Line coincidence

Let us consider a thin homogeneous isotropic beam of thickness 2s. We assume that the beam mid-line coincides with the segment (0,1) of the axis X. At the point y = 1/2, the beam has an inclined cut as a segment having the angle a with the vertical line, 0 < tana < 2e). We look for the function % = (W, w) of horizontal displacements W(x) and vertical displacements w(x) provided that the external forces g(x),f(x) are given. The condition of clamped edges... 

P. W. J. M. Boumans. Line Coincidence Tables for Inductively Coupled Plasma Atomic Emission Spectrometry. Pergamon Press, Oxford, 1980, 1984. Lists of emission lines for analysis and potentially overlapping lines with relative intensities, using spectrometers with two different resolutions. 

On this chart, the wet bulb temperatures appear as diagonal lines, coinciding with the dry bulb at the saturation line. If measurements are taken with the two thermometers of the sling psychrometer, the condition can be plotted on the psychrometric chart by taking the intersection of the dry bulb temperature, as read on the vertical line, with the wet bulb temperature, read down the diagonal wet bulb line. 

Theory predicts and experiment establishes 2/1 as the intensity ratio of Kal/Ka2 for sulfur. Where 3 lines appear in a spectrum, in Figure 1-19, the presumption is that more than one kind of sulfur atom is present, and that an intensity ratio exceeding 2 results because certain lines coincide. The observed intensities in Figure 1-19 are those to be expected on the basis of this presumption, 2 being the known value of K l/K 2 for every kind of sulfur atom. 

It is proper to point out again that the simplicity of Equations (2.87 and 2.88) is related to the fact that the plane xOy is tangent to the level surface at the point p and, correspondingly, the plumb line coincides with the z-axis. 

A laser-based detection system for NO2 gas (which is an industrial hazard and common environmental pollutant) was developed by Koybayashi et al.4. This was achieved by splitting light, from an Ar-ion multi-line laser, into two paths, one passing through a measurement gas, and the other being transmitted directly to the measurement unit as a reference signal. The detection unit contained two filters to separate the two chosen laser lines, and these were then detected on separate optical receivers. One of these chosen laser lines coincided with a strong absorption line in the NO2... 

The value of the phase constant a does not affect the size or shape of the orbit, but it does affect the position of the orbit as a function of time. If at time t = t number of different identical oscillators have q = 0 but values of q lying between v and — v the representative points will lie on a straight line coincident with the p axis. In time, these points will describe their individual orbits but remain collinear. 

Figure Jh The characteristics of the flow equation (11) for Nf = 2 light flavors. The innermost characteristic line coincides with the prediction from lattice QCD [13] for the critical line at small g, which is represented by the dashed line with the hatched error band.

The (TMTTF)2PF6 spectra can be satisfactorily decomposed into two main lines, each with their corresponding spin-orbit doublet, the lower energy line coinciding... 

In the current example, the majorin of the variation observed in two dimensions is described using one principal component <5>S%). The data essentially forms a line coincident with PCI (with some scatter). This demonstrates the ability of PCA to more efficiently describe the variation in the data set. 

If the effect of an inhibitor on an enzyme is to be investigated, the Dixon plot is recommended. To obtain data for the Dixon plot, estimate the reaction rate at constant substrate concentration and vary the inhibitor concentration [I]. At competitive inhibition, all the obtained straight lines coincide at a point with the coordinates X = -Ki, y = 1/Vmax> and at non-competitive inhibition all the straight lines have the same intercept on abscissa at x = -Ki. At... 

That the trend lines coincide almost completely is some justification of the empirical approach. 

In the example given, the line coincides exactly with the least-squares line calculated for estimating y from x. 

Figure 2-10 shows a more nearly complete pressure-volume diagram.2 The dashed line shows the locus of all bubble points and dew points. The area within the dashed line indicates conditions for which liquid and gas coexist. Often this area is called the saturation envelope. The bubble-point line and dew-point line coincide at the critical point. Notice that the isotherm at the critical temperature shows a point of horizontal inflection as it passes through the critical pressure. 

Both maximum and minimum limits exist of the solvent/feed ratio. The maximum is the value that locates the mix point M on the binodal curve near the solvent vertex, such as point Mmla on Figure 14.7(b). When an operating line coincides with a tieline, the number of stages will be infinite and will correspond to the minimum solvent/feed ratio. The pinch point is determined by the intersection of some tieline with line RnS- Depending on whether the slopes of the tielines are negative or positive, the intersection that is closest or farthest from the solvent vertex locates the operating point for minimum solvent. Figure 14.9 shows the two... 

The calculations were carried out for a shear stress at the wall of 2 X 104 dynes/cm2. The two values for ftaPP and their average were plotted together with the results from the measurements (Figure 3) against the volume fraction of the PAN solution in Figure 9. The calculated mean viscosity line coincides with the measurements on the emulsions that are rich in PU solution. On the other hand, for PAN rich... 

The dependencies of kohs [Equation (14)], kE [Equation (11)], and kK [Equation (8)] on proton concentration are usually displayed in log-log plots called pH-rate profiles, which allow one to identify the reaction paths dominating at various pH values as well as the parameters of the rate law, namely the acidity constants K and the elementary rate constants of the rate-determining steps. Figure 3 shows pH- rate profiles of kE (dashed line), kK (thin full line, coincides with kohs below pH 17), and kobs = kE + /cK (thick gray line), which were plotted using Equations (8) and (11) with the six relevant kinetic and thermodynamic parameters that have been determined for acetophenone (see Table 1 in section Examples ).4,19 23... 

It is based on the idea of electromagnetic knot, introduced in 1990 [27-29] and developed later [30-32], An electromagnetic knot is defined as a standard electromagnetic field with the property that any pair of its magnetic lines, or any pair of its electric lines, is a link with linking number i (which is a measure of the extent to which the force lines curl themselves around one another, i.e., of the helicity of the field). These lines coincide with the level curves of a pair of complex scalar fields , 0. The physical space and the complex plane are compactified to Si and S2, so that the scalars can be... 

FIGURE 2.20 Mossbauer spectra of purified pMMO recorded at 4.2 K at B = 45 mT (a) and 8.0 T (b). Solid line identifies contribution of the diiron(III) center. Arrow in (a) indicates high-energy line of ahigh-spin ferrous species its low-energy line coincides with low-energy feature of the diferric center. The simulation in (b) assumes that the center is diamagnetic. 

Fig. 9.6 Absorption spectrum of 16 along mer 16b dashed line (coincident with the line...

Fig. 5.3 The data from Examples 1, 2, and 3. Note that the solid and the dashed line coincides.

When the pressure decreases, the chemical equilibrium line intersects the liquid-liquid envelope within the physically relevant composition space. Consequently, the chemical equilibrium line coincides with one of the tie lines with the phase-splitting region (see the enlarged region of Fig. 4.9(b)). 

Finally, when the two chemical shifts are exactly equal (Av = 0), the two inner lines become one and the two outer lines have zero intensity. The two protons Ha and Hb are chemically equivalent, that is, they have the same chemical shift, and we see no splitting at all. This explains, in a way, why equivalent protons do not split each other they do split each other but the inner lines coincide and the outer lines have zero intensity. Theoretically, the pattern still has four lines, but we only observe one a singlet. The same applies to any number of chemically equivalent protons, for example, a methyl group (CH3). All three protons have the same chemical shift, and in the absence of any other coupling (e.g., CH3O or CH3Cq, where Cq is a quaternary carbon), we will see a singlet with area proportional to three. 

The condition where the reflux and stripping ratios approach infinity is termed total reflux. No feed enters the column and no product leaves. Both component balance lines coincide with the 45° diagonal line and are therefore furthest away from the equilibrium curve. Total reflux sets the minimum number of stages required for the separation. For Example 2,1, Fig. 3,lid shows that the minimum number of stages required for the separation is 6. 

Fig. 2. Plots testing simple ionic model equations for diatomic molecules in the gas phase, ionic crystals and the hydration of ions. The slopes of the lines coincide with those of the simple theory, see Phillips and Williams. U is the binding energy from free gas ions.

The third plane identified in Fig. 12.1 is the vertical one perpendicular to the composition axis and indicated by MNQRSLM. When projected on a parallel plane, the lines from several such planes present a diagram such as that shown by Fig. 12.4. This is the PT diagram lines t/C, and KC2 are vapor-pressure curves for the pure species, identified by the same letters as in Fig. 12.1. Each interior loop represents the PT behavior of saturated liquid and of saturated vapor for a mixture of fixed composition the different loops are for different compositions. Clearly, the PT relation for saturated liquid is different from that for saturated vapor of the same composition. This is in contrast with the behavior of a pure species, for which the bubble line and the dew line coincide. At points A and B in Fig. 12.4 saturated-liquid and saturated-vapor lines intersect. At such points a saturated liquid of one composition and a saturated vapor of another composition have the same T and P, and the two phases are therefore in equilibrium. The tie lines connecting the coinciding points at A and at B are perpendicular to the PT plane, as illustrated by the tie line VX in Fig. 12.1. 

In practice it appears that in a figure where log rj is plotted vs. log q and log q vs. log co, both lines coincide not only in the linear region but also, to some extent, in the region where both viscosities decrease. The agreement is even better if log rj instead of log rf is plotted vs. log co. This is demonstrated for polystyrene in Figs. 15.12 and 15.13... 

In the same way, but much more complicated, with a damping function depending on the Hencky strain, it proved to be possible to calculate the transient extensional viscosity as a function of qe. The result is illustrated in Fig. 15.30 for the same polymer. It shows that extensional viscosity remains finite and increases with increasing strain rate up to a maximum at qe = 2 s, after which it decreases again. The calculated lines coincide quite well with the experiments, but the calculated viscosities are somewhat too high. 

Trajectory is determined by the pixel s neighborhood connectivity method of image processing, where the direction of displacement is described by a number from 0 to 7. This is in function of the relative position of any of the eight neighbors of a central point. The number chosen for the neighbor that is bonded to the central point describes a line coincident with the direction of the displacement. 

Our description of an ABX spectrum in Section 6.13 pointed out that eight lines are normally observed in the AB region and either four or six lines in the X region. Frequently, however, some of the lines coincide, creating a spectrum whose appearance is not that of a typical ABX pattern. Consider, for example, an ABX system with the parameters shown in Fig. 6.10, which depicts calculated ABX spectra as a function of only one changing parameter, fB. The spectrum in... 

B. If the source contained both elements and the absorption lines of these could not be resolved by the monochromator, element B would cause an interference. In some early hollow-cathode lamps this was a well-known phenomenon. It could be overcome, however, by using an alternative absorption line, the probability of two lines coinciding again being extremely remote. Most of the interferences have now disappeared due to improvements in the purification techniques of the cathodes. 

Coupling of the C2 proton to the Cl vinylic proton occurs with 7 = 16 Hz and causes the signal of the C2 proton to be split into a doublet. The C2 proton is also coupled to the two C3 protons with 7=8 Hz. This splitting causes each leg of the C2 proton doublet to be split into a triplet, producing six lines in all. Because of the size of the coupling constants, two of the lines coincide, and a five-line multiplet is observed. 

All sodium salts impart a characteristic yellow coloration to the Bunsen flame, the test being so delicate as to detect 3xl0-10 grams. The extreme delicacy of the test and the wide distribution of traces of sodium chloride throughout the atmosphere render the persistence or otherwise of the coloration an important factor in determining the presence or absence of sodium in the substance under examination. In the spectroscope, sodium gives a yellow line, coincident with the D-line of the solar spectrum. 

It often takes time for the implications of experimental data to be understood and to be acted upon. Fraunhofer s earlier observation that the solar D-lines coincided with the spectral lines of a sodium lamp eventually prompted further important experiments. In 1849, Jean Bernard Leon Foucault (1819-1868), a Parisian physicist, made an unexpected discovery. He passed sunlight through a vapor of sodium and he found that the solar D-lines were darker. His conclusion was that the sodium vapor presents us with a medium which emits the rays D on its own account, and which absorbs them when they come from another quarter. The consequences of Foucault s experiment, however, were expressed more cogently by Sir William Thomson (later Lord Kelvin). He drew the following explicit conclusion That the double line D, whether bright or dark, is due to the vapor of sodium. . . That Fraunhofer s double dark line D, of solar and stellar spectra, is due to the presence of vapor of sodium in atmospheres surrounding the Sun and those stars in whose spectra it has been observed. ... 

The organic linkers, that have been studied, can be divided into four groups (1) symmetric linkers, in which the atoms 02-C1-C1-02 are in the same line coinciding the linkage axis [Fig. 3 (a), (g)] (2) asymmetric linkers (relative to... 

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