Sawtooth curve

The dashed lines extending downward from the sawtooth curves show the capacity of a trap at reduced Ap. Thus the capacity of a trap with a3/8-in (9.53-mm) orifice at Ap = 30 psig (207 kPa) is 6200 lb/h (2790 kg/h), read at the intersection of the 30-psig (207-kPa) ordinate and the dashed curve extended from the 3/8-in (9.53-mm) solid curve. 

FIGURE 4.10 The sawtooth curve in (a) is created by adding together the component waves in (,b). The lowest frequency component wave, which has the greatest amplitude, has exactly the same frequency, or wavelength, as the resultant wave in (a). All the component waves have frequencies of twice, three times, four times, and so on. The more high-frequency waves that are added into the summation, the more the resultant wave resembles the sawtooth curve. 

The calculated ionization potentials of Li, Na and K reproduce the drops associated with the closing of electronic shells [6]. However, the spherical jellium yields sawtoothed curves which lack fine structure between shell closings. In addition, the sawtooth rises above the experimental data before falling sharply at the next shell-closure. This behavior contrasts with the observed ionization potential curves, which remain rather flat between magic clusters, exhibiting a staircase profile. 

We shall approximate the scalar potential of the electrostatic field by the sawtooth curve of Fig. 5 that has the periodicity of the Bom von Karman zone, i.e., of the length of one unit cell times the number of k points that is used in a calculation. We have here assumed that the potential takes both positive and negative values. By doing so, the average potential from the field vanishes and we have therefore an optimal starting point for eliminating effects that are linear in the number of k points of the calculation (i.e., in the length of the Bom von Karman zone). 

Fig. 5. The left part shows the sawtooth curve z that is periodic (and linear) with the periodicity of the Bom von Karman zone. It is decomposed into the lattice-periodic part of the middle part, and the piecewise constant part shown in the right part of the figure, z = Zi+Z2-...

We considered two approximate treatments of the DC field, i.e., one where we only included Z of Fig. 5 and equations (48)-(50), and another where the full sawtooth curve z was included. Some representative results are shown in Figs 7 and 8. Since the Wannier functions can be ascribed to individual unit cells, we show in Fig. 7 the number of electrons (relative to the number, 8, for the undistorted system) of each unit cell in the case that the field operator has the symmetry of z of Fig. 5. Not surprisingly, the electrons do show an asymmetric distribution, although the flow from one end of the Born von Karman zone to the other is small. The number of electrons inside the muffin-tin spheres also gives information on the electron redistributions. Thus, for e-E = 0.0002 hartree these numbers are 3.2403 and 3.2413 for the two carbon atoms per unit cell for the operator zi of Fig. 5, and 3.2217 and 3.2575 for the operator z- Here we also see a larger effect for z than for z However, for the z all atomic spheres show the same numbers, so that the charge redistribution of Fig. 6 is restricted to the interstitial region. 

Equation (9.88) leads to the so-called sawtooth curve, for which Fig. 9.23 gives an example. 

Figure 18. Mapping of experimental time-of-arrival data for mass 27 (HCN) to volume elements in the charge. The experimental data (the series of rectangular bars) in arbitrary units, the calculated total fluid density in g/cm (sawtooth curve), and the mapping of time intervals back to volume elements (dotted lines) are shown. This calculation is an earlier one that started with somewhat less realistic conditions than those used in the most recent calculations.

According to Fig. 12.20, the intermittent addition of excess ligands extends the catalyst s lifetime in a sawtooth curve. This addition of ligand compensates for the system-immanent formation of deactivating substances which are brought into the system by the feedstocks. 

Please call to mind the sawtooth-curve. It is obvious that the maximum justified is the value of the currently valid safety stock plus an entire lot size. This maximum inventory can theoretically be achieved only if a delivery has just taken place (Fig. 57). 

PFDavg and PFH are both average values of time-dependent parameters U(t) for PFDavg and w(t) for the PFH. When the SIS components are tested periodically, U(t) and w(t) take the shape of typical sawtooth curves that increase gradually, peaking just before the tests and falhng to low values just after the tests. 

Besides the sawtooth curve on the PFD(t) at the top of the fault tree, we also show the PFD(t) of each component and the CCF in this figure. As the tests were all performed at the same time, the curves are identical for the components and the CCF is also tested every 3 months. 

The emission of light from Cepheid stars has a characteristic light curve seen in Figure 4.14 for a Cepheid in the constellation of Perseus. The sawtooth pattern is characteristic of the class and enables the period of variation to be determined. The observation, however, that the luminosity and period are related has powerful consequences. The Cepheid variables fall into two classes type I classical Cepheids have periods of 5-10 days and type II have periods of 12-20 days. The two types of Cepheids initially caused problems when determining the luminosity-period relation but the relation has now been determined. Type I Cepheids follow the expression... 

A titration curve for an acid soil suspension to which 1 mL of a calcium hydroxide titrant is added and the change in pH followed for 2.3 minutes is shown in Figure 10.3. As can be seen, the pH initially increases and then falls back toward the original pH. The curve not only has a sawtooth pattern but is also curved in the reverse direction from a standard titration of an acid with a basic solution. 

If a slow continuous addition of base is made to the same soil used in Figure 10.3, a similar titration curve without the sawtooth pattern is seen. Figure 10.5 shows the titration curve obtained by the continuous slow addition of 0.1 M NaOH. Again, the curve is not a smooth line, and irregularities seen in this titration are seen in other titrations of this same soil. Note that no distinct titration end point is seen here as there is in Figure 10.2. However, it is possible to determine the amount of base needed to bring this soil to pH 6.5, which is a typical pH desired for crop production. 

Figure 10.3 AFM single chain force-extension data, (a) An overlay of representative force-extension curves for modular polymer 8. Sawtooth patterned curves were consistently obtained, (b) One representative single chain force-extension curve for control polymer 9, in which only one peak was observed, (c) A single chain force-extension curve for modular polymer 8 shows the characteristic sawtooth pattern with three peaks. In Figure 10.2b and c, all scattered dots represent experimental data and the solid lines are results from WLC fitting. Adapted from Guan et al. (2004). Copyright 2004 American Chemical Society.

Figure 6. The I—V characteristic of the single probe without and with the styrene film cover t is the polymerization period the curve fort = 0 was obtained with no apparent contamination the probe measurements were performed in the dc glow discharge in nitrogen gas at 5 torr and 12 mA the frequency of sawtooth pulses applied to the probe was 47.6 Hz and their peak values were 26 V and...

The electrochemical properties of the Ni(lll) electrode characterized in UHV were studied using Cyclic Voltammetry, a fairly common, but powerful technique for electrochemical studies °2. In a voltammetry experiment the current I through the working electrode was recorded while its potential was cycled in a sawtooth pattern. The experimental I/V curve is called a voltammogram. Features in a voltammogram due to charge transfer provide useful information of electrochemical reaction... 

Fig. 3. The total energy per two atoms as a function of field strength for the Hiickel model with a ring of 12, 102, 204, and 306 sites (from below) with the sawtooth approximation of Fig. 1(c) together with the results for the same chains with the approximation of Fig. 1(b) (uppermost curve). The two panels differ in the scale of the ordinate.

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