ferroelectric phases a sign of the slope of function l/x(7) in the two phases is opposite and the magnitude of the slope is twice larger in the ferroelectric phase in agreement with experiment. Note that the Curie law is valid for both second and first order transitions, but the critical temperatures and T the two cases may not coincide, like in Fig. 6.8b. 

The constant ai > 0 tunes the slope of function and the constant > 0 locates the center of the function. Equation 8 returns values in [0,1]. Figure 11 shows an example of sigmoid function. As one can notice this is in accordance with what it is expected the smaller is, the smaller the membership grade of the candidate. 

Large values of the uncertainty are assigned to the generated points, because the primary purpose of the generated points is not to have them fit accurately, but rather to maintain a reasonable slope of the function in the range outside the experimental points. 

The surface elasticity E is found to vary linearly with t and with a slope of 2. Obtain the corresponding equation of state for the surface film, that is, the function relating t and a. 

Thus D(r) is given by the slope of the V versus P plot. The same distribution function can be calculated from an analysis of vapor adsorption data showing hysteresis due to capillary condensation (see Section XVII-16). Joyner and co-woikers [38] found that the two methods gave very similar results in the case of charcoal, as illustrated in Fig. XVI-2. See Refs. 36 and 39 for more recent such comparisons. There can be some question as to what the local contact angle is [31,40] an error here would shift the distribution curve. 

Procedure. Compute the slope of the function by a linear least squares procedure and obtain a value of Boltzmann s constant. How many particles do you expect to find 125 pm above the reference point Take the uncertainty you have calculated for the slope, as the uncertainty in k. Is the modem value of = 1.381 x 10 within these enor limits ... 

Expand the three detemiinants D, Dt, and for the least squares fit to a linear function not passing through the origin so as to obtain explicit algebraic expressions for b and m, the y-intercept and the slope of the best straight line representing the experimental data. 

The correlations of rates with acidity functions provide a convenient means of treatii results, and their uses will frequently be illustrated. However, their status is empirical, for whilst the acidity dependence of nitration becomes less steep with increasing temperature, the slope of... 

The applicability of the two-parameter equation and the constants devised by Brown to electrophilic aromatic substitutions was tested by plotting values of the partial rate factors for a reaction against the appropriate substituent constants. It was maintained that such comparisons yielded satisfactory linear correlations for the results of many electrophilic substitutions, the slopes of the correlations giving the values of the reaction constants. If the existence of linear free energy relationships in electrophilic aromatic substitutions were not in dispute, the above procedure would suffice, and the precision of the correlation would measure the usefulness of the p+cr+ equation. However, a point at issue was whether the effect of a substituent could be represented by a constant, or whether its nature depended on the specific reaction. To investigate the effect of a particular substituent in different reactions, the values for the various reactions of the logarithms of the partial rate factors for the substituent were plotted against the p+ values of the reactions. This procedure should show more readily whether the effect of a substituent depends on the reaction, in which case deviations from a hnear relationship would occur. It was concluded that any variation in substituent effects was random, and not a function of electron demand by the electrophile. ... 

Direct-Computation Rate Methods Rate methods for analyzing kinetic data are based on the differential form of the rate law. The rate of a reaction at time f, (rate)f, is determined from the slope of a curve showing the change in concentration for a reactant or product as a function of time (Figure 13.5). For a reaction that is first-order, or pseudo-first-order in analyte, the rate at time f is given as... 

The slope of the hysteresis loop in is also an important parameter. From this slope, the parameter S can be derived (17). In Figure 3 a part of the hysteresis loop (M as a function of the appHed field H) is given. The point at which M is constant as the function of the appHed field is defined as saturation magnetisation (M.). From the slope at can be written tan0 = = 1/ 1 — S ) ot dM/dH = — S ). Thus the S is defined in... 

Figure 36 is representative of creep and recovery curves for viscoelastic fluids. Such a curve is obtained when a stress is placed on the specimen and the deformation is monitored as a function of time. During the experiment the stress is removed, and the specimen, if it can, is free to recover. The slope of the linear portion of the creep curve gives the shear rate, and the viscosity is the appHed stress divided by the slope. A steep slope indicates a low viscosity, and a gradual slope a high viscosity. The recovery part of Figure 36 shows that the specimen was viscoelastic because relaxation took place and some of the strain was recovered. A purely viscous material would not have shown any recovery, as shown in Figure 16b. 

H2O/100 kg of adsorbent. At equilibrium and at a given adsorbed water content, the dew point that can be obtained in the treated fluid is a function only of the adsorbent temperature. The slopes of the isosteres indicate that the capacity of molecular sieves is less temperature sensitive than that of siUca gel or activated alumina. In another type of isostere plot, the natural logarithm of the vapor pressure of water in equiUbrium with the desiccant is plotted against the reciprocal of absolute temperature. The slopes of these isosteres are proportional to the isosteric heats of adsorption of water on the desiccant (see... 

Because of the steep pitch required, slides are limited in application. They are most commonly used to bridge the gap between roller-conveyor systems on two floors, because the roller conveyor can take the container off the slide rapidly and ehminate or reduce the chance for collisions. Slides may also be used when containers can be chuted from an upper floor to a manually loaded carrier. The use of several rollers at the feed point is recommended for easy dehveiy to the sloping section. If the drop is short and containers light, a roller cleanout will prevent backup of containers on the slide. The slope of gravity slides is a function of container weight, size, and friction characteristics and should be selected with care to be sure that containers do not move either too swiftly or not at all. Slides usually use flat steel sheet. 

The comparison of the results of very different methods has to be judged very precisely, as, e.g., the given thickness of a layer is a function of the limit of detection (EOD) of a method. Additionally, the detected areas vary from about 0.01 up to about 400 mm-. Therefore, the methods with a low level of detection and with a high sensitivity (high slope of the calibration function) give a higher value for the layer thickness. Furthermore, the layers are broadened with time by diffusion. 

Order 2 minimization algorithms, which use the second derivative (curvamre) as well as the first derivative (slope) of the potential function, exhibit in many cases improved rate of convergence. For a molecule of N atoms these methods require calculating the 3N X 3N Hessian matrix of second derivatives (for the coordinate set at step k)... 

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